GNU Scientific Library
Release 2.7

Mark Galassi
Jim Davies
James Theiler
Brian GoughGerard Jungman
Patrick Alken
Michael Booth
Fabrice Rossi
Rhys UlerichJun 18, 2021

CONTENTS

1

Introduction
1.1 Routines available in GSL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 GSL is Free Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Obtaining GSL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 No Warranty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Reporting Bugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Further Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Conventions used in this manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
1
2
2
3
3
3

2

Using the Library
2.1 An Example Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Compiling and Linking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Shared Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 ANSI C Compliance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Inline functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Long double . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Portability functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Alternative optimized functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Support for different numeric types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Compatibility with C++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11 Aliasing of arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12 Thread-safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.13 Deprecated Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.14 Code Reuse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5
5
5
7
7
8
8
9
9
10
11
11
11
11
12

3

Error Handling
3.1 Error Reporting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Error Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Error Handlers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Using GSL error reporting in your own functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13
13
14
14
15
16

4

Mathematical Functions
4.1 Mathematical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Infinities and Not-a-number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Elementary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Small integer powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Testing the Sign of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Testing for Odd and Even Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Maximum and Minimum functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17
17
18
18
19
20
20
20

i

4.8

Approximate Comparison of Floating Point Numbers . . . . . . . . . . . . . . . . . . . . . . . . . .

21

5Complex Numbers
5.1 Representation of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Complex number macros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Assigning complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Properties of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Complex arithmetic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Elementary Complex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Complex Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Inverse Complex Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Complex Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.10 Inverse Complex Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23
23
24
24
25
25
26
26
27
27
28
28

6

Polynomials
6.1 Polynomial Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Divided Difference Representation of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Cubic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 General Polynomial Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29
29
29
30
31
31
32
33

7

Special Functions
7.1 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 The gsl_sf_result struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Airy Functions and Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6 Clausen Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7 Coulomb Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8 Coupling Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.9 Dawson Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.10 Debye Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.11 Dilogarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.12 Elementary Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.13 Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.14 Elliptic Functions (Jacobi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.15 Error Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.16 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.17 Exponential Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.18 Fermi-Dirac Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.19 Gamma and Beta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.20 Gegenbauer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.21 Hermite Polynomials and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.22 Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.23 Laguerre Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.24 Lambert W Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.25 Legendre Functions and Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.26 Logarithm and Related Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.27 Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.28 Power Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.29 Psi (Digamma) Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35
35
36
36
37
38
45
45
47
48
48
49
49
50
52
52
53
55
57
58
61
62
65
67
67
68
73
74
75
76

ii

7.30
7.31
7.32
7.33
7.34
7.35

Synchrotron Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transport Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Zeta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77
77
78
79
81
82

8

Vectors and Matrices
8.1 Data types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83
83
84
85
94

9

Permutations
9.1 The Permutation struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Permutation allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 Accessing permutation elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4 Permutation properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5 Permutation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6 Applying Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7 Reading and writing permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.8 Permutations in cyclic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.10 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107
107
107
108
108
108
109
109
110
110
112

10 Combinations
10.1 The Combination struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Combination allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Accessing combination elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4 Combination properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.5 Combination functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6 Reading and writing combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.8 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113
113
113
114
114
114
115
115
116

11 Multisets
11.1 The Multiset struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Multiset allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 Accessing multiset elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4 Multiset properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5 Multiset functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.6 Reading and writing multisets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117
117
117
118
118
118
119
119

12 Sorting
12.1 Sorting objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Sorting vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Selecting the k smallest or largest elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4 Computing the rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.6 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123
123
124
125
125
126
127

13 BLAS Support
129
13.1 GSL BLAS Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
13.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138iii

13.3 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
14 Linear Algebra
14.1 LU Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2 QR Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3 QR Decomposition with Column Pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.4 LQ Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.5 QL Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.6 Complete Orthogonal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.7 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.8 Cholesky Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.9 Pivoted Cholesky Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.10 Modified Cholesky Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.11 LDLT Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.12 Tridiagonal Decomposition of Real Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . .
14.13 Tridiagonal Decomposition of Hermitian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.14 Hessenberg Decomposition of Real Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.15 Hessenberg-Triangular Decomposition of Real Matrices . . . . . . . . . . . . . . . . . . . . . . . .
14.16 Bidiagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.17 Givens Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.18 Householder Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.19 Householder solver for linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.20 Tridiagonal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.21 Triangular Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.22 Banded Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.23 Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.24 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.25 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141
141
143
149
150
151
152
153
154
156
157
158
158
159
159
160
160
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15 Eigensystems
15.1 Real Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.2 Complex Hermitian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.3 Real Nonsymmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.4 Real Generalized Symmetric-Definite Eigensystems . . . . . . . . . . . . . . . . . . . . . . . . . .
15.5 Complex Generalized Hermitian-Definite Eigensystems . . . . . . . . . . . . . . . . . . . . . . . .
15.6 Real Generalized Nonsymmetric Eigensystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.7 Sorting Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.9 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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175
176
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16 Fast Fourier Transforms (FFTs)
16.1 Mathematical Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.2 Overview of complex data FFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.3 Radix-2 FFT routines for complex data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.4 Mixed-radix FFT routines for complex data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.5 Overview of real data FFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.6 Radix-2 FFT routines for real data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.7 Mixed-radix FFT routines for real data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.8 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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192
193
198

17 Numerical Integration
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2 QNG non-adaptive Gauss-Kronrod integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.3 QAG adaptive integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.4 QAGS adaptive integration with singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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17.5 QAGP adaptive integration with known singular points . . . . . . . . . . . . . . . . . . . . . . . . .
17.6 QAGI adaptive integration on infinite intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.7 QAWC adaptive integration for Cauchy principal values . . . . . . . . . . . . . . . . . . . . . . . .
17.8 QAWS adaptive integration for singular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.9 QAWO adaptive integration for oscillatory functions . . . . . . . . . . . . . . . . . . . . . . . . . .
17.10 QAWF adaptive integration for Fourier integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.11 CQUAD doubly-adaptive integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.12 Romberg integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.13 Gauss-Legendre integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.14 Fixed point quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.15 Error codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.16 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.17 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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18 Random Number Generation
18.1 General comments on random numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2 The Random Number Generator Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.3 Random number generator initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.4 Sampling from a random number generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.5 Auxiliary random number generator functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.6 Random number environment variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.7 Copying random number generator state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.8 Reading and writing random number generator state . . . . . . . . . . . . . . . . . . . . . . . . . .
18.9 Random number generator algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.10 Unix random number generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.11 Other random number generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.12 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.13 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.14 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.15 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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227
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229

19 Quasi-Random Sequences
19.1 Quasi-random number generator initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.2 Sampling from a quasi-random number generator . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.3 Auxiliary quasi-random number generator functions . . . . . . . . . . . . . . . . . . . . . . . . . .
19.4 Saving and restoring quasi-random number generator state . . . . . . . . . . . . . . . . . . . . . . .
19.5 Quasi-random number generator algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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20 Random Number Distributions
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.2 The Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.3 The Gaussian Tail Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.4 The Bivariate Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.5 The Multivariate Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.6 The Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.7 The Laplace Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.8 The Exponential Power Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.9 The Cauchy Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.10 The Rayleigh Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.11 The Rayleigh Tail Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.12 The Landau Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.13 The Levy alpha-Stable Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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20.14 The Levy skew alpha-Stable Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.15 The Gamma Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.16 The Flat (Uniform) Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.17 The Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.18 The Chi-squared Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.19 The F-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.20 The t-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.21 The Beta Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.22 The Logistic Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.23 The Pareto Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.24 Spherical Vector Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.25 The Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.26 The Type-1 Gumbel Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.27 The Type-2 Gumbel Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.28 The Dirichlet Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.29 General Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.30 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.31 The Bernoulli Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.32 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.33 The Multinomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.34 The Negative Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.35 The Pascal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.36 The Geometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.37 The Hypergeometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.38 The Logarithmic Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.39 The Wishart Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.40 Shuffling and Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.41 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.42 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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21 Statistics
21.1 Mean, Standard Deviation and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.2 Absolute deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.3 Higher moments (skewness and kurtosis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.4 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.5 Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.6 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.7 Weighted Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.8 Maximum and Minimum values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.9 Median and Percentiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.10 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.11 Robust Location Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.12 Robust Scale Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.13 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.14 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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22 Running Statistics
22.1 Initializing the Accumulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22.2 Adding Data to the Accumulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22.3 Current Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22.4 Quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22.6 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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23 Moving Window Statistics
23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23.2 Handling Endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23.3 Allocation for Moving Window Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23.4 Moving Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23.5 Moving Variance and Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23.6 Moving Minimum and Maximum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23.7 Moving Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23.8 Moving Median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23.9 Robust Scale Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23.10 User-defined Moving Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23.11 Accumulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23.12 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23.13 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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24 Digital Filtering
24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24.2 Handling Endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24.3 Linear Digital Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24.4 Nonlinear Digital Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24.6 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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25 Histograms
25.1 The histogram struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.2 Histogram allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.3 Copying Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.4 Updating and accessing histogram elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.5 Searching histogram ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.6 Histogram Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.7 Histogram Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.8 Reading and writing histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.9 Resampling from histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.10 The histogram probability distribution struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.11 Example programs for histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.12 Two dimensional histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.13 The 2D histogram struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.14 2D Histogram allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.15 Copying 2D Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.16 Updating and accessing 2D histogram elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.17 Searching 2D histogram ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.18 2D Histogram Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.19 2D Histogram Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.20 Reading and writing 2D histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.21 Resampling from 2D histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25.22 Example programs for 2D histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

335
335
336
337
337
338
338
338
339
340
340
341
342
342
343
343
344
345
345
346
346
347
348

26 N-tuples
26.1 The ntuple struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26.2 Creating ntuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26.3 Opening an existing ntuple file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26.4 Writing ntuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26.5 Reading ntuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26.6 Closing an ntuple file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

351
351
351
352
352
352
352

vii

26.7 Histogramming ntuple values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
26.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
26.9 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
27 Monte Carlo Integration
27.1 Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27.2 PLAIN Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27.3 MISER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27.4 VEGAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27.6 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

357
357
358
359
361
363
366

28 Simulated Annealing
28.1 Simulated Annealing algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28.2 Simulated Annealing functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28.4 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

367
367
367
369
381

29 Ordinary Differential Equations
29.1 Defining the ODE System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29.2 Stepping Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29.3 Adaptive Step-size Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29.4 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29.5 Driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29.7 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

383
383
384
386
388
389
390
394

30 Interpolation
30.1 Introduction to 1D Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30.2 1D Interpolation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30.3 1D Interpolation Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30.4 1D Index Look-up and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30.5 1D Evaluation of Interpolating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30.6 1D Higher-level Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30.7 1D Interpolation Example Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30.8 Introduction to 2D Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30.9 2D Interpolation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30.10 2D Interpolation Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30.11 2D Interpolation Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30.12 2D Evaluation of Interpolating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30.13 2D Higher-level Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30.14 2D Interpolation Example programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30.15 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

395
395
395
396
397
397
398
399
403
403
404
404
405
406
408
409

31 Numerical Differentiation
31.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31.3 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

411
411
412
413

32 Chebyshev Approximations
32.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32.2 Creation and Calculation of Chebyshev Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32.3 Auxiliary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32.4 Chebyshev Series Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32.5 Derivatives and Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

415
415
415
416
416
416

viii

32.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
32.7 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
33 Series Acceleration
33.1 Acceleration functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33.2 Acceleration functions without error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33.4 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

419
419
420
420
422

34 Wavelet Transforms
34.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.2 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.3 Transform Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34.5 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

423
423
423
424
426
428

35 Discrete Hankel Transforms
35.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35.3 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

431
431
432
432

36 One Dimensional Root-Finding
36.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36.2 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36.3 Initializing the Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36.4 Providing the function to solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36.5 Search Bounds and Guesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36.6 Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36.7 Search Stopping Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36.8 Root Bracketing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36.9 Root Finding Algorithms using Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36.10 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36.11 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

433
433
434
434
435
437
437
438
438
439
440
445

37 One Dimensional Minimization
37.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37.2 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37.3 Initializing the Minimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37.4 Providing the function to minimize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37.5 Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37.6 Stopping Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37.7 Minimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37.9 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

447
447
448
448
449
449
450
450
451
453

38 Multidimensional Root-Finding
38.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38.2 Initializing the Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38.3 Providing the function to solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38.4 Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38.5 Search Stopping Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38.6 Algorithms using Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38.7 Algorithms without Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38.9 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

455
455
456
457
459
460
460
462
463
467
ix

39 Multidimensional Minimization
39.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39.2 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39.3 Initializing the Multidimensional Minimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39.4 Providing a function to minimize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39.5 Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39.6 Stopping Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39.7 Algorithms with Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39.8 Algorithms without Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39.10 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

469
469
470
470
471
473
473
474
475
476
480

40 Linear Least-Squares Fitting
40.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40.2 Linear regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40.3 Multi-parameter regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40.4 Regularized regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40.5 Robust linear regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40.6 Large dense linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40.7 Troubleshooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40.9 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

481
481
482
483
485
489
493
497
497
520

41 Nonlinear Least-Squares Fitting
41.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41.2 Solving the Trust Region Subproblem (TRS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41.3 Weighted Nonlinear Least-Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41.4 Tunable Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41.5 Initializing the Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41.6 Providing the Function to be Minimized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41.7 Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41.8 Testing for Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41.9 High Level Driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41.10 Covariance matrix of best fit parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41.11 Troubleshooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41.12 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41.13 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

521
521
522
525
525
529
530
532
534
534
535
536
536
565

42 Basis Splines
42.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42.2 Allocation of B-splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42.3 Initializing the knots vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42.4 Properties of B-splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42.5 Evaluation of B-splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42.6 Evaluation of B-spline derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42.7 Evaluation of B-spline integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42.8 Least Squares Fitting with B-splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42.9 Gram Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42.10 Interpolation with B-splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42.11 Projection onto the B-spline Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42.12 Greville abscissae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42.13 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42.14 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

567
567
568
569
570
570
570
571
572
574
575
576
576
576
605

43 Sparse Matrices

607x43.1 Data types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43.2 Sparse Matrix Storage Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43.4 Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43.5 Accessing Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43.6 Initializing Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43.7 Reading and Writing Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43.8 Copying Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43.9 Exchanging Rows and Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43.10 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43.11 Matrix Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43.12 Finding Maximum and Minimum Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43.13 Compressed Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43.14 Conversion Between Sparse and Dense Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43.15 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43.16 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

607
608
609
610
611
611
611
612
612
612
613
614
614
614
615
617

44 Sparse BLAS Support
619
44.1 Sparse BLAS operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619
44.2 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619
45 Sparse Linear Algebra
45.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45.2 Sparse Iterative Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45.4 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

621
621
621
623
626

46 Physical Constants
46.1 Fundamental Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46.2 Astronomy and Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46.3 Atomic and Nuclear Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46.4 Measurement of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46.5 Imperial Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46.6 Speed and Nautical Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46.7 Printers Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46.8 Volume, Area and Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46.9 Mass and Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46.10 Thermal Energy and Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46.11 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46.12 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46.13 Light and Illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46.14 Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46.15 Force and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46.16 Prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46.17 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46.18 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

627
627
628
628
629
629
630
630
630
631
631
632
632
632
633
633
633
634
635

47 IEEE floating-point arithmetic
47.1 Representation of floating point numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47.2 Setting up your IEEE environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47.3 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

637
637
639
642

48 Debugging Numerical Programs
643
48.1 Using gdb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
48.2 Examining floating point registers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644
xi48.3 Handling floating point exceptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
48.4 GCC warning options for numerical programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
48.5 References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647
49 Contributors to GSL

649

50 Autoconf Macros

653

51 GSL CBLAS Library
51.1 Level 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51.2 Level 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51.3 Level 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

655
655
657
662
665

52 GNU General Public License

667

53 GNU Free Documentation License

681

Index

691

xii

CHAPTERONEINTRODUCTION

The GNU Scientific Library (GSL) is a collection of routines for numerical computing. The routines have been writtenfrom scratch in C, and present a modern Applications Programming Interface (API) for C programmers, allowing
wrappers to be written for very high level languages. The source code is distributed under the GNU General Public
License.

1.1 Routines available in GSL
The library covers a wide range of topics in numerical computing. Routines are available for the following areas,
Complex Numbers
Vectors and Matrices
Sorting
CBLAS Library
Random Numbers
Quasi-Random Sequences
Monte Carlo Integration
Simulated Annealing
Series Acceleration
Discrete Hankel Transforms
IEEE Floating-Point
Wavelets

Roots of Polynomials
Permutations
BLAS Support
Fast Fourier Transforms
Quadrature
Histograms
N-TuplesNumerical Differentiation
Chebyshev Approximations
Least-Squares Fitting
Physical Constants
Sparse BLAS Support

Special Functions
Combinations
Linear Algebra
Eigensystems
Random Distributions
Statistics
Differential Equations
Interpolation
Root-Finding
Minimization
Basis Splines
Sparse Linear Algebra

The use of these routines is described in this manual. Each chapter provides detailed definitions of the functions,followed by example programs and references to the articles on which the algorithms are based.
Where possible the routines have been based on reliable public-domain packages such as FFTPACK and QUADPACK,
which the developers of GSL have reimplemented in C with modern coding conventions.

1.2 GSL is Free Software
The subroutines in the GNU Scientific Library are “free software”; this means that everyone is free to use them, andto redistribute them in other free programs. The library is not in the public domain; it is copyrighted and there areconditions on its distribution. These conditions are designed to permit everything that a good cooperating citizen
would want to do. What is not allowed is to try to prevent others from further sharing any version of the software thatthey might get from you.

1

GNU Scientific Library, Release 2.7

Specifically, we want to make sure that you have the right to share copies of programs that you are given which use the
GNU Scientific Library, that you receive their source code or else can get it if you want it, that you can change these
programs or use pieces of them in new free programs, and that you know you can do these things.
To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights. For example,
if you distribute copies of any code which uses the GNU Scientific Library, you must give the recipients all the rightsthat you have received. You must make sure that they, too, receive or can get the source code, both to the library andthe code which uses it. And you must tell them their rights. This means that the library should not be redistributed in
proprietary programs.
Also, for our own protection, we must make certain that everyone finds out that there is no warranty for the GNU
Scientific Library. If these programs are modified by someone else and passed on, we want their recipients to know thatwhat they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation.
The precise conditions for the distribution of software related to the GNU Scientific Library are found in the GNU
General Public License. Further information about this license is available from the GNU Project webpage Frequently
Asked Questions about the GNU GPL.
The Free Software Foundation also operates a license consulting service for commercial users (contact details available
from http://www.fsf.org.

1.3 Obtaining GSL
The source code for the library can be obtained in different ways, by copying it from a friend, purchasing it on CDROM
or downloading it from the internet. A list of public ftp servers which carry the source code can be found on the GNU
website, http://www.gnu.org/software/gsl/.
The preferred platform for the library is a GNU system, which allows it to take advantage of additional features in the
GNU C compiler and GNU C library. However, the library is fully portable and should compile on most systems witha C compiler.
Announcements of new releases, updates and other relevant events are made on the info-gsl@gnu.org mailing list. To
subscribe to this low-volume list, send an email of the following form:
To: info-gsl-request@gnu.org
Subject: subscribe
You will receive a response asking you to reply in order to confirm your subscription.

1.4 No Warranty
The software described in this manual has no warranty, it is provided “as is”. It is your responsibility to validate thebehavior of the routines and their accuracy using the source code provided, or to purchase support and warranties fromcommercial redistributors. Consult the GNU General Public License for further details.

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Chapter 1. Introduction

GNU Scientific Library, Release 2.7

1.5 Reporting Bugs
A list of known bugs can be found in the BUGS file included in the GSL distribution or online in the GSL bug tracker.1
Details of compilation problems can be found in the INSTALL file.
If you find a bug which is not listed in these files, please report it to bug-gsl@gnu.org.
All bug reports should include:• The version number of GSL• The hardware and operating system• The compiler used, including version number and compilation options• A description of the bug behavior• A short program which exercises the bug
It is useful if you can check whether the same problem occurs when the library is compiled without optimization. Thankyou.
Any errors or omissions in this manual can also be reported to the same address.

1.6 Further Information
Additional information, including online copies of this manual, links to related projects, and mailing list archives areavailable from the website mentioned above.
Any questions about the use and installation of the library can be asked on the mailing list help-gsl@gnu.org. To
subscribe to this list, send an email of the following form:
To: help-gsl-request@gnu.org
Subject: subscribe
This mailing list can be used to ask questions not covered by this manual, and to contact the developers of the library.
If you would like to refer to the GNU Scientific Library in a journal article, the recommended way is to cite this reference
manual, e.g.:
M. Galassi et al, GNU Scientific Library Reference Manual (3rd Ed.), ISBN 0954612078.
If you want to give a url, use “http://www.gnu.org/software/gsl/”.

1.7 Conventions used in this manualThis manual contains many examples which can be typed at the keyboard. A command entered at the terminal is shown
like this:$ command
The first character on the line is the terminal prompt, and should not be typed. The dollar sign $ is used as the standard
prompt in this manual, although some systems may use a different character.
1 http://savannah.gnu.org/bugs/?group=gsl

1.5. Reporting Bugs

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GNU Scientific Library, Release 2.7

The examples assume the use of the GNU operating system. There may be minor differences in the output on othersystems. The commands for setting environment variables use the Bourne shell syntax of the standard GNU shell(bash).

4

Chapter 1. Introduction

CHAPTERTWOUSING THE LIBRARY

This chapter describes how to compile programs that use GSL, and introduces its conventions.

2.1 An Example Program
The following short program demonstrates the use of the library by computing the value of the Bessel function J0 (x)for x = 5:#include <stdio.h>#include <gsl/gsl_sf_bessel.h>
intmain (void){
double x = 5.0;
double y = gsl_sf_bessel_J0 (x);printf ("J0(%g) = %.18e\n", x, y);return 0;}
The output is shown below, and should be correct to double-precision accuracy1 ,J0(5) = -1.775967713143382642e-01
The steps needed to compile this program are described in the following sections.

2.2 Compiling and Linking
The library header files are installed in their own gsl directory. You should write any preprocessor include statementswith a gsl/ directory prefix thus:#include <gsl/gsl_math.h>
If the directory is not installed on the standard search path of your compiler you will also need to provide its location
to the preprocessor as a command line flag. The default location of the gsl directory is /usr/local/include/gsl.
A typical compilation command for a source file example.c with the GNU C compiler gcc is:
1 The last few digits may vary slightly depending on the compiler and platform used—this is normal

5GNU Scientific Library, Release 2.7$ gcc -Wall -I/usr/local/include -c example.c
This results in an object file example.o. The default include path for gcc searches /usr/local/include automatically so the -I option can actually be omitted when GSL is installed in its default location.

2.2.1 Linking programs with the library
The library is installed as a single file, libgsl.a. A shared version of the library libgsl.so is also installed on
systems that support shared libraries. The default location of these files is /usr/local/lib. If this directory is noton the standard search path of your linker you will also need to provide its location as a command line flag.
To link against the library you need to specify both the main library and a supporting CBLAS library, which provides
standard basic linear algebra subroutines. A suitable CBLAS implementation is provided in the library libgslcblas.
a if your system does not provide one. The following example shows how to link an application with the library:$ gcc -L/usr/local/lib example.o -lgsl -lgslcblas -lm
The default library path for gcc searches /usr/local/lib automatically so the -L option can be omitted when GSL
is installed in its default location.
The option -lm links with the system math library. On some systems it is not needed.2
For a tutorial introduction to the GNU C Compiler and related programs, see “An Introduction to GCC” (ISBN0954161793).3

2.2.2 Linking with an alternative BLAS library
The following command line shows how you would link the same application with an alternative CBLAS librarylibcblas.a:$ gcc example.o -lgsl -lcblas -lm
For the best performance an optimized platform-specific CBLAS library should be used for -lcblas. The librarymust conform to the CBLAS standard. The ATLAS package provides a portable high-performance BLAS library witha CBLAS interface. It is free software and should be installed for any work requiring fast vector and matrix operations.
The following command line will link with the ATLAS library and its CBLAS interface:$ gcc example.o -lgsl -lcblas -latlas -lm
If the ATLAS library is installed in a non-standard directory use the -L option to add it to the search path, as describedabove.
For more information about BLAS functions see BLAS Support.
2 It is not needed on MacOS X

3 http://www.network-theory.co.uk/gcc/intro/

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Chapter 2. Using the LibraryGNU Scientific Library, Release 2.7

2.3 Shared Libraries
To run a program linked with the shared version of the library the operating system must be able to locate the corresponding .so file at runtime. If the library cannot be found, the following error will occur:$ ./a.out./a.out: error while loading shared libraries:
libgsl.so.0: cannot open shared object file: No such file or directory
To avoid this error, either modify the system dynamic linker configuration4 or define the shell variable
LD_LIBRARY_PATH to include the directory where the library is installed.
For example, in the Bourne shell (/bin/sh or /bin/bash), the library search path can be set with the following
commands:$ LD_LIBRARY_PATH=/usr/local/lib$ export LD_LIBRARY_PATH$ ./example
In the C-shell (/bin/csh or /bin/tcsh) the equivalent command is:% setenv LD_LIBRARY_PATH /usr/local/lib
The standard prompt for the C-shell in the example above is the percent character %, and should not be typed as partof the command.
To save retyping these commands each session they can be placed in an individual or system-wide login file.
To compile a statically linked version of the program, use the -static flag in gcc:
$ gcc -static example.o -lgsl -lgslcblas -lm

2.4 ANSI C Compliance
The library is written in ANSI C and is intended to conform to the ANSI C standard (C89). It should be portable toany system with a working ANSI C compiler.
The library does not rely on any non-ANSI extensions in the interface it exports to the user. Programs you write using
GSL can be ANSI compliant. Extensions which can be used in a way compatible with pure ANSI C are supported, however, via conditional compilation. This allows the library to take advantage of compiler extensions on those platforms
which support them.
When an ANSI C feature is known to be broken on a particular system the library will exclude any related functions
at compile-time. This should make it impossible to link a program that would use these functions and give incorrect
results.
To avoid namespace conflicts all exported function names and variables have the prefix gsl_, while exported macros
have the prefix GSL_.
4 /etc/ld.so.conf on GNU/Linux systems

2.3. Shared Libraries

7GNU Scientific Library, Release 2.7

2.5 Inline functions
The inline keyword is not part of the original ANSI C standard (C89) so the library does not export any inline function
definitions by default. Inline functions were introduced officially in the newer C99 standard but most C89 compilers
have also included inline as an extension for a long time.
To allow the use of inline functions, the library provides optional inline versions of performance-critical routines byconditional compilation in the exported header files. The inline versions of these functions can be included by defining
the macro HAVE_INLINE when compiling an application:$ gcc -Wall -c -DHAVE_INLINE example.c
If you use autoconf this macro can be defined automatically. If you do not define the macro HAVE_INLINE then theslower non-inlined versions of the functions will be used instead.
By default, the actual form of the inline keyword is extern inline, which is a gcc extension that eliminates unnecessary function definitions. If the form extern inline causes problems with other compilers a stricter autoconf test
can be used, see Autoconf Macros.
When compiling with gcc in C99 mode (gcc -std=c99) the header files automatically switch to C99-compatible inline function declarations instead of extern inline. With other C99 compilers, define the macro GSL_C99_INLINE
to use these declarations.

2.6 Long double
In general, the algorithms in the library are written for double precision only. The long double type is not supported
for actual computation.
One reason for this choice is that the precision of long double is platform dependent. The IEEE standard only
specifies the minimum precision of extended precision numbers, while the precision of double is the same on all
platforms.
However, it is sometimes necessary to interact with external data in long-double format, so the vector and matrix
datatypes include long-double versions.
It should be noted that in some system libraries the stdio.h formatted input/output functions printf and scanf are
not implemented correctly for long double. Undefined or incorrect results are avoided by testing these functions
during the configure stage of library compilation and eliminating certain GSL functions which depend on them ifnecessary. The corresponding line in the configure output looks like this:checking whether printf works with long double... noConsequently when long double formatted input/output does not work on a given system it should be impossible tolink a program which uses GSL functions dependent on this.
If it is necessary to work on a system which does not support formatted long double input/output then the options
are to use binary formats or to convert long double results into double for reading and writing.

8

Chapter 2. Using the LibraryGNU Scientific Library, Release 2.7

2.7 Portability functions
To help in writing portable applications GSL provides some implementations of functions that are found in otherlibraries, such as the BSD math library. You can write your application to use the native versions of these functions,and substitute the GSL versions via a preprocessor macro if they are unavailable on another platform.
For example, after determining whether the BSD function hypot() is available you can include the following macrodefinitions in a file config.h with your application:/* Substitute gsl_hypot for missing system hypot */#ifndef HAVE_HYPOT#define hypot gsl_hypot#endif
The application source files can then use the include command #include <config.h> to replace each occurrenceof hypot() by gsl_hypot() when hypot() is not available. This substitution can be made automatically if you use
autoconf, see Autoconf Macros.
In most circumstances the best strategy is to use the native versions of these functions when available, and fall back to
GSL versions otherwise, since this allows your application to take advantage of any platform-specific optimizations inthe system library. This is the strategy used within GSL itself.

2.8 Alternative optimized functions
The main implementation of some functions in the library will not be optimal on all architectures. For example, there areseveral ways to compute a Gaussian random variate and their relative speeds are platform-dependent. In cases like thisthe library provides alternative implementations of these functions with the same interface. If you write your application
using calls to the standard implementation you can select an alternative version later via a preprocessor definition. It
is also possible to introduce your own optimized functions this way while retaining portability. The following lines
demonstrate the use of a platform-dependent choice of methods for sampling from the Gaussian distribution:#ifdef SPARC#define gsl_ran_gaussian gsl_ran_gaussian_ratio_method#endif#ifdef INTEL#define gsl_ran_gaussian my_gaussian#endif
These lines would be placed in the configuration header file config.h of the application, which should then be included
by all the source files. Note that the alternative implementations will not produce bit-for-bit identical results, and in thecase of random number distributions will produce an entirely different stream of random variates.

2.7. Portability functions

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GNU Scientific Library, Release 2.7

2.9 Support for different numeric types
Many functions in the library are defined for different numeric types. This feature is implemented by varying the nameof the function with a type-related modifier—a primitive form of C++ templates. The modifier is inserted into thefunction name after the initial module prefix. The following table shows the function names defined for all the numerictypes of an imaginary module gsl_foo with function fn():
gsl_foo_fn
gsl_foo_long_double_fn
gsl_foo_float_fn
gsl_foo_long_fn
gsl_foo_ulong_fn
gsl_foo_int_fn
gsl_foo_uint_fn
gsl_foo_short_fn
gsl_foo_ushort_fn
gsl_foo_char_fn
gsl_foo_uchar_fn

doublelong double
float
long
unsigned longint
unsigned int
short
unsigned shortchar
unsigned char

The normal numeric precision double is considered the default and does not require a suffix. For example, the function
gsl_stats_mean() computes the mean of double precision numbers, while the function gsl_stats_int_mean()computes the mean of integers.
A corresponding scheme is used for library defined types, such as gsl_vector and gsl_matrix. In this case themodifier is appended to the type name. For example, if a module defines a new type-dependent struct or typedef
gsl_foo it is modified for other types in the following way:
gsl_foo
gsl_foo_long_double
gsl_foo_float
gsl_foo_long
gsl_foo_ulong
gsl_foo_int
gsl_foo_uint
gsl_foo_short
gsl_foo_ushort
gsl_foo_char
gsl_foo_uchar

doublelong double
float
long
unsigned longint
unsigned int
short
unsigned shortchar
unsigned char

When a module contains type-dependent definitions the library provides individual header files for each type. The
filenames are modified as shown in the below. For convenience the default header includes the definitions for all thetypes. To include only the double precision header file, or any other specific type, use its individual filename:#include <gsl/gsl_foo.h>#include <gsl/gsl_foo_double.h>#include <gsl/gsl_foo_long_double.h>#include <gsl/gsl_foo_float.h>#include <gsl/gsl_foo_long.h>#include <gsl/gsl_foo_ulong.h>#include <gsl/gsl_foo_int.h>#include <gsl/gsl_foo_uint.h>#include <gsl/gsl_foo_short.h>#include <gsl/gsl_foo_ushort.h>All types
doublelong double
float
long
unsigned longint
unsigned int
short
unsigned short(continues on next page)10

Chapter 2. Using the LibraryGNU Scientific Library, Release 2.7(continued from previous page)#include <gsl/gsl_foo_char.h>#include <gsl/gsl_foo_uchar.h>char
unsigned char

2.10 Compatibility with C++
The library header files automatically define functions to have extern "C" linkage when included in C++ programs.
This allows the functions to be called directly from C++.
To use C++ exception handling within user-defined functions passed to the library as parameters, the library must be
built with the additional CFLAGS compilation option -fexceptions.

2.11 Aliasing of arrays
The library assumes that arrays, vectors and matrices passed as modifiable arguments are not aliased and do not overlap
with each other. This removes the need for the library to handle overlapping memory regions as a special case, andallows additional optimizations to be used. If overlapping memory regions are passed as modifiable arguments then the
results of such functions will be undefined. If the arguments will not be modified (for example, if a function prototype
declares them as const arguments) then overlapping or aliased memory regions can be safely used.

2.12 Thread-safety
The library can be used in multi-threaded programs. All the functions are thread-safe, in the sense that they do not use
static variables. Memory is always associated with objects and not with functions. For functions which use workspace
objects as temporary storage the workspaces should be allocated on a per-thread basis. For functions which use tableobjects as read-only memory the tables can be used by multiple threads simultaneously. Table arguments are always
declared const in function prototypes, to indicate that they may be safely accessed by different threads.
There are a small number of static global variables which are used to control the overall behavior of the library (e.g.
whether to use range-checking, the function to call on fatal error, etc). These variables are set directly by the user, sothey should be initialized once at program startup and not modified by different threads.

2.13 Deprecated Functions
From time to time, it may be necessary for the definitions of some functions to be altered or removed from the library.
In these circumstances the functions will first be declared deprecated and then removed from subsequent versions of
the library. Functions that are deprecated can be disabled in the current release by setting the preprocessor definition
GSL_DISABLE_DEPRECATED. This allows existing code to be tested for forwards compatibility.

2.10. Compatibility with C++

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GNU Scientific Library, Release 2.7

2.14 Code Reuse
Where possible the routines in the library have been written to avoid dependencies between modules and files. Thisshould make it possible to extract individual functions for use in your own applications, without needing to have thewhole library installed. You may need to define certain macros such as GSL_ERROR and remove some #include
statements in order to compile the files as standalone units. Reuse of the library code in this way is encouraged, subject
to the terms of the GNU General Public License.

12

Chapter 2. Using the LibraryCHAPTERTHREE

ERROR HANDLINGThis chapter describes the way that GSL functions report and handle errors. By examining the status information
returned by every function you can determine whether it succeeded or failed, and if it failed you can find out what the
precise cause of failure was. You can also define your own error handling functions to modify the default behavior of
the library.
The functions described in this section are declared in the header file gsl_errno.h.

3.1 Error Reporting
The library follows the thread-safe error reporting conventions of the POSIX Threads library. Functions return a nonzero error code to indicate an error and 0 to indicate success:int status = gsl_function (...)
if (status) { /* an error occurred */.....
/* status value specifies the type of error */}The routines report an error whenever they cannot perform the task requested of them. For example, a root-finding
function would return a non-zero error code if could not converge to the requested accuracy, or exceeded a limit on thenumber of iterations. Situations like this are a normal occurrence when using any mathematical library and you should
check the return status of the functions that you call.
Whenever a routine reports an error the return value specifies the type of error. The return value is analogous to thevalue of the variable errno in the C library. The caller can examine the return code and decide what action to take,including ignoring the error if it is not considered serious.
In addition to reporting errors by return codes the library also has an error handler function gsl_error(). Thisfunction is called by other library functions when they report an error, just before they return to the caller. The default
behavior of the error handler is to print a message and abort the program:
gsl: file.c:67: ERROR: invalid argument supplied by user
Default GSL error handler invoked.
AbortedThe purpose of the gsl_error() handler is to provide a function where a breakpoint can be set that will catch libraryerrors when running under the debugger. It is not intended for use in production programs, which should handle any
errors using the return codes.

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GNU Scientific Library, Release 2.7

3.2 Error Codes
The error code numbers returned by library functions are defined in the file gsl_errno.h. They all have the prefix
GSL_ and expand to non-zero constant integer values. Error codes above 1024 are reserved for applications, and are notused by the library. Many of the error codes use the same base name as the corresponding error code in the C library.
Here are some of the most common error codes,
int GSL_EDOM
Domain error; used by mathematical functions when an argument value does not fall into the domain over which
the function is defined (like EDOM in the C library)
int GSL_ERANGE
Range error; used by mathematical functions when the result value is not representable because of overflow orunderflow (like ERANGE in the C library)
int GSL_ENOMEM
No memory available. The system cannot allocate more virtual memory because its capacity is full (like ENOMEM
in the C library). This error is reported when a GSL routine encounters problems when trying to allocate memory
with malloc().
int GSL_EINVAL
Invalid argument. This is used to indicate various kinds of problems with passing the wrong argument to a libraryfunction (like EINVAL in the C library).
The error codes can be converted into an error message using the function gsl_strerror().
const char *gsl_strerror(const int gsl_errno)
This function returns a pointer to a string describing the error code gsl_errno. For example:
printf ("error: %s\n", gsl_strerror (status));would print an error message like error:

output range error for a status value of GSL_ERANGE.

3.3 Error Handlers
The default behavior of the GSL error handler is to print a short message and call abort(). When this default is in use
programs will stop with a core-dump whenever a library routine reports an error. This is intended as a fail-safe default
for programs which do not check the return status of library routines (we don’t encourage you to write programs thisway).
If you turn off the default error handler it is your responsibility to check the return values of routines and handle themyourself. You can also customize the error behavior by providing a new error handler. For example, an alternative error
handler could log all errors to a file, ignore certain error conditions (such as underflows), or start the debugger andattach it to the current process when an error occurs.
All GSL error handlers have the type gsl_error_handler_t, which is defined in gsl_errno.h,
type gsl_error_handler_t
This is the type of GSL error handler functions. An error handler will be passed four arguments which specify
the reason for the error (a string), the name of the source file in which it occurred (also a string), the line numberin that file (an integer) and the error number (an integer). The source file and line number are set at compile timeusing the __FILE__ and __LINE__ directives in the preprocessor. An error handler function returns type void.
Error handler functions should be defined like this:14

Chapter 3. Error HandlingGNU Scientific Library, Release 2.7

void handler (const char * reason,
const char * file,
int line,
int gsl_errno)
To request the use of your own error handler you need to call the function gsl_set_error_handler() which is alsodeclared in gsl_errno.h,
gsl_error_handler_t *gsl_set_error_handler(gsl_error_handler_t *new_handler)
This function sets a new error handler, new_handler, for the GSL library routines. The previous handler isreturned (so that you can restore it later). Note that the pointer to a user defined error handler function is stored
in a static variable, so there can be only one error handler per program. This function should be not be usedin multi-threaded programs except to set up a program-wide error handler from a master thread. The following
example shows how to set and restore a new error handler:/* save original handler, install new handler */old_handler = gsl_set_error_handler (&my_handler);/* code uses new handler */.....
/* restore original handler */gsl_set_error_handler (old_handler);
To use the default behavior (abort() on error) set the error handler to NULL:
old_handler = gsl_set_error_handler (NULL);
gsl_error_handler_t *gsl_set_error_handler_off()
This function turns off the error handler by defining an error handler which does nothing. This will cause theprogram to continue after any error, so the return values from any library routines must be checked. This is therecommended behavior for production programs. The previous handler is returned (so that you can restore it
later).
The error behavior can be changed for specific applications by recompiling the library with a customized definition of
the GSL_ERROR macro in the file gsl_errno.h.

3.4 Using GSL error reporting in your own functions
If you are writing numerical functions in a program which also uses GSL code you may find it convenient to adopt thesame error reporting conventions as in the library.
To report an error you need to call the function gsl_error() with a string describing the error and then return anappropriate error code from gsl_errno.h, or a special value, such as NaN. For convenience the file gsl_errno.h
defines two macros which carry out these steps:
GSL_ERROR(reason, gsl_errno)
This macro reports an error using the GSL conventions and returns a status value of gsl_errno. It expands tothe following code fragment:
gsl_error (reason, __FILE__, __LINE__, gsl_errno);
return gsl_errno;

3.4. Using GSL error reporting in your own functions

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GNU Scientific Library, Release 2.7

The macro definition in gsl_errno.h actually wraps the code in a do { ... } while (0) block to prevent
possible parsing problems.
Here is an example of how the macro could be used to report that a routine did not achieve a requested tolerance. To
report the error the routine needs to return the error code GSL_ETOL:
if (residual > tolerance){
GSL_ERROR("residual exceeds tolerance", GSL_ETOL);}
GSL_ERROR_VAL(reason, gsl_errno, value)
This macro is the same as GSL_ERROR but returns a user-defined value of value instead of an error code. It canbe used for mathematical functions that return a floating point value.
The following example shows how to return a NaN at a mathematical singularity using the GSL_ERROR_VAL macro:if (x == 0){
GSL_ERROR_VAL("argument lies on singularity", GSL_ERANGE, GSL_NAN);}

3.5 Examples
Here is an example of some code which checks the return value of a function where an error might be reported:#include <stdio.h>#include <gsl/gsl_errno.h>#include <gsl/gsl_fft_complex.h>...
int status;
size_t n = 37;
gsl_set_error_handler_off();
status = gsl_fft_complex_radix2_forward (data, stride, n);if (status) {if (status == GSL_EINVAL) {fprintf (stderr, "invalid argument, n=%d\n", n);} else {fprintf (stderr, "failed, gsl_errno=%d\n", status);}
exit (-1);}...
The function gsl_fft_complex_radix2_forward() only accepts integer lengths which are a power of two. If thevariable n is not a power of two then the call to the library function will return GSL_EINVAL, indicating that thelength argument is invalid. The function call to gsl_set_error_handler_off() stops the default error handler
from aborting the program. The else clause catches any other possible errors.

16

Chapter 3. Error HandlingCHAPTER

FOUR

MATHEMATICAL FUNCTIONSThis chapter describes basic mathematical functions. Some of these functions are present in system libraries, but thealternative versions given here can be used as a substitute when the system functions are not available.
The functions and macros described in this chapter are defined in the header file gsl_math.h.

4.1 Mathematical Constants
The library ensures that the standard BSD mathematical constants are defined. For reference, here is a list of theconstants:
M_E
M_LOG2E
M_LOG10E
M_SQRT2
M_SQRT1_2
M_SQRT3
M_PI
M_PI_2
M_PI_4
M_SQRTPI
M_2_SQRTPI
M_1_PI
M_2_PI
M_LN10
M_LN2
M_LNPIM_EULERThe base of exponentials, e
The base-2 logarithm of e, log2 (e)The base-10 logarithm of√e, log10 (e)The square root of two, 2 p
The square root of one-half,√ 1/2
The square root of three, 3
The constant pi, πPi divided by two, π/2
Pi divided by four, π/4√The square root of pi, π√Two divided by the square root of pi, 2/ π
The reciprocal of pi, 1/πTwice the reciprocal of pi, 2/π
The natural logarithm of ten, ln(10)The natural logarithm of two, ln(2)The natural logarithm of pi, ln(π)
Euler’s constant, γ

17GNU Scientific Library, Release 2.7

4.2 Infinities and Not-a-number
GSL_POSINF
This macro contains the IEEE representation of positive infinity, +∞. It is computed from the expression +1.
0/0.0.
GSL_NEGINF
This macro contains the IEEE representation of negative infinity, −∞. It is computed from the expression -1.
0/0.0.
GSL_NAN
This macro contains the IEEE representation of the Not-a-Number symbol, NaN. It is computed from the ratio
0.0/0.0.
int gsl_isnan(const double x)
This function returns 1 if x is not-a-number.
int gsl_isinf(const double x)
This function returns +1 if x is positive infinity, −1 if x is negative infinity and 0 otherwise.1
int gsl_finite(const double x)
This function returns 1 if x is a real number, and 0 if it is infinite or not-a-number.

4.3 Elementary Functions
The following routines provide portable implementations of functions found in the BSD math library. When native
versions are not available the functions described here can be used instead. The substitution can be made automatically
if you use autoconf to compile your application (see Portability functions).
double gsl_log1p(const double x)
This function computes the value of log(1 + x) in a way that is accurate for small x. It provides an alternative tothe BSD math function log1p(x).
double gsl_expm1(const double x)
This function computes the value of exp(x) − 1 in a way that is accurate for small x. It provides an alternative
to the BSD math function expm1(x).
double gsl_hypot(const double x, const double y)pThis function computes the value of x2 + y 2 in a way that avoids overflow. It provides an alternative to the
BSD math function hypot(x,y).
double gsl_hypot3(const double x, const double y, const double z)pThis function computes the value of x2 + y 2 + z 2 in a way that avoids overflow.
double gsl_acosh(const double x)
This function computes the value of arccosh (x). It provides an alternative to the standard math function
acosh(x).
double gsl_asinh(const double x)
This function computes the value of arcsinh (x). It provides an alternative to the standard math function
asinh(x).
1 Note that the C99 standard only requires the system isinf() function to return a non-zero value, without the sign of the infinity. The implementation in some earlier versions of GSL used the system isinf() function and may have this behavior on some platforms. Therefore, it isadvisable to test the sign of x separately, if needed, rather than relying the sign of the return value from gsl_isinf().

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double gsl_atanh(const double x)
This function computes the value of arctanh (x). It provides an alternative to the standard math function
atanh(x).
double gsl_ldexp(double x, int e)
This function computes the value of x ∗ 2e . It provides an alternative to the standard math function ldexp(x,e).
double gsl_frexp(double x, int *e)
This function splits the number x into its normalized fraction f and exponent e, such that x = f ∗ 2e and0.5 <= f < 1. The function returns f and stores the exponent in e. If x is zero, both f and e are set to zero.
This function provides an alternative to the standard math function frexp(x, e).

4.4 Small integer powers
A common complaint about the standard C library is its lack of a function for calculating (small) integer powers. GSL
provides some simple functions to fill this gap. For reasons of efficiency, these functions do not check for overflow orunderflow conditions.
double gsl_pow_int(double x, int n)double gsl_pow_uint(double x, unsigned int n)
These routines computes the power xn for integer n. The power is computed efficiently—for example, x8 is
computed as ((x2 )2 )2 , requiring only 3 multiplications. A version of this function which also computes thenumerical error in the result is available as gsl_sf_pow_int_e().
double gsl_pow_2(const double x)
double gsl_pow_3(const double x)
double gsl_pow_4(const double x)
double gsl_pow_5(const double x)
double gsl_pow_6(const double x)
double gsl_pow_7(const double x)
double gsl_pow_8(const double x)
double gsl_pow_9(const double x)
These functions can be used to compute small integer powers x2 , x3 , etc. efficiently. The functions will beinlined when HAVE_INLINE is defined, so that use of these functions should be as efficient as explicitly writing
the corresponding product expression:#include <gsl/gsl_math.h>double y = gsl_pow_4 (3.141)

4.4. Small integer powers/* compute 3.141**4 */19

GNU Scientific Library, Release 2.7

4.5 Testing the Sign of Numbers
GSL_SIGN(x)This macro returns the sign of x. It is defined as ((x) >= 0 ? 1 :sign of zero is positive (regardless of its IEEE sign bit).

-1). Note that with this definition the

4.6 Testing for Odd and Even Numbers
GSL_IS_ODD(n)This macro evaluates to 1 if n is odd and 0 if n is even. The argument n must be of integer type.
GSL_IS_EVEN(n)This macro is the opposite of GSL_IS_ODD. It evaluates to 1 if n is even and 0 if n is odd. The argument n must
be of integer type.

4.7 Maximum and Minimum functions
Note that the following macros perform multiple evaluations of their arguments, so they should not be used with arguments that have side effects (such as a call to a random number generator).
GSL_MAX(a, b)This macro returns the maximum of a and b. It is defined as ((a) > (b) ? (a):(b)).
GSL_MIN(a, b)This macro returns the minimum of a and b. It is defined as ((a) < (b) ? (a):(b)).
extern inline double GSL_MAX_DBL(double a, double b)
This function returns the maximum of the double precision numbers a and b using an inline function. The useof a function allows for type checking of the arguments as an extra safety feature. On platforms where inline
functions are not available the macro GSL_MAX will be automatically substituted.
extern inline double GSL_MIN_DBL(double a, double b)
This function returns the minimum of the double precision numbers a and b using an inline function. The useof a function allows for type checking of the arguments as an extra safety feature. On platforms where inline
functions are not available the macro GSL_MIN will be automatically substituted.
extern inline int GSL_MAX_INT(int a, int b)
extern inline int GSL_MIN_INT(int a, int b)
These functions return the maximum or minimum of the integers a and b using an inline function. On platforms
where inline functions are not available the macros GSL_MAX or GSL_MIN will be automatically substituted.
extern inline long double GSL_MAX_LDBL(long double a, long double b)
extern inline long double GSL_MIN_LDBL(long double a, long double b)
These functions return the maximum or minimum of the long doubles a and b using an inline function. On
platforms where inline functions are not available the macros GSL_MAX or GSL_MIN will be automatically substituted.

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4.8 Approximate Comparison of Floating Point Numbers
It is sometimes useful to be able to compare two floating point numbers approximately, to allow for rounding andtruncation errors. The following function implements the approximate floating-point comparison algorithm proposedby D.E. Knuth in Section 4.2.2 of “Seminumerical Algorithms” (3rd edition).
int gsl_fcmp(double x, double y, double epsilon)
This function determines whether x and y are approximately equal to a relative accuracy epsilon.
The relative accuracy is measured using an interval of size 2δ, where δ = 2k  and k is the maximum base-2
exponent of x and y as computed by the function frexp().
If x and y lie within this interval, they are considered approximately equal and the function returns 0. Otherwise
if x < y, the function returns −1, or if x > y, the function returns +1.
Note that x and y are compared to relative accuracy, so this function is not suitable for testing whether a value isapproximately zero.
The implementation is based on the package fcmp by T.C. Belding.

4.8. Approximate Comparison of Floating Point Numbers

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22Chapter 4. Mathematical Functions

CHAPTERFIVECOMPLEX NUMBERS

The functions described in this chapter provide support for complex numbers. The algorithms take care to avoid
unnecessary intermediate underflows and overflows, allowing the functions to be evaluated over as much of the complex
plane as possible.
For multiple-valued functions the branch cuts have been chosen to follow the conventions of Abramowitz and Stegun.
The functions return principal values which are the same as those in GNU Calc, which in turn are the same as those in“Common Lisp, The Language (Second Edition)”1 and the HP-28/48 series of calculators.
The complex types are defined in the header file gsl_complex.h, while the corresponding complex functions andarithmetic operations are defined in gsl_complex_math.h.

5.1 Representation of complex numbers
Complex numbers are represented using the type gsl_complex. The default interface defines gsl_complex as:
typedef struct{
double dat[2];} gsl_complex;
The real and imaginary part are stored in contiguous elements of a two element array. This eliminates any padding
between the real and imaginary parts, dat[0] and dat[1], allowing the struct to be mapped correctly onto packed
complex arrays.
If a C compiler is available which supports the C11 standard, and the <complex.h> header file is included prior to
gsl_complex.h, then gsl_complex will be defined to be the native C complex type:typedef double complex gsl_complex
This allows users to use gsl_complex in ordinary operations such as:
gsl_complex x = 2 + 5 * I;
gsl_complex y = x + (3 - 4*I);

Important: Native C support for complex numbers was introduced in the C99 standard, and additional functionality
was added in C11. When <complex.h> is included in a user’s program prior to gsl_complex.h, GSL uses the new
C11 functionality to define the GSL_REAL and GSL_IMAG macros. It does not appear possible to properly define thesemacros using the C99 standard, and so using a C99 compiler will not define gsl_complex to the native complex type.
1 Note that the first edition uses different definitions.

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GNU Scientific Library, Release 2.7

Some compilers, such as the gcc 4.8 series implement only a portion of the C11 standard and so they may fail tocorrectly compile GSL code when a user tries to turn on native complex functionality. A workaround for this issue is toeither remove <complex.h> from the include list, or add -DGSL_COMPLEX_LEGACY to the compiler flags, which will
use the older struct-based definition of gsl_complex.

5.2 Complex number macros
The following C macros offer convenient ways to manipulate complex numbers.
GSL_REAL(z)GSL_IMAG(z)These macros return a memory location (lvalue) corresponding to the real and imaginary parts respectively of
the complex number z. This allows users to perform operations like:
gsl_complex x, y;
GSL_REAL(x) = 4;
GSL_IMAG(x) = 2;
GSL_REAL(y) = GSL_REAL(x);GSL_IMAG(y) = GSL_REAL(x);In other words, these macros can both read and write to the real and imaginary parts of a complex variable.
GSL_SET_COMPLEX(zp, x, y)This macro uses the Cartesian components (x, y) to set the real and imaginary parts of the complex number
pointed to by zp. For example:
GSL_SET_COMPLEX(&z, 3, 4)
sets z to be 3 + 4i.

5.3 Assigning complex numbers
gsl_complex gsl_complex_rect(double x, double y)This function uses the rectangular Cartesian components (x, y) to return the complex number z = x + iy. An
inline version of this function is used when HAVE_INLINE is defined.
gsl_complex gsl_complex_polar(double r, double theta)
This function returns the complex number z = r exp(iθ) = r(cos(θ) + i sin(θ)) from the polar representation(r, theta).

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5.4 Properties of complex numbers
double gsl_complex_arg(gsl_complex z)
This function returns the argument of the complex number z, arg(z), where −π < arg(z) <= π.
double gsl_complex_abs(gsl_complex z)
This function returns the magnitude of the complex number z, |z|.
double gsl_complex_abs2(gsl_complex z)
This function returns the squared magnitude of the complex number z, |z|2 .
double gsl_complex_logabs(gsl_complex z)
This function returns the natural logarithm of the magnitude of the complex number z, log |z|. It allows an
accurate evaluation of log |z| when |z| is close to one. The direct evaluation of log(gsl_complex_abs(z))
would lead to a loss of precision in this case.

5.5 Complex arithmetic operators
gsl_complex gsl_complex_add(gsl_complex a, gsl_complex b)
This function returns the sum of the complex numbers a and b, z = a + b.
gsl_complex gsl_complex_sub(gsl_complex a, gsl_complex b)
This function returns the difference of the complex numbers a and b, z = a − b.
gsl_complex gsl_complex_mul(gsl_complex a, gsl_complex b)
This function returns the product of the complex numbers a and b, z = ab.
gsl_complex gsl_complex_div(gsl_complex a, gsl_complex b)
This function returns the quotient of the complex numbers a and b, z = a/b.
gsl_complex gsl_complex_add_real(gsl_complex a, double x)
This function returns the sum of the complex number a and the real number x, z = a + x.
gsl_complex gsl_complex_sub_real(gsl_complex a, double x)
This function returns the difference of the complex number a and the real number x, z = a − x.
gsl_complex gsl_complex_mul_real(gsl_complex a, double x)
This function returns the product of the complex number a and the real number x, z = ax.
gsl_complex gsl_complex_div_real(gsl_complex a, double x)
This function returns the quotient of the complex number a and the real number x, z = a/x.
gsl_complex gsl_complex_add_imag(gsl_complex a, double y)This function returns the sum of the complex number a and the imaginary number iy, z = a + iy.
gsl_complex gsl_complex_sub_imag(gsl_complex a, double y)This function returns the difference of the complex number a and the imaginary number iy, z = a − iy.
gsl_complex gsl_complex_mul_imag(gsl_complex a, double y)This function returns the product of the complex number a and the imaginary number iy, z = a ∗ (iy).
gsl_complex gsl_complex_div_imag(gsl_complex a, double y)This function returns the quotient of the complex number a and the imaginary number iy, z = a/(iy).
gsl_complex gsl_complex_conjugate(gsl_complex z)
This function returns the complex conjugate of the complex number z, z ∗ = x − iy.
gsl_complex gsl_complex_inverse(gsl_complex z)
This function returns the inverse, or reciprocal, of the complex number z, 1/z = (x − iy)/(x2 + y 2 ).

5.4. Properties of complex numbers

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gsl_complex gsl_complex_negative(gsl_complex z)
This function returns the negative of the complex number z, −z = (−x) + i(−y).

5.6 Elementary Complex Functions
gsl_complex gsl_complex_sqrt(gsl_complex z)√
This function returns the square root of the complex number z, z. The branch cut is the negative real axis. The
result always lies in the right half of the complex plane.
gsl_complex gsl_complex_sqrt_real(double x)
This function returns the complex square root of the real number x, where x may be negative.
gsl_complex gsl_complex_pow(gsl_complex z, gsl_complex a)
The function returns the complex number z raised to the complex power a, z a . This is computed as exp(log(z) ∗a) using complex logarithms and complex exponentials.
gsl_complex gsl_complex_pow_real(gsl_complex z, double x)
This function returns the complex number z raised to the real power x, z x .
gsl_complex gsl_complex_exp(gsl_complex z)
This function returns the complex exponential of the complex number z, exp(z).
gsl_complex gsl_complex_log(gsl_complex z)
This function returns the complex natural logarithm (base e) of the complex number z, log(z). The branch cut
is the negative real axis.
gsl_complex gsl_complex_log10(gsl_complex z)
This function returns the complex base-10 logarithm of the complex number z, log10 (z).
gsl_complex gsl_complex_log_b(gsl_complex z, gsl_complex b)
This function returns the complex base-b logarithm of the complex number z, logb (z). This quantity is computed
as the ratio log(z)/ log(b).

5.7 Complex Trigonometric Functions
gsl_complex gsl_complex_sin(gsl_complex z)
This function returns the complex sine of the complex number z, sin(z) = (exp(iz) − exp(−iz))/(2i).
gsl_complex gsl_complex_cos(gsl_complex z)
This function returns the complex cosine of the complex number z, cos(z) = (exp(iz) + exp(−iz))/2.
gsl_complex gsl_complex_tan(gsl_complex z)
This function returns the complex tangent of the complex number z, tan(z) = sin(z)/ cos(z).
gsl_complex gsl_complex_sec(gsl_complex z)
This function returns the complex secant of the complex number z, sec(z) = 1/ cos(z).
gsl_complex gsl_complex_csc(gsl_complex z)
This function returns the complex cosecant of the complex number z, csc(z) = 1/ sin(z).
gsl_complex gsl_complex_cot(gsl_complex z)
This function returns the complex cotangent of the complex number z, cot(z) = 1/ tan(z).

26Chapter 5. Complex Numbers

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5.8 Inverse Complex Trigonometric Functions
gsl_complex gsl_complex_arcsin(gsl_complex z)
This function returns the complex arcsine of the complex number z, arcsin(z). The branch cuts are on the realaxis, less than −1 and greater than 1.
gsl_complex gsl_complex_arcsin_real(double z)
This function returns the complex arcsine of the real number z, arcsin(z). For z between −1 and 1, the function
returns a real value in the range [−π/2, π/2]. For z less than −1 the result has a real part of −π/2 and a positive
imaginary part. For z greater than 1 the result has a real part of π/2 and a negative imaginary part.
gsl_complex gsl_complex_arccos(gsl_complex z)
This function returns the complex arccosine of the complex number z, arccos(z). The branch cuts are on thereal axis, less than −1 and greater than 1.
gsl_complex gsl_complex_arccos_real(double z)
This function returns the complex arccosine of the real number z, arccos(z). For z between −1 and 1, thefunction returns a real value in the range [0, π]. For z less than −1 the result has a real part of π and a negative
imaginary part. For z greater than 1 the result is purely imaginary and positive.
gsl_complex gsl_complex_arctan(gsl_complex z)
This function returns the complex arctangent of the complex number z, arctan(z). The branch cuts are on theimaginary axis, below −i and above i.
gsl_complex gsl_complex_arcsec(gsl_complex z)
This function returns the complex arcsecant of the complex number z, arcsec(z) = arccos(1/z).
gsl_complex gsl_complex_arcsec_real(double z)
This function returns the complex arcsecant of the real number z, arcsec(z) = arccos(1/z).
gsl_complex gsl_complex_arccsc(gsl_complex z)
This function returns the complex arccosecant of the complex number z, arccsc(z) = arcsin(1/z).
gsl_complex gsl_complex_arccsc_real(double z)
This function returns the complex arccosecant of the real number z, arccsc(z) = arcsin(1/z).
gsl_complex gsl_complex_arccot(gsl_complex z)
This function returns the complex arccotangent of the complex number z, arccot(z) = arctan(1/z).

5.9 Complex Hyperbolic Functions
gsl_complex gsl_complex_sinh(gsl_complex z)
This function returns the complex hyperbolic sine of the complex number z, sinh(z) = (exp(z) − exp(−z))/2.
gsl_complex gsl_complex_cosh(gsl_complex z)
This function returns the complex hyperbolic cosine of the complex number z, cosh(z) = (exp(z)+exp(−z))/2.
gsl_complex gsl_complex_tanh(gsl_complex z)
This function returns the complex hyperbolic tangent of the complex number z, tanh(z) = sinh(z)/ cosh(z).
gsl_complex gsl_complex_sech(gsl_complex z)
This function returns the complex hyperbolic secant of the complex number z, sech(z) = 1/ cosh(z).
gsl_complex gsl_complex_csch(gsl_complex z)
This function returns the complex hyperbolic cosecant of the complex number z, csch(z) = 1/ sinh(z).
gsl_complex gsl_complex_coth(gsl_complex z)
This function returns the complex hyperbolic cotangent of the complex number z, coth(z) = 1/ tanh(z).

5.8. Inverse Complex Trigonometric Functions

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5.10 Inverse Complex Hyperbolic Functions
gsl_complex gsl_complex_arcsinh(gsl_complex z)
This function returns the complex hyperbolic arcsine of the complex number z, arcsinh(z). The branch cuts areon the imaginary axis, below −i and above i.
gsl_complex gsl_complex_arccosh(gsl_complex z)
This function returns the complex hyperbolic arccosine of the complex number z, arccosh(z). The branch cut ison the real axis, less than 1. Note that in this√ case we use the negative square root in formula 4.6.21 of Abramowitz& Stegun giving arccosh(z) = log(z − z 2 − 1).
gsl_complex gsl_complex_arccosh_real(double z)
This function returns the complex hyperbolic arccosine of the real number z, arccosh(z).
gsl_complex gsl_complex_arctanh(gsl_complex z)
This function returns the complex hyperbolic arctangent of the complex number z, arctanh(z). The branch cuts
are on the real axis, less than −1 and greater than 1.
gsl_complex gsl_complex_arctanh_real(double z)
This function returns the complex hyperbolic arctangent of the real number z, arctanh(z).
gsl_complex gsl_complex_arcsech(gsl_complex z)
This function returns the complex hyperbolic arcsecant of the complex number z, arcsech(z) = arccosh(1/z).
gsl_complex gsl_complex_arccsch(gsl_complex z)
This function returns the complex hyperbolic arccosecant of the complex number z, arccsch(z) = arcsinh(1/z).
gsl_complex gsl_complex_arccoth(gsl_complex z)
This function returns the complex hyperbolic arccotangent of the complex number z, arccoth(z)arctanh(1/z).

=

5.11 References and Further Reading
The implementations of the elementary and trigonometric functions are based on the following papers,• T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang, “Implementing Complex Elementary Functions Using Exception Handling”, ACM Transactions on Mathematical Software, Volume 20 (1994), pp 215–244, Corrigenda,
p553• T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang, “Implementing the complex arcsin and arccosine functions
using exception handling”, ACM Transactions on Mathematical Software, Volume 23 (1997) pp 299–335
The general formulas and details of branch cuts can be found in the following books,• Abramowitz and Stegun, Handbook of Mathematical Functions, “Circular Functions in Terms of Real and Imaginary Parts”, Formulas 4.3.55–58, “Inverse Circular Functions in Terms of Real and Imaginary Parts”, Formulas
4.4.37–39, “Hyperbolic Functions in Terms of Real and Imaginary Parts”, Formulas 4.5.49–52, “Inverse Hyperbolic Functions—relation to Inverse Circular Functions”, Formulas 4.6.14–19.
• Dave Gillespie, Calc Manual, Free Software Foundation, ISBN 1-882114-18-3

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CHAPTERSIXPOLYNOMIALSThis chapter describes functions for evaluating and solving polynomials. There are routines for finding real and complex
roots of quadratic and cubic equations using analytic methods. An iterative polynomial solver is also available forfinding the roots of general polynomials with real coefficients (of any order). The functions are declared in the header
file gsl_poly.h.

6.1 Polynomial Evaluation
The functions described here evaluate the polynomial
P (x) = c[0] + c[1]x + c[2]x2 + · · · + c[len − 1]xlen−1
using Horner’s method for stability. Inline versions of these functions are used when HAVE_INLINE is defined.
double gsl_poly_eval(const double c[], const int len, const double x)
This function evaluates a polynomial with real coefficients for the real variable x.
gsl_complex gsl_poly_complex_eval(const double c[], const int len, const gsl_complex z)
This function evaluates a polynomial with real coefficients for the complex variable z.
gsl_complex gsl_complex_poly_complex_eval(const gsl_complex c[], const int len, const gsl_complex z)
This function evaluates a polynomial with complex coefficients for the complex variable z.
int gsl_poly_eval_derivs(const double c[], const size_t lenc, const double x, double res[], const size_t lenres)This function evaluates a polynomial and its derivatives storing the results in the array res of size lenres. The
output array contains the values of dk P (x)/dxk for the specified value of x starting with k = 0.

6.2 Divided Difference Representation of Polynomials
The functions described here manipulate polynomials stored in Newton’s divided-difference representation. The use ofdivided-differences is described in Abramowitz & Stegun sections 25.1.4 and 25.2.26, and Burden and Faires, chapter
3, and discussed briefly below.
Given a function f (x), an nth degree interpolating polynomial Pn (x) can be constructed which agrees with f at
n + 1 distinct points x0 , x1 , ..., xn . This polynomial can be written in a form known as Newton’s divided-difference
representation
Pn (x) = f (x0 ) +nX[x0 , x1 , ..., xk ](x − x0 )(x − x1 ) · · · (x − xk−1 )

k=1

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where the divided differences [x0 , x1 , ..., xk ] are defined in section 25.1.4 of Abramowitz and Stegun. Additionally, it
is possible to construct an interpolating polynomial of degree 2n + 1 which also matches the first derivatives of f at
the points x0 , x1 , ..., xn . This is called the Hermite interpolating polynomial and is defined as
H2n+1 (x) = f (z0 ) +2n+1X[z0 , z1 , ..., zk ](x − z0 )(x − z1 ) · · · (x − zk−1 )k=1where the elements of z = {x0 , x0 , x1 , x1 , ..., xn , xn } are defined by z2k = z2k+1 = xk . The divided-differences[z0 , z1 , ..., zk ] are discussed in Burden and Faires, section 3.4.
int gsl_poly_dd_init(double dd[], const double xa[], const double ya[], size_t size)
This function computes a divided-difference representation of the interpolating polynomial for the points (x, y)
stored in the arrays xa and ya of length size. On output the divided-differences of (xa, ya) are stored in thearray dd, also of length size. Using the notation above, dd[k] = [x0 , x1 , ..., xk ].
double gsl_poly_dd_eval(const double dd[], const double xa[], const size_t size, const double x)
This function evaluates the polynomial stored in divided-difference form in the arrays dd and xa of length sizeat the point x. An inline version of this function is used when HAVE_INLINE is defined.
int gsl_poly_dd_taylor(double c[], double xp, const double dd[], const double xa[], size_t size, double w[])
This function converts the divided-difference representation of a polynomial to a Taylor expansion. The divideddifference representation is supplied in the arrays dd and xa of length size. On output the Taylor coefficientsof the polynomial expanded about the point xp are stored in the array c also of length size. A workspace oflength size must be provided in the array w.
int gsl_poly_dd_hermite_init(double dd[], double za[], const double xa[], const double ya[], const double dya[],const size_t size)
This function computes a divided-difference representation of the interpolating Hermite polynomial for the points(x, y) stored in the arrays xa and ya of length size. Hermite interpolation constructs polynomials which also
match first derivatives dy/dx which are provided in the array dya also of length size. The first derivatives canbe incorported into the usual divided-difference algorithm by forming a new dataset z = {x0 , x0 , x1 , x1 , ...},
which is stored in the array za of length 2*size on output. On output the divided-differences of the Hermite
representation are stored in the array dd, also of length 2*size. Using the notation above, dd[k] = [z0 , z1 , ..., zk ].
The resulting Hermite polynomial can be evaluated by calling gsl_poly_dd_eval() and using za for the input
argument xa.

6.3 Quadratic Equations
int gsl_poly_solve_quadratic(double a, double b, double c, double *x0, double *x1)
This function finds the real roots of the quadratic equation,
ax2 + bx + c = 0
The number of real roots (either zero, one or two) is returned, and their locations are stored in x0 and x1. If noreal roots are found then x0 and x1 are not modified. If one real root is found (i.e. if a = 0) then it is stored in
x0. When two real roots are found they are stored in x0 and x1 in ascending order. The case of coincident roots
is not considered special. For example (x − 1)2 = 0 will have two roots, which happen to have exactly equalvalues.
The number of roots found depends on the sign of the discriminant b2 − 4ac. This will be subject to rounding
and cancellation errors when computed in double precision, and will also be subject to errors if the coefficientsof the polynomial are inexact. These errors may cause a discrete change in the number of roots. However, forpolynomials with small integer coefficients the discriminant can always be computed exactly.

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int gsl_poly_complex_solve_quadratic(double a, double b, double c, gsl_complex *z0, gsl_complex *z1)
This function finds the complex roots of the quadratic equation,
az 2 + bz + c = 0
The number of complex roots is returned (either one or two) and the locations of the roots are stored in z0 andz1. The roots are returned in ascending order, sorted first by their real components and then by their imaginary
components. If only one real root is found (i.e. if a = 0) then it is stored in z0.

6.4 Cubic Equations
int gsl_poly_solve_cubic(double a, double b, double c, double *x0, double *x1, double *x2)
This function finds the real roots of the cubic equation,
x3 + ax2 + bx + c = 0
with a leading coefficient of unity. The number of real roots (either one or three) is returned, and their locations
are stored in x0, x1 and x2. If one real root is found then only x0 is modified. When three real roots are foundthey are stored in x0, x1 and x2 in ascending order. The case of coincident roots is not considered special. For
example, the equation (x − 1)3 = 0 will have three roots with exactly equal values. As in the quadratic case,finite precision may cause equal or closely-spaced real roots to move off the real axis into the complex plane,
leading to a discrete change in the number of real roots.
int gsl_poly_complex_solve_cubic(double a, double b, double c, gsl_complex *z0, gsl_complex *z1,
gsl_complex *z2)
This function finds the complex roots of the cubic equation,z 3 + az 2 + bz + c = 0
The number of complex roots is returned (always three) and the locations of the roots are stored in z0, z1 andz2. The roots are returned in ascending order, sorted first by their real components and then by their imaginary
components.

6.5 General Polynomial Equations
The roots of polynomial equations cannot be found analytically beyond the special cases of the quadratic, cubic andquartic equation. The algorithm described in this section uses an iterative method to find the approximate locations ofroots of higher order polynomials.
type gsl_poly_complex_workspace
This workspace contains parameters used for finding roots of general polynomials
gsl_poly_complex_workspace *gsl_poly_complex_workspace_alloc(size_t n)This function allocates space for a gsl_poly_complex_workspace struct and a workspace suitable for solving
a polynomial with n coefficients using the routine gsl_poly_complex_solve().
The function returns a pointer to the newly allocated gsl_poly_complex_workspace if no errors were detected,
and a null pointer in the case of error.
void gsl_poly_complex_workspace_free(gsl_poly_complex_workspace *w)
This function frees all the memory associated with the workspace w.

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int gsl_poly_complex_solve(const double *a, size_t n, gsl_poly_complex_workspace *w,
gsl_complex_packed_ptr z)
This function computes the roots of the general polynomial
P (x) = a0 + a1 x + a2 x2 + · · · + an−1 xn−1
using balanced-QR reduction of the companion matrix. The parameter n specifies the length of the coefficientarray. The coefficient of the highest order term must be non-zero. The function requires a workspace w of theappropriate size. The n − 1 roots are returned in the packed complex array z of length 2(n − 1), alternating realand imaginary parts.
The function returns GSL_SUCCESS if all the roots are found. If the QR reduction does not converge, the error handler is invoked with an error code of GSL_EFAILED. Note that due to finite precision, roots of higher
multiplicity are returned as a cluster of simple roots with reduced accuracy. The solution of polynomials withhigher-order roots requires specialized algorithms that take the multiplicity structure into account (see e.g. Z.
Zeng, Algorithm 835, ACM Transactions on Mathematical Software, Volume 30, Issue 2 (2004), pp 218–236).

6.6 Examples
To demonstrate the use of the general polynomial solver we will take the polynomial P (x) = x5 − 1 which has theseroots:
1, e2πi/5 , e4πi/5 , e6πi/5 , e8πi/5
The following program will find these roots.
#include <stdio.h>#include <gsl/gsl_poly.h>
intmain (void){
int i;/* coefficients of P(x) = -1 + x^5 */double a[6] = { -1, 0, 0, 0, 0, 1 };double z[10];gsl_poly_complex_workspace * w= gsl_poly_complex_workspace_alloc (6);gsl_poly_complex_solve (a, 6, w, z);
gsl_poly_complex_workspace_free (w);for (i = 0; i < 5; i++){printf ("z%d = %+.18f %+.18f \n",i, z[2*i], z[2*i+1]);}
return 0;}
The output of the program is
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z0 = -0.809016994374947673 +0.587785252292473359
z1 = -0.809016994374947673 -0.587785252292473359
z2 = +0.309016994374947507 +0.951056516295152976
z3 = +0.309016994374947507 -0.951056516295152976
z4 = +0.999999999999999889 +0.000000000000000000
which agrees with the analytic result, zn = exp(2πni/5).

6.7 References and Further Reading
The balanced-QR method and its error analysis are described in the following papers,• R.S. Martin, G. Peters and J.H. Wilkinson, “The QR Algorithm for Real Hessenberg Matrices”, Numerische
Mathematik, 14 (1970), 219–231.
• B.N. Parlett and C. Reinsch, “Balancing a Matrix for Calculation of Eigenvalues and Eigenvectors”, Numerische
Mathematik, 13 (1969), 293–304.
• A. Edelman and H. Murakami, “Polynomial roots from companion matrix eigenvalues”, Mathematics of Computation, Vol.: 64, No.: 210 (1995), 763–776.
The formulas for divided differences are given in the following texts,• Abramowitz and Stegun, Handbook of Mathematical Functions, Sections 25.1.4 and 25.2.26.
• R. L. Burden and J. D. Faires, Numerical Analysis, 9th edition, ISBN 0-538-73351-9, 2011.

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Chapter 6. Polynomials

CHAPTERSEVENSPECIAL FUNCTIONSThis chapter describes the GSL special function library. The library includes routines for calculating the values ofAiry functions, Bessel functions, Clausen functions, Coulomb wave functions, Coupling coefficients, the Dawson function, Debye functions, Dilogarithms, Elliptic integrals, Jacobi elliptic functions, Error functions, Exponential integrals,
Fermi-Dirac functions, Gamma functions, Gegenbauer functions, Hermite polynomials and functions, Hypergeometricfunctions, Laguerre functions, Legendre functions and Spherical Harmonics, the Psi (Digamma) Function, Synchrotron
functions, Transport functions, Trigonometric functions and Zeta functions. Each routine also computes an estimateof the numerical error in the calculated value of the function.
The functions in this chapter are declared in individual header files, such as gsl_sf_airy.h, gsl_sf_bessel.h, etc.
The complete set of header files can be included using the file gsl_sf.h.

7.1 Usage
The special functions are available in two calling conventions, a natural form which returns the numerical value of thefunction and an error-handling form which returns an error code. The two types of function provide alternative waysof accessing the same underlying code.
The natural form returns only the value of the function and can be used directly in mathematical expressions. For
example, the following function call will compute the value of the Bessel function J0 (x):double y = gsl_sf_bessel_J0 (x);There is no way to access an error code or to estimate the error using this method. To allow access to this information
the alternative error-handling form stores the value and error in a modifiable argument:
gsl_sf_result result;int status = gsl_sf_bessel_J0_e (x, &result);
The error-handling functions have the suffix _e. The returned status value indicates error conditions such as overflow,
underflow or loss of precision. If there are no errors the error-handling functions return GSL_SUCCESS.

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7.2 The gsl_sf_result struct
The error handling form of the special functions always calculate an error estimate along with the value of the result.
Therefore, structures are provided for amalgamating a value and error estimate. These structures are declared in theheader file gsl_sf_result.h.
The following struct contains value and error fields.
type gsl_sf_resulttypedef struct{
double val;
double err;
} gsl_sf_result;
The field val contains the value and the field err contains an estimate of the absolute error in the value.
In some cases, an overflow or underflow can be detected and handled by a function. In this case, it may be possible toreturn a scaling exponent as well as an error/value pair in order to save the result from exceeding the dynamic range ofthe built-in types. The following struct contains value and error fields as well as an exponent field such that the actual
result is obtained as result * 10^(e10).
type gsl_sf_result_e10
typedef struct{
double val;
double err;
int
e10;
} gsl_sf_result_e10;

7.3 Modes
The goal of the library is to achieve double precision accuracy wherever possible. However the cost of evaluating somespecial functions to double precision can be significant, particularly where very high order terms are required. In thesecases a mode argument, of type gsl_mode_t allows the accuracy of the function to be reduced in order to improve
performance. The following precision levels are available for the mode argument,
type gsl_mode_t
GSL_PREC_DOUBLE
Double-precision, a relative accuracy of approximately 2 ∗ 10−16 .
GSL_PREC_SINGLE
Single-precision, a relative accuracy of approximately 10−7 .
GSL_PREC_APPROXApproximate values, a relative accuracy of approximately 5 ∗ 10−4 .
The approximate mode provides the fastest evaluation at the lowest accuracy.

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7.4 Airy Functions and Derivatives
The Airy functions Ai(x) and Bi(x) are defined by the integral representations,Z
1 ∞cos(t3 /3 + xt) dt
Ai(x) =π 0Z
1 ∞ −t3 /3+xt(eBi(x) =+ sin(t3 /3 + xt)) dt
π 0
For further information see Abramowitz & Stegun, Section 10.4. The Airy functions are defined in the header file
gsl_sf_airy.h.

7.4.1 Airy Functions
double gsl_sf_airy_Ai(double x, gsl_mode_t mode)int gsl_sf_airy_Ai_e(double x, gsl_mode_t mode, gsl_sf_result *result)
These routines compute the Airy function Ai(x) with an accuracy specified by mode.
double gsl_sf_airy_Bi(double x, gsl_mode_t mode)int gsl_sf_airy_Bi_e(double x, gsl_mode_t mode, gsl_sf_result *result)
These routines compute the Airy function Bi(x) with an accuracy specified by mode.
double gsl_sf_airy_Ai_scaled(double x, gsl_mode_t mode)int gsl_sf_airy_Ai_scaled_e(double x, gsl_mode_t mode, gsl_sf_result *result)
These routines compute a scaled version of the Airy function SA (x)Ai(x). For x > 0 the scaling factor SA (x)is exp(+(2/3)x3/2 ), and is 1 for x < 0.
double gsl_sf_airy_Bi_scaled(double x, gsl_mode_t mode)int gsl_sf_airy_Bi_scaled_e(double x, gsl_mode_t mode, gsl_sf_result *result)
These routines compute a scaled version of the Airy function SB (x)Bi(x). For x > 0 the scaling factor SB (x)is exp(−(2/3)x3/2 ), and is 1 for x < 0.

7.4.2 Derivatives of Airy Functions
double gsl_sf_airy_Ai_deriv(double x, gsl_mode_t mode)int gsl_sf_airy_Ai_deriv_e(double x, gsl_mode_t mode, gsl_sf_result *result)
These routines compute the Airy function derivative Ai0 (x) with an accuracy specified by mode.
double gsl_sf_airy_Bi_deriv(double x, gsl_mode_t mode)int gsl_sf_airy_Bi_deriv_e(double x, gsl_mode_t mode, gsl_sf_result *result)
These routines compute the Airy function derivative Bi0 (x) with an accuracy specified by mode.
double gsl_sf_airy_Ai_deriv_scaled(double x, gsl_mode_t mode)int gsl_sf_airy_Ai_deriv_scaled_e(double x, gsl_mode_t mode, gsl_sf_result *result)
These routines compute the scaled Airy function derivative SA (x)Ai0 (x). For x > 0 the scaling factor SA (x) is
exp(+(2/3)x3/2 ), and is 1 for x < 0.
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double gsl_sf_airy_Bi_deriv_scaled(double x, gsl_mode_t mode)int gsl_sf_airy_Bi_deriv_scaled_e(double x, gsl_mode_t mode, gsl_sf_result *result)
These routines compute the scaled Airy function derivative SB (x)Bi0 (x). For x > 0 the scaling factor SB (x)is exp(−(2/3)x3/2 ), and is 1 for x < 0.

7.4.3 Zeros of Airy Functions
double gsl_sf_airy_zero_Ai(unsigned int s)
int gsl_sf_airy_zero_Ai_e(unsigned int s, gsl_sf_result *result)
These routines compute the location of the s-th zero of the Airy function Ai(x).
double gsl_sf_airy_zero_Bi(unsigned int s)
int gsl_sf_airy_zero_Bi_e(unsigned int s, gsl_sf_result *result)
These routines compute the location of the s-th zero of the Airy function Bi(x).

7.4.4 Zeros of Derivatives of Airy Functions
double gsl_sf_airy_zero_Ai_deriv(unsigned int s)
int gsl_sf_airy_zero_Ai_deriv_e(unsigned int s, gsl_sf_result *result)
These routines compute the location of the s-th zero of the Airy function derivative Ai0 (x).
double gsl_sf_airy_zero_Bi_deriv(unsigned int s)
int gsl_sf_airy_zero_Bi_deriv_e(unsigned int s, gsl_sf_result *result)
These routines compute the location of the s-th zero of the Airy function derivative Bi0 (x).

7.5 Bessel Functions
The routines described in this section compute the Cylindrical Bessel functions Jn (x), Yn (x), Modified cylindricalBessel functions In (x), Kn (x), Spherical Bessel functions jl (x), yl (x), and Modified Spherical Bessel functionsil (x), kl (x). For more information see Abramowitz & Stegun, Chapters 9 and 10. The Bessel functions are defined in
the header file gsl_sf_bessel.h.

7.5.1 Regular Cylindrical Bessel Functions
double gsl_sf_bessel_J0(double x)
int gsl_sf_bessel_J0_e(double x, gsl_sf_result *result)
These routines compute the regular cylindrical Bessel function of zeroth order, J0 (x).
double gsl_sf_bessel_J1(double x)
int gsl_sf_bessel_J1_e(double x, gsl_sf_result *result)
These routines compute the regular cylindrical Bessel function of first order, J1 (x).

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double gsl_sf_bessel_Jn(int n, double x)
int gsl_sf_bessel_Jn_e(int n, double x, gsl_sf_result *result)
These routines compute the regular cylindrical Bessel function of order n, Jn (x).
int gsl_sf_bessel_Jn_array(int nmin, int nmax, double x, double result_array[])This routine computes the values of the regular cylindrical Bessel functions Jn (x) for n from nmin to nmax
inclusive, storing the results in the array result_array. The values are computed using recurrence relations
for efficiency, and therefore may differ slightly from the exact values.

7.5.2 Irregular Cylindrical Bessel Functions
double gsl_sf_bessel_Y0(double x)
int gsl_sf_bessel_Y0_e(double x, gsl_sf_result *result)
These routines compute the irregular cylindrical Bessel function of zeroth order, Y0 (x), for x > 0.
double gsl_sf_bessel_Y1(double x)
int gsl_sf_bessel_Y1_e(double x, gsl_sf_result *result)
These routines compute the irregular cylindrical Bessel function of first order, Y1 (x), for x > 0.
double gsl_sf_bessel_Yn(int n, double x)
int gsl_sf_bessel_Yn_e(int n, double x, gsl_sf_result *result)
These routines compute the irregular cylindrical Bessel function of order n, Yn (x), for x > 0.
int gsl_sf_bessel_Yn_array(int nmin, int nmax, double x, double result_array[])This routine computes the values of the irregular cylindrical Bessel functions Yn (x) for n from nmin to nmax
inclusive, storing the results in the array result_array. The domain of the function is x > 0. The values arecomputed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.

7.5.3 Regular Modified Cylindrical Bessel Functions
double gsl_sf_bessel_I0(double x)
int gsl_sf_bessel_I0_e(double x, gsl_sf_result *result)
These routines compute the regular modified cylindrical Bessel function of zeroth order, I0 (x).
double gsl_sf_bessel_I1(double x)
int gsl_sf_bessel_I1_e(double x, gsl_sf_result *result)
These routines compute the regular modified cylindrical Bessel function of first order, I1 (x).
double gsl_sf_bessel_In(int n, double x)
int gsl_sf_bessel_In_e(int n, double x, gsl_sf_result *result)
These routines compute the regular modified cylindrical Bessel function of order n, In (x).
int gsl_sf_bessel_In_array(int nmin, int nmax, double x, double result_array[])This routine computes the values of the regular modified cylindrical Bessel functions In (x) for n from nmin to
nmax inclusive, storing the results in the array result_array. The start of the range nmin must be positive orzero. The values are computed using recurrence relations for efficiency, and therefore may differ slightly fromthe exact values.
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double gsl_sf_bessel_I0_scaled(double x)
int gsl_sf_bessel_I0_scaled_e(double x, gsl_sf_result *result)
These routines compute the scaled regular modified cylindrical Bessel function of zeroth order exp(−|x|)I0 (x).
double gsl_sf_bessel_I1_scaled(double x)
int gsl_sf_bessel_I1_scaled_e(double x, gsl_sf_result *result)
These routines compute the scaled regular modified cylindrical Bessel function of first order exp(−|x|)I1 (x).
double gsl_sf_bessel_In_scaled(int n, double x)
int gsl_sf_bessel_In_scaled_e(int n, double x, gsl_sf_result *result)
These routines compute the scaled regular modified cylindrical Bessel function of order n, exp(−|x|)In (x)int gsl_sf_bessel_In_scaled_array(int nmin, int nmax, double x, double result_array[])This routine computes the values of the scaled regular cylindrical Bessel functions exp(−|x|)In (x) for n fromnmin to nmax inclusive, storing the results in the array result_array. The start of the range nmin must be
positive or zero. The values are computed using recurrence relations for efficiency, and therefore may differ
slightly from the exact values.

7.5.4 Irregular Modified Cylindrical Bessel Functions
double gsl_sf_bessel_K0(double x)
int gsl_sf_bessel_K0_e(double x, gsl_sf_result *result)
These routines compute the irregular modified cylindrical Bessel function of zeroth order, K0 (x), for x > 0.
double gsl_sf_bessel_K1(double x)
int gsl_sf_bessel_K1_e(double x, gsl_sf_result *result)
These routines compute the irregular modified cylindrical Bessel function of first order, K1 (x), for x > 0.
double gsl_sf_bessel_Kn(int n, double x)
int gsl_sf_bessel_Kn_e(int n, double x, gsl_sf_result *result)
These routines compute the irregular modified cylindrical Bessel function of order n, Kn (x), for x > 0.
int gsl_sf_bessel_Kn_array(int nmin, int nmax, double x, double result_array[])This routine computes the values of the irregular modified cylindrical Bessel functions Kn (x) for n from nmin
to nmax inclusive, storing the results in the array result_array. The start of the range nmin must be positive
or zero. The domain of the function is x > 0. The values are computed using recurrence relations for efficiency,
and therefore may differ slightly from the exact values.
double gsl_sf_bessel_K0_scaled(double x)
int gsl_sf_bessel_K0_scaled_e(double x, gsl_sf_result *result)
These routines compute the scaled irregular modified cylindrical Bessel function of zeroth order exp(x)K0 (x)for x > 0.
double gsl_sf_bessel_K1_scaled(double x)
int gsl_sf_bessel_K1_scaled_e(double x, gsl_sf_result *result)
These routines compute the scaled irregular modified cylindrical Bessel function of first order exp(x)K1 (x) forx > 0.

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double gsl_sf_bessel_Kn_scaled(int n, double x)
int gsl_sf_bessel_Kn_scaled_e(int n, double x, gsl_sf_result *result)
These routines compute the scaled irregular modified cylindrical Bessel function of order n, exp(x)Kn (x), forx > 0.
int gsl_sf_bessel_Kn_scaled_array(int nmin, int nmax, double x, double result_array[])This routine computes the values of the scaled irregular cylindrical Bessel functions exp(x)Kn (x) for n fromnmin to nmax inclusive, storing the results in the array result_array. The start of the range nmin must be
positive or zero. The domain of the function is x > 0. The values are computed using recurrence relations for
efficiency, and therefore may differ slightly from the exact values.

7.5.5 Regular Spherical Bessel Functions
double gsl_sf_bessel_j0(double x)
int gsl_sf_bessel_j0_e(double x, gsl_sf_result *result)
These routines compute the regular spherical Bessel function of zeroth order, j0 (x) = sin(x)/x.
double gsl_sf_bessel_j1(double x)
int gsl_sf_bessel_j1_e(double x, gsl_sf_result *result)
These routines compute the regular spherical Bessel function of first order, j1 (x) = (sin(x)/x − cos(x))/x.
double gsl_sf_bessel_j2(double x)
int gsl_sf_bessel_j2_e(double x, gsl_sf_result *result)
These routines compute the regular spherical Bessel function of second order, j2 (x) = ((3/x2 − 1) sin(x) −3 cos(x)/x)/x.
double gsl_sf_bessel_jl(int l, double x)
int gsl_sf_bessel_jl_e(int l, double x, gsl_sf_result *result)
These routines compute the regular spherical Bessel function of order l, jl (x), for l ≥ 0 and x ≥ 0.
int gsl_sf_bessel_jl_array(int lmax, double x, double result_array[])This routine computes the values of the regular spherical Bessel functions jl (x) for l from 0 to lmax inclusive forlmax ≥ 0 and x ≥ 0, storing the results in the array result_array. The values are computed using recurrence
relations for efficiency, and therefore may differ slightly from the exact values.
int gsl_sf_bessel_jl_steed_array(int lmax, double x, double *result_array)
This routine uses Steed’s method to compute the values of the regular spherical Bessel functions jl (x) for l from0 to lmax inclusive for lmax ≥ 0 and x ≥ 0, storing the results in the array result_array. The Steed/Barnett
algorithm is described in Comp. Phys. Comm. 21, 297 (1981). Steed’s method is more stable than the recurrence
used in the other functions but is also slower.

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7.5.6 Irregular Spherical Bessel Functions
double gsl_sf_bessel_y0(double x)
int gsl_sf_bessel_y0_e(double x, gsl_sf_result *result)
These routines compute the irregular spherical Bessel function of zeroth order, y0 (x) = − cos(x)/x.
double gsl_sf_bessel_y1(double x)
int gsl_sf_bessel_y1_e(double x, gsl_sf_result *result)
These routines compute the irregular spherical Bessel function of first order, y1 (x) = −(cos(x)/x + sin(x))/x.
double gsl_sf_bessel_y2(double x)
int gsl_sf_bessel_y2_e(double x, gsl_sf_result *result)
These routines compute the irregular spherical Bessel function of second order, y2 (x) = (−3/x3 +1/x) cos(x)−(3/x2 ) sin(x).
double gsl_sf_bessel_yl(int l, double x)
int gsl_sf_bessel_yl_e(int l, double x, gsl_sf_result *result)
These routines compute the irregular spherical Bessel function of order l, yl (x), for l ≥ 0.
int gsl_sf_bessel_yl_array(int lmax, double x, double result_array[])This routine computes the values of the irregular spherical Bessel functions yl (x) for l from 0 to lmax inclusive
for lmax ≥ 0, storing the results in the array result_array. The values are computed using recurrence
relations for efficiency, and therefore may differ slightly from the exact values.

7.5.7 Regular Modified Spherical Bessel Functions
The regular
p modified spherical Bessel functions il (x) are related to the modified Bessel functions of fractional order,il (x) = π/(2x)Il+1/2 (x)double gsl_sf_bessel_i0_scaled(double x)
int gsl_sf_bessel_i0_scaled_e(double x, gsl_sf_result *result)
These routines compute the scaled regular modified spherical Bessel function of zeroth order, exp(−|x|)i0 (x).
double gsl_sf_bessel_i1_scaled(double x)
int gsl_sf_bessel_i1_scaled_e(double x, gsl_sf_result *result)
These routines compute the scaled regular modified spherical Bessel function of first order, exp(−|x|)i1 (x).
double gsl_sf_bessel_i2_scaled(double x)
int gsl_sf_bessel_i2_scaled_e(double x, gsl_sf_result *result)
These routines compute the scaled regular modified spherical Bessel function of second order, exp(−|x|)i2 (x)double gsl_sf_bessel_il_scaled(int l, double x)
int gsl_sf_bessel_il_scaled_e(int l, double x, gsl_sf_result *result)
These routines compute the scaled regular modified spherical Bessel function of order l, exp(−|x|)il (x)42Chapter 7. Special Functions

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int gsl_sf_bessel_il_scaled_array(int lmax, double x, double result_array[])This routine computes the values of the scaled regular modified spherical Bessel functions exp(−|x|)il (x) for l
from 0 to lmax inclusive for lmax ≥ 0, storing the results in the array result_array. The values are computed
using recurrence relations for efficiency, and therefore may differ slightly from the exact values.

7.5.8 Irregular Modified Spherical Bessel Functions
The irregular modified
pspherical Bessel functions kl (x) are related to the irregular modified Bessel functions of fractional order, kl (x) = π/(2x)Kl+1/2 (x).
double gsl_sf_bessel_k0_scaled(double x)
int gsl_sf_bessel_k0_scaled_e(double x, gsl_sf_result *result)
These routines compute the scaled irregular modified spherical Bessel function of zeroth order, exp(x)k0 (x),for x > 0.
double gsl_sf_bessel_k1_scaled(double x)
int gsl_sf_bessel_k1_scaled_e(double x, gsl_sf_result *result)
These routines compute the scaled irregular modified spherical Bessel function of first order, exp(x)k1 (x), forx > 0.
double gsl_sf_bessel_k2_scaled(double x)
int gsl_sf_bessel_k2_scaled_e(double x, gsl_sf_result *result)
These routines compute the scaled irregular modified spherical Bessel function of second order, exp(x)k2 (x),for x > 0.
double gsl_sf_bessel_kl_scaled(int l, double x)
int gsl_sf_bessel_kl_scaled_e(int l, double x, gsl_sf_result *result)
These routines compute the scaled irregular modified spherical Bessel function of order l, exp(x)kl (x), forx > 0.
int gsl_sf_bessel_kl_scaled_array(int lmax, double x, double result_array[])This routine computes the values of the scaled irregular modified spherical Bessel functions exp(x)kl (x) for l
from 0 to lmax inclusive for lmax ≥ 0 and x > 0, storing the results in the array result_array. The values
are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.

7.5.9 Regular Bessel Function—Fractional Orderdouble gsl_sf_bessel_Jnu(double nu, double x)
int gsl_sf_bessel_Jnu_e(double nu, double x, gsl_sf_result *result)
These routines compute the regular cylindrical Bessel function of fractional order ν, Jν (x).
int gsl_sf_bessel_sequence_Jnu_e(double nu, gsl_mode_t mode, size_t size, double v[])
This function computes the regular cylindrical Bessel function of fractional order ν, Jν (x), evaluated at a seriesof x values. The array v of length size contains the x values. They are assumed to be strictly ordered andpositive. The array is over-written with the values of Jν (xi ).

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7.5.10 Irregular Bessel Functions—Fractional Orderdouble gsl_sf_bessel_Ynu(double nu, double x)
int gsl_sf_bessel_Ynu_e(double nu, double x, gsl_sf_result *result)
These routines compute the irregular cylindrical Bessel function of fractional order ν, Yν (x).

7.5.11 Regular Modified Bessel Functions—Fractional Orderdouble gsl_sf_bessel_Inu(double nu, double x)
int gsl_sf_bessel_Inu_e(double nu, double x, gsl_sf_result *result)
These routines compute the regular modified Bessel function of fractional order ν, Iν (x) for x > 0, ν > 0.
double gsl_sf_bessel_Inu_scaled(double nu, double x)
int gsl_sf_bessel_Inu_scaled_e(double nu, double x, gsl_sf_result *result)
These routines compute the scaled regular modified Bessel function of fractional order ν, exp(−|x|)Iν (x) forx > 0, ν > 0.

7.5.12 Irregular Modified Bessel Functions—Fractional Orderdouble gsl_sf_bessel_Knu(double nu, double x)
int gsl_sf_bessel_Knu_e(double nu, double x, gsl_sf_result *result)
These routines compute the irregular modified Bessel function of fractional order ν, Kν (x) for x > 0, ν > 0.
double gsl_sf_bessel_lnKnu(double nu, double x)
int gsl_sf_bessel_lnKnu_e(double nu, double x, gsl_sf_result *result)
These routines compute the logarithm of the irregular modified Bessel function of fractional order ν, ln(Kν (x))
for x > 0, ν > 0.
double gsl_sf_bessel_Knu_scaled(double nu, double x)
int gsl_sf_bessel_Knu_scaled_e(double nu, double x, gsl_sf_result *result)
These routines compute the scaled irregular modified Bessel function of fractional order ν, exp(+|x|)Kν (x) forx > 0, ν > 0.

7.5.13 Zeros of Regular Bessel Functions
double gsl_sf_bessel_zero_J0(unsigned int s)
int gsl_sf_bessel_zero_J0_e(unsigned int s, gsl_sf_result *result)
These routines compute the location of the s-th positive zero of the Bessel function J0 (x).
double gsl_sf_bessel_zero_J1(unsigned int s)
int gsl_sf_bessel_zero_J1_e(unsigned int s, gsl_sf_result *result)
These routines compute the location of the s-th positive zero of the Bessel function J1 (x).

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double gsl_sf_bessel_zero_Jnu(double nu, unsigned int s)
int gsl_sf_bessel_zero_Jnu_e(double nu, unsigned int s, gsl_sf_result *result)
These routines compute the location of the s-th positive zero of the Bessel function Jν (x). The current implementation does not support negative values of nu.

7.6 Clausen Functions
The Clausen function is defined by the following integral,Z x
dt log (2 sin (t/2))
Cl2 (x) = −0

It is related to the dilogarithm by Cl2 (θ) = =Li2 (exp(iθ)). The Clausen functions are declared in the header file
gsl_sf_clausen.h.
double gsl_sf_clausen(double x)
int gsl_sf_clausen_e(double x, gsl_sf_result *result)
These routines compute the Clausen integral Cl2 (x).

7.7 Coulomb Functions
The prototypes of the Coulomb functions are declared in the header file gsl_sf_coulomb.h. Both bound state andscattering solutions are available.

7.7.1 Normalized Hydrogenic Bound States
double gsl_sf_hydrogenicR_1(double Z, double r)
int gsl_sf_hydrogenicR_1_e(double Z, double r, gsl_sf_result *result)
These√ routines compute the lowest-order normalized hydrogenic bound state radial wavefunction R1 :=2Z Z exp(−Zr).
double gsl_sf_hydrogenicR(int n, int l, double Z, double r)
int gsl_sf_hydrogenicR_e(int n, int l, double Z, double r, gsl_sf_result *result)
These routines compute the n-th normalized hydrogenic bound state radial wavefunction,l s
2Z 3/2 2Zr(n − l − 1)!
Rn :=exp(−Zr/n)L2l+1
n−l−1 (2Zr/n).
n2
n(n + l)!
where Lab (x) is the generalized Laguerre polynomial. The normalization is chosen such that the wavefunction
ψ is given by ψ(n, l, r) = Rn Ylm .

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7.7.2 Coulomb Wave Functions
The Coulomb wave functions FL (η, x), GL (η, x) are described in Abramowitz & Stegun, Chapter 14. Because there
can be a large dynamic range of values for these functions, overflows are handled gracefully. If an overflow occurs,
GSL_EOVRFLW is signalled and exponent(s) are returned through the modifiable parameters exp_F, exp_G. The full
solution can be reconstructed from the following relations,
FL (η, x) = f c[kL ] ∗ exp(expF )GL (η, x) = gc[kL ] ∗ exp(expG )FL0 (η, x) = f cp[kL ] ∗ exp(expF )G0L (η, x) = gcp[kL ] ∗ exp(expG )int gsl_sf_coulomb_wave_FG_e(double eta, double x, double L_F, int k, gsl_sf_result *F, gsl_sf_result *Fp,
gsl_sf_result *G, gsl_sf_result *Gp, double *exp_F, double *exp_G)
This function computes the Coulomb wave functions FL (η, x), GL−k (η, x) and their derivatives FL0 (η, x),
G0L−k (η, x) with respect to x. The parameters are restricted to L, L − k > −1/2, x > 0 and integer k. Note that
L itself is not restricted to being an integer. The results are stored in the parameters F, G for the function values
and Fp, Gp for the derivative values. If an overflow occurs, GSL_EOVRFLW is returned and scaling exponents are
stored in the modifiable parameters exp_F, exp_G.
int gsl_sf_coulomb_wave_F_array(double L_min, int kmax, double eta, double x, double fc_array[], double*F_exponent)
This function computes the Coulomb wave function FL (η, x) for L = Lmin . . . Lmin + kmax, storing theresults in fc_array. In the case of overflow the exponent is stored in F_exponent.
int gsl_sf_coulomb_wave_FG_array(double L_min, int kmax, double eta, double x, double fc_array[], double
gc_array[], double *F_exponent, double *G_exponent)
This function computes the functions FL (η, x), GL (η, x) for L = Lmin . . . Lmin + kmax storing the resultsin fc_array and gc_array. In the case of overflow the exponents are stored in F_exponent and G_exponent.
int gsl_sf_coulomb_wave_FGp_array(double L_min, int kmax, double eta, double x, double fc_array[], double
fcp_array[], double gc_array[], double gcp_array[], double *F_exponent,
double *G_exponent)
This function computes the functions FL (η, x), GL (η, x) and their derivatives FL0 (η, x), G0L (η, x) for L =Lmin . . . Lmin + kmax storing the results in fc_array, gc_array, fcp_array and gcp_array. In the caseof overflow the exponents are stored in F_exponent and G_exponent.
int gsl_sf_coulomb_wave_sphF_array(double L_min, int kmax, double eta, double x, double fc_array[], double
F_exponent[])
This function computes the Coulomb wave function divided by the argument FL (η, x)/x for L =Lmin . . . Lmin + kmax, storing the results in fc_array. In the case of overflow the exponent is stored in
F_exponent. This function reduces to spherical Bessel functions in the limit η → 0.

7.7.3 Coulomb Wave Function Normalization Constant
The Coulomb wave function normalization constant is defined in Abramowitz 14.1.7.
int gsl_sf_coulomb_CL_e(double L, double eta, gsl_sf_result *result)This function computes the Coulomb wave function normalization constant CL (η) for L > −1.
int gsl_sf_coulomb_CL_array(double Lmin, int kmax, double eta, double cl[])This function computes the Coulomb wave function normalization constant CL (η) for L = Lmin . . . Lmin +kmax, Lmin > −1.

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7.8 Coupling Coefficients
The Wigner 3-j, 6-j and 9-j symbols give the coupling coefficients for combined angular momentum vectors. Since
the arguments of the standard coupling coefficient functions are integer or half-integer, the arguments of the following functions are, by convention, integers equal to twice the actual spin value. For information on the 3-j coefficients see Abramowitz & Stegun, Section 27.9. The functions described in this section are declared in the header file
gsl_sf_coupling.h.

7.8.1 3-j Symbols
double gsl_sf_coupling_3j(int two_ja, int two_jb, int two_jc, int two_ma, int two_mb, int two_mc)
int gsl_sf_coupling_3j_e(int two_ja, int two_jb, int two_jc, int two_ma, int two_mb, int two_mc, gsl_sf_result*result)
These routines compute the Wigner 3-j coefficient,
ja jb jc
ma mb mc
where the arguments are given in half-integer units, ja = two_ja/2, ma = two_ma/2, etc.

7.8.2 6-j Symbols
double gsl_sf_coupling_6j(int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf)
int gsl_sf_coupling_6j_e(int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf, gsl_sf_result*result)
These routines compute the Wigner 6-j coefficient,
ja jb jc
jd je jf
where the arguments are given in half-integer units, ja = two_ja/2, ma = two_ma/2, etc.

7.8.3 9-j Symbols
double gsl_sf_coupling_9j(int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf, int two_jg, int
two_jh, int two_ji)
int gsl_sf_coupling_9j_e(int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf, int two_jg, int
two_jh, int two_ji, gsl_sf_result *result)
These routines compute the Wigner 9-j coefficient, ja jb jc 
jd je jf
jg jh ji
where the arguments are given in half-integer units, ja = two_ja/2, ma = two_ma/2, etc.

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7.9 Dawson Function
The Dawson integral is defined by2

Z xexp(−x )dt exp(t2 )0

A table of Dawson’s integral can be found in Abramowitz & Stegun, Table 7.5. The Dawson functions are declared in
the header file gsl_sf_dawson.h.
double gsl_sf_dawson(double x)
int gsl_sf_dawson_e(double x, gsl_sf_result *result)
These routines compute the value of Dawson’s integral for x.

7.10 Debye Functions
The Debye functions Dn (x) are defined by the following integral,
Z x
tn
n
dt t
Dn (x) = n
x 0
e −1
For further information see Abramowitz & Stegun, Section 27.1. The Debye functions are declared in the header file
gsl_sf_debye.h.
double gsl_sf_debye_1(double x)
int gsl_sf_debye_1_e(double x, gsl_sf_result *result)
These routines compute the first-order Debye function D1 (x).
double gsl_sf_debye_2(double x)
int gsl_sf_debye_2_e(double x, gsl_sf_result *result)
These routines compute the second-order Debye function D2 (x).
double gsl_sf_debye_3(double x)
int gsl_sf_debye_3_e(double x, gsl_sf_result *result)
These routines compute the third-order Debye function D3 (x).
double gsl_sf_debye_4(double x)
int gsl_sf_debye_4_e(double x, gsl_sf_result *result)
These routines compute the fourth-order Debye function D4 (x).
double gsl_sf_debye_5(double x)
int gsl_sf_debye_5_e(double x, gsl_sf_result *result)
These routines compute the fifth-order Debye function D5 (x).
double gsl_sf_debye_6(double x)
int gsl_sf_debye_6_e(double x, gsl_sf_result *result)
These routines compute the sixth-order Debye function D6 (x).

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7.11 Dilogarithm
The dilogarithm is defined as
Z z
Li2 (z) = −ds0log (1 − s)s

The functions described in this section are declared in the header file gsl_sf_dilog.h.

7.11.1 Real Argument
double gsl_sf_dilog(double x)
int gsl_sf_dilog_e(double x, gsl_sf_result *result)
These routines compute the dilogarithm for a real argument. In Lewin’s notation this is Li2 (x), the real part ofthe dilogarithm of a real x. It is defined by the integral representation
Z x
Li2 (x) = −<ds log(1 − s)/s0Note that =(Li2 (x)) = 0 for x ≤ 1, and −π log(x) for x > 1.
Note that Abramowitz & Stegun refer to the Spence integral S(x) = Li2 (1 − x) as the dilogarithm rather thanLi2 (x).

7.11.2 Complex Argument
int gsl_sf_complex_dilog_e(double r, double theta, gsl_sf_result *result_re, gsl_sf_result *result_im)
This function computes the full complex-valued dilogarithm for the complex argument z = r exp(iθ). The realand imaginary parts of the result are returned in result_re, result_im.

7.12 Elementary Operations
The following functions allow for the propagation of errors when combining quantities by multiplication. The functions
are declared in the header file gsl_sf_elementary.h.
double gsl_sf_multiply(double x, double y)
int gsl_sf_multiply_e(double x, double y, gsl_sf_result *result)This function multiplies x and y storing the product and its associated error in result.
int gsl_sf_multiply_err_e(double x, double dx, double y, double dy, gsl_sf_result *result)Thisp function multiplies x and y with associated absolute errors dx and dy. The product xy ±xy (dx/x)2 + (dy/y)2 is stored in result.

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7.13 Elliptic Integrals
The functions described in this section are declared in the header file gsl_sf_ellint.h. Further information about
the elliptic integrals can be found in Abramowitz & Stegun, Chapter 17.

7.13.1 Definition of Legendre Forms
The Legendre forms of elliptic integrals F (φ, k), E(φ, k) and Π(φ, k, n) are defined by,
Z φ
F (φ, k) =0

Z φE(φ, k) =dt q

1(1 − k 2 sin2 (t))

qdt (1 − k 2 sin2 (t))

0Z φΠ(φ, k, n) =

dt
0

1
q
(1 + n sin (t)) 1 − k 2 sin2 (t)2The complete Legendre forms are denoted by K(k) = F (π/2, k) and E(k) = E(π/2, k).
The notation used here is based on Carlson, “Numerische Mathematik” 33 (1979) 1 and differs slightly from that used
by Abramowitz & Stegun, where the functions are given in terms of the parameter m = k 2 and n is replaced by −n.

7.13.2 Definition of Carlson Forms
The Carlson symmetric forms of elliptical integrals RC(x, y), RD(x, y, z), RF (x, y, z) and RJ(x, y, z, p) are definedby,Z ∞RC(x, y) = 1/2
dt(t + x)−1/2 (t + y)−10
Z ∞RD(x, y, z) = 3/2
dt(t + x)−1/2 (t + y)−1/2 (t + z)−3/20
Z ∞RF (x, y, z) = 1/2
dt(t + x)−1/2 (t + y)−1/2 (t + z)−1/20
Z ∞RJ(x, y, z, p) = 3/2
dt(t + x)−1/2 (t + y)−1/2 (t + z)−1/2 (t + p)−107.13.3 Legendre Form of Complete Elliptic Integrals
double gsl_sf_ellint_Kcomp(double k, gsl_mode_t mode)int gsl_sf_ellint_Kcomp_e(double k, gsl_mode_t mode, gsl_sf_result *result)
These routines compute the complete elliptic integral K(k) to the accuracy specified by the mode variable mode.
Note that Abramowitz & Stegun define this function in terms of the parameter m = k 2 .
double gsl_sf_ellint_Ecomp(double k, gsl_mode_t mode)int gsl_sf_ellint_Ecomp_e(double k, gsl_mode_t mode, gsl_sf_result *result)
These routines compute the complete elliptic integral E(k) to the accuracy specified by the mode variable mode.
Note that Abramowitz & Stegun define this function in terms of the parameter m = k 2 .

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double gsl_sf_ellint_Pcomp(double k, double n, gsl_mode_t mode)int gsl_sf_ellint_Pcomp_e(double k, double n, gsl_mode_t mode, gsl_sf_result *result)
These routines compute the complete elliptic integral Π(k, n) to the accuracy specified by the mode variable
mode. Note that Abramowitz & Stegun define this function in terms of the parameters m = k 2 and sin2 (α) = k 2 ,with the change of sign n → −n.

7.13.4 Legendre Form of Incomplete Elliptic Integrals
double gsl_sf_ellint_F(double phi, double k, gsl_mode_t mode)int gsl_sf_ellint_F_e(double phi, double k, gsl_mode_t mode, gsl_sf_result *result)
These routines compute the incomplete elliptic integral F (φ, k) to the accuracy specified by the mode variable
mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k 2 .
double gsl_sf_ellint_E(double phi, double k, gsl_mode_t mode)int gsl_sf_ellint_E_e(double phi, double k, gsl_mode_t mode, gsl_sf_result *result)
These routines compute the incomplete elliptic integral E(φ, k) to the accuracy specified by the mode variable
mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k 2 .
double gsl_sf_ellint_P(double phi, double k, double n, gsl_mode_t mode)int gsl_sf_ellint_P_e(double phi, double k, double n, gsl_mode_t mode, gsl_sf_result *result)
These routines compute the incomplete elliptic integral Π(φ, k, n) to the accuracy specified by the mode variable
mode. Note that Abramowitz & Stegun define this function in terms of the parameters m = k 2 and sin2 (α) = k 2 ,with the change of sign n → −n.
double gsl_sf_ellint_D(double phi, double k, gsl_mode_t mode)int gsl_sf_ellint_D_e(double phi, double k, gsl_mode_t mode, gsl_sf_result *result)
These functions compute the incomplete elliptic integral D(φ, k) which is defined through the Carlson form
RD(x, y, z) by the following relation,D(φ, k) =1(sin φ)3 RD(1 − sin2 (φ), 1 − k 2 sin2 (φ), 1)
3

7.13.5 Carlson Forms
double gsl_sf_ellint_RC(double x, double y, gsl_mode_t mode)int gsl_sf_ellint_RC_e(double x, double y, gsl_mode_t mode, gsl_sf_result *result)
These routines compute the incomplete elliptic integral RC(x, y) to the accuracy specified by the mode variable
mode.
double gsl_sf_ellint_RD(double x, double y, double z, gsl_mode_t mode)int gsl_sf_ellint_RD_e(double x, double y, double z, gsl_mode_t mode, gsl_sf_result *result)
These routines compute the incomplete elliptic integral RD(x, y, z) to the accuracy specified by the mode variable mode.
double gsl_sf_ellint_RF(double x, double y, double z, gsl_mode_t mode)

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int gsl_sf_ellint_RF_e(double x, double y, double z, gsl_mode_t mode, gsl_sf_result *result)
These routines compute the incomplete elliptic integral RF (x, y, z) to the accuracy specified by the mode variable mode.
double gsl_sf_ellint_RJ(double x, double y, double z, double p, gsl_mode_t mode)int gsl_sf_ellint_RJ_e(double x, double y, double z, double p, gsl_mode_t mode, gsl_sf_result *result)
These routines compute the incomplete elliptic integral RJ(x, y, z, p) to the accuracy specified by the mode
variable mode.

7.14 Elliptic Functions (Jacobi)
The Jacobian Elliptic functions are defined in Abramowitz & Stegun, Chapter 16. The functions are declared in theheader file gsl_sf_elljac.h.
int gsl_sf_elljac_e(double u, double m, double *sn, double *cn, double *dn)
This function computes the Jacobian elliptic functions sn(u|m), cn(u|m), dn(u|m) by descending Landen transformations.

7.15 Error Functions
The error function is described in Abramowitz & Stegun, Chapter 7. The functions in this section are declared in theheader file gsl_sf_erf.h.

7.15.1 Error Function
double gsl_sf_erf(double x)
int gsl_sf_erf_e(double x, gsl_sf_result *result)√ Rx
These routines compute the error function erf(x), where erf(x) = (2/ π) 0 dt exp(−t2 ).

7.15.2 Complementary Error Function
double gsl_sf_erfc(double x)
int gsl_sf_erfc_e(double x, gsl_sf_result *result)√ R∞
These routines compute the complementary error function erfc(x) = 1 − erf(x) = (2/ π) x exp(−t2 )7.15.3 Log Complementary Error Function
double gsl_sf_log_erfc(double x)
int gsl_sf_log_erfc_e(double x, gsl_sf_result *result)
These routines compute the logarithm of the complementary error function log(erfc(x)).

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7.15.4 Probability functions
The probability functions for the Normal or Gaussian distribution are described in Abramowitz & Stegun, Section 26.2.
double gsl_sf_erf_Z(double x)
int gsl_sf_erf_Z_e(double x, gsl_sf_result *result)√
These routines compute the Gaussian probability density function Z(x) = (1/ 2π) exp(−x2 /2)
double gsl_sf_erf_Q(double x)
int gsl_sf_erf_Q_e(double x, gsl_sf_result *result)√
R∞
These routines compute the upper tail of the Gaussian probability function Q(x) = (1/ 2π) x dt exp(−t2 /2)
The hazard function for the normal distribution, also known as the inverse Mills’ ratio, is defined as,r
2 exp(−x2 /2)Z(x)√=h(x) =Q(x)π erfc(x/ 2)
It decreases rapidly as x approaches −∞ and asymptotes to h(x) ∼ x as x approaches +∞.
double gsl_sf_hazard(double x)
int gsl_sf_hazard_e(double x, gsl_sf_result *result)
These routines compute the hazard function for the normal distribution.

7.16 Exponential Functions
The functions described in this section are declared in the header file gsl_sf_exp.h.

7.16.1 Exponential Function
double gsl_sf_exp(double x)
int gsl_sf_exp_e(double x, gsl_sf_result *result)
These routines provide an exponential function exp(x) using GSL semantics and error checking.
int gsl_sf_exp_e10_e(double x, gsl_sf_result_e10 *result)This function computes the exponential exp(x) using the gsl_sf_result_e10 type to return a result withextended range. This function may be useful if the value of exp(x) would overflow the numeric range of double.
double gsl_sf_exp_mult(double x, double y)
int gsl_sf_exp_mult_e(double x, double y, gsl_sf_result *result)
These routines exponentiate x and multiply by the factor y to return the product y exp(x).
int gsl_sf_exp_mult_e10_e(const double x, const double y, gsl_sf_result_e10 *result)This function computes the product y exp(x) using the gsl_sf_result_e10 type to return a result with extended numeric range.

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7.16.2 Relative Exponential Functions
double gsl_sf_expm1(double x)
int gsl_sf_expm1_e(double x, gsl_sf_result *result)
These routines compute the quantity exp(x) − 1 using an algorithm that is accurate for small x.
double gsl_sf_exprel(double x)
int gsl_sf_exprel_e(double x, gsl_sf_result *result)
These routines compute the quantity (exp(x) − 1)/x using an algorithm that is accurate for small x. For small
x the algorithm is based on the expansion (exp(x) − 1)/x = 1 + x/2 + x2 /(2 ∗ 3) + x3 /(2 ∗ 3 ∗ 4) + . . ..
double gsl_sf_exprel_2(double x)
int gsl_sf_exprel_2_e(double x, gsl_sf_result *result)
These routines compute the quantity 2(exp(x) − 1 − x)/x2 using an algorithm that is accurate for small x. For
small x the algorithm is based on the expansion 2(exp(x)−1−x)/x2 = 1+x/3+x2 /(3∗4)+x3 /(3∗4∗5)+. . ..
double gsl_sf_exprel_n(int n, double x)
int gsl_sf_exprel_n_e(int n, double x, gsl_sf_result *result)
These routines compute the N -relative exponential, which is the n-th generalization of the functions
gsl_sf_exprel() and gsl_sf_exprel_2(). The N -relative exponential is given by,!
N−1
X
exprelN (x) = N !/xN exp(x) −xk /k!
k=0= 1 + x/(N + 1) + x2 /((N + 1)(N + 2)) + . . .
= 1 F1 (1, 1 + N, x)

7.16.3 Exponentiation With Error Estimate
int gsl_sf_exp_err_e(double x, double dx, gsl_sf_result *result)This function exponentiates x with an associated absolute error dx.
int gsl_sf_exp_err_e10_e(double x, double dx, gsl_sf_result_e10 *result)This function exponentiates a quantity x with an associated absolute error dx using the gsl_sf_result_e10
type to return a result with extended range.
int gsl_sf_exp_mult_err_e(double x, double dx, double y, double dy, gsl_sf_result *result)This routine computes the product y exp(x) for the quantities x, y with associated absolute errors dx, dy.
int gsl_sf_exp_mult_err_e10_e(double x, double dx, double y, double dy, gsl_sf_result_e10 *result)This routine computes the product y exp(x) for the quantities x, y with associated absolute errors dx, dy using
the gsl_sf_result_e10 type to return a result with extended range.

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7.17 Exponential Integrals
Information on the exponential integrals can be found in Abramowitz & Stegun, Chapter 5. These functions are declaredin the header file gsl_sf_expint.h.

7.17.1 Exponential Integral
double gsl_sf_expint_E1(double x)
int gsl_sf_expint_E1_e(double x, gsl_sf_result *result)
These routines compute the exponential integral E1 (x),Z ∞E1 (x) := <dt exp(−xt)/t.
1

double gsl_sf_expint_E2(double x)
int gsl_sf_expint_E2_e(double x, gsl_sf_result *result)
These routines compute the second-order exponential integral E2 (x),Z ∞E2 (x) := <dt exp(−xt)/t2
1

double gsl_sf_expint_En(int n, double x)
int gsl_sf_expint_En_e(int n, double x, gsl_sf_result *result)
These routines compute the exponential integral En (x) of order n,
Z ∞En (x) := <dt exp(−xt)/tn .
1

7.17.2 Ei(x)double gsl_sf_expint_Ei(double x)
int gsl_sf_expint_Ei_e(double x, gsl_sf_result *result)
These routines compute the exponential integral Ei(x),Z ∞
Ei(x) = −P V
dt exp(−t)/t−xwhere P V denotes the principal value of the integral.

7.17.3 Hyperbolic Integrals
double gsl_sf_Shi(double x)
int gsl_sf_Shi_e(double x, gsl_sf_result *result)
These routines compute the integral
Shi(x) =Z x
dt sinh(t)/t07.17. Exponential Integrals

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double gsl_sf_Chi(double x)
int gsl_sf_Chi_e(double x, gsl_sf_result *result)
These routines compute the integralZ xdt(cosh(t) − 1)/t
Chi(x) := < γE + log(x) +0where γE is the Euler constant (available as the macro M_EULER).

7.17.4 Ei_3(x)double gsl_sf_expint_3(double x)
int gsl_sf_expint_3_e(double x, gsl_sf_result *result)
These routines compute the third-order exponential integral
Z x
Ei3 (x) =dt exp(−t3 )0

for x ≥ 0.

7.17.5 Trigonometric Integrals
double gsl_sf_Si(const double x)
int gsl_sf_Si_e(double x, gsl_sf_result *result)
These routines compute the Sine integral
Si(x) =

Z x
dt sin(t)/t0double gsl_sf_Ci(const double x)
int gsl_sf_Ci_e(double x, gsl_sf_result *result)
These routines compute the Cosine integral
Ci(x) = −Z ∞dt cos(t)/tx

for x > 07.17.6 Arctangent Integral
double gsl_sf_atanint(double x)
int gsl_sf_atanint_e(double x, gsl_sf_result *result)
These routines compute the Arctangent integral, which is defined as
Z xAtanInt(x) =dt arctan(t)/t
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7.18 Fermi-Dirac Function
The functions described in this section are declared in the header file gsl_sf_fermi_dirac.h.

7.18.1 Complete Fermi-Dirac Integrals
The complete Fermi-Dirac integral Fj (x) is given by,
1
Fj (x) :=Γ(j + 1)

Z ∞
dt
0

tj
(exp(t − x) + 1)

Note that the Fermi-Dirac integral is sometimes defined without the normalisation factor in other texts.
double gsl_sf_fermi_dirac_m1(double x)
int gsl_sf_fermi_dirac_m1_e(double x, gsl_sf_result *result)
These routines compute the complete Fermi-Dirac integral with an index of −1. This integral is given by
F−1 (x) = ex /(1 + ex ).
double gsl_sf_fermi_dirac_0(double x)
int gsl_sf_fermi_dirac_0_e(double x, gsl_sf_result *result)
These routines compute the complete Fermi-Dirac integral with an index of 0. This integral is given by F0 (x) =ln(1 + ex ).
double gsl_sf_fermi_dirac_1(double x)
int gsl_sf_fermi_dirac_1_e(double x, gsl_sf_result *result)
R∞
These routines compute the complete Fermi-Dirac integral with an index of 1, F1 (x) = 0 dt(t/(exp(t − x) +1)).
double gsl_sf_fermi_dirac_2(double x)
int gsl_sf_fermi_dirac_2_e(double x, gsl_sf_result *result)
R∞
These routines compute the complete Fermi-Dirac integral with an index of 2, F2 (x) = (1/2) 0 dt(t2 /(exp(t−x) + 1)).
double gsl_sf_fermi_dirac_int(int j, double x)
int gsl_sf_fermi_dirac_int_e(int j, double x, gsl_sf_result *result)
These
R ∞routines compute the complete Fermi-Dirac integral with an integer index of j, Fj (x) = (1/Γ(j +1)) 0 dt(tj /(exp(t − x) + 1)).
double gsl_sf_fermi_dirac_mhalf(double x)
int gsl_sf_fermi_dirac_mhalf_e(double x, gsl_sf_result *result)
These routines compute the complete Fermi-Dirac integral F−1/2 (x).
double gsl_sf_fermi_dirac_half(double x)
int gsl_sf_fermi_dirac_half_e(double x, gsl_sf_result *result)
These routines compute the complete Fermi-Dirac integral F1/2 (x).
double gsl_sf_fermi_dirac_3half(double x)

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int gsl_sf_fermi_dirac_3half_e(double x, gsl_sf_result *result)
These routines compute the complete Fermi-Dirac integral F3/2 (x).

7.18.2 Incomplete Fermi-Dirac Integrals
The incomplete Fermi-Dirac integral Fj (x, b) is given by,
1
Fj (x, b) :=Γ(j + 1)

Z ∞
dt
b

tj
(exp(t − x) + 1)

double gsl_sf_fermi_dirac_inc_0(double x, double b)
int gsl_sf_fermi_dirac_inc_0_e(double x, double b, gsl_sf_result *result)
These routines compute the incomplete Fermi-Dirac integral with an index of zero, F0 (x, b) = ln(1 + eb−x ) −(b − x)7.19 Gamma and Beta Functions
The following routines compute the gamma and beta functions in their full and incomplete forms, as well as various
kinds of factorials. The functions described in this section are declared in the header file gsl_sf_gamma.h.

7.19.1 Gamma Functions
The Gamma function is defined by the following integral,Z ∞Γ(x) =
dttx−1 exp(−t)
0It is related to the factorial function by Γ(n) = (n − 1)! for positive integer n. Further information on the Gamma
function can be found in Abramowitz & Stegun, Chapter 6.
double gsl_sf_gamma(double x)
int gsl_sf_gamma_e(double x, gsl_sf_result *result)
These routines compute the Gamma function Γ(x), subject to x not being a negative integer or zero. The function
is computed using the real Lanczos method. The maximum value of x such that Γ(x) is not considered an
overflow is given by the macro GSL_SF_GAMMA_XMAX and is 171.0.
double gsl_sf_lngamma(double x)
int gsl_sf_lngamma_e(double x, gsl_sf_result *result)
These routines compute the logarithm of the Gamma function, log(Γ(x)), subject to x not being a negative integer
or zero. For x < 0 the real part of log(Γ(x)) is returned, which is equivalent to log(|Γ(x)|). The function iscomputed using the real Lanczos method.
int gsl_sf_lngamma_sgn_e(double x, gsl_sf_result *result_lg, double *sgn)
This routine computes the sign of the gamma function and the logarithm of its magnitude, subject to x not beinga negative integer or zero. The function is computed using the real Lanczos method. The value of the gammafunction and its error can be reconstructed using the relation Γ(x) = sgn ∗ exp(result_lg), taking into accountthe two components of result_lg.

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double gsl_sf_gammastar(double x)
int gsl_sf_gammastar_e(double x, gsl_sf_result *result)
These routines compute the regulated Gamma Function Γ∗ (x) for x > 0. The regulated gamma function is givenby,√Γ∗ (x) = Γ(x)/( 2πx(x−1/2) exp(−x))


1
= 1+
+ ...
for x → ∞12x
and is a useful suggestion of Temme.
double gsl_sf_gammainv(double x)
int gsl_sf_gammainv_e(double x, gsl_sf_result *result)
These routines compute the reciprocal of the gamma function, 1/Γ(x) using the real Lanczos method.
int gsl_sf_lngamma_complex_e(double zr, double zi, gsl_sf_result *lnr, gsl_sf_result *arg)
This routine computes log(Γ(z)) for complex z = zr +izi and z not a negative integer or zero, using the complex
Lanczos method. The returned parameters are lnr = log |Γ(z)| and arg = arg(Γ(z)) in (−π, π]. Note that thephase part (arg) is not well-determined when |z| is very large, due to inevitable roundoff in restricting to (−π, π].
This will result in a GSL_ELOSS error when it occurs. The absolute value part (lnr), however, never suffers fromloss of precision.

7.19.2 Factorials
Although factorials can be computed from the Gamma function, using the relation n! = Γ(n + 1) for non-negative
integer n, it is usually more efficient to call the functions in this section, particularly for small values of n, whose
factorial values are maintained in hardcoded tables.
double gsl_sf_fact(unsigned int n)
int gsl_sf_fact_e(unsigned int n, gsl_sf_result *result)
These routines compute the factorial n!. The factorial is related to the Gamma function by n! = Γ(n + 1). The
maximum value of n such that n! is not considered an overflow is given by the macro GSL_SF_FACT_NMAX andis 170.
double gsl_sf_doublefact(unsigned int n)
int gsl_sf_doublefact_e(unsigned int n, gsl_sf_result *result)
These routines compute the double factorial n!! = n(n − 2)(n − 4) . . .. The maximum value of n such that n!!
is not considered an overflow is given by the macro GSL_SF_DOUBLEFACT_NMAX and is 297.
double gsl_sf_lnfact(unsigned int n)
int gsl_sf_lnfact_e(unsigned int n, gsl_sf_result *result)
These routines compute the logarithm of the factorial of n, log(n!). The algorithm is faster than computingln(Γ(n + 1)) via gsl_sf_lngamma() for n < 170, but defers for larger n.
double gsl_sf_lndoublefact(unsigned int n)
int gsl_sf_lndoublefact_e(unsigned int n, gsl_sf_result *result)
These routines compute the logarithm of the double factorial of n, log(n!!).
double gsl_sf_choose(unsigned int n, unsigned int m)

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int gsl_sf_choose_e(unsigned int n, unsigned int m, gsl_sf_result *result)
These routines compute the combinatorial factor n choose m = n!/(m!(n − m)!)double gsl_sf_lnchoose(unsigned int n, unsigned int m)
int gsl_sf_lnchoose_e(unsigned int n, unsigned int m, gsl_sf_result *result)
These routines compute the logarithm of n choose m. This is equivalent to the sum log(n!) − log(m!) −log((n − m)!).
double gsl_sf_taylorcoeff(int n, double x)
int gsl_sf_taylorcoeff_e(int n, double x, gsl_sf_result *result)
These routines compute the Taylor coefficient xn /n! for x ≥ 0, n ≥ 0

7.19.3 Pochhammer Symbol
double gsl_sf_poch(double a, double x)
int gsl_sf_poch_e(double a, double x, gsl_sf_result *result)
These routines compute the Pochhammer symbol (a)x = Γ(a + x)/Γ(a). The Pochhammer symbol is also
known as the Apell symbol and sometimes written as (a, x). When a and a + x are negative integers or zero,
the limiting value of the ratio is returned.
double gsl_sf_lnpoch(double a, double x)
int gsl_sf_lnpoch_e(double a, double x, gsl_sf_result *result)
These routines compute the logarithm of the Pochhammer symbol, log((a)x ) = log(Γ(a + x)/Γ(a)).
int gsl_sf_lnpoch_sgn_e(double a, double x, gsl_sf_result *result, double *sgn)
These routines compute the sign of the Pochhammer symbol and the logarithm of its magnitude. The computed
parameters are result = log(|(a)x |) with a corresponding error term, and sgn = sgn((a)x ) where (a)x =Γ(a + x)/Γ(a).
double gsl_sf_pochrel(double a, double x)
int gsl_sf_pochrel_e(double a, double x, gsl_sf_result *result)
These routines compute the relative Pochhammer symbol ((a)x − 1)/x where (a)x = Γ(a + x)/Γ(a).

7.19.4 Incomplete Gamma Functions
double gsl_sf_gamma_inc(double a, double x)
int gsl_sf_gamma_inc_e(double a, double x, gsl_sf_result *result)
R∞
These functions compute the unnormalized incomplete Gamma Function Γ(a, x) = x dtt(a−1) exp(−t) for areal and x ≥ 0.
double gsl_sf_gamma_inc_Q(double a, double x)
int gsl_sf_gamma_inc_Q_e(double a, double x, gsl_sf_result *result)
R∞
These routines compute the normalized incomplete Gamma Function Q(a, x) = 1/Γ(a) x dtt(a−1) exp(−t)
for a > 0, x ≥ 0.
double gsl_sf_gamma_inc_P(double a, double x)

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int gsl_sf_gamma_inc_P_e(double a, double x, gsl_sf_result *result)
These routinescompute the complementary normalized incomplete Gamma Function P (a, x) = 1 − Q(a, x) =Rx
1/Γ(a) 0 dtt(a−1) exp(−t) for a > 0, x ≥ 0.
Note that Abramowitz & Stegun call P (a, x) the incomplete gamma function (section 6.5).

7.19.5 Beta Functions
double gsl_sf_beta(double a, double b)
int gsl_sf_beta_e(double a, double b, gsl_sf_result *result)
These routines compute the Beta Function, B(a, b) = Γ(a)Γ(b)/Γ(a + b) subject to a and b not being negative
integers.
double gsl_sf_lnbeta(double a, double b)
int gsl_sf_lnbeta_e(double a, double b, gsl_sf_result *result)
These routines compute the logarithm of the Beta Function, log(B(a, b)) subject to a and b not being negative
integers.

7.19.6 Incomplete Beta Function
double gsl_sf_beta_inc(double a, double b, double x)
int gsl_sf_beta_inc_e(double a, double b, double x, gsl_sf_result *result)
These routines compute the normalized incomplete Beta function Ix (a, b) = Bx (a, b)/B(a, b) whereZ x
Bx (a, b) =ta−1 (1 − t)b−1 dt0

for 0 ≤ x ≤ 1. For a > 0, b > 0 the value is computed using a continued fraction expansion. For all othervalues it is computed using the relation
Ix (a, b, x) = (1/a)xa 2 F1 (a, 1 − b, a + 1, x)/B(a, b)

7.20 Gegenbauer Functions
The Gegenbauer polynomials are defined in Abramowitz & Stegun, Chapter 22, where they are known as Ultrasphericalpolynomials. The functions described in this section are declared in the header file gsl_sf_gegenbauer.h.
double gsl_sf_gegenpoly_1(double lambda, double x)
double gsl_sf_gegenpoly_2(double lambda, double x)
double gsl_sf_gegenpoly_3(double lambda, double x)
int gsl_sf_gegenpoly_1_e(double lambda, double x, gsl_sf_result *result)int gsl_sf_gegenpoly_2_e(double lambda, double x, gsl_sf_result *result)

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int gsl_sf_gegenpoly_3_e(double lambda, double x, gsl_sf_result *result)(λ)

These functions evaluate the Gegenbauer polynomials Cn (x) using explicit representations for n = 1, 2, 3.
double gsl_sf_gegenpoly_n(int n, double lambda, double x)
int gsl_sf_gegenpoly_n_e(int n, double lambda, double x, gsl_sf_result *result)(λ)

These functions evaluate the Gegenbauer polynomial Cn (x) for a specific value of n, lambda, x subject toλ > −1/2, n ≥ 0.
int gsl_sf_gegenpoly_array(int nmax, double lambda, double x, double result_array[])(λ)

This function computes an array of Gegenbauer polynomials Cn (x) for n = 0, 1, 2, . . . , nmax, subject toλ > −1/2, nmax ≥ 0.

7.21 Hermite Polynomials and Functions
Hermite polynomials and functions are discussed in Abramowitz & Stegun, Chapter 22 and Szego, Gabor (1939, 1958,
1967), Orthogonal Polynomials, American Mathematical Society. The Hermite polynomials and functions are definedin the header file gsl_sf_hermite.h.

7.21.1 Hermite Polynomials
The Hermite polynomials exist in two variants: the physicist version Hn (x) and the probabilist version Hen (x). They
are defined by the derivatives
 n
2
d
n x2
e−x
Hn (x) = (−1) e
dx
 n
2
2
d
Hen (x) = (−1)n ex /2
e−x /2dx
They are connected via
√


2x


x
−n/2
√
Hen (x) = 2
Hn2
Hn (x) = 2n/2 Hen

and satisfy the ordinary differential equations
Hn00 (x) − 2xHn0 (x) + 2nHn (x) = 0
He00n (x) − xHe0n (x) + nHen (x) = 0
double gsl_sf_hermite(const int n, const double x)
int gsl_sf_hermite_e(const int n, const double x, gsl_sf_result *result)
These routines evaluate the physicist Hermite polynomial Hn (x) of order n at position x. If an overflow is
detected, GSL_EOVRFLW is returned without calling the error handler.
int gsl_sf_hermite_array(const int nmax, const double x, double *result_array)
This routine evaluates all physicist Hermite polynomials Hn up to order nmax at position x. The results are
stored in result_array.
double gsl_sf_hermite_series(const int n, const double x, const double *a)62

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int gsl_sf_hermite_series_e(const int
Pnn, const double x, const double *a, gsl_sf_result *result)
These routines evaluate the series j=0 aj Hj (x) with Hj being the j-th physicist Hermite polynomial using
the Clenshaw algorithm.
double gsl_sf_hermite_prob(const int n, const double x)
int gsl_sf_hermite_prob_e(const int n, const double x, gsl_sf_result *result)
These routines evaluate the probabilist Hermite polynomial Hen (x) of order n at position x. If an overflow is
detected, GSL_EOVRFLW is returned without calling the error handler.
int gsl_sf_hermite_prob_array(const int nmax, const double x, double *result_array)
This routine evaluates all probabilist Hermite polynomials Hen (x) up to order nmax at position x. The results
are stored in result_array.
double gsl_sf_hermite_prob_series(const int n, const double x, const double *a)
int gsl_sf_hermite_prob_series_e(const
Pn int n, const double x, const double *a, gsl_sf_result *result)
These routines evaluate the series j=0 aj Hej (x) with Hej being the j-th probabilist Hermite polynomialusing the Clenshaw algorithm.

7.21.2 Derivatives of Hermite Polynomials
double gsl_sf_hermite_deriv(const int m, const int n, const double x)
int gsl_sf_hermite_deriv_e(const int m, const int n, const double x, gsl_sf_result *result)
These routines evaluate the m-th derivative of the physicist Hermite polynomial Hn (x) of order n at position x.
int gsl_sf_hermite_array_deriv(const int m, const int nmax, const double x, double *result_array)
This routine evaluates the m-th derivative of all physicist Hermite polynomials Hn (x) from orders 0, . . . , nmax
at position x. The result dm /dxm Hn (x) is stored in result_array[n]. The output result_array must havelength at least nmax + 1.
int gsl_sf_hermite_deriv_array(const int mmax, const int n, const double x, double *result_array)
This routine evaluates all derivative orders from 0, . . . , mmax of the physicist Hermite polynomial of order n,
Hn , at position x. The result dm /dxm Hn (x) is stored in result_array[m]. The output result_array musthave length at least mmax + 1.
double gsl_sf_hermite_prob_deriv(const int m, const int n, const double x)
int gsl_sf_hermite_prob_deriv_e(const int m, const int n, const double x, gsl_sf_result *result)
These routines evaluate the m-th derivative of the probabilist Hermite polynomial Hen (x) of order n at positionx.
int gsl_sf_hermite_prob_array_deriv(const int m, const int nmax, const double x, double *result_array)
This routine evaluates the m-th derivative of all probabilist Hermite polynomials Hen (x) from orders
0, . . . , nmax at position x. The result dm /dxm Hen (x) is stored in result_array[n]. The output
result_array must have length at least nmax + 1.
int gsl_sf_hermite_prob_deriv_array(const int mmax, const int n, const double x, double *result_array)
This routine evaluates all derivative orders from 0, . . . , mmax of the probabilist Hermite polynomial of order n,
Hen , at position x. The result dm /dxm Hen (x) is stored in result_array[m]. The output result_arraymust have length at least mmax + 1.

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7.21.3 Hermite Functions
The Hermite functions are defined by√ −1/2 −x2 /2
ψn (x) = 2n n! πe
Hn (x)and satisfy the Schrödinger equation for a quantum mechanical harmonic oscillatorψn00 (x) + (2n + 1 − x2 )ψn (x) = 0
They are orthonormal,Z ∞ψm (x)ψn (x)dx = δmn−∞

and form an orthonormal basis of L2 (R). The Hermite functions are also eigenfunctions of the continuous Fouriertransform. GSL offers two methods for evaluating the Hermite functions. The first uses the standard three-term recurrence relation which has O(n)√complexity and is the most accurate. The second uses a Cauchy integral approach due
to Bunck (2009) which has O( n) complexity which represents a significant speed improvement for large n, although
it is slightly less accurate.
double gsl_sf_hermite_func(const int n, const double x)
int gsl_sf_hermite_func_e(const int n, const double x, gsl_sf_result *result)
These routines evaluate the Hermite function ψn (x) of order n at position x using a three term recurrence relation.
The algorithm complexity is O(n).
double gsl_sf_hermite_func_fast(const int n, const double x)
int gsl_sf_hermite_func_fast_e(const int n, const double x, gsl_sf_result *result)
These routines evaluate the Hermite function ψn (x) of√order n at position x using a the Cauchy integral algorithm
due to Bunck, 2009. The algorithm complexity is O( n).
int gsl_sf_hermite_func_array(const int nmax, const double x, double *result_array)
This routine evaluates all Hermite functions ψn (x) for orders n = 0, . . . , nmax at position x, using the recurrence
relation algorithm. The results are stored in result_array which has length at least nmax + 1.
double gsl_sf_hermite_func_series(const int n, const double x, const double *a)
int gsl_sf_hermite_func_series_e(const
Pn int n, const double x, const double *a, gsl_sf_result *result)
These routines evaluate the series j=0 aj ψj (x) with ψj being the j-th Hermite function using the Clenshaw
algorithm.

7.21.4 Derivatives of Hermite Functions
double gsl_sf_hermite_func_der(const int m, const int n, const double x)
int gsl_sf_hermite_func_der_e(const int m, const int n, const double x, gsl_sf_result *result)
These routines evaluate the m-th derivative of the Hermite function ψn (x) of order n at position x.

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7.21.5 Zeros of Hermite Polynomials and Hermite Functions
These routines calculate the s-th zero of the Hermite polynomial/function of order n. Since the zeros are symmetricalaround zero, only positive zeros are calculated, ordered from smallest to largest, starting from index 1. Only for odd
polynomial orders a zeroth zero exists, its value always being zero.
double gsl_sf_hermite_zero(const int n, const int s)
int gsl_sf_hermite_zero_e(const int n, const int s, gsl_sf_result *result)
These routines evaluate the s-th zero of the physicist Hermite polynomial Hn (x) of order n.
double gsl_sf_hermite_prob_zero(const int n, const int s)
int gsl_sf_hermite_prob_zero_e(const int n, const int s, gsl_sf_result *result)
These routines evaluate the s-th zero of the probabilist Hermite polynomial Hen (x) of order n.
double gsl_sf_hermite_func_zero(const int n, const int s)
int gsl_sf_hermite_func_zero_e(const int n, const int s, gsl_sf_result *result)
These routines evaluate the s-th zero of the Hermite function ψn (x) of order n.

7.22 Hypergeometric Functions
Hypergeometric functions are described in Abramowitz & Stegun, Chapters 13 and 15. These functions are declaredin the header file gsl_sf_hyperg.h.
double gsl_sf_hyperg_0F1(double c, double x)
int gsl_sf_hyperg_0F1_e(double c, double x, gsl_sf_result *result)
These routines compute the hypergeometric function
0 F1 (c, x)

double gsl_sf_hyperg_1F1_int(int m, int n, double x)
int gsl_sf_hyperg_1F1_int_e(int m, int n, double x, gsl_sf_result *result)
These routines compute the confluent hypergeometric function
1 F1 (m, n, x) = M (m, n, x)

for integer parameters m, n.
double gsl_sf_hyperg_1F1(double a, double b, double x)
int gsl_sf_hyperg_1F1_e(double a, double b, double x, gsl_sf_result *result)
These routines compute the confluent hypergeometric function
1 F1 (a, b, x) = M (a, b, x)

for general parameters a, b.
double gsl_sf_hyperg_U_int(int m, int n, double x)

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int gsl_sf_hyperg_U_int_e(int m, int n, double x, gsl_sf_result *result)
These routines compute the confluent hypergeometric function U (m, n, x) for integer parameters m, n.
int gsl_sf_hyperg_U_int_e10_e(int m, int n, double x, gsl_sf_result_e10 *result)This routine computes the confluent hypergeometric function U (m, n, x) for integer parameters m, n using thegsl_sf_result_e10 type to return a result with extended range.
double gsl_sf_hyperg_U(double a, double b, double x)
int gsl_sf_hyperg_U_e(double a, double b, double x, gsl_sf_result *result)
These routines compute the confluent hypergeometric function U (a, b, x).
int gsl_sf_hyperg_U_e10_e(double a, double b, double x, gsl_sf_result_e10 *result)This routine computes the confluent hypergeometric function U (a, b, x) using the gsl_sf_result_e10 type toreturn a result with extended range.
double gsl_sf_hyperg_2F1(double a, double b, double c, double x)
int gsl_sf_hyperg_2F1_e(double a, double b, double c, double x, gsl_sf_result *result)
These routines compute the Gauss hypergeometric function
2 F1 (a, b, c, x) = F (a, b, c, x)

for |x| < 1. If the arguments (a, b, c, x) are too close to a singularity then the function can return the error
code GSL_EMAXITER when the series approximation converges too slowly. This occurs in the region of x = 1,c − a − b = m for integer m.
double gsl_sf_hyperg_2F1_conj(double aR, double aI, double c, double x)
int gsl_sf_hyperg_2F1_conj_e(double aR, double aI, double c, double x, gsl_sf_result *result)
These routines compute the Gauss hypergeometric function
2 F1 (aR + iaI , aR − iaI, c, x)with complex parameters for |x| < 1.
double gsl_sf_hyperg_2F1_renorm(double a, double b, double c, double x)
int gsl_sf_hyperg_2F1_renorm_e(double a, double b, double c, double x, gsl_sf_result *result)
These routines compute the renormalized Gauss hypergeometric function
2 F1 (a, b, c, x)/Γ(c)for |x| < 1.
double gsl_sf_hyperg_2F1_conj_renorm(double aR, double aI, double c, double x)
int gsl_sf_hyperg_2F1_conj_renorm_e(double aR, double aI, double c, double x, gsl_sf_result *result)
These routines compute the renormalized Gauss hypergeometric function
2 F1 (aR + iaI , aR − iaI , c, x)/Γ(c)for |x| < 1.
double gsl_sf_hyperg_2F0(double a, double b, double x)

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int gsl_sf_hyperg_2F0_e(double a, double b, double x, gsl_sf_result *result)
These routines compute the hypergeometric function
2 F0 (a, b, x)

The series representation is a divergent hypergeometric series. However, for x < 0 we havea
2 F0 (a, b, x) = (−1/x) U (a, 1 + a − b, −1/x)

7.23 Laguerre Functions
The generalized Laguerre polynomials, sometimes referred to as associated Laguerre polynomials, are defined in termsof confluent hypergeometric functions as
Lan (x) =(a + 1)n1 F1 (−n, a + 1, x)n!

where (a)n is the Pochhammer symbol (rising factorial). They are related to the plain Laguerre polynomials Ln (x) by
L0n (x) = Ln (x) and Lkn (x) = (−1)k (dk /dxk )L(n+k) (x) For more information see Abramowitz & Stegun, Chapter
22.
The functions described in this section are declared in the header file gsl_sf_laguerre.h.
double gsl_sf_laguerre_1(double a, double x)
double gsl_sf_laguerre_2(double a, double x)
double gsl_sf_laguerre_3(double a, double x)
int gsl_sf_laguerre_1_e(double a, double x, gsl_sf_result *result)int gsl_sf_laguerre_2_e(double a, double x, gsl_sf_result *result)int gsl_sf_laguerre_3_e(double a, double x, gsl_sf_result *result)
These routines evaluate the generalized Laguerre polynomials La1 (x), La2 (x), La3 (x) using explicit representations.
double gsl_sf_laguerre_n(const int n, const double a, const double x)
int gsl_sf_laguerre_n_e(int n, double a, double x, gsl_sf_result *result)
These routines evaluate the generalized Laguerre polynomials Lan (x) for a > −1, n ≥ 0.

7.24 Lambert W Functions
Lambert’s W functions, W (x), are defined to be solutions of the equation W (x) exp(W (x)) = x. This function has
multiple branches for x < 0; however, it has only two real-valued branches. We define W0 (x) to be the principal
branch, where W > −1 for x < 0, and W−1 (x) to be the other real branch, where W < −1 for x < 0. The Lambertfunctions are declared in the header file gsl_sf_lambert.h.
double gsl_sf_lambert_W0(double x)

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int gsl_sf_lambert_W0_e(double x, gsl_sf_result *result)
These compute the principal branch of the Lambert W function, W0 (x).
double gsl_sf_lambert_Wm1(double x)
int gsl_sf_lambert_Wm1_e(double x, gsl_sf_result *result)
These compute the secondary real-valued branch of the Lambert W function, W−1 (x).

7.25 Legendre Functions and Spherical Harmonics
The Legendre Functions and Legendre Polynomials are described in Abramowitz & Stegun, Chapter 8. These functions
are declared in the header file gsl_sf_legendre.h.

7.25.1 Legendre Polynomials
double gsl_sf_legendre_P1(double x)
double gsl_sf_legendre_P2(double x)
double gsl_sf_legendre_P3(double x)
int gsl_sf_legendre_P1_e(double x, gsl_sf_result *result)int gsl_sf_legendre_P2_e(double x, gsl_sf_result *result)int gsl_sf_legendre_P3_e(double x, gsl_sf_result *result)
These functions evaluate the Legendre polynomials Pl (x) using explicit representations for l = 1, 2, 3.
double gsl_sf_legendre_Pl(int l, double x)
int gsl_sf_legendre_Pl_e(int l, double x, gsl_sf_result *result)
These functions evaluate the Legendre polynomial Pl (x) for a specific value of l, x subject to l ≥ 0 and |x| ≤ 1.
int gsl_sf_legendre_Pl_array(int lmax, double x, double result_array[])
int gsl_sf_legendre_Pl_deriv_array(int lmax, double x, double result_array[], double result_deriv_array[])
These functions compute arrays of Legendre polynomials Pl (x) and derivatives dPl (x)/dx for l = 0, . . . , lmax
and |x| ≤ 1.
double gsl_sf_legendre_Q0(double x)
int gsl_sf_legendre_Q0_e(double x, gsl_sf_result *result)
These routines compute the Legendre function Q0 (x) for x > −1 and x 6= 1.
double gsl_sf_legendre_Q1(double x)
int gsl_sf_legendre_Q1_e(double x, gsl_sf_result *result)
These routines compute the Legendre function Q1 (x) for x > −1 and x 6= 1.
double gsl_sf_legendre_Ql(int l, double x)

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int gsl_sf_legendre_Ql_e(int l, double x, gsl_sf_result *result)
These routines compute the Legendre function Ql (x) for x > −1, x 6= 1 and l ≥ 0.

7.25.2 Associated Legendre Polynomials and Spherical Harmonics
The following functions compute the associated Legendre polynomials Plm (x) which are solutions of the differential
equation


d
m2
d2
Plm (x) = 0(1 − x2 ) 2 Plm (x) − 2x Plm (x) + l(l + 1) −dx
dx
1 − x2where the degree l and order m satisfy 0 ≤ l and 0 ≤ m ≤ l. The functions Plm (x) grow combinatorially with l andcan overflow for l larger than about 150. Alternatively, one may calculate normalized associated Legendre polynomials.
There are a number of different normalization conventions, and these functions can be stably computed up to degreeand order 2700. The following normalizations are provided:• Schmidt semi-normalization
Schmidt semi-normalized associated Legendre polynomials are often used in the magnetics community and aredefined as
Sl0 (x) = Pl0 (x)sSlm (x) = (−1)m2(l − m)! m
P (x), m > 0(l + m)! l

The factor of (−1)m is called the Condon-Shortley phase factor and can be excluded if desired by setting theparameter csphase = 1 in the functions below.
• Spherical Harmonic Normalization
The associated Legendre polynomials suitable for calculating spherical harmonics are defined ass
2l + 1 (l − m)! m
m
m
P (x)
Yl (x) = (−1)
4π (l + m)! l
where again the phase factor (−1)m can be included or excluded if desired.
• Full Normalization
The fully normalized associated Legendre polynomials are defined ass
1 (l − m)! m
Nlm (x) = (−1)m (l + )P (x)2 (l + m)! l
and have the property
Z 1

Nlm (x)2 dx = 1−1

The normalized associated Legendre routines below use a recurrence relation which is stable up to a degree and orderof about 2700. Beyond this, the computed functions could suffer from underflow leading to incorrect results. Routines are provided to compute first and second derivatives dPlm (x)/dx and d2 Plm (x)/dx2 as well as their alternate
versions dPlm (cos θ)/dθ and d2 Plm (cos θ)/dθ2 . While there is a simple scaling relationship between the two forms,
the derivatives involving θ are heavily used in spherical harmonic expansions and so these routines are also provided.
In the functions below, a parameter of type gsl_sf_legendre_t specifies the type of normalization to use. The
possible values are
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type gsl_sf_legendre_t

Value
GSL_SF_LEGENDRE_NONE
GSL_SF_LEGENDRE_SCHMIDTGSL_SF_LEGENDRE_SPHARM
GSL_SF_LEGENDRE_FULL

Description
The unnormalized associated Legendre polynomials Plm (x)The Schmidt semi-normalized associated Legendre polynomials Slm (x)The spherical harmonic associated Legendre polynomials Ylm (x)The fully normalized associated Legendre polynomials Nlm (x)int gsl_sf_legendre_array(const gsl_sf_legendre_t norm, const size_t lmax, const double x, double
result_array[])
int gsl_sf_legendre_array_e(const gsl_sf_legendre_t norm, const size_t lmax, const double x, const double
csphase, double result_array[])
These functions calculate all normalized associated Legendre polynomials for 0 ≤ l ≤ lmax and 0 ≤ m ≤ lfor |x| ≤ 1. The norm parameter specifies which normalization is used. The normalized Plm (x) values are
stored in result_array, whose minimum size can be obtained from calling gsl_sf_legendre_array_n().
The array index of Plm (x) is obtained from calling gsl_sf_legendre_array_index(l, m). To include or
exclude the Condon-Shortley phase factor of (−1)m , set the parameter csphase to either −1 or 1 respectively
in the _e function. This factor is excluded by default.
int gsl_sf_legendre_deriv_array(const gsl_sf_legendre_t norm, const size_t lmax, const double x, double
result_array[], double result_deriv_array[])
int gsl_sf_legendre_deriv_array_e(const gsl_sf_legendre_t norm, const size_t lmax, const double x, const
double csphase, double result_array[], double result_deriv_array[])
These functions calculate all normalized associated Legendre functions and their first derivatives up to degreelmax for |x| < 1. The parameter norm specifies the normalization used. The normalized Plm (x) values andtheir derivatives dPlm (x)/dx are stored in result_array and result_deriv_array respectively. To include
or exclude the Condon-Shortley phase factor of (−1)m , set the parameter csphase to either −1 or 1 respectively
in the _e function. This factor is excluded by default.
int gsl_sf_legendre_deriv_alt_array(const gsl_sf_legendre_t norm, const size_t lmax, const double x, double
result_array[], double result_deriv_array[])
int gsl_sf_legendre_deriv_alt_array_e(const gsl_sf_legendre_t norm, const size_t lmax, const double x,
const double csphase, double result_array[], double
result_deriv_array[])
These functions calculate all normalized associated Legendre functions and their (alternate) first derivatives up
to degree lmax for |x| < 1. The normalized Plm (x) values and their derivatives dPlm (cos θ)/dθ are stored
in result_array and result_deriv_array respectively. To include or exclude the Condon-Shortley phasefactor of (−1)m , set the parameter csphase to either −1 or 1 respectively in the _e function. This factor is
excluded by default.
int gsl_sf_legendre_deriv2_array(const gsl_sf_legendre_t norm, const size_t lmax, const double x, double
result_array[], double result_deriv_array[], double result_deriv2_array[])
int gsl_sf_legendre_deriv2_array_e(const gsl_sf_legendre_t norm, const size_t lmax, const double x, const
double csphase, double result_array[], double result_deriv_array[],
double result_deriv2_array[])
These functions calculate all normalized associated Legendre functions and their first and second derivatives up
to degree lmax for |x| < 1. The parameter norm specifies the normalization used. The normalized Plm (x),their first derivatives dPlm (x)/dx, and their second derivatives d2 Plm (x)/dx2 are stored in result_array,
result_deriv_array, and result_deriv2_array respectively. To include or exclude the Condon-Shortley
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phase factor of (−1)m , set the parameter csphase to either −1 or 1 respectively in the _e function. This factor
is excluded by default.
int gsl_sf_legendre_deriv2_alt_array(const gsl_sf_legendre_t norm, const size_t lmax, const double x,
double result_array[], double result_deriv_array[], double
result_deriv2_array[])
int gsl_sf_legendre_deriv2_alt_array_e(const gsl_sf_legendre_t norm, const size_t lmax, const double x,
const double csphase, double result_array[], double
result_deriv_array[], double result_deriv2_array[])
These functions calculate all normalized associated Legendre functions and their (alternate) first and second
derivatives up to degree lmax for |x| < 1. The parameter norm specifies the normalization used. The normalized Plm (x), their first derivatives dPlm (cos θ)/dθ, and their second derivatives d2 Plm (cos θ)/dθ2 are stored inresult_array, result_deriv_array, and result_deriv2_array respectively. To include or exclude theCondon-Shortley phase factor of (−1)m , set the parameter csphase to either −1 or 1 respectively in the _efunction. This factor is excluded by default.
size_t gsl_sf_legendre_nlm(const size_t lmax)
This function returns the total number of associated Legendre functions Plm (x) for a given lmax. The number
is (lmax+1) * (lmax+2) / 2.
size_t gsl_sf_legendre_array_n(const size_t lmax)
This function returns the minimum array size for maximum degree lmax needed for the array versionsof the associated Legendre functions. Size is calculated as the total number of Plm (x) functions (seegsl_sf_legendre_nlm()), plus extra space for precomputing multiplicative factors used in the recurrence
relations.
size_t gsl_sf_legendre_array_index(const size_t l, const size_t m)
This function returns the index into result_array, result_deriv_array, or result_deriv2_array cor000
responding to Plm (x), Pl m (x), or Pl m (x). The index is given by l(l + 1)/2 + m.
An inline version of this function is used if HAVE_INLINE is defined.
double gsl_sf_legendre_Plm(int l, int m, double x)
int gsl_sf_legendre_Plm_e(int l, int m, double x, gsl_sf_result *result)
These routines compute the associated Legendre polynomial Plm (x) for m ≥ 0, l ≥ m, and |x| ≤ 1.
double gsl_sf_legendre_sphPlm(int l, int m, double x)
int gsl_sf_legendre_sphPlm_e(int l, int m, double x, gsl_sf_result *result)These
routinescomputethenormalized
associatedLegendre
polynomialpp(2l + 1)/(4π) (l − m)!/(l + m)!Plm (x) suitable for use in spherical harmonics.
The parametersmust satisfy m ≥ 0, l ≥ m, and |x| ≤ 1. These routines avoid the overflows that occur for the standard
normalization of Plm (x).
int gsl_sf_legendre_Plm_array(int lmax, int m, double x, double result_array[])
int gsl_sf_legendre_Plm_deriv_array(int lmax, int m, double x, double result_array[], double
result_deriv_array[])
These functions are now deprecated and will be removed in a future release; see gsl_sf_legendre_array()and gsl_sf_legendre_deriv_array().
int gsl_sf_legendre_sphPlm_array(int lmax, int m, double x, double result_array[])

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int gsl_sf_legendre_sphPlm_deriv_array(int lmax, int m, double x, double result_array[], double
result_deriv_array[])
These functions are now deprecated and will be removed in a future release; see gsl_sf_legendre_array()and gsl_sf_legendre_deriv_array().
int gsl_sf_legendre_array_size(const int lmax, const int m)
This function is now deprecated and will be removed in a future release.

7.25.3 Conical Functionsµ
The Conical Functions P−(1/2)+iλ(x) and Qµ−(1/2)+iλ are described in Abramowitz & Stegun, Section 8.12.

double gsl_sf_conicalP_half(double lambda, double x)
int gsl_sf_conicalP_half_e(double lambda, double x, gsl_sf_result *result)
1/2

These routines compute the irregular Spherical Conical Function P−1/2+iλ (x) for x > −1.
double gsl_sf_conicalP_mhalf(double lambda, double x)
int gsl_sf_conicalP_mhalf_e(double lambda, double x, gsl_sf_result *result)−1/2

These routines compute the regular Spherical Conical Function P−1/2+iλ (x) for x > −1.
double gsl_sf_conicalP_0(double lambda, double x)
int gsl_sf_conicalP_0_e(double lambda, double x, gsl_sf_result *result)
0
These routines compute the conical function P−1/2+iλ(x) for x > −1.
double gsl_sf_conicalP_1(double lambda, double x)
int gsl_sf_conicalP_1_e(double lambda, double x, gsl_sf_result *result)
1
These routines compute the conical function P−1/2+iλ(x) for x > −1.
double gsl_sf_conicalP_sph_reg(int l, double lambda, double x)
int gsl_sf_conicalP_sph_reg_e(int l, double lambda, double x, gsl_sf_result *result)−1/2−lThese routines compute the Regular Spherical Conical Function P−1/2+iλ (x) for x > −1 and l ≥ −1.
double gsl_sf_conicalP_cyl_reg(int m, double lambda, double x)
int gsl_sf_conicalP_cyl_reg_e(int m, double lambda, double x, gsl_sf_result *result)−m
These routines compute the Regular Cylindrical Conical Function P−1/2+iλ(x) for x > −1 and m ≥ −1.

7.25.4 Radial Functions for Hyperbolic Space
The following spherical functions are specializations of Legendre functions which give the regular eigenfunctions of
the Laplacian on a 3-dimensional hyperbolic space H 3 . Of particular interest is the flat limit, λ → ∞, η → 0, λη
fixed.
double gsl_sf_legendre_H3d_0(double lambda, double eta)

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int gsl_sf_legendre_H3d_0_e(double lambda, double eta, gsl_sf_result *result)
These routines compute the zeroth radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space,LH3d(λ, η) :=0sin(λη)λ sinh(η)

for η ≥ 0. In the flat limit this takes the form LH3d(λ, η) = j0 (λη).
0double gsl_sf_legendre_H3d_1(double lambda, double eta)
int gsl_sf_legendre_H3d_1_e(double lambda, double eta, gsl_sf_result *result)
These routines compute the first radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space,sin(λη)
1
√
LH3d
(λ,
η)
:=
(coth(η) − λ cot(λη))
1
λ2 + 1 λ sinh(η)
for η ≥ 0 In the flat limit this takes the form LH3d(λ, η) = j1 (λη).
1double gsl_sf_legendre_H3d(int l, double lambda, double eta)
int gsl_sf_legendre_H3d_e(int l, double lambda, double eta, gsl_sf_result *result)
These routines compute the l-th radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space(λ, η) = jl (λη).
η ≥ 0 and l ≥ 0. In the flat limit this takes the form LH3dlint gsl_sf_legendre_H3d_array(int lmax, double lambda, double eta, double result_array[])This function computes an array of radial eigenfunctions LH3d(λ, η) for 0 ≤ l ≤ lmax.
l7.26 Logarithm and Related Functions
Information on the properties of the Logarithm function can be found in Abramowitz & Stegun, Chapter 4. The functions described in this section are declared in the header file gsl_sf_log.h.
double gsl_sf_log(double x)
int gsl_sf_log_e(double x, gsl_sf_result *result)
These routines compute the logarithm of x, log(x), for x > 0.
double gsl_sf_log_abs(double x)
int gsl_sf_log_abs_e(double x, gsl_sf_result *result)
These routines compute the logarithm of the magnitude of x, log(|x|), for x 6= 0.
int gsl_sf_complex_log_e(double zr, double zi, gsl_sf_result *lnr, gsl_sf_result *theta)
This routine computes the complex logarithm of z = zr + izi . The results are returned as lnr, theta such thatexp(lnr + iθ) = zr + izi , where θ lies in the range [−π, π].
double gsl_sf_log_1plusx(double x)
int gsl_sf_log_1plusx_e(double x, gsl_sf_result *result)
These routines compute log(1 + x) for x > −1 using an algorithm that is accurate for small x.
double gsl_sf_log_1plusx_mx(double x)
int gsl_sf_log_1plusx_mx_e(double x, gsl_sf_result *result)
These routines compute log(1 + x) − x for x > −1 using an algorithm that is accurate for small x.

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7.27 Mathieu Functions
The routines described in this section compute the angular and radial Mathieu functions, and their characteristic values.
Mathieu functions are the solutions of the following two differential equations:d2 y+ (a − 2q cos 2v)y = 0
dv 2
d2 f
− (a − 2q cosh 2u)f = 0
du2
The angular Mathieu functions cer (x, q), ser (x, q) are the even and odd periodic solutions of the first equation, which
is known as Mathieu’s equation. These exist only for the discrete sequence of characteristic values a = ar (q) (evenperiodic) and a = br (q) (odd-periodic).
(j)

(j)

The radial Mathieu functions M cr (z, q) and M sr (z, q) are the solutions of the second equation, which is referred
to as Mathieu’s modified equation. The radial Mathieu functions of the first, second, third and fourth kind are denotedby the parameter j, which takes the value 1, 2, 3 or 4.
For more information on the Mathieu functions, see Abramowitz and Stegun, Chapter 20. These functions are definedin the header file gsl_sf_mathieu.h.

7.27.1 Mathieu Function Workspace
The Mathieu functions can be computed for a single order or for multiple orders, using array-based routines. The
array-based routines require a preallocated workspace.
type gsl_sf_mathieu_workspace
Workspace required for array-based routines
gsl_sf_mathieu_workspace *gsl_sf_mathieu_alloc(size_t n, double qmax)
This function returns a workspace for the array versions of the Mathieu routines. The arguments n and qmax
specify the maximum order and q-value of Mathieu functions which can be computed with this workspace.
void gsl_sf_mathieu_free(gsl_sf_mathieu_workspace *work)
This function frees the workspace work.

7.27.2 Mathieu Function Characteristic Values
int gsl_sf_mathieu_a(int n, double q)
int gsl_sf_mathieu_a_e(int n, double q, gsl_sf_result *result)int gsl_sf_mathieu_b(int n, double q)
int gsl_sf_mathieu_b_e(int n, double q, gsl_sf_result *result)
These routines compute the characteristic values an (q), bn (q) of the Mathieu functions cen (q, x) and sen (q, x),
respectively.
int gsl_sf_mathieu_a_array(int order_min, int order_max, double q, gsl_sf_mathieu_workspace *work, double
result_array[])
int gsl_sf_mathieu_b_array(int order_min, int order_max, double q, gsl_sf_mathieu_workspace *work, double
result_array[])
These routines compute a series of Mathieu characteristic values an (q), bn (q) for n from order_min toorder_max inclusive, storing the results in the array result_array.
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7.27.3 Angular Mathieu Functions
int gsl_sf_mathieu_ce(int n, double q, double x)
int gsl_sf_mathieu_ce_e(int n, double q, double x, gsl_sf_result *result)int gsl_sf_mathieu_se(int n, double q, double x)
int gsl_sf_mathieu_se_e(int n, double q, double x, gsl_sf_result *result)
These routines compute the angular Mathieu functions cen (q, x) and sen (q, x), respectively.
int gsl_sf_mathieu_ce_array(int nmin, int nmax, double q, double x, gsl_sf_mathieu_workspace *work, double
result_array[])
int gsl_sf_mathieu_se_array(int nmin, int nmax, double q, double x, gsl_sf_mathieu_workspace *work, double
result_array[])
These routines compute a series of the angular Mathieu functions cen (q, x) and sen (q, x) of order n from nmin
to nmax inclusive, storing the results in the array result_array.

7.27.4 Radial Mathieu Functions
int gsl_sf_mathieu_Mc(int j, int n, double q, double x)
int gsl_sf_mathieu_Mc_e(int j, int n, double q, double x, gsl_sf_result *result)int gsl_sf_mathieu_Ms(int j, int n, double q, double x)
int gsl_sf_mathieu_Ms_e(int j, int n, double q, double x, gsl_sf_result *result)
(j)

(j)

These routines compute the radial j-th kind Mathieu functions M cn (q, x) and M sn (q, x) of order n.
(3)

(1)

(2)

The allowed values of j are 1 and 2. The functions for j = 3, 4 can be computed as Mn = Mn + iMn and
(j)
(j)
(j)
(2)
(1)
(4)
Mn = Mn − iMn , where Mn = M cn or M sn .
int gsl_sf_mathieu_Mc_array(int j, int nmin, int nmax, double q, double x, gsl_sf_mathieu_workspace *work,
double result_array[])
int gsl_sf_mathieu_Ms_array(int j, int nmin, int nmax, double q, double x, gsl_sf_mathieu_workspace *work,
double result_array[])
These routines compute a series of the radial Mathieu functions of kind j, with order from nmin to nmax inclusive, storing the results in the array result_array.

7.28 Power Function
The following functions are equivalent to the function gsl_pow_int() with an error estimate. These functions are
declared in the header file gsl_sf_pow_int.h.
double gsl_sf_pow_int(double x, int n)

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int gsl_sf_pow_int_e(double x, int n, gsl_sf_result *result)
These routines compute the power xn for integer n. The power is computed using the minimum number ofmultiplications. For example, x8 is computed as ((x2 )2 )2 , requiring only 3 multiplications. For reasons ofefficiency, these functions do not check for overflow or underflow conditions. The following is a simple example:#include <gsl/gsl_sf_pow_int.h>/* compute 3.0**12 */double y = gsl_sf_pow_int(3.0, 12);7.29 Psi (Digamma) Function
The polygamma functions of order n are defined by
ψ (n) (x) =



d
dx

n


ψ(x) =ddx

n+1
log(Γ(x))

where ψ(x) = Γ0 (x)/Γ(x) is known as the digamma function. These functions are declared in the header file
gsl_sf_psi.h.

7.29.1 Digamma Function
double gsl_sf_psi_int(int n)
int gsl_sf_psi_int_e(int n, gsl_sf_result *result)
These routines compute the digamma function ψ(n) for positive integer n. The digamma function is also called
the Psi function.
double gsl_sf_psi(double x)
int gsl_sf_psi_e(double x, gsl_sf_result *result)
These routines compute the digamma function ψ(x) for general x, x 6= 0.
double gsl_sf_psi_1piy(double y)
int gsl_sf_psi_1piy_e(double y, gsl_sf_result *result)
These routines compute the real part of the digamma function on the line 1 + iy, <[ψ(1 + iy)].

7.29.2 Trigamma Function
double gsl_sf_psi_1_int(int n)
int gsl_sf_psi_1_int_e(int n, gsl_sf_result *result)
These routines compute the Trigamma function ψ 0 (n) for positive integer n.
double gsl_sf_psi_1(double x)
int gsl_sf_psi_1_e(double x, gsl_sf_result *result)
These routines compute the Trigamma function ψ 0 (x) for general x.

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7.29.3 Polygamma Function
double gsl_sf_psi_n(int n, double x)
int gsl_sf_psi_n_e(int n, double x, gsl_sf_result *result)
These routines compute the polygamma function ψ (n) (x) for n ≥ 0, x > 0.

7.30 Synchrotron Functions
The functions described in this section are declared in the header file gsl_sf_synchrotron.h.
double gsl_sf_synchrotron_1(double x)
int gsl_sf_synchrotron_1_e(double x, gsl_sf_result *result)
R∞
These routines compute the first synchrotron function x x dtK5/3 (t) for x ≥ 0.
double gsl_sf_synchrotron_2(double x)
int gsl_sf_synchrotron_2_e(double x, gsl_sf_result *result)
These routines compute the second synchrotron function xK2/3 (x) for x ≥ 0.

7.31 Transport Functions
The transport functions J(n, x) are defined by the integral representations
Z x
J(n, x) =tn et /(et − 1)2 dt
0They are declared in the header file gsl_sf_transport.h.
double gsl_sf_transport_2(double x)
int gsl_sf_transport_2_e(double x, gsl_sf_result *result)
These routines compute the transport function J(2, x).
double gsl_sf_transport_3(double x)
int gsl_sf_transport_3_e(double x, gsl_sf_result *result)
These routines compute the transport function J(3, x).
double gsl_sf_transport_4(double x)
int gsl_sf_transport_4_e(double x, gsl_sf_result *result)
These routines compute the transport function J(4, x).
double gsl_sf_transport_5(double x)
int gsl_sf_transport_5_e(double x, gsl_sf_result *result)
These routines compute the transport function J(5, x).

7.30. Synchrotron Functions

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7.32 Trigonometric Functions
The library includes its own trigonometric functions in order to provide consistency across platforms and reliable error
estimates. These functions are declared in the header file gsl_sf_trig.h.

7.32.1 Circular Trigonometric Functions
double gsl_sf_sin(double x)
int gsl_sf_sin_e(double x, gsl_sf_result *result)
These routines compute the sine function sin(x).
double gsl_sf_cos(double x)
int gsl_sf_cos_e(double x, gsl_sf_result *result)
These routines compute the cosine function cos(x).
double gsl_sf_hypot(double x, double y)
int gsl_sf_hypot_e(double x, double y, gsl_sf_result *result)p
These routines compute the hypotenuse function x2 + y 2 avoiding overflow and underflow.
double gsl_sf_sinc(double x)
int gsl_sf_sinc_e(double x, gsl_sf_result *result)
These routines compute sinc(x) = sin(πx)/(πx) for any value of x.

7.32.2 Trigonometric Functions for Complex Arguments
int gsl_sf_complex_sin_e(double zr, double zi, gsl_sf_result *szr, gsl_sf_result *szi)
This function computes the complex sine, sin(zr + izi ) storing the real and imaginary parts in szr, szi.
int gsl_sf_complex_cos_e(double zr, double zi, gsl_sf_result *czr, gsl_sf_result *czi)
This function computes the complex cosine, cos(zr + izi ) storing the real and imaginary parts in czr, czi.
int gsl_sf_complex_logsin_e(double zr, double zi, gsl_sf_result *lszr, gsl_sf_result *lszi)
This function computes the logarithm of the complex sine, log(sin(zr +izi )) storing the real and imaginary partsin lszr, lszi.

7.32.3 Hyperbolic Trigonometric Functions
double gsl_sf_lnsinh(double x)
int gsl_sf_lnsinh_e(double x, gsl_sf_result *result)
These routines compute log(sinh(x)) for x > 0.
double gsl_sf_lncosh(double x)
int gsl_sf_lncosh_e(double x, gsl_sf_result *result)
These routines compute log(cosh(x)) for any x.

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7.32.4 Conversion Functions
int gsl_sf_polar_to_rect(double r, double theta, gsl_sf_result *x, gsl_sf_result *y)This function converts the polar coordinates (r, theta) to rectilinear coordinates (x, y), x = r cos(θ), y =r sin(θ).
int gsl_sf_rect_to_polar(double x, double y, gsl_sf_result *r, gsl_sf_result *theta)
This function converts the rectilinear coordinates (x, y) to polar coordinates (r, theta), such that x = r cos(θ),
y = r sin(θ). The argument theta lies in the range [−π, π].

7.32.5 Restriction Functions
double gsl_sf_angle_restrict_symm(double theta)
int gsl_sf_angle_restrict_symm_e(double *theta)
These routines force the angle theta to lie in the range (−π, π].
Note that the mathematical value of π is slightly greater than M_PI, so the machine numbers M_PI and -M_PI
are included in the range.
double gsl_sf_angle_restrict_pos(double theta)
int gsl_sf_angle_restrict_pos_e(double *theta)
These routines force the angle theta to lie in the range [0, 2π).
Note that the mathematical value of 2π is slightly greater than 2*M_PI, so the machine number 2*M_PI is included in the range.

7.32.6 Trigonometric Functions With Error Estimatesint gsl_sf_sin_err_e(double x, double dx, gsl_sf_result *result)This routine computes the sine of an angle x with an associated absolute error dx, sin(x ± dx). Note that thisfunction is provided in the error-handling form only since its purpose is to compute the propagated error.
int gsl_sf_cos_err_e(double x, double dx, gsl_sf_result *result)This routine computes the cosine of an angle x with an associated absolute error dx, cos(x ± dx). Note that thisfunction is provided in the error-handling form only since its purpose is to compute the propagated error.

7.33 Zeta Functions
The Riemann zeta function is defined in Abramowitz & Stegun, Section 23.2. The functions described in this section
are declared in the header file gsl_sf_zeta.h.

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7.33.1 Riemann Zeta Function
The Riemann zeta function is defined by the infinite sum
ζ(s) =

∞
X

k −s

k=1

double gsl_sf_zeta_int(int n)
int gsl_sf_zeta_int_e(int n, gsl_sf_result *result)
These routines compute the Riemann zeta function ζ(n) for integer n, n 6= 1.
double gsl_sf_zeta(double s)
int gsl_sf_zeta_e(double s, gsl_sf_result *result)
These routines compute the Riemann zeta function ζ(s) for arbitrary s, s 6= 1.

7.33.2 Riemann Zeta Function Minus One
For large positive argument, the Riemann zeta function approaches one. In this region the fractional part is interesting,and therefore we need a function to evaluate it explicitly.
double gsl_sf_zetam1_int(int n)
int gsl_sf_zetam1_int_e(int n, gsl_sf_result *result)
These routines compute ζ(n) − 1 for integer n, n 6= 1.
double gsl_sf_zetam1(double s)
int gsl_sf_zetam1_e(double s, gsl_sf_result *result)
These routines compute ζ(s) − 1 for arbitrary s, s 6= 1.

7.33.3 Hurwitz Zeta Function
The Hurwitz zeta function is defined by
ζ(s, q) =∞X(k + q)−s0

double gsl_sf_hzeta(double s, double q)
int gsl_sf_hzeta_e(double s, double q, gsl_sf_result *result)
These routines compute the Hurwitz zeta function ζ(s, q) for s > 1, q > 0.

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7.33.4 Eta Function
The eta function is defined byη(s) = (1 − 21−s )ζ(s)double gsl_sf_eta_int(int n)
int gsl_sf_eta_int_e(int n, gsl_sf_result *result)
These routines compute the eta function η(n) for integer n.
double gsl_sf_eta(double s)
int gsl_sf_eta_e(double s, gsl_sf_result *result)
These routines compute the eta function η(s) for arbitrary s.

7.34 Examples
The following example demonstrates the use of the error handling form of the special functions, in this case to computethe Bessel function J0 (5.0),#include <stdio.h>#include <gsl/gsl_errno.h>#include <gsl/gsl_sf_bessel.h>
intmain (void){
double x = 5.0;
gsl_sf_result result;double expected = -0.17759677131433830434739701;
int status = gsl_sf_bessel_J0_e (x, &result);
printf ("status = %s\n", gsl_strerror(status));printf ("J0(5.0) = %.18f \n""+/- % .18f \n",result.val, result.err);
printf ("exact= %.18f \n", expected);
return status;}
Here are the results of running the program,status = success
J0(5.0) = -0.177596771314338264+/- 0.000000000000000193
exact= -0.177596771314338292
The next program computes the same quantity using the natural form of the function. In this case the error term
result.err and return status are not accessible.
7.34. Examples

81

GNU Scientific Library, Release 2.7#include <stdio.h>#include <gsl/gsl_sf_bessel.h>
intmain (void){
double x = 5.0;
double expected = -0.17759677131433830434739701;
double y = gsl_sf_bessel_J0 (x);printf ("J0(5.0) = %.18f \n", y);printf ("exact= %.18f \n", expected);
return 0;}
The results of the function are the same,J0(5.0) = -0.177596771314338264
exact= -0.177596771314338292

7.35 References and Further Reading
The library follows the conventions of the following book where possible,• Handbook of Mathematical Functions, edited by Abramowitz & Stegun, Dover, ISBN 0486612724.
The following papers contain information on the algorithms used to compute the special functions,• Allan J. MacLeod, MISCFUN: A software package to compute uncommon special functions. ACM Trans. Math.
Soft., vol.: 22, 1996, 288–301• Bunck, B. F., A fast algorithm for evaluation of normalized Hermite functions, BIT Numer. Math, 49: 281-295,
2009.
• G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd Edition (Cambridge University Press, 1944).
• G. Nemeth, Mathematical Approximations of Special Functions, Nova Science Publishers, ISBN 1-56072-052-2• B.C. Carlson, Special Functions of Applied Mathematics (1977)• N. M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics (1996),
ISBN 978-0471113133.
• W.J. Thompson, Atlas for Computing Mathematical Functions, John Wiley & Sons, New York (1997).
• Y.Y. Luke, Algorithms for the Computation of Mathematical Functions, Academic Press, New York (1977).
• S. A. Holmes and W. E. Featherstone, A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions, Journal of Geodesy, 76, pg.
279-299, 2002.

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Chapter 7. Special Functions

CHAPTEREIGHTVECTORS AND MATRICESThe functions described in this chapter provide a simple vector and matrix interface to ordinary C arrays. The memory
management of these arrays is implemented using a single underlying type, known as a block. By writing your functionsin terms of vectors and matrices you can pass a single structure containing both data and dimensions as an argument
without needing additional function parameters. The structures are compatible with the vector and matrix formats used
by BLAS routines.

8.1 Data types
All the functions are available for each of the standard data-types. The versions for double have the prefix
gsl_block, gsl_vector and gsl_matrix. Similarly the versions for single-precision float arrays have the prefix gsl_block_float, gsl_vector_float and gsl_matrix_float. The full list of available types is given below,
Prefixgsl_block
gsl_block_float
gsl_block_long_double
gsl_block_int
gsl_block_uint
gsl_block_long
gsl_block_ulong
gsl_block_short
gsl_block_ushort
gsl_block_char
gsl_block_uchar
gsl_block_complex
gsl_block_complex_float
gsl_block_complex_long_doubleTypedoublefloat
long double
int
unsigned int
longunsigned longshort
unsigned shortchar
unsigned charcomplex double
complex float
complex long double

Corresponding types exist for the gsl_vector and gsl_matrix functions.

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8.2 Blocks
For consistency all memory is allocated through a gsl_block structure. The structure contains two components, thesize of an area of memory and a pointer to the memory. The gsl_block structure looks like this,type gsl_block
typedef struct{
size_t size;
double * data;} gsl_block;
Vectors and matrices are made by slicing an underlying block. A slice is a set of elements formed from an initial offset
and a combination of indices and step-sizes. In the case of a matrix the step-size for the column index represents the
row-length. The step-size for a vector is known as the stride.
The functions for allocating and deallocating blocks are defined in gsl_block.h.

8.2.1 Block allocation
The functions for allocating memory to a block follow the style of malloc and free. In addition they also performtheir own error checking. If there is insufficient memory available to allocate a block then the functions call the GSL
error handler (with an error number of GSL_ENOMEM) in addition to returning a null pointer. Thus if you use the libraryerror handler to abort your program then it isn’t necessary to check every alloc.
gsl_block *gsl_block_alloc(size_t n)This function allocates memory for a block of n double-precision elements, returning a pointer to the block
struct. The block is not initialized and so the values of its elements are undefined. Use the function
gsl_block_calloc() if you want to ensure that all the elements are initialized to zero.
Zero-sized requests are valid and return a non-null result. A null pointer is returned if insufficient memory isavailable to create the block.
gsl_block *gsl_block_calloc(size_t n)This function allocates memory for a block and initializes all the elements of the block to zero.
void gsl_block_free(gsl_block *b)
This function frees the memory used by a block b previously allocated with gsl_block_alloc() or
gsl_block_calloc().

8.2.2 Reading and writing blocks
The library provides functions for reading and writing blocks to a file as binary data or formatted text.
int gsl_block_fwrite(FILE *stream, const gsl_block *b)
This function writes the elements of the block b to the stream stream in binary format. The return value is 0
for success and GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native
binary format it may not be portable between different architectures.
int gsl_block_fread(FILE *stream, gsl_block *b)
This function reads into the block b from the open stream stream in binary format. The block b must be
preallocated with the correct length since the function uses the size of b to determine how many bytes to read.
The return value is 0 for success and GSL_EFAILED if there was a problem reading from the file. The data is
assumed to have been written in the native binary format on the same architecture.
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int gsl_block_fprintf(FILE *stream, const gsl_block *b, const char *format)
This function writes the elements of the block b line-by-line to the stream stream using the format specifier
format, which should be one of the %g, %e or %f formats for floating point numbers and %d for integers. The
function returns 0 for success and GSL_EFAILED if there was a problem writing to the file.
int gsl_block_fscanf(FILE *stream, gsl_block *b)
This function reads formatted data from the stream stream into the block b. The block b must be preallocatedwith the correct length since the function uses the size of b to determine how many numbers to read. The function
returns 0 for success and GSL_EFAILED if there was a problem reading from the file.

8.2.3 Example programs for blocks
The following program shows how to allocate a block,#include <stdio.h>#include <gsl/gsl_block.h>
intmain (void){
gsl_block * b = gsl_block_alloc (100);printf ("length of block = %zu\n", b->size);printf ("block data address = %p\n", b->data);
gsl_block_free (b);return 0;}
Here is the output from the program,length of block = 100
block data address = 0x804b0d8

8.3 Vectors
Vectors are defined by a gsl_vector structure which describes a slice of a block. Different vectors can be created
which point to the same block. A vector slice is a set of equally-spaced elements of an area of memory.
The gsl_vector structure contains five components, the size, the stride, a pointer to the memory where the elements
are stored, data, a pointer to the block owned by the vector, block, if any, and an ownership flag, owner. The structure
is very simple and looks like this,type gsl_vector
typedef struct{
size_t size;
size_t stride;
double * data;
gsl_block * block;(continues on next page)8.3. Vectors

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int owner;} gsl_vector;
The size is simply the number of vector elements. The range of valid indices runs from 0 to size-1. The stride
is the step-size from one element to the next in physical memory, measured in units of the appropriate datatype. The
pointer data gives the location of the first element of the vector in memory. The pointer block stores the location of
the memory block in which the vector elements are located (if any). If the vector owns this block then the owner field
is set to one and the block will be deallocated when the vector is freed. If the vector points to a block owned by anotherobject then the owner field is zero and any underlying block will not be deallocated with the vector.
The functions for allocating and accessing vectors are defined in gsl_vector.h.

8.3.1 Vector allocation
The functions for allocating memory to a vector follow the style of malloc and free. In addition they also performtheir own error checking. If there is insufficient memory available to allocate a vector then the functions call the GSL
error handler (with an error number of GSL_ENOMEM) in addition to returning a null pointer. Thus if you use the libraryerror handler to abort your program then it isn’t necessary to check every alloc.
gsl_vector *gsl_vector_alloc(size_t n)This function creates a vector of length n, returning a pointer to a newly initialized vector struct. A new blockis allocated for the elements of the vector, and stored in the block component of the vector struct. The block is“owned” by the vector, and will be deallocated when the vector is deallocated. Zero-sized requests are valid andreturn a non-null result.
gsl_vector *gsl_vector_calloc(size_t n)This function allocates memory for a vector of length n and initializes all the elements of the vector to zero.
void gsl_vector_free(gsl_vector *v)
This function frees a previously allocated vector v. If the vector was created using gsl_vector_alloc() thenthe block underlying the vector will also be deallocated. If the vector has been created from another object thenthe memory is still owned by that object and will not be deallocated.

8.3.2 Accessing vector elements
Unlike Fortran compilers, C compilers do not usually provide support for range checking of vectors and matrices.1 The
functions gsl_vector_get() and gsl_vector_set() can perform portable range checking for you and report anerror if you attempt to access elements outside the allowed range.
The functions for accessing the elements of a vector or matrix are defined in gsl_vector.h and declared
extern inline to eliminate function-call overhead. You must compile your program with the preprocessor macro
HAVE_INLINE defined to use these functions.
GSL_RANGE_CHECK_OFF
If necessary you can turn off range checking completely without modifying any source files by recompiling your program with the preprocessor definition GSL_RANGE_CHECK_OFF. Provided your compiler supports
inline functions the effect of turning off range checking is to replace calls to gsl_vector_get(v,i) by
v->data[i*v->stride] and calls to gsl_vector_set(v,i,x) by v->data[i*v->stride]=x. Thus there
should be no performance penalty for using the range checking functions when range checking is turned off.
GSL_C99_INLINE
If you use a C99 compiler which requires inline functions in header files to be declared inline instead of extern
1 Range checking is available in the GNU C Compiler bounds-checking extension, but it is not part of the default installation of GCC. Memory
accesses can also be checked with Valgrind or the gcc -fmudflap memory protection option.

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inline, define the macro GSL_C99_INLINE (see Inline functions). With GCC this is selected automaticallywhen compiling in C99 mode (-std=c99).
int gsl_check_range
If inline functions are not used, calls to the functions gsl_vector_get() and gsl_vector_set() will link tothe compiled versions of these functions in the library itself. The range checking in these functions is controlled
by the global integer variable gsl_check_range. It is enabled by default—to disable it, set gsl_check_range
to zero. Due to function-call overhead, there is less benefit in disabling range checking here than for inline
functions.
double gsl_vector_get(const gsl_vector *v, const size_t i)This function returns the i-th element of a vector v. If i lies outside the allowed range of 0 to size - 1 thenthe error handler is invoked and 0 is returned. An inline version of this function is used when HAVE_INLINE isdefined.
void gsl_vector_set(gsl_vector *v, const size_t i, double x)
This function sets the value of the i-th element of a vector v to x. If i lies outside the allowed range of 0 to size
- 1 then the error handler is invoked. An inline version of this function is used when HAVE_INLINE is defined.
double *gsl_vector_ptr(gsl_vector *v, size_t i)const double *gsl_vector_const_ptr(const gsl_vector *v, size_t i)
These functions return a pointer to the i-th element of a vector v. If i lies outside the allowed range of 0 to size
- 1 then the error handler is invoked and a null pointer is returned. Inline versions of these functions are used
when HAVE_INLINE is defined.

8.3.3 Initializing vector elements
void gsl_vector_set_all(gsl_vector *v, double x)
This function sets all the elements of the vector v to the value x.
void gsl_vector_set_zero(gsl_vector *v)
This function sets all the elements of the vector v to zero.
int gsl_vector_set_basis(gsl_vector *v, size_t i)This function makes a basis vector by setting all the elements of the vector v to zero except for the i-th element
which is set to one.

8.3.4 Reading and writing vectors
The library provides functions for reading and writing vectors to a file as binary data or formatted text.
int gsl_vector_fwrite(FILE *stream, const gsl_vector *v)
This function writes the elements of the vector v to the stream stream in binary format. The return value is 0
for success and GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native
binary format it may not be portable between different architectures.
int gsl_vector_fread(FILE *stream, gsl_vector *v)
This function reads into the vector v from the open stream stream in binary format. The vector v must be
preallocated with the correct length since the function uses the size of v to determine how many bytes to read.
The return value is 0 for success and GSL_EFAILED if there was a problem reading from the file. The data is
assumed to have been written in the native binary format on the same architecture.
int gsl_vector_fprintf(FILE *stream, const gsl_vector *v, const char *format)
This function writes the elements of the vector v line-by-line to the stream stream using the format specifier
format, which should be one of the %g, %e or %f formats for floating point numbers and %d for integers. The

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function returns 0 for success and GSL_EFAILED if there was a problem writing to the file.
int gsl_vector_fscanf(FILE *stream, gsl_vector *v)
This function reads formatted data from the stream stream into the vector v. The vector v must be preallocatedwith the correct length since the function uses the size of v to determine how many numbers to read. The function
returns 0 for success and GSL_EFAILED if there was a problem reading from the file.

8.3.5 Vector views
In addition to creating vectors from slices of blocks it is also possible to slice vectors and create vector views. For
example, a subvector of another vector can be described with a view, or two views can be made which provide accessto the even and odd elements of a vector.
type gsl_vector_view
type gsl_vector_const_view
A vector view is a temporary object, stored on the stack, which can be used to operate on a subset of vector
elements. Vector views can be defined for both constant and non-constant vectors, using separate types that
preserve constness. A vector view has the type gsl_vector_view and a constant vector view has the typegsl_vector_const_view. In both cases the elements of the view can be accessed as a gsl_vector using thevector component of the view object. A pointer to a vector of type gsl_vector * or const gsl_vector *can be obtained by taking the address of this component with the & operator.
When using this pointer it is important to ensure that the view itself remains in scope—the simplest way to doso is by always writing the pointer as &view.vector, and never storing this value in another variable.
gsl_vector_view gsl_vector_subvector(gsl_vector *v, size_t offset, size_t n)
gsl_vector_const_view gsl_vector_const_subvector(const gsl_vector *v, size_t offset, size_t n)
These functions return a vector view of a subvector of another vector v. The start of the new vector is offset byoffset elements from the start of the original vector. The new vector has n elements. Mathematically, the i-th
element of the new vector v' is given by:
v'(i) = v->data[(offset + i)*v->stride]
where the index i runs from 0 to n - 1.
The data pointer of the returned vector struct is set to null if the combined parameters (offset, n) overrun theend of the original vector.
The new vector is only a view of the block underlying the original vector, v. The block containing the elementsof v is not owned by the new vector. When the view goes out of scope the original vector v and its block willcontinue to exist. The original memory can only be deallocated by freeing the original vector. Of course, theoriginal vector should not be deallocated while the view is still in use.
The function gsl_vector_const_subvector() is equivalent to gsl_vector_subvector() but can be usedfor vectors which are declared const.
gsl_vector_view gsl_vector_subvector_with_stride(gsl_vector *v, size_t offset, size_t stride, size_t n)
gsl_vector_const_view gsl_vector_const_subvector_with_stride(const gsl_vector *v, size_t offset, size_t
stride, size_t n)
These functions return a vector view of a subvector of another vector v with an additional stride argument. The
subvector is formed in the same way as for gsl_vector_subvector() but the new vector has n elements witha step-size of stride from one element to the next in the original vector. Mathematically, the i-th element of
the new vector v' is given by:

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GNU Scientific Library, Release 2.7v'(i) = v->data[(offset + i*stride)*v->stride]
where the index i runs from 0 to n - 1.
Note that subvector views give direct access to the underlying elements of the original vector. For example, thefollowing code will zero the even elements of the vector v of length n, while leaving the odd elements untouched:
gsl_vector_view v_even = gsl_vector_subvector_with_stride (v, 0, 2, n/2);
gsl_vector_set_zero (&v_even.vector);
A vector view can be passed to any subroutine which takes a vector argument just as a directly allocated vector
would be, using &view.vector. For example, the following code computes the norm of the odd elements of v
using the BLAS routine dnrm2:
gsl_vector_view v_odd = gsl_vector_subvector_with_stride (v, 1, 2, n/2);
double r = gsl_blas_dnrm2 (&v_odd.vector);
Thefunction
gsl_vector_const_subvector_with_stride()isequivalent
gsl_vector_subvector_with_stride() but can be used for vectors which are declared const.

to

gsl_vector_view gsl_vector_complex_real(gsl_vector_complex *v)
gsl_vector_const_view gsl_vector_complex_const_real(const gsl_vector_complex *v)
These functions return a vector view of the real parts of the complex vector v.
The function gsl_vector_complex_const_real() is equivalent to gsl_vector_complex_real() but canbe used for vectors which are declared const.
gsl_vector_view gsl_vector_complex_imag(gsl_vector_complex *v)
gsl_vector_const_view gsl_vector_complex_const_imag(const gsl_vector_complex *v)
These functions return a vector view of the imaginary parts of the complex vector v.
The function gsl_vector_complex_const_imag() is equivalent to gsl_vector_complex_imag() but canbe used for vectors which are declared const.
gsl_vector_view gsl_vector_view_array(double *base, size_t n)
gsl_vector_const_view gsl_vector_const_view_array(const double *base, size_t n)
These functions return a vector view of an array. The start of the new vector is given by base and has n elements.
Mathematically, the i-th element of the new vector v' is given by:
v'(i) = base[i]where the index i runs from 0 to n - 1.
The array containing the elements of v is not owned by the new vector view. When the view goes out of scope
the original array will continue to exist. The original memory can only be deallocated by freeing the original
pointer base. Of course, the original array should not be deallocated while the view is still in use.
The function gsl_vector_const_view_array() is equivalent to gsl_vector_view_array() but can beused for arrays which are declared const.
gsl_vector_view gsl_vector_view_array_with_stride(double *base, size_t stride, size_t n)8.3. Vectors

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gsl_vector_const_view gsl_vector_const_view_array_with_stride(const double *base, size_t stride, size_tn)These functions return a vector view of an array base with an additional stride argument. The subvector isformed in the same way as for gsl_vector_view_array() but the new vector has n elements with a step-sizeof stride from one element to the next in the original array. Mathematically, the i-th element of the new vector
v' is given by:
v'(i) = base[i*stride]
where the index i runs from 0 to n - 1.
Note that the view gives direct access to the underlying elements of the original array. A vector view can bepassed to any subroutine which takes a vector argument just as a directly allocated vector would be, using &view.
vector.
The
function
gsl_vector_const_view_array_with_stride()isequivalent
gsl_vector_view_array_with_stride() but can be used for arrays which are declared const.

to8.3.6 Copying vectors
Common operations on vectors such as addition and multiplication are available in the BLAS part of the library (seeBLAS Support). However, it is useful to have a small number of utility functions which do not require the full BLAS
code. The following functions fall into this category.
int gsl_vector_memcpy(gsl_vector *dest, const gsl_vector *src)
This function copies the elements of the vector src into the vector dest. The two vectors must have the samelength.
int gsl_vector_complex_conj_memcpy(gsl_vector_complex *dest, const gsl_vector_complex *src)
This function copies the complex conjugate of the vector src into the vector dest. The two vectors must havethe same length.
int gsl_vector_swap(gsl_vector *v, gsl_vector *w)
This function exchanges the elements of the vectors v and w by copying. The two vectors must have the samelength.

8.3.7 Exchanging elements
The following functions can be used to exchange, or permute, the elements of a vector.
int gsl_vector_swap_elements(gsl_vector *v, size_t i, size_t j)
This function exchanges the i-th and j-th elements of the vector v in-place.
int gsl_vector_reverse(gsl_vector *v)
This function reverses the order of the elements of the vector v.

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8.3.8 Vector operations
int gsl_vector_add(gsl_vector *a, const gsl_vector *b)
This function adds the elements of vector b to the elements of vector a. The result ai ← ai + bi is stored in aand b remains unchanged. The two vectors must have the same length.
int gsl_vector_sub(gsl_vector *a, const gsl_vector *b)
This function subtracts the elements of vector b from the elements of vector a. The result ai ← ai − bi is stored
in a and b remains unchanged. The two vectors must have the same length.
int gsl_vector_mul(gsl_vector *a, const gsl_vector *b)
This function multiplies the elements of vector a by the elements of vector b. The result ai ← ai ∗ bi is stored
in a and b remains unchanged. The two vectors must have the same length.
int gsl_vector_div(gsl_vector *a, const gsl_vector *b)
This function divides the elements of vector a by the elements of vector b. The result ai ← ai /bi is stored in aand b remains unchanged. The two vectors must have the same length.
int gsl_vector_scale(gsl_vector *a, const double x)
This function multiplies the elements of vector a by the constant factor x. The result ai ← xai is stored in a.
int gsl_vector_add_constant(gsl_vector *a, const double x)
This function adds the constant value x to the elements of the vector a. The result ai ← ai + x is stored in a.
double gsl_vector_sum(const gsl_vector *a)Pn
This function returns the sum of the elements of a, defined as i=1 ai
int gsl_vector_axpby(const double alpha, const gsl_vector *x, const double beta, gsl_vector *y)This function performs the operation y ← αx + βy. The vectors x and y must have the same length.

8.3.9 Finding maximum and minimum elements of vectors
The following operations are only defined for real vectors.
double gsl_vector_max(const gsl_vector *v)
This function returns the maximum value in the vector v.
double gsl_vector_min(const gsl_vector *v)
This function returns the minimum value in the vector v.
void gsl_vector_minmax(const gsl_vector *v, double *min_out, double *max_out)
This function returns the minimum and maximum values in the vector v, storing them in min_out and max_out.
size_t gsl_vector_max_index(const gsl_vector *v)
This function returns the index of the maximum value in the vector v. When there are several equal maximum
elements then the lowest index is returned.
size_t gsl_vector_min_index(const gsl_vector *v)
This function returns the index of the minimum value in the vector v. When there are several equal minimum
elements then the lowest index is returned.
void gsl_vector_minmax_index(const gsl_vector *v, size_t *imin, size_t *imax)
This function returns the indices of the minimum and maximum values in the vector v, storing them in imin andimax. When there are several equal minimum or maximum elements then the lowest indices are returned.

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8.3.10 Vector properties
The following functions are defined for real and complex vectors. For complex vectors both the real and imaginaryparts must satisfy the conditions.
int gsl_vector_isnull(const gsl_vector *v)
int gsl_vector_ispos(const gsl_vector *v)
int gsl_vector_isneg(const gsl_vector *v)
int gsl_vector_isnonneg(const gsl_vector *v)
These functions return 1 if all the elements of the vector v are zero, strictly positive, strictly negative, or nonnegative respectively, and 0 otherwise.
int gsl_vector_equal(const gsl_vector *u, const gsl_vector *v)
This function returns 1 if the vectors u and v are equal (by comparison of element values) and 0 otherwise.

8.3.11 Example programs for vectors
This program shows how to allocate, initialize and read from a vector using the functions gsl_vector_alloc(),
gsl_vector_set() and gsl_vector_get().
#include <stdio.h>#include <gsl/gsl_vector.h>intmain (void){
int i;
gsl_vector * v = gsl_vector_alloc (3);for (i = 0; i < 3; i++){gsl_vector_set (v, i, 1.23 + i);}for (i = 0; i < 100; i++) /* OUT OF RANGE ERROR */{printf ("v_%d = %g\n", i, gsl_vector_get (v, i));}gsl_vector_free (v);return 0;}
Here is the output from the program. The final loop attempts to read outside the range of the vector v, and the error istrapped by the range-checking code in gsl_vector_get().
$ ./a.out
v_0 = 1.23
v_1 = 2.23
v_2 = 3.23
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GNU Scientific Library, Release 2.7(continued from previous page)gsl: vector_source.c:12: ERROR: index out of range
Default GSL error handler invoked.
Aborted (core dumped)
The next program shows how to write a vector to a file.
#include <stdio.h>#include <gsl/gsl_vector.h>intmain (void){
int i;
gsl_vector * v = gsl_vector_alloc (100);for (i = 0; i < 100; i++){gsl_vector_set (v, i, 1.23 + i);}{
FILE * f = fopen ("test.dat", "w");gsl_vector_fprintf (f, v, "%.5g");
fclose (f);}gsl_vector_free (v);return 0;}
After running this program the file test.dat should contain the elements of v, written using the format specifier %.5g.
The vector could then be read back in using the function gsl_vector_fscanf (f, v) as follows:#include <stdio.h>#include <gsl/gsl_vector.h>intmain (void){
int i;
gsl_vector * v = gsl_vector_alloc (10);{FILE * f = fopen ("test.dat", "r");gsl_vector_fscanf (f, v);
fclose (f);}for (i = 0; i < 10; i++){printf ("%g\n", gsl_vector_get(v, i));}(continues on next page)8.3. Vectors

93GNU Scientific Library, Release 2.7(continued from previous page)gsl_vector_free (v);return 0;}

8.4 Matrices
Matrices are defined by a gsl_matrix structure which describes a generalized slice of a block. Like a vector it
represents a set of elements in an area of memory, but uses two indices instead of one.
type gsl_matrix
The gsl_matrix structure contains six components, the two dimensions of the matrix, a physical dimension, apointer to the memory where the elements of the matrix are stored, data, a pointer to the block owned by thematrix block, if any, and an ownership flag, owner. The physical dimension determines the memory layout andcan differ from the matrix dimension to allow the use of submatrices. The gsl_matrix structure is very simple
and looks like this:typedef struct{
size_t size1;
size_t size2;
size_t tda;
double * data;
gsl_block * block;int owner;} gsl_matrix;
Matrices are stored in row-major order, meaning that each row of elements forms a contiguous block in memory. This is
the standard “C-language ordering” of two-dimensional arrays. Note that Fortran stores arrays in column-major order.
The number of rows is size1. The range of valid row indices runs from 0 to size1 - 1. Similarly size2 is thenumber of columns. The range of valid column indices runs from 0 to size2 - 1. The physical row dimension tda,
or trailing dimension, specifies the size of a row of the matrix as laid out in memory.
For example, in the following matrix size1 is 3, size2 is 4, and tda is 8. The physical memory layout of the matrix
begins in the top left hand-corner and proceeds from left to right along each row in turn.
00 01 02 03 XX XX XX XX
10 11 12 13 XX XX XX XX
20 21 22 23 XX XX XX XX
Each unused memory location is represented by “XX”. The pointer data gives the location of the first element of thematrix in memory. The pointer block stores the location of the memory block in which the elements of the matrix are
located (if any). If the matrix owns this block then the owner field is set to one and the block will be deallocated whenthe matrix is freed. If the matrix is only a slice of a block owned by another object then the owner field is zero and any
underlying block will not be freed.
The functions for allocating and accessing matrices are defined in gsl_matrix.h.

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8.4.1 Matrix allocation
The functions for allocating memory to a matrix follow the style of malloc and free. They also perform their own
error checking. If there is insufficient memory available to allocate a matrix then the functions call the GSL error
handler (with an error number of GSL_ENOMEM) in addition to returning a null pointer. Thus if you use the library error
handler to abort your program then it isn’t necessary to check every alloc.
gsl_matrix *gsl_matrix_alloc(size_t n1, size_t n2)This function creates a matrix of size n1 rows by n2 columns, returning a pointer to a newly initialized matrix
struct. A new block is allocated for the elements of the matrix, and stored in the block component of the matrix
struct. The block is “owned” by the matrix, and will be deallocated when the matrix is deallocated. Requestingzero for n1 or n2 is valid and returns a non-null result.
gsl_matrix *gsl_matrix_calloc(size_t n1, size_t n2)This function allocates memory for a matrix of size n1 rows by n2 columns and initializes all the elements ofthe matrix to zero.
void gsl_matrix_free(gsl_matrix *m)This function frees a previously allocated matrix m. If the matrix was created using gsl_matrix_alloc() thenthe block underlying the matrix will also be deallocated. If the matrix has been created from another object thenthe memory is still owned by that object and will not be deallocated.

8.4.2 Accessing matrix elements
The functions for accessing the elements of a matrix use the same range checking system as vectors. You can turn off
range checking by recompiling your program with the preprocessor definition GSL_RANGE_CHECK_OFF.
The elements of the matrix are stored in “C-order”, where the second index moves continuously through memory. Moreprecisely, the element accessed by the function gsl_matrix_get(m,i,j) and gsl_matrix_set(m,i,j,x) is:m->data[i * m->tda + j]
where tda is the physical row-length of the matrix.
double gsl_matrix_get(const gsl_matrix *m, const size_t i, const size_t j)
This function returns the (i, j)-th element of a matrix m. If i or j lie outside the allowed range of 0 to n1 - 1
and 0 to n2 - 1 then the error handler is invoked and 0 is returned. An inline version of this function is used
when HAVE_INLINE is defined.
void gsl_matrix_set(gsl_matrix *m, const size_t i, const size_t j, double x)
This function sets the value of the (i, j)-th element of a matrix m to x. If i or j lies outside the allowed range of
0 to n1 - 1 and 0 to n2 - 1 then the error handler is invoked. An inline version of this function is used when
HAVE_INLINE is defined.
double *gsl_matrix_ptr(gsl_matrix *m, size_t i, size_t j)
const double *gsl_matrix_const_ptr(const gsl_matrix *m, size_t i, size_t j)
These functions return a pointer to the (i, j)-th element of a matrix m. If i or j lie outside the allowed range of
0 to n1 - 1 and 0 to n2 - 1 then the error handler is invoked and a null pointer is returned. Inline versions ofthese functions are used when HAVE_INLINE is defined.

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8.4.3 Initializing matrix elements
void gsl_matrix_set_all(gsl_matrix *m, double x)
This function sets all the elements of the matrix m to the value x.
void gsl_matrix_set_zero(gsl_matrix *m)This function sets all the elements of the matrix m to zero.
void gsl_matrix_set_identity(gsl_matrix *m)This function sets the elements of the matrix m to the corresponding elements of the identity matrix, m(i, j) =δ(i, j), i.e. a unit diagonal with all off-diagonal elements zero. This applies to both square and rectangularmatrices.

8.4.4 Reading and writing matrices
The library provides functions for reading and writing matrices to a file as binary data or formatted text.
int gsl_matrix_fwrite(FILE *stream, const gsl_matrix *m)This function writes the elements of the matrix m to the stream stream in binary format. The return value is 0
for success and GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native
binary format it may not be portable between different architectures.
int gsl_matrix_fread(FILE *stream, gsl_matrix *m)This function reads into the matrix m from the open stream stream in binary format. The matrix m must be
preallocated with the correct dimensions since the function uses the size of m to determine how many bytes toread. The return value is 0 for success and GSL_EFAILED if there was a problem reading from the file. The data
is assumed to have been written in the native binary format on the same architecture.
int gsl_matrix_fprintf(FILE *stream, const gsl_matrix *m, const char *format)
This function writes the elements of the matrix m line-by-line to the stream stream using the format specifier
format, which should be one of the %g, %e or %f formats for floating point numbers and %d for integers. The
function returns 0 for success and GSL_EFAILED if there was a problem writing to the file.
int gsl_matrix_fscanf(FILE *stream, gsl_matrix *m)This function reads formatted data from the stream stream into the matrix m. The matrix m must be preallocatedwith the correct dimensions since the function uses the size of m to determine how many numbers to read. The
function returns 0 for success and GSL_EFAILED if there was a problem reading from the file.

8.4.5 Matrix views
type gsl_matrix_view
type gsl_matrix_const_view
A matrix view is a temporary object, stored on the stack, which can be used to operate on a subset of matrix
elements. Matrix views can be defined for both constant and non-constant matrices using separate types that
preserve constness. A matrix view has the type gsl_matrix_view and a constant matrix view has the typegsl_matrix_const_view. In both cases the elements of the view can by accessed using the matrix componentof the view object. A pointer gsl_matrix * or const gsl_matrix * can be obtained by taking the addressof the matrix component with the & operator. In addition to matrix views it is also possible to create vector
views of a matrix, such as row or column views.
gsl_matrix_view gsl_matrix_submatrix(gsl_matrix *m, size_t k1, size_t k2, size_t n1, size_t n2)

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gsl_matrix_const_view gsl_matrix_const_submatrix(const gsl_matrix *m, size_t k1, size_t k2, size_t n1, size_tn2)These functions return a matrix view of a submatrix of the matrix m. The upper-left element of the submatrix isthe element (k1, k2) of the original matrix. The submatrix has n1 rows and n2 columns. The physical numberof columns in memory given by tda is unchanged. Mathematically, the (i, j)-th element of the new matrix is
given by:m'(i,j) = m->data[(k1*m->tda + k2) + i*m->tda + j]
where the index i runs from 0 to n1 - 1 and the index j runs from 0 to n2 - 1.
The data pointer of the returned matrix struct is set to null if the combined parameters (i, j, n1, n2, tda)
overrun the ends of the original matrix.
The new matrix view is only a view of the block underlying the existing matrix, m. The block containing theelements of m is not owned by the new matrix view. When the view goes out of scope the original matrix m andits block will continue to exist. The original memory can only be deallocated by freeing the original matrix. Of
course, the original matrix should not be deallocated while the view is still in use.
The function gsl_matrix_const_submatrix() is equivalent to gsl_matrix_submatrix() but can be used
for matrices which are declared const.
gsl_matrix_view gsl_matrix_view_array(double *base, size_t n1, size_t n2)
gsl_matrix_const_view gsl_matrix_const_view_array(const double *base, size_t n1, size_t n2)
These functions return a matrix view of the array base. The matrix has n1 rows and n2 columns. The physicalnumber of columns in memory is also given by n2. Mathematically, the (i, j)-th element of the new matrix is
given by:m'(i,j) = base[i*n2 + j]
where the index i runs from 0 to n1 - 1 and the index j runs from 0 to n2 - 1.
The new matrix is only a view of the array base. When the view goes out of scope the original array base willcontinue to exist. The original memory can only be deallocated by freeing the original array. Of course, theoriginal array should not be deallocated while the view is still in use.
The function gsl_matrix_const_view_array() is equivalent to gsl_matrix_view_array() but can beused for matrices which are declared const.
gsl_matrix_view gsl_matrix_view_array_with_tda(double *base, size_t n1, size_t n2, size_t tda)
gsl_matrix_const_view gsl_matrix_const_view_array_with_tda(const double *base, size_t n1, size_t n2,
size_t tda)
These functions return a matrix view of the array base with a physical number of columns tda which may differ
from the corresponding dimension of the matrix. The matrix has n1 rows and n2 columns, and the physicalnumber of columns in memory is given by tda. Mathematically, the (i, j)-th element of the new matrix is givenby:m'(i,j) = base[i*tda + j]
where the index i runs from 0 to n1 - 1 and the index j runs from 0 to n2 - 1.
The new matrix is only a view of the array base. When the view goes out of scope the original array base willcontinue to exist. The original memory can only be deallocated by freeing the original array. Of course, theoriginal array should not be deallocated while the view is still in use.
Thefunction
gsl_matrix_const_view_array_with_tda()
isequivalent
gsl_matrix_view_array_with_tda() but can be used for matrices which are declared const.
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gsl_matrix_view gsl_matrix_view_vector(gsl_vector *v, size_t n1, size_t n2)
gsl_matrix_const_view gsl_matrix_const_view_vector(const gsl_vector *v, size_t n1, size_t n2)
These functions return a matrix view of the vector v. The matrix has n1 rows and n2 columns. The vector musthave unit stride. The physical number of columns in memory is also given by n2. Mathematically, the (i, j)-thelement of the new matrix is given by:m'(i,j) = v->data[i*n2 + j]
where the index i runs from 0 to n1 - 1 and the index j runs from 0 to n2 - 1.
The new matrix is only a view of the vector v. When the view goes out of scope the original vector v willcontinue to exist. The original memory can only be deallocated by freeing the original vector. Of course, theoriginal vector should not be deallocated while the view is still in use.
The function gsl_matrix_const_view_vector() is equivalent to gsl_matrix_view_vector() but can beused for matrices which are declared const.
gsl_matrix_view gsl_matrix_view_vector_with_tda(gsl_vector *v, size_t n1, size_t n2, size_t tda)
gsl_matrix_const_view gsl_matrix_const_view_vector_with_tda(const gsl_vector *v, size_t n1, size_t n2,
size_t tda)
These functions return a matrix view of the vector v with a physical number of columns tda which may differ
from the corresponding matrix dimension. The vector must have unit stride. The matrix has n1 rows and n2
columns, and the physical number of columns in memory is given by tda. Mathematically, the (i, j)-th elementof the new matrix is given by:m'(i,j) = v->data[i*tda + j]
where the index i runs from 0 to n1 - 1 and the index j runs from 0 to n2 - 1.
The new matrix is only a view of the vector v. When the view goes out of scope the original vector v willcontinue to exist. The original memory can only be deallocated by freeing the original vector. Of course, theoriginal vector should not be deallocated while the view is still in use.
Thefunction
gsl_matrix_const_view_vector_with_tda()
isequivalent
gsl_matrix_view_vector_with_tda() but can be used for matrices which are declared const.

to8.4.6 Creating row and column views
In general there are two ways to access an object, by reference or by copying. The functions described in this section
create vector views which allow access to a row or column of a matrix by reference. Modifying elements of the view
is equivalent to modifying the matrix, since both the vector view and the matrix point to the same memory block.
gsl_vector_view gsl_matrix_row(gsl_matrix *m, size_t i)
gsl_vector_const_view gsl_matrix_const_row(const gsl_matrix *m, size_t i)
These functions return a vector view of the i-th row of the matrix m. The data pointer of the new vector is set
to null if i is out of range.
The function gsl_matrix_const_row() is equivalent to gsl_matrix_row() but can be used for matrices
which are declared const.
gsl_vector_view gsl_matrix_column(gsl_matrix *m, size_t j)

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gsl_vector_const_view gsl_matrix_const_column(const gsl_matrix *m, size_t j)
These functions return a vector view of the j-th column of the matrix m. The data pointer of the new vector isset to null if j is out of range.
The function gsl_matrix_const_column() is equivalent to gsl_matrix_column() but can be used for matrices which are declared const.
gsl_vector_view gsl_matrix_subrow(gsl_matrix *m, size_t i, size_t offset, size_t n)
gsl_vector_const_view gsl_matrix_const_subrow(const gsl_matrix *m, size_t i, size_t offset, size_t n)
These functions return a vector view of the i-th row of the matrix m beginning at offset elements past the firstcolumn and containing n elements. The data pointer of the new vector is set to null if i, offset, or n are outof range.
The function gsl_matrix_const_subrow() is equivalent to gsl_matrix_subrow() but can be used for matrices which are declared const.
gsl_vector_view gsl_matrix_subcolumn(gsl_matrix *m, size_t j, size_t offset, size_t n)
gsl_vector_const_view gsl_matrix_const_subcolumn(const gsl_matrix *m, size_t j, size_t offset, size_t n)
These functions return a vector view of the j-th column of the matrix m beginning at offset elements past thefirst row and containing n elements. The data pointer of the new vector is set to null if j, offset, or n are outof range.
The function gsl_matrix_const_subcolumn() is equivalent to gsl_matrix_subcolumn() but can be used
for matrices which are declared const.
gsl_vector_view gsl_matrix_diagonal(gsl_matrix *m)
gsl_vector_const_view gsl_matrix_const_diagonal(const gsl_matrix *m)
These functions return a vector view of the diagonal of the matrix m. The matrix m is not required to be square.
For a rectangular matrix the length of the diagonal is the same as the smaller dimension of the matrix.
The function gsl_matrix_const_diagonal() is equivalent to gsl_matrix_diagonal() but can be used for
matrices which are declared const.
gsl_vector_view gsl_matrix_subdiagonal(gsl_matrix *m, size_t k)
gsl_vector_const_view gsl_matrix_const_subdiagonal(const gsl_matrix *m, size_t k)
These functions return a vector view of the k-th subdiagonal of the matrix m. The matrix m is not required to besquare. The diagonal of the matrix corresponds to k = 0.
The function gsl_matrix_const_subdiagonal() is equivalent to gsl_matrix_subdiagonal() but can beused for matrices which are declared const.
gsl_vector_view gsl_matrix_superdiagonal(gsl_matrix *m, size_t k)
gsl_vector_const_view gsl_matrix_const_superdiagonal(const gsl_matrix *m, size_t k)
These functions return a vector view of the k-th superdiagonal of the matrix m. The matrix m is not required tobe square. The diagonal of the matrix corresponds to k = 0.
The function gsl_matrix_const_superdiagonal() is equivalent to gsl_matrix_superdiagonal() but
can be used for matrices which are declared const.

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8.4.7 Copying matrices
int gsl_matrix_memcpy(gsl_matrix *dest, const gsl_matrix *src)
This function copies the elements of the matrix src into the matrix dest. The two matrices must have the samesize.
int gsl_matrix_swap(gsl_matrix *m1, gsl_matrix *m2)
This function exchanges the elements of the matrices m1 and m2 by copying. The two matrices must have thesame size.

8.4.8 Copying rows and columns
The functions described in this section copy a row or column of a matrix into a vector. This allows the elements ofthe vector and the matrix to be modified independently. Note that if the matrix and the vector point to overlapping
regions of memory then the result will be undefined. The same effect can be achieved with more generality using
gsl_vector_memcpy() with vector views of rows and columns.
int gsl_matrix_get_row(gsl_vector *v, const gsl_matrix *m, size_t i)This function copies the elements of the i-th row of the matrix m into the vector v. The length of the vector must
be the same as the length of the row.
int gsl_matrix_get_col(gsl_vector *v, const gsl_matrix *m, size_t j)
This function copies the elements of the j-th column of the matrix m into the vector v. The length of the vector
must be the same as the length of the column.
int gsl_matrix_set_row(gsl_matrix *m, size_t i, const gsl_vector *v)
This function copies the elements of the vector v into the i-th row of the matrix m. The length of the vector must
be the same as the length of the row.
int gsl_matrix_set_col(gsl_matrix *m, size_t j, const gsl_vector *v)
This function copies the elements of the vector v into the j-th column of the matrix m. The length of the vector
must be the same as the length of the column.

8.4.9 Exchanging rows and columns
The following functions can be used to exchange the rows and columns of a matrix.
int gsl_matrix_swap_rows(gsl_matrix *m, size_t i, size_t j)
This function exchanges the i-th and j-th rows of the matrix m in-place.
int gsl_matrix_swap_columns(gsl_matrix *m, size_t i, size_t j)
This function exchanges the i-th and j-th columns of the matrix m in-place.
int gsl_matrix_swap_rowcol(gsl_matrix *m, size_t i, size_t j)
This function exchanges the i-th row and j-th column of the matrix m in-place. The matrix must be square forthis operation to be possible.
int gsl_matrix_transpose_memcpy(gsl_matrix *dest, const gsl_matrix *src)
This function makes the matrix dest the transpose of the matrix src by copying the elements of src into dest.
This function works for all matrices provided that the dimensions of the matrix dest match the transposed
dimensions of the matrix src.
int gsl_matrix_transpose(gsl_matrix *m)This function replaces the matrix m by its transpose by copying the elements of the matrix in-place. The matrix
must be square for this operation to be possible.

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int gsl_matrix_complex_conjtrans_memcpy(gsl_matrix_complex *dest, const gsl_matrix_complex *src)
This function makes the matrix dest the conjugate transpose of the matrix src by copying the complex conjugate
elements of src into dest. This function works for all complex matrices provided that the dimensions of thematrix dest match the transposed dimensions of the matrix src.

8.4.10 Matrix operations
The following operations are defined for real and complex matrices.
int gsl_matrix_add(gsl_matrix *a, const gsl_matrix *b)
This function adds the elements of matrix b to the elements of matrix a. The result a(i, j) ← a(i, j) + b(i, j) isstored in a and b remains unchanged. The two matrices must have the same dimensions.
int gsl_matrix_sub(gsl_matrix *a, const gsl_matrix *b)
This function subtracts the elements of matrix b from the elements of matrix a. The result a(i, j) ← a(i, j) −b(i, j) is stored in a and b remains unchanged. The two matrices must have the same dimensions.
int gsl_matrix_mul_elements(gsl_matrix *a, const gsl_matrix *b)
This function multiplies the elements of matrix a by the elements of matrix b. The result a(i, j) ← a(i, j)∗b(i, j)
is stored in a and b remains unchanged. The two matrices must have the same dimensions.
int gsl_matrix_div_elements(gsl_matrix *a, const gsl_matrix *b)
This function divides the elements of matrix a by the elements of matrix b. The result a(i, j) ← a(i, j)/b(i, j)
is stored in a and b remains unchanged. The two matrices must have the same dimensions.
int gsl_matrix_scale(gsl_matrix *a, const double x)
This function multiplies the elements of matrix a by the constant factor x. The result a(i, j) ← xa(i, j) is storedin a.
int gsl_matrix_scale_columns(gsl_matrix *A, const gsl_vector *x)
This function scales the columns of the M -by-N matrix A by the elements of the vector x, of length N . The j-thcolumn of A is multiplied by xj . This is equivalent to forming
A → AX
where X = diag(x).
int gsl_matrix_scale_rows(gsl_matrix *A, const gsl_vector *x)
This function scales the rows of the M -by-N matrix A by the elements of the vector x, of length M . The i-th
row of A is multiplied by xi . This is equivalent to forming
A → XA
where X = diag(x).
int gsl_matrix_add_constant(gsl_matrix *a, const double x)
This function adds the constant value x to the elements of the matrix a. The result a(i, j) ← a(i, j) + x is stored
in a.

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8.4.11 Finding maximum and minimum elements of matrices
The following operations are only defined for real matrices.
double gsl_matrix_max(const gsl_matrix *m)This function returns the maximum value in the matrix m.
double gsl_matrix_min(const gsl_matrix *m)This function returns the minimum value in the matrix m.
void gsl_matrix_minmax(const gsl_matrix *m, double *min_out, double *max_out)
This function returns the minimum and maximum values in the matrix m, storing them in min_out and max_out.
void gsl_matrix_max_index(const gsl_matrix *m, size_t *imax, size_t *jmax)
This function returns the indices of the maximum value in the matrix m, storing them in imax and jmax. When
there are several equal maximum elements then the first element found is returned, searching in row-major order.
void gsl_matrix_min_index(const gsl_matrix *m, size_t *imin, size_t *jmin)This function returns the indices of the minimum value in the matrix m, storing them in imin and jmin. When
there are several equal minimum elements then the first element found is returned, searching in row-major order.
void gsl_matrix_minmax_index(const gsl_matrix *m, size_t *imin, size_t *jmin, size_t *imax, size_t *jmax)
This function returns the indices of the minimum and maximum values in the matrix m, storing them in (imin,jmin) and (imax, jmax). When there are several equal minimum or maximum elements then the first elements
found are returned, searching in row-major order.

8.4.12 Matrix properties
The following functions are defined for real and complex matrices. For complex matrices both the real and imaginaryparts must satisfy the conditions.
int gsl_matrix_isnull(const gsl_matrix *m)
int gsl_matrix_ispos(const gsl_matrix *m)
int gsl_matrix_isneg(const gsl_matrix *m)
int gsl_matrix_isnonneg(const gsl_matrix *m)
These functions return 1 if all the elements of the matrix m are zero, strictly positive, strictly negative, or nonnegative respectively, and 0 otherwise. To test whether a matrix is positive-definite, use the Cholesky decomposition.
int gsl_matrix_equal(const gsl_matrix *a, const gsl_matrix *b)
This function returns 1 if the matrices a and b are equal (by comparison of element values) and 0 otherwise.
double gsl_matrix_norm1(const gsl_matrix *A)
This function returns the 1-norm of the m-by-n matrix A, defined as the maximum column sum,||A||1 = max1≤j≤nm
X|Aij |

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8.4.13 Example programs for matrices
The program below shows how to allocate, initialize and read from a matrix using the functions gsl_matrix_alloc(),
gsl_matrix_set() and gsl_matrix_get().
#include <stdio.h>#include <gsl/gsl_matrix.h>
intmain (void){
int i, j;
gsl_matrix * m = gsl_matrix_alloc (10, 3);
for (i = 0; i < 10; i++)for (j = 0; j < 3; j++)gsl_matrix_set (m, i, j, 0.23 + 100*i + j);for (i = 0; i < 100; i++) /* OUT OF RANGE ERROR */for (j = 0; j < 3; j++)printf ("m(%d,%d) = %g\n", i, j,
gsl_matrix_get (m, i, j));gsl_matrix_free (m);return 0;}
Here is the output from the program. The final loop attempts to read outside the range of the matrix m, and the error istrapped by the range-checking code in gsl_matrix_get().
$ ./a.outm(0,0) = 0.23m(0,1) = 1.23m(0,2) = 2.23m(1,0) = 100.23m(1,1) = 101.23m(1,2) = 102.23...
m(9,2) = 902.23
gsl: matrix_source.c:13: ERROR: first index out of range
Default GSL error handler invoked.
Aborted (core dumped)
The next program shows how to write a matrix to a file.
#include <stdio.h>#include <gsl/gsl_matrix.h>
intmain (void){
int i, j, k = 0;
gsl_matrix * m = gsl_matrix_alloc (100, 100);(continues on next page)

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103GNU Scientific Library, Release 2.7(continued from previous page)gsl_matrix * a = gsl_matrix_alloc (100, 100);for (i = 0; i < 100; i++)for (j = 0; j < 100; j++)gsl_matrix_set (m, i, j, 0.23 + i + j);{
FILE * f = fopen ("test.dat", "wb");
gsl_matrix_fwrite (f, m);fclose (f);}{FILE * f = fopen ("test.dat", "rb");gsl_matrix_fread (f, a);fclose (f);}for (i = 0; i < 100; i++)for (j = 0; j < 100; j++){double mij = gsl_matrix_get (m, i, j);double aij = gsl_matrix_get (a, i, j);if (mij != aij) k++;}gsl_matrix_free (m);gsl_matrix_free (a);printf ("differences = %d (should be zero)\n", k);return (k > 0);}
After running this program the file test.dat should contain the elements of m, written in binary format. The matrix
which is read back in using the function gsl_matrix_fread() should be exactly equal to the original matrix.
The following program demonstrates the use of vector views. The program computes the column norms of a matrix.
#include <math.h>#include <stdio.h>#include <gsl/gsl_matrix.h>#include <gsl/gsl_blas.h>intmain (void){
size_t i,j;
gsl_matrix *m = gsl_matrix_alloc (10, 10);for (i = 0; i < 10; i++)for (j = 0; j < 10; j++)gsl_matrix_set (m, i, j, sin (i) + cos (j));(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)for (j = 0; j < 10; j++){gsl_vector_view column = gsl_matrix_column (m, j);double d;d = gsl_blas_dnrm2 (&column.vector);
printf ("matrix column %zu, norm = %g\n", j, d);}
gsl_matrix_free (m);return 0;}
Here is the output of the program,matrix column 0, norm = 4.31461
matrix column 1, norm = 3.1205
matrix column 2, norm = 2.19316
matrix column 3, norm = 3.26114
matrix column 4, norm = 2.53416
matrix column 5, norm = 2.57281
matrix column 6, norm = 4.20469
matrix column 7, norm = 3.65202
matrix column 8, norm = 2.08524
matrix column 9, norm = 3.07313
The results can be confirmed using GNU octave:$ octave
GNU Octave, version 2.0.16.92
octave> m = sin(0:9)' * ones(1,10)+ ones(10,1) * cos(0:9);
octave> sqrt(sum(m.^2))
ans =4.3146 3.1205 2.1932 3.2611 2.5342
4.2047 3.6520 2.0852 3.0731

2.57288.4.14 References and Further Reading
The block, vector and matrix objects in GSL follow the valarray model of C++. A description of this model can befound in the following reference,• B. Stroustrup, The C++ Programming Language (3rd Ed), Section 22.4 Vector Arithmetic. Addison-Wesley
1997, ISBN 0-201-88954-4.

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106Chapter 8. Vectors and Matrices

CHAPTERNINEPERMUTATIONSThis chapter describes functions for creating and manipulating permutations. A permutation p is represented by anarray of n integers in the range 0 to n − 1, where each value pi occurs once and only once. The application of a
permutation p to a vector v yields a new vector v 0 where vi0 = vpi . For example, the array (0, 1, 3, 2) represents a
permutation which exchanges the last two elements of a four element vector. The corresponding identity permutationis (0, 1, 2, 3).
Note that the permutations produced by the linear algebra routines correspond to the exchange of matrix columns, andso should be considered as applying to row-vectors in the form v 0 = vP rather than column-vectors, when permuting
the elements of a vector.
The functions described in this chapter are defined in the header file gsl_permutation.h.

9.1 The Permutation structtype gsl_permutation
A permutation is defined by a structure containing two components, the size of the permutation and a pointer tothe permutation array. The elements of the permutation array are all of type size_t. The gsl_permutation
structure looks like this:typedef struct{
size_t size;
size_t * data;} gsl_permutation;

9.2 Permutation allocation
gsl_permutation *gsl_permutation_alloc(size_t n)This function allocates memory for a new permutation of size n. The permutation is not initialized and its
elements are undefined. Use the function gsl_permutation_calloc() if you want to create a permutation
which is initialized to the identity. A null pointer is returned if insufficient memory is available to create thepermutation.
gsl_permutation *gsl_permutation_calloc(size_t n)This function allocates memory for a new permutation of size n and initializes it to the identity. A null pointeris returned if insufficient memory is available to create the permutation.
void gsl_permutation_init(gsl_permutation *p)
This function initializes the permutation p to the identity, i.e. (0, 1, 2, . . . , n − 1).

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void gsl_permutation_free(gsl_permutation *p)
This function frees all the memory used by the permutation p.
int gsl_permutation_memcpy(gsl_permutation *dest, const gsl_permutation *src)
This function copies the elements of the permutation src into the permutation dest. The two permutations musthave the same size.

9.3 Accessing permutation elements
The following functions can be used to access and manipulate permutations.
size_t gsl_permutation_get(const gsl_permutation *p, const size_t i)This function returns the value of the i-th element of the permutation p. If i lies outside the allowed range of
0 to n − 1 then the error handler is invoked and 0 is returned. An inline version of this function is used when
HAVE_INLINE is defined.
int gsl_permutation_swap(gsl_permutation *p, const size_t i, const size_t j)
This function exchanges the i-th and j-th elements of the permutation p.

9.4 Permutation properties
size_t gsl_permutation_size(const gsl_permutation *p)
This function returns the size of the permutation p.
size_t *gsl_permutation_data(const gsl_permutation *p)
This function returns a pointer to the array of elements in the permutation p.
int gsl_permutation_valid(const gsl_permutation *p)
This function checks that the permutation p is valid. The n elements should contain each of the numbers 0 to n
- 1 once and only once.

9.5 Permutation functions
void gsl_permutation_reverse(gsl_permutation *p)
This function reverses the elements of the permutation p.
int gsl_permutation_inverse(gsl_permutation *inv, const gsl_permutation *p)
This function computes the inverse of the permutation p, storing the result in inv.
int gsl_permutation_next(gsl_permutation *p)
This function advances the permutation p to the next permutation in lexicographic order and returns
GSL_SUCCESS. If no further permutations are available it returns GSL_FAILURE and leaves p unmodified. Starting with the identity permutation and repeatedly applying this function will iterate through all possible permutations of a given order.
int gsl_permutation_prev(gsl_permutation *p)
This function steps backwards from the permutation p to the previous permutation in lexicographic order, returning GSL_SUCCESS. If no previous permutation is available it returns GSL_FAILURE and leaves p unmodified.

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9.6 Applying Permutations
The following functions are defined in the header files gsl_permute.h and gsl_permute_vector.h.
int gsl_permute(const size_t *p, double *data, size_t stride, size_t n)This function applies the permutation p to the array data of size n with stride stride.
int gsl_permute_inverse(const size_t *p, double *data, size_t stride, size_t n)This function applies the inverse of the permutation p to the array data of size n with stride stride.
int gsl_permute_vector(const gsl_permutation *p, gsl_vector *v)
This function applies the permutation p to the elements of the vector v, considered as a row-vector acted on by a
permutation matrix from the right, v 0 = vP . The j-th column of the permutation matrix P is given by the pj -thcolumn of the identity matrix. The permutation p and the vector v must have the same length.
int gsl_permute_vector_inverse(const gsl_permutation *p, gsl_vector *v)
This function applies the inverse of the permutation p to the elements of the vector v, considered as a row-vector
acted on by an inverse permutation matrix from the right, v 0 = vP T . Note that for permutation matrices theinverse is the same as the transpose. The j-th column of the permutation matrix P is given by the pj -th columnof the identity matrix. The permutation p and the vector v must have the same length.
int gsl_permute_matrix(const gsl_permutation *p, gsl_matrix *A)
This function applies the permutation p to the matrix A from the right, A0 = AP . The j-th column of the
permutation matrix P is given by the pj -th column of the identity matrix. This effectively permutes the columnsof A according to the permutation p, and so the number of columns of A must equal the size of the permutationp.
int gsl_permutation_mul(gsl_permutation *p, const gsl_permutation *pa, const gsl_permutation *pb)
This function combines the two permutations pa and pb into a single permutation p, where p = pa ∗ pb The
permutation p is equivalent to applying pb first and then pa.

9.7 Reading and writing permutations
The library provides functions for reading and writing permutations to a file as binary data or formatted text.
int gsl_permutation_fwrite(FILE *stream, const gsl_permutation *p)
This function writes the elements of the permutation p to the stream stream in binary format. The function
returns GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native binary
format it may not be portable between different architectures.
int gsl_permutation_fread(FILE *stream, gsl_permutation *p)
This function reads into the permutation p from the open stream stream in binary format. The permutation p
must be preallocated with the correct length since the function uses the size of p to determine how many bytes
to read. The function returns GSL_EFAILED if there was a problem reading from the file. The data is assumed
to have been written in the native binary format on the same architecture.
int gsl_permutation_fprintf(FILE *stream, const gsl_permutation *p, const char *format)
This function writes the elements of the permutation p line-by-line to the stream stream using the format specifier format, which should be suitable for a type of size_t. In ISO C99 the type modifier z represents size_t,
so "%zu\n" is a suitable format1 . The function returns GSL_EFAILED if there was a problem writing to the file.
int gsl_permutation_fscanf(FILE *stream, gsl_permutation *p)
This function reads formatted data from the stream stream into the permutation p. The permutation p must be
preallocated with the correct length since the function uses the size of p to determine how many numbers to read.
The function returns GSL_EFAILED if there was a problem reading from the file.
1 In versions of the GNU C library prior to the ISO C99 standard, the type modifier Z was used instead.

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9.8 Permutations in cyclic form
A permutation can be represented in both linear and cyclic notations. The functions described in this section convert
between the two forms. The linear notation is an index mapping, and has already been described above. The cyclic
notation expresses a permutation as a series of circular rearrangements of groups of elements, or cycles.
For example, under the cycle (1 2 3), 1 is replaced by 2, 2 is replaced by 3 and 3 is replaced by 1 in a circular fashion.
Cycles of different sets of elements can be combined independently, for example (1 2 3) (4 5) combines the cycle (1 2
3) with the cycle (4 5), which is an exchange of elements 4 and 5. A cycle of length one represents an element whichis unchanged by the permutation and is referred to as a singleton.
It can be shown that every permutation can be decomposed into combinations of cycles. The decomposition is notunique, but can always be rearranged into a standard canonical form by a reordering of elements. The library uses the
canonical form defined in Knuth’s Art of Computer Programming (Vol 1, 3rd Ed, 1997) Section 1.3.3, p.178.
The procedure for obtaining the canonical form given by Knuth is,
1. Write all singleton cycles explicitly
2. Within each cycle, put the smallest number first
3. Order the cycles in decreasing order of the first number in the cycle.
For example, the linear representation (2 4 3 0 1) is represented as (1 4) (0 2 3) in canonical form. The permutation
corresponds to an exchange of elements 1 and 4, and rotation of elements 0, 2 and 3.
The important property of the canonical form is that it can be reconstructed from the contents of each cycle without thebrackets. In addition, by removing the brackets it can be considered as a linear representation of a different permutation.
In the example given above the permutation (2 4 3 0 1) would become (1 4 0 2 3). This mapping has many applicationsin the theory of permutations.
int gsl_permutation_linear_to_canonical(gsl_permutation *q, const gsl_permutation *p)
This function computes the canonical form of the permutation p and stores it in the output argument q.
int gsl_permutation_canonical_to_linear(gsl_permutation *p, const gsl_permutation *q)
This function converts a permutation q in canonical form back into linear form storing it in the output argumentp.
size_t gsl_permutation_inversions(const gsl_permutation *p)
This function counts the number of inversions in the permutation p. An inversion is any pair of elements that are
not in order. For example, the permutation 2031 has three inversions, corresponding to the pairs (2,0) (2,1) and(3,1). The identity permutation has no inversions.
size_t gsl_permutation_linear_cycles(const gsl_permutation *p)
This function counts the number of cycles in the permutation p, given in linear form.
size_t gsl_permutation_canonical_cycles(const gsl_permutation *q)
This function counts the number of cycles in the permutation q, given in canonical form.

9.9 Examples
The example program below creates a random permutation (by shuffling the elements of the identity) and finds its
inverse.
#include <stdio.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)#include <gsl/gsl_permutation.h>int
main (void){
const size_t N = 10;
const gsl_rng_type * T;
gsl_rng * r;
gsl_permutation * p = gsl_permutation_alloc (N);gsl_permutation * q = gsl_permutation_alloc (N);gsl_rng_env_setup();
T = gsl_rng_default;r = gsl_rng_alloc (T);printf ("initial permutation:");gsl_permutation_init (p);gsl_permutation_fprintf (stdout, p, " %u");printf ("\n");printf (" random permutation:");gsl_ran_shuffle (r, p->data, N, sizeof(size_t));gsl_permutation_fprintf (stdout, p, " %u");printf ("\n");printf ("inverse permutation:");gsl_permutation_inverse (q, p);
gsl_permutation_fprintf (stdout, q, " %u");printf ("\n");gsl_permutation_free (p);gsl_permutation_free (q);gsl_rng_free (r);return 0;}
Here is the output from the program:$ ./a.out
initial permutation: 0 1 2 3 4 5 6 7 8 9
random permutation: 1 3 5 2 7 6 0 4 9 8
inverse permutation: 6 0 3 1 7 2 5 4 9 8
The random permutation p[i] and its inverse q[i] are related through the identity p[q[i]] = i, which can be verified
from the output.
The next example program steps forwards through all possible third order permutations, starting from the identity,#include <stdio.h>#include <gsl/gsl_permutation.h>(continues on next page)

9.9. Examples

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GNU Scientific Library, Release 2.7(continued from previous page)int
main (void){
gsl_permutation * p = gsl_permutation_alloc (3);gsl_permutation_init (p);do{gsl_permutation_fprintf (stdout, p, " %u");printf ("\n");}while (gsl_permutation_next(p) == GSL_SUCCESS);
gsl_permutation_free (p);return 0;}
Here is the output from the program:$ ./a.out
0 1 2
0 2 1
1 0 2
1 2 0
2 0 1
2 1 0
The permutations are generated in lexicographic order. To reverse the sequence, begin with the final permutation (whichis the reverse of the identity) and replace gsl_permutation_next() with gsl_permutation_prev().

9.10 References and Further Reading
The subject of permutations is covered extensively in the following,• Donald E. Knuth, The Art of Computer Programming: Sorting and Searching (Vol 3, 3rd Ed, 1997), AddisonWesley, ISBN 0201896850.
For the definition of the canonical form see,
• Donald E. Knuth, The Art of Computer Programming: Fundamental Algorithms (Vol 1, 3rd Ed, 1997), AddisonWesley, ISBN 0201896850. Section 1.3.3, An Unusual Correspondence, p.178–179.

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CHAPTERTENCOMBINATIONSThis chapter describes functions for creating and manipulating combinations. A combination c is represented by anarray of k integers in the range 0 to n − 1, where each value ci occurs at most once. The combination c corresponds
to indices of k elements chosen from an n element vector. Combinations are useful for iterating over all k-element
subsets of a set.
The functions described in this chapter are defined in the header file gsl_combination.h.

10.1 The Combination structtype gsl_combination
A combination is defined by a structure containing three components, the values of n and k, and a pointer to the
combination array. The elements of the combination array are all of type size_t, and are stored in increasingorder. The gsl_combination structure looks like this:typedef struct{
size_t n;size_t k;size_t *data;} gsl_combination;

10.2 Combination allocation
gsl_combination *gsl_combination_alloc(size_t n, size_t k)
This function allocates memory for a new combination with parameters n, k. The combination is not initialized
and its elements are undefined. Use the function gsl_combination_calloc() if you want to create a combination which is initialized to the lexicographically first combination. A null pointer is returned if insufficient
memory is available to create the combination.
gsl_combination *gsl_combination_calloc(size_t n, size_t k)
This function allocates memory for a new combination with parameters n, k and initializes it to the lexicographically first combination. A null pointer is returned if insufficient memory is available to create the combination.
void gsl_combination_init_first(gsl_combination *c)
This function initializes the combination c to the lexicographically first combination, i.e. (0, 1, 2, . . . , k − 1).
void gsl_combination_init_last(gsl_combination *c)
This function initializes the combination c to the lexicographically last combination, i.e. (n−k, n−k+1, . . . , n−
1).

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void gsl_combination_free(gsl_combination *c)
This function frees all the memory used by the combination c.
int gsl_combination_memcpy(gsl_combination *dest, const gsl_combination *src)
This function copies the elements of the combination src into the combination dest. The two combinations
must have the same size.

10.3 Accessing combination elements
The following function can be used to access the elements of a combination.
size_t gsl_combination_get(const gsl_combination *c, const size_t i)This function returns the value of the i-th element of the combination c. If i lies outside the allowed range of
0 to k − 1 then the error handler is invoked and 0 is returned. An inline version of this function is used when
HAVE_INLINE is defined.

10.4 Combination properties
size_t gsl_combination_n(const gsl_combination *c)
This function returns the range (n) of the combination c.
size_t gsl_combination_k(const gsl_combination *c)
This function returns the number of elements (k) in the combination c.
size_t *gsl_combination_data(const gsl_combination *c)
This function returns a pointer to the array of elements in the combination c.
int gsl_combination_valid(gsl_combination *c)
This function checks that the combination c is valid. The k elements should lie in the range 0 to n − 1, with eachvalue occurring once at most and in increasing order.

10.5 Combination functions
int gsl_combination_next(gsl_combination *c)
This function advances the combination c to the next combination in lexicographic order and returns
GSL_SUCCESS. If no further combinations are available it returns GSL_FAILURE and leaves c unmodified. Starting with the first combination and repeatedly applying this function will iterate through all possible combinationsof a given order.
int gsl_combination_prev(gsl_combination *c)
This function steps backwards from the combination c to the previous combination in lexicographic order, returning GSL_SUCCESS. If no previous combination is available it returns GSL_FAILURE and leaves c unmodified.

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10.6 Reading and writing combinations
The library provides functions for reading and writing combinations to a file as binary data or formatted text.
int gsl_combination_fwrite(FILE *stream, const gsl_combination *c)
This function writes the elements of the combination c to the stream stream in binary format. The function
returns GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native binary
format it may not be portable between different architectures.
int gsl_combination_fread(FILE *stream, gsl_combination *c)
This function reads elements from the open stream stream into the combination c in binary format. The combination c must be preallocated with correct values of n and k since the function uses the size of c to determine
how many bytes to read. The function returns GSL_EFAILED if there was a problem reading from the file. The
data is assumed to have been written in the native binary format on the same architecture.
int gsl_combination_fprintf(FILE *stream, const gsl_combination *c, const char *format)
This function writes the elements of the combination c line-by-line to the stream stream using the format
specifier format, which should be suitable for a type of size_t. In ISO C99 the type modifier z represents
size_t, so "%zu\n" is a suitable format1 . The function returns GSL_EFAILED if there was a problem writing
to the file.
int gsl_combination_fscanf(FILE *stream, gsl_combination *c)
This function reads formatted data from the stream stream into the combination c. The combination c must be
preallocated with correct values of n and k since the function uses the size of c to determine how many numbers
to read. The function returns GSL_EFAILED if there was a problem reading from the file.

10.7 Examples
The example program below prints all subsets of the set 0, 1, 2, 3 ordered by size. Subsets of the same size are ordered
lexicographically.
#include <stdio.h>#include <gsl/gsl_combination.h>int
main (void){
gsl_combination * c;size_t i;
printf ("All subsets of {0,1,2,3} by size:\n") ;for (i = 0; i <= 4; i++){c = gsl_combination_calloc (4, i);do{printf ("{");
gsl_combination_fprintf (stdout, c, " %u");printf (" }\n");}while (gsl_combination_next (c) == GSL_SUCCESS);(continues on next page)
1 In versions of the GNU C library prior to the ISO C99 standard, the type modifier Z was used instead.

10.6. Reading and writing combinations

115

GNU Scientific Library, Release 2.7(continued from previous page)gsl_combination_free (c);}return 0;}
Here is the output from the program,All subsets of {0,1,2,3} by size:
{ }
{ 0 }
{ 1 }
{ 2 }
{ 3 }
{ 0 1 }
{ 0 2 }
{ 0 3 }
{ 1 2 }
{ 1 3 }
{ 2 3 }
{ 0 1 2 }{ 0 1 3 }{ 0 2 3 }{ 1 2 3 }{ 0 1 2 3 }All 16 subsets are generated, and the subsets of each size are sorted lexicographically.

10.8 References and Further Reading
Further information on combinations can be found in,• Donald L. Kreher, Douglas R. Stinson, Combinatorial Algorithms: Generation, Enumeration and Search, 1998,
CRC Press LLC, ISBN 084933988X

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CHAPTERELEVEN

MULTISETS

This chapter describes functions for creating and manipulating multisets. A multiset c is represented by an array of k
integers in the range 0 to n − 1, where each value ci may occur more than once. The multiset c corresponds to indicesof k elements chosen from an n element vector with replacement. In mathematical terms, n is the cardinality of themultiset while k is the maximum multiplicity of any value. Multisets are useful, for example, when iterating over theindices of a k-th order symmetric tensor in n-space.
The functions described in this chapter are defined in the header file gsl_multiset.h.

11.1 The Multiset structtype gsl_multiset
A multiset is defined by a structure containing three components, the values of n and k, and a pointer to themultiset array. The elements of the multiset array are all of type size_t, and are stored in increasing order. The
gsl_multiset structure looks like this:typedef struct{
size_t n;size_t k;size_t *data;} gsl_multiset;11.2 Multiset allocation
gsl_multiset *gsl_multiset_alloc(size_t n, size_t k)
This function allocates memory for a new multiset with parameters n, k. The multiset is not initialized and its
elements are undefined. Use the function gsl_multiset_calloc() if you want to create a multiset which is
initialized to the lexicographically first multiset element. A null pointer is returned if insufficient memory isavailable to create the multiset.
gsl_multiset *gsl_multiset_calloc(size_t n, size_t k)
This function allocates memory for a new multiset with parameters n, k and initializes it to the lexicographicallyfirst multiset element. A null pointer is returned if insufficient memory is available to create the multiset.
void gsl_multiset_init_first(gsl_multiset *c)
This function initializes the multiset c to the lexicographically first multiset element, i.e. 0 repeated k times.
void gsl_multiset_init_last(gsl_multiset *c)
This function initializes the multiset c to the lexicographically last multiset element, i.e. n − 1 repeated k times.

117GNU Scientific Library, Release 2.7

void gsl_multiset_free(gsl_multiset *c)
This function frees all the memory used by the multiset c.
int gsl_multiset_memcpy(gsl_multiset *dest, const gsl_multiset *src)
This function copies the elements of the multiset src into the multiset dest. The two multisets must have thesame size.

11.3 Accessing multiset elements
The following function can be used to access the elements of a multiset.
size_t gsl_multiset_get(const gsl_multiset *c, const size_t i)This function returns the value of the i-th element of the multiset c. If i lies outside the allowed range of 0
to k − 1 then the error handler is invoked and 0 is returned. An inline version of this function is used when
HAVE_INLINE is defined.

11.4 Multiset properties
size_t gsl_multiset_n(const gsl_multiset *c)
This function returns the range (n) of the multiset c.
size_t gsl_multiset_k(const gsl_multiset *c)
This function returns the number of elements (k) in the multiset c.
size_t *gsl_multiset_data(const gsl_multiset *c)
This function returns a pointer to the array of elements in the multiset c.
int gsl_multiset_valid(gsl_multiset *c)
This function checks that the multiset c is valid. The k elements should lie in the range 0 to n − 1, with eachvalue occurring in nondecreasing order.

11.5 Multiset functions
int gsl_multiset_next(gsl_multiset *c)
This function advances the multiset c to the next multiset element in lexicographic order and returns
GSL_SUCCESS. If no further multisets elements are available it returns GSL_FAILURE and leaves c unmodified.
Starting with the first multiset and repeatedly applying this function will iterate through all possible multisets ofa given order.
int gsl_multiset_prev(gsl_multiset *c)
This function steps backwards from the multiset c to the previous multiset element in lexicographic order, returning GSL_SUCCESS. If no previous multiset is available it returns GSL_FAILURE and leaves c unmodified.

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11.6 Reading and writing multisets
The library provides functions for reading and writing multisets to a file as binary data or formatted text.
int gsl_multiset_fwrite(FILE *stream, const gsl_multiset *c)
This function writes the elements of the multiset c to the stream stream in binary format. The function returns
GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native binary format it
may not be portable between different architectures.
int gsl_multiset_fread(FILE *stream, gsl_multiset *c)
This function reads elements from the open stream stream into the multiset c in binary format. The multiset
c must be preallocated with correct values of n and k since the function uses the size of c to determine how
many bytes to read. The function returns GSL_EFAILED if there was a problem reading from the file. The data
is assumed to have been written in the native binary format on the same architecture.
int gsl_multiset_fprintf(FILE *stream, const gsl_multiset *c, const char *format)
This function writes the elements of the multiset c line-by-line to the stream stream using the format specifier
format, which should be suitable for a type of size_t. In ISO C99 the type modifier z represents size_t, so"%zu\n" is a suitable format1 . The function returns GSL_EFAILED if there was a problem writing to the file.
int gsl_multiset_fscanf(FILE *stream, gsl_multiset *c)
This function reads formatted data from the stream stream into the multiset c. The multiset c must be preallocated with correct values of n and k since the function uses the size of c to determine how many numbers to
read. The function returns GSL_EFAILED if there was a problem reading from the file.

11.7 Examples
The example program below prints all multisets elements containing the values 0, 1, 2, 3 ordered by size. Multiset
elements of the same size are ordered lexicographically.
#include <stdio.h>#include <gsl/gsl_multiset.h>
int
main (void){
gsl_multiset * c;size_t i;
printf ("All multisets of {0,1,2,3} by size:\n") ;for (i = 0; i <= 4; i++){c = gsl_multiset_calloc (4, i);do{printf ("{");
gsl_multiset_fprintf (stdout, c, " %u");printf (" }\n");}while (gsl_multiset_next (c) == GSL_SUCCESS);
gsl_multiset_free (c);(continues on next page)
1 In versions of the GNU C library prior to the ISO C99 standard, the type modifier Z was used instead.

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119GNU Scientific Library, Release 2.7(continued from previous page)}
return 0;}
Here is the output from the program,All multisets of {0,1,2,3} by size:
{ }
{ 0 }
{ 1 }
{ 2 }
{ 3 }
{ 0 0 }
{ 0 1 }
{ 0 2 }
{ 0 3 }
{ 1 1 }
{ 1 2 }
{ 1 3 }
{ 2 2 }
{ 2 3 }
{ 3 3 }
{ 0 0 0 }{ 0 0 1 }{ 0 0 2 }{ 0 0 3 }{ 0 1 1 }{ 0 1 2 }{ 0 1 3 }{ 0 2 2 }{ 0 2 3 }{ 0 3 3 }{ 1 1 1 }{ 1 1 2 }{ 1 1 3 }{ 1 2 2 }{ 1 2 3 }{ 1 3 3 }{ 2 2 2 }{ 2 2 3 }{ 2 3 3 }{ 3 3 3 }{ 0 0 0 0 }{ 0 0 0 1 }{ 0 0 0 2 }{ 0 0 0 3 }{ 0 0 1 1 }{ 0 0 1 2 }{ 0 0 1 3 }{ 0 0 2 2 }{ 0 0 2 3 }(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page){ 0 0 3 3 }{ 0 1 1 1 }{ 0 1 1 2 }{ 0 1 1 3 }{ 0 1 2 2 }{ 0 1 2 3 }{ 0 1 3 3 }{ 0 2 2 2 }{ 0 2 2 3 }{ 0 2 3 3 }{ 0 3 3 3 }{ 1 1 1 1 }{ 1 1 1 2 }{ 1 1 1 3 }{ 1 1 2 2 }{ 1 1 2 3 }{ 1 1 3 3 }{ 1 2 2 2 }{ 1 2 2 3 }{ 1 2 3 3 }{ 1 3 3 3 }{ 2 2 2 2 }{ 2 2 2 3 }{ 2 2 3 3 }{ 2 3 3 3 }{ 3 3 3 3 }All 70 multisets are generated and sorted lexicographically.

11.7. Examples

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122Chapter 11. Multisets

CHAPTERTWELVESORTING

This chapter describes functions for sorting data, both directly and indirectly (using an index). All the functions use
the heapsort algorithm. Heapsort is an O(N log N ) algorithm which operates in-place and does not require any additional storage. It also provides consistent performance, the running time for its worst-case (ordered data) being notsignificantly longer than the average and best cases. Note that the heapsort algorithm does not preserve the relative
ordering of equal elements—it is an unstable sort. However the resulting order of equal elements will be consistentacross different platforms when using these functions.

12.1 Sorting objects
The following function provides a simple alternative to the standard library function qsort(). It is intended for systems
lacking qsort(), not as a replacement for it. The function qsort() should be used whenever possible, as it will befaster and can provide stable ordering of equal elements. Documentation for qsort() is available in the GNU C Library
Reference Manual.
The functions described in this section are defined in the header file gsl_heapsort.h.
void gsl_heapsort(void *array, size_t count, size_t size, gsl_comparison_fn_t compare)This function sorts the count elements of the array array, each of size size, into ascending order using thecomparison function compare. The type of the comparison function is defined bytype gsl_comparison_fn_t
int (*gsl_comparison_fn_t) (const void * a, const void * b)
A comparison function should return a negative integer if the first argument is less than the second argument, 0
if the two arguments are equal and a positive integer if the first argument is greater than the second argument.
For example, the following function can be used to sort doubles into ascending numerical order.
int
compare_doubles (const double * a, const double * b){
if (*a > *b)
return 1;
else if (*a < *b)return -1;
else
return 0;}
The appropriate function call to perform the sort is:123GNU Scientific Library, Release 2.7

gsl_heapsort (array, count, sizeof(double), compare_doubles);Note that unlike qsort() the heapsort algorithm cannot be made into a stable sort by pointer arithmetic. The trickof comparing pointers for equal elements in the comparison function does not work for the heapsort algorithm.
The heapsort algorithm performs an internal rearrangement of the data which destroys its initial ordering.
int gsl_heapsort_index(size_t *p, const void *array, size_t count, size_t size, gsl_comparison_fn_t compare)This function indirectly sorts the count elements of the array array, each of size size, into ascending orderusing the comparison function compare. The resulting permutation is stored in p, an array of length n. The
elements of p give the index of the array element which would have been stored in that position if the array had
been sorted in place. The first element of p gives the index of the least element in array, and the last elementof p gives the index of the greatest element in array. The array itself is not changed.

12.2 Sorting vectors
The following functions will sort the elements of an array or vector, either directly or indirectly. They are defined for all real and integer types using the normal suffix rules. For example, the float versions of the array functions are gsl_sort_float() and gsl_sort_float_index(). The corresponding vector functions are
gsl_sort_vector_float() and gsl_sort_vector_float_index(). The prototypes are available in the header
files gsl_sort_float.h gsl_sort_vector_float.h. The complete set of prototypes can be included using theheader files gsl_sort.h and gsl_sort_vector.h.
There are no functions for sorting complex arrays or vectors, since the ordering of complex numbers is not uniquely
defined. To sort a complex vector by magnitude compute a real vector containing the magnitudes of the complex
elements, and sort this vector indirectly. The resulting index gives the appropriate ordering of the original complex
vector.
void gsl_sort(double *data, const size_t stride, size_t n)This function sorts the n elements of the array data with stride stride into ascending numerical order.
void gsl_sort2(double *data1, const size_t stride1, double *data2, const size_t stride2, size_t n)This function sorts the n elements of the array data1 with stride stride1 into ascending numerical order, while
making the same rearrangement of the array data2 with stride stride2, also of size n.
void gsl_sort_vector(gsl_vector *v)
This function sorts the elements of the vector v into ascending numerical order.
void gsl_sort_vector2(gsl_vector *v1, gsl_vector *v2)
This function sorts the elements of the vector v1 into ascending numerical order, while making the same rearrangement of the vector v2.
void gsl_sort_index(size_t *p, const double *data, size_t stride, size_t n)This function indirectly sorts the n elements of the array data with stride stride into ascending order, storing
the resulting permutation in p. The array p must be allocated with a sufficient length to store the n elements ofthe permutation. The elements of p give the index of the array element which would have been stored in that
position if the array had been sorted in place. The array data is not changed.
int gsl_sort_vector_index(gsl_permutation *p, const gsl_vector *v)
This function indirectly sorts the elements of the vector v into ascending order, storing the resulting permutationin p. The elements of p give the index of the vector element which would have been stored in that position if
the vector had been sorted in place. The first element of p gives the index of the least element in v, and the lastelement of p gives the index of the greatest element in v. The vector v is not changed.

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12.3 Selecting the k smallest or largest elements
The functions described in this section select the k smallest or largest elements of a data set of size N . The routines
use an O(kN ) direct insertion algorithm which is suited to subsets that are small compared with the total size of thedataset. For example, the routines are useful for selecting the 10 largest values from one million data points, but notfor selecting the largest 100,000 values. If the subset is a significant part of the total dataset it may be faster to sort all
the elements of the dataset directly with an O(N log N ) algorithm and obtain the smallest or largest values that way.
int gsl_sort_smallest(double *dest, size_t k, const double *src, size_t stride, size_t n)This function copies the k smallest elements of the array src, of size n and stride stride, in ascending numericalorder into the array dest. The size k of the subset must be less than or equal to n. The data src is not modified
by this operation.
int gsl_sort_largest(double *dest, size_t k, const double *src, size_t stride, size_t n)This function copies the k largest elements of the array src, of size n and stride stride, in descending numericalorder into the array dest. k must be less than or equal to n. The data src is not modified by this operation.
int gsl_sort_vector_smallest(double *dest, size_t k, const gsl_vector *v)
int gsl_sort_vector_largest(double *dest, size_t k, const gsl_vector *v)
These functions copy the k smallest or largest elements of the vector v into the array dest. k must be less thanor equal to the length of the vector v.
The following functions find the indices of the k smallest or largest elements of a dataset.
int gsl_sort_smallest_index(size_t *p, size_t k, const double *src, size_t stride, size_t n)This function stores the indices of the k smallest elements of the array src, of size n and stride stride, in thearray p. The indices are chosen so that the corresponding data is in ascending numerical order. k must be lessthan or equal to n. The data src is not modified by this operation.
int gsl_sort_largest_index(size_t *p, size_t k, const double *src, size_t stride, size_t n)This function stores the indices of the k largest elements of the array src, of size n and stride stride, in thearray p. The indices are chosen so that the corresponding data is in descending numerical order. k must be lessthan or equal to n. The data src is not modified by this operation.
int gsl_sort_vector_smallest_index(size_t *p, size_t k, const gsl_vector *v)
int gsl_sort_vector_largest_index(size_t *p, size_t k, const gsl_vector *v)
These functions store the indices of the k smallest or largest elements of the vector v in the array p. k must be
less than or equal to the length of the vector v.

12.4 Computing the rank
The rank of an element is its order in the sorted data. The rank is the inverse of the index permutation, p. It can becomputed using the following algorithm:for (i = 0; i < p->size; i++){size_t pi = p->data[i];rank->data[pi] = i;}This can be computed directly from the function gsl_permutation_inverse(rank,p).
The following function will print the rank of each element of the vector v:

12.3. Selecting the k smallest or largest elements

125GNU Scientific Library, Release 2.7

void
print_rank (gsl_vector * v){
size_t i;size_t n = v->size;
gsl_permutation * perm = gsl_permutation_alloc(n);gsl_permutation * rank = gsl_permutation_alloc(n);gsl_sort_vector_index (perm, v);
gsl_permutation_inverse (rank, perm);for (i = 0; i < n; i++){double vi = gsl_vector_get(v, i);printf ("element = %d, value = %g, rank = %d\n",i, vi, rank->data[i]);}gsl_permutation_free (perm);
gsl_permutation_free (rank);}

12.5 Examples
The following example shows how to use the permutation p to print the elements of the vector v in ascending order:
gsl_sort_vector_index (p, v);
for (i = 0; i < v->size; i++){double vpi = gsl_vector_get (v, p->data[i]);printf ("order = %d, value = %g\n", i, vpi);}
The next example uses the function gsl_sort_smallest() to select the 5 smallest numbers from 100000 uniform
random variates stored in an array,#include <gsl/gsl_rng.h>#include <gsl/gsl_sort_double.h>intmain (void){
const gsl_rng_type * T;
gsl_rng * r;
size_t i, k = 5, N = 100000;
double * x = malloc (N * sizeof(double));double * small = malloc (k * sizeof(double));(continues on next page)

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Chapter 12. SortingGNU Scientific Library, Release 2.7(continued from previous page)gsl_rng_env_setup();
T = gsl_rng_default;r = gsl_rng_alloc (T);for (i = 0; i < N; i++){x[i] = gsl_rng_uniform(r);}gsl_sort_smallest (small, k, x, 1, N);
printf ("%zu smallest values from %zu\n", k, N);
for (i = 0; i < k; i++){printf ("%zu: %.18f \n", i, small[i]);}free (x);free (small);gsl_rng_free (r);return 0;}
The output lists the 5 smallest values, in ascending order,
5 smallest values from 100000
0: 0.000003489200025797
1: 0.000008199829608202
2: 0.000008953968062997
3: 0.000010712770745158
4: 0.000033531803637743

12.6 References and Further Reading
The subject of sorting is covered extensively in the following,• Donald E. Knuth, The Art of Computer Programming: Sorting and Searching (Vol 3, 3rd Ed, 1997), AddisonWesley, ISBN 0201896850.
The Heapsort algorithm is described in the following book,• Robert Sedgewick, Algorithms in C, Addison-Wesley, ISBN 0201514257.

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128Chapter 12. SortingCHAPTERTHIRTEEN

BLAS SUPPORTThe Basic Linear Algebra Subprograms (BLAS) define a set of fundamental operations on vectors and matrices which
can be used to create optimized higher-level linear algebra functionality.
The library provides a low-level layer which corresponds directly to the C-language BLAS standard, referred to here as“CBLAS”, and a higher-level interface for operations on GSL vectors and matrices. Users who are interested in simple
operations on GSL vector and matrix objects should use the high-level layer described in this chapter. The functions
are declared in the file gsl_blas.h and should satisfy the needs of most users.
Note that GSL matrices are implemented using dense-storage so the interface only includes the corresponding densestorage BLAS functions. The full BLAS functionality for band-format and packed-format matrices is available throughthe low-level CBLAS interface. Similarly, GSL vectors are restricted to positive strides, whereas the low-level CBLAS
interface supports negative strides as specified in the BLAS standard1 .
The interface for the gsl_cblas layer is specified in the file gsl_cblas.h. This interface corresponds to the BLAS
Technical Forum’s standard for the C interface to legacy BLAS implementations. Users who have access to otherconforming CBLAS implementations can use these in place of the version provided by the library. Note that users
who have only a Fortran BLAS library can use a CBLAS conformant wrapper to convert it into a CBLAS library.
A reference CBLAS wrapper for legacy Fortran implementations exists as part of the CBLAS standard and can beobtained from Netlib. The complete set of CBLAS functions is listed in an appendix.
There are three levels of BLAS operations,Level 1
Level 2
Level 3Vector operations, e.g. y = αx + y
Matrix-vector operations, e.g. y = αAx + βy
Matrix-matrix operations, e.g. C = αAB + C

Each routine has a name which specifies the operation, the type of matrices involved and their precisions. Some of themost common operations and their names are given below,
DOT
AXPY
MV
SV
MM
SM

scalar product, xT yvector sum, αx + ymatrix-vector product, Ax
matrix-vector solve, inv(A)xmatrix-matrix product, AB
matrix-matrix solve, inv(A)B

The types of matrices are,
1 In the low-level CBLAS interface, a negative stride accesses the vector elements in reverse order, i.e. the i-th element is given by (N −i)∗|incx|
for incx < 0.

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GE
GB
SY
SB
SP
HE
HB
HP
TR
TB
TP

generalgeneral bandsymmetric
symmetric bandsymmetric packed
hermitianhermitian band
hermitian packed
triangulartriangular band
triangular packed

S
D
C
Z

single real
double real
single complex
double complex

Each operation is defined for four precisions,Thus, for example, the name SGEMM stands for “single-precision general matrix-matrix multiply” and ZGEMM stands
for “double-precision complex matrix-matrix multiply”.
Note that the vector and matrix arguments to BLAS functions must not be aliased, as the results are undefined when
the underlying arrays overlap (Aliasing of arrays).

13.1 GSL BLAS Interface
GSL provides dense vector and matrix objects, based on the relevant built-in types. The library provides an interface tothe BLAS operations which apply to these objects. The interface to this functionality is given in the file gsl_blas.h.

13.1.1 Level 1
int gsl_blas_sdsdot(float alpha, const gsl_vector_float *x, const gsl_vector_float *y, float *result)This function computes the sum α + xT y for the vectors x and y, returning the result in result.
int gsl_blas_sdot(const gsl_vector_float *x, const gsl_vector_float *y, float *result)int gsl_blas_dsdot(const gsl_vector_float *x, const gsl_vector_float *y, double *result)int gsl_blas_ddot(const gsl_vector *x, const gsl_vector *y, double *result)
These functions compute the scalar product xT y for the vectors x and y, returning the result in result.
int gsl_blas_cdotu(const gsl_vector_complex_float *x, const gsl_vector_complex_float *y, gsl_complex_float*dotu)
int gsl_blas_zdotu(const gsl_vector_complex *x, const gsl_vector_complex *y, gsl_complex *dotu)
These functions compute the complex scalar product xT y for the vectors x and y, returning the result in dotu
int gsl_blas_cdotc(const gsl_vector_complex_float *x, const gsl_vector_complex_float *y, gsl_complex_float*dotc)

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int gsl_blas_zdotc(const gsl_vector_complex *x, const gsl_vector_complex *y, gsl_complex *dotc)
These functions compute the complex conjugate scalar product xH y for the vectors x and y, returning the resultin dotc
float gsl_blas_snrm2(const gsl_vector_float *x)
double gsl_blas_dnrm2(const gsl_vector *x)pP
x2i of the vector x.
These functions compute the Euclidean norm ||x||2 =float gsl_blas_scnrm2(const gsl_vector_complex_float *x)
double gsl_blas_dznrm2(const gsl_vector_complex *x)
These functions compute the Euclidean norm of the complex vector x,qX||x||2 =(<(xi )2 + =(xi )2 ).
float gsl_blas_sasum(const gsl_vector_float *x)
double gsl_blas_dasum(const gsl_vector *x) P
These functions compute the absolute sum |xi | of the elements of the vector x.
float gsl_blas_scasum(const gsl_vector_complex_float *x)
double gsl_blas_dzasum(const gsl_vector_complex *x)
These functions compute the sum of the magnitudes of the real and imaginary parts of the complex vector x,
P(|<(xi )| + |=(xi )|).
CBLAS_INDEX_t gsl_blas_isamax(const gsl_vector_float *x)
CBLAS_INDEX_t gsl_blas_idamax(const gsl_vector *x)
CBLAS_INDEX_t gsl_blas_icamax(const gsl_vector_complex_float *x)
CBLAS_INDEX_t gsl_blas_izamax(const gsl_vector_complex *x)
These functions return the index of the largest element of the vector x. The largest element is determined by its
absolute magnitude for real vectors and by the sum of the magnitudes of the real and imaginary parts |<(xi )| +|=(xi )| for complex vectors. If the largest value occurs several times then the index of the first occurrence isreturned.
int gsl_blas_sswap(gsl_vector_float *x, gsl_vector_float *y)
int gsl_blas_dswap(gsl_vector *x, gsl_vector *y)
int gsl_blas_cswap(gsl_vector_complex_float *x, gsl_vector_complex_float *y)
int gsl_blas_zswap(gsl_vector_complex *x, gsl_vector_complex *y)These functions exchange the elements of the vectors x and y.
int gsl_blas_scopy(const gsl_vector_float *x, gsl_vector_float *y)
int gsl_blas_dcopy(const gsl_vector *x, gsl_vector *y)
int gsl_blas_ccopy(const gsl_vector_complex_float *x, gsl_vector_complex_float *y)

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int gsl_blas_zcopy(const gsl_vector_complex *x, gsl_vector_complex *y)These functions copy the elements of the vector x into the vector y.
int gsl_blas_saxpy(float alpha, const gsl_vector_float *x, gsl_vector_float *y)
int gsl_blas_daxpy(double alpha, const gsl_vector *x, gsl_vector *y)
int gsl_blas_caxpy(const gsl_complex_float alpha, const gsl_vector_complex_float *x, gsl_vector_complex_float*y)
int gsl_blas_zaxpy(const gsl_complex alpha, const gsl_vector_complex *x, gsl_vector_complex *y)These functions compute the sum y = αx + y for the vectors x and y.
void gsl_blas_sscal(float alpha, gsl_vector_float *x)void gsl_blas_dscal(double alpha, gsl_vector *x)
void gsl_blas_cscal(const gsl_complex_float alpha, gsl_vector_complex_float *x)void gsl_blas_zscal(const gsl_complex alpha, gsl_vector_complex *x)void gsl_blas_csscal(float alpha, gsl_vector_complex_float *x)void gsl_blas_zdscal(double alpha, gsl_vector_complex *x)
These functions rescale the vector x by the multiplicative factor alpha.
int gsl_blas_srotg(float a[], float b[], float c[], float s[])int gsl_blas_drotg(double a[], double b[], double c[], double s[])
These functions compute a Givens rotation (c, s) which zeroes the vector (a, b),    0
c s
a
r
=
−s c
b
0
The variables a and b are overwritten by the routine.
int gsl_blas_srot(gsl_vector_float *x, gsl_vector_float *y, float c, float s)
int gsl_blas_drot(gsl_vector *x, gsl_vector *y, const double c, const double s)
These functions apply a Givens rotation (x0 , y 0 ) = (cx + sy, −sx + cy) to the vectors x, y.
int gsl_blas_srotmg(float d1[], float d2[], float b1[], float b2, float P[])
int gsl_blas_drotmg(double d1[], double d2[], double b1[], double b2, double P[])
These functions compute a modified Givens transformation. The modified Givens transformation is defined in
the original Level-1 BLAS specification, given in the references.
int gsl_blas_srotm(gsl_vector_float *x, gsl_vector_float *y, const float P[])
int gsl_blas_drotm(gsl_vector *x, gsl_vector *y, const double P[])
These functions apply a modified Givens transformation.

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13.1.2 Level 2
int gsl_blas_sgemv(CBLAS_TRANSPOSE_t TransA, float alpha, const gsl_matrix_float *A, const
gsl_vector_float *x, float beta, gsl_vector_float *y)
int gsl_blas_dgemv(CBLAS_TRANSPOSE_t TransA, double alpha, const gsl_matrix *A, const gsl_vector *x,
double beta, gsl_vector *y)
int gsl_blas_cgemv(CBLAS_TRANSPOSE_t TransA, const gsl_complex_float alpha, const
gsl_matrix_complex_float *A, const gsl_vector_complex_float *x, const gsl_complex_float
beta, gsl_vector_complex_float *y)
int gsl_blas_zgemv(CBLAS_TRANSPOSE_t TransA, const gsl_complex alpha, const gsl_matrix_complex *A,
const gsl_vector_complex *x, const gsl_complex beta, gsl_vector_complex *y)These functions compute the matrix-vector product and sum y = αop(A)x + βy, where op(A) = A, AT , AH
for TransA = CblasNoTrans, CblasTrans, CblasConjTrans.
int gsl_blas_strmv(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
gsl_matrix_float *A, gsl_vector_float *x)
int gsl_blas_dtrmv(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
gsl_matrix *A, gsl_vector *x)
int gsl_blas_ctrmv(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
gsl_matrix_complex_float *A, gsl_vector_complex_float *x)
int gsl_blas_ztrmv(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
gsl_matrix_complex *A, gsl_vector_complex *x)
These functions compute the matrix-vector product x = op(A)x for the triangular matrix A, where op(A) = A,
AT , AH for TransA = CblasNoTrans, CblasTrans, CblasConjTrans. When Uplo is CblasUpper then theupper triangle of A is used, and when Uplo is CblasLower then the lower triangle of A is used. If Diag is
CblasNonUnit then the diagonal of the matrix is used, but if Diag is CblasUnit then the diagonal elements ofthe matrix A are taken as unity and are not referenced.
int gsl_blas_strsv(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
gsl_matrix_float *A, gsl_vector_float *x)
int gsl_blas_dtrsv(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
gsl_matrix *A, gsl_vector *x)
int gsl_blas_ctrsv(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
gsl_matrix_complex_float *A, gsl_vector_complex_float *x)
int gsl_blas_ztrsv(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
gsl_matrix_complex *A, gsl_vector_complex *x)
These functions compute inv(op(A))x for x, where op(A) = A, AT , AH for TransA = CblasNoTrans,
CblasTrans, CblasConjTrans. When Uplo is CblasUpper then the upper triangle of A is used, and when
Uplo is CblasLower then the lower triangle of A is used. If Diag is CblasNonUnit then the diagonal of thematrix is used, but if Diag is CblasUnit then the diagonal elements of the matrix A are taken as unity and are
not referenced.
int gsl_blas_ssymv(CBLAS_UPLO_t Uplo, float alpha, const gsl_matrix_float *A, const gsl_vector_float *x, float
beta, gsl_vector_float *y)

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int gsl_blas_dsymv(CBLAS_UPLO_t Uplo, double alpha, const gsl_matrix *A, const gsl_vector *x, double beta,
gsl_vector *y)
These functions compute the matrix-vector product and sum y = αAx + βy for the symmetric matrix A. Since
the matrix A is symmetric only its upper half or lower half need to be stored. When Uplo is CblasUpper then theupper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonalof A are used.
int gsl_blas_chemv(CBLAS_UPLO_t Uplo, const gsl_complex_float alpha, const gsl_matrix_complex_float *A,
const gsl_vector_complex_float *x, const gsl_complex_float beta, gsl_vector_complex_float*y)
int gsl_blas_zhemv(CBLAS_UPLO_t Uplo, const gsl_complex alpha, const gsl_matrix_complex *A, const
gsl_vector_complex *x, const gsl_complex beta, gsl_vector_complex *y)These functions compute the matrix-vector product and sum y = αAx + βy for the hermitian matrix A. Since
the matrix A is hermitian only its upper half or lower half need to be stored. When Uplo is CblasUpper then theupper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonal of
A are used. The imaginary elements of the diagonal are automatically assumed to be zero and are not referenced.
int gsl_blas_sger(float alpha, const gsl_vector_float *x, const gsl_vector_float *y, gsl_matrix_float *A)
int gsl_blas_dger(double alpha, const gsl_vector *x, const gsl_vector *y, gsl_matrix *A)
int gsl_blas_cgeru(const gsl_complex_float alpha, const gsl_vector_complex_float *x, const
gsl_vector_complex_float *y, gsl_matrix_complex_float *A)
int gsl_blas_zgeru(const gsl_complex alpha, const gsl_vector_complex *x, const gsl_vector_complex *y,
gsl_matrix_complex *A)
These functions compute the rank-1 update A = αxy T + A of the matrix A.
int gsl_blas_cgerc(const gsl_complex_float alpha, const gsl_vector_complex_float *x, const
gsl_vector_complex_float *y, gsl_matrix_complex_float *A)
int gsl_blas_zgerc(const gsl_complex alpha, const gsl_vector_complex *x, const gsl_vector_complex *y,
gsl_matrix_complex *A)
These functions compute the conjugate rank-1 update A = αxy H + A of the matrix A.
int gsl_blas_ssyr(CBLAS_UPLO_t Uplo, float alpha, const gsl_vector_float *x, gsl_matrix_float *A)
int gsl_blas_dsyr(CBLAS_UPLO_t Uplo, double alpha, const gsl_vector *x, gsl_matrix *A)
These functions compute the symmetric rank-1 update A = αxxT + A of the symmetric matrix A. Since thematrix A is symmetric only its upper half or lower half need to be stored. When Uplo is CblasUpper then theupper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonalof A are used.
int gsl_blas_cher(CBLAS_UPLO_t Uplo, float alpha, const gsl_vector_complex_float *x,
gsl_matrix_complex_float *A)
int gsl_blas_zher(CBLAS_UPLO_t Uplo, double alpha, const gsl_vector_complex *x, gsl_matrix_complex *A)
These functions compute the hermitian rank-1 update A = αxxH + A of the hermitian matrix A. Since thematrix A is hermitian only its upper half or lower half need to be stored. When Uplo is CblasUpper then theupper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonalof A are used. The imaginary elements of the diagonal are automatically set to zero.

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int gsl_blas_ssyr2(CBLAS_UPLO_t Uplo, float alpha, const gsl_vector_float *x, const gsl_vector_float *y,
gsl_matrix_float *A)
int gsl_blas_dsyr2(CBLAS_UPLO_t Uplo, double alpha, const gsl_vector *x, const gsl_vector *y, gsl_matrix*A)
These functions compute the symmetric rank-2 update A = αxy T + αyxT + A of the symmetric matrix A.
Since the matrix A is symmetric only its upper half or lower half need to be stored. When Uplo is CblasUpper
then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle anddiagonal of A are used.
int gsl_blas_cher2(CBLAS_UPLO_t Uplo, const gsl_complex_float alpha, const gsl_vector_complex_float *x,
const gsl_vector_complex_float *y, gsl_matrix_complex_float *A)
int gsl_blas_zher2(CBLAS_UPLO_t Uplo, const gsl_complex alpha, const gsl_vector_complex *x, const
gsl_vector_complex *y, gsl_matrix_complex *A)
These functions compute the hermitian rank-2 update A = αxy H + α∗ yxH + A of the hermitian matrix A. Since
the matrix A is hermitian only its upper half or lower half need to be stored. When Uplo is CblasUpper then theupper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonalof A are used. The imaginary elements of the diagonal are automatically set to zero.

13.1.3 Level 3
int gsl_blas_sgemm(CBLAS_TRANSPOSE_t TransA, CBLAS_TRANSPOSE_t TransB, float alpha, const
gsl_matrix_float *A, const gsl_matrix_float *B, float beta, gsl_matrix_float *C)
int gsl_blas_dgemm(CBLAS_TRANSPOSE_t TransA, CBLAS_TRANSPOSE_t TransB, double alpha, const
gsl_matrix *A, const gsl_matrix *B, double beta, gsl_matrix *C)
int gsl_blas_cgemm(CBLAS_TRANSPOSE_t TransA, CBLAS_TRANSPOSE_t TransB, const gsl_complex_floatalpha, const gsl_matrix_complex_float *A, const gsl_matrix_complex_float *B, const
gsl_complex_float beta, gsl_matrix_complex_float *C)
int gsl_blas_zgemm(CBLAS_TRANSPOSE_t TransA, CBLAS_TRANSPOSE_t TransB, const gsl_complex
alpha, const gsl_matrix_complex *A, const gsl_matrix_complex *B, const gsl_complex beta,
gsl_matrix_complex *C)
These functions compute the matrix-matrix product and sum C = αop(A)op(B) + βC where op(A) = A, AT ,AH for TransA = CblasNoTrans, CblasTrans, CblasConjTrans and similarly for the parameter TransB.
int gsl_blas_ssymm(CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, float alpha, const gsl_matrix_float *A, const
gsl_matrix_float *B, float beta, gsl_matrix_float *C)
int gsl_blas_dsymm(CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, double alpha, const gsl_matrix *A, const
gsl_matrix *B, double beta, gsl_matrix *C)
int gsl_blas_csymm(CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, const gsl_complex_float alpha, const
gsl_matrix_complex_float *A, const gsl_matrix_complex_float *B, const gsl_complex_float
beta, gsl_matrix_complex_float *C)
int gsl_blas_zsymm(CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, const gsl_complex alpha, const
gsl_matrix_complex *A, const gsl_matrix_complex *B, const gsl_complex beta,
gsl_matrix_complex *C)
These functions compute the matrix-matrix product and sum C = αAB + βC for Side is CblasLeft and13.1. GSL BLAS Interface

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C = αBA + βC for Side is CblasRight, where the matrix A is symmetric. When Uplo is CblasUpper
then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle anddiagonal of A are used.
int gsl_blas_chemm(CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, const gsl_complex_float alpha, const
gsl_matrix_complex_float *A, const gsl_matrix_complex_float *B, const gsl_complex_float
beta, gsl_matrix_complex_float *C)
int gsl_blas_zhemm(CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, const gsl_complex alpha, const
gsl_matrix_complex *A, const gsl_matrix_complex *B, const gsl_complex beta,
gsl_matrix_complex *C)
These functions compute the matrix-matrix product and sum C = αAB + βC for Side is CblasLeft andC = αBA + βC for Side is CblasRight, where the matrix A is hermitian. When Uplo is CblasUpper thenthe upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonalof A are used. The imaginary elements of the diagonal are automatically set to zero.
int gsl_blas_strmm(CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA,
CBLAS_DIAG_t Diag, float alpha, const gsl_matrix_float *A, gsl_matrix_float *B)
int gsl_blas_dtrmm(CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA,
CBLAS_DIAG_t Diag, double alpha, const gsl_matrix *A, gsl_matrix *B)
int gsl_blas_ctrmm(CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA,
CBLAS_DIAG_t Diag, const gsl_complex_float alpha, const gsl_matrix_complex_float *A,
gsl_matrix_complex_float *B)
int gsl_blas_ztrmm(CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA,
CBLAS_DIAG_t Diag, const gsl_complex alpha, const gsl_matrix_complex *A,
gsl_matrix_complex *B)
These functions compute the matrix-matrix product B = αop(A)B for Side is CblasLeft and B = αBop(A)for Side is CblasRight. The matrix A is triangular and op(A) = A, AT , AH for TransA = CblasNoTrans,
CblasTrans, CblasConjTrans. When Uplo is CblasUpper then the upper triangle of A is used, and when
Uplo is CblasLower then the lower triangle of A is used. If Diag is CblasNonUnit then the diagonal of A
is used, but if Diag is CblasUnit then the diagonal elements of the matrix A are taken as unity and are notreferenced.
int gsl_blas_strsm(CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA,
CBLAS_DIAG_t Diag, float alpha, const gsl_matrix_float *A, gsl_matrix_float *B)
int gsl_blas_dtrsm(CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA,
CBLAS_DIAG_t Diag, double alpha, const gsl_matrix *A, gsl_matrix *B)
int gsl_blas_ctrsm(CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA,
CBLAS_DIAG_t Diag, const gsl_complex_float alpha, const gsl_matrix_complex_float *A,
gsl_matrix_complex_float *B)
int gsl_blas_ztrsm(CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA,
CBLAS_DIAG_t Diag, const gsl_complex alpha, const gsl_matrix_complex *A,
gsl_matrix_complex *B)
These functions compute the inverse-matrix matrix product B = αop(inv(A))B for Side is CblasLeft andB = αBop(inv(A)) for Side is CblasRight. The matrix A is triangular and op(A) = A, AT , AH for TransA= CblasNoTrans, CblasTrans, CblasConjTrans. When Uplo is CblasUpper then the upper triangle of A isused, and when Uplo is CblasLower then the lower triangle of A is used. If Diag is CblasNonUnit then the136

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diagonal of A is used, but if Diag is CblasUnit then the diagonal elements of the matrix A are taken as unity
and are not referenced.
int gsl_blas_ssyrk(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, float alpha, const gsl_matrix_float*A, float beta, gsl_matrix_float *C)
int gsl_blas_dsyrk(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, double alpha, const gsl_matrix *A,
double beta, gsl_matrix *C)
int gsl_blas_csyrk(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, const gsl_complex_float alpha, const
gsl_matrix_complex_float *A, const gsl_complex_float beta, gsl_matrix_complex_float *C)
int gsl_blas_zsyrk(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, const gsl_complex alpha, const
gsl_matrix_complex *A, const gsl_complex beta, gsl_matrix_complex *C)
These functions compute a rank-k update of the symmetric matrix C, C = αAAT + βC when Trans is
CblasNoTrans and C = αAT A + βC when Trans is CblasTrans. Since the matrix C is symmetric only
its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonalof C are used, and when Uplo is CblasLower then the lower triangle and diagonal of C are used.
int gsl_blas_cherk(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, float alpha, const
gsl_matrix_complex_float *A, float beta, gsl_matrix_complex_float *C)
int gsl_blas_zherk(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, double alpha, const
gsl_matrix_complex *A, double beta, gsl_matrix_complex *C)
These functions compute a rank-k update of the hermitian matrix C, C = αAAH + βC when Trans is
CblasNoTrans and C = αAH A + βC when Trans is CblasConjTrans. Since the matrix C is hermitian
only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of C are used, and when Uplo is CblasLower then the lower triangle and diagonal of C are used. The
imaginary elements of the diagonal are automatically set to zero.
int gsl_blas_ssyr2k(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, float alpha, const gsl_matrix_float*A, const gsl_matrix_float *B, float beta, gsl_matrix_float *C)
int gsl_blas_dsyr2k(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, double alpha, const gsl_matrix *A,
const gsl_matrix *B, double beta, gsl_matrix *C)
int gsl_blas_csyr2k(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, const gsl_complex_float alpha,const gsl_matrix_complex_float *A, const gsl_matrix_complex_float *B, const
gsl_complex_float beta, gsl_matrix_complex_float *C)
int gsl_blas_zsyr2k(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, const gsl_complex alpha, const
gsl_matrix_complex *A, const gsl_matrix_complex *B, const gsl_complex beta,
gsl_matrix_complex *C)
These functions compute a rank-2k update of the symmetric matrix C, C = αAB T + αBAT + βC whenTrans is CblasNoTrans and C = αAT B + αB T A + βC when Trans is CblasTrans. Since the matrix C issymmetric only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle
and diagonal of C are used, and when Uplo is CblasLower then the lower triangle and diagonal of C are used.
int gsl_blas_cher2k(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, const gsl_complex_float alpha,const gsl_matrix_complex_float *A, const gsl_matrix_complex_float *B, float beta,
gsl_matrix_complex_float *C)

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int gsl_blas_zher2k(CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, const gsl_complex alpha, const
gsl_matrix_complex *A, const gsl_matrix_complex *B, double beta, gsl_matrix_complex*C)
These functions compute a rank-2k update of the hermitian matrix C, C = αAB H + α∗ BAH + βC when Transis CblasNoTrans and C = αAH B + α∗ B H A + βC when Trans is CblasConjTrans. Since the matrix C is
hermitian only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle
and diagonal of C are used, and when Uplo is CblasLower then the lower triangle and diagonal of C are used.
The imaginary elements of the diagonal are automatically set to zero.

13.2 Examples
The following program computes the product of two matrices using the Level-3 BLAS function DGEMM, 1011 1012

0.11 0.12 0.13 
367.76 368.12
1021 1022 =0.21 0.22 0.23
674.06 674.72
1031 1031
The matrices are stored in row major order, according to the C convention for arrays.
#include <stdio.h>#include <gsl/gsl_blas.h>intmain (void){
double a[] = { 0.11, 0.12, 0.13,
0.21, 0.22, 0.23 };double b[] = { 1011, 1012,
1021, 1022,
1031, 1032 };double c[] = { 0.00, 0.00,
0.00, 0.00 };gsl_matrix_view A = gsl_matrix_view_array(a, 2, 3);
gsl_matrix_view B = gsl_matrix_view_array(b, 3, 2);
gsl_matrix_view C = gsl_matrix_view_array(c, 2, 2);/* Compute C = A B */gsl_blas_dgemm (CblasNoTrans, CblasNoTrans,
1.0, &A.matrix, &B.matrix,
0.0, &C.matrix);
printf ("[ %g, %g\n", c[0], c[1]);printf (" %g, %g ]\n", c[2], c[3]);return 0;}
Here is the output from the program,

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674.06, 674.72 ]13.3 References and Further Reading
Information on the BLAS standards, including both the legacy and updated interface standards, is available online fromthe BLAS Homepage and BLAS Technical Forum web-site.
• BLAS Homepage, http://www.netlib.org/blas/• BLAS Technical Forum, http://www.netlib.org/blas/blast-forum/
The following papers contain the specifications for Level 1, Level 2 and Level 3 BLAS.
• C. Lawson, R. Hanson, D. Kincaid, F. Krogh, “Basic Linear Algebra Subprograms for Fortran Usage”, ACM
Transactions on Mathematical Software, Vol.: 5 (1979), Pages 308–325.
• J.J. Dongarra, J. DuCroz, S. Hammarling, R. Hanson, “An Extended Set of Fortran Basic Linear Algebra Subprograms”, ACM Transactions on Mathematical Software, Vol.: 14, No.: 1 (1988), Pages 1–32.
• J.J. Dongarra, I. Duff, J. DuCroz, S. Hammarling, “A Set of Level 3 Basic Linear Algebra Subprograms”, ACM
Transactions on Mathematical Software, Vol.: 16 (1990), Pages 1–28.
Postscript versions of the latter two papers are available from http://www.netlib.org/blas/. A CBLAS wrapper for
Fortran BLAS libraries is available from the same location.

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140Chapter 13. BLAS Support

CHAPTERFOURTEEN

LINEAR ALGEBRAThis chapter describes functions for solving linear systems. The library provides linear algebra operations which operate
directly on the gsl_vector and gsl_matrix objects. These routines use the standard algorithms from Golub & Van
Loan’s Matrix Computations with Level-1 and Level-2 BLAS calls for efficiency.
The functions described in this chapter are declared in the header file gsl_linalg.h.

14.1 LU Decomposition
A general M -by-N matrix A has an LU decomposition
P A = LU
where P is an M -by-M permutation matrix, L is M -by-min(M, N ) and U is min(M, N )-by-N . For square matrices,
L is a lower unit triangular matrix and U is upper triangular. For M > N , L is a unit lower trapezoidal matrix of size
M -by-N . For M < N , U is upper trapezoidal of size M -by-N . For square matrices this decomposition can be used
to convert the linear system Ax = b into a pair of triangular systems (Ly = P b, U x = y), which can be solved byforward and back-substitution. Note that the LU decomposition is valid for singular matrices.
int gsl_linalg_LU_decomp(gsl_matrix *A, gsl_permutation *p, int *signum)
int gsl_linalg_complex_LU_decomp(gsl_matrix_complex *A, gsl_permutation *p, int *signum)
These functions factorize the matrix A into the LU decomposition P A = LU . On output the diagonal and upper
triangular (or trapezoidal) part of the input matrix A contain the matrix U . The lower triangular (or trapezoidal)
part of the input matrix (excluding the diagonal) contains L. The diagonal elements of L are unity, and are notstored.
The permutation matrix P is encoded in the permutation p on output. The j-th column of the matrix P is givenby the k-th column of the identity matrix, where k = pj the j-th element of the permutation vector. The signof the permutation is given by signum. It has the value (−1)n , where n is the number of interchanges in the
permutation.
The algorithm used in the decomposition is Gaussian Elimination with partial pivoting (Golub & Van Loan,
Matrix Computations, Algorithm 3.4.1), combined with a recursive algorithm based on Level 3 BLAS (Peiseand Bientinesi, 2016).
int gsl_linalg_LU_solve(const gsl_matrix *LU, const gsl_permutation *p, const gsl_vector *b, gsl_vector *x)
int gsl_linalg_complex_LU_solve(const gsl_matrix_complex *LU, const gsl_permutation *p, const
gsl_vector_complex *b, gsl_vector_complex *x)
These functions solve the square system Ax = b using the LU decomposition of A into (LU, p) given by
gsl_linalg_LU_decomp() or gsl_linalg_complex_LU_decomp() as input.

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int gsl_linalg_LU_svx(const gsl_matrix *LU, const gsl_permutation *p, gsl_vector *x)
int gsl_linalg_complex_LU_svx(const gsl_matrix_complex *LU, const gsl_permutation *p, gsl_vector_complex*x)
These functions solve the square system Ax = b in-place using the precomputed LU decomposition of A into
(LU, p). On input x should contain the right-hand side b, which is replaced by the solution on output.
int gsl_linalg_LU_refine(const gsl_matrix *A, const gsl_matrix *LU, const gsl_permutation *p, const
gsl_vector *b, gsl_vector *x, gsl_vector *work)
int gsl_linalg_complex_LU_refine(const gsl_matrix_complex *A, const gsl_matrix_complex *LU, const
gsl_permutation *p, const gsl_vector_complex *b, gsl_vector_complex *x,
gsl_vector_complex *work)
These functions apply an iterative improvement to x, the solution of Ax = b, from the precomputed LU decomposition of A into (LU, p). Additional workspace of length N is required in work.
int gsl_linalg_LU_invert(const gsl_matrix *LU, const gsl_permutation *p, gsl_matrix *inverse)
int gsl_linalg_complex_LU_invert(const gsl_matrix_complex *LU, const gsl_permutation *p,
gsl_matrix_complex *inverse)
These functions compute the inverse of a matrix A from its LU decomposition (LU, p), storing the result in thematrix inverse. The inverse is computed by computing the inverses U −1 , L−1 and finally forming the productA−1 = U −1 L−1 P . Each step is based on Level 3 BLAS calls.
It is preferable to avoid direct use of the inverse whenever possible, as the linear solver functions can obtainthe same result more efficiently and reliably (consult any introductory textbook on numerical linear algebra for
details).
int gsl_linalg_LU_invx(gsl_matrix *LU, const gsl_permutation *p)
int gsl_linalg_complex_LU_invx(gsl_matrix_complex *LU, const gsl_permutation *p)
These functions compute the inverse of a matrix A from its LU decomposition (LU, p), storing the result in-placein the matrix LU. The inverse is computed by computing the inverses U −1 , L−1 and finally forming the productA−1 = U −1 L−1 P . Each step is based on Level 3 BLAS calls.
It is preferable to avoid direct use of the inverse whenever possible, as the linear solver functions can obtainthe same result more efficiently and reliably (consult any introductory textbook on numerical linear algebra for
details).
double gsl_linalg_LU_det(gsl_matrix *LU, int signum)
gsl_complex gsl_linalg_complex_LU_det(gsl_matrix_complex *LU, int signum)
These functions compute the determinant of a matrix A from its LU decomposition, LU. The determinant iscomputed as the product of the diagonal elements of U and the sign of the row permutation signum.
double gsl_linalg_LU_lndet(gsl_matrix *LU)
double gsl_linalg_complex_LU_lndet(gsl_matrix_complex *LU)
These functions compute the logarithm of the absolute value of the determinant of a matrix A, ln | det(A)|,from its LU decomposition, LU. This function may be useful if the direct computation of the determinant would
overflow or underflow.
int gsl_linalg_LU_sgndet(gsl_matrix *LU, int signum)
gsl_complex gsl_linalg_complex_LU_sgndet(gsl_matrix_complex *LU, int signum)
These functions compute the sign or phase factor of the determinant of a matrix A, det(A)/| det(A)|, from its
LU decomposition, LU.
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14.2 QR Decomposition
A general rectangular M -by-N matrix A has a QR decomposition into the product of a unitary M -by-M square matrix
Q (where Q† Q = I) and an M -by-N right-triangular matrix R,
A = QR
This decomposition can be used to convert the square linear system Ax = b into the triangular system Rx = Q† b,which can be solved by back-substitution. Another use of the QR decomposition is to compute an orthonormal basis
for a set of vectors. The first N columns of Q form an orthonormal basis for the range of A, ran(A), when A has fullcolumn rank.
When M > N , the bottom M − N rows of R are zero, and so A can be naturally partioned as  R1
A = Q1 Q2= Q1 R10
where R1 is N -by-N upper triangular, Q1 is M -by-N , and Q2 is M -by-(M − N ). Q1 R1 is sometimes called the thinor reduced QR decomposition. The solution of the least squares problem minx ||b − Ax||2 when A has full rank is:
x = R1−1 c1
where c1 is the first N elements of Q† b. If A is rank deficient, see QR Decomposition with Column Pivoting andComplete Orthogonal Decomposition.
GSL offers two interfaces for the QR decomposition. The first proceeds by zeroing out columns below the diagonalof A, one column at a time using Householder transforms. In this method, the factor Q is represented as a product ofHouseholder reflectors:
Q = Hn · · · H2 H1
where each Hi = I − τi vi vi† for a scalar τi and column vector vi . In this method, functions which compute the fullmatrix Q or apply Q† to a right hand side vector operate by applying the Householder matrices one at a time using
Level 2 BLAS.
The second interface is based on a Level 3 BLAS block recursive algorithm developed by Elmroth and Gustavson. In
this case, Q is written in block form as
Q = I − V TV †where V is an M -by-N matrix of the column vectors vi and T is an N -by-N upper triangular matrix, whose diagonal
elements are the τi . Computing the full T , while requiring more flops than the Level 2 approach, offers the advantagethat all standard operations can take advantage of cache efficient Level 3 BLAS operations, and so this method often
performs faster than the Level 2 approach. The functions for the recursive block algorithm have a _r suffix, and it isrecommended to use this interface for performance critical applications.
int gsl_linalg_QR_decomp_r(gsl_matrix *A, gsl_matrix *T)
int gsl_linalg_complex_QR_decomp_r(gsl_matrix_complex *A, gsl_matrix_complex *T)
These functions factor the M -by-N matrix A into the QR decomposition A = QR using the recursive Level 3
BLAS algorithm of Elmroth and Gustavson. On output the diagonal and upper triangular part of A contain thematrix R. The N -by-N matrix T stores the upper triangular factor appearing in Q. The matrix Q is given by
Q = I − V T V † , where the elements below the diagonal of A contain the columns of V on output.
This algorithm requires M ≥ N and performs best for “tall-skinny” matrices, i.e. M  N .
int gsl_linalg_QR_solve_r(const gsl_matrix *QR, const gsl_matrix *T, const gsl_vector *b, gsl_vector *x)

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int gsl_linalg_complex_QR_solve_r(const gsl_matrix_complex *QR, const gsl_matrix_complex *T, const
gsl_vector_complex *b, gsl_vector_complex *x)
These functions solve the square system Ax = b using the QR decomposition of A held in (QR, T).
The least-squares solution for rectangular systems can be found using gsl_linalg_QR_lssolve_r() or
gsl_linalg_complex_QR_lssolve_r().
int gsl_linalg_QR_lssolve_r(const gsl_matrix *QR, const gsl_matrix *T, const gsl_vector *b, gsl_vector *x,
gsl_vector *work)
int gsl_linalg_complex_QR_lssolve_r(const gsl_matrix_complex *QR, const gsl_matrix_complex *T, const
gsl_vector_complex *b, gsl_vector_complex *x, gsl_vector_complex*work)
These functions find the least squares solution to the overdetermined system Ax = b where the matrix A has
more rows than columns. The least squares solution minimizes the Euclidean norm of the residual, ||b − Ax||.
The routine requires as input the QR decomposition of A into (QR, T) given by gsl_linalg_QR_decomp_r()
or gsl_linalg_complex_QR_decomp_r(). The parameter x is of length M . The solution x is returned in thefirst N rows of x, i.e. x = x[0], x[1], ..., x[N-1]. The last M − N rows of x contain a vector whose
norm is equal to the residual norm ||b − Ax||. This similar to the behavior of LAPACK DGELS. Additional
workspace of length N is required in work.
int gsl_linalg_QR_QTvec_r(const gsl_matrix *QR, const gsl_matrix *T, gsl_vector *v, gsl_vector *work)
int gsl_linalg_complex_QR_QHvec_r(const gsl_matrix_complex *QR, const gsl_matrix_complex *T,
gsl_vector_complex *v, gsl_vector_complex *work)
These functions apply the matrix QT (or Q† ) encoded in the decomposition (QR, T) to the vector v, storing theresult QT v (or Q† v) in v. The matrix multiplication is carried out directly using the encoding of the Householder
vectors without needing to form the full matrix QT (or Q† ). Additional workspace of size N is required in work.
int gsl_linalg_QR_QTmat_r(const gsl_matrix *QR, const gsl_matrix *T, gsl_matrix *B, gsl_matrix *work)
This function applies the matrix QT encoded in the decomposition (QR, T) to the M -by-K matrix B, storing theresult QT B in B. The matrix multiplication is carried out directly using the encoding of the Householder vectors
without needing to form the full matrix QT . Additional workspace of size N -by-K is required in work.
int gsl_linalg_QR_unpack_r(const gsl_matrix *QR, const gsl_matrix *T, gsl_matrix *Q, gsl_matrix *R)
int gsl_linalg_complex_QR_unpack_r(const gsl_matrix_complex *QR, const gsl_matrix_complex *T,
gsl_matrix_complex *Q, gsl_matrix_complex *R)
These functions unpack the encoded QR decomposition (QR, T) as output from gsl_linalg_QR_decomp_r()
or gsl_linalg_complex_QR_decomp_r() into the matrices Q and R, where Q is M -by-M and R is N -by-N .
Note that the full R matrix is M -by-N , however the lower trapezoidal portion is zero, so only the upper triangularfactor is stored.
int gsl_linalg_QR_rcond(const gsl_matrix *QR, double *rcond, gsl_vector *work)
This function estimates the reciprocal condition number (using the 1-norm) of the R factor, stored in the upper
triangle of QR. The reciprocal condition number estimate, defined as 1/(||R||1 · ||R−1 ||1 ), is stored in rcond.
Additional workspace of size 3N is required in work.

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14.2.1 Level 2 Interface
The functions below are for the slower Level 2 interface to the QR decomposition. It is recommended to use thesefunctions only for M < N , since the Level 3 interface above performs much faster for M ≥ N .
int gsl_linalg_QR_decomp(gsl_matrix *A, gsl_vector *tau)
int gsl_linalg_complex_QR_decomp(gsl_matrix_complex *A, gsl_vector_complex *tau)
These functions factor the M -by-N matrix A into the QR decomposition A = QR. On output the diagonal andupper triangular part of the input matrix contain the matrix R. The vector tau and the columns of the lower
triangular part of the matrix A contain the Householder coefficients and Householder vectors which encode theorthogonal matrix Q. The vector tau must be of length N . The matrix Q is related to these components by
the product of k = min(M, N ) reflector matrices, Q = Hk ...H2 H1 where Hi = I − τi vi vi† and vi is theHouseholder vector vi = (0, ..., 1, A(i + 1, i), A(i + 2, i), ..., A(m, i)). This is the same storage scheme as usedby LAPACK.
The algorithm used to perform the decomposition is Householder QR (Golub & Van Loan, “Matrix Computations”, Algorithm 5.2.1).
int gsl_linalg_QR_solve(const gsl_matrix *QR, const gsl_vector *tau, const gsl_vector *b, gsl_vector *x)
int gsl_linalg_complex_QR_solve(const gsl_matrix_complex *QR, const gsl_vector_complex *tau, const
gsl_vector_complex *b, gsl_vector_complex *x)
These functions solve the square system Ax = b using the QR decomposition of A held in (QR, tau). The
least-squares solution for rectangular systems can be found using gsl_linalg_QR_lssolve().
int gsl_linalg_QR_svx(const gsl_matrix *QR, const gsl_vector *tau, gsl_vector *x)
int gsl_linalg_complex_QR_svx(const gsl_matrix_complex *QR, const gsl_vector_complex *tau,
gsl_vector_complex *x)
These functions solve the square system Ax = b in-place using the QR decomposition of A held in (QR, tau).
On input x should contain the right-hand side b, which is replaced by the solution on output.
int gsl_linalg_QR_lssolve(const gsl_matrix *QR, const gsl_vector *tau, const gsl_vector *b, gsl_vector *x,
gsl_vector *residual)
int gsl_linalg_complex_QR_lssolve(const gsl_matrix_complex *QR, const gsl_vector_complex *tau, const
gsl_vector_complex *b, gsl_vector_complex *x, gsl_vector_complex*residual)
These functions find the least squares solution to the overdetermined system Ax = b where the matrix A has morerows than columns. The least squares solution minimizes the Euclidean norm of the residual, ||Ax − b||.The
routine requires as input the QR decomposition of A into (QR, tau) given by gsl_linalg_QR_decomp() or
gsl_linalg_complex_QR_decomp(). The solution is returned in x. The residual is computed as a by-productand stored in residual.
int gsl_linalg_QR_QTvec(const gsl_matrix *QR, const gsl_vector *tau, gsl_vector *v)
int gsl_linalg_complex_QR_QHvec(const gsl_matrix_complex *QR, const gsl_vector_complex *tau,
gsl_vector_complex *v)
These functions apply the matrix QT (or Q† ) encoded in the decomposition (QR, tau) to the vector v, storing theresult QT v (or Q† v) in v. The matrix multiplication is carried out directly using the encoding of the Householder
vectors without needing to form the full matrix QT (or Q† ).
int gsl_linalg_QR_Qvec(const gsl_matrix *QR, const gsl_vector *tau, gsl_vector *v)

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int gsl_linalg_complex_QR_Qvec(const gsl_matrix_complex *QR, const gsl_vector_complex *tau,
gsl_vector_complex *v)
These functions apply the matrix Q encoded in the decomposition (QR, tau) to the vector v, storing the result
Qv in v. The matrix multiplication is carried out directly using the encoding of the Householder vectors without
needing to form the full matrix Q.
int gsl_linalg_QR_QTmat(const gsl_matrix *QR, const gsl_vector *tau, gsl_matrix *B)
This function applies the matrix QT encoded in the decomposition (QR, tau) to the M -by-K matrix B, storing
the result QT B in B. The matrix multiplication is carried out directly using the encoding of the Householder
vectors without needing to form the full matrix QT .
int gsl_linalg_QR_Rsolve(const gsl_matrix *QR, const gsl_vector *b, gsl_vector *x)
This function solves the triangular system Rx = b for x. It may be useful if the product b0 = QT b has alreadybeen computed using gsl_linalg_QR_QTvec().
int gsl_linalg_QR_Rsvx(const gsl_matrix *QR, gsl_vector *x)
This function solves the triangular system Rx = b for x in-place. On input x should contain the right-hand sideb and is replaced by the solution on output. This function may be useful if the product b0 = QT b has alreadybeen computed using gsl_linalg_QR_QTvec().
int gsl_linalg_QR_unpack(const gsl_matrix *QR, const gsl_vector *tau, gsl_matrix *Q, gsl_matrix *R)
This function unpacks the encoded QR decomposition (QR, tau) into the matrices Q and R, where Q is M -by-M
and R is M -by-N .
int gsl_linalg_QR_QRsolve(gsl_matrix *Q, gsl_matrix *R, const gsl_vector *b, gsl_vector *x)
This function solves the system Rx = QT b for x. It can be used when the QR decomposition of a matrix isavailable in unpacked form as (Q , R).
int gsl_linalg_QR_update(gsl_matrix *Q, gsl_matrix *R, gsl_vector *w, const gsl_vector *v)
This function performs a rank-1 update wv T of the QR decomposition (Q , R). The update is given by Q0 R0 =Q(R+wv T ) where the output matrices Q and R are also orthogonal and right triangular. Note that w is destroyedby the update.
int gsl_linalg_R_solve(const gsl_matrix *R, const gsl_vector *b, gsl_vector *x)
This function solves the triangular system Rx = b for the N -by-N matrix R.
int gsl_linalg_R_svx(const gsl_matrix *R, gsl_vector *x)
This function solves the triangular system Rx = b in-place. On input x should contain the right-hand side b,which is replaced by the solution on output.

14.2.2 Triangle on Top of Rectangle
This section provides routines for computing the QR decomposition of the specialized matrix
 
U
= QR
A
where U is an N -by-N upper triangular matrix, and A is an M -by-N dense matrix. This type of matrix arises, forexample, in the sequential TSQR algorithm. The Elmroth and Gustavson algorithm is used to efficiently factor thismatrix. Due to the upper triangular factor, the Q matrix takes the form
Q = I − V TV T
with
V =

 
I
Y

and Y is dense and of the same dimensions as A.
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int gsl_linalg_QR_UR_decomp(gsl_matrix *U, gsl_matrix *A, gsl_matrix *T)
This function computes the QR decomposition of the matrix (U ; A), where U is N -by-N upper triangular and
A is M -by-N dense. On output, U is replaced by the R factor, and A is replaced by Y . The N -by-N upper
triangular block reflector is stored in T on output.

14.2.3 Triangle on Top of Triangle
This section provides routines for computing the QR decomposition of the specialized matrix
 
U1
= QR
U2
where U1 , U2 are N -by-N upper triangular matrices. The Elmroth and Gustavson algorithm is used to efficiently factor
this matrix. The Q matrix takes the form
Q = I − V TV T
with
V =

 
I
Y

and Y is N -by-N upper triangular.
int gsl_linalg_QR_UU_decomp(gsl_matrix *U1, gsl_matrix *U2, gsl_matrix *T)
This function computes the QR decomposition of the matrix (U1 ; U2 ), where U1 , U2 are N -by-N upper triangular. On output, U1 is replaced by the R factor, and U2 is replaced by Y . The N -by-N upper triangular blockreflector is stored in T on output.
int gsl_linalg_QR_UU_lssolve(const gsl_matrix *R, const gsl_matrix *Y, const gsl_matrix *T, const gsl_vector*b, gsl_vector *x, gsl_vector *work)
This function find the least squares solution to the overdetermined system,  2
U1
min b −
xU2
x
where U1 , U2 are N -by-N upper triangular matrices. The routine requires as input the QR decomposition of(U1 ; U2 ) into (R, Y, T) given by gsl_linalg_QR_UU_decomp(). The parameter x is of length 2N . The solution
x is returned in the first N rows of x, i.e. x = x[0], x[1], ..., x[N-1]. The last N rows of x contain avector whose norm is equal to the residual norm ||b − (U1 ; U2 )x||. This similar to the behavior of LAPACKDGELS. Additional workspace of length N is required in work.
int gsl_linalg_QR_UU_QTec(const gsl_matrix *Y, const gsl_matrix *T, gsl_vector *b, gsl_vector *work)
This function computes QT b using the decomposition (Y, T) previously computed by
gsl_linalg_QR_UU_decomp(). On input, b contains the vector b, and on output it will contain QT b.
Additional workspace of length N is required in work.

14.2.4 Triangle on Top of Trapezoidal
This section provides routines for computing the QR decomposition of the specialized matrix
 
U
= QR
A
where U is an N -by-N upper triangular matrix, and A is an M -by-N upper trapezoidal matrix with M ≥ N . A has
the structure,
 
Ad
A=
Au
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where Ad is (M − N )-by-N dense, and Au is N -by-N upper triangular. The Elmroth and Gustavson algorithm is used
to efficiently factor this matrix. The Q matrix takes the form
Q = I − V TV T
with
V =

 
I
Y

and Y is upper trapezoidal and of the same dimensions as A.
int gsl_linalg_QR_UZ_decomp(gsl_matrix *U, gsl_matrix *A, gsl_matrix *T)
This function computes the QR decomposition of the matrix (U ; A), where U is N -by-N upper triangular and
A is M -by-N upper trapezoidal. On output, U is replaced by the R factor, and A is replaced by Y . The N -by-N
upper triangular block reflector is stored in T on output.

14.2.5 Triangle on Top of Diagonal
This section provides routines for computing the QR decomposition of the specialized matrix
 
U
= QR
D
where U is an N -by-N upper triangular matrix and D is an N -by-N diagonal matrix. This type of matrix arises inregularized least squares problems. The Elmroth and Gustavson algorithm is used to efficiently factor this matrix. The
Q matrix takes the form
Q = I − V TV T
with
 
I
V =
Y
and Y is N -by-N upper triangular.
int gsl_linalg_QR_UD_decomp(gsl_matrix *U, const gsl_vector *D, gsl_matrix *Y, gsl_matrix *T)
This function computes the QR decomposition of the matrix (U ; D), where U is N -by-N upper triangular and D
is N -by-N diagonal. On output, U is replaced by the R factor and Y is stored in Y. The N -by-N upper triangular
block reflector is stored in T on output.
int gsl_linalg_QR_UD_lssolve(const gsl_matrix *R, const gsl_matrix *Y, const gsl_matrix *T, const gsl_vector*b, gsl_vector *x, gsl_vector *work)
This function find the least squares solution to the overdetermined system,  2Umin b −xDxwhere U is N -by-N upper triangular and D is N -by-N diagonal. The routine requires as input the QR decomposition of (U ; D) into (R, Y, T) given by gsl_linalg_QR_UD_decomp(). The parameter x is of length 2N .
The solution x is returned in the first N rows of x, i.e. x = x[0], x[1], ..., x[N-1]. The last N rows ofx contain a vector whose norm is equal to the residual norm ||b − (U ; D)x||. This similar to the behavior of
LAPACK DGELS. Additional workspace of length N is required in work.

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14.3 QR Decomposition with Column Pivoting
The QR decomposition of an M -by-N matrix A can be extended to the rank deficient case by introducing a column
permutation P ,AP = QR
The first r columns of Q form an orthonormal basis for the range of A for a matrix with column rank r. This decomposition can also be used to convert the square linear system Ax = b into the triangular system Ry = QT b, x = P y,which can be solved by back-substitution and permutation. We denote the QR decomposition with column pivoting by
QRP T since A = QRP T . When A is rank deficient with r = rank(A), the matrix R can be partitioned as 
R11 R12R11 R12
R=
≈
0
R22
0
0
where R11 is r-by-r and nonsingular. In this case, a basic least squares solution for the overdetermined system Ax = b
can be obtained as −1 
R11 c1x=P
0where c1 consists of the first r elements of QT b. This basic solution is not guaranteed to be the minimum norm solution
unless R12 = 0 (see Complete Orthogonal Decomposition).
int gsl_linalg_QRPT_decomp(gsl_matrix *A, gsl_vector *tau, gsl_permutation *p, int *signum, gsl_vector*norm)
This function factorizes the M -by-N matrix A into the QRP T decomposition A = QRP T . On output the
diagonal and upper triangular part of the input matrix contain the matrix R. The permutation matrix P is stored
in the permutation p. The sign of the permutation is given by signum. It has the value (−1)n , where n is thenumber of interchanges in the permutation. The vector tau and the columns of the lower triangular part of thematrix A contain the Householder coefficients and vectors which encode the orthogonal matrix Q. The vector tau
must be of length k = min(M, N ). The matrix Q is related to these components by, Q = Qk ...Q2 Q1 where
Qi = I − τi vi viT and vi is the Householder vector
vi = (0, ..., 1, A(i + 1, i), A(i + 2, i), ..., A(m, i))This is the same storage scheme as used by LAPACK. The vector norm is a workspace of length N used forcolumn pivoting.
The algorithm used to perform the decomposition is Householder QR with column pivoting (Golub & Van Loan,“Matrix Computations”, Algorithm 5.4.1).
int gsl_linalg_QRPT_decomp2(const gsl_matrix *A, gsl_matrix *q, gsl_matrix *r, gsl_vector *tau,
gsl_permutation *p, int *signum, gsl_vector *norm)
This function factorizes the matrix A into the decomposition A = QRP T without modifying A itself and storing
the output in the separate matrices q and r.
int gsl_linalg_QRPT_solve(const gsl_matrix *QR, const gsl_vector *tau, const gsl_permutation *p, const
gsl_vector *b, gsl_vector *x)
This function solves the square system Ax = b using the QRP T decomposition of A held in (QR, tau, p) whichmust have been computed previously by gsl_linalg_QRPT_decomp().
int gsl_linalg_QRPT_svx(const gsl_matrix *QR, const gsl_vector *tau, const gsl_permutation *p, gsl_vector *x)
This function solves the square system Ax = b in-place using the QRP T decomposition of A held in (QR, tau,p). On input x should contain the right-hand side b, which is replaced by the solution on output.

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int gsl_linalg_QRPT_lssolve(const gsl_matrix *QR, const gsl_vector *tau, const gsl_permutation *p, const
gsl_vector *b, gsl_vector *x, gsl_vector *residual)This function finds the least squares solution to the overdetermined system Ax = b where the matrix A has morerows than columns and is assumed to have full rank. The least squares solution minimizes the Euclidean normof the residual, ||b − Ax||. The routine requires as input the QR decomposition of A into (QR, tau, p) given
by gsl_linalg_QRPT_decomp(). The solution is returned in x. The residual is computed as a by-product andstored in residual. For rank deficient matrices, gsl_linalg_QRPT_lssolve2() should be used instead.
int gsl_linalg_QRPT_lssolve2(const gsl_matrix *QR, const gsl_vector *tau, const gsl_permutation *p, const
gsl_vector *b, const size_t rank, gsl_vector *x, gsl_vector *residual)This function finds the least squares solution to the overdetermined system Ax = b where the matrix A has morerows than columns and has rank given by the input rank. If the user does not know the rank of A, the routine
gsl_linalg_QRPT_rank() can be called to estimate it. The least squares solution is the so-called “basic”
solution discussed above and may not be the minimum norm solution. The routine requires as input the QR
decomposition of A into (QR, tau, p) given by gsl_linalg_QRPT_decomp(). The solution is returned in x.
The residual is computed as a by-product and stored in residual.
int gsl_linalg_QRPT_QRsolve(const gsl_matrix *Q, const gsl_matrix *R, const gsl_permutation *p, const
gsl_vector *b, gsl_vector *x)
This function solves the square system RP T x = QT b for x. It can be used when the QR decomposition of amatrix is available in unpacked form as (Q , R).
int gsl_linalg_QRPT_update(gsl_matrix *Q, gsl_matrix *R, const gsl_permutation *p, gsl_vector *w, const
gsl_vector *v)
This function performs a rank-1 update wv T of the QRP T decomposition (Q , R, p). The update is given by
Q0 R0 = Q(R + wv T P ) where the output matrices Q0 and R0 are also orthogonal and right triangular. Note thatw is destroyed by the update. The permutation p is not changed.
int gsl_linalg_QRPT_Rsolve(const gsl_matrix *QR, const gsl_permutation *p, const gsl_vector *b, gsl_vector*x)This function solves the triangular system RP T x = b for the N -by-N matrix R contained in QR.
int gsl_linalg_QRPT_Rsvx(const gsl_matrix *QR, const gsl_permutation *p, gsl_vector *x)
This function solves the triangular system RP T x = b in-place for the N -by-N matrix R contained in QR. On
input x should contain the right-hand side b, which is replaced by the solution on output.
size_t gsl_linalg_QRPT_rank(const gsl_matrix *QR, const double tol)This function estimates the rank of the triangular matrix R contained in QR. The algorithm simply counts thenumber of diagonal elements of R whose absolute value is greater than the specified tolerance tol. If the input
tol is negative, a default value of 20(M + N )eps(max(|diag(R)|)) is used.
int gsl_linalg_QRPT_rcond(const gsl_matrix *QR, double *rcond, gsl_vector *work)
This function estimates the reciprocal condition number (using the 1-norm) of the R factor, stored in the upper
triangle of QR. The reciprocal condition number estimate, defined as 1/(||R||1 · ||R−1 ||1 ), is stored in rcond.
Additional workspace of size 3N is required in work.

14.4 LQ Decomposition
A general rectangular M -by-N matrix A has a LQ decomposition into the product of a lower trapezoidal M -by-N
matrix L and an orthogonal N -by-N square matrix Q:
A = LQ
If M ≤ N , then L can be written as L = (L10) where L1 is M -by-M lower triangular, and  Q1
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where Q1 consists of the first M rows of Q, and Q2 contains the remaining N − M rows. The LQ factorization of A
is essentially the same as the QR factorization of AT .
The LQ factorization may be used to find the minimum norm solution of an underdetermined system of equations
Ax = b, where A is M -by-N and M ≤ N . The solution is −1 
L1 b
x = QT0
int gsl_linalg_LQ_decomp(gsl_matrix *A, gsl_vector *tau)
This function factorizes the M -by-N matrix A into the LQ decomposition A = LQ. On output the diagonal
and lower trapezoidal part of the input matrix contain the matrix L. The vector tau and the elements above
the diagonal of the matrix A contain the Householder coefficients and Householder vectors which encode theorthogonal matrix Q. The vector tau must be of length k = min(M, N ). The matrix Q is related to thesecomponents by, Q = Qk ...Q2 Q1 where Qi = I −τi vi viT and vi is the Householder vector vi = (0, ..., 1, A(i, i+1), A(i, i + 2), ..., A(i, N )). This is the same storage scheme as used by LAPACK.
int gsl_linalg_LQ_lssolve(const gsl_matrix *LQ, const gsl_vector *tau, const gsl_vector *b, gsl_vector *x,
gsl_vector *residual)This function finds the minimum norm least squares solution to the underdetermined system Ax = b, where
the M -by-N matrix A has M ≤ N . The routine requires as input the LQ decomposition of A into (LQ , tau)given by gsl_linalg_LQ_decomp(). The solution is returned in x. The residual, b − Ax, is computed as aby-product and stored in residual.
int gsl_linalg_LQ_unpack(const gsl_matrix *LQ, const gsl_vector *tau, gsl_matrix *Q, gsl_matrix *L)This function unpacks the encoded LQ decomposition (LQ , tau) into the matrices Q and L, where Q is N -by-N
and L is M -by-N .
int gsl_linalg_LQ_QTvec(const gsl_matrix *LQ, const gsl_vector *tau, gsl_vector *v)
This function applies QT to the vector v, storing the result QT v in v on output.

14.5 QL Decomposition
A general rectangular M -by-N matrix A has a QL decomposition into the product of an orthogonal M -by-M squarematrix Q (where QT Q = I) and an M -by-N left-triangular matrix L.
When M ≥ N , the decomposition is given by


0
A=Q
L1



where L1 is N -by-N lower triangular. When M ≤ N , the decomposition is given by
A = Q L1

L2



where L1 is a dense M -by-N − M matrix and L2 is a lower triangular M -by-M matrix.
int gsl_linalg_QL_decomp(gsl_matrix *A, gsl_vector *tau)
This function factorizes the M -by-N matrix A into the QL decomposition A = QL. The vector tau must be oflength N and contains the Householder coefficients on output. The matrix Q is stored in packed form in A on
output, using the same storage scheme as LAPACK.
int gsl_linalg_QL_unpack(const gsl_matrix *QL, const gsl_vector *tau, gsl_matrix *Q, gsl_matrix *L)This function unpacks the encoded QL decomposition (QL, tau) into the matrices Q and L, where Q is M -by-M
and L is M -by-N .

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14.6 Complete Orthogonal Decomposition
The complete orthogonal decomposition of a M -by-N matrix A is a generalization of the QR decomposition withcolumn pivoting, given by


R11 0
ZT
AP = Q00where P is a N -by-N permutation matrix, Q is M -by-M orthogonal, R11 is r-by-r upper triangular, with r = rank(A),and Z is N -by-N orthogonal. If A has full rank, then R11 = R, Z = I and this reduces to the QR decomposition withcolumn pivoting.
For a rank deficient least squares problem, minx ||b − Ax||2 , the solution vector x is not unique. However if we further
require that ||x||2 is minimized, then the complete orthogonal decomposition gives us the ability to compute the uniqueminimum norm solution, which is given by −1 
R11 c1x = PZ0and the vector c1 is the first r elements of QT b.
The COD also enables a straightforward solution of regularized least squares problems in Tikhonov standard form,
written as
min ||b − Ax||2 + λ2 ||x||2xwhere λ > 0 is a regularization parameter which represents a tradeoff between minimizing the residual norm ||b − Ax||and the solution norm ||x||. For this system, the solution is given by yx = PZ 10
where y1 is a vector of length r which is found by solving


 
R11
c
y1 = 1λIr
0and c1 is defined above. The equation above can be solved efficiently for different values of λ using QR factorizationsof the left hand side matrix.
int gsl_linalg_COD_decomp(gsl_matrix *A, gsl_vector *tau_Q, gsl_vector *tau_Z, gsl_permutation *p, size_t*rank, gsl_vector *work)
int gsl_linalg_COD_decomp_e(gsl_matrix *A, gsl_vector *tau_Q, gsl_vector *tau_Z, gsl_permutation *p, double
tol, size_t *rank, gsl_vector *work)
These functions factor the M -by-N matrix A into the decomposition A = QRZP T . The rank of A is computed
as the number of diagonal elements of R greater than the tolerance tol and output in rank. If tol is not
specified, a default value is used (see gsl_linalg_QRPT_rank()). On output, the permutation matrix P isstored in p. The matrix R11 is stored in the upper rank-by-rank block of A. The matrices Q and Z are encodedin packed storage in A on output. The vectors tau_Q and tau_Z contain the Householder scalars corresponding
to the matrices Q and Z respectively and must be of length k = min(M, N ). The vector work is additional
workspace of length N .
int gsl_linalg_COD_lssolve(const gsl_matrix *QRZT, const gsl_vector *tau_Q, const gsl_vector *tau_Z, const
gsl_permutation *p, const size_t rank, const gsl_vector *b, gsl_vector *x,
gsl_vector *residual)This function finds the unique minimum norm least squares solution to the overdetermined system Ax = b
where the matrix A has more rows than columns. The least squares solution minimizes the Euclidean norm
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GNU Scientific Library, Release 2.7of the residual, ||b − Ax|| as well as the norm of the solution ||x||. The routine requires as input the QRZT
decomposition of A into (QRZT, tau_Q , tau_Z, p, rank) given by gsl_linalg_COD_decomp(). The solution
is returned in x. The residual, b − Ax, is computed as a by-product and stored in residual.
int gsl_linalg_COD_lssolve2(const double lambda, const gsl_matrix *QRZT, const gsl_vector *tau_Q, const
gsl_vector *tau_Z, const gsl_permutation *p, const size_t rank, const gsl_vector*b, gsl_vector *x, gsl_vector *residual, gsl_matrix *S, gsl_vector *work)
This function finds the solution to the regularized least squares problem in Tikhonov standard form, minx ||b −Ax||2 + λ2 ||x||2 . The routine requires as input the QRZT decomposition of A into (QRZT, tau_Q , tau_Z, p,rank) given by gsl_linalg_COD_decomp(). The parameter λ is supplied in lambda. The solution is returnedin x. The residual, b − Ax, is stored in residual on output. S is additional workspace of size rank-by-rank.
work is additional workspace of length rank.
int gsl_linalg_COD_unpack(const gsl_matrix *QRZT, const gsl_vector *tau_Q, const gsl_vector *tau_Z, const
size_t rank, gsl_matrix *Q, gsl_matrix *R, gsl_matrix *Z)
This function unpacks the encoded QRZT decomposition (QRZT, tau_Q , tau_Z, rank) into the matrices Q , R,
and Z, where Q is M -by-M , R is M -by-N , and Z is N -by-N .
int gsl_linalg_COD_matZ(const gsl_matrix *QRZT, const gsl_vector *tau_Z, const size_t rank, gsl_matrix *A,
gsl_vector *work)
This function multiplies the input matrix A on the right by Z, A0 = AZ using the encoded QRZT decomposition(QRZT, tau_Z, rank). A must have N columns but may have any number of rows. Additional workspace oflength M is provided in work.

14.7 Singular Value Decomposition
A general rectangular M -by-N matrix A has a singular value decomposition (SVD) into the product of an M -by-N
orthogonal matrix U , an N -by-N diagonal matrix of singular values S and the transpose of an N -by-N orthogonalsquare matrix V ,A = U SV T
The singular values σi = Sii are all non-negative and are generally chosen to form a non-increasing sequence
σ1 ≥ σ2 ≥ ... ≥ σN ≥ 0
The singular value decomposition of a matrix has many practical uses. The condition number of the matrix is given by
the ratio of the largest singular value to the smallest singular value. The presence of a zero singular value indicates thatthe matrix is singular. The number of non-zero singular values indicates the rank of the matrix. In practice singularvalue decomposition of a rank-deficient matrix will not produce exact zeroes for singular values, due to finite numericalprecision. Small singular values should be edited by choosing a suitable tolerance.
For a rank-deficient matrix, the null space of A is given by the columns of V corresponding to the zero singular values.
Similarly, the range of A is given by columns of U corresponding to the non-zero singular values.
Note that the routines here compute the “thin” version of the SVD with U as M -by-N orthogonal matrix. This allows
in-place computation and is the most commonly-used form in practice. Mathematically, the “full” SVD is defined with
U as an M -by-M orthogonal matrix and S as an M -by-N diagonal matrix (with additional rows of zeros).
int gsl_linalg_SV_decomp(gsl_matrix *A, gsl_matrix *V, gsl_vector *S, gsl_vector *work)
This function factorizes the M -by-N matrix A into the singular value decomposition A = U SV T for M ≥ N .
On output the matrix A is replaced by U . The diagonal elements of the singular value matrix S are stored in thevector S. The singular values are non-negative and form a non-increasing sequence from S1 to SN . The matrix V
contains the elements of V in untransposed form. To form the product U SV T it is necessary to take the transposeof V. A workspace of length N is required in work.
This routine uses the Golub-Reinsch SVD algorithm.
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int gsl_linalg_SV_decomp_mod(gsl_matrix *A, gsl_matrix *X, gsl_matrix *V, gsl_vector *S, gsl_vector *work)
This function computes the SVD using the modified Golub-Reinsch algorithm, which is faster for M  N . It
requires the vector work of length N and the N -by-N matrix X as additional working space.
int gsl_linalg_SV_decomp_jacobi(gsl_matrix *A, gsl_matrix *V, gsl_vector *S)
This function computes the SVD of the M -by-N matrix A using one-sided Jacobi orthogonalization for M ≥ N .
The Jacobi method can compute singular values to higher relative accuracy than Golub-Reinsch algorithms (seereferences for details).
int gsl_linalg_SV_solve(const gsl_matrix *U, const gsl_matrix *V, const gsl_vector *S, const gsl_vector *b,
gsl_vector *x)
This function solves the system Ax = b using the singular value decomposition (U, S, V) of A which must havebeen computed previously with gsl_linalg_SV_decomp().
Only non-zero singular values are used in computing the solution. The parts of the solution corresponding tosingular values of zero are ignored. Other singular values can be edited out by setting them to zero before calling
this function.
In the over-determined case where A has more rows than columns the system is solved in the least squares sense,
returning the solution x which minimizes ||Ax − b||2 .
int gsl_linalg_SV_leverage(const gsl_matrix *U, gsl_vector *h)This function computes the statistical leverage values hi of a matrix A using its singular value decomposition (U, S, V) previously computed with gsl_linalg_SV_decomp(). hi are the diagonal values of the matrix
A(AT A)−1 AT and depend only on the matrix U which is the input to this function.

14.8 Cholesky Decomposition
A symmetric, positive definite square matrix A has a Cholesky decomposition into a product of a lower triangularmatrix L and its transpose LT ,A = LLT
This is sometimes referred to as taking the square-root of a matrix. The Cholesky decomposition can only be carriedout when all the eigenvalues of the matrix are positive. This decomposition can be used to convert the linear system
Ax = b into a pair of triangular systems (Ly = b, LT x = y), which can be solved by forward and back-substitution.
If the matrix A is near singular, it is sometimes possible to reduce the condition number and recover a more accurate
solution vector x by scaling as(SAS) S −1 x = Sb√
where S is a diagonal matrix whose elements are given by Sii = 1/ Aii . This scaling is also known as Jacobi
preconditioning. There are routines below to solve both the scaled and unscaled systems.
int gsl_linalg_cholesky_decomp1(gsl_matrix *A)
int gsl_linalg_complex_cholesky_decomp(gsl_matrix_complex *A)
These functions factorize the symmetric, positive-definite square matrix A into the Cholesky decomposition
A = LLT (or A = LL† for the complex case). On input, the values from the diagonal and lower-triangular partof the matrix A are used (the upper triangular part is ignored). On output the diagonal and lower triangular partof the input matrix A contain the matrix L, while the upper triangular part contains the original matrix. If thematrix is not positive-definite then the decomposition will fail, returning the error code GSL_EDOM.
When testing whether a matrix is positive-definite, disable the error handler first to avoid triggering an error.
These functions use Level 3 BLAS to compute the Cholesky factorization (Peise and Bientinesi, 2016).

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int gsl_linalg_cholesky_decomp(gsl_matrix *A)
This function is now deprecated and is provided only for backward compatibility.
int gsl_linalg_cholesky_solve(const gsl_matrix *cholesky, const gsl_vector *b, gsl_vector *x)
int gsl_linalg_complex_cholesky_solve(const gsl_matrix_complex *cholesky, const gsl_vector_complex *b,
gsl_vector_complex *x)
These functions solve the system Ax = b using the Cholesky decomposition of A held in the matrix cholesky which must have been previously computed by gsl_linalg_cholesky_decomp() or
gsl_linalg_complex_cholesky_decomp().
int gsl_linalg_cholesky_svx(const gsl_matrix *cholesky, gsl_vector *x)
int gsl_linalg_complex_cholesky_svx(const gsl_matrix_complex *cholesky, gsl_vector_complex *x)
These functions solve the system Ax = b in-place using the Cholesky decomposition of A held in thematrix cholesky which must have been previously computed by gsl_linalg_cholesky_decomp() or
gsl_linalg_complex_cholesky_decomp(). On input x should contain the right-hand side b, which is replaced by the solution on output.
int gsl_linalg_cholesky_invert(gsl_matrix *cholesky)
int gsl_linalg_complex_cholesky_invert(gsl_matrix_complex *cholesky)
These functions compute the inverse of a matrix from its Cholesky decomposition cholesky,
which must have been previously computed by gsl_linalg_cholesky_decomp() or
gsl_linalg_complex_cholesky_decomp(). On output, the inverse is stored in-place in cholesky.
int gsl_linalg_cholesky_decomp2(gsl_matrix *A, gsl_vector *S)
This function calculates a diagonal scaling transformation S for the symmetric, positive-definite square matrix A,
and then computes the Cholesky decomposition SAS = LLT . On input, the values from the diagonal and lowertriangular part of the matrix A are used (the upper triangular part is ignored). On output the diagonal and lower
triangular part of the input matrix A contain the matrix L, while the upper triangular part of the input matrix isoverwritten with LT (the diagonal terms being identical for both L and LT ). If the matrix is not positive-definite
then the decomposition will fail, returning the error code GSL_EDOM. The diagonal scale factors are stored in S
on output.
When testing whether a matrix is positive-definite, disable the error handler first to avoid triggering an error.
int gsl_linalg_cholesky_solve2(const gsl_matrix *cholesky, const gsl_vector *S, const gsl_vector *b,
gsl_vector *x)
This function solves the system (SAS)(S −1 x) = Sb using the Cholesky decomposition of SAS held in thematrix cholesky which must have been previously computed by gsl_linalg_cholesky_decomp2().
int gsl_linalg_cholesky_svx2(const gsl_matrix *cholesky, const gsl_vector *S, gsl_vector *x)
This function solves the system (SAS)(S −1 x) = Sb in-place using the Cholesky decomposition of SAS heldin the matrix cholesky which must have been previously computed by gsl_linalg_cholesky_decomp2().
On input x should contain the right-hand side b, which is replaced by the solution on output.
int gsl_linalg_cholesky_scale(const gsl_matrix *A, gsl_vector *S)
This function calculates a diagonal scaling transformation of the symmetric, positive definite matrix A, such thatSAS has a condition number within a factor of N of the matrix of smallest possible√condition number over all
possible diagonal scalings. On output, S contains the scale factors, given by Si = 1/ Aii . For any Aii ≤ 0, thecorresponding scale factor Si is set to 1.
int gsl_linalg_cholesky_scale_apply(gsl_matrix *A, const gsl_vector *S)
This function applies the scaling transformation S to the matrix A. On output, A is replaced by SAS.
int gsl_linalg_cholesky_rcond(const gsl_matrix *cholesky, double *rcond, gsl_vector *work)
This function estimates the reciprocal condition number (using the 1-norm) of the symmetric positive definite

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matrix A, using its Cholesky decomposition provided in cholesky. The reciprocal condition number estimate,defined as 1/(||A||1 · ||A−1 ||1 ), is stored in rcond. Additional workspace of size 3N is required in work.

14.9 Pivoted Cholesky Decomposition
A symmetric positive semi-definite square matrix A has an alternate Cholesky decomposition into a product of a lower
unit triangular matrix L, a diagonal matrix D and LT , given by LDLT . For postive definite matrices, this is equivalent1
to the Cholesky formulation discussed above, with the standard Cholesky lower triangular factor given by LD 2 . For
ill-conditioned matrices, it can help to use a pivoting strategy to prevent the entries of D and L from growing too large,
and also ensure D1 ≥ D2 ≥ · · · ≥ Dn > 0, where Di are the diagonal entries of D. The final decomposition is givenby
P AP T = LDLT
where P is a permutation matrix.
int gsl_linalg_pcholesky_decomp(gsl_matrix *A, gsl_permutation *p)
This function factors the symmetric, positive-definite square matrix A into the Pivoted Cholesky decomposition
P AP T = LDLT . On input, the values from the diagonal and lower-triangular part of the matrix A are used toconstruct the factorization. On output the diagonal of the input matrix A stores the diagonal elements of D, andthe lower triangular portion of A contains the matrix L. Since L has ones on its diagonal these do not need to be
explicitely stored. The upper triangular portion of A is unmodified. The permutation matrix P is stored in p on
output.
int gsl_linalg_pcholesky_solve(const gsl_matrix *LDLT, const gsl_permutation *p, const gsl_vector *b,
gsl_vector *x)
This function solves the system Ax = b using the Pivoted Cholesky decomposition of A held in the matrix LDLT
and permutation p which must have been previously computed by gsl_linalg_pcholesky_decomp().
int gsl_linalg_pcholesky_svx(const gsl_matrix *LDLT, const gsl_permutation *p, gsl_vector *x)
This function solves the system Ax = b in-place using the Pivoted Cholesky decomposition of
A held in the matrix LDLT and permutation p which must have been previously computed by
gsl_linalg_pcholesky_decomp(). On input, x contains the right hand side vector b which is replaced by thesolution vector on output.
int gsl_linalg_pcholesky_decomp2(gsl_matrix *A, gsl_permutation *p, gsl_vector *S)
This function computes the pivoted Cholesky factorization of the matrix SAS, where the input matrix A issymmetric and positive definite, and the diagonal scaling matrix S is computed to reduce the condition numberof A as much as possible. See Cholesky Decomposition for more information on the matrix S. The Pivoted
Cholesky decomposition satisfies P SASP T = LDLT . On input, the values from the diagonal and lowertriangular part of the matrix A are used to construct the factorization. On output the diagonal of the input matrix
A stores the diagonal elements of D, and the lower triangular portion of A contains the matrix L. Since L has
ones on its diagonal these do not need to be explicitely stored. The upper triangular portion of A is unmodified.
The permutation matrix P is stored in p on output. The diagonal scaling transformation is stored in S on output.
int gsl_linalg_pcholesky_solve2(const gsl_matrix *LDLT, const gsl_permutation *p, const gsl_vector *S,
const gsl_vector *b, gsl_vector *x)
This function solves the system (SAS)(S −1 x) = Sb using the Pivoted Cholesky decomposition of SAS
held in the matrix LDLT, permutation p, and vector S, which must have been previously computed by
gsl_linalg_pcholesky_decomp2().
int gsl_linalg_pcholesky_svx2(const gsl_matrix *LDLT, const gsl_permutation *p, const gsl_vector *S,
gsl_vector *x)
This function solves the system (SAS)(S −1 x) = Sb in-place using the Pivoted Cholesky decomposition of
SAS held in the matrix LDLT, permutation p and vector S, which must have been previously computed by

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gsl_linalg_pcholesky_decomp2(). On input, x contains the right hand side vector b which is replaced by
the solution vector on output.
int gsl_linalg_pcholesky_invert(const gsl_matrix *LDLT, const gsl_permutation *p, gsl_matrix *Ainv)
This function computes the inverse of the matrix A, using the Pivoted Cholesky decomposition stored in LDLT
and p. On output, the matrix Ainv contains A−1 .
int gsl_linalg_pcholesky_rcond(const gsl_matrix *LDLT, const gsl_permutation *p, double *rcond, gsl_vector*work)
This function estimates the reciprocal condition number (using the 1-norm) of the symmetric positive definite
matrix A, using its pivoted Cholesky decomposition provided in LDLT. The reciprocal condition number estimate,defined as 1/(||A||1 · ||A−1 ||1 ), is stored in rcond. Additional workspace of size 3N is required in work.

14.10 Modified Cholesky Decomposition
The modified Cholesky decomposition is suitable for solving systems Ax = b where A is a symmetric indefinite matrix.
Such matrices arise in nonlinear optimization algorithms. The standard Cholesky decomposition requires a positive
definite matrix and would fail in this case. Instead of resorting to a method like QR or SVD, which do not take into
account the symmetry of the matrix, we can instead introduce a small perturbation to the matrix A to make it positive
definite, and then use a Cholesky decomposition on the perturbed matrix. The resulting decomposition satisfies
P (A + E)P T = LDLT
where P is a permutation matrix, E is a diagonal perturbation matrix, L is unit lower triangular, and D is diagonal.
If A is sufficiently positive definite, then the perturbation matrix E will be zero and this method is equivalent to thepivoted Cholesky algorithm. For indefinite matrices, the perturbation matrix E is computed to ensure that A + E ispositive definite and well conditioned.
int gsl_linalg_mcholesky_decomp(gsl_matrix *A, gsl_permutation *p, gsl_vector *E)
This function factors the symmetric, indefinite square matrix A into the Modified Cholesky decomposition P (A+E)P T = LDLT . On input, the values from the diagonal and lower-triangular part of the matrix A are used toconstruct the factorization. On output the diagonal of the input matrix A stores the diagonal elements of D, andthe lower triangular portion of A contains the matrix L. Since L has ones on its diagonal these do not need to be
explicitely stored. The upper triangular portion of A is unmodified. The permutation matrix P is stored in p on
output. The diagonal perturbation matrix is stored in E on output. The parameter E may be set to NULL if it is
not required.
int gsl_linalg_mcholesky_solve(const gsl_matrix *LDLT, const gsl_permutation *p, const gsl_vector *b,
gsl_vector *x)
This function solves the perturbed system (A + E)x = b using the Cholesky decomposition of
A + E held in the matrix LDLT and permutation p which must have been previously computed by
gsl_linalg_mcholesky_decomp().
int gsl_linalg_mcholesky_svx(const gsl_matrix *LDLT, const gsl_permutation *p, gsl_vector *x)
This function solves the perturbed system (A + E)x = b in-place using the Cholesky decompositionof A + E held in the matrix LDLT and permutation p which must have been previously computed by
gsl_linalg_mcholesky_decomp(). On input, x contains the right hand side vector b which is replaced by thesolution vector on output.
int gsl_linalg_mcholesky_rcond(const gsl_matrix *LDLT, const gsl_permutation *p, double *rcond, gsl_vector*work)
This function estimates the reciprocal condition number (using the 1-norm) of the perturbed matrix A + E,
using its pivoted Cholesky decomposition provided in LDLT. The reciprocal condition number estimate, definedas 1/(||A + E||1 · ||(A + E)−1 ||1 ), is stored in rcond. Additional workspace of size 3N is required in work.

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14.11 LDLT Decomposition
If A is a symmetric, nonsingular square matrix, then it has a unique factorization of the form
A = LDLT
where L is a unit lower triangular matrix and D is diagonal. If A is positive definite, then this factorization is equivalent1
to the Cholesky factorization, where the lower triangular Cholesky factor is LD 2 . Some indefinite matrices for which
no Cholesky decomposition exists have an LDLT decomposition with negative entries in D. The LDLT algorithm
is sometimes referred to as the square root free Cholesky decomposition, as the algorithm does not require the computation of square roots. The algorithm is stable for positive definite matrices, but is not guaranteed to be stable forindefinite matrices.
int gsl_linalg_ldlt_decomp(gsl_matrix *A)
This function factorizes the symmetric, non-singular square matrix A into the decomposition A = LDLT . On
input, the values from the diagonal and lower-triangular part of the matrix A are used. The upper triangle of A
is used as temporary workspace. On output the diagonal of A contains the matrix D and the lower triangle of
A contains the unit lower triangular matrix L. The matrix 1-norm, ||A||1 is stored in the upper right corner on
output, for later use by gsl_linalg_ldlt_rcond().
If the matrix is detected to be singular, the function returns the error code GSL_EDOM.
int gsl_linalg_ldlt_solve(const gsl_matrix *LDLT, const gsl_vector *b, gsl_vector *x)
This function solves the system Ax = b using the LDLT decomposition of A held in the matrix LDLT whichmust have been previously computed by gsl_linalg_ldlt_decomp().
int gsl_linalg_ldlt_svx(const gsl_matrix *LDLT, gsl_vector *x)
This function solves the system Ax = b in-place using the LDLT decomposition of A held in the matrix LDLT
which must have been previously computed by gsl_linalg_ldlt_decomp(). On input x should contain theright-hand side b, which is replaced by the solution on output.
int gsl_linalg_ldlt_rcond(const gsl_matrix *LDLT, double *rcond, gsl_vector *work)
This function estimates the reciprocal condition number (using the 1-norm) of the symmetric nonsingular matrix
A, using its LDLT decomposition provided in LDLT. The reciprocal condition number estimate, defined as
1/(||A||1 · ||A−1 ||1 ), is stored in rcond. Additional workspace of size 3N is required in work.

14.12 Tridiagonal Decomposition of Real Symmetric Matrices
A symmetric matrix A can be factorized by similarity transformations into the form,
A = QT QT
where Q is an orthogonal matrix and T is a symmetric tridiagonal matrix.
int gsl_linalg_symmtd_decomp(gsl_matrix *A, gsl_vector *tau)
This function factorizes the symmetric square matrix A into the symmetric tridiagonal decomposition QT QT . On
output the diagonal and subdiagonal part of the input matrix A contain the tridiagonal matrix T . The remaining
lower triangular part of the input matrix contains the Householder vectors which, together with the Householder
coefficients tau, encode the orthogonal matrix Q. This storage scheme is the same as used by LAPACK. The
upper triangular part of A is not referenced.
int gsl_linalg_symmtd_unpack(const gsl_matrix *A, const gsl_vector *tau, gsl_matrix *Q, gsl_vector *diag,
gsl_vector *subdiag)This function unpacks the encoded symmetric tridiagonal decomposition (A, tau) obtained from
gsl_linalg_symmtd_decomp() into the orthogonal matrix Q , the vector of diagonal elements diag and thevector of subdiagonal elements subdiag.
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int gsl_linalg_symmtd_unpack_T(const gsl_matrix *A, gsl_vector *diag, gsl_vector *subdiag)This function unpacks the diagonal and subdiagonal of the encoded symmetric tridiagonal decomposition (A,tau) obtained from gsl_linalg_symmtd_decomp() into the vectors diag and subdiag.

14.13 Tridiagonal Decomposition of Hermitian Matrices
A hermitian matrix A can be factorized by similarity transformations into the form,
A = UTUT
where U is a unitary matrix and T is a real symmetric tridiagonal matrix.
int gsl_linalg_hermtd_decomp(gsl_matrix_complex *A, gsl_vector_complex *tau)
This function factorizes the hermitian matrix A into the symmetric tridiagonal decomposition U T U T . On output
the real parts of the diagonal and subdiagonal part of the input matrix A contain the tridiagonal matrix T . The
remaining lower triangular part of the input matrix contains the Householder vectors which, together with theHouseholder coefficients tau, encode the unitary matrix U . This storage scheme is the same as used by LAPACK.
The upper triangular part of A and imaginary parts of the diagonal are not referenced.
int gsl_linalg_hermtd_unpack(const gsl_matrix_complex *A, const gsl_vector_complex *tau,
gsl_matrix_complex *U, gsl_vector *diag, gsl_vector *subdiag)This function unpacks the encoded tridiagonal decomposition (A, tau) obtained from
gsl_linalg_hermtd_decomp() into the unitary matrix U, the real vector of diagonal elements diag
and the real vector of subdiagonal elements subdiag.
int gsl_linalg_hermtd_unpack_T(const gsl_matrix_complex *A, gsl_vector *diag, gsl_vector *subdiag)This function unpacks the diagonal and subdiagonal of the encoded tridiagonal decomposition (A, tau) obtained
from the gsl_linalg_hermtd_decomp() into the real vectors diag and subdiag.

14.14 Hessenberg Decomposition of Real Matrices
A general real matrix A can be decomposed by orthogonal similarity transformations into the form
A = U HU T
where U is orthogonal and H is an upper Hessenberg matrix, meaning that it has zeros below the first subdiagonal.
The Hessenberg reduction is the first step in the Schur decomposition for the nonsymmetric eigenvalue problem, but
has applications in other areas as well.
int gsl_linalg_hessenberg_decomp(gsl_matrix *A, gsl_vector *tau)
This function computes the Hessenberg decomposition of the matrix A by applying the similarity transformation
H = U T AU . On output, H is stored in the upper portion of A. The information required to construct the matrix
U is stored in the lower triangular portion of A. U is a product of N − 2 Householder matrices. The Householder
vectors are stored in the lower portion of A (below the subdiagonal) and the Householder coefficients are stored
in the vector tau. tau must be of length N.
int gsl_linalg_hessenberg_unpack(gsl_matrix *H, gsl_vector *tau, gsl_matrix *U)
This function constructs the orthogonal matrix U from the information stored in the Hessenberg matrix H alongwith the vector tau. H and tau are outputs from gsl_linalg_hessenberg_decomp().
int gsl_linalg_hessenberg_unpack_accum(gsl_matrix *H, gsl_vector *tau, gsl_matrix *V)This function is similar to gsl_linalg_hessenberg_unpack(), except it accumulates the matrix U into V, so
that V 0 = V U . The matrix V must be initialized prior to calling this function. Setting V to the identity matrix
provides the same result as gsl_linalg_hessenberg_unpack(). If H is order N, then V must have N columns
but may have any number of rows.
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int gsl_linalg_hessenberg_set_zero(gsl_matrix *H)
This function sets the lower triangular portion of H, below the subdiagonal, to zero. It is useful for clearing out
the Householder vectors after calling gsl_linalg_hessenberg_decomp().

14.15 Hessenberg-Triangular Decomposition of Real Matrices
A general real matrix pair (A, B) can be decomposed by orthogonal similarity transformations into the form
A = U HV T
B = U RV T
where U and V are orthogonal, H is an upper Hessenberg matrix, and R is upper triangular. The Hessenberg-Triangularreduction is the first step in the generalized Schur decomposition for the generalized eigenvalue problem.
int gsl_linalg_hesstri_decomp(gsl_matrix *A, gsl_matrix *B, gsl_matrix *U, gsl_matrix *V, gsl_vector*work)
This function computes the Hessenberg-Triangular decomposition of the matrix pair (A, B). On output, H isstored in A, and R is stored in B. If U and V are provided (they may be null), the similarity transformations are
stored in them. Additional workspace of length N is needed in work.

14.16 Bidiagonalization
A general matrix A can be factorized by similarity transformations into the form,
A = U BV T
where U and V are orthogonal matrices and B is a N -by-N bidiagonal matrix with non-zero entries only on the
diagonal and superdiagonal. The size of U is M -by-N and the size of V is N -by-N .
int gsl_linalg_bidiag_decomp(gsl_matrix *A, gsl_vector *tau_U, gsl_vector *tau_V)
This function factorizes the M -by-N matrix A into bidiagonal form U BV T . The diagonal and superdiagonal ofthe matrix B are stored in the diagonal and superdiagonal of A. The orthogonal matrices U and V are stored ascompressed Householder vectors in the remaining elements of A. The Householder coefficients are stored in thevectors tau_U and tau_V. The length of tau_U must equal the number of elements in the diagonal of A and thelength of tau_V should be one element shorter.
int gsl_linalg_bidiag_unpack(const gsl_matrix *A, const gsl_vector *tau_U, gsl_matrix *U, const gsl_vector*tau_V, gsl_matrix *V, gsl_vector *diag, gsl_vector *superdiag)This function unpacks the bidiagonal decomposition of A produced by gsl_linalg_bidiag_decomp(), (A,tau_U, tau_V) into the separate orthogonal matrices U, V and the diagonal vector diag and superdiagonal
superdiag. Note that U is stored as a compact M -by-N orthogonal matrix satisfying U T U = I for efficiency.
int gsl_linalg_bidiag_unpack2(gsl_matrix *A, gsl_vector *tau_U, gsl_vector *tau_V, gsl_matrix *V)This function unpacks the bidiagonal decomposition of A produced by gsl_linalg_bidiag_decomp(), (A,tau_U, tau_V) into the separate orthogonal matrices U, V and the diagonal vector diag and superdiagonal
superdiag. The matrix U is stored in-place in A.
int gsl_linalg_bidiag_unpack_B(const gsl_matrix *A, gsl_vector *diag, gsl_vector *superdiag)This function unpacks the diagonal and superdiagonal of the bidiagonal decomposition of A from
gsl_linalg_bidiag_decomp(), into the diagonal vector diag and superdiagonal vector superdiag.

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14.17 Givens Rotations
A Givens rotation is a rotation in the plane acting on two elements of a given vector. It can be represented in matrix
form as


1 ...
0
...
0
... 0
 .. . .
..
..
.. 
.
.
.
.
.


0 . . . cos θ . . . − sin θ . . . 0



..
..
.. 
..
G(i, j, θ) =  ...
.
.
.
.


0 . . . sin θ . . . cos θ . . . 0


.
..
..
.. 
..
 ..
.
.
.
.
0 ...
0
...
0
... 1
where the cos θ and sin θ appear at the intersection of the i-th and j-th rows and columns. When acting on a vectorx, G(i, j, θ)x performs a rotation of the (i, j) elements of x. Givens rotations are typically used to introduce zeros invectors, such as during the QR decomposition of a matrix. In this case, it is typically desired to find c and s such that   
c −s
a
r
=
s c
b
0
with r =√a2 + b2 .

void gsl_linalg_givens(const double a, const double b, double *c, double *s)
This function computes c√= cos θ and s = sin θ so that the Givens matrix G(θ) acting on the vector (a, b)
produces (r, 0), with r = a2 + b2 .
void gsl_linalg_givens_gv(gsl_vector *v, const size_t i, const size_t j, const double c, const double s)This function applies the Givens rotation defined by c = cos θ and s = sin θ to the i and j elements of v. On
output, (v(i), v(j)) ← G(θ)(v(i), v(j)).

14.18 Householder Transformations
A Householder transformation is a rank-1 modification of the identity matrix which can be used to zero out selected
elements of a vector. A Householder matrix H takes the form,
H = I − τ vv T
where v is a vector (called the Householder vector) and τ = 2/(v T v). The functions described in this section use therank-1 structure of the Householder matrix to create and apply Householder transformations efficiently.
double gsl_linalg_householder_transform(gsl_vector *w)
gsl_complex gsl_linalg_complex_householder_transform(gsl_vector_complex *w)
This function prepares a Householder transformation H = I − τ vv T which can be used to zero all the elementsof the input vector w except the first. On output the Householder vector v is stored in w and the scalar τ isreturned. The householder vector v is normalized so that v[0] = 1, however this 1 is not stored in the output
vector. Instead, w[0] is set to the first element of the transformed vector, so that if u = Hw, w[0] = u[0] on
output and the remainder of u is zero.
int gsl_linalg_householder_hm(double tau, const gsl_vector *v, gsl_matrix *A)

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int gsl_linalg_complex_householder_hm(gsl_complex tau, const gsl_vector_complex *v, gsl_matrix_complex*A)
This function applies the Householder matrix H defined by the scalar tau and the vector v to the left-hand sideof the matrix A. On output the result HA is stored in A.
int gsl_linalg_householder_mh(double tau, const gsl_vector *v, gsl_matrix *A)
int gsl_linalg_complex_householder_mh(gsl_complex tau, const gsl_vector_complex *v, gsl_matrix_complex*A)
This function applies the Householder matrix H defined by the scalar tau and the vector v to the right-hand sideof the matrix A. On output the result AH is stored in A.
int gsl_linalg_householder_hv(double tau, const gsl_vector *v, gsl_vector *w)
int gsl_linalg_complex_householder_hv(gsl_complex tau, const gsl_vector_complex *v, gsl_vector_complex*w)
This function applies the Householder transformation H defined by the scalar tau and the vector v to the vector
w. On output the result Hw is stored in w.

14.19 Householder solver for linear systems
int gsl_linalg_HH_solve(gsl_matrix *A, const gsl_vector *b, gsl_vector *x)
This function solves the system Ax = b directly using Householder transformations. On output the solution isstored in x and b is not modified. The matrix A is destroyed by the Householder transformations.
int gsl_linalg_HH_svx(gsl_matrix *A, gsl_vector *x)
This function solves the system Ax = b in-place using Householder transformations. On input x should containthe right-hand side b, which is replaced by the solution on output. The matrix A is destroyed by the Householdertransformations.

14.20 Tridiagonal Systems
The functions described in this section efficiently solve symmetric, non-symmetric and cyclic tridiagonal systems withminimal storage. Note that the current implementations of these functions use a variant of Cholesky decomposition, so
the tridiagonal matrix must be positive definite. For non-positive definite matrices, the functions return the error code
GSL_ESING.
int gsl_linalg_solve_tridiag(const gsl_vector *diag, const gsl_vector *e, const gsl_vector *f, const gsl_vector*b, gsl_vector *x)
This function solves the general N -by-N system Ax = b where A is tridiagonal (N ≥ 2). The super-diagonal
and sub-diagonal vectors e and f must be one element shorter than the diagonal vector diag. The form of A for
the 4-by-4 case is shown below,
d0 e0 0 0
f0 d1 e1 0 
A= 0 f1 d2 e2 
0 0 f2 d3
int gsl_linalg_solve_symm_tridiag(const gsl_vector *diag, const gsl_vector *e, const gsl_vector *b,
gsl_vector *x)
This function solves the general N -by-N system Ax = b where A is symmetric tridiagonal (N ≥ 2). The offdiagonal vector e must be one element shorter than the diagonal vector diag. The form of A for the 4-by-4 case
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is shown below,

d0
 e0
A=
0
0

e0
d1
e1
0

0
e1
d2
e2


0
0

e2 
d3

int gsl_linalg_solve_cyc_tridiag(const gsl_vector *diag, const gsl_vector *e, const gsl_vector *f, const
gsl_vector *b, gsl_vector *x)
This function solves the general N -by-N system Ax = b where A is cyclic tridiagonal (N ≥ 3). The cyclicsuper-diagonal and sub-diagonal vectors e and f must have the same number of elements as the diagonal vector
diag. The form of A for the 4-by-4 case is shown below,
d0 e0 0 f3
f0 d1 e1 0 
A= 0 f1 d 2 e 2 
e3 0 f2 d3
int gsl_linalg_solve_symm_cyc_tridiag(const gsl_vector *diag, const gsl_vector *e, const gsl_vector *b,
gsl_vector *x)
This function solves the general N -by-N system Ax = b where A is symmetric cyclic tridiagonal (N ≥ 3). The
cyclic off-diagonal vector e must have the same number of elements as the diagonal vector diag. The form of A
for the 4-by-4 case is shown below,
d0 e0 0 e3
e0 d1 e1 0 A= 0 e1 d2 e2 
e3 0 e2 d3

14.21 Triangular Systems
int gsl_linalg_tri_invert(CBLAS_UPLO_t Uplo, CBLAS_DIAG_t Diag, gsl_matrix *T)
int gsl_linalg_complex_tri_invert(CBLAS_UPLO_t Uplo, CBLAS_DIAG_t Diag, gsl_matrix_complex *T)
These functions compute the in-place inverse of the triangular matrix T, stored in the lower triangle when Uplo =CblasLower and upper triangle when Uplo = CblasUpper. The parameter Diag = CblasUnit, CblasNonUnit
specifies whether the matrix is unit triangular.
int gsl_linalg_tri_LTL(gsl_matrix *L)
int gsl_linalg_complex_tri_LHL(gsl_matrix_complex *L)
These functions compute the product LT L (or L† L) in-place and stores it in the lower triangle of L on output.
int gsl_linalg_tri_UL(gsl_matrix *LU)
int gsl_linalg_complex_tri_UL(gsl_matrix_complex *LU)
These functions compute the product U L where U is upper triangular and L is unit lower triangular, stored
in LU, as computed by gsl_linalg_LU_decomp() or gsl_linalg_complex_LU_decomp(). The product iscomputed in-place using Level 3 BLAS.
int gsl_linalg_tri_rcond(CBLAS_UPLO_t Uplo, const gsl_matrix *A, double *rcond, gsl_vector *work)
This function estimates the 1-norm reciprocal condition number of the triangular matrix A, using the lower
triangle when Uplo is CblasLowerand upper triangle when Uplo is CblasUpper. The reciprocal condition
number 1/ ||A||1 A−1 1 is stored in rcond on output. Additional workspace of size 3N is required inwork.
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14.22 Banded Systems
Band matrices are sparse matrices whose non-zero entries are confined to a diagonal band. From a storage point ofview, significant savings can be achieved by storing only the non-zero diagonals of a banded matrix. Algorithms such as
LU and Cholesky factorizations preserve the band structure of these matrices. Computationally, working with compact
banded matrices is preferable to working on the full dense matrix with many zero entries.

14.22.1 General Banded Format
An example of a general banded matrix is given below.

α1 β1 γ1 δ1 α2 β2 0 δ2 α3
A=
0
0 δ3

0
0
0
0
0
0

0
γ2
β3
α4
δ4
0

0
0
γ3
β4
α5
δ5


0
0

0

γ4 

β5 
α6

This matrix has a lower bandwidth of 1 and an upper bandwidth of 2. The lower bandwidth is the number of non-zero
subdiagonals, and the upper bandwidth is the number of non-zero superdiagonals. A (p, q) banded matrix has a lower
bandwidth p and upper bandwidth q. For example, diagonal matrices are (0, 0), tridiagonal matrices are (1, 1), andupper triangular matrices are (0, N − 1) banded matrices.
The corresponding 6-by-4 packed banded matrix looks like∗ ∗
 ∗ β1γ1 β2
AB = 
γ2 β3γ3 β4γ 4 β5

α1
α2
α3
α4
α5
α6


δ1
δ2 

δ3 

δ4 

δ5 
∗

where the superdiagonals are stored in columns, followed by the diagonal, followed by the subdiagonals. The entries
marked by ∗ are not referenced by the banded routines. With this format, each row of AB corresponds to the non-zero
entries of the corresponding column of A. For an N -by-N matrix A, the dimension of AB will be N -by-(p + q + 1).

14.22.2 Symmetric Banded Format
Symmetric banded matrices allow for additional storage savings. As an example, consider the following 6×6 symmetricbanded matrix with lower bandwidth p = 2:
α1 β1 γ1 000
 β1 α 2 β2 γ 2 00 γ1 β2 α3 β3 γ3 0 A= 0 γ2 β3 α4 β4 γ4 
00 γ3 β4 α5 β5 000 γ4 β5 α6The packed symmetric banded 6 × 3 matrix will look like:

α1
α2

α3
AB = 
α4

α5
α6
164

β1
β2
β3
β4
β5
∗


γ1
γ2 

γ3 

γ4 

∗
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The entries marked by ∗ are not referenced by the symmetric banded routines. The relationship between the packed
format and original matrix is,
AB(i, j) = A(i, i + j) = A(i + j, i)
for i = 0, . . . , N − 1, j = 0, . . . , p. Conversely,
A(i, j) = AB(j, i − j)for i = 0, . . . , N − 1, j = max(0, i − p), . . . , i.
Warning: Note that this format is the transpose of the symmetric banded format used by LAPACK. In order todevelop efficient routines for symmetric banded matrices, it helps to have the nonzero elements in each column incontiguous memory locations. Since C uses row-major order, GSL stores the columns in the rows of the packed
banded format, while LAPACK, written in Fortran, uses the transposed format.

14.22.3 Banded LU Decomposition
The routines in this section are designed to factor banded M -by-N matrices with an LU factorization, P A = LU . The
matrix A is banded of type (p, q), i.e. a lower bandwidth of p and an upper bandwidth of q. See LU Decomposition
for more information on the factorization. For banded (p, q) matrices, the U factor will have an upper bandwidth ofp + q, while the L factor will have a lower bandwidth of at most p. Therefore, additional storage is needed to store thep additional bands of U .
As an example, consider the M = N = 7 matrix with lower bandwidth p = 3 and upper bandwidth q = 2,
α1 β1 γ1 0000 δ1 α2 β2 γ2 0
0
0



 1 δ2 α3 β3 γ3 0
0



ζ

δ
α
β
γ
0
A=
2
3
4
4
4

 1
 0 ζ2 3 δ4 α5 β5 γ5 00 ζ3 4 δ5 α6 β6 000 ζ4 5 δ6 α7
The corresponding N -by-2p + q + 1 packed banded matrix looks like∗ ∗ ∗ ∗ ∗ α1 δ1 1 ζ1
 ∗ ∗ ∗ ∗ β1 α2 δ2 2 ζ2  ∗ ∗ ∗ γ1 β2 α3 δ3 3 ζ3 
AB =  ∗ ∗ − γ2 β3 α4 δ4 4 ζ4  ∗ − − γ3 β4 α5 δ5 5 ∗ 
− − − γ4 β5 α6 δ6 ∗ ∗ − − − γ5 β6 α7 ∗ ∗ ∗| {z }| {z } | {z }pqEntries marked with − are used to store the additional p diagonals of thepU factor. Entries marked with ∗ are notreferenced by the banded routines.
int gsl_linalg_LU_band_decomp(const size_t M, const size_t lb, const size_t ub, gsl_matrix *AB, gsl_vector_uint*piv)This function computes the LU factorization of the banded matrix AB which is stored in packed band format (seeabove) and has dimension N -by-2p + q + 1. The number of rows M of the original matrix is provided in M. The
lower bandwidth p is provided in lb and the upper bandwidth q is provided in ub. The vector piv has lengthmin(M, N ) and stores the pivot indices on output (for 0 ≤ i < min(M, N ), row i of the matrix was interchanged
with row piv[i]). On output, AB contains both the L and U factors in packed format.
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int gsl_linalg_LU_band_solve(const size_t lb, const size_t ub, const gsl_matrix *LUB, const gsl_vector_uint*piv, const gsl_vector *b, gsl_vector *x)
This function solves the square system Ax = b using the banded LU factorization (LUB, piv) computed by
gsl_linalg_LU_band_decomp(). The lower and upper bandwidths are provided in lb and ub respectively.
The right hand side vector is provided in b. The solution vector is stored in x on output.
int gsl_linalg_LU_band_svx(const size_t lb, const size_t ub, const gsl_matrix *LUB, const gsl_vector_uint *piv,
gsl_vector *x)
This function solves the square system Ax = b in-place, using the banded LU factorization (LUB, piv) computedby gsl_linalg_LU_band_decomp(). The lower and upper bandwidths are provided in lb and ub respectively.
On input, the right hand side vector b is provided in x, which is replaced by the solution vector x on output.
int gsl_linalg_LU_band_unpack(const size_t M, const size_t lb, const size_t ub, const gsl_matrix *LUB, const
gsl_vector_uint *piv, gsl_matrix *L, gsl_matrix *U)
This function unpacks the banded LU factorization (LUB, piv) previously computed by
gsl_linalg_LU_band_decomp() into the matrices L and U. The matrix U has dimension min(M, N )by-N and stores the upper triangular factor on output. The matrix L has dimension M -by-min(M, N ) and storesthe matrix P T L on output.

14.22.4 Banded Cholesky Decomposition
The routines in this section are designed to factor and solve N -by-N linear systems of the form Ax = b where A
is a banded, symmetric, and positive definite matrix with lower bandwidth p. See Cholesky Decomposition for more
information on the factorization. The lower triangular factor of the Cholesky decomposition preserves the same banded
structure as the matrix A, enabling an efficient algorithm which overwrites the original matrix with the L factor.
int gsl_linalg_cholesky_band_decomp(gsl_matrix *A)
This function factorizes the symmetric, positive-definite square matrix A into the Cholesky decomposition A =LLT . The input matrix A is given in symmetric banded format, and has dimensions N -by-(p + 1), where pis the lower bandwidth of the matrix. On output, the entries of A are replaced by the entries of the matrix L
in the same format. In addition, the lower right element of A is used to store the matrix 1-norm, used later by
gsl_linalg_cholesky_band_rcond() to calculate the reciprocal condition number.
If the matrix is not positive-definite then the decomposition will fail, returning the error code GSL_EDOM. Whentesting whether a matrix is positive-definite, disable the error handler first to avoid triggering an error.
int gsl_linalg_cholesky_band_solve(const gsl_matrix *LLT, const gsl_vector *b, gsl_vector *x)
int gsl_linalg_cholesky_band_solvem(const gsl_matrix *LLT, const gsl_matrix *B, gsl_matrix *X)
This function solves the symmetric banded system Ax = b (or AX = B) using the Cholesky
decomposition of A held in the matrix LLT which must have been previously computed by
gsl_linalg_cholesky_band_decomp().
int gsl_linalg_cholesky_band_svx(const gsl_matrix *LLT, gsl_vector *x)
int gsl_linalg_cholesky_band_svxm(const gsl_matrix *LLT, gsl_matrix *X)
This function solves the symmetric banded system Ax = b (or AX = B) in-place using theCholesky decomposition of A held in the matrix LLT which must have been previously computed by
gsl_linalg_cholesky_band_decomp(). On input x (or X) should contain the right-hand side b (or B), whichis replaced by the solution on output.
int gsl_linalg_cholesky_band_invert(const gsl_matrix *LLT, gsl_matrix *Ainv)
This function computes the inverse of a symmetric banded matrix from its Cholesky decomposition LLT, whichmust have been previously computed by gsl_linalg_cholesky_band_decomp(). On output, the inverse isstored in Ainv, using both the lower and upper portions.

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int gsl_linalg_cholesky_band_unpack(const gsl_matrix *LLT, gsl_matrix *L)This function unpacks the lower triangular Cholesky factor from LLT and stores it in the lower triangular portionof the N -by-N matrix L. The upper triangular portion of L is not referenced.
int gsl_linalg_cholesky_band_scale(const gsl_matrix *A, gsl_vector *S)
This function calculates a diagonal scaling transformation of the symmetric, positive definite banded matrix A,
such that SAS has a condition number within a factor of N of the matrix of smallest possible condition
number√over all possible diagonal scalings. On output, S contains the scale factors, given by Si = 1/ Aii . For any
Aii ≤ 0, the corresponding scale factor Si is set to 1.
int gsl_linalg_cholesky_band_scale_apply(gsl_matrix *A, const gsl_vector *S)
This function applies the scaling transformation S to the banded symmetric positive definite matrix A. On output,
A is replaced by SAS.
int gsl_linalg_cholesky_band_rcond(const gsl_matrix *LLT, double *rcond, gsl_vector *work)
This function estimates the reciprocal condition number (using the 1-norm) of the symmetric banded positive
definite matrix A, using its Cholesky decomposition provided in LLT. The reciprocal condition number estimate,defined as 1/(||A||1 · ||A−1 ||1 ), is stored in rcond. Additional workspace of size 3N is required in work.

14.22.5 Banded LDLT Decomposition
The routines in this section are designed to factor and solve N -by-N linear systems of the form Ax = b where A is abanded, symmetric, and non-singular matrix with lower bandwidth p. See LDLT Decomposition for more informationon the factorization. The lower triangular factor of the LDLT decomposition preserves the same banded structure as
the matrix A, enabling an efficient algorithm which overwrites the original matrix with the L and D factors.
int gsl_linalg_ldlt_band_decomp(gsl_matrix *A)
This function factorizes the symmetric, non-singular square matrix A into the decomposition A = LDLT . The
input matrix A is given in symmetric banded format, and has dimensions N -by-(p + 1), where p is the lower
bandwidth of the matrix. On output, the entries of A are replaced by the entries of the matrices D and L in thesame format.
If the matrix is singular then the decomposition will fail, returning the error code GSL_EDOM.
int gsl_linalg_ldlt_band_solve(const gsl_matrix *LDLT, const gsl_vector *b, gsl_vector *x)
This function solves the symmetric banded system Ax = b using the LDLT decomposition of A held in thematrix LDLT which must have been previously computed by gsl_linalg_ldlt_band_decomp().
int gsl_linalg_ldlt_band_svx(const gsl_matrix *LDLT, gsl_vector *x)
This function solves the symmetric banded system Ax = b in-place using the LDLT decomposition of A heldin the matrix LDLT which must have been previously computed by gsl_linalg_ldlt_band_decomp(). On
input x should contain the right-hand side b, which is replaced by the solution on output.
int gsl_linalg_ldlt_band_unpack(const gsl_matrix *LDLT, gsl_matrix *L, gsl_vector *D)
This function unpacks the unit lower triangular factor L from LDLT and stores it in the lower triangular portionof the N -by-N matrix L. The upper triangular portion of L is not referenced. The diagonal matrix D is stored in
the vector D.
int gsl_linalg_ldlt_band_rcond(const gsl_matrix *LDLT, double *rcond, gsl_vector *work)
This function estimates the reciprocal condition number (using the 1-norm) of the symmetric banded nonsingularmatrix A, using its LDLT decomposition provided in LDLT. The reciprocal condition number estimate, definedas 1/(||A||1 · ||A−1 ||1 ), is stored in rcond. Additional workspace of size 3N is required in work.

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14.23 Balancing
The process of balancing a matrix applies similarity transformations to make the rows and columns have comparable
norms. This is useful, for example, to reduce roundoff errors in the solution of eigenvalue problems. Balancing a matrix
A consists of replacing A with a similar matrix
A0 = D−1 AD
where D is a diagonal matrix whose entries are powers of the floating point radix.
int gsl_linalg_balance_matrix(gsl_matrix *A, gsl_vector *D)
This function replaces the matrix A with its balanced counterpart and stores the diagonal elements of the similarity
transformation into the vector D.

14.24 Examples
The following program solves the linear system Ax = b. The system to be solved is,   
x00.18 0.60 0.57 0.96
1.0
0.41 0.24 0.99 0.58 x1  2.0   0.14 0.30 0.97 0.66 x2  = 3.0
0.51 0.13 0.19 0.85
4.0x3
and the solution is found using LU decomposition of the matrix A.
#include <stdio.h>#include <gsl/gsl_linalg.h>
int
main (void){
double a_data[] = { 0.18, 0.60, 0.57, 0.96,
0.41, 0.24, 0.99, 0.58,
0.14, 0.30, 0.97, 0.66,
0.51, 0.13, 0.19, 0.85 };double b_data[] = { 1.0, 2.0, 3.0, 4.0 };gsl_matrix_view m= gsl_matrix_view_array (a_data, 4, 4);
gsl_vector_view b= gsl_vector_view_array (b_data, 4);
gsl_vector *x = gsl_vector_alloc (4);int s;
gsl_permutation * p = gsl_permutation_alloc (4);gsl_linalg_LU_decomp (&m.matrix, p, &s);
(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)gsl_linalg_LU_solve (&m.matrix, p, &b.vector, x);
printf ("x = \n");gsl_vector_fprintf (stdout, x, "%g");
gsl_permutation_free (p);gsl_vector_free (x);return 0;}
Here is the output from the program,x =-4.05205-12.6056
1.66091
8.69377
This can be verified by multiplying the solution x by the original matrix A using GNU octave,
octave> A = [ 0.18, 0.60, 0.57, 0.96;
0.41, 0.24, 0.99, 0.58;
0.14, 0.30, 0.97, 0.66;
0.51, 0.13, 0.19, 0.85 ];octave> x = [ -4.05205; -12.6056; 1.66091; 8.69377];
octave> A * x
ans =1.00002.00003.00004.0000This reproduces the original right-hand side vector, b, in accordance with the equation Ax = b.

14.25 References and Further Reading
Further information on the algorithms described in this section can be found in the following book,• G. H. Golub, C. F. Van Loan, “Matrix Computations” (3rd Ed, 1996), Johns Hopkins University Press, ISBN
0-8018-5414-8.
The LAPACK library is described in the following manual,• LAPACK Users’ Guide (Third Edition, 1999), Published by SIAM, ISBN 0-89871-447-8
The LAPACK source code can be found at http://www.netlib.org/lapack, along with an online copy of the users guide.
Further information on recursive Level 3 BLAS algorithms may be found in the following paper,• E. Peise and P. Bientinesi, “Recursive algorithms for dense linear algebra: the ReLAPACK collection”, http://arxiv.org/abs/1602.06763, 2016.
The recursive Level 3 BLAS QR decomposition is described in the following paper,
14.25. References and Further Reading169

GNU Scientific Library, Release 2.7• E. Elmroth and F. G. Gustavson, 2000. Applying recursion to serial and parallel QR factorization leads to better
performance. IBM Journal of Research and Development, 44(4), pp.605-624.
The Modified Golub-Reinsch algorithm is described in the following paper,• T.F. Chan, “An Improved Algorithm for Computing the Singular Value Decomposition”, ACM Transactions onMathematical Software, 8 (1982), pp 72–83.
The Jacobi algorithm for singular value decomposition is described in the following papers,• J.C. Nash, “A one-sided transformation method for the singular value decomposition and algebraic eigenproblem”, Computer Journal, Volume 18, Number 1 (1975), p 74–76• J.C. Nash and S. Shlien “Simple algorithms for the partial singular value decomposition”, Computer Journal,
Volume 30 (1987), p 268–275.
• J. Demmel, K. Veselic, “Jacobi’s Method is more accurate than QR”, Lapack Working Note 15 (LAWN-15),
October 1989. Available from netlib, http://www.netlib.org/lapack/ in the lawns or lawnspdf directories.
The algorithm for estimating a matrix condition number is described in the following paper,• N. J. Higham, “FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications tocondition estimation”, ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

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CHAPTERFIFTEEN

EIGENSYSTEMSThis chapter describes functions for computing eigenvalues and eigenvectors of matrices. There are routines forreal symmetric, real nonsymmetric, complex hermitian, real generalized symmetric-definite, complex generalizedhermitian-definite, and real generalized nonsymmetric eigensystems. Eigenvalues can be computed with or without
eigenvectors. The hermitian and real symmetric matrix algorithms are symmetric bidiagonalization followed by QR
reduction. The nonsymmetric algorithm is the Francis QR double-shift. The generalized nonsymmetric algorithm is
the QZ method due to Moler and Stewart.
The functions described in this chapter are declared in the header file gsl_eigen.h.

15.1 Real Symmetric Matrices
For real symmetric matrices, the library uses the symmetric bidiagonalization and QR reduction method. This is
described in Golub & van Loan, section 8.3. The computed eigenvalues are accurate to an absolute accuracy of ||A||2 ,where  is the machine precision.
type gsl_eigen_symm_workspace
This workspace contains internal parameters used for solving symmetric eigenvalue problems.
gsl_eigen_symm_workspace *gsl_eigen_symm_alloc(const size_t n)This function allocates a workspace for computing eigenvalues of n-by-n real symmetric matrices. The size of
the workspace is O(2n).
void gsl_eigen_symm_free(gsl_eigen_symm_workspace *w)
This function frees the memory associated with the workspace w.
int gsl_eigen_symm(gsl_matrix *A, gsl_vector *eval, gsl_eigen_symm_workspace *w)
This function computes the eigenvalues of the real symmetric matrix A. Additional workspace of the appropriate
size must be provided in w. The diagonal and lower triangular part of A are destroyed during the computation, but
the strict upper triangular part is not referenced. The eigenvalues are stored in the vector eval and are unordered.
type gsl_eigen_symmv_workspace
This workspace contains internal parameters used for solving symmetric eigenvalue and eigenvector problems.
gsl_eigen_symmv_workspace *gsl_eigen_symmv_alloc(const size_t n)This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real symmetric matrices. The size of the workspace is O(4n).
void gsl_eigen_symmv_free(gsl_eigen_symmv_workspace *w)
This function frees the memory associated with the workspace w.
int gsl_eigen_symmv(gsl_matrix *A, gsl_vector *eval, gsl_matrix *evec, gsl_eigen_symmv_workspace *w)
This function computes the eigenvalues and eigenvectors of the real symmetric matrix A. Additional workspaceof the appropriate size must be provided in w. The diagonal and lower triangular part of A are destroyed during
the computation, but the strict upper triangular part is not referenced. The eigenvalues are stored in the vector
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eval and are unordered. The corresponding eigenvectors are stored in the columns of the matrix evec. For
example, the eigenvector in the first column corresponds to the first eigenvalue. The eigenvectors are guaranteed
to be mutually orthogonal and normalised to unit magnitude.

15.2 Complex Hermitian Matrices
For hermitian matrices, the library uses the complex form of the symmetric bidiagonalization and QR reduction method.
type gsl_eigen_herm_workspace
This workspace contains internal parameters used for solving hermitian eigenvalue problems.
gsl_eigen_herm_workspace *gsl_eigen_herm_alloc(const size_t n)This function allocates a workspace for computing eigenvalues of n-by-n complex hermitian matrices. The sizeof the workspace is O(3n).
void gsl_eigen_herm_free(gsl_eigen_herm_workspace *w)
This function frees the memory associated with the workspace w.
int gsl_eigen_herm(gsl_matrix_complex *A, gsl_vector *eval, gsl_eigen_herm_workspace *w)
This function computes the eigenvalues of the complex hermitian matrix A. Additional workspace of the appropriate size must be provided in w. The diagonal and lower triangular part of A are destroyed during the computation,
but the strict upper triangular part is not referenced. The imaginary parts of the diagonal are assumed to be zeroand are not referenced. The eigenvalues are stored in the vector eval and are unordered.
type gsl_eigen_hermv_workspace
This workspace contains internal parameters used for solving hermitian eigenvalue and eigenvector problems.
gsl_eigen_hermv_workspace *gsl_eigen_hermv_alloc(const size_t n)This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n complex hermitianmatrices. The size of the workspace is O(5n).
void gsl_eigen_hermv_free(gsl_eigen_hermv_workspace *w)
This function frees the memory associated with the workspace w.
int gsl_eigen_hermv(gsl_matrix_complex *A, gsl_vector *eval, gsl_matrix_complex *evec,
gsl_eigen_hermv_workspace *w)
This function computes the eigenvalues and eigenvectors of the complex hermitian matrix A. Additional
workspace of the appropriate size must be provided in w. The diagonal and lower triangular part of A are destroyed during the computation, but the strict upper triangular part is not referenced. The imaginary parts of the
diagonal are assumed to be zero and are not referenced. The eigenvalues are stored in the vector eval and areunordered. The corresponding complex eigenvectors are stored in the columns of the matrix evec. For example,
the eigenvector in the first column corresponds to the first eigenvalue. The eigenvectors are guaranteed to be
mutually orthogonal and normalised to unit magnitude.

15.3 Real Nonsymmetric Matrices
The solution of the real nonsymmetric eigensystem problem for a matrix A involves computing the Schur decomposition
A = ZT Z T
where Z is an orthogonal matrix of Schur vectors and T , the Schur form, is quasi upper triangular with diagonal 1by-1 blocks which are real eigenvalues of A, and diagonal 2-by-2 blocks whose eigenvalues are complex conjugate
eigenvalues of A. The algorithm used is the double-shift Francis method.
type gsl_eigen_nonsymm_workspace
This workspace contains internal parameters used for solving nonsymmetric eigenvalue problems.
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gsl_eigen_nonsymm_workspace *gsl_eigen_nonsymm_alloc(const size_t n)This function allocates a workspace for computing eigenvalues of n-by-n real nonsymmetric matrices. The sizeof the workspace is O(2n).
void gsl_eigen_nonsymm_free(gsl_eigen_nonsymm_workspace *w)
This function frees the memory associated with the workspace w.
void gsl_eigen_nonsymm_params(const int compute_t, const int balance, gsl_eigen_nonsymm_workspace *w)
This function sets some parameters which determine how the eigenvalue problem is solved in subsequent calls
to gsl_eigen_nonsymm().
If compute_t is set to 1, the full Schur form T will be computed by gsl_eigen_nonsymm(). If it is set to 0, T
will not be computed (this is the default setting). Computing the full Schur form T requires approximately 1.5–2
times the number of flops.
If balance is set to 1, a balancing transformation is applied to the matrix prior to computing eigenvalues. Thistransformation is designed to make the rows and columns of the matrix have comparable norms, and can resultin more accurate eigenvalues for matrices whose entries vary widely in magnitude. See Balancing for more
information. Note that the balancing transformation does not preserve the orthogonality of the Schur vectors, so
if you wish to compute the Schur vectors with gsl_eigen_nonsymm_Z() you will obtain the Schur vectors of
the balanced matrix instead of the original matrix. The relationship will be
T = QT D−1 ADQ
where Q is the matrix of Schur vectors for the balanced matrix, and D is the balancing transformation. Thengsl_eigen_nonsymm_Z() will compute a matrix Z which satisfies
T = Z −1 AZ
with Z = DQ. Note that Z will not be orthogonal. For this reason, balancing is not performed by default.
int gsl_eigen_nonsymm(gsl_matrix *A, gsl_vector_complex *eval, gsl_eigen_nonsymm_workspace *w)
This function computes the eigenvalues of the real nonsymmetric matrix A and stores them in the vector eval. If
T is desired, it is stored in the upper portion of A on output. Otherwise, on output, the diagonal of A will containthe 1-by-1 real eigenvalues and 2-by-2 complex conjugate eigenvalue systems, and the rest of A is destroyed. In
rare cases, this function may fail to find all eigenvalues. If this happens, an error code is returned and the numberof converged eigenvalues is stored in w->n_evals. The converged eigenvalues are stored in the beginning ofeval.
int gsl_eigen_nonsymm_Z(gsl_matrix *A, gsl_vector_complex *eval, gsl_matrix *Z,
gsl_eigen_nonsymm_workspace *w)
This function is identical to gsl_eigen_nonsymm() except that it also computes the Schur vectors and storesthem into Z.
type gsl_eigen_nonsymmv_workspace
This workspace contains internal parameters used for solving nonsymmetric eigenvalue and eigenvector problems.
gsl_eigen_nonsymmv_workspace *gsl_eigen_nonsymmv_alloc(const size_t n)This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real nonsymmetricmatrices. The size of the workspace is O(5n).
void gsl_eigen_nonsymmv_free(gsl_eigen_nonsymmv_workspace *w)
This function frees the memory associated with the workspace w.
void gsl_eigen_nonsymmv_params(const int balance, gsl_eigen_nonsymm_workspace *w)
This function sets parameters which determine how the eigenvalue problem is solved in subsequent calls to
gsl_eigen_nonsymmv(). If balance is set to 1, a balancing transformation is applied to the matrix. See
gsl_eigen_nonsymm_params() for more information. Balancing is turned off by default since it does notpreserve the orthogonality of the Schur vectors.
15.3. Real Nonsymmetric Matrices

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int gsl_eigen_nonsymmv(gsl_matrix *A, gsl_vector_complex *eval, gsl_matrix_complex *evec,
gsl_eigen_nonsymmv_workspace *w)
This function computes eigenvalues and right eigenvectors of the n-by-n real nonsymmetric matrix A. It first calls
gsl_eigen_nonsymm() to compute the eigenvalues, Schur form T , and Schur vectors. Then it finds eigenvectorsof T and backtransforms them using the Schur vectors. The Schur vectors are destroyed in the process, but can besaved by using gsl_eigen_nonsymmv_Z(). The computed eigenvectors are normalized to have unit magnitude.
On output, the upper portion of A contains the Schur form T . If gsl_eigen_nonsymm() fails, no eigenvectors
are computed, and an error code is returned.
int gsl_eigen_nonsymmv_Z(gsl_matrix *A, gsl_vector_complex *eval, gsl_matrix_complex *evec, gsl_matrix *Z,
gsl_eigen_nonsymmv_workspace *w)
This function is identical to gsl_eigen_nonsymmv() except that it also saves the Schur vectors into Z.

15.4 Real Generalized Symmetric-Definite Eigensystems
The real generalized symmetric-definite eigenvalue problem is to find eigenvalues λ and eigenvectors x such that
Ax = λBx
where A and B are symmetric matrices, and B is positive-definite. This problem reduces to the standard symmetriceigenvalue problem by applying the Cholesky decomposition to B:
Ax = λBx
Ax = λLLT x

L−1 AL−T LT x = λLT x
Therefore, the problem becomes Cy = λy where C = L−1 AL−T is symmetric, and y = LT x. The standard
symmetric eigensolver can be applied to the matrix C. The resulting eigenvectors are backtransformed to find thevectors of the original problem. The eigenvalues and eigenvectors of the generalized symmetric-definite eigenproblem
are always real.
type gsl_eigen_gensymm_workspace
This workspace contains internal parameters used for solving generalized symmetric eigenvalue problems.
gsl_eigen_gensymm_workspace *gsl_eigen_gensymm_alloc(const size_t n)This function allocates a workspace for computing eigenvalues of n-by-n real generalized symmetric-definite
eigensystems. The size of the workspace is O(2n).
void gsl_eigen_gensymm_free(gsl_eigen_gensymm_workspace *w)
This function frees the memory associated with the workspace w.
int gsl_eigen_gensymm(gsl_matrix *A, gsl_matrix *B, gsl_vector *eval, gsl_eigen_gensymm_workspace *w)
This function computes the eigenvalues of the real generalized symmetric-definite matrix pair (A, B), and storesthem in eval, using the method outlined above. On output, B contains its Cholesky decomposition and A isdestroyed.
type gsl_eigen_gensymmv_workspace
This workspace contains internal parameters used for solving generalized symmetric eigenvalue and eigenvector
problems.
gsl_eigen_gensymmv_workspace *gsl_eigen_gensymmv_alloc(const size_t n)This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real generalized
symmetric-definite eigensystems. The size of the workspace is O(4n).
void gsl_eigen_gensymmv_free(gsl_eigen_gensymmv_workspace *w)
This function frees the memory associated with the workspace w.
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int gsl_eigen_gensymmv(gsl_matrix *A, gsl_matrix *B, gsl_vector *eval, gsl_matrix *evec,
gsl_eigen_gensymmv_workspace *w)
This function computes the eigenvalues and eigenvectors of the real generalized symmetric-definite matrix pair(A, B), and stores them in eval and evec respectively. The computed eigenvectors are normalized to have unit
magnitude. On output, B contains its Cholesky decomposition and A is destroyed.

15.5 Complex Generalized Hermitian-Definite Eigensystems
The complex generalized hermitian-definite eigenvalue problem is to find eigenvalues λ and eigenvectors x such that
Ax = λBx
where A and B are hermitian matrices, and B is positive-definite. Similarly to the real case, this can be reduced to
Cy = λy where C = L−1 AL−† is hermitian, and y = L† x. The standard hermitian eigensolver can be applied to thematrix C. The resulting eigenvectors are backtransformed to find the vectors of the original problem. The eigenvaluesof the generalized hermitian-definite eigenproblem are always real.
type gsl_eigen_genherm_workspace
This workspace contains internal parameters used for solving generalized hermitian eigenvalue problems.
gsl_eigen_genherm_workspace *gsl_eigen_genherm_alloc(const size_t n)This function allocates a workspace for computing eigenvalues of n-by-n complex generalized hermitian-definite
eigensystems. The size of the workspace is O(3n).
void gsl_eigen_genherm_free(gsl_eigen_genherm_workspace *w)
This function frees the memory associated with the workspace w.
int gsl_eigen_genherm(gsl_matrix_complex *A, gsl_matrix_complex *B, gsl_vector *eval,
gsl_eigen_genherm_workspace *w)
This function computes the eigenvalues of the complex generalized hermitian-definite matrix pair (A, B), andstores them in eval, using the method outlined above. On output, B contains its Cholesky decomposition and A
is destroyed.
type gsl_eigen_genhermv_workspace
This workspace contains internal parameters used for solving generalized hermitian eigenvalue and eigenvector
problems.
gsl_eigen_genhermv_workspace *gsl_eigen_genhermv_alloc(const size_t n)This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n complex generalizedhermitian-definite eigensystems. The size of the workspace is O(5n).
void gsl_eigen_genhermv_free(gsl_eigen_genhermv_workspace *w)
This function frees the memory associated with the workspace w.
int gsl_eigen_genhermv(gsl_matrix_complex *A, gsl_matrix_complex *B, gsl_vector *eval, gsl_matrix_complex*evec, gsl_eigen_genhermv_workspace *w)
This function computes the eigenvalues and eigenvectors of the complex generalized hermitian-definite matrix
pair (A, B), and stores them in eval and evec respectively. The computed eigenvectors are normalized to haveunit magnitude. On output, B contains its Cholesky decomposition and A is destroyed.

15.5. Complex Generalized Hermitian-Definite Eigensystems

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15.6 Real Generalized Nonsymmetric Eigensystems
Given two square matrices (A, B), the generalized nonsymmetric eigenvalue problem is to find eigenvalues λ andeigenvectors x such that
Ax = λBx
We may also define the problem as finding eigenvalues µ and eigenvectors y such that
µAy = By
Note that these two problems are equivalent (with λ = 1/µ) if neither λ nor µ is zero. If say, λ is zero, then it is stilla well defined eigenproblem, but its alternate problem involving µ is not. Therefore, to allow for zero (and infinite)eigenvalues, the problem which is actually solved isβAx = αBx
The eigensolver routines below will return two values α and β and leave it to the user to perform the divisions λ = α/βand µ = β/α.
If the determinant of the matrix pencil A − λB is zero for all λ, the problem is said to be singular; otherwise it is called
regular. Singularity normally leads to some α = β = 0 which means the eigenproblem is ill-conditioned and generally
does not have well defined eigenvalue solutions. The routines below are intended for regular matrix pencils and could
yield unpredictable results when applied to singular pencils.
The solution of the real generalized nonsymmetric eigensystem problem for a matrix pair (A, B) involves computing
the generalized Schur decomposition
A = QSZ T
B = QT Z T
where Q and Z are orthogonal matrices of left and right Schur vectors respectively, and (S, T ) is the generalized Schurform whose diagonal elements give the α and β values. The algorithm used is the QZ method due to Moler and Stewart(see references).
type gsl_eigen_gen_workspace
This workspace contains internal parameters used for solving generalized eigenvalue problems.
gsl_eigen_gen_workspace *gsl_eigen_gen_alloc(const size_t n)This function allocates a workspace for computing eigenvalues of n-by-n real generalized nonsymmetric eigensystems. The size of the workspace is O(n).
void gsl_eigen_gen_free(gsl_eigen_gen_workspace *w)
This function frees the memory associated with the workspace w.
void gsl_eigen_gen_params(const int compute_s, const int compute_t, const int balance,
gsl_eigen_gen_workspace *w)
This function sets some parameters which determine how the eigenvalue problem is solved in subsequent calls
to gsl_eigen_gen().
If compute_s is set to 1, the full Schur form S will be computed by gsl_eigen_gen(). If it is set to 0, S
will not be computed (this is the default setting). S is a quasi upper triangular matrix with 1-by-1 and 2-by-2
blocks on its diagonal. 1-by-1 blocks correspond to real eigenvalues, and 2-by-2 blocks correspond to complex
eigenvalues.
If compute_t is set to 1, the full Schur form T will be computed by gsl_eigen_gen(). If it is set to 0, T will
not be computed (this is the default setting). T is an upper triangular matrix with non-negative elements on its
diagonal. Any 2-by-2 blocks in S will correspond to a 2-by-2 diagonal block in T .
The balance parameter is currently ignored, since generalized balancing is not yet implemented.
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int gsl_eigen_gen(gsl_matrix *A, gsl_matrix *B, gsl_vector_complex *alpha, gsl_vector *beta,
gsl_eigen_gen_workspace *w)
This function computes the eigenvalues of the real generalized nonsymmetric matrix pair (A, B), and stores themas pairs in (alpha, beta), where alpha is complex and beta is real. If βi is non-zero, then λ = αi /βi is aneigenvalue. Likewise, if αi is non-zero, then µ = βi /αi is an eigenvalue of the alternate problem µAy = By.
The elements of beta are normalized to be non-negative.
If S is desired, it is stored in A on output. If T is desired, it is stored in B on output. The ordering of eigenvaluesin (alpha, beta) follows the ordering of the diagonal blocks in the Schur forms S and T . In rare cases, thisfunction may fail to find all eigenvalues. If this occurs, an error code is returned.
int gsl_eigen_gen_QZ(gsl_matrix *A, gsl_matrix *B, gsl_vector_complex *alpha, gsl_vector *beta, gsl_matrix*Q, gsl_matrix *Z, gsl_eigen_gen_workspace *w)
This function is identical to gsl_eigen_gen() except that it also computes the left and right Schur vectors andstores them into Q and Z respectively.
type gsl_eigen_genv_workspace
This workspace contains internal parameters used for solving generalized eigenvalue and eigenvector problems.
gsl_eigen_genv_workspace *gsl_eigen_genv_alloc(const size_t n)This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real generalized nonsymmetric eigensystems. The size of the workspace is O(7n).
void gsl_eigen_genv_free(gsl_eigen_genv_workspace *w)
This function frees the memory associated with the workspace w.
int gsl_eigen_genv(gsl_matrix *A, gsl_matrix *B, gsl_vector_complex *alpha, gsl_vector *beta,
gsl_matrix_complex *evec, gsl_eigen_genv_workspace *w)
This function computes eigenvalues and right eigenvectors of the n-by-n real generalized nonsymmetric matrix
pair (A, B). The eigenvalues are stored in (alpha, beta) and the eigenvectors are stored in evec. It first calls
gsl_eigen_gen() to compute the eigenvalues, Schur forms, and Schur vectors. Then it finds eigenvectorsof the Schur forms and backtransforms them using the Schur vectors. The Schur vectors are destroyed in theprocess, but can be saved by using gsl_eigen_genv_QZ(). The computed eigenvectors are normalized to haveunit magnitude. On output, (A, B) contains the generalized Schur form (S, T ). If gsl_eigen_gen() fails, noeigenvectors are computed, and an error code is returned.
int gsl_eigen_genv_QZ(gsl_matrix *A, gsl_matrix *B, gsl_vector_complex *alpha, gsl_vector *beta,
gsl_matrix_complex *evec, gsl_matrix *Q, gsl_matrix *Z, gsl_eigen_genv_workspace*w)
This function is identical to gsl_eigen_genv() except that it also computes the left and right Schur vectors
and stores them into Q and Z respectively.

15.7 Sorting Eigenvalues and Eigenvectors
int gsl_eigen_symmv_sort(gsl_vector *eval, gsl_matrix *evec, gsl_eigen_sort_t sort_type)This function simultaneously sorts the eigenvalues stored in the vector eval and the corresponding real eigenvectors stored in the columns of the matrix evec into ascending or descending order according to the value ofthe parameter sort_type,
type gsl_eigen_sort_t

GSL_EIGEN_SORT_VAL_ASC
GSL_EIGEN_SORT_VAL_DESC
GSL_EIGEN_SORT_ABS_ASC
GSL_EIGEN_SORT_ABS_DESC
15.7. Sorting Eigenvalues and Eigenvectors

ascending order in numerical value
descending order in numerical value
ascending order in magnitude
descending order in magnitude177GNU Scientific Library, Release 2.7

int gsl_eigen_hermv_sort(gsl_vector *eval, gsl_matrix_complex *evec, gsl_eigen_sort_t sort_type)This function simultaneously sorts the eigenvalues stored in the vector eval and the corresponding complex
eigenvectors stored in the columns of the matrix evec into ascending or descending order according to the valueof the parameter sort_type as shown above.
int gsl_eigen_nonsymmv_sort(gsl_vector_complex *eval, gsl_matrix_complex *evec, gsl_eigen_sort_t
sort_type)This function simultaneously sorts the eigenvalues stored in the vector eval and the corresponding complex eigenvectors stored in the columns of the matrix evec into ascending or descending order according to the value of the parameter sort_type as shown above. Only GSL_EIGEN_SORT_ABS_ASC andGSL_EIGEN_SORT_ABS_DESC are supported due to the eigenvalues being complex.
int gsl_eigen_gensymmv_sort(gsl_vector *eval, gsl_matrix *evec, gsl_eigen_sort_t sort_type)This function simultaneously sorts the eigenvalues stored in the vector eval and the corresponding real eigenvectors stored in the columns of the matrix evec into ascending or descending order according to the value ofthe parameter sort_type as shown above.
int gsl_eigen_genhermv_sort(gsl_vector *eval, gsl_matrix_complex *evec, gsl_eigen_sort_t sort_type)This function simultaneously sorts the eigenvalues stored in the vector eval and the corresponding complex
eigenvectors stored in the columns of the matrix evec into ascending or descending order according to the valueof the parameter sort_type as shown above.
int gsl_eigen_genv_sort(gsl_vector_complex *alpha, gsl_vector *beta, gsl_matrix_complex *evec,
gsl_eigen_sort_t sort_type)This function simultaneously sorts the eigenvalues stored in the vectors (alpha, beta) and the corresponding complex eigenvectors stored in the columns of the matrix evec into ascending or descending order according to the value of the parameter sort_type as shown above. Only GSL_EIGEN_SORT_ABS_ASC andGSL_EIGEN_SORT_ABS_DESC are supported due to the eigenvalues being complex.

15.8 Examples
The following program computes the eigenvalues and eigenvectors of the 4-th order Hilbert matrix, H(i, j) = 1/(i +j + 1).
#include <stdio.h>#include <gsl/gsl_math.h>#include <gsl/gsl_eigen.h>
intmain (void){
double data[] = { 1.0 , 1/2.0, 1/3.0, 1/4.0,
1/2.0, 1/3.0, 1/4.0, 1/5.0,
1/3.0, 1/4.0, 1/5.0, 1/6.0,
1/4.0, 1/5.0, 1/6.0, 1/7.0 };gsl_matrix_view m= gsl_matrix_view_array (data, 4, 4);
gsl_vector *eval = gsl_vector_alloc (4);gsl_matrix *evec = gsl_matrix_alloc (4, 4);
gsl_eigen_symmv_workspace * w =gsl_eigen_symmv_alloc (4);(continues on next page)

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gsl_eigen_symmv (&m.matrix, eval, evec, w);
gsl_eigen_symmv_free (w);gsl_eigen_symmv_sort (eval, evec,
GSL_EIGEN_SORT_ABS_ASC);{
int i;for (i = 0; i < 4; i++){double eval_i= gsl_vector_get (eval, i);
gsl_vector_view evec_i= gsl_matrix_column (evec, i);printf ("eigenvalue = %g\n", eval_i);
printf ("eigenvector = \n");gsl_vector_fprintf (stdout,&evec_i.vector, "%g");}}
gsl_vector_free (eval);
gsl_matrix_free (evec);
return 0;}
Here is the beginning of the output from the program:$ ./a.out
eigenvalue = 9.67023e-05
eigenvector =-0.0291933
0.328712-0.791411
0.514553...
This can be compared with the corresponding output from GNU octave:
octave> [v,d] = eig(hilb(4));
octave> diag(d)ans =
9.6702e-05
6.7383e-03
1.6914e-01
1.5002e+00(continues on next page)

15.8. Examples

179

GNU Scientific Library, Release 2.7(continued from previous page)octave> v
v =0.029193-0.328712
0.791411-0.514553

0.179186-0.741918
0.100228
0.638283-0.582076
0.370502
0.509579
0.514048

0.792608
0.451923
0.322416
0.252161

Note that the eigenvectors can differ by a change of sign, since the sign of an eigenvector is arbitrary.
The following program illustrates the use of the nonsymmetric eigensolver, by computing the eigenvalues and eigenwith x = (−1, −2, 3, 4).
vectors of the Vandermonde matrix V (x; i, j) = xn−ji#include <stdio.h>#include <gsl/gsl_math.h>#include <gsl/gsl_eigen.h>
intmain (void){
double data[] = { -1.0, 1.0, -1.0, 1.0,
-8.0, 4.0, -2.0, 1.0,
27.0, 9.0, 3.0, 1.0,
64.0, 16.0, 4.0, 1.0 };gsl_matrix_view m= gsl_matrix_view_array (data, 4, 4);
gsl_vector_complex *eval = gsl_vector_complex_alloc (4);gsl_matrix_complex *evec = gsl_matrix_complex_alloc (4, 4);
gsl_eigen_nonsymmv_workspace * w =gsl_eigen_nonsymmv_alloc (4);gsl_eigen_nonsymmv (&m.matrix, eval, evec, w);
gsl_eigen_nonsymmv_free (w);gsl_eigen_nonsymmv_sort (eval, evec,
GSL_EIGEN_SORT_ABS_DESC);{
int i, j;for (i = 0; i < 4; i++){gsl_complex eval_i= gsl_vector_complex_get (eval, i);
gsl_vector_complex_view evec_i= gsl_matrix_complex_column (evec, i);printf ("eigenvalue = %g + %gi\n",(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)

GSL_REAL(eval_i), GSL_IMAG(eval_i));
printf ("eigenvector = \n");for (j = 0; j < 4; ++j){
gsl_complex z =gsl_vector_complex_get(&evec_i.vector, j);printf("%g + %gi\n", GSL_REAL(z), GSL_IMAG(z));}}}
gsl_vector_complex_free(eval);
gsl_matrix_complex_free(evec);
return 0;}
Here is the beginning of the output from the program:$ ./a.out
eigenvalue = -6.41391 + 0i
eigenvector =-0.0998822 + 0i
-0.111251 + 0i
0.292501 + 0i
0.944505 + 0i
eigenvalue = 5.54555 + 3.08545i
eigenvector =-0.043487 + -0.0076308i
0.0642377 + -0.142127i-0.515253 + 0.0405118i-0.840592 + -0.00148565i...
This can be compared with the corresponding output from GNU octave:
octave> [v,d] = eig(vander([-1 -2 3 4]));octave> diag(d)ans =-6.4139 + 0.0000i
5.5456 + 3.0854i
5.5456 - 3.0854i
2.3228 + 0.0000i
octave> v
v =Columns 1 through 3:-0.09988 + 0.00000i-0.11125 + 0.00000i-0.04350 - 0.00755i
0.06399 - 0.14224i-0.04350 + 0.00755i
0.06399 + 0.14224i(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)

0.29250 + 0.00000i
0.94451 + 0.00000i-0.51518 + 0.04142i-0.84059 + 0.00000i-0.51518 - 0.04142i-0.84059 - 0.00000i

Column 4:-0.14493 + 0.00000i
0.35660 + 0.00000i
0.91937 + 0.00000i
0.08118 + 0.00000i
Note that the eigenvectors corresponding to the eigenvalue 5.54555 + 3.08545i differ by the multiplicative constant0.9999984 + 0.0017674i which is an arbitrary phase factor of magnitude 1.

15.9 References and Further Reading
Further information on the algorithms described in this section can be found in the following book,• G. H. Golub, C. F. Van Loan, “Matrix Computations” (3rd Ed, 1996), Johns Hopkins University Press, ISBN
0-8018-5414-8.
Further information on the generalized eigensystems QZ algorithm can be found in this paper,• C. Moler, G. Stewart, “An Algorithm for Generalized Matrix Eigenvalue Problems”, SIAM J. Numer. Anal., Vol
10, No 2, 1973.
Eigensystem routines for very large matrices can be found in the Fortran library LAPACK. The LAPACK library is
described in,• LAPACK Users’ Guide (Third Edition, 1999), Published by SIAM, ISBN 0-89871-447-8.
The LAPACK source code can be found at the website http://www.netlib.org/lapack along with an online copy of theusers guide.

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CHAPTERSIXTEEN

FAST FOURIER TRANSFORMS (FFTS)

This chapter describes functions for performing Fast Fourier Transforms (FFTs). The library includes radix-2 routines(for lengths which are a power of two) and mixed-radix routines (which work for any length). For efficiency there are
separate versions of the routines for real data and for complex data. The mixed-radix routines are a reimplementation of
the FFTPACK library of Paul Swarztrauber. Fortran code for FFTPACK is available on Netlib (FFTPACK also includes
some routines for sine and cosine transforms but these are currently not available in GSL). For details and derivationsof the underlying algorithms consult the document “GSL FFT Algorithms” (see References and Further Reading)

16.1 Mathematical Definitions
Fast Fourier Transforms are efficient algorithms for calculating the discrete Fourier transform (DFT),
xj =

n−1
X

zk exp(−2πijk/n)

k=0

The DFT usually arises as an approximation to the continuous Fourier transform when functions are sampled at discreteintervals in space or time. The naive evaluation of the discrete Fourier transform is a matrix-vector multiplication W ~z.
A general matrix-vector multiplication takes O(n2 ) operations for n data-points. Fast Fourier transform algorithms
use a divide-and-conquer strategy to factorize the matrix W into smaller sub-matrices, corresponding to the integer
factors
Pof the length n. If n can be factorized into a product of integers f1 f2 . . . fm then the DFT can be computed inO(n fi ) operations. For a radix-2 FFT this gives an operation count of O(n log2 n).
All the FFT functions offer three types of transform: forwards, inverse and backwards, based on the same mathematicaldefinitions. The definition of the forward Fourier transform, x = FFT(z), is,
xj =

n−1
X

zk exp(−2πijk/n)

k=0

and the definition of the inverse Fourier transform, x = IFFT(z), is,
n−1

zj =

1X
xk exp(2πijk/n).
n
k=0

The factor of 1/n makes this a true inverse. For example, a call to gsl_fft_complex_forward() followed by a callto gsl_fft_complex_inverse() should return the original data (within numerical errors).
In general there are two possible choices for the sign of the exponential in the transform/ inverse-transform pair. GSL
follows the same convention as FFTPACK, using a negative exponential for the forward transform. The advantage ofthis convention is that the inverse transform recreates the original function with simple Fourier synthesis. Numerical
Recipes uses the opposite convention, a positive exponential in the forward transform.

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The backwards FFT is simply our terminology for an unscaled version of the inverse FFT,zjbackwards =n−1
Xxk exp(2πijk/n)

k=0

When the overall scale of the result is unimportant it is often convenient to use the backwards FFT instead of the inverse
to save unnecessary divisions.

16.2 Overview of complex data FFTs
The inputs and outputs for the complex FFT routines are packed arrays of floating point numbers. In a packed array
the real and imaginary parts of each complex number are placed in alternate neighboring elements. For example, thefollowing definition of a packed array of length 6:
double x[3*2];
gsl_complex_packed_array data = x;
can be used to hold an array of three complex numbers, z[3], in the following way:data[0] = Re(z[0])data[1] = Im(z[0])data[2] = Re(z[1])data[3] = Im(z[1])data[4] = Re(z[2])data[5] = Im(z[2])The array indices for the data have the same ordering as those in the definition of the DFT—i.e. there are no index
transformations or permutations of the data.
A stride parameter allows the user to perform transforms on the elements z[stride*i] instead of z[i]. A stride
greater than 1 can be used to take an in-place FFT of the column of a matrix. A stride of 1 accesses the array without
any additional spacing between elements.
To perform an FFT on a vector argument, such as gsl_vector_complex * v, use the following definitions (or theirequivalents) when calling the functions described in this chapter:
gsl_complex_packed_array data = v->data;
size_t stride = v->stride;size_t n = v->size;
For physical applications it is important to remember that the index appearing in the DFT does not correspond directly
to a physical frequency. If the time-step of the DFT is ∆ then the frequency-domain includes both positive and negative
frequencies, ranging from −1/(2∆) through 0 to +1/(2∆). The positive frequencies are stored from the beginning of
the array up to the middle, and the negative frequencies are stored backwards from the end of the array.
Here is a table which shows the layout of the array data, and the correspondence between the time-domain data z, andthe frequency-domain data x:
index

z

x = FFT(z)

0
1
2
.

z(t = 0)z(t = 1)z(t = 2)........

x(f = 0)x(f = 1/(n Delta))
x(f = 2/(n Delta))..................
(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)n/2z(t = n/2)

.
n-3
n-2
n-1

........
z(t = n-3)z(t = n-2)z(t = n-1)x(f = +1/(2 Delta),-1/(2 Delta))..................
x(f = -3/(n Delta))
x(f = -2/(n Delta))
x(f = -1/(n Delta))

When n is even the location n/2 contains the most positive and negative frequencies (+1/(2∆), −1/(2∆)) which areequivalent. If n is odd then general structure of the table above still applies, but n/2 does not appear.

16.3 Radix-2 FFT routines for complex data
The radix-2 algorithms described in this section are simple and compact, although not necessarily the most efficient.
They use the Cooley-Tukey algorithm to compute in-place complex FFTs for lengths which are a power of 2—noadditional storage is required. The corresponding self-sorting mixed-radix routines offer better performance at theexpense of requiring additional working space.
All the functions described in this section are declared in the header file gsl_fft_complex.h.
int gsl_fft_complex_radix2_forward(gsl_complex_packed_array data, size_t stride, size_t n)
int gsl_fft_complex_radix2_transform(gsl_complex_packed_array data, size_t stride, size_t n,
gsl_fft_direction sign)
int gsl_fft_complex_radix2_backward(gsl_complex_packed_array data, size_t stride, size_t n)
int gsl_fft_complex_radix2_inverse(gsl_complex_packed_array data, size_t stride, size_t n)
These functions compute forward, backward and inverse FFTs of length n with stride stride, on the packed
complex array data using an in-place radix-2 decimation-in-time algorithm. The length of the transform n is
restricted to powers of two. For the transform version of the function the sign argument can be either forward(−1) or backward (+1).
The functions return a value of GSL_SUCCESS if no errors were detected, or GSL_EDOM if the length of the datan is not a power of two.
int gsl_fft_complex_radix2_dif_forward(gsl_complex_packed_array data, size_t stride, size_t n)
int gsl_fft_complex_radix2_dif_transform(gsl_complex_packed_array data, size_t stride, size_t n,
gsl_fft_direction sign)
int gsl_fft_complex_radix2_dif_backward(gsl_complex_packed_array data, size_t stride, size_t n)
int gsl_fft_complex_radix2_dif_inverse(gsl_complex_packed_array data, size_t stride, size_t n)
These are decimation-in-frequency versions of the radix-2 FFT functions.
Here is an example program which computes the FFT of a short pulse in a sample of length 128. To make the resulting
Fourier transform real the pulse is defined for equal positive and negative times (−10 . . . 10), where the negative times
wrap around the end of the array.
#include <stdio.h>#include <math.h>#include <gsl/gsl_errno.h>(continues on next page)

16.3. Radix-2 FFT routines for complex data

185

GNU Scientific Library, Release 2.7(continued from previous page)#include <gsl/gsl_fft_complex.h>#define REAL(z,i) ((z)[2*(i)])#define IMAG(z,i) ((z)[2*(i)+1])
int
main (void){
int i; double data[2*128];for (i = 0; i < 128; i++){
REAL(data,i) = 0.0; IMAG(data,i) = 0.0;}
REAL(data,0) = 1.0;
for (i = 1; i <= 10; i++){
REAL(data,i) = REAL(data,128-i) = 1.0;}
for (i = 0; i < 128; i++){printf ("%d %e %e\n", i,
REAL(data,i), IMAG(data,i));}printf ("\n\n");gsl_fft_complex_radix2_forward (data, 1, 128);for (i = 0; i < 128; i++){printf ("%d %e %e\n", i,
REAL(data,i)/sqrt(128),
IMAG(data,i)/sqrt(128));}
return 0;}
Note that we have assumed that the program is using the default error handler (which calls abort() forany errors).
If you are not using a safe error handler you would need to check the return status of
gsl_fft_complex_radix2_forward().
√
The transformed data is rescaled by 1/ n so that it fits on the same plot as the input. Only the real part is shown, by
the choice of the input data the imaginary part is zero. Allowing for the wrap-around of negative times at t = 128, andworking in units of k/n, the DFT approximates the continuum Fourier transform, giving a modulated sine function.
Z +ae−2πikx dx =−asin(2πka)
πkThe output of the example program is plotted in Fig. 16.1.

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Fig. 16.1: A pulse and its discrete Fourier transform, output from the example program.

16.3. Radix-2 FFT routines for complex data187

GNU Scientific Library, Release 2.7

16.4 Mixed-radix FFT routines for complex data
This section describes mixed-radix FFT algorithms for complex data. The mixed-radix functions work for FFTs of any
length. They are a reimplementation of Paul Swarztrauber’s Fortran FFTPACK library. The theory is explained in thereview article “Self-sorting Mixed-radix FFTs” by Clive Temperton. The routines here use the same indexing scheme
and basic algorithms as FFTPACK.
The mixed-radix algorithm is based on sub-transform modules—highly optimized small length FFTs which are combined to create larger FFTs. There are efficient modules for factors of 2, 3, 4, 5, 6 and 7. The modules for the composite
factors of 4 and 6 are faster than combining the modules for 2 ∗ 2 and 2 ∗ 3.
For factors which are not implemented as modules there is a fall-back to a general length-n module which uses Singleton’s method for efficiently computing a DFT. This module is O(n2 ), and slower than a dedicated module would be but
works for any length n. Of course, lengths which use the general length-n module will still be factorized as much as
possible. For example, a length of 143 will be factorized into 11 ∗ 13. Large prime factors are the worst case scenario,
e.g. as found in n = 2 ∗ 3 ∗ 99991, and should be avoided because their O(n2 ) scaling will dominate the run-time(consult the document “GSL FFT Algorithms” included in the GSL distribution if you encounter this problem).
The mixed-radix initialization function gsl_fft_complex_wavetable_alloc() returns the list of factors chosen
by the library for a given length n. It can be used to checkPhow well the length has been factorized, and estimate therun-time. To a first approximation the run-time scales as n fi , where the fi are the factors of n. For programs underuser control you may wish to issue a warning that the transform will be slow when the length is poorly factorized. If
you frequently encounter data lengths which cannot be factorized using the existing small-prime modules consult “GSLFFT Algorithms” for details on adding support for other factors.
All the functions described in this section are declared in the header file gsl_fft_complex.h.
gsl_fft_complex_wavetable *gsl_fft_complex_wavetable_alloc(size_t n)This function prepares a trigonometric lookup table for a complex FFT of length n. The function returns a pointerto the newly allocated gsl_fft_complex_wavetable if no errors were detected, and a null pointer in the caseof error. The length n is factorized into a product of subtransforms, and the factors and their trigonometriccoefficients are stored in the wavetable. The trigonometric coefficients are computed using direct calls to sin
and cos, for accuracy. Recursion relations could be used to compute the lookup table faster, but if an application
performs many FFTs of the same length then this computation is a one-off overhead which does not affect the
final throughput.
The wavetable structure can be used repeatedly for any transform of the same length. The table is not modified
by calls to any of the other FFT functions. The same wavetable can be used for both forward and backward (orinverse) transforms of a given length.
void gsl_fft_complex_wavetable_free(gsl_fft_complex_wavetable *wavetable)
This function frees the memory associated with the wavetable wavetable. The wavetable can be freed if nofurther FFTs of the same length will be needed.
These functions operate on a gsl_fft_complex_wavetable structure which contains internal parameters for the FFT.
It is not necessary to set any of the components directly but it can sometimes be useful to examine them. For example,
the chosen factorization of the FFT length is given and can be used to provide an estimate of the run-time or numericalerror. The wavetable structure is declared in the header file gsl_fft_complex.h.
type gsl_fft_complex_wavetable
This is a structure that holds the factorization and trigonometric lookup tables for the mixed radix fft algorithm.
It has the following components:

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GNU Scientific Library, Release 2.7

size_t nsize_t nf
size_t factor[64]gsl_complex * trig
gsl_complex *twiddle[64]This is the number of complex data points
This is the number of factors that the length n was decomposed into.
This is the array of factors. Only the first nf elements are used.
This is a pointer to a preallocated trigonometric lookup table of n complex
elements.
This is an array of pointers into trig, giving the twiddle factors for eachpass.

type gsl_fft_complex_workspace
The mixed radix algorithms require additional working space to hold the intermediate steps of the transform.
gsl_fft_complex_workspace *gsl_fft_complex_workspace_alloc(size_t n)This function allocates a workspace for a complex transform of length n.
void gsl_fft_complex_workspace_free(gsl_fft_complex_workspace *workspace)
This function frees the memory associated with the workspace workspace. The workspace can be freed if nofurther FFTs of the same length will be needed.
The following functions compute the transform,
int gsl_fft_complex_forward(gsl_complex_packed_array data, size_t stride, size_t n, const
gsl_fft_complex_wavetable *wavetable, gsl_fft_complex_workspace *work)
int gsl_fft_complex_transform(gsl_complex_packed_array data, size_t stride, size_t n, const
gsl_fft_complex_wavetable *wavetable, gsl_fft_complex_workspace *work,
gsl_fft_direction sign)
int gsl_fft_complex_backward(gsl_complex_packed_array data, size_t stride, size_t n, const
gsl_fft_complex_wavetable *wavetable, gsl_fft_complex_workspace *work)
int gsl_fft_complex_inverse(gsl_complex_packed_array data, size_t stride, size_t n, const
gsl_fft_complex_wavetable *wavetable, gsl_fft_complex_workspace *work)
These functions compute forward, backward and inverse FFTs of length n with stride stride, on the packed complex array data, using a mixed radix decimation-in-frequency algorithm. There is no restriction on the lengthn. Efficient modules are provided for subtransforms of length 2, 3, 4, 5, 6 and 7. Any remaining factors arecomputed with a slow, O(n2 ), general-n module. The caller must supply a wavetable containing the trigonometric lookup tables and a workspace work. For the transform version of the function the sign argument canbe either forward (−1) or backward (+1).
The functions return a value of 0 if no errors were detected. The following gsl_errno conditions are definedfor these functions:
GSL_EDOM
GSL_EINVALThe length of the data n is not a positive integer (i.e. n is zero).
The length of the data n and the length used to compute the given wavetable do not match.

Here is an example program which computes the FFT of a short pulse in a sample of length 630 (= 2 ∗ 3 ∗ 3 ∗ 5 ∗ 7)
using the mixed-radix algorithm.
#include <stdio.h>#include <math.h>#include <gsl/gsl_errno.h>#include <gsl/gsl_fft_complex.h>(continues on next page)

16.4. Mixed-radix FFT routines for complex data

189

GNU Scientific Library, Release 2.7(continued from previous page)#define REAL(z,i) ((z)[2*(i)])#define IMAG(z,i) ((z)[2*(i)+1])
int
main (void){
int i;
const int n = 630;
double data[2*n];gsl_fft_complex_wavetable * wavetable;
gsl_fft_complex_workspace * workspace;for (i = 0; i < n; i++){
REAL(data,i) = 0.0;
IMAG(data,i) = 0.0;}
data[0] = 1.0;
for (i = 1; i <= 10; i++){
REAL(data,i) = REAL(data,n-i) = 1.0;}
for (i = 0; i < n; i++){printf ("%d: %e %e\n", i, REAL(data,i),
IMAG(data,i));}printf ("\n");wavetable = gsl_fft_complex_wavetable_alloc (n);workspace = gsl_fft_complex_workspace_alloc (n);for (i = 0; i < (int) wavetable->nf; i++){printf ("# factor %d: %zu\n", i,wavetable->factor[i]);}gsl_fft_complex_forward (data, 1, n,
wavetable, workspace);for (i = 0; i < n; i++){printf ("%d: %e %e\n", i, REAL(data,i),
IMAG(data,i));}gsl_fft_complex_wavetable_free (wavetable);(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)gsl_fft_complex_workspace_free (workspace);
return 0;}
Note that we have assumed that the program is using the default gsl error handler (which calls abort() for any errors).
If you are not using a safe error handler you would need to check the return status of all the gsl routines.

16.5 Overview of real data FFTs
The functions for real data are similar to those for complex data. However, there is an important difference between
forward and inverse transforms. The Fourier transform of a real sequence is not real. It is a complex sequence with a
special symmetry:∗zk = zn−k

A sequence with this symmetry is called conjugate-complex or half-complex. This different structure requires different
storage layouts for the forward transform (from real to half-complex) and inverse transform (from half-complex backto real). As a consequence the routines are divided into two sets: functions in gsl_fft_real which operate on real
sequences and functions in gsl_fft_halfcomplex which operate on half-complex sequences.
Functions in gsl_fft_real compute the frequency coefficients of a real sequence. The half-complex coefficients c
of a real sequence x are given by Fourier analysis,
ck =

n−1
X

xj exp(−2πijk/n)j=0Functions in gsl_fft_halfcomplex compute inverse or backwards transforms. They reconstruct real sequences by
Fourier synthesis from their half-complex frequency coefficients, c,
n−1

xj =

1X
ck exp(2πijk/n)n
k=0

The symmetry of the half-complex sequence implies that only half of the complex numbers in the output need to be
stored. The remaining half can be reconstructed using the half-complex symmetry condition. This works for all lengths,
even and odd—when the length is even the middle value where k = n/2 is also real. Thus only n real numbers are
required to store the half-complex sequence, and the transform of a real sequence can be stored in the same size arrayas the original data.
The precise storage arrangements depend on the algorithm, and are different for radix-2 and mixed-radix routines. The
radix-2 function operates in-place, which constrains the locations where each element can be stored. The restriction
forces real and imaginary parts to be stored far apart. The mixed-radix algorithm does not have this restriction, and it
stores the real and imaginary parts of a given term in neighboring locations (which is desirable for better locality ofmemory accesses).

16.5. Overview of real data FFTs

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16.6 Radix-2 FFT routines for real data
This section describes radix-2 FFT algorithms for real data. They use the Cooley-Tukey algorithm to compute in-place
FFTs for lengths which are a power of 2.
The radix-2 FFT functions for real data are declared in the header files gsl_fft_real.h
int gsl_fft_real_radix2_transform(double data[], size_t stride, size_t n)This function computes an in-place radix-2 FFT of length n and stride stride on the real array data. The
output is a half-complex sequence, which is stored in-place. The arrangement of the half-complex terms uses
the following scheme: for k < n/2 the real part of the k-th term is stored in location k, and the corresponding
imaginary part is stored in location n − k. Terms with k > n/2 can be reconstructed using the symmetry∗zk = zn−k. The terms for k = 0 and k = n/2 are both purely real, and count as a special case. Their real parts
are stored in locations 0 and n/2 respectively, while their imaginary parts which are zero are not stored.
The following table shows the correspondence between the output data and the equivalent results obtained byconsidering the input data as a complex sequence with zero imaginary part (assuming stride = 1}):complex[0].real
complex[0].imag
complex[1].real
complex[1].imag...............
complex[k].real
complex[k].imag...............
complex[n/2].real
complex[n/2].imag...............
complex[k'].real
complex[k'].imag...............
complex[n-1].real
complex[n-1].imag

=
=
=
=
=
=
=
=
=
=
=
=

data[0]0data[1]
data[n-1]................
data[k]data[n-k]................
data[n/2]0................
data[k]k' = n - k-data[n-k]................
data[1]-data[n-1]Note that the output data can be converted into the full complex sequence using the function
gsl_fft_halfcomplex_radix2_unpack() described below.
The radix-2 FFT functions for halfcomplex data are declared in the header file gsl_fft_halfcomplex.h.
int gsl_fft_halfcomplex_radix2_inverse(double data[], size_t stride, size_t n)
int gsl_fft_halfcomplex_radix2_backward(double data[], size_t stride, size_t n)
These functions compute the inverse or backwards in-place radix-2 FFT of length n and stride stride on thehalf-complex sequence data stored according the output scheme used by gsl_fft_real_radix2(). The result
is a real array stored in natural order.
int gsl_fft_halfcomplex_radix2_unpack(const double halfcomplex_coefficient[], gsl_complex_packed_arraycomplex_coefficient, size_t stride, size_t n)This function converts halfcomplex_coefficient, an array of half-complex coefficients as returned by
gsl_fft_real_radix2_transform(), into an ordinary complex array, complex_coefficient. It fills in∗
the complex array using the symmetry zk = zn−k
to reconstruct the redundant elements. The algorithm for theconversion is:complex_coefficient[0].real = halfcomplex_coefficient[0];complex_coefficient[0].imag = 0.0;
(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)for (i = 1; i < n - i; i++){double hc_real = halfcomplex_coefficient[i*stride];
double hc_imag = halfcomplex_coefficient[(n-i)*stride];complex_coefficient[i*stride].real = hc_real;complex_coefficient[i*stride].imag = hc_imag;
complex_coefficient[(n - i)*stride].real = hc_real;complex_coefficient[(n - i)*stride].imag = -hc_imag;}
if (i == n - i){complex_coefficient[i*stride].real = halfcomplex_coefficient[(n - 1)*stride];complex_coefficient[i*stride].imag = 0.0;}

16.7 Mixed-radix FFT routines for real data
This section describes mixed-radix FFT algorithms for real data. The mixed-radix functions work for FFTs of any
length. They are a reimplementation of the real-FFT routines in the Fortran FFTPACK library by Paul Swarztrauber.
The theory behind the algorithm is explained in the article “Fast Mixed-Radix Real Fourier Transforms” by Clive
Temperton. The routines here use the same indexing scheme and basic algorithms as FFTPACK.
The functions use the FFTPACK storage convention for half-complex sequences. In this convention the half-complex
transform of a real sequence is stored with frequencies in increasing order, starting at zero, with the real and imaginaryparts of each frequency in neighboring locations. When a value is known to be real the imaginary part is not stored.
The imaginary part of the zero-frequency component is never stored. It is known to be zero (since the zero frequency
component is simply the sum of the input data (all real)). For a sequence of even length the imaginary part of the∗frequency n/2 is not stored either, since the symmetry zk = zn−k
implies that this is purely real too.
The storage scheme is best shown by some examples. The table below shows the output for an odd-length sequence,n = 5. The two columns give the correspondence between the 5 values in the half-complex sequence returned by
gsl_fft_real_transform(), halfcomplex[] and the values complex[] that would be returned if the same real
input sequence were passed to gsl_fft_complex_backward() as a complex sequence (with imaginary parts set to0):complex[0].real
complex[0].imag
complex[1].real
complex[1].imag
complex[2].real
complex[2].imag
complex[3].real
complex[3].imag
complex[4].real
complex[4].imag= halfcomplex[0]= 0= halfcomplex[1]= halfcomplex[2]= halfcomplex[3]= halfcomplex[4]= halfcomplex[3]= -halfcomplex[4]= halfcomplex[1]= -halfcomplex[2]The upper elements of the complex array, complex[3] and complex[4] are filled in using the symmetry condition.
The imaginary part of the zero-frequency term complex[0].imag is known to be zero by the symmetry.
The next table shows the output for an even-length sequence, n = 6. In the even case there are two values which are

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purely real:complex[0].real
complex[0].imag
complex[1].real
complex[1].imag
complex[2].real
complex[2].imag
complex[3].real
complex[3].imag
complex[4].real
complex[4].imag
complex[5].real
complex[5].imag= halfcomplex[0]= 0= halfcomplex[1]= halfcomplex[2]= halfcomplex[3]= halfcomplex[4]= halfcomplex[5]= 0= halfcomplex[3]= -halfcomplex[4]= halfcomplex[1]= -halfcomplex[2]The upper elements of the complex array, complex[4] and complex[5] are filled in using the symmetry condition.
Both complex[0].imag and complex[3].imag are known to be zero.
All these functions are declared in the header files gsl_fft_real.h and gsl_fft_halfcomplex.h.
type gsl_fft_real_wavetable
type gsl_fft_halfcomplex_wavetable
These data structures contain lookup tables for an FFT of a fixed size.
gsl_fft_real_wavetable *gsl_fft_real_wavetable_alloc(size_t n)
gsl_fft_halfcomplex_wavetable *gsl_fft_halfcomplex_wavetable_alloc(size_t n)
These functions prepare trigonometric lookup tables for an FFT of size n real elements. The functions return apointer to the newly allocated struct if no errors were detected, and a null pointer in the case of error. The lengthn is factorized into a product of subtransforms, and the factors and their trigonometric coefficients are stored
in the wavetable. The trigonometric coefficients are computed using direct calls to sin and cos, for accuracy.
Recursion relations could be used to compute the lookup table faster, but if an application performs many FFTsof the same length then computing the wavetable is a one-off overhead which does not affect the final throughput.
The wavetable structure can be used repeatedly for any transform of the same length. The table is not modified
by calls to any of the other FFT functions. The appropriate type of wavetable must be used for forward real or
inverse half-complex transforms.
void gsl_fft_real_wavetable_free(gsl_fft_real_wavetable *wavetable)
void gsl_fft_halfcomplex_wavetable_free(gsl_fft_halfcomplex_wavetable *wavetable)
These functions free the memory associated with the wavetable wavetable. The wavetable can be freed if nofurther FFTs of the same length will be needed.
The mixed radix algorithms require additional working space to hold the intermediate steps of the transform,
type gsl_fft_real_workspace
This workspace contains parameters needed to compute a real FFT.
gsl_fft_real_workspace *gsl_fft_real_workspace_alloc(size_t n)This function allocates a workspace for a real transform of length n. The same workspace can be used for bothforward real and inverse halfcomplex transforms.
void gsl_fft_real_workspace_free(gsl_fft_real_workspace *workspace)
This function frees the memory associated with the workspace workspace. The workspace can be freed if nofurther FFTs of the same length will be needed.
The following functions compute the transforms of real and half-complex data,

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int gsl_fft_real_transform(double data[], size_t stride, size_t n, const gsl_fft_real_wavetable *wavetable,
gsl_fft_real_workspace *work)
int gsl_fft_halfcomplex_transform(double data[], size_t stride, size_t n, const gsl_fft_halfcomplex_wavetable*wavetable, gsl_fft_real_workspace *work)
These functions compute the FFT of data, a real or half-complex array of length n, using a mixed radix
decimation-in-frequency algorithm. For gsl_fft_real_transform() data is an array of time-ordered realdata. For gsl_fft_halfcomplex_transform() data contains Fourier coefficients in the half-complex ordering described above. There is no restriction on the length n. Efficient modules are provided for subtransformsof length 2, 3, 4 and 5. Any remaining factors are computed with a slow, O(n2 ), general-n module. The caller
must supply a wavetable containing trigonometric lookup tables and a workspace work.
int gsl_fft_real_unpack(const double real_coefficient[], gsl_complex_packed_array complex_coefficient, size_t
stride, size_t n)This function converts a single real array, real_coefficient into an equivalent complex array,complex_coefficient, (with imaginary part set to zero), suitable for gsl_fft_complex routines. The algorithm for the conversion is simply:
for (i = 0; i < n; i++){complex_coefficient[i*stride].real = real_coefficient[i*stride];complex_coefficient[i*stride].imag = 0.0;}
int gsl_fft_halfcomplex_unpack(const double halfcomplex_coefficient[], gsl_complex_packed_arraycomplex_coefficient, size_t stride, size_t n)This function converts halfcomplex_coefficient, an array of half-complex coefficients as returned by
gsl_fft_real_transform(), into an ordinary complex array, complex_coefficient. It fills in the complex∗
array using the symmetry zk = zn−k
to reconstruct the redundant elements. The algorithm for the conversion
is:complex_coefficient[0].real = halfcomplex_coefficient[0];complex_coefficient[0].imag = 0.0;
for (i = 1; i < n - i; i++){double hc_real = halfcomplex_coefficient[(2 * i - 1)*stride];
double hc_imag = halfcomplex_coefficient[(2 * i)*stride];complex_coefficient[i*stride].real = hc_real;complex_coefficient[i*stride].imag = hc_imag;
complex_coefficient[(n - i)*stride].real = hc_real;complex_coefficient[(n - i)*stride].imag = -hc_imag;}
if (i == n - i){complex_coefficient[i*stride].real = halfcomplex_coefficient[(n - 1)*stride];complex_coefficient[i*stride].imag = 0.0;}
Here is an example program using gsl_fft_real_transform() and gsl_fft_halfcomplex_inverse(). It
generates a real signal in the shape of a square pulse. The pulse is Fourier transformed to frequency space,and all but the lowest ten frequency components are removed from the array of Fourier coefficients returned by
gsl_fft_real_transform().

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The remaining Fourier coefficients are transformed back to the time-domain, to give a filtered version of the squarepulse. Since Fourier coefficients are stored using the half-complex symmetry both positive and negative frequencies
are removed and the final filtered signal is also real.
#include <stdio.h>#include <math.h>#include <gsl/gsl_errno.h>#include <gsl/gsl_fft_real.h>#include <gsl/gsl_fft_halfcomplex.h>int
main (void){
int i, n = 100;
double data[n];gsl_fft_real_wavetable * real;
gsl_fft_halfcomplex_wavetable * hc;
gsl_fft_real_workspace * work;for (i = 0; i < n; i++){data[i] = 0.0;}
for (i = n / 3; i < 2 * n / 3; i++){data[i] = 1.0;}
for (i = 0; i < n; i++){printf ("%d: %e\n", i, data[i]);}printf ("\n");work = gsl_fft_real_workspace_alloc (n);real = gsl_fft_real_wavetable_alloc (n);gsl_fft_real_transform (data, 1, n,real, work);
gsl_fft_real_wavetable_free (real);for (i = 11; i < n; i++){data[i] = 0;}hc = gsl_fft_halfcomplex_wavetable_alloc (n);gsl_fft_halfcomplex_inverse (data, 1, n,
hc, work);(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)gsl_fft_halfcomplex_wavetable_free (hc);for (i = 0; i < n; i++){printf ("%d: %e\n", i, data[i]);}gsl_fft_real_workspace_free (work);
return 0;}
The program output is shown in Fig. 16.2.

Fig. 16.2: Low-pass filtered version of a real pulse, output from the example program.

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16.8 References and Further Reading
A good starting point for learning more about the FFT is the following review article,• P. Duhamel and M. Vetterli. Fast Fourier transforms: A tutorial review and a state of the art. Signal Processing,
19:259–299, 1990.
To find out about the algorithms used in the GSL routines you may want to consult the document “GSL FFT Algorithms”(it is included in GSL, as doc/fftalgorithms.tex). This has general information on FFTs and explicit derivationsof the implementation for each routine. There are also references to the relevant literature. For convenience some of
the more important references are reproduced below.
There are several introductory books on the FFT with example programs, such as “The Fast Fourier Transform” by
Brigham and “DFT/FFT and Convolution Algorithms” by Burrus and Parks,•

E. Oran Brigham. “The Fast Fourier Transform”. Prentice Hall, 1974.

• C. S. Burrus and T. W. Parks. “DFT/FFT and Convolution Algorithms”, Wiley, 1984.
Both these introductory books cover the radix-2 FFT in some detail. The mixed-radix algorithm at the heart of the
FFTPACK routines is reviewed in Clive Temperton’s paper,• Clive Temperton, Self-sorting mixed-radix fast Fourier transforms, Journal of Computational Physics,
52(1):1–23, 1983.
The derivation of FFTs for real-valued data is explained in the following two articles,• Henrik V. Sorenson, Douglas L. Jones, Michael T. Heideman, and C. Sidney Burrus. Real-valued fast Fouriertransform algorithms. “IEEE Transactions on Acoustics, Speech, and Signal Processing”, ASSP-35(6):849–863,
1987.
• Clive Temperton. Fast mixed-radix real Fourier transforms. “Journal of Computational Physics”, 52:340–350,
1983.
In 1979 the IEEE published a compendium of carefully-reviewed Fortran FFT programs in “Programs for Digital Signal
Processing”. It is a useful reference for implementations of many different FFT algorithms,• Digital Signal Processing Committee and IEEE Acoustics, Speech, and Signal Processing Committee, editors.
Programs for Digital Signal Processing. IEEE Press, 1979.
For large-scale FFT work we recommend the use of the dedicated FFTW library by Frigo and Johnson. The FFTW
library is self-optimizing—it automatically tunes itself for each hardware platform in order to achieve maximum performance. It is available under the GNU GPL.
• FFTW Website, http://www.fftw.org/
The source code for FFTPACK is available from http://www.netlib.org/fftpack/198

Chapter 16. Fast Fourier Transforms (FFTs)

CHAPTERSEVENTEEN

NUMERICAL INTEGRATIONThis chapter describes routines for performing numerical integration (quadrature) of a function in one dimension. There
are routines for adaptive and non-adaptive integration of general functions, with specialised routines for specific cases.
These include integration over infinite and semi-infinite ranges, singular integrals, including logarithmic singularities,
computation of Cauchy principal values and oscillatory integrals. The library reimplements the algorithms used in
QUADPACK, a numerical integration package written by Piessens, de Doncker-Kapenga, Ueberhuber and Kahaner.
Fortran code for QUADPACK is available on Netlib. Also included are non-adaptive, fixed-order Gauss-Legendre
integration routines with high precision coefficients, as well as fixed-order quadrature rules for a variety of weighting
functions from IQPACK.
The functions described in this chapter are declared in the header file gsl_integration.h.

17.1 Introduction
Each algorithm computes an approximation to a definite integral of the form,
Z b
I=

f (x)w(x)dxawhere w(x) is a weight function (for general integrands w(x) = 1). The user provides absolute and relative error
bounds (epsabs, epsrel) which specify the following accuracy requirement,|RESU LT − I| ≤ max (epsabs, epsrel|I|)
where RESU LT is the numerical approximation obtained by the algorithm. The algorithms attempt to estimate theabsolute error ABSERR = |RESU LT − I| in such a way that the following inequality holds,|RESU LT − I| ≤ ABSERR ≤ max (epsabs, epsrel|I|)
In short, the routines return the first approximation which has an absolute error smaller than epsabs or a relative error
smaller than epsrel.
Note that this is an either-or constraint, not simultaneous. To compute to a specified absolute error, set epsrel to zero.
To compute to a specified relative error, set epsabs to zero. The routines will fail to converge if the error bounds are
too stringent, but always return the best approximation obtained up to that stage.
The algorithms in QUADPACK use a naming convention based on the following letters:
Q - quadrature routine
N - non-adaptive integrator
A - adaptive integrator(continues on next page)199

GNU Scientific Library, Release 2.7(continued from previous page)G - general integrand (user-defined)
W - weight function with integrandS - singularities can be more readily integrated
P - points of special difficulty can be supplied
I - infinite range of integration
O - oscillatory weight function, cos or sin
F - Fourier integral
C - Cauchy principal value
The algorithms are built on pairs of quadrature rules, a higher order rule and a lower order rule. The higher order ruleis used to compute the best approximation to an integral over a small range. The difference between the results of thehigher order rule and the lower order rule gives an estimate of the error in the approximation.

17.1.1 Integrands without weight functions
The algorithms for general functions (without a weight function) are based on Gauss-Kronrod rules.
A Gauss-Kronrod rule begins with a classical Gaussian quadrature rule of order m. This is extended with additional
points between each of the abscissae to give a higher order Kronrod rule of order 2m + 1. The Kronrod rule is efficientbecause it reuses existing function evaluations from the Gaussian rule.
The higher order Kronrod rule is used as the best approximation to the integral, and the difference between the tworules is used as an estimate of the error in the approximation.

17.1.2 Integrands with weight functions
For integrands with weight functions the algorithms use Clenshaw-Curtis quadrature rules.
A Clenshaw-Curtis rule begins with an n-th order Chebyshev polynomial approximation to the integrand. This polynomial can be integrated exactly to give an approximation to the integral of the original function. The Chebyshev
expansion can be extended to higher orders to improve the approximation and provide an estimate of the error.

17.1.3 Integrands with singular weight functions
The presence of singularities (or other behavior) in the integrand can cause slow convergence in the Chebyshev approximation. The modified Clenshaw-Curtis rules used in QUADPACK separate out several common weight functions
which cause slow convergence.
These weight functions are integrated analytically against the Chebyshev polynomials to precompute modified Chebyshev moments. Combining the moments with the Chebyshev approximation to the function gives the desired integral.
The use of analytic integration for the singular part of the function allows exact cancellations and substantially improves
the overall convergence behavior of the integration.

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17.2 QNG non-adaptive Gauss-Kronrod integration
The QNG algorithm is a non-adaptive procedure which uses fixed Gauss-Kronrod-Patterson abscissae to sample theintegrand at a maximum of 87 points. It is provided for fast integration of smooth functions.
int gsl_integration_qng(const gsl_function *f, double a, double b, double epsabs, double epsrel, double *result,double *abserr, size_t *neval)This function applies the Gauss-Kronrod 10-point, 21-point, 43-point and 87-point integration rules in succession
until an estimate of the integral of f over (a, b) is achieved within the desired absolute and relative error limits,
epsabs and epsrel. The function returns the final approximation, result, an estimate of the absolute error,
abserr and the number of function evaluations used, neval. The Gauss-Kronrod rules are designed in sucha way that each rule uses all the results of its predecessors, in order to minimize the total number of function
evaluations.

17.3 QAG adaptive integration
The QAG algorithm is a simple adaptive integration procedure. The integration region is divided into subintervals, andon each iteration the subinterval with the largest estimated error is bisected. This reduces the overall error rapidly, as
the subintervals become concentrated around local difficulties in the integrand. These subintervals are managed by thefollowing struct,
type gsl_integration_workspace
This workspace handles the memory for the subinterval ranges, results and error estimates.
gsl_integration_workspace *gsl_integration_workspace_alloc(size_t n)This function allocates a workspace sufficient to hold n double precision intervals, their integration results anderror estimates. One workspace may be used multiple times as all necessary reinitialization is performed automatically by the integration routines.
void gsl_integration_workspace_free(gsl_integration_workspace *w)
This function frees the memory associated with the workspace w.
int gsl_integration_qag(const gsl_function *f, double a, double b, double epsabs, double epsrel, size_t limit, int
key, gsl_integration_workspace *workspace, double *result, double *abserr)
This function applies an integration rule adaptively until an estimate of the integral of f over (a, b) is achieved
within the desired absolute and relative error limits, epsabs and epsrel. The function returns the final approximation, result, and an estimate of the absolute error, abserr. The integration rule is determined by the valueof key, which should be chosen from the following symbolic names,
Symbolic NameGSL_INTEG_GAUSS15
GSL_INTEG_GAUSS21
GSL_INTEG_GAUSS31
GSL_INTEG_GAUSS41
GSL_INTEG_GAUSS51
GSL_INTEG_GAUSS61

Key
1
2
3
4
5
6

corresponding to the 15, 21, 31, 41, 51 and 61 point Gauss-Kronrod rules. The higher-order rules give better
accuracy for smooth functions, while lower-order rules save time when the function contains local difficulties,
such as discontinuities.
On each iteration the adaptive integration strategy bisects the interval with the largest error estimate. The subintervals and their results are stored in the memory provided by workspace. The maximum number of subintervals
is given by limit, which may not exceed the allocated size of the workspace.
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17.4 QAGS adaptive integration with singularities
The presence of an integrable singularity in the integration region causes an adaptive routine to concentrate new subintervals around the singularity. As the subintervals decrease in size the successive approximations to the integral converge in a limiting fashion. This approach to the limit can be accelerated using an extrapolation procedure. The QAGS
algorithm combines adaptive bisection with the Wynn epsilon-algorithm to speed up the integration of many types ofintegrable singularities.
int gsl_integration_qags(const gsl_function *f, double a, double b, double epsabs, double epsrel, size_t limit,
gsl_integration_workspace *workspace, double *result, double *abserr)
This function applies the Gauss-Kronrod 21-point integration rule adaptively until an estimate of the integral off over (a, b) is achieved within the desired absolute and relative error limits, epsabs and epsrel. The results
are extrapolated using the epsilon-algorithm, which accelerates the convergence of the integral in the presence of
discontinuities and integrable singularities. The function returns the final approximation from the extrapolation,
result, and an estimate of the absolute error, abserr. The subintervals and their results are stored in thememory provided by workspace. The maximum number of subintervals is given by limit, which may notexceed the allocated size of the workspace.

17.5 QAGP adaptive integration with known singular points
int gsl_integration_qagp(const gsl_function *f, double *pts, size_t npts, double epsabs, double epsrel, size_t
limit, gsl_integration_workspace *workspace, double *result, double *abserr)
This function applies the adaptive integration algorithm QAGS taking account of the user-supplied locations of
singular points. The array pts of length npts should contain the endpoints of the integration ranges definedby the integration region and locations of the singularities. For example, to integrate over the region (a, b) withbreak-points at x1 , x2 , x3 (where a < x1 < x2 < x3 < b) the following pts array should be used:
pts[0] = a
pts[1] = x_1
pts[2] = x_2
pts[3] = x_3
pts[4] = b
with npts = 5.
If you know the locations of the singular points in the integration region then this routine will be faster thangsl_integration_qags().

17.6 QAGI adaptive integration on infinite intervals
int gsl_integration_qagi(gsl_function *f, double epsabs, double epsrel, size_t limit, gsl_integration_workspace*workspace, double *result, double *abserr)
This function computes the integral of the function f over the infinite interval (−∞, +∞). The integral ismapped onto the semi-open interval (0, 1] using the transformation x = (1 − t)/t,
Z +∞

Z 1
dxf (x) =−∞dt(f ((1 − t)/t) + f (−(1 − t)/t))/t2 .

0

It is then integrated using the QAGS algorithm. The normal 21-point Gauss-Kronrod rule of QAGS is replaced
by a 15-point rule, because the transformation can generate an integrable singularity at the origin. In this case alower-order rule is more efficient.

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int gsl_integration_qagiu(gsl_function *f, double a, double epsabs, double epsrel, size_t limit,
gsl_integration_workspace *workspace, double *result, double *abserr)
This function computes the integral of the function f over the semi-infinite interval (a, +∞). The integral ismapped onto the semi-open interval (0, 1] using the transformation x = a + (1 − t)/t,
Z +∞

Z 1
dxf (x) =a

dtf (a + (1 − t)/t)/t2

0

and then integrated using the QAGS algorithm.
int gsl_integration_qagil(gsl_function *f, double b, double epsabs, double epsrel, size_t limit,
gsl_integration_workspace *workspace, double *result, double *abserr)
This function computes the integral of the function f over the semi-infinite interval (−∞, b). The integral ismapped onto the semi-open interval (0, 1] using the transformation x = b − (1 − t)/t,
Z b

Z 1
dxf (x) =−∞dtf (b − (1 − t)/t)/t2

0

and then integrated using the QAGS algorithm.

17.7 QAWC adaptive integration for Cauchy principal values
int gsl_integration_qawc(gsl_function *f, double a, double b, double c, double epsabs, double epsrel, size_t
limit, gsl_integration_workspace *workspace, double *result, double *abserr)
This function computes the Cauchy principal value of the integral of f over (a, b), with a singularity at c,
(Z
)
Z b
Z b
c−
f (x)
f (x)
f (x)
= lim
dx
+
dx
I=
dx
x − c →0
x−c
x−c
a
c+
a
The adaptive bisection algorithm of QAG is used, with modifications to ensure that subdivisions do not occur atthe singular point x = c. When a subinterval contains the point x = c or is close to it then a special 25-point
modified Clenshaw-Curtis rule is used to control the singularity. Further away from the singularity the algorithm
uses an ordinary 15-point Gauss-Kronrod integration rule.

17.8 QAWS adaptive integration for singular functions
The QAWS algorithm is designed for integrands with algebraic-logarithmic singularities at the end-points of an integration region. In order to work efficiently the algorithm requires a precomputed table of Chebyshev moments.
type gsl_integration_qaws_table
This structure contains precomputed quantities for the QAWS algorithm.
gsl_integration_qaws_table *gsl_integration_qaws_table_alloc(double alpha, double beta, int mu, int nu)
This function allocates space for a gsl_integration_qaws_table struct describing a singular weight function
w(x) with the parameters (α, β, µ, ν),w(x) = (x − a)α (b − x)β logµ (x − a) logν (b − x)
where α > −1, β > −1, and µ = 0, 1, ν = 0, 1. The weight function can take four different forms dependingon the values of µ and ν,

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Weight function w(x)(x − a)α (b − x)β(x − a)α (b − x)β log (x − a)(x − a)α (b − x)β log (b − x)(x − a)α (b − x)β log (x − a) log (b − x)(µ, ν)(0, 0)(1, 0)(0, 1)(1, 1)The singular points (a, b) do not have to be specified until the integral is computed, where they are the endpointsof the integration range.
The function returns a pointer to the newly allocated table gsl_integration_qaws_table if no errors weredetected, and 0 in the case of error.
int gsl_integration_qaws_table_set(gsl_integration_qaws_table *t, double alpha, double beta, int mu, int nu)
This function modifies the parameters (α, β, µ, ν) of an existing gsl_integration_qaws_table struct t.
void gsl_integration_qaws_table_free(gsl_integration_qaws_table *t)
This function frees all the memory associated with the gsl_integration_qaws_table struct t.
int gsl_integration_qaws(gsl_function *f, const double a, const double b, gsl_integration_qaws_table *t, const
double epsabs, const double epsrel, const size_t limit, gsl_integration_workspace*workspace, double *result, double *abserr)
This function computes the integral of the function f (x) over the interval (a, b) with the singular weight function(x − a)α (b − x)β logµ (x − a) logν (b − x). The parameters of the weight function (α, β, µ, ν) are taken fromthe table t. The integral is,Z bI=dxf (x)(x − a)α (b − x)β logµ (x − a) logν (b − x).

a

The adaptive bisection algorithm of QAG is used. When a subinterval contains one of the endpoints then aspecial 25-point modified Clenshaw-Curtis rule is used to control the singularities. For subintervals which do
not include the endpoints an ordinary 15-point Gauss-Kronrod integration rule is used.

17.9 QAWO adaptive integration for oscillatory functions
The QAWO algorithm is designed for integrands with an oscillatory factor, sin(ωx) or cos(ωx). In order to work
efficiently the algorithm requires a table of Chebyshev moments which must be pre-computed with calls to the functions
below.
gsl_integration_qawo_table *gsl_integration_qawo_table_alloc(double omega, double L, enum
gsl_integration_qawo_enum sine, size_t n)This function allocates space for a gsl_integration_qawo_table struct and its associated workspace describing a sine or cosine weight function w(x) with the parameters (ω, L),sin (ωx)w(x) =cos (ωx)
The parameter L must be the length of the interval over which the function will be integrated L = b − a. The
choice of sine or cosine is made with the parameter sine which should be chosen from one of the two following
symbolic values:
GSL_INTEG_COSINE
GSL_INTEG_SINE
The gsl_integration_qawo_table is a table of the trigonometric coefficients required in the integration process. The parameter n determines the number of levels of coefficients that are computed. Each level corresponds
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to one bisection of the interval L, so that n levels are sufficient for subintervals down to the length L/2n . The integration routine gsl_integration_qawo() returns the error GSL_ETABLE if the number of levels is insufficientfor the requested accuracy.
int gsl_integration_qawo_table_set(gsl_integration_qawo_table *t, double omega, double L, enum
gsl_integration_qawo_enum sine)This function changes the parameters omega, L and sine of the existing workspace t.
int gsl_integration_qawo_table_set_length(gsl_integration_qawo_table *t, double L)
This function allows the length parameter L of the workspace t to be changed.
void gsl_integration_qawo_table_free(gsl_integration_qawo_table *t)
This function frees all the memory associated with the workspace t.
int gsl_integration_qawo(gsl_function *f, const double a, const double epsabs, const double epsrel, const size_t
limit, gsl_integration_workspace *workspace, gsl_integration_qawo_table *wf,double *result, double *abserr)
This function uses an adaptive algorithm to compute the integral of f over (a, b) with the weight function sin(ωx)
or cos(ωx) defined by the table wf ,
Z b
I=


dxf (x)asin (ωx)cos (ωx)

The results are extrapolated using the epsilon-algorithm to accelerate the convergence of the integral. The
function returns the final approximation from the extrapolation, result, and an estimate of the absolute error,
abserr. The subintervals and their results are stored in the memory provided by workspace. The maximum
number of subintervals is given by limit, which may not exceed the allocated size of the workspace.
Those subintervals with “large” widths d where dω > 4 are computed using a 25-point Clenshaw-Curtis integration rule, which handles the oscillatory behavior. Subintervals with a “small” widths where dω < 4 arecomputed using a 15-point Gauss-Kronrod integration.

17.10 QAWF adaptive integration for Fourier integrals
int gsl_integration_qawf(gsl_function *f, const double a, const double epsabs, const size_t limit,
gsl_integration_workspace *workspace, gsl_integration_workspace*cycle_workspace, gsl_integration_qawo_table *wf, double *result, double *abserr)
This function attempts to compute a Fourier integral of the function f over the semi-infinite interval [a, +∞)Z +∞sin (ωx)I=
dxf (x)cos (ωx)a
The parameter ω and choice of sin or cos is taken from the table wf (the length L can take any value, since it isoverridden by this function to a value appropriate for the Fourier integration). The integral is computed using
the QAWO algorithm over each of the subintervals,
C1 = [a, a + c]
C2 = [a + c, a + 2c]... = ...
Ck = [a + (k − 1)c, a + kc]
where c = (2f loor(|ω|)+1)π/|ω|. The width c is chosen to cover an odd number of periods so that the contributions from the intervals alternate in sign and are monotonically decreasing when f is positive and monotonicallydecreasing. The sum of this sequence of contributions is accelerated using the epsilon-algorithm.

17.10. QAWF adaptive integration for Fourier integrals

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This function works to an overall absolute tolerance of abserr. The following strategy is used: on each interval
Ck the algorithm tries to achieve the tolerance
T OLk = uk abserr
where uk = (1 − p)pk−1 and p = 9/10. The sum of the geometric series of contributions from each interval
gives an overall tolerance of abserr.
If the integration of a subinterval leads to difficulties then the accuracy requirement for subsequent intervals is
relaxed,T OLk = uk max (abserr, max (Ei ))i<kwhere Ek is the estimated error on the interval Ck .
The subintervals and their results are stored in the memory provided by workspace. The maximum number ofsubintervals is given by limit, which may not exceed the allocated size of the workspace. The integration overeach subinterval uses the memory provided by cycle_workspace as workspace for the QAWO algorithm.

17.11 CQUAD doubly-adaptive integration
CQUAD is a new doubly-adaptive general-purpose quadrature routine which can handle most types of singularities,
non-numerical function values such as Inf or NaN, as well as some divergent integrals. It generally requires morefunction evaluations than the integration routines in QUADPACK, yet fails less often for difficult integrands.
The underlying algorithm uses a doubly-adaptive scheme in which Clenshaw-Curtis quadrature rules of increasing
degree are used to compute the integral in each interval. The L2 -norm of the difference between the underlying interpolatory polynomials of two successive rules is used as an error estimate. The interval is subdivided if the difference
between two successive rules is too large or a rule of maximum degree has been reached.
gsl_integration_cquad_workspace *gsl_integration_cquad_workspace_alloc(size_t n)This function allocates a workspace sufficient to hold the data for n intervals. The number n is not the maximum
number of intervals that will be evaluated. If the workspace is full, intervals with smaller error estimates will bediscarded. A minimum of 3 intervals is required and for most functions, a workspace of size 100 is sufficient.
void gsl_integration_cquad_workspace_free(gsl_integration_cquad_workspace *w)
This function frees the memory associated with the workspace w.
int gsl_integration_cquad(const gsl_function *f, double a, double b, double epsabs, double epsrel,
gsl_integration_cquad_workspace *workspace, double *result, double *abserr, size_t*nevals)
This function computes the integral of f over (a, b) within the desired absolute and relative error limits, epsabs
and epsrel using the CQUAD algorithm. The function returns the final approximation, result, an estimate of
the absolute error, abserr, and the number of function evaluations required, nevals.
The CQUAD algorithm divides the integration region into subintervals, and in each iteration, the subinterval withthe largest estimated error is processed. The algorithm uses Clenshaw-Curtis quadrature rules of degree 4, 8, 16
and 32 over 5, 9, 17 and 33 nodes respectively. Each interval is initialized with the lowest-degree rule. Whenan interval is processed, the next-higher degree rule is evaluated and an error estimate is computed based on theL2 -norm of the difference between the underlying interpolating polynomials of both rules. If the highest-degree
rule has already been used, or the interpolatory polynomials differ significantly, the interval is bisected.
The subintervals and their results are stored in the memory provided by workspace. If the error estimate or thenumber of function evaluations is not needed, the pointers abserr and nevals can be set to NULL.

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17.12 Romberg integration
The Romberg integration method estimates the definite integral
Z bI=f (x)dxaby applying Richardson extrapolation on the trapezoidal rule, using equally spaced points with spacing
hk = (b − a)2−k
for k = 1, . . . , n. For each k, Richardson extrapolation is used k − 1 times on previous approximations to improve theorder of accuracy as much as possible. Romberg integration typically works well (and converges quickly) for smooth
integrands with no singularities in the interval or at the end points.
gsl_integration_romberg_workspace *gsl_integration_romberg_alloc(const size_t n)This function allocates a workspace for Romberg integration, specifying a maximum of n iterations, or divisionsof the interval. Since the number of divisions is 2n + 1, n can be kept relatively small (i.e. 10 or 20). It is capped
at a maximum value of 30 to prevent overflow. The size of the workspace is O(2n).
void gsl_integration_romberg_free(gsl_integration_romberg_workspace *w)
This function frees the memory associated with the workspace w.
int gsl_integration_romberg(const gsl_function *f, const double a, const double b, const double epsabs, const
double epsrel, double *result, size_t *neval, gsl_integration_romberg_workspace*w)
This function integrates f (x), specified by f , from a to b, storing the answer in result. At each step in theiteration, convergence is tested by checking:|Ik − Ik−1 | ≤ max (epsabs, epsrel × |Ik |)where Ik is the current approximation and Ik−1 is the approximation of the previous iteration. If the method does
not converge within the previously specified n iterations, the function stores the best current estimate in result
and returns GSL_EMAXITER. If the method converges, the function returns GSL_SUCCESS. The total number offunction evaluations is returned in neval.

17.13 Gauss-Legendre integration
The fixed-order Gauss-Legendre integration routines are provided for fast integration of smooth functions with known
polynomial order. The n-point Gauss-Legendre rule is exact for polynomials of order 2n − 1 or less. For example,
these rules are useful when integrating basis functions to form mass matrices for the Galerkin method. Unlike othernumerical integration routines within the library, these routines do not accept absolute or relative error bounds.
gsl_integration_glfixed_table *gsl_integration_glfixed_table_alloc(size_t n)This function determines the Gauss-Legendre abscissae and weights necessary for an n-point fixed order integration scheme. If possible, high precision precomputed coefficients are used. If precomputed weights are notavailable, lower precision coefficients are computed on the fly.
double gsl_integration_glfixed(const gsl_function *f, double a, double b, const gsl_integration_glfixed_table*t)This function applies the Gauss-Legendre integration rule contained in table t and returns the result.
int gsl_integration_glfixed_point(double a, double b, size_t i, double *xi, double *wi, const
gsl_integration_glfixed_table *t)
For i in [0, . . . , n − 1], this function obtains the i-th Gauss-Legendre point xi and weight wi on the interval [a,b]. The points and weights are ordered by increasing point value. A function f may be integrated on [a, b] bysumming wi ∗ f (xi) over i.
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void gsl_integration_glfixed_table_free(gsl_integration_glfixed_table *t)
This function frees the memory associated with the table t.

17.14 Fixed point quadratures
The routines in this section approximate an integral by the sum
Z bw(x)f (x)dx =anXwi f (xi )i=1

where f (x) is the function to be integrated and w(x) is a weighting function. The n weights wi and nodes xi arecarefully chosen so that the result is exact when f (x) is a polynomial of degree 2n − 1 or less. Once the user chooses
the order n and weighting function w(x), the weights wi and nodes xi can be precomputed and used to efficiently
evaluate integrals for any number of functions f (x).
This method works best when f (x) is well approximated by a polynomial on the interval (a, b), and so is not suitablefor functions with singularities. Since the user specifies ahead of time how many quadrature nodes will be used,
these routines do not accept absolute or relative error bounds. The table below lists the weighting functions currently
supported.
Name
Legendre
Chebyshev Type 1
Gegenbauer
Jacobi
Laguerre
HermiteExponential
Rational
Chebyshev Type 2

Interval(a, b)(a, b)(a, b)(a, b)(a, ∞)(−∞, ∞)(a, b)(a, ∞)(a, b)

Weighting function w(x)1 p
1/ (b − x)(x − a)((b − x)(x − a))α(b − x)α (x − a)β(x − a)α exp (−b(x − a))|x − a|α exp (−b(x − a)2 )|x − (a + b)/2|ααβ(xp − a) (x + b)
(b − x)(x − a)Constraintsb>ab>aα > −1, b > aα, β > −1, b > a
α > −1, b > 0
α > −1, b > 0
α > −1, b > a
α > −1, α + β + 2n < 0, a + b > 0
b>aThe fixed point quadrature routines use the following workspace to store the nodes and weights, as well as additionalvariables for intermediate calculations:type gsl_integration_fixed_workspace
This workspace is used for fixed point quadrature rules and looks like this:typedef struct{
size_t n;/* number of nodes/weights */double *weights; /* quadrature weights */double *x;/* quadrature nodes */double *diag;/* diagonal of Jacobi matrix */double *subdiag; /* subdiagonal of Jacobi matrix */const gsl_integration_fixed_type * type;} gsl_integration_fixed_workspace;
gsl_integration_fixed_workspace *gsl_integration_fixed_alloc(const gsl_integration_fixed_type *T, const
size_t n, const double a, const double b, const
double alpha, const double beta)
This function allocates a workspace for computing integrals with interpolating quadratures using n quadrature
nodes. The parameters a, b, alpha, and beta specify the integration interval and/or weighting function for the

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various quadrature types. See the table above for constraints on these parameters. The size of the workspace isO(4n).
type gsl_integration_fixed_type
The type of quadrature used is specified by T which can be set to the following choices:
gsl_integration_fixed_type *gsl_integration_fixed_legendre
This specifies Legendre quadrature integration. The parameters alpha and beta are ignored for thistype.
gsl_integration_fixed_type *gsl_integration_fixed_chebyshev
This specifies Chebyshev type 1 quadrature integration. The parameters alpha and beta are ignored
for this type.
gsl_integration_fixed_type *gsl_integration_fixed_gegenbauer
This specifies Gegenbauer quadrature integration. The parameter beta is ignored for this type.
gsl_integration_fixed_type *gsl_integration_fixed_jacobi
This specifies Jacobi quadrature integration.
gsl_integration_fixed_type *gsl_integration_fixed_laguerre
This specifies Laguerre quadrature integration. The parameter beta is ignored for this type.
gsl_integration_fixed_type *gsl_integration_fixed_hermite
This specifies Hermite quadrature integration. The parameter beta is ignored for this type.
gsl_integration_fixed_type *gsl_integration_fixed_exponential
This specifies exponential quadrature integration. The parameter beta is ignored for this type.
gsl_integration_fixed_type *gsl_integration_fixed_rational
This specifies rational quadrature integration.
gsl_integration_fixed_type *gsl_integration_fixed_chebyshev2
This specifies Chebyshev type 2 quadrature integration. The parameters alpha and beta are ignored
for this type.
void gsl_integration_fixed_free(gsl_integration_fixed_workspace *w)
This function frees the memory assocated with the workspace w
size_t gsl_integration_fixed_n(const gsl_integration_fixed_workspace *w)
This function returns the number of quadrature nodes and weights.
double *gsl_integration_fixed_nodes(const gsl_integration_fixed_workspace *w)
This function returns a pointer to an array of size n containing the quadrature nodes xi .
double *gsl_integration_fixed_weights(const gsl_integration_fixed_workspace *w)
This function returns a pointer to an array of size n containing the quadrature weights wi .
int gsl_integration_fixed(const gsl_function *func, double *result, const gsl_integration_fixed_workspace *w)
This function integrates the function f (x) provided in func using previously computed fixed quadrature rules.
The integral is approximated asnX

wi f (xi )i=1

where wi are the quadrature weights and xi are the quadrature nodes computed previously by
gsl_integration_fixed_alloc(). The sum is stored in result on output.

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17.15 Error codes
In addition to the standard error codes for invalid arguments the functions can return the following values,
GSL_EMAXITER the maximum number of subdivisions was exceeded.
GSL_EROUND
cannot reach tolerance because of roundoff error, or roundoff error was detected in the extrapolation table.
GSL_ESING
a non-integrable singularity or other bad integrand behavior was found in the integration interval.
GSL_EDIVERGE the integral is divergent, or too slowly convergent to be integrated numerically.
GSL_EDOM
error in the values of the input arguments

17.16 Examples
17.16.1 Adaptive integration example
The integrator QAGS will handle a large class of definite integrals. For example, consider the following integral, which
has an algebraic-logarithmic singularity at the origin,
Z 1
x−1/2 log(x)dx = −40

The program below computes this integral to a relative accuracy bound of 1e-7.
#include <stdio.h>#include <math.h>#include <gsl/gsl_integration.h>double f (double x, void * params) {double alpha = *(double *) params;
double f = log(alpha*x) / sqrt(x);return f;}
int
main (void){
gsl_integration_workspace * w= gsl_integration_workspace_alloc (1000);
double result, error;
double expected = -4.0;
double alpha = 1.0;
gsl_function F;
F.function = &f;F.params = &alpha;gsl_integration_qags (&F, 0, 1, 0, 1e-7, 1000,
w, &result, &error);printf ("result= % .18f \n", result);(continues on next page)

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printf ("exact result= % .18f \n", expected);
printf ("estimated error = % .18f \n", error);printf ("actual error= % .18f \n", result - expected);
printf ("intervals= %zu\n", w->size);
gsl_integration_workspace_free (w);return 0;}
The results below show that the desired accuracy is achieved after 8 subdivisions.
result= -4.000000000000085265
exact result= -4.000000000000000000
estimated error = 0.000000000000135447
actual error= -0.000000000000085265
intervals= 8
In fact, the extrapolation procedure used by QAGS produces an accuracy of almost twice as many digits. The error
estimate returned by the extrapolation procedure is larger than the actual error, giving a margin of safety of one orderof magnitude.

17.16.2 Fixed-point quadrature example
In this example, we use a fixed-point quadrature rule to integrate the integral

 √
Z ∞
2
π +√
Γ m+1
, m even2e−x (xm + 1) dx =
π,
m odd
−∞
for integer m. Consulting our table of fixed point quadratures, we see that this integral can be evaluated with a Hermite
quadrature rule, setting α = 0, a = 0, b = 1. Since we are integrating a polynomial of degree m, we need to choose
the number of nodes n ≥ (m + 1)/2 to achieve the best results.
First we will try integrating for m = 10, n = 5, which does not satisfy our criteria above:$ ./integration2 10 5
The output is,mintervals
resultexact result
actual error

= 10
= 5
= 47.468529694563351029= 54.115231635459025483= -6.646701940895674454

So, we find a large error. Now we try integrating for m = 10, n = 6 which does satisfy the criteria above:$ ./integration2 10 6
The output is,mintervals= 10= 6(continues on next page)

17.16. Examples

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resultexact result
actual error===

54.115231635459096537
54.115231635459025483
0.000000000000071054

The program is given below.
#include <stdio.h>#include <math.h>#include <gsl/gsl_integration.h>#include <gsl/gsl_sf_gamma.h>
doublef(double x, void * params){
int m = *(int *) params;
double f = gsl_pow_int(x, m) + 1.0;
return f;}
int
main (int argc, char *argv[]){
gsl_integration_fixed_workspace * w;
const gsl_integration_fixed_type * T = gsl_integration_fixed_hermite;
int m = 10;
int n = 6;
double expected, result;
gsl_function F;
if (argc > 1)
m = atoi(argv[1]);if (argc > 2)n = atoi(argv[2]);w = gsl_integration_fixed_alloc(T, n, 0.0, 1.0, 0.0, 0.0);
F.function = &f;F.params = &m;
gsl_integration_fixed(&F, &result, w);
if (m % 2 == 0)
expected = M_SQRTPI + gsl_sf_gamma(0.5*(1.0 + m));else
expected = M_SQRTPI;
printf ("mprintf ("intervals
printf ("result
printf ("exact result
printf ("actual error= %d\n", m);= %zu\n", gsl_integration_fixed_n(w));= % .18f \n", result);= % .18f \n", expected);= % .18f \n", result - expected);(continues on next page)212

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gsl_integration_fixed_free (w);return 0;}

17.17 References and Further Reading
The following book is the definitive reference for QUADPACK, and was written by the original authors. It provides descriptions of the algorithms, program listings, test programs and examples. It also includes useful advice on numericalintegration and many references to the numerical integration literature used in developing QUADPACK.
• R. Piessens, E. de Doncker-Kapenga, C.W. Ueberhuber, D.K. Kahaner. QUADPACK A subroutine package for
automatic integration Springer Verlag, 1983.
The CQUAD integration algorithm is described in the following paper:• P. Gonnet, “Increasing the Reliability of Adaptive Quadrature Using Explicit Interpolants”, ACM Transactions
on Mathematical Software, Volume 37 (2010), Issue 3, Article 26.
The fixed-point quadrature routines are based on IQPACK, described in the following papers:• S. Elhay, J. Kautsky, Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of Interpolatory Quadrature, ACM Transactions on Mathematical Software, Volume 13, Number 4, December 1987, pages 399-415.
• J. Kautsky, S. Elhay, Calculation of the Weights of Interpolatory Quadratures, Numerische Mathematik, Volume
40, Number 3, October 1982, pages 407-422.

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214Chapter 17. Numerical Integration

CHAPTEREIGHTEENRANDOM NUMBER GENERATION

The library provides a large collection of random number generators which can be accessed through a uniform interface.
Environment variables allow you to select different generators and seeds at runtime, so that you can easily switch
between generators without needing to recompile your program. Each instance of a generator keeps track of its own
state, allowing the generators to be used in multi-threaded programs. Additional functions are available for transforming
uniform random numbers into samples from continuous or discrete probability distributions such as the Gaussian, lognormal or Poisson distributions.
These functions are declared in the header file gsl_rng.h.

18.1 General comments on random numbers
In 1988, Park and Miller wrote a paper entitled “Random number generators: good ones are hard to find.” [Commun.:
ACM, 31, 1192–1201]. Fortunately, some excellent random number generators are available, though poor ones arestill in common use. You may be happy with the system-supplied random number generator on your computer, but
you should be aware that as computers get faster, requirements on random number generators increase. Nowadays, asimulation that calls a random number generator millions of times can often finish before you can make it down thehall to the coffee machine and back.
A very nice review of random number generators was written by Pierre L’Ecuyer, as Chapter 4 of the book: Handbook
on Simulation, Jerry Banks, ed. (Wiley, 1997). The chapter is available in postscript from L’Ecuyer’s ftp site (seereferences). Knuth’s volume on Seminumerical Algorithms (originally published in 1968) devotes 170 pages to random
number generators, and has recently been updated in its 3rd edition (1997). It is brilliant, a classic. If you don’t own it,
you should stop reading right now, run to the nearest bookstore, and buy it.
A good random number generator will satisfy both theoretical and statistical properties. Theoretical properties areoften hard to obtain (they require real math!), but one prefers a random number generator with a long period, low serial
correlation, and a tendency not to “fall mainly on the planes.” Statistical tests are performed with numerical simulations.
Generally, a random number generator is used to estimate some quantity for which the theory of probability provides
an exact answer. Comparison to this exact answer provides a measure of “randomness”.

18.2 The Random Number Generator Interface
It is important to remember that a random number generator is not a “real” function like sine or cosine. Unlike realfunctions, successive calls to a random number generator yield different return values. Of course that is just what you
want for a random number generator, but to achieve this effect, the generator must keep track of some kind of “state”
variable. Sometimes this state is just an integer (sometimes just the value of the previously generated random number),but often it is more complicated than that and may involve a whole array of numbers, possibly with some indices thrown
in. To use the random number generators, you do not need to know the details of what comprises the state, and besides
that varies from algorithm to algorithm.

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type gsl_rng_type
type gsl_rng
The random number generator library uses two special structs, gsl_rng_type which holds static information
about each type of generator and gsl_rng which describes an instance of a generator created from a given
gsl_rng_type.
The functions described in this section are declared in the header file gsl_rng.h.

18.3 Random number generator initialization
gsl_rng *gsl_rng_alloc(const gsl_rng_type *T)This function returns a pointer to a newly-created instance of a random number generator of type T. For example,
the following code creates an instance of the Tausworthe generator:
gsl_rng * r = gsl_rng_alloc (gsl_rng_taus);
If there is insufficient memory to create the generator then the function returns a null pointer and the error handler
is invoked with an error code of GSL_ENOMEM.
The generator is automatically initialized with the default seed, gsl_rng_default_seed. This is zero by default
but can be changed either directly or by using the environment variable GSL_RNG_SEED.
The details of the available generator types are described later in this chapter.
void gsl_rng_set(const gsl_rng *r, unsigned long int s)
This function initializes (or “seeds”) the random number generator. If the generator is seeded with the samevalue of s on two different runs, the same stream of random numbers will be generated by successive calls to theroutines below. If different values of s ≥ 1 are supplied, then the generated streams of random numbers should
be completely different. If the seed s is zero then the standard seed from the original implementation is usedinstead. For example, the original Fortran source code for the ranlux generator used a seed of 314159265, andso choosing s equal to zero reproduces this when using gsl_rng_ranlux.
When using multiple seeds with the same generator, choose seed values greater than zero to avoid collisions withthe default setting.
Note that the most generators only accept 32-bit seeds, with higher values being reduced modulo 232 . For generators with smaller ranges the maximum seed value will typically be lower.
void gsl_rng_free(gsl_rng *r)
This function frees all the memory associated with the generator r.

18.4 Sampling from a random number generator
The following functions return uniformly distributed random numbers, either as integers or double precision floating
point numbers. Inline versions of these functions are used when HAVE_INLINE is defined. To obtain non-uniform
distributions, see Random Number Distributions.
unsigned long int gsl_rng_get(const gsl_rng *r)
This function returns a random integer from the generator r. The minimum and maximum values depend on thealgorithm used, but all integers in the range [min, max] are equally likely. The values of min and max can be
determined using the auxiliary functions gsl_rng_max() and gsl_rng_min().
double gsl_rng_uniform(const gsl_rng *r)
This function returns a double precision floating point number uniformly distributed in the range [0,1). The
range includes 0.0 but excludes 1.0. The value is typically obtained by dividing the result of gsl_rng_get(r)216

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by gsl_rng_max(r) + 1.0 in double precision. Some generators compute this ratio internally so that they canprovide floating point numbers with more than 32 bits of randomness (the maximum number of bits that can be
portably represented in a single unsigned long int).
double gsl_rng_uniform_pos(const gsl_rng *r)
This function returns a positive double precision floating point number uniformly distributed in the range(0,1), excluding both 0.0 and 1.0. The number is obtained by sampling the generator with the algorithm of
gsl_rng_uniform() until a non-zero value is obtained. You can use this function if you need to avoid a singularity at 0.0.
unsigned long int gsl_rng_uniform_int(const gsl_rng *r, unsigned long int n)This function returns a random integer from 0 to n − 1 inclusive by scaling down and/or discarding samples fromthe generator r. All integers in the range [0, n − 1] are produced with equal probability. For generators with a
non-zero minimum value an offset is applied so that zero is returned with the correct probability.
Note that this function is designed for sampling from ranges smaller than the range of the underlying generator.
The parameter n must be less than or equal to the range of the generator r. If n is larger than the range of thegenerator then the function calls the error handler with an error code of GSL_EINVAL and returns zero.
In particular, this function is not intended for generating the full range of unsigned integer values [0, 232 − 1].
Instead choose a generator with the maximal integer range and zero minimum value, such as gsl_rng_ranlxd1,
gsl_rng_mt19937 or gsl_rng_taus, and sample it directly using gsl_rng_get(). The range of each generator can be found using the auxiliary functions described in the next section.

18.5 Auxiliary random number generator functions
The following functions provide information about an existing generator. You should use them in preference to hardcoding the generator parameters into your own code.
const char *gsl_rng_name(const gsl_rng *r)
This function returns a pointer to the name of the generator. For example:
printf ("r is a '%s' generator\n", gsl_rng_name (r));
would print something like:r is a 'taus' generator
unsigned long int gsl_rng_max(const gsl_rng *r)
This function returns the largest value that gsl_rng_get() can return.
unsigned long int gsl_rng_min(const gsl_rng *r)
This function returns the smallest value that gsl_rng_get() can return. Usually this value is zero. There aresome generators with algorithms that cannot return zero, and for these generators the minimum value is 1.
void *gsl_rng_state(const gsl_rng *r)
size_t gsl_rng_size(const gsl_rng *r)
These functions return a pointer to the state of generator r and its size. You can use this information to accessthe state directly. For example, the following code will write the state of a generator to a stream:
void * state = gsl_rng_state (r);size_t n = gsl_rng_size (r);fwrite (state, n, 1, stream);

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const gsl_rng_type **gsl_rng_types_setup(void)
This function returns a pointer to an array of all the available generator types, terminated by a null pointer. The
function should be called once at the start of the program, if needed. The following code fragment shows how
to iterate over the array of generator types to print the names of the available algorithms:
const gsl_rng_type **t, **t0;
t0 = gsl_rng_types_setup ();printf ("Available generators:\n");for (t = t0; *t != 0; t++){printf ("%s\n", (*t)->name);}

18.6 Random number environment variables
The library allows you to choose a default generator and seed from the environment variables GSL_RNG_TYPE andGSL_RNG_SEED and the function gsl_rng_env_setup(). This makes it easy try out different generators and seeds
without having to recompile your program.
GSL_RNG_TYPE
This environment variable specifies the default random number generator. It should be the name of a generator,
such as taus or mt19937.
GSL_RNG_SEEDThis environment variable specifies the default seed for the random number generator
gsl_rng_type *gsl_rng_default
This global library variable specifies the default random number generator, and can be initialized from
GSL_RNG_TYPE using gsl_rng_env_setup(). It is defined as follows:
extern const gsl_rng_type *gsl_rng_default
unsigned long int gsl_rng_default_seed
This global library variable specifies the seed for the default random number generator, and can be initialized
from GSL_RNG_SEED using gsl_rng_env_setup(). It is set to zero by default and is defined as follows:
extern unsigned long int gsl_rng_default_seed
const gsl_rng_type *gsl_rng_env_setup(void)
This function reads the environment variables GSL_RNG_TYPE and GSL_RNG_SEED and uses their values to set
the corresponding library variables gsl_rng_default and gsl_rng_default_seed.
The value of GSL_RNG_SEED is converted to an unsigned long int using the C library function strtoul().
If you don’t specify a generator for GSL_RNG_TYPE then gsl_rng_mt19937 is used as the default. The initialvalue of gsl_rng_default_seed is zero.
Here is a short program which shows how to create a global generator using the environment variables GSL_RNG_TYPE
and GSL_RNG_SEED,#include <stdio.h>#include <gsl/gsl_rng.h>(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)gsl_rng * r;/* global generator */intmain (void){
const gsl_rng_type * T;
gsl_rng_env_setup();
T = gsl_rng_default;r = gsl_rng_alloc (T);printf ("generator type: %s\n", gsl_rng_name (r));
printf ("seed = %lu\n", gsl_rng_default_seed);printf ("first value = %lu\n", gsl_rng_get (r));
gsl_rng_free (r);return 0;}
Running the program without any environment variables uses the initial defaults, an mt19937 generator with a seed of0,
generator type: mt19937
seed = 0
first value = 4293858116
By setting the two variables on the command line we can change the default generator and the seed:$ GSL_RNG_TYPE="taus" GSL_RNG_SEED=123 ./a.out
GSL_RNG_TYPE=taus
GSL_RNG_SEED=123
generator type: taus
seed = 123
first value = 2720986350

18.7 Copying random number generator state
The above methods do not expose the random number state which changes from call to call. It is often useful to be able
to save and restore the state. To permit these practices, a few somewhat more advanced functions are supplied. These
include:int gsl_rng_memcpy(gsl_rng *dest, const gsl_rng *src)
This function copies the random number generator src into the pre-existing generator dest, making dest into
an exact copy of src. The two generators must be of the same type.
gsl_rng *gsl_rng_clone(const gsl_rng *r)
This function returns a pointer to a newly created generator which is an exact copy of the generator r.

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18.8 Reading and writing random number generator state
The library provides functions for reading and writing the random number state to a file as binary data.
int gsl_rng_fwrite(FILE *stream, const gsl_rng *r)
This function writes the random number state of the random number generator r to the stream stream in binary
format. The return value is 0 for success and GSL_EFAILED if there was a problem writing to the file. Since thedata is written in the native binary format it may not be portable between different architectures.
int gsl_rng_fread(FILE *stream, gsl_rng *r)
This function reads the random number state into the random number generator r from the open stream stream inbinary format. The random number generator r must be preinitialized with the correct random number generator
type since type information is not saved. The return value is 0 for success and GSL_EFAILED if there was a
problem reading from the file. The data is assumed to have been written in the native binary format on the samearchitecture.

18.9 Random number generator algorithms
The functions described above make no reference to the actual algorithm used. This is deliberate so that you can switch
algorithms without having to change any of your application source code. The library provides a large number ofgenerators of different types, including simulation quality generators, generators provided for compatibility with otherlibraries and historical generators from the past.
The following generators are recommended for use in simulation. They have extremely long periods, low correlationand pass most statistical tests. For the most reliable source of uncorrelated numbers, the second-generation RANLUX
generators have the strongest proof of randomness.
gsl_rng_type *gsl_rng_mt19937
The MT19937 generator of Makoto Matsumoto and Takuji Nishimura is a variant of the twisted generalized
feedback shift-register algorithm, and is known as the “Mersenne Twister” generator. It has a Mersenne prime
period of 219937 − 1 (about 106000 ) and is equi-distributed in 623 dimensions. It has passed the DIEHARD
statistical tests. It uses 624 words of state per generator and is comparable in speed to the other generators. The
original generator used a default seed of 4357 and choosing s equal to zero in gsl_rng_set() reproduces this.
Later versions switched to 5489 as the default seed, you can choose this explicitly via gsl_rng_set() instead
if you require it.
For more information see,
• Makoto Matsumoto and Takuji Nishimura, “Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator”. ACM Transactions on Modeling and Computer Simulation, Vol.:
8, No.: 1 (Jan. 1998), Pages 3–30
The generator gsl_rng_mt19937 uses the second revision of the seeding procedure published by the two authors
above in 2002. The original seeding procedures could cause spurious artifacts for some seed values. They arestill available through the alternative generators gsl_rng_mt19937_1999 and gsl_rng_mt19937_1998.
gsl_rng_type *gsl_rng_ranlxs0
gsl_rng_type *gsl_rng_ranlxs1
gsl_rng_type *gsl_rng_ranlxs2
The generator ranlxs0 is a second-generation version of the RANLUX algorithm of Luscher, which produces“luxury random numbers”. This generator provides single precision output (24 bits) at three luxury levels
ranlxs0, ranlxs1 and ranlxs2, in increasing order of strength. It uses double-precision floating point arithmetic internally and can be significantly faster than the integer version of ranlux, particularly on 64-bit architectures. The period of the generator is about 10171 . The algorithm has mathematically proven properties and can

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provide truly decorrelated numbers at a known level of randomness. The higher luxury levels provide increased
decorrelation between samples as an additional safety margin.
Note that the range of allowed seeds for this generator is [0, 231 − 1]. Higher seed values are wrapped modulo
231 .
gsl_rng_type *gsl_rng_ranlxd1
gsl_rng_type *gsl_rng_ranlxd2
These generators produce double precision output (48 bits) from the RANLXS generator. The library provides
two luxury levels ranlxd1 and ranlxd2, in increasing order of strength.
gsl_rng_type *gsl_rng_ranlux
gsl_rng_type *gsl_rng_ranlux389
The ranlux generator is an implementation of the original algorithm developed by Luscher. It uses a laggedfibonacci-with-skipping algorithm to produce “luxury random numbers”. It is a 24-bit generator, originallydesigned for single-precision IEEE floating point numbers. This implementation is based on integer arithmetic,
while the second-generation versions RANLXS and RANLXD described above provide floating-point implementations which will be faster on many platforms. The period of the generator is about 10171 . The algorithm
has mathematically proven properties and it can provide truly decorrelated numbers at a known level of randomness. The default level of decorrelation recommended by Luscher is provided by gsl_rng_ranlux, while
gsl_rng_ranlux389 gives the highest level of randomness, with all 24 bits decorrelated. Both types of generator use 24 words of state per generator.
For more information see,• M. Luscher, “A portable high-quality random number generator for lattice field theory calculations”, Computer Physics Communications, 79 (1994) 100–110.
• F. James, “RANLUX: A Fortran implementation of the high-quality pseudo-random number generator of
Luscher”, Computer Physics Communications, 79 (1994) 111–114
gsl_rng_type *gsl_rng_cmrg
This is a combined multiple recursive generator by L’Ecuyer. Its sequence is,zn = (xn − yn )mod m1

where the two underlying generators xn and yn are,xn = (a1 xn−1 + a2 xn−2 + a3 xn−3 )mod m1yn = (b1 yn−1 + b2 yn−2 + b3 yn−3 )mod m2

with coefficients a1 = 0, a2 = 63308, a3 = −183326, b1 = 86098, b2 = 0, b3 = −539608, and moduli
m1 = 231 − 1 = 2147483647 and m2 = 2145483479.
The period of this generator is lcm(m31 − 1, m32 − 1), which is approximately 2185 (about 1056 ). It uses 6 wordsof state per generator. For more information see,• P. L’Ecuyer, “Combined Multiple Recursive Random Number Generators”, Operations Research, 44, 5(1996), 816–822.
gsl_rng_type *gsl_rng_mrg
This is a fifth-order multiple recursive generator by L’Ecuyer, Blouin and Coutre. Its sequence is,xn = (a1 xn−1 + a5 xn−5 )mod mwith a1 = 107374182, a2 = a3 = a4 = 0, a5 = 104480 and m = 231 − 1.
The period of this generator is about 1046 . It uses 5 words of state per generator. More information can be foundin the following paper,

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GNU Scientific Library, Release 2.7• P. L’Ecuyer, F. Blouin, and R. Coutre, “A search for good multiple recursive random number generators”,
ACM Transactions on Modeling and Computer Simulation 3, 87–98 (1993).
gsl_rng_type *gsl_rng_taus
gsl_rng_type *gsl_rng_taus2
This is a maximally equidistributed combined Tausworthe generator by L’Ecuyer. The sequence is,xn = (s1n ⊕ s2n ⊕ s3n )where,
s1n+1 = (((s1n &4294967294)  12) ⊕ (((s1n  13) ⊕ s1n )  19))s2n+1 = (((s2n &4294967288)  4) ⊕ (((s2n  2) ⊕ s2n )  25))
s3n+1 = (((s3n &4294967280)  17) ⊕ (((s3n  3) ⊕ s3n )  11))
computed modulo 232 . In the formulas above ⊕ denotes exclusive-or. Note that the algorithm relies on theproperties of 32-bit unsigned integers and has been implemented using a bitmask of 0xFFFFFFFF to make itwork on 64 bit machines.
The period of this generator is 288 (about 1026 ). It uses 3 words of state per generator. For more information see,• P. L’Ecuyer, “Maximally Equidistributed Combined Tausworthe Generators”, Mathematics of Computation, 65, 213 (1996), 203–213.
The generator gsl_rng_taus2 uses the same algorithm as gsl_rng_taus but with an improved seeding procedure described in the paper,• P. L’Ecuyer, “Tables of Maximally Equidistributed Combined LFSR Generators”, Mathematics of Computation, 68, 225 (1999), 261–269
The generator gsl_rng_taus2 should now be used in preference to gsl_rng_taus.
gsl_rng_type *gsl_rng_gfsr4
The gfsr4 generator is like a lagged-fibonacci generator, and produces each number as an xor’d sum of four
previous values.
rn = rn−A ⊕ rn−B ⊕ rn−C ⊕ rn−D
Ziff (ref below) notes that “it is now widely known” that two-tap registers (such as R250, which is described
below) have serious flaws, the most obvious one being the three-point correlation that comes from the definitionof the generator. Nice mathematical properties can be derived for GFSR’s, and numerics bears out the claim that
4-tap GFSR’s with appropriately chosen offsets are as random as can be measured, using the author’s test.
This implementation uses the values suggested the example on p392 of Ziff’s article: A = 471, B = 1586,
C = 6988, D = 9689.
If the offsets are appropriately chosen (such as the one ones in this implementation), then the sequence is said to
be maximal; that means that the period is 2D − 1, where D is the longest lag. (It is one less than 2D because it
is not permitted to have all zeros in the ra[] array.) For this implementation with D = 9689 that works out toabout 102917 .
Note that the implementation of this generator using a 32-bit integer amounts to 32 parallel implementationsof one-bit generators. One consequence of this is that the period of this 32-bit generator is the same as for the
one-bit generator. Moreover, this independence means that all 32-bit patterns are equally likely, and in particularthat 0 is an allowed random value. (We are grateful to Heiko Bauke for clarifying for us these properties of GFSR
random number generators.)
For more information see,
• Robert M. Ziff, “Four-tap shift-register-sequence random-number generators”, Computers in Physics,
12(4), Jul/Aug 1998, pp 385–392.
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18.10 Unix random number generators
The standard Unix random number generators rand, random and rand48 are provided as part of GSL. Althoughthese generators are widely available individually often they aren’t all available on the same platform. This makes it
difficult to write portable code using them and so we have included the complete set of Unix generators in GSL for
convenience. Note that these generators don’t produce high-quality randomness and aren’t suitable for work requiring
accurate statistics. However, if you won’t be measuring statistical quantities and just want to introduce some variation
into your program then these generators are quite acceptable.
gsl_rng_type *gsl_rng_rand
This is the BSD rand generator. Its sequence isxn+1 = (axn + c)

mod mwith a = 1103515245, c = 12345 and m = 231 . The seed specifies the initial value, x1 . The period of thisgenerator is 231 , and it uses 1 word of storage per generator.
gsl_rng_type *gsl_rng_random_bsd
gsl_rng_type *gsl_rng_random_libc5
gsl_rng_type *gsl_rng_random_glibc2
These generators implement the random family of functions, a set of linear feedback shift register generators
originally used in BSD Unix. There are several versions of random in use today: the original BSD version (e.g.
on SunOS4), a libc5 version (found on older GNU/Linux systems) and a glibc2 version. Each version uses a
different seeding procedure, and thus produces different sequences.
The original BSD routines accepted a variable length buffer for the generator state, with longer buffers providing
higher-quality randomness. The random function implemented algorithms for buffer lengths of 8, 32, 64, 128
and 256 bytes, and the algorithm with the largest length that would fit into the user-supplied buffer was used. To
support these algorithms additional generators are available with the following names:
gsl_rng_random8_bsd
gsl_rng_random32_bsd
gsl_rng_random64_bsd
gsl_rng_random128_bsd
gsl_rng_random256_bsd
where the numeric suffix indicates the buffer length. The original BSD random function used a 128-byte default
buffer and so gsl_rng_random_bsd has been made equivalent to gsl_rng_random128_bsd. Corresponding
versions of the libc5 and glibc2 generators are also available, with the names gsl_rng_random8_libc5,
gsl_rng_random8_glibc2, etc.
gsl_rng_type *gsl_rng_rand48
This is the Unix rand48 generator. Its sequence isxn+1 = (axn + c)

mod m

defined on 48-bit unsigned integers with a = 25214903917, c = 11 and m = 248 . The seed specifies the upper 32
bits of the initial value, x1 , with the lower 16 bits set to 0x330E. The function gsl_rng_get() returns the upper
32 bits from each term of the sequence. This does not have a direct parallel in the original rand48 functions,but forcing the result to type long int reproduces the output of mrand48. The function gsl_rng_uniform()
uses the full 48 bits of internal state to return the double precision number xn /m, which is equivalent to thefunction drand48. Note that some versions of the GNU C Library contained a bug in mrand48 function whichcaused it to produce different results (only the lower 16-bits of the return value were set).

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18.11 Other random number generators
The generators in this section are provided for compatibility with existing libraries. If you are converting an existing
program to use GSL then you can select these generators to check your new implementation against the original one,using the same random number generator. After verifying that your new program reproduces the original results you
can then switch to a higher-quality generator.
Note that most of the generators in this section are based on single linear congruence relations, which are the least
sophisticated type of generator. In particular, linear congruences have poor properties when used with a non-prime
modulus, as several of these routines do (e.g. with a power of two modulus, 231 or 232 ). This leads to periodicity in theleast significant bits of each number, with only the higher bits having any randomness. Thus if you want to produce a
random bitstream it is best to avoid using the least significant bits.
gsl_rng_type *gsl_rng_ranf
This is the CRAY random number generator RANF. Its sequence isxn+1 = (axn )mod mdefined on 48-bit unsigned integers with a = 44485709377909 and m = 248 . The seed specifies the lower 32
bits of the initial value, x1 , with the lowest bit set to prevent the seed taking an even value. The upper 16 bitsof x1 are set to 0. A consequence of this procedure is that the pairs of seeds 2 and 3, 4 and 5, etc.: produce thesame sequences.
The generator compatible with the CRAY MATHLIB routine RANF. It produces double precision floating point
numbers which should be identical to those from the original RANF.
There is a subtlety in the implementation of the seeding. The initial state is reversed through one step, by multiplying by the modular inverse of a mod m. This is done for compatibility with the original CRAY implementation.
Note that you can only seed the generator with integers up to 232 , while the original CRAY implementation uses
non-portable wide integers which can cover all 248 states of the generator.
The function gsl_rng_get() returns the upper 32 bits from each term of the sequence. The function
gsl_rng_uniform() uses the full 48 bits to return the double precision number xn /m.
The period of this generator is 246 .
gsl_rng_type *gsl_rng_ranmar
This is the RANMAR lagged-fibonacci generator of Marsaglia, Zaman and Tsang. It is a 24-bit generator, originally designed for single-precision IEEE floating point numbers. It was included in the CERNLIB high-energy
physics library.
gsl_rng_type *gsl_rng_r250
This is the shift-register generator of Kirkpatrick and Stoll. The sequence is based on the recurrence
xn = xn−103 ⊕ xn−250
where ⊕ denotes exclusive-or, defined on 32-bit words. The period of this generator is about 2250 and it uses
250 words of state per generator.
For more information see,
• S. Kirkpatrick and E. Stoll, “A very fast shift-register sequence random number generator”, Journal ofComputational Physics, 40, 517–526 (1981)
gsl_rng_type *gsl_rng_tt800
This is an earlier version of the twisted generalized feedback shift-register generator, and has been supersededby the development of MT19937. However, it is still an acceptable generator in its own right. It has a period of
2800 and uses 33 words of storage per generator.
For more information see,
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GNU Scientific Library, Release 2.7• Makoto Matsumoto and Yoshiharu Kurita, “Twisted GFSR Generators II”, ACM Transactions on Modelling
and Computer Simulation, Vol.: 4, No.: 3, 1994, pages 254–266.
gsl_rng_type *gsl_rng_vax
This is the VAX generator MTH$RANDOM. Its sequence is,xn+1 = (axn + c)

mod mwith a = 69069, c = 1 and m = 232 . The seed specifies the initial value, x1 . The period of this generator is 232
and it uses 1 word of storage per generator.
gsl_rng_type *gsl_rng_transputerThis is the random number generator from the INMOS Transputer Development system. Its sequence is,xn+1 = (axn )mod mwith a = 1664525 and m = 232 . The seed specifies the initial value, x1 .
gsl_rng_type *gsl_rng_randu
This is the IBM RANDU generator. Its sequence isxn+1 = (axn )mod mwith a = 65539 and m = 231 . The seed specifies the initial value, x1 . The period of this generator was only
229 . It has become a textbook example of a poor generator.
gsl_rng_type *gsl_rng_minstd
This is Park and Miller’s “minimal standard” MINSTD generator, a simple linear congruence which takes careto avoid the major pitfalls of such algorithms. Its sequence is,xn+1 = (axn )mod mwith a = 16807 and m = 231 − 1 = 2147483647. The seed specifies the initial value, x1 . The period of thisgenerator is about 231 .
This generator was used in the IMSL Library (subroutine RNUN) and in MATLAB (the RAND function) in thepast. It is also sometimes known by the acronym “GGL” (I’m not sure what that stands for).
For more information see,• Park and Miller, “Random Number Generators: Good ones are hard to find”, Communications of the ACM,
October 1988, Volume 31, No 10, pages 1192–1201.
gsl_rng_type *gsl_rng_uni
gsl_rng_type *gsl_rng_uni32
This is a reimplementation of the 16-bit SLATEC random number generator RUNIF. A generalization of thegenerator to 32 bits is provided by gsl_rng_uni32. The original source code is available from NETLIB.
gsl_rng_type *gsl_rng_slatec
This is the SLATEC random number generator RAND. It is ancient. The original source code is available fromNETLIB.
gsl_rng_type *gsl_rng_zuf
This is the ZUFALL lagged Fibonacci series generator of Peterson. Its sequence is,t = un−273 + un−607
un = t − floor(t)The original source code is available from NETLIB. For more information see,18.11. Other random number generators

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GNU Scientific Library, Release 2.7• W. Petersen, “Lagged Fibonacci Random Number Generators for the NEC SX-3”, International Journal ofHigh Speed Computing (1994).
gsl_rng_type *gsl_rng_knuthran2
This is a second-order multiple recursive generator described by Knuth in Seminumerical Algorithms, 3rd Ed.,
page 108. Its sequence is,xn = (a1 xn−1 + a2 xn−2 )mod mwith a1 = 271828183, a2 = 314159269, and m = 231 − 1.
gsl_rng_type *gsl_rng_knuthran2002
gsl_rng_type *gsl_rng_knuthranThis is a second-order multiple recursive generator described by Knuth in Seminumerical Algorithms, 3rd Ed.,
Section 3.6. Knuth provides its C code. The updated routine gsl_rng_knuthran2002 is from the revised 9th
printing and corrects some weaknesses in the earlier version, which is implemented as gsl_rng_knuthran.
gsl_rng_type *gsl_rng_borosh13
gsl_rng_type *gsl_rng_fishman18
gsl_rng_type *gsl_rng_fishman20
gsl_rng_type *gsl_rng_lecuyer21
gsl_rng_type *gsl_rng_waterman14
These multiplicative generators are taken from Knuth’s Seminumerical Algorithms, 3rd Ed., pages 106–108.
Their sequence is,xn+1 = (axn )mod mwhere the seed specifies the initial value, x1 . The parameters a and m are as follows, Borosh-Niederreiter:a = 1812433253, m = 232 , Fishman18: a = 62089911, m = 231 − 1, Fishman20: a = 48271, m = 231 − 1,
L’Ecuyer: a = 40692, m = 231 − 249, Waterman: a = 1566083941, m = 232 .
gsl_rng_type *gsl_rng_fishman2x
This is the L’Ecuyer–Fishman random number generator. It is taken from Knuth’s Seminumerical Algorithms,
3rd Ed., page 108. Its sequence is,zn+1 = (xn − yn )mod mwith m = 231 − 1. xn and yn are given by the fishman20 and lecuyer21 algorithms. The seed specifies theinitial value, x1 .
gsl_rng_type *gsl_rng_coveyou
This is the Coveyou random number generator. It is taken from Knuth’s Seminumerical Algorithms, 3rd Ed.,
Section 3.2.2. Its sequence is,xn+1 = (xn (xn + 1))

mod mwith m = 232 . The seed specifies the initial value, x1 .

18.12 Performance
The following table shows the relative performance of a selection the available random number generators. The fastest
simulation quality generators are taus, gfsr4 and mt19937. The generators which offer the best mathematicallyproven quality are those based on the RANLUX algorithm:

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1754 k ints/sec,
1613 k ints/sec,
1370 k ints/sec,
565 k ints/sec,
400 k ints/sec,
490 k ints/sec,
407 k ints/sec,
243 k ints/sec,
251 k ints/sec,
238 k ints/sec,
247 k ints/sec,
141 k ints/sec,

870 k doubles/sec, taus
855 k doubles/sec, gfsr4
769 k doubles/sec, mt19937
571 k doubles/sec, ranlxs0
405 k doubles/sec, ranlxs1
389 k doubles/sec, mrg
297 k doubles/sec, ranlux
254 k doubles/sec, ranlxd1
253 k doubles/sec, ranlxs2
215 k doubles/sec, cmrg
198 k doubles/sec, ranlux389
140 k doubles/sec, ranlxd2

18.13 Examples
The following program demonstrates the use of a random number generator to produce uniform random numbers in
the range [0.0, 1.0),#include <stdio.h>#include <gsl/gsl_rng.h>
int
main (void){
const gsl_rng_type * T;
gsl_rng * r;
int i, n = 10;
gsl_rng_env_setup();
T = gsl_rng_default;r = gsl_rng_alloc (T);for (i = 0; i < n; i++){double u = gsl_rng_uniform (r);printf ("%.5f \n", u);}
gsl_rng_free (r);return 0;}
Here is the output of the program,
0.99974
0.162910.282620.94720(continues on next page)

18.13. Examples

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GNU Scientific Library, Release 2.7(continued from previous page)0.23166
0.48497
0.957480.744310.54004
0.73995
The numbers depend on the seed used by the generator. The default seed can be changed with the GSL_RNG_SEED
environment variable to produce a different stream of numbers. The generator itself can be changed using the environment variable GSL_RNG_TYPE. Here is the output of the program using a seed value of 123 and the multiple-recursive
generator mrg:
$ GSL_RNG_SEED=123 GSL_RNG_TYPE=mrg ./a.out
0.33050
0.866310.32982
0.676200.53391
0.06457
0.16847
0.70229
0.04371
0.86374

18.14 References and Further Reading
The subject of random number generation and testing is reviewed extensively in Knuth’s Seminumerical Algorithms.
• Donald E. Knuth, The Art of Computer Programming: Seminumerical Algorithms (Vol 2, 3rd Ed, 1997),
Addison-Wesley, ISBN 0201896842.
Further information is available in the review paper written by Pierre L’Ecuyer,• P. L’Ecuyer, “Random Number Generation”, Chapter 4 of the Handbook on Simulation, Jerry Banks Ed., Wiley,
1998, 93–137.
• http://www.iro.umontreal.ca/~lecuyer/papers.html in the file handsim.ps.
The source code for the DIEHARD random number generator tests is also available online,• DIEHARD source code, G. Marsaglia, http://stat.fsu.edu/pub/diehard/
A comprehensive set of random number generator tests is available from NIST,• NIST Special Publication 800-22, “A Statistical Test Suite for the Validation of Random Number Generators andPseudo Random Number Generators for Cryptographic Applications”.
• http://csrc.nist.gov/rng/

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18.15 Acknowledgements
Thanks to Makoto Matsumoto, Takuji Nishimura and Yoshiharu Kurita for making the source code to their generators(MT19937, MM&TN; TT800, MM&YK) available under the GNU General Public License. Thanks to Martin Luscher
for providing notes and source code for the RANLXS and RANLXD generators.

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CHAPTERNINETEENQUASI-RANDOM SEQUENCES

This chapter describes functions for generating quasi-random sequences in arbitrary dimensions. A quasi-random
sequence progressively covers a d-dimensional space with a set of points that are uniformly distributed. Quasi-random
sequences are also known as low-discrepancy sequences. The quasi-random sequence generators use an interface thatis similar to the interface for random number generators, except that seeding is not required—each generator produces
a single sequence.
The functions described in this section are declared in the header file gsl_qrng.h.

19.1 Quasi-random number generator initialization
type gsl_qrng
This is a workspace for computing quasi-random sequences.
gsl_qrng *gsl_qrng_alloc(const gsl_qrng_type *T, unsigned int d)
This function returns a pointer to a newly-created instance of a quasi-random sequence generator of type T anddimension d. If there is insufficient memory to create the generator then the function returns a null pointer andthe error handler is invoked with an error code of GSL_ENOMEM.
void gsl_qrng_free(gsl_qrng *q)
This function frees all the memory associated with the generator q.
void gsl_qrng_init(gsl_qrng *q)
This function reinitializes the generator q to its starting point. Note that quasi-random sequences do not use aseed and always produce the same set of values.

19.2 Sampling from a quasi-random number generator
int gsl_qrng_get(const gsl_qrng *q, double x[])This function stores the next point from the sequence generator q in the array x. The space available for x mustmatch the dimension of the generator. The point x will lie in the range 0 < xi < 1 for each xi . An inline versionof this function is used when HAVE_INLINE is defined.

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19.3 Auxiliary quasi-random number generator functions
const char *gsl_qrng_name(const gsl_qrng *q)
This function returns a pointer to the name of the generator.
size_t gsl_qrng_size(const gsl_qrng *q)
void *gsl_qrng_state(const gsl_qrng *q)
These functions return a pointer to the state of generator r and its size. You can use this information to accessthe state directly. For example, the following code will write the state of a generator to a stream:
void * state = gsl_qrng_state (q);size_t n = gsl_qrng_size (q);fwrite (state, n, 1, stream);

19.4 Saving and restoring quasi-random number generator state
int gsl_qrng_memcpy(gsl_qrng *dest, const gsl_qrng *src)
This function copies the quasi-random sequence generator src into the pre-existing generator dest, making
dest into an exact copy of src. The two generators must be of the same type.
gsl_qrng *gsl_qrng_clone(const gsl_qrng *q)
This function returns a pointer to a newly created generator which is an exact copy of the generator q.

19.5 Quasi-random number generator algorithms
The following quasi-random sequence algorithms are available,
type gsl_qrng_type
gsl_qrng_type *gsl_qrng_niederreiter_2
This generator uses the algorithm described in Bratley, Fox, Niederreiter, ACM Trans. Model. Comp. Sim.
2, 195 (1992). It is valid up to 12 dimensions.
gsl_qrng_type *gsl_qrng_sobol
This generator uses the Sobol sequence described in Antonov, Saleev, USSR Comput. Maths. Math. Phys.
19, 252 (1980). It is valid up to 40 dimensions.
gsl_qrng_type *gsl_qrng_halton
gsl_qrng_type *gsl_qrng_reversehalton
These generators use the Halton and reverse Halton sequences described in J.H. Halton, Numerische Mathematik, 2, 84-90 (1960) and B. Vandewoestyne and R. Cools Computational and Applied Mathematics,
189, 1&2, 341-361 (2006). They are valid up to 1229 dimensions.

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19.6 Examples
The following program prints the first 1024 points of the 2-dimensional Sobol sequence.
#include <stdio.h>#include <gsl/gsl_qrng.h>
int
main (void){
int i;
gsl_qrng * q = gsl_qrng_alloc (gsl_qrng_sobol, 2);
for (i = 0; i < 1024; i++){double v[2];gsl_qrng_get (q, v);
printf ("%.5f %.5f \n", v[0], v[1]);}gsl_qrng_free (q);return 0;}
Here is the output from the program:$ ./a.out
0.50000 0.50000
0.75000 0.25000
0.25000 0.75000
0.37500 0.37500
0.87500 0.87500
0.62500 0.12500
0.12500 0.62500....
It can be seen that successive points progressively fill-in the spaces between previous points.
Fig. 19.1 shows the distribution in the x-y plane of the first 1024 points from the Sobol sequence,

19.7 References
The implementations of the quasi-random sequence routines are based on the algorithms described in the following
paper,• P. Bratley and B.L. Fox and H. Niederreiter, “Algorithm 738: Programs to Generate Niederreiter’s Lowdiscrepancy Sequences”, ACM Transactions on Mathematical Software, Vol.: 20, No.: 4, December, 1994,
p.: 494–495.

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Chapter 19. Quasi-Random Sequences

CHAPTERTWENTYRANDOM NUMBER DISTRIBUTIONSThis chapter describes functions for generating random variates and computing their probability distributions. Samples
from the distributions described in this chapter can be obtained using any of the random number generators in the libraryas an underlying source of randomness.
In the simplest cases a non-uniform distribution can be obtained analytically from the uniform distribution of a random
number generator by applying an appropriate transformation. This method uses one call to the random number generator. More complicated distributions are created by the acceptance-rejection method, which compares the desired
distribution against a distribution which is similar and known analytically. This usually requires several samples fromthe generator.
The library also provides cumulative distribution functions and inverse cumulative distribution functions, sometimes
referred to as quantile functions. The cumulative distribution functions and their inverses are computed separately forthe upper and lower tails of the distribution, allowing full accuracy to be retained for small results.
The functions for random variates and probability density functions described in this section are declared in
gsl_randist.h. The corresponding cumulative distribution functions are declared in gsl_cdf.h.
Note that the discrete random variate functions always return a value of type unsigned int, and on most platforms
this has a maximum value of
232 − 1 ≈ 4.29 × 109
They should only be called with a safe range of parameters (where there is a negligible probability of a variate exceeding
this limit) to prevent incorrect results due to overflow.

20.1 Introduction
Continuous random number distributions are defined by a probability density function, p(x), such that the probabilityof x occurring in the infinitesimal range x to x + dx is p(x)dx.
The cumulative distribution function for the lower tail P (x) is defined by the integral,
Z x
P (x) =
dx0 p(x0 )−∞and gives the probability of a variate taking a value less than x.
The cumulative distribution function for the upper tail Q(x) is defined by the integral,Z +∞Q(x) =

dx0 p(x0 )xand gives the probability of a variate taking a value greater than x.

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The upper and lower cumulative distribution functions are related by P (x) + Q(x) = 1 and satisfy 0 ≤ P (x) ≤ 1,
0 ≤ Q(x) ≤ 1.
The inverse cumulative distributions, x = P −1 (P ) and x = Q−1 (Q) give the values of x which correspond to a
specific value of P or Q. They can be used to find confidence limits from probability values.
P
For discrete distributions the probability of sampling the integer value k is given by p(k), where k p(k) = 1. The
cumulative distribution for the lower tail P (k) of a discrete distribution is defined as,X
P (k) =p(i)i≤kwhere the sum is over the allowed range of the distribution less than or equal to k.
The cumulative distribution for the upper tail of a discrete distribution Q(k) is defined as
X
Q(k) =p(i)i>kgiving the sum of probabilities for all values greater than k. These two definitions satisfy the identity P (k)+Q(k) = 1.
If the range of the distribution is 1 to n inclusive then P (n) = 1, Q(n) = 0 while P (1) = p(1), Q(1) = 1 − p(1).

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20.2 The Gaussian Distribution
double gsl_ran_gaussian(const gsl_rng *r, double sigma)
This function returns a Gaussian random variate, with mean zero and standard deviation sigma. The probability
distribution for Gaussian random variates is,p(x)dx = √

1
2πσ 2

exp(−x2 /2σ 2 )dx

for x in the range −∞ to +∞. Use the transformation z = µ + x on the numbers returned by
gsl_ran_gaussian() to obtain a Gaussian distribution with mean µ. This function uses the Box-Muller algorithm which requires two calls to the random number generator r.
double gsl_ran_gaussian_pdf(double x, double sigma)
This function computes the probability density p(x) at x for a Gaussian distribution with standard deviation
sigma, using the formula given above.

double gsl_ran_gaussian_ziggurat(const gsl_rng *r, double sigma)
double gsl_ran_gaussian_ratio_method(const gsl_rng *r, double sigma)
This function computes a Gaussian random variate using the alternative Marsaglia-Tsang ziggurat andKinderman-Monahan-Leva ratio methods. The Ziggurat algorithm is the fastest available algorithm in mostcases.
double gsl_ran_ugaussian(const gsl_rng *r)

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double gsl_ran_ugaussian_pdf(double x)
double gsl_ran_ugaussian_ratio_method(const gsl_rng *r)
These functions compute results for the unit Gaussian distribution. They are equivalent to the functions above
with a standard deviation of one, sigma = 1.
double gsl_cdf_gaussian_P(double x, double sigma)
double gsl_cdf_gaussian_Q(double x, double sigma)
double gsl_cdf_gaussian_Pinv(double P, double sigma)
double gsl_cdf_gaussian_Qinv(double Q, double sigma)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the Gaussian
distribution with standard deviation sigma.
double gsl_cdf_ugaussian_P(double x)
double gsl_cdf_ugaussian_Q(double x)
double gsl_cdf_ugaussian_Pinv(double P)
double gsl_cdf_ugaussian_Qinv(double Q)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the unit Gaussian distribution.

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20.3 The Gaussian Tail Distribution
double gsl_ran_gaussian_tail(const gsl_rng *r, double a, double sigma)
This function provides random variates from the upper tail of a Gaussian distribution with standard deviation
sigma. The values returned are larger than the lower limit a, which must be positive. The method is based on
Marsaglia’s famous rectangle-wedge-tail algorithm (Ann. Math. Stat. 32, 894–899 (1961)), with this aspect
explained in Knuth, v2, 3rd ed, p139,586 (exercise 11).
The probability distribution for Gaussian tail random variates is,p(x)dx =1√exp(−x2 /2σ 2 )dx
2N (a; σ) 2πσ

for x > a where N (a; σ) is the normalization constant,
N (a; σ) =

1
erfc
2


√

a
2σ 2


.

double gsl_ran_gaussian_tail_pdf(double x, double a, double sigma)
This function computes the probability density p(x) at x for a Gaussian tail distribution with standard deviation
sigma and lower limit a, using the formula given above.

double gsl_ran_ugaussian_tail(const gsl_rng *r, double a)double gsl_ran_ugaussian_tail_pdf(double x, double a)
These functions compute results for the tail of a unit Gaussian distribution. They are equivalent to the functions
above with a standard deviation of one, sigma = 1.
20.3. The Gaussian Tail Distribution

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20.4 The Bivariate Gaussian Distribution
void gsl_ran_bivariate_gaussian(const gsl_rng *r, double sigma_x, double sigma_y, double rho, double *x,
double *y)This function generates a pair of correlated Gaussian variates, with mean zero, correlation coefficient rho andstandard deviations sigma_x and sigma_y in the x and y directions. The probability distribution for bivariate
Gaussian random variates is,!
(x2 /σx2 + y 2 /σy2 − 2ρxy/(σx σy ))
1
p
exp −
dxdy
p(x, y)dxdy =2(1 − ρ2 )2πσx σy 1 − ρ2for x, y in the range −∞ to +∞. The correlation coefficient rho should lie between 1 and −1.
double gsl_ran_bivariate_gaussian_pdf(double x, double y, double sigma_x, double sigma_y, double rho)This function computes the probability density p(x, y) at (x, y) for a bivariate Gaussian distribution with standard
deviations sigma_x, sigma_y and correlation coefficient rho, using the formula given above.

20.4. The Bivariate Gaussian Distribution

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20.5 The Multivariate Gaussian Distribution
int gsl_ran_multivariate_gaussian(const gsl_rng *r, const gsl_vector *mu, const gsl_matrix *L, gsl_vector*result)
This function generates a random vector satisfying the k-dimensional multivariate Gaussian distribution withmean µ and variance-covariance matrix Σ. On input, the k-vector µ is given in mu, and the Cholesky factor of the
k-by-k matrix Σ = LLT is given in the lower triangle of L, as output from gsl_linalg_cholesky_decomp().
The random vector is stored in result on output. The probability distribution for multivariate Gaussian randomvariates is11
exp − (x − µ)T Σ−1 (x − µ) dx1 . . . dxk
p(x1 , . . . , xk )dx1 . . . dxk = p2(2π)k |Σ|int gsl_ran_multivariate_gaussian_pdf(const gsl_vector *x, const gsl_vector *mu, const gsl_matrix *L,
double *result, gsl_vector *work)
int gsl_ran_multivariate_gaussian_log_pdf(const gsl_vector *x, const gsl_vector *mu, const gsl_matrix *L,
double *result, gsl_vector *work)
These functions compute p(x) or log p(x) at the point x, using mean vector mu and variance-covariance matrix
specified by its Cholesky factor L using the formula above. Additional workspace of length k is required in work.
int gsl_ran_multivariate_gaussian_mean(const gsl_matrix *X, gsl_vector *mu_hat)
Given a set of n samples Xj from a k-dimensional multivariate Gaussian distribution, this function computes
the maximum likelihood estimate of the mean of the distribution, given by
n

1X
Xj
µ̂ =
n j=1
The samples X1 , X2 , . . . , Xn are given in the n-by-k matrix X, and the maximum likelihood estimate of themean is stored in mu_hat on output.
int gsl_ran_multivariate_gaussian_vcov(const gsl_matrix *X, gsl_matrix *sigma_hat)
Given a set of n samples Xj from a k-dimensional multivariate Gaussian distribution, this function computes
the maximum likelihood estimate of the variance-covariance matrix of the distribution, given by
n

Σ̂ =

1X
T
(Xj − µ̂) (Xj − µ̂)
n j=1

The samples X1 , X2 , . . . , Xn are given in the n-by-k matrix X and the maximum likelihood estimate of thevariance-covariance matrix is stored in sigma_hat on output.

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20.6 The Exponential Distribution
double gsl_ran_exponential(const gsl_rng *r, double mu)
This function returns a random variate from the exponential distribution with mean mu. The distribution is,p(x)dx =1exp(−x/µ)dxµ

for x ≥ 0.
double gsl_ran_exponential_pdf(double x, double mu)
This function computes the probability density p(x) at x for an exponential distribution with mean mu, using theformula given above.

double gsl_cdf_exponential_P(double x, double mu)
double gsl_cdf_exponential_Q(double x, double mu)
double gsl_cdf_exponential_Pinv(double P, double mu)
double gsl_cdf_exponential_Qinv(double Q, double mu)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the exponential
distribution with mean mu.

20.6. The Exponential Distribution

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20.7 The Laplace Distribution
double gsl_ran_laplace(const gsl_rng *r, double a)This function returns a random variate from the Laplace distribution with width a. The distribution is,p(x)dx =1exp(−|x/a|)dx
2afor −∞ < x < ∞.
double gsl_ran_laplace_pdf(double x, double a)This function computes the probability density p(x) at x for a Laplace distribution with width a, using theformula given above.

double gsl_cdf_laplace_P(double x, double a)double gsl_cdf_laplace_Q(double x, double a)double gsl_cdf_laplace_Pinv(double P, double a)double gsl_cdf_laplace_Qinv(double Q, double a)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the Laplace
distribution with width a.

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20.8 The Exponential Power Distribution
double gsl_ran_exppow(const gsl_rng *r, double a, double b)
This function returns a random variate from the exponential power distribution with scale parameter a and exponent b. The distribution is,p(x)dx =1exp(−|x/a|b )dx
2aΓ(1 + 1/b)

for x ≥ 0. For b = 1 this√reduces to the Laplace distribution. For b = 2 it has the same form as a Gaussian
distribution, but with a = 2σ.
double gsl_ran_exppow_pdf(double x, double a, double b)
This function computes the probability density p(x) at x for an exponential power distribution with scale parameter a and exponent b, using the formula given above.

double gsl_cdf_exppow_P(double x, double a, double b)
double gsl_cdf_exppow_Q(double x, double a, double b)
These functions compute the cumulative distribution functions P (x), Q(x) for the exponential power distribution
with parameters a and b.

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20.9 The Cauchy Distribution
double gsl_ran_cauchy(const gsl_rng *r, double a)This function returns a random variate from the Cauchy distribution with scale parameter a. The probability
distribution for Cauchy random variates is,p(x)dx =1dxaπ(1 + (x/a)2 )for x in the range −∞ to +∞. The Cauchy distribution is also known as the Lorentz distribution.
double gsl_ran_cauchy_pdf(double x, double a)This function computes the probability density p(x) at x for a Cauchy distribution with scale parameter a, using
the formula given above.

double gsl_cdf_cauchy_P(double x, double a)double gsl_cdf_cauchy_Q(double x, double a)double gsl_cdf_cauchy_Pinv(double P, double a)double gsl_cdf_cauchy_Qinv(double Q, double a)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the Cauchy
distribution with scale parameter a.

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20.10 The Rayleigh Distribution
double gsl_ran_rayleigh(const gsl_rng *r, double sigma)
This function returns a random variate from the Rayleigh distribution with scale parameter sigma. The distribution is,p(x)dx =xexp(−x2 /(2σ 2 ))dx
σ2

for x > 0.
double gsl_ran_rayleigh_pdf(double x, double sigma)
This function computes the probability density p(x) at x for a Rayleigh distribution with scale parameter sigma,
using the formula given above.

double gsl_cdf_rayleigh_P(double x, double sigma)
double gsl_cdf_rayleigh_Q(double x, double sigma)
double gsl_cdf_rayleigh_Pinv(double P, double sigma)
double gsl_cdf_rayleigh_Qinv(double Q, double sigma)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the Rayleighdistribution with scale parameter sigma.

20.10. The Rayleigh Distribution

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20.11 The Rayleigh Tail Distribution
double gsl_ran_rayleigh_tail(const gsl_rng *r, double a, double sigma)
This function returns a random variate from the tail of the Rayleigh distribution with scale parameter sigma anda lower limit of a. The distribution is,p(x)dx =xexp((a2 − x2 )/(2σ 2 ))dx
σ2

for x > a.
double gsl_ran_rayleigh_tail_pdf(double x, double a, double sigma)
This function computes the probability density p(x) at x for a Rayleigh tail distribution with scale parametersigma and lower limit a, using the formula given above.

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20.12 The Landau Distribution
double gsl_ran_landau(const gsl_rng *r)
This function returns a random variate from the Landau distribution. The probability distribution for Landaurandom variates is defined analytically by the complex integral,p(x) =

12πi

Z c+i∞
ds exp(s log(s) + xs)
c−i∞

For numerical purposes it is more convenient to use the following equivalent form of the integral,Z ∞dt exp(−t log(t) − xt) sin(πt).
p(x) = (1/π)
0double gsl_ran_landau_pdf(double x)
This function computes the probability density p(x) at x for the Landau distribution using an approximation to
the formula given above.

20.12. The Landau Distribution

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20.13 The Levy alpha-Stable Distributions
double gsl_ran_levy(const gsl_rng *r, double c, double alpha)This function returns a random variate from the Levy symmetric stable distribution with scale c and exponent
alpha. The symmetric stable probability distribution is defined by a Fourier transform,
p(x) =

12π

Z +∞dt exp(−itx − |ct|α )−∞There is no explicit solution for the form of p(x) and the library does not define a corresponding pdf function.
For α√ = 1 the distribution reduces to the Cauchy distribution. For α = 2 it is a Gaussian distribution withσ = 2c. For α < 1 the tails of the distribution become extremely wide.
The algorithm only works for 0 < α ≤ 2.

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20.14 The Levy skew alpha-Stable Distribution
double gsl_ran_levy_skew(const gsl_rng *r, double c, double alpha, double beta)
This function returns a random variate from the Levy skew stable distribution with scale c, exponent alphaand skewness parameter beta. The skewness parameter must lie in the range [−1, 1]. The Levy skew stableprobability distribution is defined by a Fourier transform,
p(x) =

12π

Z +∞dt exp(−itx − |ct|α (1 − iβ sgn(t) tan(πα/2)))−∞

When α = 1 the term tan(πα/2) is replaced by −(2/π) log |t|. There is no explicit solution for the form of p(x)and the library does not√define a corresponding pdf function. For α = 2 the distribution reduces to a Gaussian
distribution with σ = 2c and the skewness parameter has no effect. For α < 1 the tails of the distribution
become extremely wide. The symmetric distribution corresponds to β = 0.
The algorithm only works for 0 < α ≤ 2.
The Levy alpha-stable distributions have the property that if N alpha-stable variates are drawn from the distributionp(c, α, β) then the sum Y = X1 + X2 + · · · + XN will also be distributed as an alpha-stable variate, p(N 1/α c, α, β).

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20.15 The Gamma Distribution
double gsl_ran_gamma(const gsl_rng *r, double a, double b)
This function returns a random variate from the gamma distribution. The distribution function is,p(x)dx =1
xa−1 e−x/b dx
Γ(a)bafor x > 0.
The gamma distribution with an integer parameter a is known as the Erlang distribution.
The variates are computed using the Marsaglia-Tsang fast gamma method. This function for this method waspreviously called gsl_ran_gamma_mt() and can still be accessed using this name.
double gsl_ran_gamma_knuth(const gsl_rng *r, double a, double b)
This function returns a gamma variate using the algorithms from Knuth (vol 2).
double gsl_ran_gamma_pdf(double x, double a, double b)
This function computes the probability density p(x) at x for a gamma distribution with parameters a and b, using
the formula given above.

double gsl_cdf_gamma_P(double x, double a, double b)
double gsl_cdf_gamma_Q(double x, double a, double b)
double gsl_cdf_gamma_Pinv(double P, double a, double b)

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double gsl_cdf_gamma_Qinv(double Q, double a, double b)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the gamma
distribution with parameters a and b.

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20.16 The Flat (Uniform) Distribution
double gsl_ran_flat(const gsl_rng *r, double a, double b)
This function returns a random variate from the flat (uniform) distribution from a to b. The distribution is,p(x)dx =1dx(b − a)if a ≤ x < b and 0 otherwise.
double gsl_ran_flat_pdf(double x, double a, double b)
This function computes the probability density p(x) at x for a uniform distribution from a to b, using the formula
given above.

double gsl_cdf_flat_P(double x, double a, double b)
double gsl_cdf_flat_Q(double x, double a, double b)
double gsl_cdf_flat_Pinv(double P, double a, double b)
double gsl_cdf_flat_Qinv(double Q, double a, double b)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for a uniform
distribution from a to b.

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20.17 The Lognormal Distribution
double gsl_ran_lognormal(const gsl_rng *r, double zeta, double sigma)
This function returns a random variate from the lognormal distribution. The distribution function is,1
exp(−(ln(x) − ζ)2 /2σ 2 )dx
p(x)dx = √x 2πσ 2for x > 0.
double gsl_ran_lognormal_pdf(double x, double zeta, double sigma)
This function computes the probability density p(x) at x for a lognormal distribution with parameters zeta andsigma, using the formula given above.

double gsl_cdf_lognormal_P(double x, double zeta, double sigma)
double gsl_cdf_lognormal_Q(double x, double zeta, double sigma)
double gsl_cdf_lognormal_Pinv(double P, double zeta, double sigma)
double gsl_cdf_lognormal_Qinv(double Q, double zeta, double sigma)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the lognormal
distribution with parameters zeta and sigma.

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20.18 The Chi-squared Distribution
The chi-squared distribution arises in statistics. If Yi are n independent Gaussian random variates with unit variancethen the sum-of-squares,
X
Xi =
Yi2
i

has a chi-squared distribution with n degrees of freedom.
double gsl_ran_chisq(const gsl_rng *r, double nu)
This function returns a random variate from the chi-squared distribution with nu degrees of freedom. The distribution function is,p(x)dx =1(x/2)ν/2−1 exp(−x/2)dx2Γ(ν/2)for x ≥ 0.
double gsl_ran_chisq_pdf(double x, double nu)
This function computes the probability density p(x) at x for a chi-squared distribution with nu degrees of freedom, using the formula given above.

double gsl_cdf_chisq_P(double x, double nu)
double gsl_cdf_chisq_Q(double x, double nu)

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double gsl_cdf_chisq_Pinv(double P, double nu)
double gsl_cdf_chisq_Qinv(double Q, double nu)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the chi-squared
distribution with nu degrees of freedom.

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20.19 The F-distribution
The F-distribution arises in statistics. If Y1 and Y2 are chi-squared deviates with ν1 and ν2 degrees of freedom then theratio,
X=(Y1 /ν1 )(Y2 /ν2 )has an F-distribution F (x; ν1 , ν2 ).
double gsl_ran_fdist(const gsl_rng *r, double nu1, double nu2)
This function returns a random variate from the F-distribution with degrees of freedom nu1 and nu2. The
distribution function is,p(x)dx =Γ((ν1 + ν2 )/2) ν1 /2 ν2 /2 ν1 /2−1ν
ν2 x
(ν2 + ν1 x)−ν1 /2−ν2 /2
Γ(ν1 /2)Γ(ν2 /2) 1

for x ≥ 0.
double gsl_ran_fdist_pdf(double x, double nu1, double nu2)
This function computes the probability density p(x) at x for an F-distribution with nu1 and nu2 degrees of
freedom, using the formula given above.

double gsl_cdf_fdist_P(double x, double nu1, double nu2)
double gsl_cdf_fdist_Q(double x, double nu1, double nu2)

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double gsl_cdf_fdist_Pinv(double P, double nu1, double nu2)
double gsl_cdf_fdist_Qinv(double Q, double nu1, double nu2)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the Fdistribution with nu1 and nu2 degrees of freedom.

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20.20 The t-distribution
The t-distribution arises in statistics. If Y1 has a normal distribution and Y2 has a chi-squared distribution with ν degrees
of freedom then the ratio,
X=p

Y1
Y2 /ν

has a t-distribution t(x; ν) with ν degrees of freedom.
double gsl_ran_tdist(const gsl_rng *r, double nu)
This function returns a random variate from the t-distribution. The distribution function is,Γ((ν + 1)/2)
p(x)dx = √(1 + x2 /ν)−(ν+1)/2 dxπνΓ(ν/2)for −∞ < x < +∞.
double gsl_ran_tdist_pdf(double x, double nu)
This function computes the probability density p(x) at x for a t-distribution with nu degrees of freedom, using
the formula given above.

double gsl_cdf_tdist_P(double x, double nu)
double gsl_cdf_tdist_Q(double x, double nu)
double gsl_cdf_tdist_Pinv(double P, double nu)

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double gsl_cdf_tdist_Qinv(double Q, double nu)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the t-distribution
with nu degrees of freedom.

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20.21 The Beta Distribution
double gsl_ran_beta(const gsl_rng *r, double a, double b)
This function returns a random variate from the beta distribution. The distribution function is,p(x)dx =Γ(a + b) a−1x(1 − x)b−1 dx
Γ(a)Γ(b)for 0 ≤ x ≤ 1.
double gsl_ran_beta_pdf(double x, double a, double b)
This function computes the probability density p(x) at x for a beta distribution with parameters a and b, using
the formula given above.

double gsl_cdf_beta_P(double x, double a, double b)
double gsl_cdf_beta_Q(double x, double a, double b)
double gsl_cdf_beta_Pinv(double P, double a, double b)
double gsl_cdf_beta_Qinv(double Q, double a, double b)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the beta distribution with parameters a and b.

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20.22 The Logistic Distribution
double gsl_ran_logistic(const gsl_rng *r, double a)This function returns a random variate from the logistic distribution. The distribution function is,p(x)dx =exp(−x/a)dxa(1 + exp(−x/a))2

for −∞ < x < +∞.
double gsl_ran_logistic_pdf(double x, double a)This function computes the probability density p(x) at x for a logistic distribution with scale parameter a, using
the formula given above.

double gsl_cdf_logistic_P(double x, double a)double gsl_cdf_logistic_Q(double x, double a)double gsl_cdf_logistic_Pinv(double P, double a)double gsl_cdf_logistic_Qinv(double Q, double a)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the logistic
distribution with scale parameter a.

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20.23 The Pareto Distribution
double gsl_ran_pareto(const gsl_rng *r, double a, double b)
This function returns a random variate from the Pareto distribution of order a. The distribution function is,p(x)dx = (a/b)/(x/b)a+1 dx
for x ≥ b.
double gsl_ran_pareto_pdf(double x, double a, double b)
This function computes the probability density p(x) at x for a Pareto distribution with exponent a and scale b,using the formula given above.

double gsl_cdf_pareto_P(double x, double a, double b)
double gsl_cdf_pareto_Q(double x, double a, double b)
double gsl_cdf_pareto_Pinv(double P, double a, double b)
double gsl_cdf_pareto_Qinv(double Q, double a, double b)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the Pareto
distribution with exponent a and scale b.

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20.24 Spherical Vector Distributions
The spherical distributions generate random vectors, located on a spherical surface. They can be used as random
directions, for example in the steps of a random walk.
void gsl_ran_dir_2d(const gsl_rng *r, double *x, double *y)void gsl_ran_dir_2d_trig_method(const gsl_rng *r, double *x, double *y)This function returns a random direction vector v = (x, y) in two dimensions. The vector is normalized such that|v|2 = x2 + y 2 = 1. The obvious way to do this is to take a uniform random number between 0 and 2π and let
x and y be the sine and cosine respectively. Two trig functions would have been expensive in the old days, butwith modern hardware implementations, this is sometimes the fastest way to go. This is the case for the Pentium(but not the case for the Sun Sparcstation). One can avoid the trig evaluations by choosing x and y in the interiorof a unit circle (choose them at random fromp
the interior of the enclosing square, and then reject those that areoutside the unit circle), and then dividing by x2 + y 2 . A much cleverer approach, attributed to von Neumann(See Knuth, v2, 3rd ed, p140, exercise 23), requires neither trig nor a square root. In this approach, u and v arechosen at random from the interior of a unit circle, and then x = (u2 − v 2 )/(u2 + v 2 ) and y = 2uv/(u2 + v 2 ).
void gsl_ran_dir_3d(const gsl_rng *r, double *x, double *y, double *z)
This function returns a random direction vector v = (x, y, z) in three dimensions. The vector is normalized suchthat |v|2 = x2 + y 2 + z 2 = 1. The method employed is due to Robert E. Knop (CACM 13, 326 (1970)), andexplained in Knuth, v2, 3rd ed, p136. It uses the surprising fact that the distribution projected along any axis isactually uniform (this is only true for 3 dimensions).
void gsl_ran_dir_nd(const gsl_rng *r, size_t n, double *x)
This function returns a random direction vector v = (x1 , x2 , . . . , xn ) in n dimensions. The vector is normalized
such that |v|2 = x21 + x22 + · · · + x2n = 1. The method uses the fact that a multivariate Gaussian distribution isspherically symmetric. Each component is generated to have a Gaussian distribution, and then the components
are normalized. The method is described by Knuth, v2, 3rd ed, p135–136, and attributed to G. W. Brown, Modern
Mathematics for the Engineer (1956).

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20.25 The Weibull Distribution
double gsl_ran_weibull(const gsl_rng *r, double a, double b)
This function returns a random variate from the Weibull distribution. The distribution function is,p(x)dx =b b−1
xexp(−(x/a)b )dx
abfor x ≥ 0.
double gsl_ran_weibull_pdf(double x, double a, double b)
This function computes the probability density p(x) at x for a Weibull distribution with scale a and exponent b,
using the formula given above.

double gsl_cdf_weibull_P(double x, double a, double b)
double gsl_cdf_weibull_Q(double x, double a, double b)
double gsl_cdf_weibull_Pinv(double P, double a, double b)
double gsl_cdf_weibull_Qinv(double Q, double a, double b)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the Weibull
distribution with scale a and exponent b.

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20.26 The Type-1 Gumbel Distribution
double gsl_ran_gumbel1(const gsl_rng *r, double a, double b)
This function returns a random variate from the Type-1 Gumbel distribution. The Type-1 Gumbel distributionfunction is,p(x)dx = ab exp(−(b exp(−ax) + ax))dx
for −∞ < x < ∞.
double gsl_ran_gumbel1_pdf(double x, double a, double b)
This function computes the probability density p(x) at x for a Type-1 Gumbel distribution with parameters aand b, using the formula given above.

double gsl_cdf_gumbel1_P(double x, double a, double b)
double gsl_cdf_gumbel1_Q(double x, double a, double b)
double gsl_cdf_gumbel1_Pinv(double P, double a, double b)
double gsl_cdf_gumbel1_Qinv(double Q, double a, double b)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the Type-1
Gumbel distribution with parameters a and b.

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20.27 The Type-2 Gumbel Distribution
double gsl_ran_gumbel2(const gsl_rng *r, double a, double b)
This function returns a random variate from the Type-2 Gumbel distribution. The Type-2 Gumbel distributionfunction is,p(x)dx = abx−a−1 exp(−bx−a )dx
for 0 < x < ∞.
double gsl_ran_gumbel2_pdf(double x, double a, double b)
This function computes the probability density p(x) at x for a Type-2 Gumbel distribution with parameters aand b, using the formula given above.

double gsl_cdf_gumbel2_P(double x, double a, double b)
double gsl_cdf_gumbel2_Q(double x, double a, double b)
double gsl_cdf_gumbel2_Pinv(double P, double a, double b)
double gsl_cdf_gumbel2_Qinv(double Q, double a, double b)
These functions compute the cumulative distribution functions P (x), Q(x) and their inverses for the Type-2
Gumbel distribution with parameters a and b.

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20.28 The Dirichlet Distribution
void gsl_ran_dirichlet(const gsl_rng *r, size_t K, const double alpha[], double theta[])This function returns an array of K random variates from a Dirichlet distribution of order K-1. The distributionfunction isKKX
1 Y αi −1θδ(1 −p(θ1 , . . . , θK ) dθ1 · · · dθK =θi )dθ1 · · · dθK
Z i=1 ii=1

for θi ≥ 0 and αi > 0. The delta function ensures thatP

θi = 1. The normalization factor Z is

QKZ=

i=1 Γ(αi )PK
Γ( i=1 αi )The random variates are generated by sampling K values from gamma distributions with parameters a = αi ,b = 1, and renormalizing. See A.M. Law, W.D. Kelton, Simulation Modeling and Analysis (1991).
double gsl_ran_dirichlet_pdf(size_t K, const double alpha[], const double theta[])This function computes the probability density p(θ1 , . . . , θK ) at theta[K] for a Dirichlet distribution with parameters alpha[K], using the formula given above.
double gsl_ran_dirichlet_lnpdf(size_t K, const double alpha[], const double theta[])This function computes the logarithm of the probability density p(θ1 , . . . , θK ) for a Dirichlet distribution withparameters alpha[K].

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20.29 General Discrete Distributions
Given K discrete events with different probabilities P [k], produce a random value k consistent with its probability.
The obvious way to do this is to preprocess the probability list by generating a cumulative probability array with K + 1
elements:
C[0] = 0
C[k + 1] = C[k] + P [k]Note that this construction produces C[K] = 1. Now choose a uniform deviate u between 0 and 1, and find the valueof k such that C[k] ≤ u < C[k + 1]. Although this in principle requires of order log K steps per random number
generation, they are fast steps, and if you use something like buKc as a starting point, you can often do pretty well.
But faster methods have been devised. Again, the idea is to preprocess the probability list, and save the result in someform of lookup table; then the individual calls for a random discrete event can go rapidly. An approach invented by
G. Marsaglia (Generating discrete random variables in a computer, Comm ACM 6, 37–38 (1963)) is very clever, andreaders interested in examples of good algorithm design are directed to this short and well-written paper. Unfortunately,
for large K, Marsaglia’s lookup table can be quite large.
A much better approach is due to Alastair J. Walker (An efficient method for generating discrete random variables
with general distributions, ACM Trans on Mathematical Software 3, 253–256 (1977); see also Knuth, v2, 3rd ed,
p120–121,139). This requires two lookup tables, one floating point and one integer, but both only of size K. After
preprocessing, the random numbers are generated in O(1) time, even for large K. The preprocessing suggested byWalker requires O(K 2 ) effort, but that is not actually necessary, and the implementation provided here only takes O(K)effort. In general, more preprocessing leads to faster generation of the individual random numbers, but a diminishingreturn is reached pretty early. Knuth points out that the optimal preprocessing is combinatorially difficult for large K.
This method can be used to speed up some of the discrete random number generators below, such as the binomial
distribution. To use it for something like the Poisson Distribution, a modification would have to be made, since it only
takes a finite set of K outcomes.
type gsl_ran_discrete_t
This structure contains the lookup table for the discrete random number generator.
gsl_ran_discrete_t *gsl_ran_discrete_preproc(size_t K, const double *P)This function returns a pointer to a structure that contains the lookup table for the discrete random number
generator. The array P contains the probabilities of the discrete events; these array elements must all be positive,
but they needn’t add up to one (so you can think of them more generally as “weights”)—the preprocessor will
normalize appropriately. This return value is used as an argument for the gsl_ran_discrete() function below.
size_t gsl_ran_discrete(const gsl_rng *r, const gsl_ran_discrete_t *g)
After the preprocessor, above, has been called, you use this function to get the discrete random numbers.
double gsl_ran_discrete_pdf(size_t k, const gsl_ran_discrete_t *g)
Returns the probability P [k] of observing the variable k. Since P [k] is not stored as part of the lookup table, it
must be recomputed; this computation takes O(K), so if K is large and you care about the original array P [k]used to create the lookup table, then you should just keep this original array P [k] around.
void gsl_ran_discrete_free(gsl_ran_discrete_t *g)De-allocates the lookup table pointed to by g.

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20.30 The Poisson Distribution
unsigned int gsl_ran_poisson(const gsl_rng *r, double mu)
This function returns a random integer from the Poisson distribution with mean mu. The probability distribution
for Poisson variates is,p(k) =µk
exp(−µ)k!

for k ≥ 0.
double gsl_ran_poisson_pdf(unsigned int k, double mu)
This function computes the probability p(k) of obtaining k from a Poisson distribution with mean mu, using theformula given above.

double gsl_cdf_poisson_P(unsigned int k, double mu)
double gsl_cdf_poisson_Q(unsigned int k, double mu)
These functions compute the cumulative distribution functions P (k), Q(k) for the Poisson distribution withparameter mu.

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20.31 The Bernoulli Distribution
unsigned int gsl_ran_bernoulli(const gsl_rng *r, double p)
This function returns either 0 or 1, the result of a Bernoulli trial with probability p. The probability distribution
for a Bernoulli trial is,p(0) = 1 − pp(1) = pdouble gsl_ran_bernoulli_pdf(unsigned int k, double p)
This function computes the probability p(k) of obtaining k from a Bernoulli distribution with probability parameter p, using the formula given above.

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20.32 The Binomial Distribution
unsigned int gsl_ran_binomial(const gsl_rng *r, double p, unsigned int n)This function returns a random integer from the binomial distribution, the number of successes in n independent
trials with probability p. The probability distribution for binomial variates is,p(k) =n!
pk (1 − p)n−k
k!(n − k)!

for 0 ≤ k ≤ n.
double gsl_ran_binomial_pdf(unsigned int k, double p, unsigned int n)This function computes the probability p(k) of obtaining k from a binomial distribution with parameters p andn, using the formula given above.

double gsl_cdf_binomial_P(unsigned int k, double p, unsigned int n)double gsl_cdf_binomial_Q(unsigned int k, double p, unsigned int n)
These functions compute the cumulative distribution functions P (k), Q(k) for the binomial distribution withparameters p and n.

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20.33 The Multinomial Distribution
void gsl_ran_multinomial(const gsl_rng *r, size_t K, unsigned int N, const double p[], unsigned int n[])This function computes a random sample n from the multinomial distribution formed by N trials from an underlying distribution p[K]. The distribution function for n is,N!
pn1 pn2 · · · pnKK
n1 !n2 ! · · · nK ! 1 2PK
where (n1 , n2 , . . P. , nK ) are nonnegative integers with k=1 nk = N , and (p1 , p2 , . . . , pK ) is a probability
distribution withpi = 1. If the array p[K] is not normalized then its entries will be treated as weights andnormalized appropriately. The arrays n and p must both be of length K.
P (n1 , n2 , · · · , nK ) =Random variates are generated using the conditional binomial method (see C.S. Davis, The computer generationof multinomial random variates, Comp. Stat. Data Anal. 16 (1993) 205–217 for details).
double gsl_ran_multinomial_pdf(size_t K, const double p[], const unsigned int n[])This function computes the probability P (n1 , n2 , . . . , nK ) of sampling n[K] from a multinomial distribution
with parameters p[K], using the formula given above.
double gsl_ran_multinomial_lnpdf(size_t K, const double p[], const unsigned int n[])This function returns the logarithm of the probability for the multinomial distribution P (n1 , n2 , . . . , nK ) withparameters p[K].

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20.34 The Negative Binomial Distribution
unsigned int gsl_ran_negative_binomial(const gsl_rng *r, double p, double n)This function returns a random integer from the negative binomial distribution, the number of failures occurring
before n successes in independent trials with probability p of success. The probability distribution for negative
binomial variates is,p(k) =

Γ(n + k)pn (1 − p)kΓ(k + 1)Γ(n)Note that n is not required to be an integer.
double gsl_ran_negative_binomial_pdf(unsigned int k, double p, double n)This function computes the probability p(k) of obtaining k from a negative binomial distribution with parameters
p and n, using the formula given above.

double gsl_cdf_negative_binomial_P(unsigned int k, double p, double n)double gsl_cdf_negative_binomial_Q(unsigned int k, double p, double n)
These functions compute the cumulative distribution functions P (k), Q(k) for the negative binomial distribution
with parameters p and n.

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20.35 The Pascal Distribution
unsigned int gsl_ran_pascal(const gsl_rng *r, double p, unsigned int n)This function returns a random integer from the Pascal distribution. The Pascal distribution is simply a negative
binomial distribution with an integer value of n.
p(k) =(n + k − 1)! n
p (1 − p)kk!(n − 1)!

for k ≥ 0.
double gsl_ran_pascal_pdf(unsigned int k, double p, unsigned int n)This function computes the probability p(k) of obtaining k from a Pascal distribution with parameters p and n,
using the formula given above.

double gsl_cdf_pascal_P(unsigned int k, double p, unsigned int n)double gsl_cdf_pascal_Q(unsigned int k, double p, unsigned int n)
These functions compute the cumulative distribution functions P (k), Q(k) for the Pascal distribution with parameters p and n.

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20.36 The Geometric Distribution
unsigned int gsl_ran_geometric(const gsl_rng *r, double p)
This function returns a random integer from the geometric distribution, the number of independent trials withprobability p until the first success. The probability distribution for geometric variates is,p(k) = p(1 − p)k−1
for k ≥ 1. Note that the distribution begins with k = 1 with this definition. There is another convention in which
the exponent k − 1 is replaced by k.
double gsl_ran_geometric_pdf(unsigned int k, double p)
This function computes the probability p(k) of obtaining k from a geometric distribution with probability parameter p, using the formula given above.

double gsl_cdf_geometric_P(unsigned int k, double p)
double gsl_cdf_geometric_Q(unsigned int k, double p)
These functions compute the cumulative distribution functions P (k), Q(k) for the geometric distribution withparameter p.

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20.37 The Hypergeometric Distribution
unsigned int gsl_ran_hypergeometric(const gsl_rng *r, unsigned int n1, unsigned int n2, unsigned int t)
This function returns a random integer from the hypergeometric distribution. The probability distribution for
hypergeometric random variates is,p(k) = C(n1 , k)C(n2 , t − k)/C(n1 + n2 , t)where C(a, b) = a!/(b!(a − b)!) and t ≤ n1 + n2 . The domain of k is max(0, t − n2 ), . . . , min(t, n1 )If a population contains n1 elements of “type 1” and n2 elements of “type 2” then the hypergeometric distribution
gives the probability of obtaining k elements of “type 1” in t samples from the population without replacement.
double gsl_ran_hypergeometric_pdf(unsigned int k, unsigned int n1, unsigned int n2, unsigned int t)
This function computes the probability p(k) of obtaining k from a hypergeometric distribution with parameters
n1, n2, t, using the formula given above.

double gsl_cdf_hypergeometric_P(unsigned int k, unsigned int n1, unsigned int n2, unsigned int t)
double gsl_cdf_hypergeometric_Q(unsigned int k, unsigned int n1, unsigned int n2, unsigned int t)
These functions compute the cumulative distribution functions P (k), Q(k) for the hypergeometric distribution
with parameters n1, n2 and t.

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20.38 The Logarithmic Distribution
unsigned int gsl_ran_logarithmic(const gsl_rng *r, double p)
This function returns a random integer from the logarithmic distribution. The probability distribution for logarithmic random variates is,
 k
p
−1
p(k) =
log(1 − p) k
for k ≥ 1.
double gsl_ran_logarithmic_pdf(unsigned int k, double p)
This function computes the probability p(k) of obtaining k from a logarithmic distribution with probability
parameter p, using the formula given above.

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20.39 The Wishart Distribution
int gsl_ran_wishart(const gsl_rng *r, const double n, const gsl_matrix *L, gsl_matrix *result, gsl_matrix *work)
This function computes a random symmetric p-by-p matrix from the Wishart distribution. The probability distribution for Wishart random variates is,p(X) =|X|(n−p−1)/2 e−tr(V
2

np
2

|V |

n/2

−1

X )/2

Γp ( n2 )Here, n > p − 1 is the number of degrees of freedom, V is a symmetric positive definite p-by-p scale matrix,
whose Cholesky factor is specified by L, and work is p-by-p workspace. The p-by-p Wishart distributed matrix
X is stored in result on output.
int gsl_ran_wishart_pdf(const gsl_matrix *X, const gsl_matrix *L_X, const double n, const gsl_matrix *L,
double *result, gsl_matrix *work)
int gsl_ran_wishart_log_pdf(const gsl_matrix *X, const gsl_matrix *L_X, const double n, const gsl_matrix *L,
double *result, gsl_matrix *work)
These functions compute p(X) or log p(X) for the p-by-p matrix X, whose Cholesky factor is specified in L_X.
The degrees of freedom is given by n, the Cholesky factor of the scale matrix V is specified in L, and work is
p-by-p workspace. The probably density value is returned in result.

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20.40 Shuffling and Sampling
The following functions allow the shuffling and sampling of a set of objects. The algorithms rely on a random number
generator as a source of randomness and a poor quality generator can lead to correlations in the output. In particular
it is important to avoid generators with a short period. For more information see Knuth, v2, 3rd ed, Section 3.4.2,“Random Sampling and Shuffling”.
void gsl_ran_shuffle(const gsl_rng *r, void *base, size_t n, size_t size)
This function randomly shuffles the order of n objects, each of size size, stored in the array base[0..n-1].
The output of the random number generator r is used to produce the permutation. The algorithm generates all
possible n! permutations with equal probability, assuming a perfect source of random numbers.
The following code shows how to shuffle the numbers from 0 to 51:
int a[52];for (i = 0; i < 52; i++){a[i] = i;}gsl_ran_shuffle (r, a, 52, sizeof (int));
int gsl_ran_choose(const gsl_rng *r, void *dest, size_t k, void *src, size_t n, size_t size)
This function fills the array dest[k] with k objects taken randomly from the n elements of the array src[0.
.n-1]. The objects are each of size size. The output of the random number generator r is used to make theselection. The algorithm ensures all possible samples are equally likely, assuming a perfect source of randomness.
The objects are sampled without replacement, thus each object can only appear once in dest. It is required thatk be less than or equal to n. The objects in dest will be in the same relative order as those in src. You willneed to call gsl_ran_shuffle(r, dest, n, size) if you want to randomize the order.
The following code shows how to select a random sample of three unique numbers from the set 0 to 99:
double a[3], b[100];for (i = 0; i < 100; i++){b[i] = (double) i;}
gsl_ran_choose (r, a, 3, b, 100, sizeof (double));void gsl_ran_sample(const gsl_rng *r, void *dest, size_t k, void *src, size_t n, size_t size)
This function is like gsl_ran_choose() but samples k items from the original array of n items src withreplacement, so the same object can appear more than once in the output sequence dest. There is no requirementthat k be less than n in this case.

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20.41 Examples
The following program demonstrates the use of a random number generator to produce variates from a distribution. It
prints 10 samples from the Poisson distribution with a mean of 3.
#include <stdio.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>intmain (void){
const gsl_rng_type * T;
gsl_rng * r;
int i, n = 10;
double mu = 3.0;/* create a generator chosen by theenvironment variable GSL_RNG_TYPE */gsl_rng_env_setup();
T = gsl_rng_default;r = gsl_rng_alloc (T);/* print n random variates chosen fromthe poisson distribution with mean
parameter mu */for (i = 0; i < n; i++){unsigned int k = gsl_ran_poisson (r, mu);printf (" %u", k);}printf ("\n");gsl_rng_free (r);return 0;}
If the library and header files are installed under /usr/local (the default location) then the program can be compiled
with these options:$ gcc -Wall demo.c -lgsl -lgslcblas -lm
Here is the output of the program,
2 5 5 2 1 0 3 4 1 1The variates depend on the seed used by the generator. The seed for the default generator type gsl_rng_default canbe changed with the GSL_RNG_SEED environment variable to produce a different stream of variates:

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GNU Scientific Library, Release 2.7$ GSL_RNG_SEED=123 ./a.out
giving output
4 5 6 3 3 1 4 2 5 5The following program generates a random walk in two dimensions.
#include <stdio.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>intmain (void){
int i;
double x = 0, y = 0, dx, dy;const gsl_rng_type * T;
gsl_rng * r;
gsl_rng_env_setup();
T = gsl_rng_default;r = gsl_rng_alloc (T);printf ("%g %g\n", x, y);for (i = 0; i < 10; i++){gsl_ran_dir_2d (r, &dx, &dy);
x += dx; y += dy;
printf ("%g %g\n", x, y);}gsl_rng_free (r);return 0;}
Fig. 20.1 shows the output from the program.
The following program computes the upper and lower cumulative distribution functions for the standard normal distribution at x = 2.
#include <stdio.h>#include <gsl/gsl_cdf.h>
intmain (void){
double P, Q;
double x = 2.0;
P = gsl_cdf_ugaussian_P (x);printf ("prob(x < %f ) = %f \n", x, P);(continues on next page)

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Fig. 20.1: Four 10-step random walks from the origin.
(continued from previous page)

Q = gsl_cdf_ugaussian_Q (x);printf ("prob(x > %f ) = %f \n", x, Q);
x = gsl_cdf_ugaussian_Pinv (P);printf ("Pinv(%f ) = %f \n", P, x);x = gsl_cdf_ugaussian_Qinv (Q);printf ("Qinv(%f ) = %f \n", Q, x);
return 0;}
Here is the output of the program,prob(x < 2.000000) = 0.977250
prob(x > 2.000000) = 0.022750
Pinv(0.977250) = 2.000000
Qinv(0.022750) = 2.000000

20.42 References and Further Reading
For an encyclopaedic coverage of the subject readers are advised to consult the book “Non-Uniform Random Variate
Generation” by Luc Devroye. It covers every imaginable distribution and provides hundreds of algorithms.
• Luc Devroye, “Non-Uniform Random Variate Generation”, Springer-Verlag, ISBN 0-387-96305-7. Available
online at http://cg.scs.carleton.ca/~luc/rnbookindex.html.
The subject of random variate generation is also reviewed by Knuth, who describes algorithms for all the major distributions.
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GNU Scientific Library, Release 2.7• Donald E. Knuth, “The Art of Computer Programming: Seminumerical Algorithms” (Vol 2, 3rd Ed, 1997),
Addison-Wesley, ISBN 0201896842.
The Particle Data Group provides a short review of techniques for generating distributions of random numbers in the“Monte Carlo” section of its Annual Review of Particle Physics.
• Review of Particle Properties, R.M. Barnett et al., Physical Review D54, 1 (1996) http://pdg.lbl.gov/.
The Review of Particle Physics is available online in postscript and pdf format.
An overview of methods used to compute cumulative distribution functions can be found in Statistical Computing byW.J. Kennedy and J.E. Gentle. Another general reference is Elements of Statistical Computing by R.A. Thisted.
• William E. Kennedy and James E. Gentle, Statistical Computing (1980), Marcel Dekker, ISBN 0-8247-6898-1.
• Ronald A. Thisted, Elements of Statistical Computing (1988), Chapman & Hall, ISBN 0-412-01371-1.
The cumulative distribution functions for the Gaussian distribution are based on the following papers,• Rational Chebyshev Approximations Using Linear Equations, W.J. Cody, W. Fraser, J.F. Hart. Numerische
Mathematik 12, 242–251 (1968).
• Rational Chebyshev Approximations for the Error Function, W.J. Cody. Mathematics of Computation 23, n107,
631–637 (July 1969).

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CHAPTERTWENTYONESTATISTICS

This chapter describes the statistical functions in the library. The basic statistical functions include routines to computethe mean, variance and standard deviation. More advanced functions allow you to calculate absolute deviations, skewness, and kurtosis as well as the median and arbitrary percentiles. The algorithms use recurrence relations to computeaverage quantities in a stable way, without large intermediate values that might overflow.
The functions are available in versions for datasets in the standard floating-point and integer types. The versions for double precision floating-point data have the prefix gsl_stats and are declared in the header file
gsl_statistics_double.h. The versions for integer data have the prefix gsl_stats_int and are declared in theheader file gsl_statistics_int.h. All the functions operate on C arrays with a stride parameter specifying the
spacing between elements.

21.1 Mean, Standard Deviation and Variancedouble gsl_stats_mean(const double data[], size_t stride, size_t n)This function returns the arithmetic mean of data, a dataset of length n with stride stride. The arithmetic
mean, or sample mean, is denoted by µ̂ and defined as,
µ̂ =

1 X
xi
N

where xi are the elements of the dataset data. For samples drawn from a gaussian distribution the variance ofµ̂ is σ 2 /N .
double gsl_stats_variance(const double data[], size_t stride, size_t n)This function returns the estimated, or sample, variance of data, a dataset of length n with stride stride. The
estimated variance is denoted by σ̂ 2 and is defined by,σ̂ 2 =

X1(xi − µ̂)2(N − 1)

where xi are the elements of the dataset data. Note that the normalization factor of 1/(N − 1) results from thederivation of σ̂ 2 as an unbiased estimator of the population variance σ 2 . For samples drawn from a Gaussian
distribution the variance of σ̂ 2 itself is 2σ 4 /N .
This function computes the mean via a call to gsl_stats_mean(). If you have already computed the mean then
you can pass it directly to gsl_stats_variance_m().
double gsl_stats_variance_m(const double data[], size_t stride, size_t n, double mean)
This function returns the sample variance of data relative to the given value of mean. The function is computedwith µ̂ replaced by the value of mean that you supply,σ̂ 2 =

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double gsl_stats_sd(const double data[], size_t stride, size_t n)double gsl_stats_sd_m(const double data[], size_t stride, size_t n, double mean)
The standard deviation is defined as the square root of the variance. These functions return the square root of thecorresponding variance functions above.
double gsl_stats_tss(const double data[], size_t stride, size_t n)double gsl_stats_tss_m(const double data[], size_t stride, size_t n, double mean)
These functions return the total sum of squares (TSS) of data about the mean. For gsl_stats_tss_m() theuser-supplied value of mean is used, and for gsl_stats_tss() it is computed using gsl_stats_mean().
XTSS =(xi − mean)2
double gsl_stats_variance_with_fixed_mean(const double data[], size_t stride, size_t n, double mean)
This function computes an unbiased estimate of the variance of data when the population mean mean of theunderlying distribution is known a priori. In this case the estimator for the variance uses the factor 1/N and thesample mean µ̂ is replaced by the known population mean µ,σ̂ 2 =

1 X(xi − µ)2
N

double gsl_stats_sd_with_fixed_mean(const double data[], size_t stride, size_t n, double mean)
This function calculates the standard deviation of data for a fixed population mean mean. The result is thesquare root of the corresponding variance function.

21.2 Absolute deviation
double gsl_stats_absdev(const double data[], size_t stride, size_t n)This function computes the absolute deviation from the mean of data, a dataset of length n with stride stride.
The absolute deviation from the mean is defined as,
absdev =1 X|xi − µ̂|N

where xi are the elements of the dataset data. The absolute deviation from the mean provides a more robust
measure of the width of a distribution than the variance. This function computes the mean of data via a call to
gsl_stats_mean().
double gsl_stats_absdev_m(const double data[], size_t stride, size_t n, double mean)
This function computes the absolute deviation of the dataset data relative to the given value of mean,absdev =1 X|xi − mean|
N

This function is useful if you have already computed the mean of data (and want to avoid recomputing it), or
wish to calculate the absolute deviation relative to another value (such as zero, or the median).

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21.3 Higher moments (skewness and kurtosis)
double gsl_stats_skew(const double data[], size_t stride, size_t n)This function computes the skewness of data, a dataset of length n with stride stride. The skewness is definedas,
3
1 X xi − µ̂
skew =
Nσ̂where xi are the elements of the dataset data. The skewness measures the asymmetry of the tails of a distribution.
The function computes the mean and estimated standard deviation of data via calls to gsl_stats_mean() and
gsl_stats_sd().
double gsl_stats_skew_m_sd(const double data[], size_t stride, size_t n, double mean, double sd)
This function computes the skewness of the dataset data using the given values of the mean mean and standard
deviation sd,3
1 X xi − mean
skew =
N
sd
These functions are useful if you have already computed the mean and standard deviation of data and want toavoid recomputing them.
double gsl_stats_kurtosis(const double data[], size_t stride, size_t n)This function computes the kurtosis of data, a dataset of length n with stride stride. The kurtosis is defined
as,

4 !
1 X xi − µ̂
kurtosis =−3Nσ̂The kurtosis measures how sharply peaked a distribution is, relative to its width. The kurtosis is normalized tozero for a Gaussian distribution.
double gsl_stats_kurtosis_m_sd(const double data[], size_t stride, size_t n, double mean, double sd)
This function computes the kurtosis of the dataset data using the given values of the mean mean and standard
deviation sd,4 !
1 X xi − mean
kurtosis =−3N
sdThis function is useful if you have already computed the mean and standard deviation of data and want to avoid
recomputing them.

21.4 Autocorrelation
double gsl_stats_lag1_autocorrelation(const double data[], const size_t stride, const size_t n)This function computes the lag-1 autocorrelation of the dataset data.
Pn
i=2 (xi − µ̂)(xi−1 − µ̂)
a1 = Pni=1 (xi − µ̂)(xi − µ̂)
double gsl_stats_lag1_autocorrelation_m(const double data[], const size_t stride, const size_t n, const
double mean)
This function computes the lag-1 autocorrelation of the dataset data using the given value of the mean mean.
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21.5 Covariancedouble gsl_stats_covariance(const double data1[], const size_t stride1, const double data2[], const size_t
stride2, const size_t n)This function computes the covariance of the datasets data1 and data2 which must both be of the same lengthn.
covar =nX1(xi − x̂)(yi − ŷ)(n − 1) i=1

double gsl_stats_covariance_m(const double data1[], const size_t stride1, const double data2[], const size_t
stride2, const size_t n, const double mean1, const double mean2)This function computes the covariance of the datasets data1 and data2 using the given values of the means,mean1 and mean2. This is useful if you have already computed the means of data1 and data2 and want to avoid
recomputing them.

21.6 Correlation
double gsl_stats_correlation(const double data1[], const size_t stride1, const double data2[], const size_t
stride2, const size_t n)This function efficiently computes the Pearson correlation coefficient between the datasets data1 and data2
which must both be of the same length n.
P1(xi − x̂)(yi − ŷ)
cov(x, y)
n−1
q
=q
r=
P
P
σ̂x σ̂y
1
1
2
(x
−
x̂)
(yi − ŷ)2
i
n−1
n−1
double gsl_stats_spearman(const double data1[], const size_t stride1, const double data2[], const size_t stride2,
const size_t n, double work[])
This function computes the Spearman rank correlation coefficient between the datasets data1 and data2 whichmust both be of the same length n. Additional workspace of size 2 * n is required in work. The Spearman rankcorrelation between vectors x and y is equivalent to the Pearson correlation between the ranked vectors xR andyR , where ranks are defined to be the average of the positions of an element in the ascending order of the values.

21.7 Weighted Samples
The functions described in this section allow the computation of statistics for weighted samples. The functions accept
an array of samples, xi , with associated weights, wi . Each sample xi is considered as having been drawn from a
Gaussian distribution with variance σi2 . The sample weight wi is defined as the reciprocal of this variance, wi = 1/σi2 .
Setting a weight to zero corresponds to removing a sample from a dataset.
double gsl_stats_wmean(const double w[], size_t wstride, const double data[], size_t stride, size_t n)This function returns the weighted mean of the dataset data with stride stride and length n, using the set ofweights w with stride wstride and length n. The weighted mean is defined as,P
wi xiµ̂ = Pwi

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double gsl_stats_wvariance(const double w[], size_t wstride, const double data[], size_t stride, size_t n)This function returns the estimated variance of the dataset data with stride stride and length n, using the setof weights w with stride wstride and length n. The estimated variance of a weighted dataset is calculated as,
P
X
wi
P 2
wi (xi − µ̂)2σ̂ 2 = P2( wi ) − (wi )Note that this expression reduces to an unweighted variance with the familiar 1/(N − 1) factor when there are
N equal non-zero weights.
double gsl_stats_wvariance_m(const double w[], size_t wstride, const double data[], size_t stride, size_t n,double wmean)
This function returns the estimated variance of the weighted dataset data using the given weighted mean wmean.
double gsl_stats_wsd(const double w[], size_t wstride, const double data[], size_t stride, size_t n)
The standard deviation is defined as the square root of the variance. This function returns the square root of thecorresponding variance function gsl_stats_wvariance() above.
double gsl_stats_wsd_m(const double w[], size_t wstride, const double data[], size_t stride, size_t n, double
wmean)
This function returns the square root of the corresponding variance function gsl_stats_wvariance_m()
above.
double gsl_stats_wvariance_with_fixed_mean(const double w[], size_t wstride, const double data[], size_t
stride, size_t n, const double mean)
This function computes an unbiased estimate of the variance of the weighted dataset data when the population
mean mean of the underlying distribution is known a priori. In this case the estimator for the variance replaces
the sample mean µ̂ by the known population mean µ,Pwi (xi − µ)2
2
P
σ̂ =
wi
double gsl_stats_wsd_with_fixed_mean(const double w[], size_t wstride, const double data[], size_t stride,size_t n, const double mean)
The standard deviation is defined as the square root of the variance. This function returns the square root of thecorresponding variance function above.
double gsl_stats_wtss(const double w[], const size_t wstride, const double data[], size_t stride, size_t n)double gsl_stats_wtss_m(const double w[], const size_t wstride, const double data[], size_t stride, size_t n,double wmean)
These functions return the weighted total sum of squares (TSS) of data about the weighted mean. For
gsl_stats_wtss_m() the user-supplied value of wmean is used, and for gsl_stats_wtss() it is computed
using gsl_stats_wmean().
XTSS =wi (xi − wmean)2
double gsl_stats_wabsdev(const double w[], size_t wstride, const double data[], size_t stride, size_t n)This function computes the weighted absolute deviation from the weighted mean of data. The absolute deviationfrom the mean is defined as,P
wi |xi − µ̂|Pabsdev =widouble gsl_stats_wabsdev_m(const double w[], size_t wstride, const double data[], size_t stride, size_t n, double
wmean)
This function computes the absolute deviation of the weighted dataset data about the given weighted mean
wmean.
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double gsl_stats_wskew(const double w[], size_t wstride, const double data[], size_t stride, size_t n)This function computes the weighted skewness of the dataset data.
Pwi ((xi − x̂)/σ̂)3P
skew =
widouble gsl_stats_wskew_m_sd(const double w[], size_t wstride, const double data[], size_t stride, size_t n, double
wmean, double wsd)
This function computes the weighted skewness of the dataset data using the given values of the weighted mean
and weighted standard deviation, wmean and wsd.
double gsl_stats_wkurtosis(const double w[], size_t wstride, const double data[], size_t stride, size_t n)This function computes the weighted kurtosis of the dataset data.
Pwi ((xi − x̂)/σ̂)4P−3kurtosis =widouble gsl_stats_wkurtosis_m_sd(const double w[], size_t wstride, const double data[], size_t stride, size_t n,double wmean, double wsd)
This function computes the weighted kurtosis of the dataset data using the given values of the weighted mean
and weighted standard deviation, wmean and wsd.

21.8 Maximum and Minimum values
The following functions find the maximum and minimum values of a dataset (or their indices). If the data contains
NaN-s then a NaN will be returned, since the maximum or minimum value is undefined. For functions which return anindex, the location of the first NaN in the array is returned.
double gsl_stats_max(const double data[], size_t stride, size_t n)This function returns the maximum value in data, a dataset of length n with stride stride. The maximum
value is defined as the value of the element xi which satisfies xi ≥ xj for all j.
If you want instead to find the element with the largest absolute magnitude you will need to apply fabs() orabs() to your data before calling this function.
double gsl_stats_min(const double data[], size_t stride, size_t n)This function returns the minimum value in data, a dataset of length n with stride stride. The minimum value
is defined as the value of the element xi which satisfies xi ≤ xj for all j.
If you want instead to find the element with the smallest absolute magnitude you will need to apply fabs() orabs() to your data before calling this function.
void gsl_stats_minmax(double *min, double *max, const double data[], size_t stride, size_t n)This function finds both the minimum and maximum values min, max in data in a single pass.
size_t gsl_stats_max_index(const double data[], size_t stride, size_t n)This function returns the index of the maximum value in data, a dataset of length n with stride stride. The
maximum value is defined as the value of the element xi which satisfies xi ≥ xj for all j. When there are several
equal maximum elements then the first one is chosen.
size_t gsl_stats_min_index(const double data[], size_t stride, size_t n)This function returns the index of the minimum value in data, a dataset of length n with stride stride. The
minimum value is defined as the value of the element xi which satisfies xi ≥ xj for all j. When there are several
equal minimum elements then the first one is chosen.
void gsl_stats_minmax_index(size_t *min_index, size_t *max_index, const double data[], size_t stride, size_t n)This function returns the indexes min_index, max_index of the minimum and maximum values in data in asingle pass.
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21.9 Median and Percentiles
The median and percentile functions described in this section operate on sorted data in O(1) time. There is also aroutine for computing the median of an unsorted input array in average O(n) time using the quickselect algorithm. For
convenience we use quantiles, measured on a scale of 0 to 1, instead of percentiles (which use a scale of 0 to 100).
double gsl_stats_median_from_sorted_data(const double sorted_data[], const size_t stride, const size_t n)This function returns the median value of sorted_data, a dataset of length n with stride stride. The elementsof the array must be in ascending numerical order. There are no checks to see whether the data are sorted, so thefunction gsl_sort() should always be used first.
When the dataset has an odd number of elements the median is the value of element (n − 1)/2. When the dataset
has an even number of elements the median is the mean of the two nearest middle values, elements (n − 1)/2
and n/2. Since the algorithm for computing the median involves interpolation this function always returns afloating-point number, even for integer data types.
double gsl_stats_median(double data[], const size_t stride, const size_t n)This function returns the median value of data, a dataset of length n with stride stride. The median is foundusing the quickselect algorithm. The input array does not need to be sorted, but note that the algorithm rearranges
the array and so the input is not preserved on output.
double gsl_stats_quantile_from_sorted_data(const double sorted_data[], size_t stride, size_t n, double f)
This function returns a quantile value of sorted_data, a double-precision array of length n with stride stride.
The elements of the array must be in ascending numerical order. The quantile is determined by the f , a fraction
between 0 and 1. For example, to compute the value of the 75th percentile f should have the value 0.75.
There are no checks to see whether the data are sorted, so the function gsl_sort() should always be used first.
The quantile is found by interpolation, using the formulaquantile = (1 − δ)xi + δxi+1
where i is floor((n - 1)f) and δ is (n − 1)f − i.
Thus the minimum value of the array (data[0*stride]) is given by f equal to zero, the maximum value(data[(n-1)*stride]) is given by f equal to one and the median value is given by f equal to 0.5. Since thealgorithm for computing quantiles involves interpolation this function always returns a floating-point number,even for integer data types.

21.10 Order Statistics
The k-th order statistic of a sample is equal to its k-th smallest value. The k-th order statistic of a set of n values
x = {xi } , 1 ≤ i ≤ n is denoted x(k) . The median of the set x is equal to x( n ) if n is odd, or the average of x( n )22and x( n +1) if n is even. The k-th smallest element of a length n vector can be found in average O(n) time using the2
quickselect algorithm.

double gsl_stats_select(double data[], const size_t stride, const size_t n, const size_t k)
This function finds the k-th smallest element of the input array data of length n and stride stride using thequickselect method. The algorithm rearranges the elements of data and so the input array is not preserved on
output.

21.9. Median and Percentiles

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21.11 Robust Location Estimates
A location estimate refers to a typical or central value which best describes a given dataset. The mean and median areboth examples of location estimators. However, the mean has a severe sensitivity to data outliers and can give erroneous
values when even a small number of outliers are present. The median on the other hand, has a strong insensitivity todata outliers, but due to its non-smoothness it can behave unexpectedly in certain situations. GSL offers the following
alternative location estimators, which are robust to the presence of outliers.

21.11.1 Trimmed Mean
The trimmed mean, or truncated mean, discards a certain number of smallest and largest samples from the input vector
before computing the mean of the remaining samples. The amount of trimming is specified by a factor α ∈ [0, 0.5].
Then the number of samples discarded from both ends of the input vector is bαnc, where n is the length of the input.
So to discard 25% of the samples from each end, one would set α = 0.25.
double gsl_stats_trmean_from_sorted_data(const double alpha, const double sorted_data[], const size_t
stride, const size_t n)This function returns the trimmed mean of sorted_data, a dataset of length n with stride stride. The elementsof the array must be in ascending numerical order. There are no checks to see whether the data are sorted, so thefunction gsl_sort() should always be used first. The trimming factor α is given in alpha. If α ≥ 0.5, thenthe median of the input is returned.

21.11.2 Gastwirth Estimator
Gastwirth’s location estimator is a weighted sum of three order statistics,
gastwirth = 0.3 × Q 31 + 0.4 × Q 21 + 0.3 × Q 23
where Q 13 is the one-third quantile, Q 21 is the one-half quantile (i.e. median), and Q 32 is the two-thirds quantile.
double gsl_stats_gastwirth_from_sorted_data(const double sorted_data[], const size_t stride, const size_t n)This function returns the Gastwirth location estimator of sorted_data, a dataset of length n with stride stride.
The elements of the array must be in ascending numerical order. There are no checks to see whether the data aresorted, so the function gsl_sort() should always be used first.

21.12 Robust Scale Estimates
A robust scale estimate, also known as a robust measure of scale, attempts to quantify the statistical dispersion (variability, scatter, spread) in a set of data which may contain outliers. In such datasets, the usual variance or standard
deviation scale estimate can be rendered useless by even a single outlier.

21.12.1 Median Absolute Deviation (MAD)The median absolute deviation (MAD) is defined as
M AD = 1.4826 × median {|xi − median (x)|}In words, first the median of all samples is computed. Then the median is subtracted from all samples in the input
to find the deviation of each sample from the median. The median of all absolute deviations is then the MAD. The
factor 1.4826 makes the MAD an unbiased estimator of the standard deviation for Gaussian data. The median absolutedeviation has an asymptotic efficiency of 37%.
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double gsl_stats_mad0(const double data[], const size_t stride, const size_t n, double work[])
double gsl_stats_mad(const double data[], const size_t stride, const size_t n, double work[])
These functions return the median absolute deviation of data, a dataset of length n and stride stride. The
mad0 function calculates median {|xi − median (x)|} (i.e. the M AD statistic without the bias correction scale
factor). These functions require additional workspace of size n provided in work.

21.12.2 Sn Statistic
The Sn statistic developed by Croux and Rousseeuw is defined as
Sn = 1.1926 × cn × mediani {medianj (|xi − xj |)}For each sample xi , 1 ≤ i ≤ n, the median of the values |xi − xj | is computed for all xj , 1 ≤ j ≤ n. This yields
n values, whose median then gives the final Sn . The factor 1.1926 makes Sn an unbiased estimate of the standard
deviation for Gaussian data. The factor cn is a correction factor to correct bias in small sample sizes. Sn has an
asymptotic efficiency of 58%.
double gsl_stats_Sn0_from_sorted_data(const double sorted_data[], const size_t stride, const size_t n, double
work[])
double gsl_stats_Sn_from_sorted_data(const double sorted_data[], const size_t stride, const size_t n, double
work[])
These functions return the Sn statistic of sorted_data, a dataset of length n with stride stride. The elementsof the array must be in ascending numerical order. There are no checks to see whether the data are sorted, so thefunction gsl_sort() should always be used first. The Sn0 function calculates mediani {medianj (|xi − xj |)}(i.e. the Sn statistic without the bias correction scale factors). These functions require additional workspace ofsize n provided in work.

21.12.3 Qn Statistic
The Qn statistic developed by Croux and Rousseeuw is defined as
Qn = 2.21914 × dn × {|xi − xj | , i < j}(k)The factor 2.21914 makes Qn an unbiased estimate of the standard deviation for Gaussian data. The factor dn is acorrection factor to correct bias in small sample sizes. The order statistic is

 n
2 +1
k=
2
Qn has an asymptotic efficiency of 82%.
double gsl_stats_Qn0_from_sorted_data(const double sorted_data[], const size_t stride, const size_t n, double
work[], int work_int[])
double gsl_stats_Qn_from_sorted_data(const double sorted_data[], const size_t stride, const size_t n, double
work[], int work_int[])
These functions return the Qn statistic of sorted_data, a dataset of length n with stride stride. The elementsof the array must be in ascending numerical order. There are no checks to see whether the data are sorted, so thefunction gsl_sort() should always be used first. The Qn0 function calculates {|xi − xj | , i < j}(k) (i.e. Qn
without the bias correction scale factors). These functions require additional workspace of size 3n provided inwork and integer workspace of size 5n provided in work_int.
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21.13 Examples
Here is a basic example of how to use the statistical functions:#include <stdio.h>#include <gsl/gsl_statistics.h>
int
main(void){
double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
double mean, variance, largest, smallest;mean= gsl_stats_mean(data, 1, 5);variance = gsl_stats_variance(data, 1, 5);largest = gsl_stats_max(data, 1, 5);smallest = gsl_stats_min(data, 1, 5);
printf ("The dataset is %g, %g, %g, %g, %g\n",data[0], data[1], data[2], data[3], data[4]);printf ("The sample mean is %g\n", mean);printf ("The estimated variance is %g\n", variance);
printf ("The largest value is %g\n", largest);
printf ("The smallest value is %g\n", smallest);
return 0;}
The program should produce the following output,
The dataset is 17.2, 18.1, 16.5, 18.3, 12.6
The sample mean is 16.54
The estimated variance is 5.373
The largest value is 18.3
The smallest value is 12.6
Here is an example using sorted data,#include <stdio.h>#include <gsl/gsl_sort.h>#include <gsl/gsl_statistics.h>
int
main(void){
double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
double median, upperq, lowerq;
printf ("Original dataset: %g, %g, %g, %g, %g\n",data[0], data[1], data[2], data[3], data[4]);gsl_sort (data, 1, 5);(continues on next page)

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printf ("Sorted dataset: %g, %g, %g, %g, %g\n",data[0], data[1], data[2], data[3], data[4]);median= gsl_stats_median_from_sorted_data (data,1, 5);
upperq= gsl_stats_quantile_from_sorted_data (data,
1, 5,0.75);
lowerq= gsl_stats_quantile_from_sorted_data (data,
1, 5,0.25);
printf ("The median is %g\n", median);printf ("The upper quartile is %g\n", upperq);
printf ("The lower quartile is %g\n", lowerq);
return 0;}
This program should produce the following output,
Original dataset: 17.2, 18.1, 16.5, 18.3, 12.6
Sorted dataset: 12.6, 16.5, 17.2, 18.1, 18.3
The median is 17.2
The upper quartile is 18.1
The lower quartile is 16.5

21.14 References and Further Reading
The standard reference for almost any topic in statistics is the multi-volume Advanced Theory of Statistics by Kendall
and Stuart.
• Maurice Kendall, Alan Stuart, and J. Keith Ord. The Advanced Theory of Statistics (multiple volumes) reprinted
as Kendall’s Advanced Theory of Statistics. Wiley, ISBN 047023380X.
Many statistical concepts can be more easily understood by a Bayesian approach. The following book by Gelman,
Carlin, Stern and Rubin gives a comprehensive coverage of the subject.
• Andrew Gelman, John B. Carlin, Hal S. Stern, Donald B. Rubin. Bayesian Data Analysis. Chapman & Hall,ISBN 0412039915.
For physicists the Particle Data Group provides useful reviews of Probability and Statistics in the “Mathematical Tools”
section of its Annual Review of Particle Physics.
• Review of Particle Properties, R.M. Barnett et al., Physical Review D54, 1 (1996)
The Review of Particle Physics is available online at the website http://pdg.lbl.gov/.
The following papers describe robust scale estimation,• C. Croux and P. J. Rousseeuw, Time-Efficient algorithms for two highly robust estimators of scale, Comp. Stat.,Physica, Heidelberg, 1992.
21.14. References and Further Reading297

GNU Scientific Library, Release 2.7• P. J. Rousseeuw and C. Croux, Explicit scale estimators with high breakdown point, L1-Statistical Analysis andRelated Methods, pp. 77-92, 1992.

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CHAPTERTWENTYTWORUNNING STATISTICS

This chapter describes routines for computing running statistics, also known as online statistics, of data. These routines
are suitable for handling large datasets for which it may be inconvenient or impractical to store in memory all at once.
The data can be processed in a single pass, one point at a time. Each time a data point is added to the accumulator,
internal parameters are updated in order to compute the current mean, variance, standard deviation, skewness, andkurtosis. These statistics are exact, and are updated with numerically stable single-pass algorithms. The median andarbitrary quantiles are also available, however these calculations use algorithms which provide approximations, andgrow more accurate as more data is added to the accumulator.
The functions described in this chapter are declared in the header file gsl_rstat.h.

22.1 Initializing the Accumulator
type gsl_rstat_workspace
This workspace contains parameters used to calculate various statistics and are updated after each data point is
added to the accumulator.
gsl_rstat_workspace *gsl_rstat_alloc(void)
This function allocates a workspace for computing running statistics. The size of the workspace is O(1).
void gsl_rstat_free(gsl_rstat_workspace *w)
This function frees the memory associated with the workspace w.
int gsl_rstat_reset(gsl_rstat_workspace *w)
This function resets the workspace w to its initial state, so it can begin working on a new set of data.

22.2 Adding Data to the Accumulator
int gsl_rstat_add(const double x, gsl_rstat_workspace *w)
This function adds the data point x to the statistical accumulator, updating calculations of the mean, variance,
standard deviation, skewness, kurtosis, and median.
size_t gsl_rstat_n(const gsl_rstat_workspace *w)
This function returns the number of data so far added to the accumulator.

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22.3 Current Statistics
double gsl_rstat_min(const gsl_rstat_workspace *w)
This function returns the minimum value added to the accumulator.
double gsl_rstat_max(const gsl_rstat_workspace *w)
This function returns the maximum value added to the accumulator.
double gsl_rstat_mean(const gsl_rstat_workspace *w)
This function returns the mean of all data added to the accumulator, defined as
µ̂ =

1 X
xi
N

double gsl_rstat_variance(const gsl_rstat_workspace *w)
This function returns the variance of all data added to the accumulator, defined asσ̂ 2 =

X1(xi − µ̂)2(N − 1)

double gsl_rstat_sd(const gsl_rstat_workspace *w)
This function returns the standard deviation of all data added to the accumulator, defined as the square root ofthe variance given above.
double gsl_rstat_sd_mean(const gsl_rstat_workspace *w)
This function returns the standard deviation of the mean, defined asσ̂
σ̂µ^ = √Ndouble gsl_rstat_rms(const gsl_rstat_workspace *w)
This function returns the root mean square of all data added to the accumulator, defined as
r
1 X 2
xi
rms =
N
double gsl_rstat_skew(const gsl_rstat_workspace *w)
This function returns the skewness of all data added to the accumulator, defined as3
1 X xi − µ̂
skew =
Nσ̂
double gsl_rstat_kurtosis(const gsl_rstat_workspace *w)
This function returns the kurtosis of all data added to the accumulator, defined as4 !
1 X xi − µ̂
−3kurtosis =Nσ̂double gsl_rstat_median(gsl_rstat_workspace *w)
This function returns an estimate of the median of the data added to the accumulator.

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22.4 Quantiles
The functions in this section estimate quantiles dynamically without storing the entire dataset, using the algorithm ofJain and Chlamtec, 1985. Only five points (markers) are stored which represent the minimum and maximum of thedata, as well as current estimates of the p/2-, p-, and (1 + p)/2-quantiles. Each time a new data point is added, themarker positions and heights are updated.
type gsl_rstat_quantile_workspace
This workspace contains parameters for estimating quantiles of the current dataset
gsl_rstat_quantile_workspace *gsl_rstat_quantile_alloc(const double p)
This function allocates a workspace for the dynamic estimation of p-quantiles, where p is between 0 and 1. The
median corresponds to p = 0.5. The size of the workspace is O(1).
void gsl_rstat_quantile_free(gsl_rstat_quantile_workspace *w)
This function frees the memory associated with the workspace w.
int gsl_rstat_quantile_reset(gsl_rstat_quantile_workspace *w)
This function resets the workspace w to its initial state, so it can begin working on a new set of data.
int gsl_rstat_quantile_add(const double x, gsl_rstat_quantile_workspace *w)
This function updates the estimate of the p-quantile with the new data point x.
double gsl_rstat_quantile_get(gsl_rstat_quantile_workspace *w)
This function returns the current estimate of the p-quantile.

22.5 Examples
Here is a basic example of how to use the statistical functions:#include <stdio.h>#include <gsl/gsl_rstat.h>intmain(void){
double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
double mean, variance, largest, smallest, sd,rms, sd_mean, median, skew, kurtosis;
gsl_rstat_workspace *rstat_p = gsl_rstat_alloc();size_t i, n;/* add data to rstat accumulator */for (i = 0; i < 5; ++i)
gsl_rstat_add(data[i], rstat_p);mean= gsl_rstat_mean(rstat_p);variance = gsl_rstat_variance(rstat_p);
largest = gsl_rstat_max(rstat_p);
smallest = gsl_rstat_min(rstat_p);
median= gsl_rstat_median(rstat_p);sd= gsl_rstat_sd(rstat_p);
sd_mean = gsl_rstat_sd_mean(rstat_p);skew= gsl_rstat_skew(rstat_p);(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)rms= gsl_rstat_rms(rstat_p);kurtosis = gsl_rstat_kurtosis(rstat_p);n= gsl_rstat_n(rstat_p);
printf ("The dataset is %g, %g, %g, %g, %g\n",data[0], data[1], data[2], data[3], data[4]);printf ("The sample mean is %g\n", mean);printf ("The estimated variance is %g\n", variance);
printf ("The largest value is %g\n", largest);
printf ("The smallest value is %g\n", smallest);
printf( "The median is %g\n", median);printf( "The standard deviation is %g\n", sd);
printf( "The root mean square is %g\n", rms);
printf( "The standard devation of the mean is %g\n", sd_mean);printf( "The skew is %g\n", skew);
printf( "The kurtosis %g\n", kurtosis);
printf( "There are %zu items in the accumulator\n", n);gsl_rstat_reset(rstat_p);n = gsl_rstat_n(rstat_p);
printf( "There are %zu items in the accumulator\n", n);gsl_rstat_free(rstat_p);
return 0;}
The program should produce the following output,
The dataset is 17.2, 18.1, 16.5, 18.3, 12.6
The sample mean is 16.54
The estimated variance is 5.373
The largest value is 18.3
The smallest value is 12.6
The median is 17.2
The standard deviation is 2.31797
The root mean square is 16.6694
The standard devation of the mean is 1.03663
The skew is -0.829058
The kurtosis -1.2217
There are 5 items in the accumulator
There are 0 items in the accumulator
This next program estimates the lower quartile, median and upper quartile from 10,000 samples of a random Rayleighdistribution, using the P 2 algorithm of Jain and Chlamtec. For comparison, the exact values are also computed fromthe sorted dataset.
#include <stdio.h>#include <stdlib.h>#include <gsl/gsl_rstat.h>#include <gsl/gsl_statistics.h>(continues on next page)302

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GNU Scientific Library, Release 2.7(continued from previous page)#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>#include <gsl/gsl_sort.h>int
main(void){
const size_t N = 10000;
double *data = malloc(N * sizeof(double));gsl_rstat_quantile_workspace *work_25 = gsl_rstat_quantile_alloc(0.25);
gsl_rstat_quantile_workspace *work_50 = gsl_rstat_quantile_alloc(0.5);
gsl_rstat_quantile_workspace *work_75 = gsl_rstat_quantile_alloc(0.75);
gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
double exact_p25, exact_p50, exact_p75;
double val_p25, val_p50, val_p75;
size_t i;/* add data to quantile accumulators; also store data for exact* comparisons */for (i = 0; i < N; ++i){
data[i] = gsl_ran_rayleigh(r, 1.0);
gsl_rstat_quantile_add(data[i], work_25);
gsl_rstat_quantile_add(data[i], work_50);
gsl_rstat_quantile_add(data[i], work_75);}/* exact values */gsl_sort(data, 1, N);
exact_p25 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.25);
exact_p50 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.5);
exact_p75 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.75);/* estimated values */val_p25 = gsl_rstat_quantile_get(work_25);
val_p50 = gsl_rstat_quantile_get(work_50);
val_p75 = gsl_rstat_quantile_get(work_75);
printf ("The dataset is %g, %g, %g, %g, %g, ...\n",data[0], data[1], data[2], data[3], data[4]);printf ("0.25 quartile: exact = %.5f , estimated = %.5f , error = %.6e\n",exact_p25, val_p25, (val_p25 - exact_p25) / exact_p25);
printf ("0.50 quartile: exact = %.5f , estimated = %.5f , error = %.6e\n",exact_p50, val_p50, (val_p50 - exact_p50) / exact_p50);
printf ("0.75 quartile: exact = %.5f , estimated = %.5f , error = %.6e\n",exact_p75, val_p75, (val_p75 - exact_p75) / exact_p75);
gsl_rstat_quantile_free(work_25);
gsl_rstat_quantile_free(work_50);
gsl_rstat_quantile_free(work_75);
gsl_rng_free(r);(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)free(data);
return 0;}
The program should produce the following output,
The dataset is 0.00645272, 0.0074002, 0.0120706, 0.0207256, 0.0227282, ...
0.25 quartile: exact = 0.75766, estimated = 0.75580, error = -2.450209e-03
0.50 quartile: exact = 1.17508, estimated = 1.17438, error = -5.995912e-04
0.75 quartile: exact = 1.65347, estimated = 1.65696, error = 2.110571e-03

22.6 References and Further Reading
The algorithm used to dynamically estimate p-quantiles is described in the paper,• R. Jain and I. Chlamtac. The P^2 algorithm for dynamic calculation of quantiles and histograms without storing
observations, Communications of the ACM, Volume 28 (October), Number 10, 1985, p. 1076-1085.

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CHAPTERTWENTYTHREE

MOVING WINDOW STATISTICS

This chapter describes routines for computing moving window statistics (also called rolling statistics and running statistics), using a window around a sample which is used to calculate various local statistical properties of an input data
stream. The window is then slid forward by one sample to process the next data point and so on.
The functions described in this chapter are declared in the header file gsl_movstat.h.

23.1 Introduction
This chapter is concerned with calculating various statistics from subsets of a given dataset. The main idea is to compute
statistics in the vicinity of a given data sample by defining a window which includes the sample itself as well as somespecified number of samples before and after the sample in question. For a sample xi , we define a window WiH,J as
WiH,J = {xi−H , . . . , xi , . . . , xi+J }The parameters H and J are non-negative integers specifying the number of samples to include before and after the
sample xi . Statistics such as the mean and standard deviation of the window WiH,J may be computed, and then thewindow is shifted forward by one sample to focus on xi+1 . The total number of samples in the window is K = H+J +1.
To define a symmetric window centered on xi , one would set H = J = bK/2c.

23.2 Handling Endpoints
When processing samples near the ends of the input signal, there will not be enough samples to fill the window WiH,J
defined above. Therefore the user must specify how to construct the windows near the end points. This is done bypassing an input argument of type gsl_movstat_end_t:type gsl_movstat_end_t
This data type specifies how to construct windows near end points and can be selected from the following choices:
GSL_MOVSTAT_END_PADZEROWith this option, a full window of length K will be constructed by inserting zeros into the window near
the signal end points. Effectively, the input signal is modified tox̃ = {0, . . . , 0, x1 , x2 , . . . , xn−1 , xn , 0, . . . , 0}
| {z }| {z }
H zeros

J zerosto ensure a well-defined window for all xi .

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GSL_MOVSTAT_END_PADVALUEWith this option, a full window of length K will be constructed by padding the window with the first andlast sample in the input signal. Effectively, the input signal is modified tox̃ = {x1 , . . . , x1 , x1 , x2 , . . . , xn−1 , xn , xn , . . . , xn }| {z }| {z }HJ

GSL_MOVSTAT_END_TRUNCATE
With this option, no padding is performed, and the windows are simply truncated as the end points areapproached.

23.3 Allocation for Moving Window Statistics
type gsl_movstat_workspace
The moving window statistical routines use a common workspace.
gsl_movstat_workspace *gsl_movstat_alloc(const size_t K)
This function allocates a workspace for computing symmetric, centered moving statistics with a window lengthof K samples. In this case, H = J = bK/2c. The size of the workspace is O(7K).
gsl_movstat_workspace *gsl_movstat_alloc2(const size_t H, const size_t J)
This function allocates a workspace for computing moving statistics using a window with H samples prior to thecurrent sample, and J samples after the current sample. The total window size is K = H + J + 1. The size of
the workspace is O(7K).
void *gsl_movstat_free(gsl_movstat_workspace *w)
This function frees the memory associated with w.

23.4 Moving Mean
The moving window mean calculates the mean of the values of each window WiH,J .
µ̂i =

1

X

WiH,J

xm ∈WiH,J

xm

Here, WiH,J represents the number of elements in the window WiH,J . This will normally be K, unless the
GSL_MOVSTAT_END_TRUNCATE option is selected, in which case it could be less than K near the signal end points.
int gsl_movstat_mean(const gsl_movstat_end_t endtype, const gsl_vector *x, gsl_vector *y,
gsl_movstat_workspace *w)
This function computes the moving window mean of the input vector x, storing the output in y. The parameter
endtype specifies how windows near the ends of the input should be handled. It is allowed to have x = y for anin-place moving mean.

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23.5 Moving Variance and Standard Deviation
The moving window variance calculates the sample variance of the values of each window WiH,J , defined byσ̂i2 = 1X2
(xm − µ̂i )H,J− 1 xm ∈W H,JWi
iwhere µ̂i is the mean of WiH,J defined above. The standard deviation σ̂i is the square root of the variance.
int gsl_movstat_variance(const gsl_movstat_end_t endtype, const gsl_vector *x, gsl_vector *y,
gsl_movstat_workspace *w)
This function computes the moving window variance of the input vector x, storing the output in y. The parameter
endtype specifies how windows near the ends of the input should be handled. It is allowed to have x = y for anin-place moving variance.
int gsl_movstat_sd(const gsl_movstat_end_t endtype, const gsl_vector *x, gsl_vector *y, gsl_movstat_workspace*w)
This function computes the moving window standard deviation of the input vector x, storing the output in y. The
parameter endtype specifies how windows near the ends of the input should be handled. It is allowed to have x= y for an in-place moving standard deviation.

23.6 Moving Minimum and Maximum
The moving minimum/maximum calculates the minimum and maximum values of each window WiH,J .
yimin = min WiH,J
yimax = max WiH,J
int gsl_movstat_min(const gsl_movstat_end_t endtype, const gsl_vector *x, gsl_vector *y, gsl_movstat_workspace*w)
This function computes the moving minimum of the input vector x, storing the result in y. The parameter
endtype specifies how windows near the ends of the input should be handled. It is allowed to have x = y for anin-place moving minimum.
int gsl_movstat_max(const gsl_movstat_end_t endtype, const gsl_vector *x, gsl_vector *y, gsl_movstat_workspace*w)
This function computes the moving maximum of the input vector x, storing the result in y. The parameter
endtype specifies how windows near the ends of the input should be handled. It is allowed to have x = y for anin-place moving maximum.
int gsl_movstat_minmax(const gsl_movstat_end_t endtype, const gsl_vector *x, gsl_vector *y_min, gsl_vector*y_max, gsl_movstat_workspace *w)
This function computes the moving minimum and maximum of the input vector x, storing the window minimums
in y_min and the window maximums in y_max. The parameter endtype specifies how windows near the endsof the input should be handled.

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23.7 Moving Sum
The moving window sum calculates the sum of the values of each window WiH,J .
X
yi =
xm
xm ∈WiH,J

int gsl_movstat_sum(const gsl_movstat_end_t endtype, const gsl_vector *x, gsl_vector *y,
gsl_movstat_workspace *w)
This function computes the moving window sum of the input vector x, storing the output in y. The parameter
endtype specifies how windows near the ends of the input should be handled. It is allowed to have x = y for anin-place moving sum.

23.8 Moving Median
The moving median calculates the median of the window WiH,J for each sample xi :yi = median WiH,J

int gsl_movstat_median(const gsl_movstat_end_t endtype, const gsl_vector *x, gsl_vector *y,
gsl_movstat_workspace *w)
This function computes the moving median of the input vector x, storing the output in y. The parameter endtype
specifies how windows near the ends of the input should be handled. It is allowed for x = y for an in-place moving
window median.

23.9 Robust Scale Estimation
A common problem in statistics is to quantify the dispersion (also known as the variability, scatter, and spread) of aset of data. Often this is done by calculating the variance or standard deviation. However these statistics are strongly
influenced by outliers, and can often provide erroneous results when even a small number of outliers are present.
Several useful statistics have emerged to provide robust estimates of scale which are not as susceptible to data outliers.
A few of these statistical scale estimators are described below.

23.9.1 Moving MAD
The median absolute deviation (MAD) for the window WiH,J is defined to be the median of the absolute deviations
from the window’s median:M ADi = 1.4826 × median WiH,J − median WiH,J
The factor of 1.4826 makes the MAD an unbiased estimator of the standard deviation for Gaussian data. The MAD
has an efficiency of 37%. See here for more information.
int gsl_movstat_mad0(const gsl_movstat_end_t endtype, const gsl_vector *x, gsl_vector *xmedian, gsl_vector*xmad, gsl_movstat_workspace *w)

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int gsl_movstat_mad(const gsl_movstat_end_t endtype, const gsl_vector *x, gsl_vector *xmedian, gsl_vector*xmad, gsl_movstat_workspace *w)
These functions compute the moving MAD of the input vector x and store the result in xmad. The mediansof each window WiH,J are stored in xmedian on output. The inputs x, xmedian, and xmad must all be thesame length. The parameter endtype specifies how windows near the ends of the input should be handled. The
function mad0 does not include the scale factor of 1.4826, while the function mad does include this factor.

23.9.2 Moving QQR
The q-quantile range (QQR) is the difference between the (1 − q) and q quantiles of a set of data,
QQR = Q1−q − Qq
The case q = 0.25 corresponds to the well-known interquartile range (IQR), which is the difference between the 75thand 25th percentiles of a set of data. The QQR is a trimmed estimator, the main idea being to discard the largest andsmallest values in a data window and compute a scale estimate from the remaining middle values. In the case of the
IQR, the largest and smallest 25% of the data are discarded and the scale is estimated from the remaining (middle)
50%.
int gsl_movstat_qqr(const gsl_movstat_end_t endtype, const gsl_vector *x, const double q, gsl_vector *xqqr,
gsl_movstat_workspace *w)
This function computes the moving QQR of the input vector x and stores the q-quantile ranges of each window
WiH,J in xqqr. The quantile parameter q must be between 0 and 0.5. The input q = 0.25 corresponds to the
IQR. The inputs x and xqqr must be the same length. The parameter endtype specifies how windows near theends of the input should be handled.

23.9.3 Moving Sn
The Sn statistic proposed by Croux and Rousseeuw is based on pairwise differences between all samples in the window.
It has an efficiency of 58%, significantly higher than the MAD. See here for more information.
int gsl_movstat_Sn(const gsl_movstat_end_t endtype, const gsl_vector *x, gsl_vector *xscale,
gsl_movstat_workspace *w)
This function computes the moving Sn of the input vector x and stores the output in xscale. The inputs x andxscale must be the same length. The parameter endtype specifies how windows near the ends of the input
should be handled. It is allowed for x = xscale for an in-place moving window Sn .

23.9.4 Moving Qn
The Qn statistic proposed by Croux and Rousseeuw is loosely based on the Hodges-Lehmann location estimator. It
has a relatively high efficiency of 82%. See here for more information.
int gsl_movstat_Qn(const gsl_movstat_end_t endtype, const gsl_vector *x, gsl_vector *xscale,
gsl_movstat_workspace *w)
This function computes the moving Qn of the input vector x and stores the output in xscale. The inputs x andxscale must be the same length. The parameter endtype specifies how windows near the ends of the input
should be handled. It is allowed for x = xscale for an in-place moving window Qn .

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23.10 User-defined Moving Statistics
GSL offers an interface for users to define their own moving window statistics functions, without needing to implement
the edge-handling and accumulator machinery. This can be done by explicitly constructing the windows WiH,J for agiven input signal (gsl_movstat_fill()), or by calculating a user-defined function for each window automatically.
In order to apply a user-defined function to each window, users must define a variable of type gsl_movstat_function
to pass into gsl_movstat_apply(). This structure is defined as follows.
type gsl_movstat_function
Structure specifying user-defined moving window statistical function:
typedef struct{
double (* function) (const size_t n, double x[], void * params);
void * params;} gsl_movstat_function;
This structure contains a pointer to the user-defined function as well as possible parameters to pass to the function.
double (*function)(const size_t n, double x[], void *params)
This function returns the user-defined statistic of the array x of length n. User-specified parameters are
passed in via params. It is allowed to modify the array x.
void *params
User-specified parameters to be passed into the function.
int gsl_movstat_apply(const gsl_movstat_end_t endtype, const gsl_movstat_function *F, const gsl_vector *x,
gsl_vector *y, gsl_movstat_workspace *w)
This function applies the user-defined moving window statistic specified in F to the input vector x, storing theoutput in y. The parameter endtype specifies how windows near the ends of the input should be handled. It isallowed for x = y for an in-place moving window calculation.
size_t gsl_movstat_fill(const gsl_movstat_end_t endtype, const gsl_vector *x, const size_t idx, const size_t H,
const size_t J, double *window)
This function explicitly constructs the sliding window for the input vector x which is centered on the sample idx.
H,JOn output, the array window will contain Widx
. The number of samples to the left and right of the sample idx
are specified by H and J respectively. The parameter endtype specifies how windows near the ends of the input
should be handled. The function returns the size of the window.

23.11 Accumulators
Many of the algorithms of this chapter are based on an accumulator design, which process the input vector one sample ata time, updating calculations of the desired statistic for the current window. Each accumulator is stored in the following
structure:
type gsl_movstat_accum
Structure specifying accumulator for moving window statistics:
typedef struct{
size_t (* size) (const size_t n);
int (* init) (const size_t n, void * vstate);int (* insert) (const double x, void * vstate);(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)int (* delete) (void * vstate);int (* get) (void * params, double * result, const void * vstate);} gsl_movstat_accum;
The structure contains function pointers responsible for performing different tasks for the accumulator.
size_t (*size)(const size_t n)This function returns the size of the workspace (in bytes) needed by the accumulator for a moving windowof length n.
int (*init)(const size_t n, void *vstate)This function initializes the workspace vstate for a moving window of length n.
int (*insert)(const double x, void *vstate)This function inserts a single sample x into the accumulator, updating internal calculations of the desired
statistic. If the accumulator is full (i.e. n samples have already been inserted), then the oldest sample isdeleted from the accumulator.
int (*delete)(void *vstate)This function deletes the oldest sample from the accumulator, updating internal calculations of the desired
statistic.
int (*get)(void *params, double *result, const void *vstate)This function stores the desired statistic for the current window in result. The input params specifies
optional parameters for calculating the statistic.
The following accumulators of type gsl_movstat_accum are defined by GSL to perform moving window statistics
calculations.
gsl_movstat_accum *gsl_movstat_accum_min
gsl_movstat_accum *gsl_movstat_accum_max
gsl_movstat_accum *gsl_movstat_accum_minmax
These accumulators calculate moving window minimum/maximums efficiently, using the algorithm of D.
Lemire.
gsl_movstat_accum *gsl_movstat_accum_mean
gsl_movstat_accum *gsl_movstat_accum_sd
gsl_movstat_accum *gsl_movstat_accum_variance
These accumulators calculate the moving window mean, standard deviation, and variance, using the algorithmof B. P. Welford.
gsl_movstat_accum *gsl_movstat_accum_medianThis accumulator calculates the moving window median using the min/max heap algorithm of Härdle and Steiger.
gsl_movstat_accum *gsl_movstat_accum_Sn
gsl_movstat_accum *gsl_movstat_accum_Qn
These accumulators calculate the moving window Sn and Qn statistics developed by Croux and Rousseeuw.
gsl_movstat_accum *gsl_movstat_accum_sum
This accumulator calculates the moving window sum.
gsl_movstat_accum *gsl_movstat_accum_qqr
This accumulator calculates the moving window q-quantile range.

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23.12 Examples
23.12.1 Example 1
The following example program computes the moving mean, minimum and maximum of a noisy sinusoid signal oflength N = 500 with a symmetric moving window of size K = 11.

Fig. 23.1: Original signal time series (gray) with moving mean (green), moving minimum (blue), and moving maximum(orange).
The program is given below.
#include <stdio.h>#include <stdlib.h>#include <gsl/gsl_math.h>#include <gsl/gsl_movstat.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>#include <gsl/gsl_vector.h>intmain(void){
const size_t N = 500;/* length of time series */const size_t K = 11;/* window size */gsl_movstat_workspace * w = gsl_movstat_alloc(K);gsl_vector *x = gsl_vector_alloc(N);gsl_vector *xmean = gsl_vector_alloc(N);gsl_vector *xmin = gsl_vector_alloc(N);gsl_vector *xmax = gsl_vector_alloc(N);gsl_rng *r = gsl_rng_alloc(gsl_rng_default);(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)size_t i;for (i = 0; i < N; ++i){
double xi = cos(4.0 * M_PI * i / (double) N);
double ei = gsl_ran_gaussian(r, 0.1);
gsl_vector_set(x, i, xi + ei);}/* compute moving statistics */gsl_movstat_mean(GSL_MOVSTAT_END_PADVALUE, x, xmean, w);
gsl_movstat_minmax(GSL_MOVSTAT_END_PADVALUE, x, xmin, xmax, w);/* print results */for (i = 0; i < N; ++i){
printf("%zu %f %f %f %f \n",i,gsl_vector_get(x, i),
gsl_vector_get(xmean, i),
gsl_vector_get(xmin, i),
gsl_vector_get(xmax, i));}gsl_vector_free(x);gsl_vector_free(xmean);gsl_rng_free(r);gsl_movstat_free(w);return 0;}23.12.2 Example 2: Robust Scale
The following example program analyzes a time series of length N = 1000 composed of Gaussian random variateswith zero mean whose standard deviation changes in a piecewise constant fashion as shown in the table below.
Sample Range
1-200
201-450451-600
601-850
851-1000

σ
1.0
5.0
1.0
3.0
5.0

Additionally, about 1% of the samples are perturbed to represent outliers by adding ±15 to the random Gaussian variate.
The program calculates the moving statistics MAD, IQR, Sn , Qn , and the standard deviation using a symmetric moving
window of length K = 41. The results are shown in Fig. 23.2.
The robust statistics follow the true standard deviation piecewise changes well, without being influenced by the outliers.
The moving standard deviation (gray curve) is heavily influenced by the presence of the outliers. The program is given

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Fig. 23.2: Top: time series of piecewise constant variance. Bottom: scale estimates using a moving window; the truesigma value is in light blue, MAD in green, IQR in red, Sn in yellow, and Qn in dark blue. The moving standard
deviation is shown in gray.
below.
#include <stdio.h>#include <stdlib.h>#include <gsl/gsl_math.h>#include <gsl/gsl_movstat.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>#include <gsl/gsl_vector.h>intmain(void){
const size_t N = 1000;/* length of time series */const double sigma[] = { 1.0, 5.0, 1.0, 3.0, 5.0 };/* variances */const size_t N_sigma[] = { 200, 450, 600, 850, 1000 }; /* samples where variance␣,→changes */const size_t K = 41;/* window size */gsl_vector *x = gsl_vector_alloc(N);gsl_vector *xmedian = gsl_vector_alloc(N);gsl_vector *xmad = gsl_vector_alloc(N);gsl_vector *xiqr = gsl_vector_alloc(N);gsl_vector *xSn = gsl_vector_alloc(N);gsl_vector *xQn = gsl_vector_alloc(N);gsl_vector *xsd = gsl_vector_alloc(N);gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
gsl_movstat_workspace * w = gsl_movstat_alloc(K);(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)size_t idx = 0;
size_t i;for (i = 0; i < N; ++i){
double gi = gsl_ran_gaussian(r, sigma[idx]);
double u = gsl_rng_uniform(r);double outlier = (u < 0.01) ? 15.0*GSL_SIGN(gi) : 0.0;
double xi = gi + outlier;gsl_vector_set(x, i, xi);
if (i == N_sigma[idx] - 1)++idx;}/* compute moving statistics */gsl_movstat_mad(GSL_MOVSTAT_END_TRUNCATE, x, xmedian, xmad, w);
gsl_movstat_qqr(GSL_MOVSTAT_END_TRUNCATE, x, 0.25, xiqr, w);
gsl_movstat_Sn(GSL_MOVSTAT_END_TRUNCATE, x, xSn, w);
gsl_movstat_Qn(GSL_MOVSTAT_END_TRUNCATE, x, xQn, w);
gsl_movstat_sd(GSL_MOVSTAT_END_TRUNCATE, x, xsd, w);/* scale IQR by factor to approximate standard deviation */gsl_vector_scale(xiqr, 0.7413);/* print results */idx = 0;for (i = 0; i < N; ++i){
printf("%zu %f %f %f %f %f %f %f\n",i,gsl_vector_get(x, i),sigma[idx],
gsl_vector_get(xmad, i),
gsl_vector_get(xiqr, i),
gsl_vector_get(xSn, i),
gsl_vector_get(xQn, i),
gsl_vector_get(xsd, i));
if (i == N_sigma[idx] - 1)++idx;}
gsl_vector_free(x);gsl_vector_free(xmedian);
gsl_vector_free(xmad);gsl_vector_free(xiqr);
gsl_vector_free(xSn);
gsl_vector_free(xQn);
gsl_vector_free(xsd);
gsl_rng_free(r);(continues on next page)

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315

GNU Scientific Library, Release 2.7(continued from previous page)gsl_movstat_free(w);return 0;}23.12.3 Example 3: User-defined Moving Window
This example program illustrates how a user can define their own moving window function to apply to an input vector.
It constructs a random noisy time series of length N = 1000 with some outliers added. Then it applies a moving
window trimmed mean to the time series with trim parameter α = 0.1. The length of the moving window is K = 11,
so the smallest and largest sample of each window is discarded prior to computing the mean. The results are shown in
Fig. 23.3.

Fig. 23.3: Noisy time series data (black) with moving window trimmed mean (red)The program is given below.
#include <stdio.h>#include <stdlib.h>#include <gsl/gsl_math.h>#include <gsl/gsl_movstat.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>#include <gsl/gsl_vector.h>#include <gsl/gsl_sort.h>#include <gsl/gsl_statistics.h>doublefunc(const size_t n, double x[], void * params)(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page){
const double alpha = *(double *) params;
gsl_sort(x, 1, n);return gsl_stats_trmean_from_sorted_data(alpha, x, 1, n);}intmain(void){
const size_t N = 1000;/* length of time series */const size_t K = 11;/* window size */double alpha = 0.1;/* trimmed mean parameter */gsl_vector *x = gsl_vector_alloc(N); /* input vector */gsl_vector *y = gsl_vector_alloc(N); /* filtered output vector for alpha1 */gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
gsl_movstat_workspace *w = gsl_movstat_alloc(K);gsl_movstat_function F;
size_t i;
double sum = 0.0;/* generate input signal */for (i = 0; i < N; ++i){
double ui = gsl_ran_gaussian(r, 1.0);
double outlier = (gsl_rng_uniform(r) < 0.01) ? 10.0*GSL_SIGN(ui) : 0.0;
sum += ui;
gsl_vector_set(x, i, sum + outlier);}/* apply moving window function */F.function = func;
F.params = &alpha;gsl_movstat_apply(GSL_MOVSTAT_END_PADVALUE, &F, x, y, w);/* print results */for (i = 0; i < N; ++i){
double xi = gsl_vector_get(x, i);double yi = gsl_vector_get(y, i);printf("%f %f\n", xi, yi);}gsl_vector_free(x);gsl_vector_free(y);gsl_rng_free(r);gsl_movstat_free(w);return 0;}23.12. Examples

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23.13 References and Further Reading
The following publications are relevant to the algorithms described in this chapter,• W.Hardle and W. Steiger, Optimal Median Smoothing, Appl. Statist., 44 (2), 1995.
• D. Lemire, Streaming Maximum-Minimum Filter Using No More than Three Comparisons per Element, Nordic
Journal of Computing, 13 (4), 2006 (https://arxiv.org/abs/cs/0610046).
• B. P. Welford, Note on a method for calculating corrected sums of squares and products, Technometrics, 4 (3),
1962.

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CHAPTERTWENTYFOURDIGITAL FILTERING

24.1 Introduction
The filters discussed in this chapter are based on the following moving data window which is centered on i-th sample:
WiH = {xi−H , . . . , xi , . . . , xi+H }Here, H is a non-negative integer called the window half-length, which represents the number of samples before andafter sample i. The total window length is K = 2H + 1.

24.2 Handling Endpoints
When processing samples near the ends of the input signal, there will not be enough samples to fill the window WiH
defined above. Therefore the user must specify how to construct the windows near the end points. This is done bypassing an input argument of type gsl_filter_end_t:type gsl_filter_end_t
This data type specifies how to construct windows near end points and can be selected from the following choices:
GSL_FILTER_END_PADZEROWith this option, a full window of length K will be constructed by inserting zeros into the window near
the signal end points. Effectively, the input signal is modified tox̃ = {0, . . . , 0, x1 , x2 , . . . , xn−1 , xn , 0, . . . , 0}
| {z }| {z }
H zeros

H zerosto ensure a well-defined window for all xi .
GSL_FILTER_END_PADVALUEWith this option, a full window of length K will be constructed by padding the window with the first andlast sample in the input signal. Effectively, the input signal is modified tox̃ = {x1 , . . . , x1 , x1 , x2 , . . . , xn−1 , xn , xn , . . . , xn }| {z }| {z }H

H

GSL_FILTER_END_TRUNCATE
With this option, no padding is performed, and the windows are simply truncated as the end points areapproached.

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24.3 Linear Digital Filters
24.3.1 Gaussian Filter
The Gaussian filter convolves the input signal with a Gaussian kernel or window. This filter is often used as a smoothingor noise reduction filter. The Gaussian kernel is defined by
2

G(k) = e− 2 (α (K−1)/2 ) = e−k /2σ
k

1

2

2

for −(K − 1)/2 ≤ k ≤ (K − 1)/2, and K is the size of the kernel. The parameter α specifies the number ofstandard deviations σ desired in the kernel. So for example setting α = 3 would define a Gaussian window of length
K which spans ±3σ. It is often more convenient to specify the parameter α rather than the standard deviation σ whenconstructing the kernel, since a fixed value of α would correspond to the same shape of Gaussian regardless of the size
K. The appropriate value of the standard deviation depends on K and is related to α as
σ=

K −1
2α

The routines below accept α as an input argument instead of σ.
The Gaussian filter offers a convenient way of differentiating and smoothing an input signal in a single pass. Using thederivative property of a convolution,ddG(G ∗ x) =∗x
dt
dt
the input signal x(t) can be smoothed and differentiated at the same time by convolution with a derivative Gaussian
kernel, which can be readily computed from the analytic expression above. The same principle applies to higher orderderivatives.
gsl_filter_gaussian_workspace *gsl_filter_gaussian_alloc(const size_t K)
This function initializes a workspace for Gaussian filtering using a kernel of size K. Here, H = K/2. If K iseven, it is rounded up to the next odd integer to ensure a symmetric window. The size of the workspace is O(K).
void gsl_filter_gaussian_free(gsl_filter_gaussian_workspace *w)
This function frees the memory associated with w.
int gsl_filter_gaussian(const gsl_filter_end_t endtype, const double alpha, const size_t order, const gsl_vector*x, gsl_vector *y, gsl_filter_gaussian_workspace *w)
This function applies a Gaussian filter parameterized by alpha to the input vector x, storing the output in y. The
derivative order is specified by order, with 0 corresponding to a Gaussian, 1 corresponding to a first derivative
Gaussian, and so on. The parameter endtype specifies how the signal end points are handled. It is allowed for
x = y for an in-place filter.
int gsl_filter_gaussian_kernel(const double alpha, const size_t order, const int normalize, gsl_vector *kernel)
This function constructs a Gaussian kernel parameterized by alpha and stores the output in kernel. The parameter order specifies the derivative order, with 0 corresponding to a Gaussian, 1 corresponding to a first
derivative Gaussian, and so on. If normalize is set to 1, then the kernel will be normalized to sum to one on
output. If normalize is set to 0, no normalization is performed.

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24.4 Nonlinear Digital Filters
The nonlinear digital filters described below are based on the window median, which is given bymi = median WiH = median {xi−H , . . . , xi , . . . , xi+H }The median is considered robust to local outliers, unlike the mean. Median filters can preserve sharp edges while atthe same removing signal noise, and are used in a wide range of applications.

24.4.1 Standard Median Filter
The standard median filter (SMF) simply replaces the sample xi by the median mi of the window WiH : This filter has
one tuning parameter given by H. The standard median filter is considered highly resistant to local outliers and local
noise in the data sequence {xi }.
gsl_filter_median_workspace *gsl_filter_median_alloc(const size_t K)
This function initializes a workspace for standard median filtering using a symmetric centered moving windowof size K. Here, H = K/2. If K is even, it is rounded up to the next odd integer to ensure a symmetric window.
The size of the workspace is O(7K).
void gsl_filter_median_free(gsl_filter_median_workspace *w)
This function frees the memory associated with w.
int gsl_filter_median(const gsl_filter_end_t endtype, const gsl_vector *x, gsl_vector *y,
gsl_filter_median_workspace *w)
This function applies a standard median filter to the input x, storing the output in y. The parameter endtype
specifies how the signal end points are handled. It is allowed to have x = y for an in-place filter.

24.4.2 Recursive Median Filter
The recursive median filter (RMF) is a modification of the SMF to include previous filter outputs in the window before
computing the median. The filter’s response isyi = median (yi−H , . . . , yi−1 , xi , xi+1 , . . . , xi+H )Sometimes, the SMF must be applied several times in a row to achieve adequate smoothing (i.e. a cascade filter). The
RMF, on the other hand, converges to a root sequence in one pass, and can sometimes provide a smoother result thanseveral passes of the SMF. A root sequence is an input which is left unchanged by the filter. So there is no need to apply
a recursive median filter twice to an input vector.
gsl_filter_rmedian_workspace *gsl_filter_rmedian_alloc(const size_t K)
This function initializes a workspace for recursive median filtering using a symmetric centered moving windowof size K. Here, H = K/2. If K is even, it is rounded up to the next odd integer to ensure a symmetric window.
The size of the workspace is O(K).
void gsl_filter_rmedian_free(gsl_filter_rmedian_workspace *w)
This function frees the memory associated with w.
int gsl_filter_rmedian(const gsl_filter_end_t endtype, const gsl_vector *x, gsl_vector *y,
gsl_filter_rmedian_workspace *w)
This function applies a recursive median filter to the input x, storing the output in y. The parameter endtype
specifies how the signal end points are handled. It is allowed to have x = y for an in-place filter.

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24.4.3 Impulse Detection Filter
Impulsive noise is characterized by short sequences of data points distinct from those in the surrounding neighborhood.
This section describes a powerful class of filters, also known as impulse rejection filters and decision-based filters,
designed to detect and remove such outliers from data. The filter’s response is given byxi , |xi − mi | ≤ tSi
yi =mi , |xi − mi | > tSi
where mi is the median value of the window WiH , Si is a robust estimate of the scatter or dispersion for the window
WiH , and t is a tuning parameter specifying the number of scale factors needed to determine that a point is an outlier.
The main idea is that the median mi will be unaffected by a small number of outliers in the window, and so a given
sample xi is tested to determine how far away it is from the median in terms of the local scale estimate Si . Samples
which are more than t scale estimates away from the median are labeled as outliers and replaced by the window medianmi . Samples which are less than t scale estimates from the median are left unchanged by the filter.
Note that when t = 0, the impulse detection filter is equivalent to the standard median filter. When t → ∞, it becomesthe identity filter. This means the impulse detection filter can be viewed as a “less aggressive” version of the standard
median filter, becoming less aggressive as t is increased. Note that this filter modifies only samples identified as outliers,
while the standard median filter changes all samples to the local median, regardless of whether they are outliers. Thisfact, plus the additional flexibility offered by the additional tuning parameter t can make the impulse detection filter abetter choice for some applications.
It is important to have a robust and accurate scale estimate Si in order to detect impulse outliers even in the presence ofnoise. The window standard deviation is not typically a good choice, as it can be significantly perturbed by the presenceof even one outlier. GSL offers the following choices (specified by a parameter of type gsl_filter_scale_t) forcomputing the scale estimate Si , all of which are robust to the presence of impulse outliers.
type gsl_filter_scale_t
This type specifies how the scale estimate Si of the window WiH is calculated.
GSL_FILTER_SCALE_MAD
This option specifies the median absolute deviation (MAD) scale estimate, defined bySi = 1.4826 × median |WiH − mi |This choice of scale estimate is also known as the Hampel filter in the statistical literature. See here formore information.
GSL_FILTER_SCALE_IQR
This option specifies the interquartile range (IQR) scale estimate, defined as the difference between the 75thand 25th percentiles of the window WiH ,Si = 0.7413 (Q0.75 − Q0.25 )where Qp is the p-quantile of the window WiH . The idea is to throw away the largest and smallest 25% of
the window samples (where the outliers would be), and estimate a scale from the middle 50%. The factor
0.7413 provides an unbiased estimate of the standard deviation for Gaussian data.
GSL_FILTER_SCALE_SN
This option specifies the so-called Sn statistic proposed by Croux and Rousseeuw. See here for more
information.
GSL_FILTER_SCALE_QN
This option specifies the so-called Qn statistic proposed by Croux and Rousseeuw. See here for more
information.

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Warning: While the scale estimates defined above are much less sensitive to outliers than the standard deviation,
they can suffer from an effect called implosion. The standard deviation of a window WiH will be zero if and only
if all samples in the window are equal. However, it is possible for the MAD of a window to be zero even if all thesamples in the window are not equal. For example, if K/2 + 1 or more of the K samples in the window are equal
to some value x∗ , then the window median will be equal to x∗ . Consequently, at least K/2 + 1 of the absolutedeviations |xj − x∗ | will be zero, and so the MAD will be zero. In such a case, the Hampel filter will act like thestandard median filter regardless of the value of t. Caution should also be exercised if dividing by Si .
gsl_filter_impulse_workspace *gsl_filter_impulse_alloc(const size_t K)
This function initializes a workspace for impulse detection filtering using a symmetric moving window of size
K. Here, H = K/2. If K is even, it is rounded up to the next odd integer to ensure a symmetric window. The
size of the workspace is O(6K).
void gsl_filter_impulse_free(gsl_filter_impulse_workspace *w)
This function frees the memory associated with w.
int gsl_filter_impulse(const gsl_filter_end_t endtype, const gsl_filter_scale_t scale_type, const double t, const
gsl_vector *x, gsl_vector *y, gsl_vector *xmedian, gsl_vector *xsigma, size_t *noutlier,gsl_vector_int *ioutlier, gsl_filter_impulse_workspace *w)
These functions apply an impulse detection filter to the input vector x, storing the filtered output in y. The tuning
parameter t is provided in t. The window medians mi are stored in xmedian and the Si are stored in xsigma on
output. The number of outliers detected is stored in noutlier on output, while the locations of flagged outliers
are stored in the boolean array ioutlier. The input ioutlier may be NULL if not desired. It is allowed to havex = y for an in-place filter.

24.5 Examples
24.5.1 Gaussian Example 1
This example program illustrates the Gaussian filter applied to smoothing a time series of length N = 500 with a kernel
size of K = 51. Three filters are applied with parameters α = 0.5, 3, 10. The results are shown in Fig. 24.1.
We see that the filter corresponding to α = 0.5 applies the most smoothing, while α = 10 corresponds to the least
amount of smoothing. The program is given below.
#include <stdio.h>#include <stdlib.h>#include <gsl/gsl_math.h>#include <gsl/gsl_filter.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>#include <gsl/gsl_vector.h>intmain(void){
const size_t N = 500;
const size_t K = 51;
const double alpha[3] = { 0.5, 3.0, 10.0 };gsl_vector *x = gsl_vector_alloc(N);gsl_vector *y1 = gsl_vector_alloc(N);/* length of time series *//* window size *//* alpha values *//* input vector *//* filtered output vector for alpha1 */(continues on next page)

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Fig. 24.1: Top panel: Gaussian kernels (unnormalized) for α = 0.5, 3, 10. Bottom panel: Time series (gray) with
Gaussian filter output for same α values.

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GNU Scientific Library, Release 2.7(continued from previous page)gsl_vector *y2 = gsl_vector_alloc(N);/* filtered output vector for alpha2 */gsl_vector *y3 = gsl_vector_alloc(N);/* filtered output vector for alpha3 */gsl_vector *k1 = gsl_vector_alloc(K);/* Gaussian kernel for alpha1 */gsl_vector *k2 = gsl_vector_alloc(K);/* Gaussian kernel for alpha2 */gsl_vector *k3 = gsl_vector_alloc(K);/* Gaussian kernel for alpha3 */gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
gsl_filter_gaussian_workspace *gauss_p = gsl_filter_gaussian_alloc(K);size_t i;
double sum = 0.0;/* generate input signal */for (i = 0; i < N; ++i){
double ui = gsl_ran_gaussian(r, 1.0);
sum += ui;
gsl_vector_set(x, i, sum);}/* compute kernels without normalization */gsl_filter_gaussian_kernel(alpha[0], 0, 0, k1);
gsl_filter_gaussian_kernel(alpha[1], 0, 0, k2);
gsl_filter_gaussian_kernel(alpha[2], 0, 0, k3);/* apply filters */gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha[0], 0, x, y1, gauss_p);
gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha[1], 0, x, y2, gauss_p);
gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha[2], 0, x, y3, gauss_p);/* print kernels */for (i = 0; i < K; ++i){
double k1i = gsl_vector_get(k1, i);double k2i = gsl_vector_get(k2, i);double k3i = gsl_vector_get(k3, i);printf("%e %e %e\n", k1i, k2i, k3i);}
printf("\n\n");/* print filter results */for (i = 0; i < N; ++i){
double xi = gsl_vector_get(x, i);double y1i = gsl_vector_get(y1, i);double y2i = gsl_vector_get(y2, i);double y3i = gsl_vector_get(y3, i);printf("%.12e %.12e %.12e %.12e\n", xi, y1i, y2i, y3i);}
gsl_vector_free(x);(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)gsl_vector_free(y1);gsl_vector_free(y2);gsl_vector_free(y3);gsl_vector_free(k1);gsl_vector_free(k2);gsl_vector_free(k3);gsl_rng_free(r);gsl_filter_gaussian_free(gauss_p);
return 0;}

24.5.2 Gaussian Example 2
A common application of the Gaussian filter is to detect edges, or sudden jumps, in a noisy input signal. It is used bothfor 1D edge detection in time series, as well as 2D edge detection in images. Here we will examine a noisy time seriesof length N = 1000 with a single edge. The input signal is defined as
0, n ≤ N/2
x(n) = e(n) +0.5, n > N/2
where e(n) is Gaussian random noise. The program smooths the input signal with order 0, 1, and 2 Gaussian filters oflength K = 61 with α = 3. For comparison, the program also computes finite differences of the input signal withoutsmoothing. The results are shown in Fig. 24.2.
The finite difference approximation of the first derivative (second row) shows the common problem with differentiatinga noisy signal. The noise is amplified and makes it extremely difficult to detect the sharp gradient at sample 500. The
third row shows the first order Gaussian smoothed signal with a clear peak at the location of the edge. Alternatively,
one could examine the second order Gaussian smoothed signal (fourth row) and look for zero crossings to determine
the edge location.
The program is given below.
#include <stdio.h>#include <stdlib.h>#include <gsl/gsl_math.h>#include <gsl/gsl_filter.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>#include <gsl/gsl_vector.h>intmain(void){
const size_t N = 1000;
const size_t K = 61;
const double alpha = 3.0;,→deviations */gsl_vector *x = gsl_vector_alloc(N);gsl_vector *y = gsl_vector_alloc(N);gsl_vector *dy = gsl_vector_alloc(N);/* length of time series *//* window size *//* Gaussian kernel has +/- 3 standard␣/* input vector *//* filtered output vector *//* first derivative filtered vector */(continues on next page)

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Fig. 24.2: Top row: original input signal x(n) (black) with Gaussian smoothed signal in red. Second row: First finite
differences of input signal. Third row: Input signal smoothed with a first order Gaussian filter. Fourth row: Input signal
smoothed with a second order Gaussian filter.

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GNU Scientific Library, Release 2.7(continued from previous page)gsl_vector *d2y = gsl_vector_alloc(N);/* second derivative filtered vector */gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
gsl_filter_gaussian_workspace *gauss_p = gsl_filter_gaussian_alloc(K);size_t i;/* generate input signal */for (i = 0; i < N; ++i){
double xi = (i > N / 2) ? 0.5 : 0.0;
double ei = gsl_ran_gaussian(r, 0.1);
gsl_vector_set(x, i, xi + ei);}/* apply filters */gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha, 0, x, y, gauss_p);
gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha, 1, x, dy, gauss_p);
gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha, 2, x, d2y, gauss_p);/* print results */for (i = 0; i < N; ++i){
double xi = gsl_vector_get(x, i);double yi = gsl_vector_get(y, i);double dyi = gsl_vector_get(dy, i);double d2yi = gsl_vector_get(d2y, i);double dxi;/* compute finite difference of x vector */if (i == 0)
dxi = gsl_vector_get(x, i + 1) - xi;
else if (i == N - 1)
dxi = gsl_vector_get(x, i) - gsl_vector_get(x, i - 1);
else
dxi = 0.5 * (gsl_vector_get(x, i + 1) - gsl_vector_get(x, i - 1));
printf("%.12e %.12e %.12e %.12e %.12e\n",
xi,
yi,
dxi,
dyi,
d2yi);}
gsl_vector_free(x);gsl_vector_free(y);gsl_vector_free(dy);gsl_vector_free(d2y);gsl_rng_free(r);gsl_filter_gaussian_free(gauss_p);
return 0;(continues on next page)

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24.5.3 Square Wave Signal Example
The following example program illustrates the median filters on a noisy square wave signal. Median filters are wellknown for preserving sharp edges in the input signal while reducing noise. The program constructs a 5 Hz square wave
signal with Gaussian noise added. Then the signal is filtered with a standard median filter and recursive median filter
using a symmetric window of length K = 7. The results are shown in Fig. 24.3.

Fig. 24.3: Original time series is in gray. The standard median filter output is in green and the recursive median filter
output is in red.
Both filters preserve the sharp signal edges while reducing the noise. The recursive median filter achieves a smootherresult than the standard median filter. The “blocky” nature of the output is characteristic of all median filters. The
program is given below.
#include <stdio.h>#include <stdlib.h>#include <gsl/gsl_math.h>#include <gsl/gsl_filter.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>#include <gsl/gsl_vector.h>intmain(void){
const size_t N = 1000;
const size_t K = 7;/* length of time series *//* window size */(continues on next page)

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const double f = 5.0;/* frequency of square wave in Hz␣
,→*/
gsl_filter_median_workspace *median_p = gsl_filter_median_alloc(K);gsl_filter_rmedian_workspace *rmedian_p = gsl_filter_rmedian_alloc(K);gsl_vector *t = gsl_vector_alloc(N);/* time */gsl_vector *x = gsl_vector_alloc(N);/* input vector */gsl_vector *y_median = gsl_vector_alloc(N);/* median filtered output */gsl_vector *y_rmedian = gsl_vector_alloc(N);/* recursive median filtered␣,→output */gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
size_t i;/* generate input signal */for (i = 0; i < N; ++i){
double ti = (double) i / (N - 1.0);
double tmp = sin(2.0 * M_PI * f * ti);double xi = (tmp >= 0.0) ? 1.0 : -1.0;
double ei = gsl_ran_gaussian(r, 0.1);
gsl_vector_set(t, i, ti);
gsl_vector_set(x, i, xi + ei);}
gsl_filter_median(GSL_FILTER_END_PADVALUE, x, y_median, median_p);
gsl_filter_rmedian(GSL_FILTER_END_PADVALUE, x, y_rmedian, rmedian_p);/* print results */for (i = 0; i < N; ++i){
double ti = gsl_vector_get(t, i);double xi = gsl_vector_get(x, i);double medi = gsl_vector_get(y_median, i);double rmedi = gsl_vector_get(y_rmedian, i);printf("%f %f %f %f\n",
ti,
xi,
medi,
rmedi);}
gsl_vector_free(t);gsl_vector_free(x);gsl_vector_free(y_median);
gsl_vector_free(y_rmedian);
gsl_rng_free(r);gsl_filter_median_free(median_p);
return 0;}

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24.5.4 Impulse Detection Example
The following example program illustrates the impulse detection filter. First, it constructs a sinusoid signal of length
N = 1000 with Gaussian noise added. Then, about 1% of the data are perturbed to represent large outliers. An impulse
detecting filter is applied with a window size K = 25 and tuning parameter t = 4, using the Qn statistic as the robust
measure of scale. The results are plotted in Fig. 24.4.

Fig. 24.4: Original time series is in blue, filter output is in green, upper and lower intervals for detecting outliers are in
red and yellow respectively. Detected outliers are marked with squares.
The program is given below.
#include <stdio.h>#include <stdlib.h>#include <gsl/gsl_math.h>#include <gsl/gsl_filter.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>#include <gsl/gsl_vector.h>intmain(void){
const size_t N = 1000;
const size_t K = 25;
const double t = 4.0;,→outlier detection */gsl_vector *x = gsl_vector_alloc(N);gsl_vector *y = gsl_vector_alloc(N);gsl_vector *xmedian = gsl_vector_alloc(N);gsl_vector *xsigma = gsl_vector_alloc(N);gsl_vector_int *ioutlier = gsl_vector_int_alloc(N);/* length of time series *//* window size *//* number of scale factors for␣/* input vector *//* output (filtered) vector *//* window medians *//* window scale estimates *//* outlier detected? */(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)gsl_filter_impulse_workspace * w = gsl_filter_impulse_alloc(K);gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
size_t noutlier;size_t i;/* generate input signal */for (i = 0; i < N; ++i){
double xi = 10.0 * sin(2.0 * M_PI * i / (double) N);
double ei = gsl_ran_gaussian(r, 2.0);
double u = gsl_rng_uniform(r);double outlier = (u < 0.01) ? 15.0*GSL_SIGN(ei) : 0.0;
gsl_vector_set(x, i, xi + ei + outlier);}/* apply impulse detection filter */gsl_filter_impulse(GSL_FILTER_END_TRUNCATE, GSL_FILTER_SCALE_QN, t, x, y,
xmedian, xsigma, &noutlier, ioutlier, w);/* print results */for (i = 0; i < N; ++i){
double xi = gsl_vector_get(x, i);double yi = gsl_vector_get(y, i);double xmedi = gsl_vector_get(xmedian, i);double xsigmai = gsl_vector_get(xsigma, i);int outlier = gsl_vector_int_get(ioutlier, i);printf("%zu %f %f %f %f %d\n",
i,
xi,
yi,
xmedi + t * xsigmai,
xmedi - t * xsigmai,
outlier);}
gsl_vector_free(x);gsl_vector_free(y);gsl_vector_free(xmedian);
gsl_vector_free(xsigma);
gsl_vector_int_free(ioutlier);
gsl_filter_impulse_free(w);gsl_rng_free(r);return 0;}332

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24.6 References and Further Reading
The following publications are relevant to the algorithms described in this chapter,• F. J. Harris, On the use of windows for harmonic analysis with the discrete Fourier transform, Proceedings ofthe IEEE, 66 (1), 1978.
• S-J. Ko, Y-H. Lee, and A. T. Fam. Efficient implementation of one-dimensional recursive median filters, IEEE
transactions on circuits and systems 37.11 (1990): 1447-1450.
• R. K. Pearson and M. Gabbouj, Nonlinear Digital Filtering with Python: An Introduction. CRC Press, 2015.

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CHAPTERTWENTYFIVEHISTOGRAMSThis chapter describes functions for creating histograms. Histograms provide a convenient way of summarizing the
distribution of a set of data. A histogram consists of a set of bins which count the number of events falling into a given
range of a continuous variable x. In GSL the bins of a histogram contain floating-point numbers, so they can be used
to record both integer and non-integer distributions. The bins can use arbitrary sets of ranges (uniformly spaced bins
are the default). Both one and two-dimensional histograms are supported.
Once a histogram has been created it can also be converted into a probability distribution function. The library provides efficient routines for selecting random samples from probability distributions. This can be useful for generating
simulations based on real data.
The functions are declared in the header files gsl_histogram.h and gsl_histogram2d.h.

25.1 The histogram struct
A histogram is defined by the following struct,
type gsl_histogram

size_t ndouble* rangedouble* binThis is the number of histogram bins
The ranges of the bins are stored in an array of n+1 elements pointed to by range.
The counts for each bin are stored in an array of n elements pointed to by bin. The bins arefloating-point numbers, so you can increment them by non-integer values if necessary.

The range for bin[i] is given by range[i] to range[i+1]. For n bins there are n+1 entries in the array range.
Each bin is inclusive at the lower end and exclusive at the upper end. Mathematically this means that the bins
are defined by the following inequality,bin[i] corresponds to range[i] ≤ x < range[i+1]Here is a diagram of the correspondence between ranges and bins on the number-line for x:[ bin[0] )[ bin[1] )[ bin[2] )[ bin[3] )[ bin[4] )---|---------|---------|---------|---------|---------|--r[0]
r[1]
r[2]
r[3]
r[4]
r[5]

x

In this picture the values of the range array are denoted by r. On the left-hand side of each bin the square
bracket [ denotes an inclusive lower bound (r ≤ x), and the round parentheses ) on the right-hand side denote
an exclusive upper bound (x < r). Thus any samples which fall on the upper end of the histogram are excluded.
If you want to include this value for the last bin you will need to add an extra bin to your histogram.
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The gsl_histogram struct and its associated functions are defined in the header file gsl_histogram.h.

25.2 Histogram allocation
The functions for allocating memory to a histogram follow the style of malloc() and free(). In addition they also
perform their own error checking. If there is insufficient memory available to allocate a histogram then the functions
call the error handler (with an error number of GSL_ENOMEM) in addition to returning a null pointer. Thus if you use
the library error handler to abort your program then it isn’t necessary to check every histogram alloc.
gsl_histogram *gsl_histogram_alloc(size_t n)This function allocates memory for a histogram with n bins, and returns a pointer to a newly createdgsl_histogram struct. If insufficient memory is available a null pointer is returned and the error handler isinvoked with an error code of GSL_ENOMEM. The bins and ranges are not initialized, and should be preparedusing one of the range-setting functions below in order to make the histogram ready for use.
int gsl_histogram_set_ranges(gsl_histogram *h, const double range[], size_t size)
This function sets the ranges of the existing histogram h using the array range of size size. The values of
the histogram bins are reset to zero. The range array should contain the desired bin limits. The ranges can bearbitrary, subject to the restriction that they are monotonically increasing.
The following example shows how to create a histogram with logarithmic bins with ranges [1,10), [10,100) and[100,1000):
gsl_histogram * h = gsl_histogram_alloc (3);/* bin[0] covers the range 1 <= x < 10 *//* bin[1] covers the range 10 <= x < 100 *//* bin[2] covers the range 100 <= x < 1000 */double range[4] = { 1.0, 10.0, 100.0, 1000.0 };gsl_histogram_set_ranges (h, range, 4);Note that the size of the range array should be defined to be one element bigger than the number of bins. The
additional element is required for the upper value of the final bin.
int gsl_histogram_set_ranges_uniform(gsl_histogram *h, double xmin, double xmax)
This function sets the ranges of the existing histogram h to cover the range xmin to xmax uniformly. The valuesof the histogram bins are reset to zero. The bin ranges are shown in the table below,bin[0]bin[1]...
bin[n-1]

corresponds toxmin ≤ x < xmin + dcorresponds toxmin + d ≤ x < xmin + 2d...
...
corresponds to xmin + (n − 1)d ≤ x < xmax

where d is the bin spacing, d = (xmax − xmin)/n.
void gsl_histogram_free(gsl_histogram *h)This function frees the histogram h and all of the memory associated with it.

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25.3 Copying Histograms
int gsl_histogram_memcpy(gsl_histogram *dest, const gsl_histogram *src)
This function copies the histogram src into the pre-existing histogram dest, making dest into an exact copyof src. The two histograms must be of the same size.
gsl_histogram *gsl_histogram_clone(const gsl_histogram *src)
This function returns a pointer to a newly created histogram which is an exact copy of the histogram src.

25.4 Updating and accessing histogram elements
There are two ways to access histogram bins, either by specifying an x coordinate or by using the bin-index directly.
The functions for accessing the histogram through x coordinates use a binary search to identify the bin which covers
the appropriate range.
int gsl_histogram_increment(gsl_histogram *h, double x)
This function updates the histogram h by adding one (1.0) to the bin whose range contains the coordinate x.
If x lies in the valid range of the histogram then the function returns zero to indicate success. If x is less than thelower limit of the histogram then the function returns GSL_EDOM, and none of bins are modified. Similarly, if thevalue of x is greater than or equal to the upper limit of the histogram then the function returns GSL_EDOM, andnone of the bins are modified. The error handler is not called, however, since it is often necessary to computehistograms for a small range of a larger dataset, ignoring the values outside the range of interest.
int gsl_histogram_accumulate(gsl_histogram *h, double x, double weight)
This function is similar to gsl_histogram_increment() but increases the value of the appropriate bin in thehistogram h by the floating-point number weight.
double gsl_histogram_get(const gsl_histogram *h, size_t i)This function returns the contents of the i-th bin of the histogram h . If i lies outside the valid range of indicesfor the histogram then the error handler is called with an error code of GSL_EDOM and the function returns 0.
int gsl_histogram_get_range(const gsl_histogram *h, size_t i, double *lower, double *upper)
This function finds the upper and lower range limits of the i-th bin of the histogram h . If the index i is valid
then the corresponding range limits are stored in lower and upper. The lower limit is inclusive (i.e. events withthis coordinate are included in the bin) and the upper limit is exclusive (i.e. events with the coordinate of theupper limit are excluded and fall in the neighboring higher bin, if it exists). The function returns 0 to indicate
success. If i lies outside the valid range of indices for the histogram then the error handler is called and thefunction returns an error code of GSL_EDOM.
double gsl_histogram_max(const gsl_histogram *h)double gsl_histogram_min(const gsl_histogram *h)size_t gsl_histogram_bins(const gsl_histogram *h)
These functions return the maximum upper and minimum lower range limits and the number of bins of thehistogram h . They provide a way of determining these values without accessing the gsl_histogram structdirectly.
void gsl_histogram_reset(gsl_histogram *h)This function resets all the bins in the histogram h to zero.

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25.5 Searching histogram ranges
The following functions are used by the access and update routines to locate the bin which corresponds to a given x
coordinate.
int gsl_histogram_find(const gsl_histogram *h, double x, size_t *i)
This function finds and sets the index i to the bin number which covers the coordinate x in the histogram h . The
bin is located using a binary search. The search includes an optimization for histograms with uniform range, andwill return the correct bin immediately in this case. If x is found in the range of the histogram then the function
sets the index i and returns GSL_SUCCESS. If x lies outside the valid range of the histogram then the function
returns GSL_EDOM and the error handler is invoked.

25.6 Histogram Statistics
double gsl_histogram_max_val(const gsl_histogram *h)This function returns the maximum value contained in the histogram bins.
size_t gsl_histogram_max_bin(const gsl_histogram *h)This function returns the index of the bin containing the maximum value. In the case where several bins containthe same maximum value the smallest index is returned.
double gsl_histogram_min_val(const gsl_histogram *h)This function returns the minimum value contained in the histogram bins.
size_t gsl_histogram_min_bin(const gsl_histogram *h)This function returns the index of the bin containing the minimum value. In the case where several bins containthe same minimum value the smallest index is returned.
double gsl_histogram_mean(const gsl_histogram *h)This function returns the mean of the histogrammed variable, where the histogram is regarded as a probability
distribution. Negative bin values are ignored for the purposes of this calculation. The accuracy of the result is
limited by the bin width.
double gsl_histogram_sigma(const gsl_histogram *h)This function returns the standard deviation of the histogrammed variable, where the histogram is regarded asa probability distribution. Negative bin values are ignored for the purposes of this calculation. The accuracy of
the result is limited by the bin width.
double gsl_histogram_sum(const gsl_histogram *h)This function returns the sum of all bin values. Negative bin values are included in the sum.

25.7 Histogram Operations
int gsl_histogram_equal_bins_p(const gsl_histogram *h1, const gsl_histogram *h2)
This function returns 1 if the all of the individual bin ranges of the two histograms are identical, and 0 otherwise.
int gsl_histogram_add(gsl_histogram *h1, const gsl_histogram *h2)
This function adds the contents of the bins in histogram h2 to the corresponding bins of histogram h1, i.e.
h01 (i) = h1 (i) + h2 (i). The two histograms must have identical bin ranges.
int gsl_histogram_sub(gsl_histogram *h1, const gsl_histogram *h2)
This function subtracts the contents of the bins in histogram h2 from the corresponding bins of histogram h1,
i.e. h01 (i) = h1 (i) − h2 (i). The two histograms must have identical bin ranges.

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int gsl_histogram_mul(gsl_histogram *h1, const gsl_histogram *h2)
This function multiplies the contents of the bins of histogram h1 by the contents of the corresponding bins inhistogram h2, i.e. h01 (i) = h1 (i) ∗ h2 (i). The two histograms must have identical bin ranges.
int gsl_histogram_div(gsl_histogram *h1, const gsl_histogram *h2)
This function divides the contents of the bins of histogram h1 by the contents of the corresponding bins inhistogram h2, i.e. h01 (i) = h1 (i)/h2 (i). The two histograms must have identical bin ranges.
int gsl_histogram_scale(gsl_histogram *h, double scale)This function multiplies the contents of the bins of histogram h by the constant scale, i.e.
h01 (i) = h1 (i) ∗ scale
int gsl_histogram_shift(gsl_histogram *h, double offset)
This function shifts the contents of the bins of histogram h by the constant offset, i.e.
h01 (i) = h1 (i) + offset

25.8 Reading and writing histograms
The library provides functions for reading and writing histograms to a file as binary data or formatted text.
int gsl_histogram_fwrite(FILE *stream, const gsl_histogram *h)This function writes the ranges and bins of the histogram h to the stream stream in binary format. The return
value is 0 for success and GSL_EFAILED if there was a problem writing to the file. Since the data is written in
the native binary format it may not be portable between different architectures.
int gsl_histogram_fread(FILE *stream, gsl_histogram *h)This function reads into the histogram h from the open stream stream in binary format. The histogram h must
be preallocated with the correct size since the function uses the number of bins in h to determine how many
bytes to read. The return value is 0 for success and GSL_EFAILED if there was a problem reading from the file.
The data is assumed to have been written in the native binary format on the same architecture.
int gsl_histogram_fprintf(FILE *stream, const gsl_histogram *h, const char *range_format, const char*bin_format)
This function writes the ranges and bins of the histogram h line-by-line to the stream stream using the format
specifiers range_format and bin_format. These should be one of the %g, %e or %f formats for floating point
numbers. The function returns 0 for success and GSL_EFAILED if there was a problem writing to the file. The
histogram output is formatted in three columns, and the columns are separated by spaces, like this:range[0] range[1] bin[0]range[1] range[2] bin[1]range[2] range[3] bin[2]....
range[n-1] range[n] bin[n-1]The values of the ranges are formatted using range_format and the value of the bins are formatted using
bin_format. Each line contains the lower and upper limit of the range of the bins and the value of the bin itself.
Since the upper limit of one bin is the lower limit of the next there is duplication of these values between lines
but this allows the histogram to be manipulated with line-oriented tools.
int gsl_histogram_fscanf(FILE *stream, gsl_histogram *h)This function reads formatted data from the stream stream into the histogram h . The data is assumed to be in
the three-column format used by gsl_histogram_fprintf(). The histogram h must be preallocated with thecorrect length since the function uses the size of h to determine how many numbers to read. The function returns
0 for success and GSL_EFAILED if there was a problem reading from the file.
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25.9 Resampling from histograms
A histogram made by counting events can be regarded as a measurement of a probability distribution. Allowing forstatistical error, the height of each bin represents the probability of an event where the value of x falls in the range ofthat bin. The probability distribution function has the one-dimensional form p(x)dx where,p(x) = ni /(N wi )In this equation ni is the number of events in the bin which contains x, wi is the width of the bin and N is the total
number of events. The distribution of events within each bin is assumed to be uniform.

25.10 The histogram probability distribution struct
The probability distribution function for a histogram consists of a set of bins which measure the probability of an event
falling into a given range of a continuous variable x. A probability distribution function is defined by the following
struct, which actually stores the cumulative probability distribution function. This is the natural quantity for generating
samples via the inverse transform method, because there is a one-to-one mapping between the cumulative probability
distribution and the range [0,1]. It can be shown that by taking a uniform random number in this range and finding
its corresponding coordinate in the cumulative probability distribution we obtain samples with the desired probability
distribution.
type gsl_histogram_pdf

size_t ndouble *range
double * sum

This is the number of bins used to approximate the probability distribution function.
The ranges of the bins are stored in an array of n + 1 elements pointed to by range.
The cumulative probability for the bins is stored in an array of n elements pointed to bysum.

The following functions allow you to create a gsl_histogram_pdf struct which represents this probability distribution
and generate random samples from it.
gsl_histogram_pdf *gsl_histogram_pdf_alloc(size_t n)This function allocates memory for a probability distribution with n bins and returns a pointer to a newly initialized gsl_histogram_pdf struct. If insufficient memory is available a null pointer is returned and the error
handler is invoked with an error code of GSL_ENOMEM.
int gsl_histogram_pdf_init(gsl_histogram_pdf *p, const gsl_histogram *h)This function initializes the probability distribution p with the contents of the histogram h . If any of the bins of hare negative then the error handler is invoked with an error code of GSL_EDOM because a probability distribution
cannot contain negative values.
void gsl_histogram_pdf_free(gsl_histogram_pdf *p)
This function frees the probability distribution function p and all of the memory associated with it.
double gsl_histogram_pdf_sample(const gsl_histogram_pdf *p, double r)
This function uses r, a uniform random number between zero and one, to compute a single random sample fromthe probability distribution p. The algorithm used to compute the sample s is given by the following formula,s = range[i] + δ ∗ (range[i + 1] − range[i])where i is the index which satisfies sum[i] ≤ r < sum[i + 1] and delta is (r − sum[i])/(sum[i + 1] − sum[i]).

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25.11 Example programs for histograms
The following program shows how to make a simple histogram of a column of numerical data supplied on stdin. The
program takes three arguments, specifying the upper and lower bounds of the histogram and the number of bins. It thenreads numbers from stdin, one line at a time, and adds them to the histogram. When there is no more data to read it
prints out the accumulated histogram using gsl_histogram_fprintf().
#include <stdio.h>#include <stdlib.h>#include <gsl/gsl_histogram.h>
intmain (int argc, char **argv){
double a, b;
size_t n;
if (argc != 4){printf ("Usage: gsl-histogram xmin xmax n\n""Computes a histogram of the data ""on stdin using n bins from xmin ""to xmax\n");exit (0);}a = atof (argv[1]);b = atof (argv[2]);n = atoi (argv[3]);{double x;
gsl_histogram * h = gsl_histogram_alloc (n);gsl_histogram_set_ranges_uniform (h, a, b);while (fscanf (stdin, "%lg", &x) == 1){
gsl_histogram_increment (h, x);}
gsl_histogram_fprintf (stdout, h, "%g", "%g");
gsl_histogram_free (h);}exit (0);}Here is an example of the program in use. We generate 10000 random samples from a Cauchy distribution with a widthof 30 and histogram them over the range -100 to 100, using 200 bins:$ gsl-randist 0 10000 cauchy 30| gsl-histogram -- -100 100 200 > histogram.dat
Fig. 25.1 shows the familiar shape of the Cauchy distribution and the fluctuations caused by the finite sample size.

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Fig. 25.1: Histogram output from example program

25.12 Two dimensional histograms
A two dimensional histogram consists of a set of bins which count the number of events falling in a given area of the(x, y) plane. The simplest way to use a two dimensional histogram is to record two-dimensional position information,n(x, y). Another possibility is to form a joint distribution by recording related variables. For example a detector mightrecord both the position of an event (x) and the amount of energy it deposited E. These could be histogrammed as thejoint distribution n(x, E).

25.13 The 2D histogram struct
Two dimensional histograms are defined by the following struct,
type gsl_histogram2d

size_t This is the number of histogram bins in the x and y directions.
nx,nydouble The ranges of the bins in the x-direction are stored in an array of nx + 1 elements pointed to by*
xrange.
xrangedouble The ranges of the bins in the y-direction are stored in an array of ny + 1 elements pointed to by*yrange.
yrangedouble The counts for each bin are stored in an array pointed to by bin. The bins are floating-point num*bers, so you can increment them by non-integer values if necessary. The array bin stores the twobindimensional array of bins in a single block of memory according to the mapping bin(i,j) = bin[i* ny + j].

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The range for bin(i,j) is given by xrange[i] to xrange[i+1] in the x-direction and yrange[j] to yrange[j+1]in the y-direction. Each bin is inclusive at the lower end and exclusive at the upper end. Mathematically this meansthat the bins are defined by the following inequality,bin(i,j) corresponds to xrange[i] ≤ x < xrange[i + 1]
andyrange[j] ≤ y < yrange[j + 1]
Note that any samples which fall on the upper sides of the histogram are excluded. If you want to include these values
for the side bins you will need to add an extra row or column to your histogram.
The gsl_histogram2d struct and its associated functions are defined in the header file gsl_histogram2d.h.

25.14 2D Histogram allocation
The functions for allocating memory to a 2D histogram follow the style of malloc() and free(). In addition they also
perform their own error checking. If there is insufficient memory available to allocate a histogram then the functions
call the error handler (with an error number of GSL_ENOMEM) in addition to returning a null pointer. Thus if you use
the library error handler to abort your program then it isn’t necessary to check every 2D histogram alloc.
gsl_histogram2d *gsl_histogram2d_alloc(size_t nx, size_t ny)
This function allocates memory for a two-dimensional histogram with nx bins in the x direction and ny binsin the y direction. The function returns a pointer to a newly created gsl_histogram2d struct. If insufficient
memory is available a null pointer is returned and the error handler is invoked with an error code of GSL_ENOMEM.
The bins and ranges must be initialized with one of the functions below before the histogram is ready for use.
int gsl_histogram2d_set_ranges(gsl_histogram2d *h, const double xrange[], size_t xsize, const double
yrange[], size_t ysize)
This function sets the ranges of the existing histogram h using the arrays xrange and yrange of size xsize andysize respectively. The values of the histogram bins are reset to zero.
int gsl_histogram2d_set_ranges_uniform(gsl_histogram2d *h, double xmin, double xmax, double ymin,double ymax)
This function sets the ranges of the existing histogram h to cover the ranges xmin to xmax and ymin to ymax
uniformly. The values of the histogram bins are reset to zero.
void gsl_histogram2d_free(gsl_histogram2d *h)This function frees the 2D histogram h and all of the memory associated with it.

25.15 Copying 2D Histograms
int gsl_histogram2d_memcpy(gsl_histogram2d *dest, const gsl_histogram2d *src)
This function copies the histogram src into the pre-existing histogram dest, making dest into an exact copyof src. The two histograms must be of the same size.
gsl_histogram2d *gsl_histogram2d_clone(const gsl_histogram2d *src)
This function returns a pointer to a newly created histogram which is an exact copy of the histogram src.

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25.16 Updating and accessing 2D histogram elements
You can access the bins of a two-dimensional histogram either by specifying a pair of (x, y) coordinates or by using
the bin indices (i, j) directly. The functions for accessing the histogram through (x, y) coordinates use binary searches
in the x and y directions to identify the bin which covers the appropriate range.
int gsl_histogram2d_increment(gsl_histogram2d *h, double x, double y)This function updates the histogram h by adding one (1.0) to the bin whose x and y ranges contain the coordinates(x, y).
If the point (x, y) lies inside the valid ranges of the histogram then the function returns zero to indicate success.
If (x, y) lies outside the limits of the histogram then the function returns GSL_EDOM, and none of the bins are
modified. The error handler is not called, since it is often necessary to compute histograms for a small range ofa larger dataset, ignoring any coordinates outside the range of interest.
int gsl_histogram2d_accumulate(gsl_histogram2d *h, double x, double y, double weight)
This function is similar to gsl_histogram2d_increment() but increases the value of the appropriate bin in
the histogram h by the floating-point number weight.
double gsl_histogram2d_get(const gsl_histogram2d *h, size_t i, size_t j)
This function returns the contents of the (i, j)-th bin of the histogram h . If (i, j) lies outside the valid range
of indices for the histogram then the error handler is called with an error code of GSL_EDOM and the function
returns 0.
int gsl_histogram2d_get_xrange(const gsl_histogram2d *h, size_t i, double *xlower, double *xupper)
int gsl_histogram2d_get_yrange(const gsl_histogram2d *h, size_t j, double *ylower, double *yupper)
These functions find the upper and lower range limits of the i-th and j-th bins in the x and y directions of thehistogram h . The range limits are stored in xlower and xupper or ylower and yupper. The lower limits
are inclusive (i.e. events with these coordinates are included in the bin) and the upper limits are exclusive (i.e.
events with the value of the upper limit are not included and fall in the neighboring higher bin, if it exists). The
functions return 0 to indicate success. If i or j lies outside the valid range of indices for the histogram then theerror handler is called with an error code of GSL_EDOM.
double gsl_histogram2d_xmax(const gsl_histogram2d *h)double gsl_histogram2d_xmin(const gsl_histogram2d *h)size_t gsl_histogram2d_nx(const gsl_histogram2d *h)double gsl_histogram2d_ymax(const gsl_histogram2d *h)double gsl_histogram2d_ymin(const gsl_histogram2d *h)size_t gsl_histogram2d_ny(const gsl_histogram2d *h)
These functions return the maximum upper and minimum lower range limits and the number of bins for thex and y directions of the histogram h . They provide a way of determining these values without accessing the
gsl_histogram2d struct directly.
void gsl_histogram2d_reset(gsl_histogram2d *h)This function resets all the bins of the histogram h to zero.

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25.17 Searching 2D histogram ranges
The following functions are used by the access and update routines to locate the bin which corresponds to a given (x, y)
coordinate.
int gsl_histogram2d_find(const gsl_histogram2d *h, double x, double y, size_t *i, size_t *j)
This function finds and sets the indices i and j to the bin which covers the coordinates (x, y). The bin is locatedusing a binary search. The search includes an optimization for histograms with uniform ranges, and will returnthe correct bin immediately in this case. If (x, y) is found then the function sets the indices (i, j) and returns
GSL_SUCCESS. If (x, y) lies outside the valid range of the histogram then the function returns GSL_EDOM and theerror handler is invoked.

25.18 2D Histogram Statistics
double gsl_histogram2d_max_val(const gsl_histogram2d *h)This function returns the maximum value contained in the histogram bins.
void gsl_histogram2d_max_bin(const gsl_histogram2d *h, size_t *i, size_t *j)
This function finds the indices of the bin containing the maximum value in the histogram h and stores the resultin (i, j). In the case where several bins contain the same maximum value the first bin found is returned.
double gsl_histogram2d_min_val(const gsl_histogram2d *h)This function returns the minimum value contained in the histogram bins.
void gsl_histogram2d_min_bin(const gsl_histogram2d *h, size_t *i, size_t *j)
This function finds the indices of the bin containing the minimum value in the histogram h and stores the resultin (i, j). In the case where several bins contain the same maximum value the first bin found is returned.
double gsl_histogram2d_xmean(const gsl_histogram2d *h)This function returns the mean of the histogrammed x variable, where the histogram is regarded as a probability
distribution. Negative bin values are ignored for the purposes of this calculation.
double gsl_histogram2d_ymean(const gsl_histogram2d *h)This function returns the mean of the histogrammed y variable, where the histogram is regarded as a probability
distribution. Negative bin values are ignored for the purposes of this calculation.
double gsl_histogram2d_xsigma(const gsl_histogram2d *h)This function returns the standard deviation of the histogrammed x variable, where the histogram is regarded asa probability distribution. Negative bin values are ignored for the purposes of this calculation.
double gsl_histogram2d_ysigma(const gsl_histogram2d *h)This function returns the standard deviation of the histogrammed y variable, where the histogram is regarded asa probability distribution. Negative bin values are ignored for the purposes of this calculation.
double gsl_histogram2d_cov(const gsl_histogram2d *h)This function returns the covariance of the histogrammed x and y variables, where the histogram is regarded asa probability distribution. Negative bin values are ignored for the purposes of this calculation.
double gsl_histogram2d_sum(const gsl_histogram2d *h)This function returns the sum of all bin values. Negative bin values are included in the sum.

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25.19 2D Histogram Operations
int gsl_histogram2d_equal_bins_p(const gsl_histogram2d *h1, const gsl_histogram2d *h2)
This function returns 1 if all the individual bin ranges of the two histograms are identical, and 0 otherwise.
int gsl_histogram2d_add(gsl_histogram2d *h1, const gsl_histogram2d *h2)
This function adds the contents of the bins in histogram h2 to the corresponding bins of histogram h1, i.e.
h01 (i, j) = h1 (i, j) + h2 (i, j). The two histograms must have identical bin ranges.
int gsl_histogram2d_sub(gsl_histogram2d *h1, const gsl_histogram2d *h2)
This function subtracts the contents of the bins in histogram h2 from the corresponding bins of histogram h1,
i.e. h01 (i, j) = h1 (i, j) − h2 (i, j). The two histograms must have identical bin ranges.
int gsl_histogram2d_mul(gsl_histogram2d *h1, const gsl_histogram2d *h2)
This function multiplies the contents of the bins of histogram h1 by the contents of the corresponding bins inhistogram h2, i.e. h01 (i, j) = h1 (i, j) ∗ h2 (i, j). The two histograms must have identical bin ranges.
int gsl_histogram2d_div(gsl_histogram2d *h1, const gsl_histogram2d *h2)
This function divides the contents of the bins of histogram h1 by the contents of the corresponding bins inhistogram h2, i.e. h01 (i, j) = h1 (i, j)/h2 (i, j). The two histograms must have identical bin ranges.
int gsl_histogram2d_scale(gsl_histogram2d *h, double scale)This function multiplies the contents of the bins of histogram h by the constant scale, i.e.
h01 (i, j) = h1 (i, j) ∗ scale
int gsl_histogram2d_shift(gsl_histogram2d *h, double offset)
This function shifts the contents of the bins of histogram h by the constant offset, i.e.
h01 (i, j) = h1 (i, j) + offset

25.20 Reading and writing 2D histograms
The library provides functions for reading and writing two dimensional histograms to a file as binary data or formatted
text.
int gsl_histogram2d_fwrite(FILE *stream, const gsl_histogram2d *h)This function writes the ranges and bins of the histogram h to the stream stream in binary format. The return
value is 0 for success and GSL_EFAILED if there was a problem writing to the file. Since the data is written in
the native binary format it may not be portable between different architectures.
int gsl_histogram2d_fread(FILE *stream, gsl_histogram2d *h)This function reads into the histogram h from the stream stream in binary format. The histogram h must be
preallocated with the correct size since the function uses the number of x and y bins in h to determine how many
bytes to read. The return value is 0 for success and GSL_EFAILED if there was a problem reading from the file.
The data is assumed to have been written in the native binary format on the same architecture.
int gsl_histogram2d_fprintf(FILE *stream, const gsl_histogram2d *h, const char *range_format, const char*bin_format)
This function writes the ranges and bins of the histogram h line-by-line to the stream stream using the format
specifiers range_format and bin_format. These should be one of the %g, %e or %f formats for floating point
numbers. The function returns 0 for success and GSL_EFAILED if there was a problem writing to the file. The
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xrange[0] xrange[1] yrange[0] yrange[1] bin(0,0)
xrange[0] xrange[1] yrange[1] yrange[2] bin(0,1)
xrange[0] xrange[1] yrange[2] yrange[3] bin(0,2)....
xrange[0] xrange[1] yrange[ny-1] yrange[ny] bin(0,ny-1)
xrange[1] xrange[2] yrange[0] yrange[1] bin(1,0)
xrange[1] xrange[2] yrange[1] yrange[2] bin(1,1)
xrange[1] xrange[2] yrange[1] yrange[2] bin(1,2)....
xrange[1] xrange[2] yrange[ny-1] yrange[ny] bin(1,ny-1)....
xrange[nx-1] xrange[nx] yrange[0] yrange[1] bin(nx-1,0)
xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,1)
xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,2)....
xrange[nx-1] xrange[nx] yrange[ny-1] yrange[ny] bin(nx-1,ny-1)
Each line contains the lower and upper limits of the bin and the contents of the bin. Since the upper limits ofthe each bin are the lower limits of the neighboring bins there is duplication of these values but this allows thehistogram to be manipulated with line-oriented tools.
int gsl_histogram2d_fscanf(FILE *stream, gsl_histogram2d *h)This function reads formatted data from the stream stream into the histogram h . The data is assumed to be in
the five-column format used by gsl_histogram2d_fprintf(). The histogram h must be preallocated withthe correct lengths since the function uses the sizes of h to determine how many numbers to read. The function
returns 0 for success and GSL_EFAILED if there was a problem reading from the file.

25.21 Resampling from 2D histograms
As in the one-dimensional case, a two-dimensional histogram made by counting events can be regarded as a measurement of a probability distribution. Allowing for statistical error, the height of each bin represents the probability of anevent where (x, y) falls in the range of that bin. For a two-dimensional histogram the probability distribution takes theform p(x, y)dxdy where,p(x, y) = nij /(N Aij )In this equation nij is the number of events in the bin which contains (x, y), Aij is the area of the bin and N is the total
number of events. The distribution of events within each bin is assumed to be uniform.
type gsl_histogram2d_pdf

size_t nx,
nydouble *xrangedouble *yrangedouble *sum

This is the number of histogram bins used to approximate the probability distribution function in the x and y directions.
The ranges of the bins in the x-direction are stored in an array of nx + 1 elements pointed
to by xrange.
The ranges of the bins in the y-direction are stored in an array of ny + 1 pointed to byyrange.
The cumulative probability for the bins is stored in an array of nx * ny elements pointed toby sum.

25.21. Resampling from 2D histograms

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The following functions allow you to create a gsl_histogram2d_pdf struct which represents a two dimensional
probability distribution and generate random samples from it.
gsl_histogram2d_pdf *gsl_histogram2d_pdf_alloc(size_t nx, size_t ny)
This function allocates memory for a two-dimensional probability distribution of size nx-by-ny and returns apointer to a newly initialized gsl_histogram2d_pdf struct. If insufficient memory is available a null pointeris returned and the error handler is invoked with an error code of GSL_ENOMEM.
int gsl_histogram2d_pdf_init(gsl_histogram2d_pdf *p, const gsl_histogram2d *h)This function initializes the two-dimensional probability distribution calculated p from the histogram h . If any ofthe bins of h are negative then the error handler is invoked with an error code of GSL_EDOM because a probability
distribution cannot contain negative values.
void gsl_histogram2d_pdf_free(gsl_histogram2d_pdf *p)
This function frees the two-dimensional probability distribution function p and all of the memory associatedwith it.
int gsl_histogram2d_pdf_sample(const gsl_histogram2d_pdf *p, double r1, double r2, double *x, double *y)This function uses two uniform random numbers between zero and one, r1 and r2, to compute a single random
sample from the two-dimensional probability distribution p.

25.22 Example programs for 2D histograms
This program demonstrates two features of two-dimensional histograms. First a 10-by-10 two-dimensional histogramis created with x and y running from 0 to 1. Then a few sample points are added to the histogram, at (0.3,0.3) with aheight of 1, at (0.8,0.1) with a height of 5 and at (0.7,0.9) with a height of 0.5. This histogram with three events is used
to generate a random sample of 1000 simulated events, which are printed out.
#include <stdio.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_histogram2d.h>
intmain (void){
const gsl_rng_type * T;
gsl_rng * r;
gsl_histogram2d * h = gsl_histogram2d_alloc (10, 10);
gsl_histogram2d_set_ranges_uniform (h,0.0, 1.0,
0.0, 1.0);
gsl_histogram2d_accumulate (h, 0.3, 0.3, 1);
gsl_histogram2d_accumulate (h, 0.8, 0.1, 5);
gsl_histogram2d_accumulate (h, 0.7, 0.9, 0.5);
gsl_rng_env_setup ();T = gsl_rng_default;r = gsl_rng_alloc (T);{(continues on next page)

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int i;
gsl_histogram2d_pdf * p= gsl_histogram2d_pdf_alloc (h->nx, h->ny);gsl_histogram2d_pdf_init (p, h);for (i = 0; i < 1000; i++) {double x, y;
double u = gsl_rng_uniform (r);double v = gsl_rng_uniform (r);gsl_histogram2d_pdf_sample (p, u, v, &x, &y);printf ("%g %g\n", x, y);}gsl_histogram2d_pdf_free (p);}gsl_histogram2d_free (h);gsl_rng_free (r);return 0;}
The following plot shows the distribution of the simulated events. Using a higher resolution grid we can see the originalunderlying histogram and also the statistical fluctuations caused by the events being uniformly distributed over the areaof the original bins.

Fig. 25.2: Distribution of simulated events from example program

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CHAPTERTWENTYSIXN-TUPLESThis chapter describes functions for creating and manipulating ntuples, sets of values associated with events. The
ntuples are stored in files. Their values can be extracted in any combination and booked in a histogram using a selection
function.
The values to be stored are held in a user-defined data structure, and an ntuple is created associating this data structurewith a file. The values are then written to the file (normally inside a loop) using the ntuple functions described below.
A histogram can be created from ntuple data by providing a selection function and a value function. The selection
function specifies whether an event should be included in the subset to be analyzed or not. The value function computes
the entry to be added to the histogram for each event.
All the ntuple functions are defined in the header file gsl_ntuple.h.

26.1 The ntuple structtype gsl_ntuple
Ntuples are manipulated using the gsl_ntuple struct. This struct contains information on the file where thentuple data is stored, a pointer to the current ntuple data row and the size of the user-defined ntuple data struct:typedef struct{
FILE * file;void * ntuple_data;
size_t size;} gsl_ntuple;

26.2 Creating ntuples
gsl_ntuple *gsl_ntuple_create(char *filename, void *ntuple_data, size_t size)
This function creates a new write-only ntuple file filename for ntuples of size size and returns a pointer to the
newly created ntuple struct. Any existing file with the same name is truncated to zero length and overwritten. A
pointer to memory for the current ntuple row ntuple_data must be supplied—this is used to copy ntuples inand out of the file.

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26.3 Opening an existing ntuple file
gsl_ntuple *gsl_ntuple_open(char *filename, void *ntuple_data, size_t size)
This function opens an existing ntuple file filename for reading and returns a pointer to a corresponding ntuple
struct. The ntuples in the file must have size size. A pointer to memory for the current ntuple row ntuple_data
must be supplied—this is used to copy ntuples in and out of the file.

26.4 Writing ntuples
int gsl_ntuple_write(gsl_ntuple *ntuple)
This function writes the current ntuple ntuple->ntuple_data of size ntuple->size to the corresponding
file.
int gsl_ntuple_bookdata(gsl_ntuple *ntuple)
This function is a synonym for gsl_ntuple_write().

26.5 Reading ntuples
int gsl_ntuple_read(gsl_ntuple *ntuple)
This function reads the current row of the ntuple file for ntuple and stores the values in ntuple->data.

26.6 Closing an ntuple file
int gsl_ntuple_close(gsl_ntuple *ntuple)
This function closes the ntuple file ntuple and frees its associated allocated memory.

26.7 Histogramming ntuple values
Once an ntuple has been created its contents can be histogrammed in various ways using the function
gsl_ntuple_project(). Two user-defined functions must be provided, a function to select events and a function
to compute scalar values. The selection function and the value function both accept the ntuple row as a first argument
and other parameters as a second argument.
type gsl_ntuple_select_fn
The selection function determines which ntuple rows are selected for histogramming. It is defined by the following struct:typedef struct{
int (* function) (void * ntuple_data, void * params);
void * params;} gsl_ntuple_select_fn;
The struct component function should return a non-zero value for each ntuple row that is to be included in thehistogram.
type gsl_ntuple_value_fn
The value function computes scalar values for those ntuple rows selected by the selection function:
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typedef struct{
double (* function) (void * ntuple_data, void * params);
void * params;} gsl_ntuple_value_fn;
In this case the struct component function should return the value to be added to the histogram for the ntuplerow.
int gsl_ntuple_project(gsl_histogram *h, gsl_ntuple *ntuple, gsl_ntuple_value_fn *value_func,
gsl_ntuple_select_fn *select_func)
This function updates the histogram h from the ntuple ntuple using the functions value_func andselect_func. For each ntuple row where the selection function select_func is non-zero the corresponding value of that row is computed using the function value_func and added to the histogram. Those ntuplerows where select_func returns zero are ignored. New entries are added to the histogram, so subsequent calls
can be used to accumulate further data in the same histogram.

26.8 Examples
The following example programs demonstrate the use of ntuples in managing a large dataset. The first program createsa set of 10,000 simulated “events”, each with 3 associated values (x, y, z). These are generated from a Gaussian
distribution with unit variance, for demonstration purposes, and written to the ntuple file test.dat.
#include <gsl/gsl_ntuple.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>
struct data{
double x;
double y;
double z;};
intmain (void){
const gsl_rng_type * T;
gsl_rng * r;
struct data ntuple_row;
int i;
gsl_ntuple *ntuple= gsl_ntuple_create ("test.dat", &ntuple_row,
sizeof (ntuple_row));
gsl_rng_env_setup ();T = gsl_rng_default;r = gsl_rng_alloc (T);(continues on next page)

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353

GNU Scientific Library, Release 2.7(continued from previous page)for (i = 0; i < 10000; i++){ntuple_row.x = gsl_ran_ugaussian (r);ntuple_row.y = gsl_ran_ugaussian (r);ntuple_row.z = gsl_ran_ugaussian (r);gsl_ntuple_write (ntuple);}
gsl_ntuple_close (ntuple);
gsl_rng_free (r);return 0;}
The next program analyses the ntuple data in the file test.dat. The analysis procedure is to compute the squaredmagnitude of each event, E 2 = x2 + y 2 + z 2 , and select only those which exceed a lower limit of 1.5. The selected
events are then histogrammed using their E 2 values.
#include <math.h>#include <gsl/gsl_ntuple.h>#include <gsl/gsl_histogram.h>
struct data{
double x;
double y;
double z;};
int sel_func (void *ntuple_data, void *params);
double val_func (void *ntuple_data, void *params);
int
main (void){
struct data ntuple_row;
gsl_ntuple *ntuple= gsl_ntuple_open ("test.dat", &ntuple_row,
sizeof (ntuple_row));double lower = 1.5;
gsl_ntuple_select_fn S;
gsl_ntuple_value_fn V;
gsl_histogram *h = gsl_histogram_alloc (100);
gsl_histogram_set_ranges_uniform(h, 0.0, 10.0);
S.function = &sel_func;
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V.function = &val_func;
V.params = 0;
gsl_ntuple_project (h, ntuple, &V, &S);
gsl_histogram_fprintf (stdout, h, "%f ", "%f ");gsl_histogram_free (h);gsl_ntuple_close (ntuple);return 0;}
int
sel_func (void *ntuple_data, void *params){
struct data * data = (struct data *) ntuple_data;
double x, y, z, E2, scale;scale = *(double *) params;
x = data->x;
y = data->y;z = data->z;
E2 = x * x + y * y + z * z;return E2 > scale;}doubleval_func (void *ntuple_data, void *params){(void)(params); /* avoid unused parameter warning */struct data * data = (struct data *) ntuple_data;
double x, y, z;
x = data->x;
y = data->y;z = data->z;
return x * x + y * y + z * z;}
Fig. 26.1 shows the distribution of the selected events. Note the cut-off at the lower bound.

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Fig. 26.1: Distribution of selected events

26.9 References and Further Reading
Further information on the use of ntuples can be found in the documentation for the CERN packages PAW and HBOOK(available online).

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CHAPTERTWENTYSEVEN

MONTE CARLO INTEGRATIONThis chapter describes routines for multidimensional Monte Carlo integration. These include the traditional Monte
Carlo method and adaptive algorithms such as VEGAS and MISER which use importance sampling and stratified
sampling techniques. Each algorithm computes an estimate of a multidimensional definite integral of the form,
Z xu
Z yu
I=
dx
dy...f (x, y, ...)
xlyl

over a hypercubic region ((xl , xu ), (yl , yu ), ...) using a fixed number of function calls. The routines also provide astatistical estimate of the error on the result. This error estimate should be taken as a guide rather than as a strict error
bound—random sampling of the region may not uncover all the important features of the function, resulting in anunderestimate of the error.
The functions are defined in separate header files for each routine, gsl_monte_plain.h, gsl_monte_miser.h andgsl_monte_vegas.h.

27.1 Interface
All of the Monte Carlo integration routines use the same general form of interface. There is an allocator to allocate
memory for control variables and workspace, a routine to initialize those control variables, the integrator itself, and afunction to free the space when done.
Each integration function requires a random number generator to be supplied, and returns an estimate of the integral
and its standard deviation. The accuracy of the result is determined by the number of function calls specified by theuser. If a known level of accuracy is required this can be achieved by calling the integrator several times and averaging
the individual results until the desired accuracy is obtained.
Random sample points used within the Monte Carlo routines are always chosen strictly within the integration region,
so that endpoint singularities are automatically avoided.
The function to be integrated has its own datatype, defined in the header file gsl_monte.h.
type gsl_monte_function
This data type defines a general function with parameters for Monte Carlo integration.
double (* f) (double* x, size_t dim,void * params)
size_t dim
void * params

this function should return the value f (x, params) for the argument x and parameters params, where x is an array of size dim giving the coordinates of thepoint where the function is to be evaluated.
the number of dimensions for x.
a pointer to the parameters of the function.

Here is an example for a quadratic function in two dimensions,f (x, y) = ax2 + bxy + cy 2357

GNU Scientific Library, Release 2.7

with a = 3, b = 2, c = 1. The following code defines a gsl_monte_function F which you could pass to an integrator:
struct my_f_params { double a; double b; double c; };doublemy_f (double x[], size_t dim, void * p) {struct my_f_params * fp = (struct my_f_params *)p;if (dim != 2){
fprintf (stderr, "error: dim != 2");
abort ();}
returnfp->a * x[0] * x[0]+ fp->b * x[0] * x[1]+ fp->c * x[1] * x[1];}gsl_monte_function F;
struct my_f_params params = { 3.0, 2.0, 1.0 };F.f = &my_f;F.dim = 2;
F.params = &params;
The function f (x) can be evaluated using the following macro:#define GSL_MONTE_FN_EVAL(F,x)(*((F)->f))(x,(F)->dim,(F)->params)

27.2 PLAIN Monte Carlo
The plain Monte Carlo algorithm samples points randomly from the integration region to estimate the integral and its
error. Using this algorithm the estimate of the integral E(f ; N ) for N randomly distributed points xi is given by,N

E(f ; N ) = V hf i =V X
f (xi )
N i

where V is the volume of the integration region. The error on this estimate σ(E; N ) is calculated from the estimatedvariance of the mean,Nσ 2 (E; N ) =V2 X(f (xi ) − hf i)2 .
N2 i

For large N this variance decreases asymptotically as Var(f )/N , where Var(f√ ) is the true variance of the function overthe integration
region.
Theerror
estimate
itselfshoulddecreaseasσ(f)/
N . The familiar law of errors decreasing√
as 1/ N applies—to reduce the error by a factor of 10 requires a 100-fold increase in the number of sample points.
The functions described in this section are declared in the header file gsl_monte_plain.h.
type gsl_monte_plain_stateThis is a workspace for plain Monte Carlo integration
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gsl_monte_plain_state *gsl_monte_plain_alloc(size_t dim)This function allocates and initializes a workspace for Monte Carlo integration in dim dimensions.
int gsl_monte_plain_init(gsl_monte_plain_state *s)
This function initializes a previously allocated integration state. This allows an existing workspace to be reusedfor different integrations.
int gsl_monte_plain_integrate(gsl_monte_function *f, const double xl[], const double xu[], size_t dim, size_t
calls, gsl_rng *r, gsl_monte_plain_state *s, double *result, double *abserr)
This routines uses the plain Monte Carlo algorithm to integrate the function f over the dim-dimensional hypercubic region defined by the lower and upper limits in the arrays xl and xu, each of size dim. The integration uses
a fixed number of function calls calls, and obtains random sampling points using the random number generator
r. A previously allocated workspace s must be supplied. The result of the integration is returned in result,with an estimated absolute error abserr.
void gsl_monte_plain_free(gsl_monte_plain_state *s)
This function frees the memory associated with the integrator state s.

27.3 MISER
The MISER algorithm of Press and Farrar is based on recursive stratified sampling. This technique aims to reduce theoverall integration error by concentrating integration points in the regions of highest variance.
The idea of stratified sampling begins with the observation that for two disjoint regions a and b with Monte Carloestimates of the integral Ea (f ) and Eb (f ) and variances σa2 (f ) and σb2 (f ), the variance Var(f ) of the combined
estimate E(f ) = 12 (Ea (f ) + Eb (f )) is given by,Var(f ) =σa2 (f ) σb2 (f )
+
.
4Na
4Nb

It can be shown that this variance is minimized by distributing the points such that,
Na
σa
=
.
Na + Nb
σa + σbHence the smallest error estimate is obtained by allocating sample points in proportion to the standard deviation of thefunction in each sub-region.
The MISER algorithm proceeds by bisecting the integration region along one coordinate axis to give two sub-regions
at each step. The direction is chosen by examining all d possible bisections and selecting the one which will minimize
the combined variance of the two sub-regions. The variance in the sub-regions is estimated by sampling with a fractionof the total number of points available to the current step. The same procedure is then repeated recursively for eachof the two half-spaces from the best bisection. The remaining sample points are allocated to the sub-regions using theformula for Na and Nb . This recursive allocation of integration points continues down to a user-specified depth where
each sub-region is integrated using a plain Monte Carlo estimate. These individual values and their error estimates arethen combined upwards to give an overall result and an estimate of its error.
The functions described in this section are declared in the header file gsl_monte_miser.h.
type gsl_monte_miser_stateThis workspace is used for MISER Monte Carlo integration
gsl_monte_miser_state *gsl_monte_miser_alloc(size_t dim)This function allocates and initializes a workspace for Monte Carlo integration in dim dimensions. The
workspace is used to maintain the state of the integration.
int gsl_monte_miser_init(gsl_monte_miser_state *s)
This function initializes a previously allocated integration state. This allows an existing workspace to be reusedfor different integrations.
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int gsl_monte_miser_integrate(gsl_monte_function *f, const double xl[], const double xu[], size_t dim, size_t
calls, gsl_rng *r, gsl_monte_miser_state *s, double *result, double *abserr)
This routines uses the MISER Monte Carlo algorithm to integrate the function f over the dim-dimensional
hypercubic region defined by the lower and upper limits in the arrays xl and xu, each of size dim. The integration
uses a fixed number of function calls calls, and obtains random sampling points using the random number
generator r. A previously allocated workspace s must be supplied. The result of the integration is returned inresult, with an estimated absolute error abserr.
void gsl_monte_miser_free(gsl_monte_miser_state *s)
This function frees the memory associated with the integrator state s.
The MISER algorithm has several configurable parameters which can be changed using the following two functions1 .
void gsl_monte_miser_params_get(const gsl_monte_miser_state *s, gsl_monte_miser_params *params)
This function copies the parameters of the integrator state into the user-supplied params structure.
void gsl_monte_miser_params_set(gsl_monte_miser_state *s, const gsl_monte_miser_params *params)
This function sets the integrator parameters based on values provided in the params structure.
Typically the values of the parameters are first read using gsl_monte_miser_params_get(), the necessary changes
are made to the fields of the params structure, and the values are copied back into the integrator state using
gsl_monte_miser_params_set(). The functions use the gsl_monte_miser_params structure which contains thefollowing fields:
type gsl_monte_miser_params
double estimate_fracThis parameter specifies the fraction of the currently available number of function calls which are allocatedto estimating the variance at each recursive step. The default value is 0.1.
size_t min_calls
This parameter specifies the minimum number of function calls required for each estimate of the variance.
If the number of function calls allocated to the estimate using estimate_frac falls below min_calls thenmin_calls are used instead. This ensures that each estimate maintains a reasonable level of accuracy. The
default value of min_calls is 16 * dim.
size_t min_calls_per_bisection
This parameter specifies the minimum number of function calls required to proceed with a bisection step.
When a recursive step has fewer calls available than min_calls_per_bisection it performs a plain
Monte Carlo estimate of the current sub-region and terminates its branch of the recursion. The default
value of this parameter is 32 * min_calls.
double alphaThis parameter controls how the estimated variances for the two sub-regions of a bisection are combined
when allocating points. With recursive sampling the overall variance should scale better than 1/N , since
the values from the sub-regions will be obtained using a procedure which explicitly minimizes their variance. To accommodate this behavior the MISER algorithm allows the total variance to depend on a scaling
parameter α,Var(f ) =

σb
σa
+ α.
Naα
Nb

The authors of the original paper describing MISER recommend the value α = 2 as a good choice, obtained
from numerical experiments, and this is used as the default value in this implementation.
double dither
This parameter introduces a random fractional variation of size dither into each bisection, which can beused to break the symmetry of integrands which are concentrated near the exact center of the hypercubic
1 The previous method of accessing these fields directly through the gsl_monte_miser_state struct is now deprecated.

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integration region. The default value of dither is zero, so no variation is introduced. If needed, a typicalvalue of dither is 0.1.

27.4 VEGAS
The VEGAS algorithm of Lepage is based on importance sampling. It samples points from the probability distribution
described by the function |f |, so that the points are concentrated in the regions that make the largest contribution to theintegral.
In general, if the Monte Carlo integral of f is sampled with points distributed according to a probability distribution
described by the function g, we obtain an estimate Eg (f ; N ),Eg (f ; N ) = E(f /g; N )with a corresponding variance,
Varg (f ; N ) = Var(f /g; N )If the probability distribution is chosen as g = |f |/I(|f |) then it can be shown that the variance Vg (f ; N ) vanishes, andthe error in the estimate will be zero. In practice it is not possible to sample from the exact distribution g for an arbitrary
function, so importance sampling algorithms aim to produce efficient approximations to the desired distribution.
The VEGAS algorithm approximates the exact distribution by making a number of passes over the integration region
while histogramming the function f . Each histogram is used to define a sampling distribution for the next pass. Asymptotically this procedure converges to the desired distribution. In order to avoid the number of histogram bins growing
like K d the probability distribution is approximated by a separable function: g(x1 , x2 , . . .) = g1 (x1 )g2 (x2 ) . . . so thatthe number of bins required is only Kd. This is equivalent to locating the peaks of the function from the projectionsof the integrand onto the coordinate axes. The efficiency of VEGAS depends on the validity of this assumption. It is
most efficient when the peaks of the integrand are well-localized. If an integrand can be rewritten in a form which isapproximately separable this will increase the efficiency of integration with VEGAS.
VEGAS incorporates a number of additional features, and combines both stratified sampling and importance sampling.
The integration region is divided into a number of “boxes”, with each box getting a fixed number of points (the goal is2). Each box can then have a fractional number of bins, but if the ratio of bins-per-box is less than two, Vegas switches
to a kind variance reduction (rather than importance sampling).
type gsl_monte_vegas_stateThis workspace is used for VEGAS Monte Carlo integration
gsl_monte_vegas_state *gsl_monte_vegas_alloc(size_t dim)This function allocates and initializes a workspace for Monte Carlo integration in dim dimensions. The
workspace is used to maintain the state of the integration.
int gsl_monte_vegas_init(gsl_monte_vegas_state *s)
This function initializes a previously allocated integration state. This allows an existing workspace to be reusedfor different integrations.
int gsl_monte_vegas_integrate(gsl_monte_function *f, double xl[], double xu[], size_t dim, size_t calls,
gsl_rng *r, gsl_monte_vegas_state *s, double *result, double *abserr)
This routines uses the VEGAS Monte Carlo algorithm to integrate the function f over the dim-dimensional
hypercubic region defined by the lower and upper limits in the arrays xl and xu, each of size dim. The integration
uses a fixed number of function calls calls, and obtains random sampling points using the random number
generator r. A previously allocated workspace s must be supplied. The result of the integration is returned inresult, with an estimated absolute error abserr. The result and its error estimate are based on a weighted
average of independent samples. The chi-squared per degree of freedom for the weighted average is returned via
the state struct component, s->chisq, and must be consistent with 1 for the weighted average to be reliable.
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void gsl_monte_vegas_free(gsl_monte_vegas_state *s)
This function frees the memory associated with the integrator state s.
The VEGAS algorithm computes a number of independent estimates of the integral internally, according to theiterations parameter described below, and returns their weighted average. Random sampling of the integrand canoccasionally produce an estimate where the error is zero, particularly if the function is constant in some regions. An
estimate with zero error causes the weighted average to break down and must be handled separately. In the original Fortran implementations of VEGAS the error estimate is made non-zero by substituting a small value (typically 1e-30).
The implementation in GSL differs from this and avoids the use of an arbitrary constant—it either assigns the value aweight which is the average weight of the preceding estimates or discards it according to the following procedure,• current estimate has zero error, weighted average has finite error
The current estimate is assigned a weight which is the average weight of the preceding estimates.
• current estimate has finite error, previous estimates had zero error
The previous estimates are discarded and the weighted averaging procedure begins with the current estimate.
• current estimate has zero error, previous estimates had zero error
The estimates are averaged using the arithmetic mean, but no error is computed.
The convergence of the algorithm can be tested using the overall chi-squared value of the results, which is available
from the following function:
double gsl_monte_vegas_chisq(const gsl_monte_vegas_state *s)
This function returns the chi-squared per degree of freedom for the weighted estimate of the integral. The returned
value should be close to 1. A value which differs significantly from 1 indicates that the values from different
iterations are inconsistent. In this case the weighted error will be under-estimated, and further iterations of thealgorithm are needed to obtain reliable results.
void gsl_monte_vegas_runval(const gsl_monte_vegas_state *s, double *result, double *sigma)
This function returns the raw (unaveraged) values of the integral result and its error sigma from the mostrecent iteration of the algorithm.
The VEGAS algorithm is highly configurable. Several parameters can be changed using the following two functions.
void gsl_monte_vegas_params_get(const gsl_monte_vegas_state *s, gsl_monte_vegas_params *params)
This function copies the parameters of the integrator state into the user-supplied params structure.
void gsl_monte_vegas_params_set(gsl_monte_vegas_state *s, const gsl_monte_vegas_params *params)
This function sets the integrator parameters based on values provided in the params structure.
Typically the values of the parameters are first read using gsl_monte_vegas_params_get(), the necessary changes
are made to the fields of the params structure, and the values are copied back into the integrator state using
gsl_monte_vegas_params_set(). The functions use the gsl_monte_vegas_params structure which contains thefollowing fields:
type gsl_monte_vegas_params
double alphaThe parameter alpha controls the stiffness of the rebinning algorithm. It is typically set between one andtwo. A value of zero prevents rebinning of the grid. The default value is 1.5.
size_t iterations
The number of iterations to perform for each call to the routine. The default value is 5 iterations.
int stage
Setting this determines the stage of the calculation. Normally, stage = 0 which begins with a new uniform
grid and empty weighted average. Calling VEGAS with stage = 1 retains the grid from the previous run
but discards the weighted average, so that one can “tune” the grid using a relatively small number of points
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and then do a large run with stage = 1 on the optimized grid. Setting stage = 2 keeps the grid and theweighted average from the previous run, but may increase (or decrease) the number of histogram bins in
the grid depending on the number of calls available. Choosing stage = 3 enters at the main loop, so that
nothing is changed, and is equivalent to performing additional iterations in a previous call.
int mode
The possible choices are GSL_VEGAS_MODE_IMPORTANCE, GSL_VEGAS_MODE_STRATIFIED,
GSL_VEGAS_MODE_IMPORTANCE_ONLY. This determines whether VEGAS will use importance sampling or stratified sampling, or whether it can pick on its own. In low dimensions VEGAS uses strict
stratified sampling (more precisely, stratified sampling is chosen if there are fewer than 2 bins per box).
int verbose
FILE *ostream
These parameters set the level of information printed by VEGAS. All information is written to the stream
ostream. The default setting of verbose is -1, which turns off all output. A verbose value of 0 prints
summary information about the weighted average and final result, while a value of 1 also displays the gridcoordinates. A value of 2 prints information from the rebinning procedure for each iteration.
The above fields and the chisq value can also be accessed directly in the gsl_monte_vegas_state but such use isdeprecated.

27.5 Examples
The example program below uses the Monte Carlo routines to estimate the value of the following 3-dimensional integral
from the theory of random walks,
Z
Z
Z +π
1
dkx +π dky +π dkz
.
I=
2π
2π
2π
(1
−
cos(k
)
cos(k
x
y ) cos(kz ))−π−π−πThe analytic value of this integral can be shown to be I = Γ(1/4)4 /(4π 3 ) = 1.393203929685676859.... The integral
gives the mean time spent at the origin by a random walk on a body-centered cubic lattice in three dimensions.
For simplicity we will compute the integral over the region (0, 0, 0) to (π, π, π) and multiply by 8 to obtain the full
result. The integral is slowly varying in the middle of the region but has integrable singularities at the corners (0, 0, 0),(0, π, π), (π, 0, π) and (π, π, 0). The Monte Carlo routines only select points which are strictly within the integration
region and so no special measures are needed to avoid these singularities.
#include <stdlib.h>#include <gsl/gsl_math.h>#include <gsl/gsl_monte.h>#include <gsl/gsl_monte_plain.h>#include <gsl/gsl_monte_miser.h>#include <gsl/gsl_monte_vegas.h>/* Computation of the integral,
I = int (dx dy dz)/(2pi)^3

1/(1-cos(x)cos(y)cos(z))

over (-pi,-pi,-pi) to (+pi, +pi, +pi). The exact answer
is Gamma(1/4)^4/(4 pi^3). This example is taken from
C.Itzykson, J.M.Drouffe, "Statistical Field Theory Volume 1", Section 1.1, p21, which cites the original
paper M.L.Glasser, I.J.Zucker, Proc.Natl.Acad.Sci.USA 74
1800 (1977) */(continues on next page)

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363

GNU Scientific Library, Release 2.7(continued from previous page)/* For simplicity we compute the integral over the region(0,0,0) -> (pi,pi,pi) and multiply by 8 */double exact = 1.3932039296856768591842462603255;
doubleg (double *k, size_t dim, void *params){(void)(dim); /* avoid unused parameter warnings */(void)(params);
double A = 1.0 / (M_PI * M_PI * M_PI);
return A / (1.0 - cos (k[0]) * cos (k[1]) * cos (k[2]));}
void
display_results (char *title, double result, double error){
printf ("%s ==================\n", title);
printf ("result = % .6f \n", result);
printf ("sigma = % .6f \n", error);printf ("exact = % .6f \n", exact);
printf ("error = % .6f = %.2g sigma\n", result - exact,
fabs (result - exact) / error);}int
main (void){
double res, err;
double xl[3] = { 0, 0, 0 };double xu[3] = { M_PI, M_PI, M_PI };const gsl_rng_type *T;
gsl_rng *r;
gsl_monte_function G = { &g, 3, 0 };size_t calls = 500000;
gsl_rng_env_setup ();T = gsl_rng_default;r = gsl_rng_alloc (T);{gsl_monte_plain_state *s = gsl_monte_plain_alloc (3);gsl_monte_plain_integrate (&G, xl, xu, 3, calls, r, s,&res, &err);
gsl_monte_plain_free (s);(continues on next page)364

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GNU Scientific Library, Release 2.7(continued from previous page)

display_results ("plain", res, err);}{
gsl_monte_miser_state *s = gsl_monte_miser_alloc (3);gsl_monte_miser_integrate (&G, xl, xu, 3, calls, r, s,&res, &err);
gsl_monte_miser_free (s);display_results ("miser", res, err);}{
gsl_monte_vegas_state *s = gsl_monte_vegas_alloc (3);gsl_monte_vegas_integrate (&G, xl, xu, 3, 10000, r, s,&res, &err);
display_results ("vegas warm-up", res, err);
printf ("converging...\n");do{gsl_monte_vegas_integrate (&G, xl, xu, 3, calls/5, r, s,&res, &err);
printf ("result = % .6f sigma = % .6f ""chisq/dof = %.1f \n", res, err, gsl_monte_vegas_chisq (s));}
while (fabs (gsl_monte_vegas_chisq (s) - 1.0) > 0.5);
display_results ("vegas final", res, err);
gsl_monte_vegas_free (s);}gsl_rng_free (r);return 0;}
With 500,000 function calls the plain Monte Carlo algorithm achieves a fractional error of 1%. The estimated error
sigma is roughly consistent with the actual error–the computed result differs from the true result by about 1.4 standard
deviations:plain ==================
result = 1.412209
sigma = 0.013436
exact = 1.393204
error = 0.019005 = 1.4 sigma
The MISER algorithm reduces the error by a factor of four, and also correctly estimates the error:

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miser ==================
result = 1.391322
sigma = 0.003461
exact = 1.393204
error = -0.001882 = 0.54 sigma
In the case of the VEGAS algorithm the program uses an initial warm-up run of 10,000 function calls to prepare, or“warm up”, the grid. This is followed by a main run with five iterations of 100,000 function calls. The chi-squared per
degree of freedom for the five iterations are checked for consistency with 1, and the run is repeated if the results havenot converged. In this case the estimates are consistent on the first pass:vegas warm-up ==================
result = 1.392673
sigma = 0.003410
exact = 1.393204
error = -0.000531 = 0.16 sigma
converging...
result = 1.393281 sigma = 0.000362 chisq/dof = 1.5
vegas final ==================
result = 1.393281
sigma = 0.000362
exact = 1.393204
error = 0.000077 = 0.21 sigma
If the value of chisq had differed significantly from 1 it would indicate inconsistent results, with a correspondingly
underestimated error. The final estimate from VEGAS (using a similar number of function calls) is significantly moreaccurate than the other two algorithms.

27.6 References and Further Reading
The MISER algorithm is described in the following article by Press and Farrar,• W.H. Press, G.R. Farrar, Recursive Stratified Sampling for Multidimensional Monte Carlo Integration, Computers
in Physics, v4 (1990), pp190–195.
The VEGAS algorithm is described in the following papers,• G.P. Lepage, A New Algorithm for Adaptive Multidimensional Integration, Journal of Computational Physics 27,
192–203, (1978)• G.P. Lepage, VEGAS: An Adaptive Multi-dimensional Integration Program, Cornell preprint CLNS 80-447,
March 1980

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CHAPTERTWENTYEIGHTSIMULATED ANNEALING

Stochastic search techniques are used when the structure of a space is not well understood or is not smooth, so thattechniques like Newton’s method (which requires calculating Jacobian derivative matrices) cannot be used. In particular, these techniques are frequently used to solve combinatorial optimization problems, such as the traveling salesman
problem.
The goal is to find a point in the space at which a real valued energy function (or cost function) is minimized. Simulated
annealing is a minimization technique which has given good results in avoiding local minima; it is based on the idea
of taking a random walk through the space at successively lower temperatures, where the probability of taking a stepis given by a Boltzmann distribution.
The functions described in this chapter are declared in the header file gsl_siman.h.

28.1 Simulated Annealing algorithm
The simulated annealing algorithm takes random walks through the problem space, looking for points with low energies;
in these random walks, the probability of taking a step is determined by the Boltzmann distribution,p = e−(Ei+1 −Ei )/(kT )if Ei+1 > Ei , and p = 1 when Ei+1 ≤ Ei .
In other words, a step will occur if the new energy is lower. If the new energy is higher, the transition can still occur,and its likelihood is proportional to the temperature T and inversely proportional to the energy difference Ei+1 − Ei .
The temperature T is initially set to a high value, and a random walk is carried out at that temperature. Then thetemperature is lowered very slightly according to a cooling schedule, for example: T → T /µT where µT is slightly
greater than 1.
The slight probability of taking a step that gives higher energy is what allows simulated annealing to frequently get outof local minima.

28.2 Simulated Annealing functions
void gsl_siman_solve(const gsl_rng *r, void *x0_p, gsl_siman_Efunc_t Ef, gsl_siman_step_t take_step,
gsl_siman_metric_t distance, gsl_siman_print_t print_position, gsl_siman_copy_t
copyfunc, gsl_siman_copy_construct_t copy_constructor, gsl_siman_destroy_t destructor,
size_t element_size, gsl_siman_params_t params)
This function performs a simulated annealing search through a given space. The space is specified by providing
the functions Ef and distance. The simulated annealing steps are generated using the random number generatorr and the function take_step.

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The starting configuration of the system should be given by x0_p. The routine offers two modes for updating
configurations, a fixed-size mode and a variable-size mode. In the fixed-size mode the configuration is stored
as a single block of memory of size element_size. Copies of this configuration are created, copied and destroyed internally using the standard library functions malloc(), memcpy() and free(). The function pointers
copyfunc, copy_constructor and destructor should be null pointers in fixed-size mode. In the variablesize mode the functions copyfunc, copy_constructor and destructor are used to create, copy and destroy
configurations internally. The variable element_size should be zero in the variable-size mode.
The params structure (described below) controls the run by providing the temperature schedule and other tunable
parameters to the algorithm.
On exit the best result achieved during the search is placed in x0_p. If the annealing process has been successful
this should be a good approximation to the optimal point in the space.
If the function pointer print_position is not null, a debugging log will be printed to stdout with the following
columns:#-iter#-evals

temperature

positionenergybest_energy

and the output of the function print_position itself. If print_position is null then no information isprinted.
The simulated annealing routines require several user-specified functions to define the configuration space and energyfunction. The prototypes for these functions are given below.
type gsl_siman_Efunc_t
This function type should return the energy of a configuration xp:
double (*gsl_siman_Efunc_t) (void *xp)
type gsl_siman_step_t
This function type should modify the configuration xp using a random step taken from the generator r, up to a
maximum distance of step_size:
void (*gsl_siman_step_t) (const gsl_rng *r, void *xp,
double step_size)
type gsl_siman_metric_t
This function type should return the distance between two configurations xp and yp:
double (*gsl_siman_metric_t) (void *xp, void *yp)type gsl_siman_print_t
This function type should print the contents of the configuration xp:
void (*gsl_siman_print_t) (void *xp)
type gsl_siman_copy_t
This function type should copy the configuration source into dest:
void (*gsl_siman_copy_t) (void *source, void *dest)
type gsl_siman_copy_construct_t
This function type should create a new copy of the configuration xp:
void * (*gsl_siman_copy_construct_t) (void *xp)

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type gsl_siman_destroy_t
This function type should destroy the configuration xp, freeing its memory:
void (*gsl_siman_destroy_t) (void *xp)
type gsl_siman_params_t
These are the parameters that control a run of gsl_siman_solve(). This structure contains all the informationneeded to control the search, beyond the energy function, the step function and the initial guess.
int n_tries
int iters_fixed_T
double step_size
double k, t_initial, mu_t,t_min

The number of points to try for each step.
The number of iterations at each temperature.
The maximum step size in the random walk.
The parameters of the Boltzmann distribution and cooling schedule.

28.3 Examples
The simulated annealing package is clumsy, and it has to be because it is written in C, for C callers, and tries to bepolymorphic at the same time. But here we provide some examples which can be pasted into your application withlittle change and should make things easier.

28.3.1 Trivial example
The first example, in one dimensional Cartesian space, sets up an energy function which is a damped sine wave; thishas many local minima, but only one global minimum, somewhere between 1.0 and 1.5. The initial guess given is 15.5,
which is several local minima away from the global minimum.
#include <math.h>#include <stdlib.h>#include <string.h>#include <gsl/gsl_siman.h>/* set up parameters for this simulated annealing run *//* how many points do we try before stepping */#define N_TRIES 200/* how many iterations for each T? */#define ITERS_FIXED_T 1000/* max step size in random walk */#define STEP_SIZE 1.0/* Boltzmann constant */#define K 1.0/* initial temperature */#define T_INITIAL 0.008/* damping factor for temperature */(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)#define MU_T 1.003#define T_MIN 2.0e-6
gsl_siman_params_t params= {N_TRIES, ITERS_FIXED_T, STEP_SIZE,
K, T_INITIAL, MU_T, T_MIN};/* now some functions to test in one dimension */double E1(void *xp){
double x = * ((double *) xp);
return exp(-pow((x-1.0),2.0))*sin(8*x);}
double M1(void *xp, void *yp){
double x = *((double *) xp);
double y = *((double *) yp);
return fabs(x - y);}void S1(const gsl_rng * r, void *xp, double step_size){
double old_x = *((double *) xp);
double new_x;
double u = gsl_rng_uniform(r);new_x = u * 2 * step_size - step_size + old_x;
memcpy(xp, &new_x, sizeof(new_x));}
void P1(void *xp){
printf ("%12g", *((double *) xp));}
int
main(void){
const gsl_rng_type * T;
gsl_rng * r;
double x_initial = 15.5;
gsl_rng_env_setup();
T = gsl_rng_default;r = gsl_rng_alloc(T);(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)gsl_siman_solve(r, &x_initial, E1, S1, M1, P1,
NULL, NULL, NULL,
sizeof(double), params);
gsl_rng_free (r);return 0;}
Fig. 28.1 is generated by running siman_test in the following way:$ ./siman_test | awk '!/^#/ {print $1, $4}'
| graph -y 1.34 1.4 -W0 -X generation -Y position| plot -Tps > siman-test.eps
Fig. 28.2 is generated by running siman_test in the following way:$ ./siman_test | awk '!/^#/ {print $1, $5}'
| graph -y -0.88 -0.83 -W0 -X generation -Y energy| plot -Tps > siman-energy.eps

Fig. 28.1: Example of a simulated annealing run: at higher temperatures (early in the plot) you see that the solution
can fluctuate, but at lower temperatures it converges.

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GNU Scientific Library, Release 2.7

Fig. 28.2: Simulated annealing energy vs generation

28.3.2 Traveling Salesman Problem
The TSP (Traveling Salesman Problem) is the classic combinatorial optimization problem. I have provided a very
simple version of it, based on the coordinates of twelve cities in the southwestern United States. This should maybe becalled the Flying Salesman Problem, since I am using the great-circle distance between cities, rather than the driving
distance. Also: I assume the earth is a sphere, so I don’t use geoid distances.
The gsl_siman_solve() routine finds a route which is 3490.62 Kilometers long; this is confirmed by an exhaustive
search of all possible routes with the same initial city.
The full code is given below.
/* siman/siman_tsp.c** Copyright (C) 1996, 1997, 1998, 1999, 2000 Mark Galassi** This program is free software; you can redistribute it and/or modify* it under the terms of the GNU General Public License as published by* the Free Software Foundation; either version 3 of the License, or (at* your option) any later version.
** This program is distributed in the hope that it will be useful, but* WITHOUT ANY WARRANTY; without even the implied warranty of* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU* General Public License for more details.
** You should have received a copy of the GNU General Public License* along with this program; if not, write to the Free Software* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/#include <config.h>#include <math.h>372(continues on next page)

Chapter 28. Simulated Annealing

GNU Scientific Library, Release 2.7(continued from previous page)#include <string.h>#include <stdio.h>#include <gsl/gsl_math.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_siman.h>#include <gsl/gsl_ieee_utils.h>/* set up parameters for this simulated annealing run */#define N_TRIES 200#define ITERS_FIXED_T 2000
#define STEP_SIZE 1.0#define K 1.0#define T_INITIAL 5000.0#define MU_T 1.002#define T_MIN 5.0e-1/* how many points do we try before stepping *//* how many iterations for each T? *//* max step size in random walk *//* Boltzmann constant *//* initial temperature *//* damping factor for temperature */gsl_siman_params_t params = {N_TRIES, ITERS_FIXED_T, STEP_SIZE,
K, T_INITIAL, MU_T, T_MIN};
struct s_tsp_city {const char * name;double lat, longitude;/* coordinates */};typedef struct s_tsp_city Stsp_city;
void prepare_distance_matrix(void);void exhaustive_search(void);void print_distance_matrix(void);
double city_distance(Stsp_city c1, Stsp_city c2);
double Etsp(void *xp);
double Mtsp(void *xp, void *yp);void Stsp(const gsl_rng * r, void *xp, double step_size);
void Ptsp(void *xp);/* in this table, latitude and longitude are obtained from the US
Census Bureau, at http://www.census.gov/cgi-bin/gazetteer */Stsp_city cities[] = {{"Santa Fe",35.68,{"Phoenix",
33.54,{"Albuquerque", 35.12,{"Clovis",34.41,{"Durango",
37.29,{"Dallas",
32.79,{"Tesuque",35.77,{"Grants",
35.15,{"Los Alamos", 35.89,{"Las Cruces", 32.34,{"Cortez",
37.35,{"Gallup",
35.52,

105.95},112.07},106.62},103.20},107.87},96.77},
105.92},107.84},106.28},106.76},108.58},108.74}};#define N_CITIES (sizeof(cities)/sizeof(Stsp_city))(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)

double distance_matrix[N_CITIES][N_CITIES];/* distance between two cities */double city_distance(Stsp_city c1, Stsp_city c2){
const double earth_radius = 6375.000; /* 6000KM approximately *//* sin and cos of lat and long; must convert to radians */double sla1 = sin(c1.lat*M_PI/180), cla1 = cos(c1.lat*M_PI/180),
slo1 = sin(c1.longitude*M_PI/180), clo1 = cos(c1.longitude*M_PI/180);
double sla2 = sin(c2.lat*M_PI/180), cla2 = cos(c2.lat*M_PI/180),
slo2 = sin(c2.longitude*M_PI/180), clo2 = cos(c2.longitude*M_PI/180);
double x1 = cla1*clo1;
double x2 = cla2*clo2;
double y1 = cla1*slo1;
double y2 = cla2*slo2;
double z1 = sla1;
double z2 = sla2;
double dot_product = x1*x2 + y1*y2 + z1*z2;
double angle = acos(dot_product);/* distance is the angle (in radians) times the earth radius */return angle*earth_radius;}/* energy for the travelling salesman problem */double Etsp(void *xp){/* an array of N_CITIES integers describing the order */int *route = (int *) xp;
double E = 0;
unsigned int i;for (i = 0; i < N_CITIES; ++i) {/* use the distance_matrix to optimize this calculation; it had
better be allocated!! */E += distance_matrix[route[i]][route[(i + 1) % N_CITIES]];}
return E;}
double Mtsp(void *xp, void *yp){
int *route1 = (int *) xp, *route2 = (int *) yp;
double distance = 0;
unsigned int i;(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)for (i = 0; i < N_CITIES; ++i) {distance += ((route1[i] == route2[i]) ? 0 : 1);}
return distance;}/* take a step through the TSP space */void Stsp(const gsl_rng * r, void *xp, double step_size){
int x1, x2, dummy;
int *route = (int *) xp;
step_size = 0 ; /* prevent warnings about unused parameter *//* pick the two cities to swap in the matrix; we leave the firstcity fixed */x1 = (gsl_rng_get (r) % (N_CITIES-1)) + 1;
do {x2 = (gsl_rng_get (r) % (N_CITIES-1)) + 1;
} while (x2 == x1);
dummy = route[x1];route[x1] = route[x2];route[x2] = dummy;}
void Ptsp(void *xp){
unsigned int i;
int *route = (int *) xp;
printf(" [");for (i = 0; i < N_CITIES; ++i) {printf(" %d ", route[i]);}printf("] ");}int main(void){
int x_initial[N_CITIES];
unsigned int i;const gsl_rng * r = gsl_rng_alloc (gsl_rng_env_setup()) ;gsl_ieee_env_setup ();prepare_distance_matrix();/* set up a trivial initial route */printf("# initial order of cities:\n");(continues on next page)

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375

GNU Scientific Library, Release 2.7(continued from previous page)for (i = 0; i < N_CITIES; ++i) {printf("# \"%s\"\n", cities[i].name);
x_initial[i] = i;}printf("# distance matrix is:\n");print_distance_matrix();
printf("# initial coordinates of cities (longitude and latitude)\n");/* this can be plotted with *//* ./siman_tsp > hhh ; grep city_coord hhh | awk '{print $2 "" $3}' | xyplot -ps -d,→"xy" > c.eps */for (i = 0; i < N_CITIES+1; ++i) {printf("###initial_city_coord: %g %g \"%s\"\n",-cities[x_initial[i % N_CITIES]].longitude,
cities[x_initial[i % N_CITIES]].lat,
cities[x_initial[i % N_CITIES]].name);}/*

exhaustive_search(); */gsl_siman_solve(r, x_initial, Etsp, Stsp, Mtsp, Ptsp, NULL, NULL, NULL,
N_CITIES*sizeof(int), params);
printf("# final order of cities:\n");for (i = 0; i < N_CITIES; ++i) {printf("# \"%s\"\n", cities[x_initial[i]].name);}printf("# final coordinates of cities (longitude and latitude)\n");/* this can be plotted with *//* ./siman_tsp > hhh ; grep city_coord hhh | awk '{print $2 "" $3}' | xyplot -ps -d,→"xy" > c.eps */for (i = 0; i < N_CITIES+1; ++i) {printf("###final_city_coord: %g %g %s\n",-cities[x_initial[i % N_CITIES]].longitude,
cities[x_initial[i % N_CITIES]].lat,
cities[x_initial[i % N_CITIES]].name);}printf("# ");fflush(stdout);#if 0
system("date");#endif /* 0 */fflush(stdout);
return 0;}
void prepare_distance_matrix(){(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)

unsigned int i, j;
double dist;for (i = 0; i < N_CITIES; ++i) {for (j = 0; j < N_CITIES; ++j) {if (i == j) {dist = 0;} else {dist = city_distance(cities[i], cities[j]);}distance_matrix[i][j] = dist;}}}
void print_distance_matrix(){
unsigned int i, j;for (i = 0; i < N_CITIES; ++i) {printf("# ");for (j = 0; j < N_CITIES; ++j) {printf("%15.8f", distance_matrix[i][j]);}printf("\n");}}/* [only works for 12] search the entire space for solutions */static double best_E = 1.0e100, second_E = 1.0e100, third_E = 1.0e100;
static int best_route[N_CITIES];
static int second_route[N_CITIES];
static int third_route[N_CITIES];
static void do_all_perms(int *route, int n);void exhaustive_search(){static int initial_route[N_CITIES] = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11};
printf("\n# ");fflush(stdout);#if 0
system("date");#endiffflush(stdout);
do_all_perms(initial_route, 1);
printf("\n# ");fflush(stdout);#if 0
system("date");#endif /* 0 */fflush(stdout);(continues on next page)

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printf("# exhaustive best route: ");Ptsp(best_route);
printf("\n# its energy is: %g\n", best_E);
printf("# exhaustive second_best route: ");Ptsp(second_route);
printf("\n# its energy is: %g\n", second_E);
printf("# exhaustive third_best route: ");Ptsp(third_route);
printf("\n# its energy is: %g\n", third_E);}/* James Theiler's recursive algorithm for generating all routes */static void do_all_perms(int *route, int n){
if (n == (N_CITIES-1)) {/* do it! calculate the energy/cost for that route */double E;
E = Etsp(route);/* TSP energy function *//* now save the best 3 energies and routes */if (E < best_E) {third_E = second_E;
memcpy(third_route, second_route, N_CITIES*sizeof(*route));second_E = best_E;
memcpy(second_route, best_route, N_CITIES*sizeof(*route));best_E = E;
memcpy(best_route, route, N_CITIES*sizeof(*route));} else if (E < second_E) {third_E = second_E;
memcpy(third_route, second_route, N_CITIES*sizeof(*route));second_E = E;
memcpy(second_route, route, N_CITIES*sizeof(*route));} else if (E < third_E) {third_E = E;
memcpy(route, third_route, N_CITIES*sizeof(*route));}} else {int new_route[N_CITIES];
unsigned int j;
int swap_tmp;
memcpy(new_route, route, N_CITIES*sizeof(*route));for (j = n; j < N_CITIES; ++j) {swap_tmp = new_route[j];new_route[j] = new_route[n];new_route[n] = swap_tmp;
do_all_perms(new_route, n+1);}}}
Below are some plots generated in the following way:378

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GNU Scientific Library, Release 2.7$ ./siman_tsp > tsp.output$ grep -v "^#" tsp.output| awk '{print $1, $NF}'| graph -y 3300 6500 -W0 -X generation -Y distance-L "TSP - 12 southwest cities"| plot -Tps > 12-cities.eps$ grep initial_city_coord tsp.output| awk '{print $2, $3}'
| graph -X "longitude (- means west)" -Y "latitude"-L "TSP - initial-order" -f 0.03 -S 1 0.1| plot -Tps > initial-route.eps$ grep final_city_coord tsp.output| awk '{print $2, $3}'
| graph -X "longitude (- means west)" -Y "latitude"-L "TSP - final-order" -f 0.03 -S 1 0.1| plot -Tps > final-route.eps
This is the output showing the initial order of the cities; longitude is negative, since it is west and I want the plot tolook like a map:
# initial coordinates of cities (longitude and latitude)###initial_city_coord: -105.95 35.68 Santa Fe###initial_city_coord: -112.07 33.54 Phoenix###initial_city_coord: -106.62 35.12 Albuquerque###initial_city_coord: -103.2 34.41 Clovis###initial_city_coord: -107.87 37.29 Durango###initial_city_coord: -96.77 32.79 Dallas###initial_city_coord: -105.92 35.77 Tesuque###initial_city_coord: -107.84 35.15 Grants###initial_city_coord: -106.28 35.89 Los Alamos###initial_city_coord: -106.76 32.34 Las Cruces###initial_city_coord: -108.58 37.35 Cortez###initial_city_coord: -108.74 35.52 Gallup###initial_city_coord: -105.95 35.68 Santa Fe
The optimal route turns out to be:# final coordinates of cities (longitude and latitude)###final_city_coord: -105.95 35.68 Santa Fe###final_city_coord: -103.2 34.41 Clovis###final_city_coord: -96.77 32.79 Dallas###final_city_coord: -106.76 32.34 Las Cruces###final_city_coord: -112.07 33.54 Phoenix###final_city_coord: -108.74 35.52 Gallup###final_city_coord: -108.58 37.35 Cortez###final_city_coord: -107.87 37.29 Durango###final_city_coord: -107.84 35.15 Grants###final_city_coord: -106.62 35.12 Albuquerque###final_city_coord: -106.28 35.89 Los Alamos###final_city_coord: -105.92 35.77 Tesuque###final_city_coord: -105.95 35.68 Santa Fe
Here’s a plot of the cost function (energy) versus generation (point in the calculation at which a new temperature is set)
for this problem:
28.3. Examples

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Fig. 28.3: Initial route for the 12 southwestern cities Flying Salesman Problem.

Fig. 28.4: Final (optimal) route for the 12 southwestern cities Flying Salesman Problem.

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Fig. 28.5: Example of a simulated annealing run for the 12 southwestern cities Flying Salesman Problem.

28.4 References and Further Reading
Further information is available in the following book,• Modern Heuristic Techniques for Combinatorial Problems, Colin R. Reeves (ed.), McGraw-Hill, 1995 (ISBN0-07-709239-2).

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CHAPTERTWENTYNINE

ORDINARY DIFFERENTIAL EQUATIONSThis chapter describes functions for solving ordinary differential equation (ODE) initial value problems. The libraryprovides a variety of low-level methods, such as Runge-Kutta and Bulirsch-Stoer routines, and higher-level components
for adaptive step-size control. The components can be combined by the user to achieve the desired solution, with fullaccess to any intermediate steps. A driver object can be used as a high level wrapper for easy use of low level functions.
These functions are declared in the header file gsl_odeiv2.h. This is a new interface in version 1.15 and uses theprefix gsl_odeiv2 for all functions. It is recommended over the previous gsl_odeiv implementation defined in
gsl_odeiv.h The old interface has been retained under the original name for backwards compatibility.

29.1 Defining the ODE System
The routines solve the general n-dimensional first-order system,dyi (t)= fi (t, y1 (t), . . . yn (t))
dt
for i = 1, . . . , n. The stepping functions rely on the vector of derivatives fi and the Jacobian matrix,
Jij = ∂fi (t, y(t))/∂yj
A system of equations is defined using the gsl_odeiv2_system datatype.
type gsl_odeiv2_system
This data type defines a general ODE system with arbitrary parameters.
int (* function) (double t, const double y[], double dydt[], void * params)
This function should store the vector elements fi (t, y, params) in the array dydt, for arguments (t,y) and parameters params.
The function should return GSL_SUCCESS if the calculation was completed successfully. Anyother return value indicates an error. A special return value GSL_EBADFUNC causes gsl_odeiv2
routines to immediately stop and return. If function is modified (for example contentsof params), the user must call an appropriate reset function (gsl_odeiv2_driver_reset(),gsl_odeiv2_evolve_reset() or gsl_odeiv2_step_reset()) before continuing. Use return
values distinct from standard GSL error codes to distinguish your function as the source of the error.
int (* jacobian) (double t, const double y[], double * dfdy, double dfdt[], void *params)This function should store the vector of derivative elements∂fi (t, y, params)/∂t
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in the array dfdt and the Jacobian matrix Jij in the array dfdy, regarded as a row-ordered matrix
J(i,j) = dfdy[i * dimension + j] where dimension is the dimension of the system.
Not all of the stepper algorithms of gsl_odeiv2 make use of the Jacobian matrix, so it may not
be necessary to provide this function (the jacobian element of the struct can be replaced by a nullpointer for those algorithms).
The function should return GSL_SUCCESS if the calculation was completed successfully. Anyother return value indicates an error. A special return value GSL_EBADFUNC causes gsl_odeiv2
routines to immediately stop and return. If jacobian is modified (for example contentsof params), the user must call an appropriate reset function (gsl_odeiv2_driver_reset(),gsl_odeiv2_evolve_reset() or gsl_odeiv2_step_reset()) before continuing. Use return
values distinct from standard GSL error codes to distinguish your function as the source of the error.
size_t dimension
This is the dimension of the system of equations.
void * params
This is a pointer to the arbitrary parameters of the system.

29.2 Stepping Functions
The lowest level components are the stepping functions which advance a solution from time t to t + h for a fixedstep-size h and estimate the resulting local error.
type gsl_odeiv2_step
This contains internal parameters for a stepping function.
gsl_odeiv2_step *gsl_odeiv2_step_alloc(const gsl_odeiv2_step_type *T, size_t dim)This function returns a pointer to a newly allocated instance of a stepping function of type T for a system of dim
dimensions. Please note that if you use a stepper method that requires access to a driver object, it is advisable touse a driver allocation method, which automatically allocates a stepper, too.
int gsl_odeiv2_step_reset(gsl_odeiv2_step *s)
This function resets the stepping function s. It should be used whenever the next use of s will not be a continuation of a previous step.
void gsl_odeiv2_step_free(gsl_odeiv2_step *s)
This function frees all the memory associated with the stepping function s.
const char *gsl_odeiv2_step_name(const gsl_odeiv2_step *s)
This function returns a pointer to the name of the stepping function. For example:
printf ("step method is '%s'\n", gsl_odeiv2_step_name (s));
would print something like step method is 'rkf45'.
unsigned int gsl_odeiv2_step_order(const gsl_odeiv2_step *s)
This function returns the order of the stepping function on the previous step. The order can vary if the stepping
function itself is adaptive.
int gsl_odeiv2_step_set_driver(gsl_odeiv2_step *s, const gsl_odeiv2_driver *d)
This function sets a pointer of the driver object d for stepper s, to allow the stepper to access control (and evolve)
object through the driver object. This is a requirement for some steppers, to get the desired error level for internal
iteration of stepper. Allocation of a driver object calls this function automatically.

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int gsl_odeiv2_step_apply(gsl_odeiv2_step *s, double t, double h, double y[], double yerr[], const double
dydt_in[], double dydt_out[], const gsl_odeiv2_system *sys)
This function applies the stepping function s to the system of equations defined by sys, using the step-size hto advance the system from time t and state y to time t + h . The new state of the system is stored in y on
output, with an estimate of the absolute error in each component stored in yerr. If the argument dydt_in is
not null it should point an array containing the derivatives for the system at time t on input. This is optional as
the derivatives will be computed internally if they are not provided, but allows the reuse of existing derivative
information. On output the new derivatives of the system at time t + h will be stored in dydt_out if it is notnull.
The stepping function returns GSL_FAILURE if it is unable to compute the requested step. Also, if the usersupplied functions defined in the system sys return a status other than GSL_SUCCESS the step will be aborted.
In that case, the elements of y will be restored to their pre-step values and the error code from the user-supplied
function will be returned. Failure may be due to a singularity in the system or too large step-size h . In that casethe step should be attempted again with a smaller step-size, e.g. h / 2.
If the driver object is not appropriately set via gsl_odeiv2_step_set_driver() for those steppers that needit, the stepping function returns GSL_EFAULT. If the user-supplied functions defined in the system sys returns
GSL_EBADFUNC, the function returns immediately with the same return code. In this case the user must call
gsl_odeiv2_step_reset() before calling this function again.
The following algorithms are available. Please note that algorithms which use step doubling for error estimation apply
the more accurate values from two half steps instead of values from a single step for the new state y.
type gsl_odeiv2_step_type
gsl_odeiv2_step_type *gsl_odeiv2_step_rk2
Explicit embedded Runge-Kutta (2, 3) method.
gsl_odeiv2_step_type *gsl_odeiv2_step_rk4
Explicit 4th order (classical) Runge-Kutta. Error estimation is carried out by the step doubling method. For
more efficient estimate of the error, use the embedded methods described below.
gsl_odeiv2_step_type *gsl_odeiv2_step_rkf45
Explicit embedded Runge-Kutta-Fehlberg (4, 5) method. This method is a good general-purpose integrator.
gsl_odeiv2_step_type *gsl_odeiv2_step_rkck
Explicit embedded Runge-Kutta Cash-Karp (4, 5) method.
gsl_odeiv2_step_type *gsl_odeiv2_step_rk8pd
Explicit embedded Runge-Kutta Prince-Dormand (8, 9) method.
gsl_odeiv2_step_type *gsl_odeiv2_step_rk1imp
Implicit Gaussian first order Runge-Kutta. Also known as implicit Euler or backward Euler method. Error
estimation is carried out by the step doubling method. This algorithm requires the Jacobian and access tothe driver object via gsl_odeiv2_step_set_driver().
gsl_odeiv2_step_type *gsl_odeiv2_step_rk2imp
Implicit Gaussian second order Runge-Kutta. Also known as implicit mid-point rule. Error estimation iscarried out by the step doubling method. This stepper requires the Jacobian and access to the driver object
via gsl_odeiv2_step_set_driver().
gsl_odeiv2_step_type *gsl_odeiv2_step_rk4imp
Implicit Gaussian 4th order Runge-Kutta. Error estimation is carried out by the step doubling method. Thisalgorithm requires the Jacobian and access to the driver object via gsl_odeiv2_step_set_driver().
gsl_odeiv2_step_type *gsl_odeiv2_step_bsimp
Implicit Bulirsch-Stoer method of Bader and Deuflhard. The method is generally suitable for stiff problems.
This stepper requires the Jacobian.

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gsl_odeiv2_step_type *gsl_odeiv2_step_msadams
A variable-coefficient linear multistep Adams method in Nordsieck form. This stepper uses explicit AdamsBashforth (predictor) and implicit Adams-Moulton (corrector) methods in P (EC)m functional iteration
mode. Method order varies dynamically between 1 and 12. This stepper requires the access to the driverobject via gsl_odeiv2_step_set_driver().
gsl_odeiv2_step_type *gsl_odeiv2_step_msbdf
A variable-coefficient linear multistep backward differentiation formula (BDF) method in Nordsieck form.
This stepper uses the explicit BDF formula as predictor and implicit BDF formula as corrector. A modified Newton iteration method is used to solve the system of non-linear equations. Method order varies
dynamically between 1 and 5. The method is generally suitable for stiff problems. This stepper requires
the Jacobian and the access to the driver object via gsl_odeiv2_step_set_driver().

29.3 Adaptive Step-size Control
The control function examines the proposed change to the solution produced by a stepping function and attempts todetermine the optimal step-size for a user-specified level of error.
type gsl_odeiv2_control
This is a workspace for controlling step size.
type gsl_odeiv2_control_type
This specifies the control type.
gsl_odeiv2_control *gsl_odeiv2_control_standard_new(double eps_abs, double eps_rel, double a_y, double
a_dydt)The standard control object is a four parameter heuristic based on absolute and relative errors eps_abs andeps_rel, and scaling factors a_y and a_dydt for the system state y(t) and derivatives y 0 (t) respectively.
The step-size adjustment procedure for this method begins by computing the desired error level Di for eachcomponent,
Di = abs + rel ∗ (ay |yi | + adydt h|y0i |)and comparing it with the observed error Ei = |yerri |. If the observed error E exceeds the desired error level D
by more than 10% for any component then the method reduces the step-size by an appropriate factor,
hnew = hold ∗ S ∗ (E/D)−1/q
where q is the consistency order of the method (e.g. q = 4 for 4(5) embedded RK), and S is a safety factor of
0.9. The ratio E/D is taken to be the maximum of the ratios Ei /Di .
If the observed error E is less than 50% of the desired error level D for the maximum ratio Ei /Di then thealgorithm takes the opportunity to increase the step-size to bring the error in line with the desired level,hnew = hold ∗ S ∗ (E/D)−1/(q+1)
This encompasses all the standard error scaling methods. To avoid uncontrolled changes in the stepsize, theoverall scaling factor is limited to the range 1/5 to 5.
gsl_odeiv2_control *gsl_odeiv2_control_y_new(double eps_abs, double eps_rel)
This function creates a new control object which will keep the local error on each step within an absolute errorof eps_abs and relative error of eps_rel with respect to the solution yi (t). This is equivalent to the standard
control object with a_y = 1 and a_dydt = 0.

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gsl_odeiv2_control *gsl_odeiv2_control_yp_new(double eps_abs, double eps_rel)
This function creates a new control object which will keep the local error on each step within an absolute errorof eps_abs and relative error of eps_rel with respect to the derivatives of the solution yi0 (t). This is equivalent
to the standard control object with a_y = 0 and a_dydt = 1.
gsl_odeiv2_control *gsl_odeiv2_control_scaled_new(double eps_abs, double eps_rel, double a_y, double
a_dydt, const double scale_abs[], size_t dim)This function creates a new control object which uses the same algorithm as
gsl_odeiv2_control_standard_new() but with an absolute error which is scaled for each component by
the array scale_abs. The formula for Di for this control object is,
Di = abs si + rel ∗ (ay |yi | + adydt h|y0i |)where si is the i-th component of the array scale_abs. The same error control heuristic is used by the Matlab
ODE suite.
gsl_odeiv2_control *gsl_odeiv2_control_alloc(const gsl_odeiv2_control_type *T)This function returns a pointer to a newly allocated instance of a control function of type T. This function is only
needed for defining new types of control functions. For most purposes the standard control functions describedabove should be sufficient.
int gsl_odeiv2_control_init(gsl_odeiv2_control *c, double eps_abs, double eps_rel, double a_y, double
a_dydt)This function initializes the control function c with the parameters eps_abs (absolute error), eps_rel (relative
error), a_y (scaling factor for y) and a_dydt (scaling factor for derivatives).
void gsl_odeiv2_control_free(gsl_odeiv2_control *c)
This function frees all the memory associated with the control function c.
int gsl_odeiv2_control_hadjust(gsl_odeiv2_control *c, gsl_odeiv2_step *s, const double y[], const double
yerr[], const double dydt[], double *h)This function adjusts the step-size h using the control function c, and the current values of y, yerr and dydt.
The stepping function step is also needed to determine the order of the method. If the error in the y-values yerris found to be too large then the step-size h is reduced and the function returns GSL_ODEIV_HADJ_DEC. If theerror is sufficiently small then h may be increased and GSL_ODEIV_HADJ_INC is returned. The function returns
GSL_ODEIV_HADJ_NIL if the step-size is unchanged. The goal of the function is to estimate the largest step-size
which satisfies the user-specified accuracy requirements for the current point.
const char *gsl_odeiv2_control_name(const gsl_odeiv2_control *c)
This function returns a pointer to the name of the control function. For example:
printf ("control method is '%s'\n", gsl_odeiv2_control_name (c));
would print something like control method is 'standard'
int gsl_odeiv2_control_errlevel(gsl_odeiv2_control *c, const double y, const double dydt, const double h,const size_t ind, double *errlev)
This function calculates the desired error level of the ind-th component to errlev. It requires the value (y) andvalue of the derivative (dydt) of the component, and the current step size h .
int gsl_odeiv2_control_set_driver(gsl_odeiv2_control *c, const gsl_odeiv2_driver *d)
This function sets a pointer of the driver object d for control object c.

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29.4 Evolution
The evolution function combines the results of a stepping function and control function to reliably advance the solution
forward one step using an acceptable step-size.
type gsl_odeiv2_evolve
This workspace contains parameters for controlling the evolution function
gsl_odeiv2_evolve *gsl_odeiv2_evolve_alloc(size_t dim)This function returns a pointer to a newly allocated instance of an evolution function for a system of dim dimensions.
int gsl_odeiv2_evolve_apply(gsl_odeiv2_evolve *e, gsl_odeiv2_control *con, gsl_odeiv2_step *step, const
gsl_odeiv2_system *sys, double *t, double t1, double *h, double y[])This function advances the system (e, sys) from time t and position y using the stepping function step. The
new time and position are stored in t and y on output.
The initial step-size is taken as h . The control function con is applied to check whether the local error estimatedby the stepping function step using step-size h exceeds the required error tolerance. If the error is too high, thestep is retried by calling step with a decreased step-size. This process is continued until an acceptable step-size
is found. An estimate of the local error for the step can be obtained from the components of the array e->yerr[].
If the user-supplied functions defined in the system sys returns GSL_EBADFUNC, the function returns immediately with the same return code. In this case the user must call gsl_odeiv2_step_reset() and
gsl_odeiv2_evolve_reset() before calling this function again.
Otherwise, if the user-supplied functions defined in the system sys or the stepping function step return a status
other than GSL_SUCCESS, the step is retried with a decreased step-size. If the step-size decreases below machine
precision, a status of GSL_FAILURE is returned if the user functions returned GSL_SUCCESS. Otherwise the value
returned by user function is returned. If no acceptable step can be made, t and y will be restored to their pre-stepvalues and h contains the final attempted step-size.
If the step is successful the function returns a suggested step-size for the next step in h . The maximum time t1
is guaranteed not to be exceeded by the time-step. On the final time-step the value of t will be set to t1 exactly.
int gsl_odeiv2_evolve_apply_fixed_step(gsl_odeiv2_evolve *e, gsl_odeiv2_control *con, gsl_odeiv2_step*step, const gsl_odeiv2_system *sys, double *t, const double h,double y[])This function advances the ODE-system (e, sys, con) from time t and position y using the stepping function
step by a specified step size h . If the local error estimated by the stepping function exceeds the desired error
level, the step is not taken and the function returns GSL_FAILURE. Otherwise the value returned by user function
is returned.
int gsl_odeiv2_evolve_reset(gsl_odeiv2_evolve *e)This function resets the evolution function e. It should be used whenever the next use of e will not be a continuation of a previous step.
void gsl_odeiv2_evolve_free(gsl_odeiv2_evolve *e)This function frees all the memory associated with the evolution function e.
int gsl_odeiv2_evolve_set_driver(gsl_odeiv2_evolve *e, const gsl_odeiv2_driver *d)
This function sets a pointer of the driver object d for evolve object e.
If a system has discontinuous changes in the derivatives at known points, it is advisable to evolve the system between
each discontinuity in sequence. For example, if a step-change in an external driving force occurs at times ta , tb and tc
then evolution should be carried out over the ranges (t0 , ta ), (ta , tb ), (tb , tc ), and (tc , t1 ) separately and not directly
over the range (t0 , t1 ).

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29.5 Driver
The driver object is a high level wrapper that combines the evolution, control and stepper objects for easy use.
gsl_odeiv2_driver *gsl_odeiv2_driver_alloc_y_new(const gsl_odeiv2_system *sys, const gsl_odeiv2_step_type*T, const double hstart, const double epsabs, const double
epsrel)gsl_odeiv2_driver *gsl_odeiv2_driver_alloc_yp_new(const gsl_odeiv2_system *sys, const
gsl_odeiv2_step_type *T, const double hstart, const
double epsabs, const double epsrel)
gsl_odeiv2_driver *gsl_odeiv2_driver_alloc_standard_new(const gsl_odeiv2_system *sys, const
gsl_odeiv2_step_type *T, const double hstart,
const double epsabs, const double epsrel, const
double a_y, const double a_dydt)
gsl_odeiv2_driver *gsl_odeiv2_driver_alloc_scaled_new(const gsl_odeiv2_system *sys, const
gsl_odeiv2_step_type *T, const double hstart, const
double epsabs, const double epsrel, const double
a_y, const double a_dydt, const double
scale_abs[])
These functions return a pointer to a newly allocated instance of a driver object. The functions automaticallyallocate and initialise the evolve, control and stepper objects for ODE system sys using stepper type T. The initial
step size is given in hstart. The rest of the arguments follow the syntax and semantics of the control functions
with same name (gsl_odeiv2_control_*_new).
int gsl_odeiv2_driver_set_hmin(gsl_odeiv2_driver *d, const double hmin)
The function sets a minimum for allowed step size hmin for driver d. Default value is 0.
int gsl_odeiv2_driver_set_hmax(gsl_odeiv2_driver *d, const double hmax)
The function sets a maximum for allowed step size hmax for driver d. Default value is GSL_DBL_MAX.
int gsl_odeiv2_driver_set_nmax(gsl_odeiv2_driver *d, const unsigned long int nmax)
The function sets a maximum for allowed number of steps nmax for driver d. Default value of 0 sets no limit forsteps.
int gsl_odeiv2_driver_apply(gsl_odeiv2_driver *d, double *t, const double t1, double y[])This function evolves the driver system d from t to t1. Initially vector y should contain the values ofdependent variables at point t. If the function is unable to complete the calculation, an error code from
gsl_odeiv2_evolve_apply() is returned, and t and y contain the values from last successful step.
If maximum number of steps is reached, a value of GSL_EMAXITER is returned. If the step size drops below
minimum value, the function returns with GSL_ENOPROG. If the user-supplied functions defined in the systemsys returns GSL_EBADFUNC, the function returns immediately with the same return code. In this case the user
must call gsl_odeiv2_driver_reset() before calling this function again.
int gsl_odeiv2_driver_apply_fixed_step(gsl_odeiv2_driver *d, double *t, const double h, const unsigned
long int n, double y[])This function evolves the driver system d from t with n steps of size h . If the function is unable to complete thecalculation, an error code from gsl_odeiv2_evolve_apply_fixed_step() is returned, and t and y containthe values from last successful step.
int gsl_odeiv2_driver_reset(gsl_odeiv2_driver *d)
This function resets the evolution and stepper objects.

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int gsl_odeiv2_driver_reset_hstart(gsl_odeiv2_driver *d, const double hstart)
The routine resets the evolution and stepper objects and sets new initial step size to hstart. This function canbe used e.g. to change the direction of integration.
int gsl_odeiv2_driver_free(gsl_odeiv2_driver *d)
This function frees the driver object, and the related evolution, stepper and control objects.

29.6 Examples
The following program solves the second-order nonlinear Van der Pol oscillator equation,
u00 (t) + µu0 (t)(u(t)2 − 1) + u(t) = 0
This can be converted into a first order system suitable for use with the routines described in this chapter by introducing
a separate variable for the velocity, v = u0 (t),u0 = v
v 0 = −u + µv(1 − u2 )The program begins by defining functions for these derivatives and their Jacobian. The main function uses driver level
functions to solve the problem. The program evolves the solution from (u, v) = (1, 0) at t = 0 to t = 100. The
step-size h is automatically adjusted by the controller to maintain an absolute accuracy of 10−6 in the function values(u, v). The loop in the example prints the solution at the points ti = 1, 2, . . . , 100.
#include <stdio.h>#include <gsl/gsl_errno.h>#include <gsl/gsl_matrix.h>#include <gsl/gsl_odeiv2.h>intfunc (double t, const double y[], double f[],void *params){(void)(t); /* avoid unused parameter warning */double mu = *(double *)params;
f[0] = y[1];f[1] = -y[0] - mu*y[1]*(y[0]*y[0] - 1);
return GSL_SUCCESS;}
int
jac (double t, const double y[], double *dfdy,
double dfdt[], void *params){(void)(t); /* avoid unused parameter warning */double mu = *(double *)params;
gsl_matrix_view dfdy_mat= gsl_matrix_view_array (dfdy, 2, 2);
gsl_matrix * m = &dfdy_mat.matrix;
gsl_matrix_set (m, 0, 0, 0.0);
gsl_matrix_set (m, 0, 1, 1.0);
gsl_matrix_set (m, 1, 0, -2.0*mu*y[0]*y[1] - 1.0);
gsl_matrix_set (m, 1, 1, -mu*(y[0]*y[0] - 1.0));(continues on next page)390

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GNU Scientific Library, Release 2.7(continued from previous page)dfdt[0] = 0.0;
dfdt[1] = 0.0;
return GSL_SUCCESS;}
int
main (void){
double mu = 10;
gsl_odeiv2_system sys = {func, jac, 2, &mu};
gsl_odeiv2_driver * d =gsl_odeiv2_driver_alloc_y_new (&sys, gsl_odeiv2_step_rk8pd,
1e-6, 1e-6, 0.0);
int i;
double t = 0.0, t1 = 100.0;
double y[2] = { 1.0, 0.0 };for (i = 1; i <= 100; i++){double ti = i * t1 / 100.0;
int status = gsl_odeiv2_driver_apply (d, &t, ti, y);if (status != GSL_SUCCESS){
printf ("error, return value=%d\n", status);
break;}
printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);}gsl_odeiv2_driver_free (d);return 0;}
The user can work with the lower level functions directly, as in the following example. In this case an intermediate
result is printed after each successful step instead of equidistant time points.
int
main (void){
const gsl_odeiv2_step_type * T= gsl_odeiv2_step_rk8pd;
gsl_odeiv2_step * s= gsl_odeiv2_step_alloc (T, 2);
gsl_odeiv2_control * c= gsl_odeiv2_control_y_new (1e-6, 0.0);
gsl_odeiv2_evolve * e= gsl_odeiv2_evolve_alloc (2);(continues on next page)

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double mu = 10;
gsl_odeiv2_system sys = {func, jac, 2, &mu};double t = 0.0, t1 = 100.0;
double h = 1e-6;
double y[2] = { 1.0, 0.0 };while (t < t1){
int status = gsl_odeiv2_evolve_apply (e, c, s,&sys,&t, t1,&h, y);
if (status != GSL_SUCCESS)
break;
printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);}gsl_odeiv2_evolve_free (e);gsl_odeiv2_control_free (c);gsl_odeiv2_step_free (s);return 0;}
For functions with multiple parameters, the appropriate information can be passed in through the params argument in
gsl_odeiv2_system definition (mu in this example) by using a pointer to a struct.
It is also possible to work with a non-adaptive integrator, using only the stepping function itself,
gsl_odeiv2_driver_apply_fixed_step() or gsl_odeiv2_evolve_apply_fixed_step().
The following program uses the driver level function, with fourth-order Runge-Kutta stepping function with a fixed stepsize of
0.001.
int
main (void){
double mu = 10;
gsl_odeiv2_system sys = { func, jac, 2, &mu };gsl_odeiv2_driver *d =gsl_odeiv2_driver_alloc_y_new (&sys, gsl_odeiv2_step_rk4,
1e-3, 1e-8, 1e-8);
double t = 0.0;
double y[2] = { 1.0, 0.0 };int i, s;for (i = 0; i < 100; i++){s = gsl_odeiv2_driver_apply_fixed_step (d, &t, 1e-3, 1000, y);if (s != GSL_SUCCESS)(continues on next page)392

Chapter 29. Ordinary Differential Equations

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Fig. 29.1: Numerical solution of the Van der Pol oscillator equation using Prince-Dormand 8th order Runge-Kutta.

29.6. Examples

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GNU Scientific Library, Release 2.7(continued from previous page){
printf ("error: driver returned %d\n", s);break;}
printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);}gsl_odeiv2_driver_free (d);return s;}

29.7 References and Further Reading• Ascher, U.M., Petzold, L.R., Computer Methods for Ordinary Differential and Differential-Algebraic Equations,
SIAM, Philadelphia, 1998.
• Hairer, E., Norsett, S. P., Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems, Springer,
Berlin, 1993.
• Hairer, E., Wanner, G., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems,
Springer, Berlin, 1996.
Many of the basic Runge-Kutta formulas can be found in the Handbook of Mathematical Functions,• Abramowitz & Stegun (eds.), Handbook of Mathematical Functions, Section 25.5.
The implicit Bulirsch-Stoer algorithm bsimp is described in the following paper,• G. Bader and P. Deuflhard, “A Semi-Implicit Mid-Point Rule for Stiff Systems of Ordinary Differential Equations.”, Numer.: Math.: 41, 373–398, 1983.
The Adams and BDF multistep methods msadams and msbdf are based on the following articles,• G. D. Byrne and A. C. Hindmarsh, “A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations.”, ACM Trans. Math. Software, 1, 71–96, 1975.
• P. N. Brown, G. D. Byrne and A. C. Hindmarsh, “VODE: A Variable-coefficient ODE Solver.”, SIAM J. Sci.
Stat. Comput. 10, 1038–1051, 1989.
• A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker and C. S. Woodward, “SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers.”, ACM Trans. Math. Software 31,
363–396, 2005.

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CHAPTERTHIRTYINTERPOLATIONThis chapter describes functions for performing interpolation. The library provides a variety of interpolation methods,
including Cubic, Akima, and Steffen splines. The interpolation types are interchangeable, allowing different methods
to be used without recompiling. Interpolations can be defined for both normal and periodic boundary conditions.
Additional functions are available for computing derivatives and integrals of interpolating functions. Routines areprovided for interpolating both one and two dimensional datasets.
These interpolation methods produce curves that pass through each datapoint. To interpolate noisy data with a smoothing curve see Basis Splines.
The functions described in this section are declared in the header files gsl_interp.h and gsl_spline.h.

30.1 Introduction to 1D Interpolation
Given a set of data points (x1 , y1 ) . . . (xn , yn ) the routines described in this section compute a continuous interpolating
function y(x) such that y(xi ) = yi . The interpolation is piecewise smooth, and its behavior at the end-points isdetermined by the type of interpolation used.

30.2 1D Interpolation Functions
The interpolation function for a given dataset is stored in a gsl_interp object. These are created by the following
functions.
type gsl_interp
Workspace for 1D interpolation
gsl_interp *gsl_interp_alloc(const gsl_interp_type *T, size_t size)
This function returns a pointer to a newly allocated interpolation object of type T for size data-points.
int gsl_interp_init(gsl_interp *interp, const double xa[], const double ya[], size_t size)
This function initializes the interpolation object interp for the data (xa, ya) where xa and ya are arrays ofsize size. The interpolation object (gsl_interp) does not save the data arrays xa and ya and only stores thestatic state computed from the data. The xa data array is always assumed to be strictly ordered, with increasing
x values; the behavior for other arrangements is not defined.
void gsl_interp_free(gsl_interp *interp)
This function frees the interpolation object interp.

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30.3 1D Interpolation Types
The interpolation library provides the following interpolation types:
type gsl_interp_type
gsl_interp_type *gsl_interp_linear
Linear interpolation. This interpolation method does not require any additional memory.
gsl_interp_type *gsl_interp_polynomial
Polynomial interpolation. This method should only be used for interpolating small numbers of points
because polynomial interpolation introduces large oscillations, even for well-behaved datasets. The numberof terms in the interpolating polynomial is equal to the number of points.
gsl_interp_type *gsl_interp_cspline
Cubic spline with natural boundary conditions. The resulting curve is piecewise cubic on each interval,
with matching first and second derivatives at the supplied data-points. The second derivative is chosen to
be zero at the first point and last point.
gsl_interp_type *gsl_interp_cspline_periodic
Cubic spline with periodic boundary conditions. The resulting curve is piecewise cubic on each interval,
with matching first and second derivatives at the supplied data-points. The derivatives at the first and lastpoints are also matched. Note that the last point in the data must have the same y-value as the first point,
otherwise the resulting periodic interpolation will have a discontinuity at the boundary.
gsl_interp_type *gsl_interp_akima
Non-rounded Akima spline with natural boundary conditions. This method uses the non-rounded corneralgorithm of Wodicka.
gsl_interp_type *gsl_interp_akima_periodic
Non-rounded Akima spline with periodic boundary conditions. This method uses the non-rounded corneralgorithm of Wodicka.
gsl_interp_type *gsl_interp_steffen
Steffen’s method guarantees the monotonicity of the interpolating function between the given data points.
Therefore, minima and maxima can only occur exactly at the data points, and there can never be spurious oscillations between data points. The interpolated function is piecewise cubic in each interval. The
resulting curve and its first derivative are guaranteed to be continuous, but the second derivative may bediscontinuous.
The following related functions are available:
const char *gsl_interp_name(const gsl_interp *interp)
This function returns the name of the interpolation type used by interp. For example:
printf ("interp uses '%s' interpolation.\n", gsl_interp_name (interp));would print something like:interp uses 'cspline' interpolation.
unsigned int gsl_interp_min_size(const gsl_interp *interp)
unsigned int gsl_interp_type_min_size(const gsl_interp_type *T)
These functions return the minimum number of points required by the interpolation object interp or interpolation type T. For example, Akima spline interpolation requires a minimum of 5 points.

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30.4 1D Index Look-up and Acceleration
The state of searches can be stored in a gsl_interp_accel object, which is a kind of iterator for interpolation lookups.
type gsl_interp_accel
This workspace stores state variables for interpolation lookups. It caches the previous value of an index lookup.
When the subsequent interpolation point falls in the same interval its index value can be returned immediately.
size_t gsl_interp_bsearch(const double x_array[], double x, size_t index_lo, size_t index_hi)
This function returns the index i of the array x_array such that x_array[i] <= x < x_array[i+1]. The
index is searched for in the range [index_lo, index_hi]. An inline version of this function is used when
HAVE_INLINE is defined.
gsl_interp_accel *gsl_interp_accel_alloc(void)
This function returns a pointer to an accelerator object, which is a kind of iterator for interpolation lookups.
It tracks the state of lookups, thus allowing for application of various acceleration strategies. When multiple
interpolants are in use, the same accelerator object may be used for all datasets with the same domain (x_array),but different accelerators should be used for data defined on different domains.
size_t gsl_interp_accel_find(gsl_interp_accel *a, const double x_array[], size_t size, double x)
This function performs a lookup action on the data array x_array of size size, using the given accelerator a.
This is how lookups are performed during evaluation of an interpolation. The function returns an index i suchthat x_array[i] <= x < x_array[i+1]. An inline version of this function is used when HAVE_INLINE isdefined.
int gsl_interp_accel_reset(gsl_interp_accel *acc);
This function reinitializes the accelerator object acc. It should be used when the cached information is no longer
applicable—for example, when switching to a new dataset.
void gsl_interp_accel_free(gsl_interp_accel *acc)
This function frees the accelerator object acc.

30.5 1D Evaluation of Interpolating Functions
double gsl_interp_eval(const gsl_interp *interp, const double xa[], const double ya[], double x, gsl_interp_accel*acc)
int gsl_interp_eval_e(const gsl_interp *interp, const double xa[], const double ya[], double x, gsl_interp_accel*acc, double *y)These functions return the interpolated value of y for a given point x, using the interpolation object interp,
data arrays xa and ya and the accelerator acc. When x is outside the range of xa, the error code GSL_EDOM isreturned with a value of GSL_NAN for y.
double gsl_interp_eval_deriv(const gsl_interp *interp, const double xa[], const double ya[], double x,
gsl_interp_accel *acc)
int gsl_interp_eval_deriv_e(const gsl_interp *interp, const double xa[], const double ya[], double x,
gsl_interp_accel *acc, double *d)These functions return the derivative d of an interpolated function for a given point x, using the interpolation
object interp, data arrays xa and ya and the accelerator acc.
double gsl_interp_eval_deriv2(const gsl_interp *interp, const double xa[], const double ya[], double x,
gsl_interp_accel *acc)

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int gsl_interp_eval_deriv2_e(const gsl_interp *interp, const double xa[], const double ya[], double x,
gsl_interp_accel *acc, double *d2)
These functions return the second derivative d2 of an interpolated function for a given point x, using the interpolation object interp, data arrays xa and ya and the accelerator acc.
double gsl_interp_eval_integ(const gsl_interp *interp, const double xa[], const double ya[], double a, double b,
gsl_interp_accel *acc)
int gsl_interp_eval_integ_e(const gsl_interp *interp, const double xa[], const double ya[], double a, double b,
gsl_interp_accel *acc, double *result)
These functions return the numerical integral result of an interpolated function over the range [a, b], using theinterpolation object interp, data arrays xa and ya and the accelerator acc.

30.6 1D Higher-level Interface
The functions described in the previous sections required the user to supply pointers to the x and y arrays on each call.
The following functions are equivalent to the corresponding gsl_interp functions but maintain a copy of this datain the gsl_spline object. This removes the need to pass both xa and ya as arguments on each evaluation. These
functions are defined in the header file gsl_spline.h.
type gsl_spline
This workspace provides a higher level interface for the gsl_interp object
gsl_spline *gsl_spline_alloc(const gsl_interp_type *T, size_t size)
int gsl_spline_init(gsl_spline *spline, const double xa[], const double ya[], size_t size)
void gsl_spline_free(gsl_spline *spline)
const char *gsl_spline_name(const gsl_spline *spline)
unsigned int gsl_spline_min_size(const gsl_spline *spline)
double gsl_spline_eval(const gsl_spline *spline, double x, gsl_interp_accel *acc)
int gsl_spline_eval_e(const gsl_spline *spline, double x, gsl_interp_accel *acc, double *y)double gsl_spline_eval_deriv(const gsl_spline *spline, double x, gsl_interp_accel *acc)
int gsl_spline_eval_deriv_e(const gsl_spline *spline, double x, gsl_interp_accel *acc, double *d)double gsl_spline_eval_deriv2(const gsl_spline *spline, double x, gsl_interp_accel *acc)
int gsl_spline_eval_deriv2_e(const gsl_spline *spline, double x, gsl_interp_accel *acc, double *d2)
double gsl_spline_eval_integ(const gsl_spline *spline, double a, double b, gsl_interp_accel *acc)

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int gsl_spline_eval_integ_e(const gsl_spline *spline, double a, double b, gsl_interp_accel *acc, double*result)

30.7 1D Interpolation Example Programs
The following program demonstrates the use of the interpolation and spline functions. It computes a cubic splineinterpolation of the 10-point dataset (xi , yi ) where xi = i + sin(i)/2 and yi = i + cos(i2 ) for i = 0 . . . 9.
#include <stdlib.h>#include <stdio.h>#include <math.h>#include <gsl/gsl_errno.h>#include <gsl/gsl_spline.h>intmain (void){
int i;
double xi, yi, x[10], y[10];printf ("#m=0,S=17\n");for (i = 0; i < 10; i++){x[i] = i + 0.5 * sin (i);y[i] = i + cos (i * i);printf ("%g %g\n", x[i], y[i]);}printf ("#m=1,S=0\n");{gsl_interp_accel *acc
= gsl_interp_accel_alloc ();gsl_spline *spline= gsl_spline_alloc (gsl_interp_cspline, 10);
gsl_spline_init (spline, x, y, 10);for (xi = x[0]; xi < x[9]; xi += 0.01){
yi = gsl_spline_eval (spline, xi, acc);
printf ("%g %g\n", xi, yi);}gsl_spline_free (spline);
gsl_interp_accel_free (acc);}
return 0;}
The output is designed to be used with the GNU plotutils graph program:

30.7. 1D Interpolation Example Programs399

GNU Scientific Library, Release 2.7$ ./a.out > interp.dat$ graph -T ps < interp.dat > interp.ps

Fig. 30.1: Cubic spline interpolation
Fig. 30.1 shows a smooth interpolation of the original points. The interpolation method can be changed simply byvarying the first argument of gsl_spline_alloc().
The next program demonstrates a periodic cubic spline with 4 data points. Note that the first and last points must be
supplied with the same y-value for a periodic spline.
#include <stdlib.h>#include <stdio.h>#include <math.h>#include <gsl/gsl_errno.h>#include <gsl/gsl_spline.h>intmain (void){
int N = 4;
double x[4] = {0.00, 0.10, 0.27, 0.30};
double y[4] = {0.15, 0.70, -0.10, 0.15};/* Note: y[0] == y[3] for periodic data */gsl_interp_accel *acc = gsl_interp_accel_alloc ();const gsl_interp_type *t = gsl_interp_cspline_periodic;
gsl_spline *spline = gsl_spline_alloc (t, N);
int i; double xi, yi;(continues on next page)

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printf ("#m=0,S=5\n");for (i = 0; i < N; i++){printf ("%g %g\n", x[i], y[i]);}printf ("#m=1,S=0\n");gsl_spline_init (spline, x, y, N);
for (i = 0; i <= 100; i++){xi = (1 - i / 100.0) * x[0] + (i / 100.0) * x[N-1];
yi = gsl_spline_eval (spline, xi, acc);
printf ("%g %g\n", xi, yi);}gsl_spline_free (spline);
gsl_interp_accel_free (acc);
return 0;}
The output can be plotted with GNU graph:$ ./a.out > interp.dat$ graph -T ps < interp.dat > interp.ps

Fig. 30.2: Periodic cubic spline interpolation
Fig. 30.2 shows a periodic interpolation of the original points. The slope of the fitted curve is the same at the beginningand end of the data, and the second derivative is also.
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The next program illustrates the difference between the cubic spline, Akima, and Steffen interpolation types on a difficult
dataset.
#include <stdio.h>#include <stdlib.h>#include <math.h>#include <gsl/gsl_math.h>#include <gsl/gsl_spline.h>intmain(void){
size_t i;const size_t N = 9;/* this dataset is taken from* J. M. Hyman, Accurate Monotonicity preserving cubic interpolation,* SIAM J. Sci. Stat. Comput. 4, 4, 1983. */const double x[] = { 7.99, 8.09, 8.19, 8.7, 9.2,
10.0, 12.0, 15.0, 20.0 };const double y[] = { 0.0, 2.76429e-5, 4.37498e-2,
0.169183, 0.469428, 0.943740,
0.998636, 0.999919, 0.999994 };gsl_interp_accel *acc = gsl_interp_accel_alloc();
gsl_spline *spline_cubic = gsl_spline_alloc(gsl_interp_cspline, N);
gsl_spline *spline_akima = gsl_spline_alloc(gsl_interp_akima, N);
gsl_spline *spline_steffen = gsl_spline_alloc(gsl_interp_steffen, N);
gsl_spline_init(spline_cubic, x, y, N);
gsl_spline_init(spline_akima, x, y, N);
gsl_spline_init(spline_steffen, x, y, N);
for (i = 0; i < N; ++i)
printf("%g %g\n", x[i], y[i]);printf("\n\n");for (i = 0; i <= 100; ++i){
double xi = (1 - i / 100.0) * x[0] + (i / 100.0) * x[N-1];
double yi_cubic = gsl_spline_eval(spline_cubic, xi, acc);
double yi_akima = gsl_spline_eval(spline_akima, xi, acc);
double yi_steffen = gsl_spline_eval(spline_steffen, xi, acc);
printf("%g %g %g %g\n", xi, yi_cubic, yi_akima, yi_steffen);}
gsl_spline_free(spline_cubic);
gsl_spline_free(spline_akima);
gsl_spline_free(spline_steffen);
gsl_interp_accel_free(acc);(continues on next page)402

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return 0;}

Fig. 30.3: Comparison of different 1D interpolation methods
The output is shown in Fig. 30.3. The cubic method exhibits a local maxima between the 6th and 7th data points andcontinues oscillating for the rest of the data. Akima also shows a local maxima but recovers and follows the data wellafter the 7th grid point. Steffen preserves monotonicity in all intervals and does not exhibit oscillations, at the expenseof having a discontinuous second derivative.

30.8 Introduction to 2D Interpolation
Given a set of x coordinates x1 , ..., xm and a set of y coordinates y1 , ..., yn , each in increasing order, plus a set offunction values zij for each grid point (xi , yj ), the routines described in this section compute a continuous interpolation
function z(x, y) such that z(xi , yj ) = zij .

30.9 2D Interpolation Functions
The interpolation function for a given dataset is stored in a gsl_interp2d object. These are created by the following
functions.
type gsl_interp2d
Workspace for 2D interpolation
gsl_interp2d *gsl_interp2d_alloc(const gsl_interp2d_type *T, const size_t xsize, const size_t ysize)
This function returns a pointer to a newly allocated interpolation object of type T for xsize grid points in the x
direction and ysize grid points in the y direction.
int gsl_interp2d_init(gsl_interp2d *interp, const double xa[], const double ya[], const double za[], const size_t
xsize, const size_t ysize)
This function initializes the interpolation object interp for the data (xa, ya, za) where xa and ya are arrays
30.8. Introduction to 2D Interpolation

403

GNU Scientific Library, Release 2.7of the x and y grid points of size xsize and ysize respectively, and za is an array of function values of size
xsize * ysize. The interpolation object (gsl_interp2d) does not save the data arrays xa, ya, and za andonly stores the static state computed from the data. The xa and ya data arrays are always assumed to be strictly
ordered, with increasing x, y values; the behavior for other arrangements is not defined.
void gsl_interp2d_free(gsl_interp2d *interp)
This function frees the interpolation object interp.

30.10 2D Interpolation Grids
The 2D interpolation routines access the function values zij with the following ordering:zij = za[j ∗ xsize + i]with i = 0, ..., xsize − 1 and j = 0, ..., ysize − 1. However, for ease of use, the following functions are provided to
add and retrieve elements from the function grid without requiring knowledge of the internal ordering.
int gsl_interp2d_set(const gsl_interp2d *interp, double za[], const size_t i, const size_t j, const double z)
This function sets the value zij for grid point (i, j) of the array za to z.
double gsl_interp2d_get(const gsl_interp2d *interp, const double za[], const size_t i, const size_t j)
This function returns the value zij for grid point (i, j) stored in the array za.
size_t gsl_interp2d_idx(const gsl_interp2d *interp, const size_t i, const size_t j)
This function returns the index corresponding to the grid point (i, j). The index is given by j ∗ xsize + i.

30.11 2D Interpolation Types
type gsl_interp2d_type
The interpolation library provides the following 2D interpolation types:
gsl_interp2d_type *gsl_interp2d_bilinear
Bilinear interpolation. This interpolation method does not require any additional memory.
gsl_interp2d_type *gsl_interp2d_bicubic
Bicubic interpolation.
const char *gsl_interp2d_name(const gsl_interp2d *interp)
This function returns the name of the interpolation type used by interp. For example:
printf ("interp uses '%s' interpolation.\n", gsl_interp2d_name (interp));would print something like:interp uses 'bilinear' interpolation.
unsigned int gsl_interp2d_min_size(const gsl_interp2d *interp)
unsigned int gsl_interp2d_type_min_size(const gsl_interp2d_type *T)
These functions return the minimum number of points required by the interpolation object interp or interpolation type T. For example, bicubic interpolation requires a minimum of 4 points.

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30.12 2D Evaluation of Interpolating Functions
double gsl_interp2d_eval(const gsl_interp2d *interp, const double xa[], const double ya[], const double za[],const double x, const double y, gsl_interp_accel *xacc, gsl_interp_accel *yacc)
int gsl_interp2d_eval_e(const gsl_interp2d *interp, const double xa[], const double ya[], const double za[], const
double x, const double y, gsl_interp_accel *xacc, gsl_interp_accel *yacc, double *z)
These functions return the interpolated value of z for a given point (x, y), using the interpolation object interp,
data arrays xa, ya, and za and the accelerators xacc and yacc. When x is outside the range of xa or y is outside
the range of ya, the error code GSL_EDOM is returned.
double gsl_interp2d_eval_extrap(const gsl_interp2d *interp, const double xa[], const double ya[], const double
za[], const double x, const double y, gsl_interp_accel *xacc,
gsl_interp_accel *yacc)
int gsl_interp2d_eval_extrap_e(const gsl_interp2d *interp, const double xa[], const double ya[], const double
za[], const double x, const double y, gsl_interp_accel *xacc, gsl_interp_accel*yacc, double *z)
These functions return the interpolated value of z for a given point (x, y), using the interpolation object interp,
data arrays xa, ya, and za and the accelerators xacc and yacc. The functions perform no bounds checking, sowhen x is outside the range of xa or y is outside the range of ya, extrapolation is performed.
double gsl_interp2d_eval_deriv_x(const gsl_interp2d *interp, const double xa[], const double ya[], const
double za[], const double x, const double y, gsl_interp_accel *xacc,
gsl_interp_accel *yacc)
int gsl_interp2d_eval_deriv_x_e(const gsl_interp2d *interp, const double xa[], const double ya[], const double
za[], const double x, const double y, gsl_interp_accel *xacc,
gsl_interp_accel *yacc, double *d)These functions return the interpolated value d = ∂z/∂x for a given point (x, y), using the interpolation object
interp, data arrays xa, ya, and za and the accelerators xacc and yacc. When x is outside the range of xa or
y is outside the range of ya, the error code GSL_EDOM is returned.
double gsl_interp2d_eval_deriv_y(const gsl_interp2d *interp, const double xa[], const double ya[], const
double za[], const double x, const double y, gsl_interp_accel *xacc,
gsl_interp_accel *yacc)
int gsl_interp2d_eval_deriv_y_e(const gsl_interp2d *interp, const double xa[], const double ya[], const double
za[], const double x, const double y, gsl_interp_accel *xacc,
gsl_interp_accel *yacc, double *d)These functions return the interpolated value d = ∂z/∂y for a given point (x, y), using the interpolation object
interp, data arrays xa, ya, and za and the accelerators xacc and yacc. When x is outside the range of xa or
y is outside the range of ya, the error code GSL_EDOM is returned.
double gsl_interp2d_eval_deriv_xx(const gsl_interp2d *interp, const double xa[], const double ya[], const
double za[], const double x, const double y, gsl_interp_accel *xacc,
gsl_interp_accel *yacc)
int gsl_interp2d_eval_deriv_xx_e(const gsl_interp2d *interp, const double xa[], const double ya[], const
double za[], const double x, const double y, gsl_interp_accel *xacc,
gsl_interp_accel *yacc, double *d)These functions return the interpolated value d = ∂ 2 z/∂x2 for a given point (x, y), using the interpolation object
interp, data arrays xa, ya, and za and the accelerators xacc and yacc. When x is outside the range of xa or
y is outside the range of ya, the error code GSL_EDOM is returned.
30.12. 2D Evaluation of Interpolating Functions

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double gsl_interp2d_eval_deriv_yy(const gsl_interp2d *interp, const double xa[], const double ya[], const
double za[], const double x, const double y, gsl_interp_accel *xacc,
gsl_interp_accel *yacc)
int gsl_interp2d_eval_deriv_yy_e(const gsl_interp2d *interp, const double xa[], const double ya[], const
double za[], const double x, const double y, gsl_interp_accel *xacc,
gsl_interp_accel *yacc, double *d)These functions return the interpolated value d = ∂ 2 z/∂y 2 for a given point (x, y), using the interpolation object
interp, data arrays xa, ya, and za and the accelerators xacc and yacc. When x is outside the range of xa or
y is outside the range of ya, the error code GSL_EDOM is returned.
double gsl_interp2d_eval_deriv_xy(const gsl_interp2d *interp, const double xa[], const double ya[], const
double za[], const double x, const double y, gsl_interp_accel *xacc,
gsl_interp_accel *yacc)
int gsl_interp2d_eval_deriv_xy_e(const gsl_interp2d *interp, const double xa[], const double ya[], const
double za[], const double x, const double y, gsl_interp_accel *xacc,
gsl_interp_accel *yacc, double *d)These functions return the interpolated value d = ∂ 2 z/∂x∂y for a given point (x, y), using the interpolation
object interp, data arrays xa, ya, and za and the accelerators xacc and yacc. When x is outside the range ofxa or y is outside the range of ya, the error code GSL_EDOM is returned.

30.13 2D Higher-level Interface
The functions described in the previous sections required the user to supply pointers to the x, y, and z arrays on eachcall. The following functions are equivalent to the corresponding gsl_interp2d functions but maintain a copy of thisdata in the gsl_spline2d object. This removes the need to pass xa, ya, and za as arguments on each evaluation.
These functions are defined in the header file gsl_spline2d.h.
type gsl_spline2d
This workspace provides a higher level interface for the gsl_interp2d object
gsl_spline2d *gsl_spline2d_alloc(const gsl_interp2d_type *T, size_t xsize, size_t ysize)
int gsl_spline2d_init(gsl_spline2d *spline, const double xa[], const double ya[], const double za[], size_t xsize,
size_t ysize)
void gsl_spline2d_free(gsl_spline2d *spline)
const char *gsl_spline2d_name(const gsl_spline2d *spline)
unsigned int gsl_spline2d_min_size(const gsl_spline2d *spline)
double gsl_spline2d_eval(const gsl_spline2d *spline, const double x, const double y, gsl_interp_accel *xacc,
gsl_interp_accel *yacc)
int gsl_spline2d_eval_e(const gsl_spline2d *spline, const double x, const double y, gsl_interp_accel *xacc,
gsl_interp_accel *yacc, double *z)

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double gsl_spline2d_eval_extrap(const gsl_spline2d *spline, const double x, const double y, gsl_interp_accel*xacc, gsl_interp_accel *yacc)
int gsl_spline2d_eval_extrap_e(const gsl_spline2d *spline, const double x, const double y, gsl_interp_accel*xacc, gsl_interp_accel *yacc, double *z)
double gsl_spline2d_eval_deriv_x(const gsl_spline2d *spline, const double x, const double y, gsl_interp_accel*xacc, gsl_interp_accel *yacc)
int gsl_spline2d_eval_deriv_x_e(const gsl_spline2d *spline, const double x, const double y, gsl_interp_accel*xacc, gsl_interp_accel *yacc, double *d)double gsl_spline2d_eval_deriv_y(const gsl_spline2d *spline, const double x, const double y, gsl_interp_accel*xacc, gsl_interp_accel *yacc)
int gsl_spline2d_eval_deriv_y_e(const gsl_spline2d *spline, const double x, const double y, gsl_interp_accel*xacc, gsl_interp_accel *yacc, double *d)double gsl_spline2d_eval_deriv_xx(const gsl_spline2d *spline, const double x, const double y,
gsl_interp_accel *xacc, gsl_interp_accel *yacc)
int gsl_spline2d_eval_deriv_xx_e(const gsl_spline2d *spline, const double x, const double y, gsl_interp_accel*xacc, gsl_interp_accel *yacc, double *d)double gsl_spline2d_eval_deriv_yy(const gsl_spline2d *spline, const double x, const double y,
gsl_interp_accel *xacc, gsl_interp_accel *yacc)
int gsl_spline2d_eval_deriv_yy_e(const gsl_spline2d *spline, const double x, const double y, gsl_interp_accel*xacc, gsl_interp_accel *yacc, double *d)double gsl_spline2d_eval_deriv_xy(const gsl_spline2d *spline, const double x, const double y,
gsl_interp_accel *xacc, gsl_interp_accel *yacc)
int gsl_spline2d_eval_deriv_xy_e(const gsl_spline2d *spline, const double x, const double y, gsl_interp_accel*xacc, gsl_interp_accel *yacc, double *d)
int gsl_spline2d_set(const gsl_spline2d *spline, double za[], const size_t i, const size_t j, const double z)
double gsl_spline2d_get(const gsl_spline2d *spline, const double za[], const size_t i, const size_t j)
This function returns the value zij for grid point (i, j) stored in the array za.

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30.14 2D Interpolation Example programs
The following example performs bilinear interpolation on the unit square, using z values of (0, 1, 0.5, 1) going clockwise around the square.
#include <stdio.h>#include <stdlib.h>#include <gsl/gsl_math.h>#include <gsl/gsl_interp2d.h>#include <gsl/gsl_spline2d.h>intmain(){
const gsl_interp2d_type *T = gsl_interp2d_bilinear;const size_t N = 100;/* number of points to interpolate */const double xa[] = { 0.0, 1.0 }; /* define unit square */const double ya[] = { 0.0, 1.0 };const size_t nx = sizeof(xa) / sizeof(double); /* x grid points */const size_t ny = sizeof(ya) / sizeof(double); /* y grid points */double *za = malloc(nx * ny * sizeof(double));gsl_spline2d *spline = gsl_spline2d_alloc(T, nx, ny);gsl_interp_accel *xacc = gsl_interp_accel_alloc();
gsl_interp_accel *yacc = gsl_interp_accel_alloc();size_t i, j;/* set z grid values */gsl_spline2d_set(spline, za, 0, 0, 0.0);
gsl_spline2d_set(spline, za, 0, 1, 1.0);
gsl_spline2d_set(spline, za, 1, 1, 0.5);
gsl_spline2d_set(spline, za, 1, 0, 1.0);/* initialize interpolation */gsl_spline2d_init(spline, xa, ya, za, nx, ny);/* interpolate N values in x and y and print out grid for plotting */for (i = 0; i < N; ++i){
double xi = i / (N - 1.0);
for (j = 0; j < N; ++j){
double yj = j / (N - 1.0);
double zij = gsl_spline2d_eval(spline, xi, yj, xacc, yacc);
printf("%f %f %f \n", xi, yj, zij);}
printf("\n");}gsl_spline2d_free(spline);
gsl_interp_accel_free(xacc);(continues on next page)

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gsl_interp_accel_free(yacc);
free(za);
return 0;}
The results of the interpolation are shown in Fig. 30.4, where the corners are labeled with their fixed z values.

Fig. 30.4: 2D interpolation example

30.15 References and Further Reading
Descriptions of the interpolation algorithms and further references can be found in the following publications:• C.W. Ueberhuber, Numerical Computation (Volume 1), Chapter 9 “Interpolation”, Springer (1997), ISBN 3540-62058-3.
• D.M. Young, R.T. Gregory, A Survey of Numerical Mathematics (Volume 1), Chapter 6.8, Dover (1988), ISBN
0-486-65691-8.
• M. Steffen, A simple method for monotonic interpolation in one dimension, Astron. Astrophys. 239, 443-450,
1990.

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CHAPTERTHIRTYONENUMERICAL DIFFERENTIATION

The functions described in this chapter compute numerical derivatives by finite differencing. An adaptive algorithm isused to find the best choice of finite difference and to estimate the error in the derivative. These functions are declaredin the header file gsl_deriv.h.

31.1 Functions
int gsl_deriv_central(const gsl_function *f, double x, double h, double *result, double *abserr)
This function computes the numerical derivative of the function f at the point x using an adaptive central difference algorithm with a step-size of h . The derivative is returned in result and an estimate of its absolute error
is returned in abserr.
The initial value of h is used to estimate an optimal step-size, based on the scaling of the truncation error andround-off error in the derivative calculation. The derivative is computed using a 5-point rule for equally spaced
abscissae at x − h, x − h/2, x, x + h/2, x + h, with an error estimate taken from the difference between the
5-point rule and the corresponding 3-point rule x − h, x, x + h. Note that the value of the function at x does notcontribute to the derivative calculation, so only 4-points are actually used.
int gsl_deriv_forward(const gsl_function *f, double x, double h, double *result, double *abserr)
This function computes the numerical derivative of the function f at the point x using an adaptive forward
difference algorithm with a step-size of h . The function is evaluated only at points greater than x, and never at
x itself. The derivative is returned in result and an estimate of its absolute error is returned in abserr. Thisfunction should be used if f (x) has a discontinuity at x, or is undefined for values less than x.
The initial value of h is used to estimate an optimal step-size, based on the scaling of the truncation error andround-off error in the derivative calculation. The derivative at x is computed using an “open” 4-point rule for
equally spaced abscissae at x + h/4, x + h/2, x + 3h/4, x + h, with an error estimate taken from the difference
between the 4-point rule and the corresponding 2-point rule x + h/2, x + h.
int gsl_deriv_backward(const gsl_function *f, double x, double h, double *result, double *abserr)
This function computes the numerical derivative of the function f at the point x using an adaptive backward
difference algorithm with a step-size of h . The function is evaluated only at points less than x, and never at x
itself. The derivative is returned in result and an estimate of its absolute error is returned in abserr. Thisfunction should be used if f (x) has a discontinuity at x, or is undefined for values greater than x.
This function is equivalent to calling gsl_deriv_forward() with a negative step-size.

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31.2 Examples
The following code estimates the derivative of the function f (x) = x3/2 at x = 2 and at x = 0. The function f (x) is
undefined for x < 0 so the derivative at x = 0 is computed using gsl_deriv_forward().
#include <stdio.h>#include <gsl/gsl_math.h>#include <gsl/gsl_deriv.h>double f (double x, void * params){(void)(params); /* avoid unused parameter warning */return pow (x, 1.5);}
int
main (void){
gsl_function F;
double result, abserr;
F.function = &f;F.params = 0;
printf ("f(x) = x^(3/2)\n");gsl_deriv_central (&F, 2.0, 1e-8, &result, &abserr);
printf ("x = 2.0\n");printf ("f'(x) = %.10f +/- %.10f \n", result, abserr);
printf ("exact = %.10f \n\n", 1.5 * sqrt(2.0));
gsl_deriv_forward (&F, 0.0, 1e-8, &result, &abserr);
printf ("x = 0.0\n");printf ("f'(x) = %.10f +/- %.10f \n", result, abserr);
printf ("exact = %.10f \n", 0.0);
return 0;}
Here is the output of the program,f(x) = x^(3/2)
x = 2.0
f'(x) = 2.1213203120 +/- 0.0000005006
exact = 2.1213203436
x = 0.0
f'(x) = 0.0000000160 +/- 0.0000000339
exact = 0.0000000000

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31.3 References and Further Reading
The algorithms used by these functions are described in the following sources:• Abramowitz and Stegun, Handbook of Mathematical Functions, Section 25.3.4, and Table 25.5 (Coefficients forDifferentiation).
• S.D. Conte and Carl de Boor, Elementary Numerical Analysis: An Algorithmic Approach, McGraw-Hill, 1972.

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CHAPTERTHIRTYTWO

CHEBYSHEV APPROXIMATIONSThis chapter describes routines for computingP
Chebyshev approximations to univariate functions. A Chebyshev approximation is a truncation of the series f (x) =cn Tn (x), where the Chebyshev polynomials Tn√(x) = cos(n arccos x)
provide an orthogonal basis of polynomials on the interval [−1, 1] with the weight function 1/ 1 − x2 . The first few
Chebyshev polynomials are, T0 (x) = 1, T1 (x) = x, T2 (x) = 2x2 − 1. For further information see Abramowitz &Stegun, Chapter 22.
The functions described in this chapter are declared in the header file gsl_chebyshev.h.

32.1 Definitions
type gsl_cheb_series
A Chebyshev series is stored using the following structure:
typedef struct{
double * c;/* coefficients c[0] .. c[order] */int order;/* order of expansion*/
double a;/* lower interval point*/
double b;/* upper interval point*/...
} gsl_cheb_series
The approximation is made over the range [a, b] using order + 1 terms, including the coefficient c[0]. The series iscomputed using the following convention,
c0 X
f (x) =+
cn Tn (x)2n=1which is needed when accessing the coefficients directly.

32.2 Creation and Calculation of Chebyshev Series
gsl_cheb_series *gsl_cheb_alloc(const size_t n)This function allocates space for a Chebyshev series of order n and returns a pointer to a new gsl_cheb_series
struct.
void gsl_cheb_free(gsl_cheb_series *cs)
This function frees a previously allocated Chebyshev series cs.

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int gsl_cheb_init(gsl_cheb_series *cs, const gsl_function *f, const double a, const double b)
This function computes the Chebyshev approximation cs for the function f over the range (a, b) to the previously
specified order. The computation of the Chebyshev approximation is an O(n2 ) process, and requires n function
evaluations.

32.3 Auxiliary Functions
The following functions provide information about an existing Chebyshev series.
size_t gsl_cheb_order(const gsl_cheb_series *cs)
This function returns the order of Chebyshev series cs.
size_t gsl_cheb_size(const gsl_cheb_series *cs)
double *gsl_cheb_coeffs(const gsl_cheb_series *cs)
These functions return the size of the Chebyshev coefficient array c[] and a pointer to its location in memory
for the Chebyshev series cs.

32.4 Chebyshev Series Evaluation
double gsl_cheb_eval(const gsl_cheb_series *cs, double x)
This function evaluates the Chebyshev series cs at a given point x.
int gsl_cheb_eval_err(const gsl_cheb_series *cs, const double x, double *result, double *abserr)
This function computes the Chebyshev series cs at a given point x, estimating both the series result and its
absolute error abserr. The error estimate is made from the first neglected term in the series.
double gsl_cheb_eval_n(const gsl_cheb_series *cs, size_t order, double x)
This function evaluates the Chebyshev series cs at a given point x, to (at most) the given order order.
int gsl_cheb_eval_n_err(const gsl_cheb_series *cs, const size_t order, const double x, double *result, double*abserr)
This function evaluates a Chebyshev series cs at a given point x, estimating both the series result and its
absolute error abserr, to (at most) the given order order. The error estimate is made from the first neglected
term in the series.

32.5 Derivatives and Integrals
The following functions allow a Chebyshev series to be differentiated or integrated, producing a new Chebyshev series.
Note that the error estimate produced by evaluating the derivative series will be underestimated due to the contributionof higher order terms being neglected.
int gsl_cheb_calc_deriv(gsl_cheb_series *deriv, const gsl_cheb_series *cs)
This function computes the derivative of the series cs, storing the derivative coefficients in the previously allocated deriv. The two series cs and deriv must have been allocated with the same order.
int gsl_cheb_calc_integ(gsl_cheb_series *integ, const gsl_cheb_series *cs)
This function computes the integral of the series cs, storing the integral coefficients in the previously allocatedinteg. The two series cs and integ must have been allocated with the same order. The lower limit of theintegration is taken to be the left hand end of the range a.

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32.6 Examples
The following example program computes Chebyshev approximations to a step function. This is an extremely difficult
approximation to make, due to the discontinuity, and was chosen as an example where approximation error is visible.
For smooth functions the Chebyshev approximation converges extremely rapidly and errors would not be visible.
#include <stdio.h>#include <gsl/gsl_math.h>#include <gsl/gsl_chebyshev.h>doublef (double x, void *p){(void)(p); /* avoid unused parameter warning */if (x < 0.5)
return 0.25;
else
return 0.75;}
int
main (void){
int i, n = 10000;
gsl_cheb_series *cs = gsl_cheb_alloc (40);gsl_function F;
F.function = f;F.params = 0;
gsl_cheb_init (cs, &F, 0.0, 1.0);
for (i = 0; i < n; i++){double x = i / (double)n;double r10 = gsl_cheb_eval_n (cs, 10, x);
double r40 = gsl_cheb_eval (cs, x);
printf ("%g %g %g %g\n",x, GSL_FN_EVAL (&F, x), r10, r40);}
gsl_cheb_free (cs);return 0;}
Fig. 32.1 shows output from the program with the original function, 10-th order approximation and 40-th order approximation, all sampled at intervals of 0.001 in x.

32.6. Examples

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Fig. 32.1: Chebyshev approximations to a step function

32.7 References and Further Reading
The following paper describes the use of Chebyshev series,• R. Broucke, “Ten Subroutines for the Manipulation of Chebyshev Series [C1] (Algorithm 446)”. Communications of the ACM 16(4), 254–256 (1973)

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CHAPTERTHIRTYTHREE

SERIES ACCELERATION

The functions described in this chapter accelerate the convergence of a series using the Levin u-transform. This method
takes a small number of terms from the start of a series and uses a systematic approximation to compute an extrapolated
value and an estimate of its error. The u-transform works for both convergent and divergent series, including asymptotic
series.
These functions are declared in the header file gsl_sum.h.

33.1 Acceleration functions
The following functions compute the full Levin u-transform of a series with its error estimate. The error estimate iscomputed by propagating rounding errors from each term through to the final extrapolation.
These functions are intended for summing analytic series where each term is known to high accuracy, and the rounding
errors are assumed to originate from finite precision. They are taken to be relative errors of order GSL_DBL_EPSILON
for each term.
The calculation of the error in the extrapolated value is an O(N 2 ) process, which is expensive in time and memory. A
faster but less reliable method which estimates the error from the convergence of the extrapolated value is describedin the next section. For the method described here a full table of intermediate values and derivatives through to O(N )must be computed and stored, but this does give a reliable error estimate.
type gsl_sum_levin_u_workspace
Workspace for a Leven u-transform.
gsl_sum_levin_u_workspace *gsl_sum_levin_u_alloc(size_t n)This function allocates a workspace for a Levin u-transform of n terms. The size of the workspace is O(2n2 +3n).
void gsl_sum_levin_u_free(gsl_sum_levin_u_workspace *w)
This function frees the memory associated with the workspace w.
int gsl_sum_levin_u_accel(const double *array, size_t array_size, gsl_sum_levin_u_workspace *w, double*sum_accel, double *abserr)
This function takes the terms of a series in array of size array_size and computes the extrapolated limit of
the series using a Levin u-transform. Additional working space must be provided in w. The extrapolated sum isstored in sum_accel, with an estimate of the absolute error stored in abserr. The actual term-by-term sum isreturned in w->sum_plain. The algorithm calculates the truncation error (the difference between two successive
extrapolations) and round-off error (propagated from the individual terms) to choose an optimal number of terms
for the extrapolation. All the terms of the series passed in through array should be non-zero.

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33.2 Acceleration functions without error estimation
The functions described in this section compute the Levin u-transform of series and attempt to estimate the error fromthe “truncation error” in the extrapolation, the difference between the final two approximations. Using this method
avoids the need to compute an intermediate table of derivatives because the error is estimated from the behavior of theextrapolated value itself. Consequently this algorithm is an O(N ) process and only requires O(N ) terms of storage.
If the series converges sufficiently fast then this procedure can be acceptable. It is appropriate to use this method
when there is a need to compute many extrapolations of series with similar convergence properties at high-speed. For
example, when numerically integrating a function defined by a parameterized series where the parameter varies only
slightly. A reliable error estimate should be computed first using the full algorithm described above in order to verify
the consistency of the results.
type gsl_sum_levin_utrunc_workspace
Workspace for a Levin u-transform without error estimation
gsl_sum_levin_utrunc_workspace *gsl_sum_levin_utrunc_alloc(size_t n)This function allocates a workspace for a Levin u-transform of n terms, without error estimation. The size of
the workspace is O(3n).
void gsl_sum_levin_utrunc_free(gsl_sum_levin_utrunc_workspace *w)
This function frees the memory associated with the workspace w.
int gsl_sum_levin_utrunc_accel(const double *array, size_t array_size, gsl_sum_levin_utrunc_workspace *w,
double *sum_accel, double *abserr_trunc)
This function takes the terms of a series in array of size array_size and computes the extrapolated limit of
the series using a Levin u-transform. Additional working space must be provided in w. The extrapolated sum
is stored in sum_accel. The actual term-by-term sum is returned in w->sum_plain. The algorithm terminates
when the difference between two successive extrapolations reaches a minimum or is sufficiently small. The
difference between these two values is used as estimate of the error and is stored in abserr_trunc. To improve
the reliability of the algorithm the extrapolated values are replaced by moving averages when calculating thetruncation error, smoothing out any fluctuations.

33.3 Examples
The following code calculates an estimate of ζ(2) = π 2 /6 using the series,ζ(2) = 1 + 1/22 + 1/32 + 1/42 + . . .
After N terms the error in the sum is O(1/N ), making direct summation of the series converge slowly.
#include <stdio.h>#include <gsl/gsl_math.h>#include <gsl/gsl_sum.h>#define N 20
intmain (void){
double t[N];double sum_accel, err;
double sum = 0;
int n;(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)gsl_sum_levin_u_workspace * w= gsl_sum_levin_u_alloc (N);const double zeta_2 = M_PI * M_PI / 6.0;/* terms for zeta(2) = \sum_{n=1}^{\infty} 1/n^2 */for (n = 0; n < N; n++){double np1 = n + 1.0;t[n] = 1.0 / (np1 * np1);
sum += t[n];}gsl_sum_levin_u_accel (t, N, w, &sum_accel, &err);
printf ("term-by-term sum = % .16f using %d terms\n",sum, N);
printf ("term-by-term sum = % .16f using %zu terms\n",w->sum_plain, w->terms_used);printf ("exact value= % .16f \n", zeta_2);
printf ("accelerated sum = % .16f using %zu terms\n",sum_accel, w->terms_used);printf ("estimated error = % .16f \n", err);
printf ("actual error= % .16f \n",sum_accel - zeta_2);
gsl_sum_levin_u_free (w);return 0;}
The output below shows that the Levin u-transform is able to obtain an estimate of the sum to 1 part in 1010 using thefirst eleven terms of the series. The error estimate returned by the function is also accurate, giving the correct numberof significant digits.
term-by-term sum =term-by-term sum =exact value=
accelerated sum =estimated error =actual error=

1.5961632439130233 using 20 terms
1.5759958390005426 using 13 terms
1.6449340668482264
1.6449340669228176 using 13 terms
0.0000000000888360
0.0000000000745912

Note that a direct summation of this series would require 1010 terms to achieve the same precision as the accelerated
sum does in 13 terms.

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33.4 References and Further Reading
The algorithms used by these functions are described in the following papers,• T. Fessler, W.F. Ford, D.A. Smith, HURRY: An acceleration algorithm for scalar sequences and series ACM
Transactions on Mathematical Software, 9(3):346–354, 1983. and Algorithm 602 9(3):355–357, 1983.
The theory of the u-transform was presented by Levin,• D. Levin, Development of Non-Linear Transformations for Improving Convergence of Sequences, Intern.: J.:
Computer Math. B3:371–388, 1973.
A review paper on the Levin Transform is available online,• Herbert H. H. Homeier, Scalar Levin-Type Sequence Transformations, http://arxiv.org/abs/math/0005209

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CHAPTERTHIRTYFOURWAVELET TRANSFORMS

This chapter describes functions for performing Discrete Wavelet Transforms (DWTs). The library includes wavelets
for real data in both one and two dimensions. The wavelet functions are declared in the header files gsl_wavelet.hand gsl_wavelet2d.h.

34.1 Definitions
The continuous wavelet transform and its inverse are defined by the relations,Z ∞∗w(s, τ ) =f (t) ∗ ψs,τ
(t)dt
−∞

and,
Z ∞
f (t) =Z ∞w(s, τ ) ∗ ψs,τ (t)dτ

ds
0

−∞

where the basis functions ψs,τ are obtained by scaling and translation from a single function, referred to as the motherwavelet.
The discrete version of the wavelet transform acts on equally-spaced samples, with fixed scaling and translation steps(s, τ ). The frequency and time axes are sampled dyadically on scales of 2j through a level parameter j.
The resulting family of functions {ψj,n } constitutes an orthonormal basis for square-integrable signals. The discrete
wavelet transform is an O(N ) algorithm, and is also referred to as the fast wavelet transform.

34.2 Initialization
type gsl_waveletThis structure contains the filter coefficients defining the wavelet and any associated offset parameters.
gsl_wavelet *gsl_wavelet_alloc(const gsl_wavelet_type *T, size_t k)
This function allocates and initializes a wavelet object of type T. The parameter k selects the specific memberof the wavelet family. A null pointer is returned if insufficient memory is available or if a unsupported member
is selected.
The following wavelet types are implemented:
type gsl_wavelet_type
gsl_wavelet_type *gsl_wavelet_daubechies423

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gsl_wavelet_type *gsl_wavelet_daubechies_centered
This is the Daubechies wavelet family of maximum phase with k/2 vanishing moments. The implemented
wavelets are k = 4, 6, . . . , 20, with k even.
gsl_wavelet_type *gsl_wavelet_haar
gsl_wavelet_type *gsl_wavelet_haar_centered
This is the Haar wavelet. The only valid choice of k for the Haar wavelet is k = 2.
gsl_wavelet_type *gsl_wavelet_bspline
gsl_wavelet_type *gsl_wavelet_bspline_centered
This is the biorthogonal B-spline wavelet family of order (i, j). The implemented values of k = 100 ∗ i + j
are 103, 105, 202, 204, 206, 208, 301, 303, 305 307, 309.
The centered forms of the wavelets align the coefficients of the various sub-bands on edges. Thus the resulting visualization of the coefficients of the wavelet transform in the phase plane is easier to understand.
const char *gsl_wavelet_name(const gsl_wavelet *w)
This function returns a pointer to the name of the wavelet family for w.
void gsl_wavelet_free(gsl_wavelet *w)
This function frees the wavelet object w.
type gsl_wavelet_workspace
This structure contains scratch space of the same size as the input data and is used to hold intermediate results
during the transform.
gsl_wavelet_workspace *gsl_wavelet_workspace_alloc(size_t n)This function allocates a workspace for the discrete wavelet transform. To perform a one-dimensional transformon n elements, a workspace of size n must be provided. For two-dimensional transforms of n-by-n matrices it
is sufficient to allocate a workspace of size n, since the transform operates on individual rows and columns. A
null pointer is returned if insufficient memory is available.
void gsl_wavelet_workspace_free(gsl_wavelet_workspace *work)
This function frees the allocated workspace work.

34.3 Transform Functions
This sections describes the actual functions performing the discrete wavelet transform. Note that the transforms use
periodic boundary conditions. If the signal is not periodic in the sample length then spurious coefficients will appear
at the beginning and end of each level of the transform.

34.3.1 Wavelet transforms in one dimension
int gsl_wavelet_transform(const gsl_wavelet *w, double *data, size_t stride, size_t n, gsl_wavelet_direction dir,
gsl_wavelet_workspace *work)
int gsl_wavelet_transform_forward(const gsl_wavelet *w, double *data, size_t stride, size_t n,
gsl_wavelet_workspace *work)
int gsl_wavelet_transform_inverse(const gsl_wavelet *w, double *data, size_t stride, size_t n,
gsl_wavelet_workspace *work)
These functions compute in-place forward and inverse discrete wavelet transforms of length n with stride stride
on the array data. The length of the transform n is restricted to powers of two. For the transform version of
the function the argument dir can be either forward (+1) or backward (−1). A workspace work of length nmust be provided.
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For the forward transform, the elements of the original array are replaced by the discrete wavelet transformfi → wj,k in a packed triangular storage layout, where j is the index of the level j = 0 . . . J − 1 and k is theindex of the coefficient within each level, k = 0 . . . 2j − 1. The total number of levels is J = log2 (n). The
output data has the following form,(s−1,0 , d0,0 , d1,0 , d1,1 , d2,0 , · · · , dj,k , · · · , dJ−1,2J−1 −1 )where the first element is the smoothing coefficient s−1,0 , followed by the detail coefficients dj,k for each level
j. The backward transform inverts these coefficients to obtain the original data.
These functions return a status of GSL_SUCCESS upon successful completion. GSL_EINVAL is returned if n is
not an integer power of 2 or if insufficient workspace is provided.

34.3.2 Wavelet transforms in two dimension
The library provides functions to perform two-dimensional discrete wavelet transforms on square matrices. The matrix
dimensions must be an integer power of two. There are two possible orderings of the rows and columns in the twodimensional wavelet transform, referred to as the “standard” and “non-standard” forms.
The “standard” transform performs a complete discrete wavelet transform on the rows of the matrix, followed by aseparate complete discrete wavelet transform on the columns of the resulting row-transformed matrix. This procedureuses the same ordering as a two-dimensional Fourier transform.
The “non-standard” transform is performed in interleaved passes on the rows and columns of the matrix for each levelof the transform. The first level of the transform is applied to the matrix rows, and then to the matrix columns. This
procedure is then repeated across the rows and columns of the data for the subsequent levels of the transform, until thefull discrete wavelet transform is complete. The non-standard form of the discrete wavelet transform is typically usedin image analysis.
The functions described in this section are declared in the header file gsl_wavelet2d.h.
int gsl_wavelet2d_transform(const gsl_wavelet *w, double *data, size_t tda, size_t size1, size_t size2,
gsl_wavelet_direction dir, gsl_wavelet_workspace *work)
int gsl_wavelet2d_transform_forward(const gsl_wavelet *w, double *data, size_t tda, size_t size1, size_t size2,
gsl_wavelet_workspace *work)
int gsl_wavelet2d_transform_inverse(const gsl_wavelet *w, double *data, size_t tda, size_t size1, size_t size2,
gsl_wavelet_workspace *work)
These functions compute two-dimensional in-place forward and inverse discrete wavelet transforms in standard
form on the array data stored in row-major form with dimensions size1 and size2 and physical row lengthtda. The dimensions must be equal (square matrix) and are restricted to powers of two. For the transformversion of the function the argument dir can be either forward (+1) or backward (−1). A workspace work of
the appropriate size must be provided. On exit, the appropriate elements of the array data are replaced by their
two-dimensional wavelet transform.
The functions return a status of GSL_SUCCESS upon successful completion. GSL_EINVAL is returned if size1
and size2 are not equal and integer powers of 2, or if insufficient workspace is provided.
int gsl_wavelet2d_transform_matrix(const gsl_wavelet *w, gsl_matrix *m, gsl_wavelet_direction dir,
gsl_wavelet_workspace *work)
int gsl_wavelet2d_transform_matrix_forward(const gsl_wavelet *w, gsl_matrix *m, gsl_wavelet_workspace*work)

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int gsl_wavelet2d_transform_matrix_inverse(const gsl_wavelet *w, gsl_matrix *m, gsl_wavelet_workspace*work)
These functions compute the two-dimensional in-place wavelet transform on a matrix m.
int gsl_wavelet2d_nstransform(const gsl_wavelet *w, double *data, size_t tda, size_t size1, size_t size2,
gsl_wavelet_direction dir, gsl_wavelet_workspace *work)
int gsl_wavelet2d_nstransform_forward(const gsl_wavelet *w, double *data, size_t tda, size_t size1, size_t
size2, gsl_wavelet_workspace *work)
int gsl_wavelet2d_nstransform_inverse(const gsl_wavelet *w, double *data, size_t tda, size_t size1, size_t
size2, gsl_wavelet_workspace *work)
These functions compute the two-dimensional wavelet transform in non-standard form.
int gsl_wavelet2d_nstransform_matrix(const gsl_wavelet *w, gsl_matrix *m, gsl_wavelet_direction dir,
gsl_wavelet_workspace *work)
int gsl_wavelet2d_nstransform_matrix_forward(const gsl_wavelet *w, gsl_matrix *m,
gsl_wavelet_workspace *work)
int gsl_wavelet2d_nstransform_matrix_inverse(const gsl_wavelet *w, gsl_matrix *m,
gsl_wavelet_workspace *work)
These functions compute the non-standard form of the two-dimensional in-place wavelet transform on a matrixm.

34.4 Examples
The following program demonstrates the use of the one-dimensional wavelet transform functions. It computes anapproximation to an input signal (of length 256) using the 20 largest components of the wavelet transform, whilesetting the others to zero.
#include <stdio.h>#include <math.h>#include <gsl/gsl_sort.h>#include <gsl/gsl_wavelet.h>
int
main (int argc, char **argv){(void)(argc); /* avoid unused parameter warning */int i, n = 256, nc = 20;
double *orig_data = malloc (n * sizeof (double));double *data = malloc (n * sizeof (double));double *abscoeff = malloc (n * sizeof (double));size_t *p = malloc (n * sizeof (size_t));
FILE * f;
gsl_wavelet *w;
gsl_wavelet_workspace *work;
w = gsl_wavelet_alloc (gsl_wavelet_daubechies, 4);work = gsl_wavelet_workspace_alloc (n);(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)f = fopen (argv[1], "r");for (i = 0; i < n; i++){fscanf (f, "%lg", &orig_data[i]);data[i] = orig_data[i];}fclose (f);gsl_wavelet_transform_forward (w, data, 1, n, work);for (i = 0; i < n; i++){abscoeff[i] = fabs (data[i]);}gsl_sort_index (p, abscoeff, 1, n);for (i = 0; (i + nc) < n; i++)data[p[i]] = 0;
gsl_wavelet_transform_inverse (w, data, 1, n, work);for (i = 0; i < n; i++){printf ("%g %g\n", orig_data[i], data[i]);}gsl_wavelet_free (w);
gsl_wavelet_workspace_free (work);
free (data);
free (orig_data);
free (abscoeff);free (p);return 0;}
The output can be used with the GNU plotutils graph program:$ ./a.out ecg.dat > dwt.txt$ graph -T ps -x 0 256 32 -h 0.3 -a dwt.txt > dwt.ps
Fig. 34.1 shows an original and compressed version of a sample ECG recording from the MIT-BIH Arrhythmia
Database, part of the PhysioNet archive of public-domain of medical datasets.

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Fig. 34.1: Original (upper) and wavelet-compressed (lower) ECG signals, using the 20 largest components of the
Daubechies(4) discrete wavelet transform.

34.5 References and Further Reading
The mathematical background to wavelet transforms is covered in the original lectures by Daubechies,• Ingrid Daubechies. Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics(1992), SIAM, ISBN 0898712742.
An easy to read introduction to the subject with an emphasis on the application of the wavelet transform in various
branches of science is,• Paul S. Addison. The Illustrated Wavelet Transform Handbook. Institute of Physics Publishing (2002), ISBN0750306920.
For extensive coverage of signal analysis by wavelets, wavelet packets and local cosine bases see,
• S. G. Mallat. A wavelet tour of signal processing (Second edition). Academic Press (1999), ISBN 012466606X.
The concept of multiresolution analysis underlying the wavelet transform is described in,• S. G. Mallat. Multiresolution Approximations and Wavelet Orthonormal Bases of L^2(R). Transactions of theAmerican Mathematical Society, 315(1), 1989, 69–87.
• S. G. Mallat. A Theory for Multiresolution Signal Decomposition—The Wavelet Representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, 1989, 674–693.
The coefficients for the individual wavelet families implemented by the library can be found in the following papers,• I. Daubechies. Orthonormal Bases of Compactly Supported Wavelets. Communications on Pure and Applied
Mathematics, 41 (1988) 909–996.
• A. Cohen, I. Daubechies, and J.-C. Feauveau. Biorthogonal Bases of Compactly Supported Wavelets. Communications on Pure and Applied Mathematics, 45 (1992) 485–560.
The PhysioNet archive of physiological datasets can be found online at http://www.physionet.org/ and is described in
the following paper,

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GNU Scientific Library, Release 2.7• Goldberger et al. PhysioBank, PhysioToolkit, and PhysioNet: Components of a New Research Resource forComplex Physiologic Signals. Circulation 101(23):e215-e220 2000.

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CHAPTERTHIRTYFIVEDISCRETE HANKEL TRANSFORMS

This chapter describes functions for performing Discrete Hankel Transforms (DHTs). The functions are declared in
the header file gsl_dht.h.

35.1 Definitions
The discrete Hankel transform acts on a vector of sampled data, where the samples are assumed to have been taken atpoints related to the zeros of a Bessel function of fixed order; compare this to the case of the discrete Fourier transform,
where samples are taken at points related to the zeroes of the sine or cosine function.
Starting with its definition, the Hankel transform (or Bessel transform) of order ν of a function f with ν > −1/2 isdefined as (see Johnson, 1987 and Lemoine, 1994)
Z ∞Fν (u) =f (t)Jν (ut)tdt
0

If the integral exists, Fν is called the Hankel transformation of f . The reverse transform is given byZ ∞f (t) =Fν (u)Jν (ut)udu
0R∞

where 0 f (t)t1/2 dt must exist and be absolutely convergent, and where f (t) satisfies Dirichlet’s conditions (of limited
total fluctuations) in the interval [0, ∞].
Now the discrete Hankel transform works on a discrete function f , which is sampled on points n = 1...M located at
positions tn = (jν,n /jν,M )X in real space and at un = jν,n /X in reciprocal space. Here, jν,m are the m-th zerosof the Bessel function Jν (x) arranged in ascending order. Moreover, the discrete functions are assumed to be band
limited, so f (tn ) = 0 and F (un ) = 0 for n > M . Accordingly, the function f is defined on the interval [0, X].
Following the work of Johnson, 1987 and Lemoine, 1994, the discrete Hankel transform is given by
M −1Fν (um ) =2X 2 X
f
2
jν,M
k=1



jν,k Xjν,MJν (jν,m jν,k /jν,M ).
Jν+1 (jν,k )2

It is this discrete expression which defines the discrete Hankel transform calculated by GSL. In GSL, forward andbackward transforms are defined equally and calculate Fν (um ). Following Johnson, the backward transform reads
M −1jν,m Jν (jν,m jν,k /jν,M )2 X
F.
f (tk ) = 2
X m=1
XJν+1 (jν,m )22
Obviously, using the forward transform instead of the backward transform gives an additional factor X 4 /jν,M=2
2
tm /um .

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The kernel in the summation above defines the matrix of the ν-Hankel transform of size M − 1. The coefficientsof this matrix, being dependent on ν and M , must be precomputed and stored; the gsl_dht object encapsulates
this data. The allocation function gsl_dht_alloc() returns a gsl_dht object which must be properly initializedwith gsl_dht_init() before it can be used to perform transforms on data sample vectors, for fixed ν and M , using
the gsl_dht_apply() function. The implementation allows to define the length X of the fundamental interval, forconvenience, while discrete Hankel transforms are often defined on the unit interval instead of [0, X].
Notice that by assumption f (t) vanishes at the endpoints of the interval, consistent with the inversion formula and thesampling formula given above. Therefore, this transform corresponds to an orthogonal expansion in eigenfunctions of
the Dirichlet problem for the Bessel differential equation.

35.2 Functions
type gsl_dht
Workspace for computing discrete Hankel transforms
gsl_dht *gsl_dht_alloc(size_t size)
This function allocates a Discrete Hankel transform object of size size.
int gsl_dht_init(gsl_dht *t, double nu, double xmax)
This function initializes the transform t for the given values of nu and xmax.
gsl_dht *gsl_dht_new(size_t size, double nu, double xmax)
This function allocates a Discrete Hankel transform object of size size and initializes it for the given values ofnu and xmax.
void gsl_dht_free(gsl_dht *t)This function frees the transform t.
int gsl_dht_apply(const gsl_dht *t, double *f_in, double *f_out)
This function applies the transform t to the array f_in whose size is equal to the size of the transform. The
result is stored in the array f_out which must be of the same length.
Applying this function to its output gives the original data multiplied by (X 2 /jν,M )2 , up to numerical errors.
double gsl_dht_x_sample(const gsl_dht *t, int n)This function returns the value of the n-th sample point in the unit interval, (jν,n+1 /jν,M )X. These are thepoints where the function f (t) is assumed to be sampled.
double gsl_dht_k_sample(const gsl_dht *t, int n)This function returns the value of the n-th sample point in “k-space”, jν,n+1 /X.

35.3 References and Further Reading
The algorithms used by these functions are described in the following papers,•

H. Fisk Johnson, Comp.: Phys.: Comm.: 43, 181 (1987).

•

D. Lemoine, J. Chem.: Phys.: 101, 3936 (1994).

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CHAPTERTHIRTYSIXONE DIMENSIONAL ROOT-FINDING

This chapter describes routines for finding roots of arbitrary one-dimensional functions. The library provides low levelcomponents for a variety of iterative solvers and convergence tests. These can be combined by the user to achieve
the desired solution, with full access to the intermediate steps of the iteration. Each class of methods uses the same
framework, so that you can switch between solvers at runtime without needing to recompile your program. Eachinstance of a solver keeps track of its own state, allowing the solvers to be used in multi-threaded programs.
The header file gsl_roots.h contains prototypes for the root finding functions and related declarations.

36.1 Overview
One-dimensional root finding algorithms can be divided into two classes, root bracketing and root polishing. Algorithms which proceed by bracketing a root are guaranteed to converge. Bracketing algorithms begin with a boundedregion known to contain a root. The size of this bounded region is reduced, iteratively, until it encloses the root to adesired tolerance. This provides a rigorous error estimate for the location of the root.
The technique of root polishing attempts to improve an initial guess to the root. These algorithms converge only ifstarted “close enough” to a root, and sacrifice a rigorous error bound for speed. By approximating the behavior of afunction in the vicinity of a root they attempt to find a higher order improvement of an initial guess. When the behaviorof the function is compatible with the algorithm and a good initial guess is available a polishing algorithm can provide
rapid convergence.
In GSL both types of algorithm are available in similar frameworks. The user provides a high-level driver for thealgorithms, and the library provides the individual functions necessary for each of the steps. There are three main
phases of the iteration. The steps are,• initialize solver state, s, for algorithm T• update s using the iteration T• test s for convergence, and repeat iteration if necessary
The state for bracketing solvers is held in a gsl_root_fsolver struct. The updating procedure uses only function
evaluations (not derivatives). The state for root polishing solvers is held in a gsl_root_fdfsolver struct. The updates
require both the function and its derivative (hence the name fdf) to be supplied by the user.

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36.2 Caveats
Note that root finding functions can only search for one root at a time. When there are several roots in the search area,
the first root to be found will be returned; however it is difficult to predict which of the roots this will be. In most cases,
no error will be reported if you try to find a root in an area where there is more than one.
Care must be taken when a function may have a multiple root (such as f (x) = (x − x0 )2 or f (x) = (x − x0 )3 . It is
not possible to use root-bracketing algorithms on even-multiplicity roots. For these algorithms the initial interval must
contain a zero-crossing, where the function is negative at one end of the interval and positive at the other end. Roots
with even-multiplicity do not cross zero, but only touch it instantaneously. Algorithms based on root bracketing will
still work for odd-multiplicity roots (e.g. cubic, quintic, . . . ). Root polishing algorithms generally work with higher
multiplicity roots, but at a reduced rate of convergence. In these cases the Steffenson algorithm can be used to accelerate
the convergence of multiple roots.
While it is not absolutely required that f have a root within the search region, numerical root finding functions should
not be used haphazardly to check for the existence of roots. There are better ways to do this. Because it is easy to create
situations where numerical root finders can fail, it is a bad idea to throw a root finder at a function you do not knowmuch about. In general it is best to examine the function visually by plotting before searching for a root.

36.3 Initializing the Solver
type gsl_root_fsolverThis is a workspace for finding roots using methods which do not require derivatives.
type gsl_root_fdfsolverThis is a workspace for finding roots using methods which require derivatives.
gsl_root_fsolver *gsl_root_fsolver_alloc(const gsl_root_fsolver_type *T)This function returns a pointer to a newly allocated instance of a solver of type T. For example, the following
code creates an instance of a bisection solver:
const gsl_root_fsolver_type * T = gsl_root_fsolver_bisection;
gsl_root_fsolver * s = gsl_root_fsolver_alloc (T);If there is insufficient memory to create the solver then the function returns a null pointer and the error handler
is invoked with an error code of GSL_ENOMEM.
gsl_root_fdfsolver *gsl_root_fdfsolver_alloc(const gsl_root_fdfsolver_type *T)This function returns a pointer to a newly allocated instance of a derivative-based solver of type T. For example,
the following code creates an instance of a Newton-Raphson solver:
const gsl_root_fdfsolver_type * T = gsl_root_fdfsolver_newton;
gsl_root_fdfsolver * s = gsl_root_fdfsolver_alloc (T);If there is insufficient memory to create the solver then the function returns a null pointer and the error handler
is invoked with an error code of GSL_ENOMEM.
int gsl_root_fsolver_set(gsl_root_fsolver *s, gsl_function *f, double x_lower, double x_upper)
This function initializes, or reinitializes, an existing solver s to use the function f and the initial search interval[x_lower, x_upper].
int gsl_root_fdfsolver_set(gsl_root_fdfsolver *s, gsl_function_fdf *fdf, double root)
This function initializes, or reinitializes, an existing solver s to use the function and derivative fdf and the initial
guess root.

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void gsl_root_fsolver_free(gsl_root_fsolver *s)
void gsl_root_fdfsolver_free(gsl_root_fdfsolver *s)
These functions free all the memory associated with the solver s.
const char *gsl_root_fsolver_name(const gsl_root_fsolver *s)
const char *gsl_root_fdfsolver_name(const gsl_root_fdfsolver *s)
These functions return a pointer to the name of the solver. For example:
printf ("s is a '%s' solver\n", gsl_root_fsolver_name (s));
would print something like s is a 'bisection' solver.

36.4 Providing the function to solve
You must provide a continuous function of one variable for the root finders to operate on, and, sometimes, its first
derivative. In order to allow for general parameters the functions are defined by the following data types:
type gsl_function
This data type defines a general function with parameters.
double (* function) (double x, void * params)
this function should return the value f (x, params) for argument x and parameters params
void * params
a pointer to the parameters of the function
Here is an example for the general quadratic function,f (x) = ax2 + bx + c
with a = 3, b = 2, c = 1. The following code defines a gsl_function F which you could pass to a root finder as afunction pointer:struct my_f_params { double a; double b; double c; };doublemy_f (double x, void * p){
struct my_f_params * params = (struct my_f_params *)p;
double a = (params->a);double b = (params->b);double c = (params->c);return(a * x + b) * x + c;}
gsl_function F;
struct my_f_params params = { 3.0, 2.0, 1.0 };F.function = &my_f;F.params = &params;

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The function f (x) can be evaluated using the macro GSL_FN_EVAL(&F,x) defined in gsl_math.h.
type gsl_function_fdf
This data type defines a general function with parameters and its first derivative.
double (* f) (double x, void * params)
this function should return the value of f (x, params) for argument x and parameters params
double (* df) (double x, void * params)
this function should return the value of the derivative of f with respect to x, f 0 (x, params), forargument x and parameters params
void (* fdf) (double x, void * params, double * f, double * df)
this function should set the values of the function f to f (x, params) and its derivative df to
f 0 (x, params) for argument x and parameters params. This function provides an optimization of
the separate functions for f (x) and f 0 (x)—it is always faster to compute the function and its derivative at the same time.
void * params
a pointer to the parameters of the function
Here is an example where f (x) = exp(2x):doublemy_f (double x, void * params){
return exp (2 * x);}
doublemy_df (double x, void * params){
return 2 * exp (2 * x);}void
my_fdf (double x, void * params,
double * f, double * df){
double t = exp (2 * x);*f = t;*df = 2 * t;/* uses existing value */}gsl_function_fdf FDF;
FDF.f = &my_f;
FDF.df = &my_df;
FDF.fdf = &my_fdf;
FDF.params = 0;
The function f (x) can be evaluated using the macro GSL_FN_FDF_EVAL_F(&FDF,x) and the derivative f 0 (x) can beevaluated using the macro GSL_FN_FDF_EVAL_DF(&FDF,x). Both the function y = f (x) and its derivative dy = f 0 (x)436

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can be evaluated at the same time using the macro GSL_FN_FDF_EVAL_F_DF(&FDF,x,y,dy). The macro stores f (x)in its y argument and f 0 (x) in its dy argument—both of these should be pointers to double.

36.5 Search Bounds and Guesses
You provide either search bounds or an initial guess; this section explains how search bounds and guesses work andhow function arguments control them.
A guess is simply an x value which is iterated until it is within the desired precision of a root. It takes the form of adouble.
Search bounds are the endpoints of an interval which is iterated until the length of the interval is smaller than therequested precision. The interval is defined by two values, the lower limit and the upper limit. Whether the endpoints
are intended to be included in the interval or not depends on the context in which the interval is used.

36.6 Iteration
The following functions drive the iteration of each algorithm. Each function performs one iteration to update the stateof any solver of the corresponding type. The same functions work for all solvers so that different methods can besubstituted at runtime without modifications to the code.
int gsl_root_fsolver_iterate(gsl_root_fsolver *s)
int gsl_root_fdfsolver_iterate(gsl_root_fdfsolver *s)
These functions perform a single iteration of the solver s. If the iteration encounters an unexpected problem thenan error code will be returned,
GSL_EBADFUNC
the iteration encountered a singular point where the function or its derivative evaluated to Inf or NaN.
GSL_EZERODIV
the derivative of the function vanished at the iteration point, preventing the algorithm from continuing
without a division by zero.
The solver maintains a current best estimate of the root at all times. The bracketing solvers also keep track of the current
best interval bounding the root. This information can be accessed with the following auxiliary functions,double gsl_root_fsolver_root(const gsl_root_fsolver *s)
double gsl_root_fdfsolver_root(const gsl_root_fdfsolver *s)
These functions return the current estimate of the root for the solver s.
double gsl_root_fsolver_x_lower(const gsl_root_fsolver *s)
double gsl_root_fsolver_x_upper(const gsl_root_fsolver *s)
These functions return the current bracketing interval for the solver s.

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36.7 Search Stopping Parameters
A root finding procedure should stop when one of the following conditions is true:• A root has been found to within the user-specified precision.
• A user-specified maximum number of iterations has been reached.
• An error has occurred.
The handling of these conditions is under user control. The functions below allow the user to test the precision of thecurrent result in several standard ways.
int gsl_root_test_interval(double x_lower, double x_upper, double epsabs, double epsrel)This function tests for the convergence of the interval [x_lower, x_upper] with absolute error epsabs andrelative error epsrel. The test returns GSL_SUCCESS if the following condition is achieved,|a − b| < epsabs + epsrel min(|a|, |b|)when the interval x = [a, b] does not include the origin. If the interval includes the origin then min(|a|, |b|) is
replaced by zero (which is the minimum value of |x| over the interval). This ensures that the relative error isaccurately estimated for roots close to the origin.
This condition on the interval also implies that any estimate of the root r in the interval satisfies the same condition
with respect to the true root r∗ ,|r − r∗ | < epsabs + epsrel r∗
assuming that the true root r∗ is contained within the interval.
int gsl_root_test_delta(double x1, double x0, double epsabs, double epsrel)This function tests for the convergence of the sequence x0, x1 with absolute error epsabs and relative error
epsrel. The test returns GSL_SUCCESS if the following condition is achieved,|x1 − x0 | < epsabs + epsrel |x1 |and returns GSL_CONTINUE otherwise.
int gsl_root_test_residual(double f, double epsabs)
This function tests the residual value f against the absolute error bound epsabs. The test returns GSL_SUCCESS
if the following condition is achieved,|f | < epsabs
and returns GSL_CONTINUE otherwise. This criterion is suitable for situations where the precise location of theroot, x, is unimportant provided a value can be found where the residual, |f (x)|, is small enough.

36.8 Root Bracketing Algorithms
The root bracketing algorithms described in this section require an initial interval which is guaranteed to contain aroot—if a and b are the endpoints of the interval then f (a) must differ in sign from f (b). This ensures that the function
crosses zero at least once in the interval. If a valid initial interval is used then these algorithm cannot fail, provided thefunction is well-behaved.
Note that a bracketing algorithm cannot find roots of even degree, since these do not cross the x-axis.
type gsl_root_fsolver_type

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gsl_root_fsolver_type *gsl_root_fsolver_bisection
The bisection algorithm is the simplest method of bracketing the roots of a function. It is the slowest
algorithm provided by the library, with linear convergence.
On each iteration, the interval is bisected and the value of the function at the midpoint is calculated. The
sign of this value is used to determine which half of the interval does not contain a root. That half isdiscarded to give a new, smaller interval containing the root. This procedure can be continued indefinitely
until the interval is sufficiently small.
At any time the current estimate of the root is taken as the midpoint of the interval.
gsl_root_fsolver_type *gsl_root_fsolver_falsepos
The false position algorithm is a method of finding roots based on linear interpolation. Its convergence islinear, but it is usually faster than bisection.
On each iteration a line is drawn between the endpoints (a, f (a)) and (b, f (b)) and the point where thisline crosses the x-axis taken as a “midpoint”. The value of the function at this point is calculated and its
sign is used to determine which side of the interval does not contain a root. That side is discarded to givea new, smaller interval containing the root. This procedure can be continued indefinitely until the interval
is sufficiently small.
The best estimate of the root is taken from the linear interpolation of the interval on the current iteration.
gsl_root_fsolver_type *gsl_root_fsolver_brent
The Brent-Dekker method (referred to here as Brent’s method) combines an interpolation strategy with thebisection algorithm. This produces a fast algorithm which is still robust.
On each iteration Brent’s method approximates the function using an interpolating curve. On the first
iteration this is a linear interpolation of the two endpoints. For subsequent iterations the algorithm uses aninverse quadratic fit to the last three points, for higher accuracy. The intercept of the interpolating curvewith the x-axis is taken as a guess for the root. If it lies within the bounds of the current interval then theinterpolating point is accepted, and used to generate a smaller interval. If the interpolating point is not
accepted then the algorithm falls back to an ordinary bisection step.
The best estimate of the root is taken from the most recent interpolation or bisection.

36.9 Root Finding Algorithms using Derivatives
The root polishing algorithms described in this section require an initial guess for the location of the root. There is
no absolute guarantee of convergence—the function must be suitable for this technique and the initial guess must be
sufficiently close to the root for it to work. When these conditions are satisfied then convergence is quadratic.
These algorithms make use of both the function and its derivative.
type gsl_root_fdfsolver_type
gsl_root_fdfsolver_type *gsl_root_fdfsolver_newton
Newton’s Method is the standard root-polishing algorithm. The algorithm begins with an initial guess forthe location of the root. On each iteration, a line tangent to the function f is drawn at that position. The
point where this line crosses the x-axis becomes the new guess. The iteration is defined by the following
sequence,xi+1 = xi −f (xi )
f 0 (xi )Newton’s method converges quadratically for single roots, and linearly for multiple roots.

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gsl_root_fdfsolver_type *gsl_root_fdfsolver_secant
The secant method is a simplified version of Newton’s method which does not require the computation of
the derivative on every step.
On its first iteration the algorithm begins with Newton’s method, using the derivative to compute a firststep,
x1 = x0 −f (x0 )
f 0 (x0 )Subsequent iterations avoid the evaluation of the derivative by replacing it with a numerical estimate, theslope of the line through the previous two points,
xi+1 = xi −f (xi )f (xi ) − f (xi−1 )0where fest=
0
fest
xi − xi−1

When the derivative does not change significantly in the vicinity of the root the secant method gives a usefulsaving. Asymptotically the secant method is faster than Newton’s method whenever the cost of evaluating
the derivative is more than 0.44 times the cost of evaluating the function itself. As with all methods ofcomputing a numerical derivative the estimate can suffer from cancellation errors if the separation of thepoints becomes too small.
√
On single roots, the method has a convergence of order (1 + 5)/2 (approximately 1.62). It converges
linearly for multiple roots.
gsl_root_fdfsolver_type *gsl_root_fdfsolver_steffenson
The Steffenson Method 1 provides the fastest convergence of all the routines. It combines the basic Newton
algorithm with an Aitken “delta-squared” acceleration. If the Newton iterates are xi then the acceleration
procedure generates a new sequence Ri ,Ri = xi −(xi+1 − xi )2(xi+2 − 2xi+1 + xi )which converges faster than the original sequence under reasonable conditions. The new sequence requires
three terms before it can produce its first value so the method returns accelerated values on the second andsubsequent iterations. On the first iteration it returns the ordinary Newton estimate. The Newton iterate isalso returned if the denominator of the acceleration term ever becomes zero.
As with all acceleration procedures this method can become unstable if the function is not well-behaved.

36.10 Examples
For any root finding algorithm we need to prepare the function to be solved. For this example we will use the general
quadratic equation described earlier. We first need a header file (demo_fn.h) to define the function parameters,
struct quadratic_params{
double a, b, c;};
double quadratic (double x, void *params);
double quadratic_deriv (double x, void *params);
void quadratic_fdf (double x, void *params,
double *y, double *dy);
1 J.F. Steffensen (1873–1961). The spelling used in the name of the function is slightly incorrect, but has been preserved to avoid incompatibility.

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We place the function definitions in a separate file (demo_fn.c),doublequadratic (double x, void *params){
struct quadratic_params *p= (struct quadratic_params *) params;
double a = p->a;double b = p->b;
double c = p->c;
return (a * x + b) * x + c;}
doublequadratic_deriv (double x, void *params){
struct quadratic_params *p= (struct quadratic_params *) params;
double a = p->a;double b = p->b;
return 2.0 * a * x + b;}
void
quadratic_fdf (double x, void *params,
double *y, double *dy){
struct quadratic_params *p= (struct quadratic_params *) params;
double a = p->a;double b = p->b;
double c = p->c;*y = (a * x + b) * x + c;*dy = 2.0 * a * x + b;}
The first program uses the function solver gsl_root_fsolver_brent for Brent’s method and the general quadratic
defined above to solve the following equation,x2 − 5 = 0
with solution x =√5 = 2.236068...

#include <stdio.h>#include <gsl/gsl_errno.h>#include <gsl/gsl_math.h>#include <gsl/gsl_roots.h>(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)#include "demo_fn.h"#include "demo_fn.c"
int
main (void){
int status;
int iter = 0, max_iter = 100;
const gsl_root_fsolver_type *T;
gsl_root_fsolver *s;
double r = 0, r_expected = sqrt (5.0);
double x_lo = 0.0, x_hi = 5.0;
gsl_function F;
struct quadratic_params params = {1.0, 0.0, -5.0};
F.function = &quadratic;
F.params = &params;
T = gsl_root_fsolver_brent;s = gsl_root_fsolver_alloc (T);gsl_root_fsolver_set (s, &F, x_lo, x_hi);
printf ("using %s method\n",gsl_root_fsolver_name (s));
printf ("%5s [%9s, %9s] %9s %10s %9s\n","iter", "lower", "upper", "root","err", "err(est)");do{iter++;
status = gsl_root_fsolver_iterate (s);r = gsl_root_fsolver_root (s);x_lo = gsl_root_fsolver_x_lower (s);x_hi = gsl_root_fsolver_x_upper (s);status = gsl_root_test_interval (x_lo, x_hi,
0, 0.001);
if (status == GSL_SUCCESS)
printf ("Converged:\n");printf ("%5d [%.7f , %.7f ] %.7f %+.7f %.7f \n",iter, x_lo, x_hi,
r, r - r_expected,
x_hi - x_lo);}while (status == GSL_CONTINUE && iter < max_iter);
gsl_root_fsolver_free (s);return status;(continues on next page)

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Here are the results of the iterations:$ ./a.out
using brent method
iter [
lower,
upper]
root
err err(est)
1 [1.0000000, 5.0000000] 1.0000000 -1.2360680 4.0000000
2 [1.0000000, 3.0000000] 3.0000000 +0.7639320 2.0000000
3 [2.0000000, 3.0000000] 2.0000000 -0.2360680 1.0000000
4 [2.2000000, 3.0000000] 2.2000000 -0.0360680 0.8000000
5 [2.2000000, 2.2366300] 2.2366300 +0.0005621 0.0366300
Converged:
6 [2.2360634, 2.2366300] 2.2360634 -0.0000046 0.0005666
If the program is modified to use the bisection solver instead of Brent’s method, by changing
gsl_root_fsolver_brent to gsl_root_fsolver_bisection the slower convergence of the Bisection method
can be observed:$ ./a.out
using bisection method
iter [
lower,
upper]
root
err err(est)
1 [0.0000000, 2.5000000] 1.2500000 -0.9860680 2.5000000
2 [1.2500000, 2.5000000] 1.8750000 -0.3610680 1.2500000
3 [1.8750000, 2.5000000] 2.1875000 -0.0485680 0.6250000
4 [2.1875000, 2.5000000] 2.3437500 +0.1076820 0.3125000
5 [2.1875000, 2.3437500] 2.2656250 +0.0295570 0.1562500
6 [2.1875000, 2.2656250] 2.2265625 -0.0095055 0.0781250
7 [2.2265625, 2.2656250] 2.2460938 +0.0100258 0.0390625
8 [2.2265625, 2.2460938] 2.2363281 +0.0002601 0.0195312
9 [2.2265625, 2.2363281] 2.2314453 -0.0046227 0.0097656
10 [2.2314453, 2.2363281] 2.2338867 -0.0021813 0.0048828
11 [2.2338867, 2.2363281] 2.2351074 -0.0009606 0.0024414
Converged:
12 [2.2351074, 2.2363281] 2.2357178 -0.0003502 0.0012207
The next program solves the same function using a derivative solver instead.
#include <stdio.h>#include <gsl/gsl_errno.h>#include <gsl/gsl_math.h>#include <gsl/gsl_roots.h>#include "demo_fn.h"#include "demo_fn.c"
int
main (void){
int status;
int iter = 0, max_iter = 100;
const gsl_root_fdfsolver_type *T;
gsl_root_fdfsolver *s;(continues on next page)

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double x0, x = 5.0, r_expected = sqrt (5.0);
gsl_function_fdf FDF;
struct quadratic_params params = {1.0, 0.0, -5.0};
FDF.f = &quadratic;
FDF.df = &quadratic_deriv;
FDF.fdf = &quadratic_fdf;
FDF.params = &params;
T = gsl_root_fdfsolver_newton;s = gsl_root_fdfsolver_alloc (T);gsl_root_fdfsolver_set (s, &FDF, x);
printf ("using %s method\n",gsl_root_fdfsolver_name (s));
printf ("%-5s %10s %10s %10s\n","iter", "root", "err", "err(est)");do{iter++;
status = gsl_root_fdfsolver_iterate (s);x0 = x;x = gsl_root_fdfsolver_root (s);status = gsl_root_test_delta (x, x0, 0, 1e-3);
if (status == GSL_SUCCESS)
printf ("Converged:\n");printf ("%5d %10.7f %+10.7f %10.7f \n",iter, x, x - r_expected, x - x0);}
while (status == GSL_CONTINUE && iter < max_iter);
gsl_root_fdfsolver_free (s);return status;}
Here are the results for Newton’s method:$ ./a.out
using newton method
iter
root
err
err(est)1 3.0000000 +0.7639320 -2.0000000
2 2.3333333 +0.0972654 -0.6666667
3 2.2380952 +0.0020273 -0.0952381
Converged:
4 2.2360689 +0.0000009 -0.0020263
Note that the error can be estimated more accurately by taking the difference between the current iterate andnext iterate rather than the previous iterate. The other derivative solvers can be investigated by changing
gsl_root_fdfsolver_newton to gsl_root_fdfsolver_secant or gsl_root_fdfsolver_steffenson.

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36.11 References and Further Reading
For information on the Brent-Dekker algorithm see the following two papers,• R. P. Brent, “An algorithm with guaranteed convergence for finding a zero of a function”, Computer Journal, 14(1971) 422–425• J. C. P. Bus and T. J. Dekker, “Two Efficient Algorithms with Guaranteed Convergence for Finding a Zero of a
Function”, ACM Transactions of Mathematical Software, Vol.: 1 No.: 4 (1975) 330–345

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CHAPTERTHIRTYSEVENONE DIMENSIONAL MINIMIZATIONThis chapter describes routines for finding minima of arbitrary one-dimensional functions. The library provides low
level components for a variety of iterative minimizers and convergence tests. These can be combined by the user
to achieve the desired solution, with full access to the intermediate steps of the algorithms. Each class of methods
uses the same framework, so that you can switch between minimizers at runtime without needing to recompile yourprogram. Each instance of a minimizer keeps track of its own state, allowing the minimizers to be used in multi-threaded
programs.
The header file gsl_min.h contains prototypes for the minimization functions and related declarations. To use theminimization algorithms to find the maximum of a function simply invert its sign.

37.1 Overview
The minimization algorithms begin with a bounded region known to contain a minimum. The region is described by alower bound a and an upper bound b, with an estimate of the location of the minimum x, as shown in Fig. 37.1.

Fig. 37.1: Function with lower and upper bounds with an estimate of the minimum.
The value of the function at x must be less than the value of the function at the ends of the interval,
f (a) > f (x) < f (b)This condition guarantees that a minimum is contained somewhere within the interval. On each iteration a new point
x0 is selected using one of the available algorithms. If the new point is a better estimate of the minimum, i.e.: wheref (x0 ) < f (x), then the current estimate of the minimum x is updated. The new point also allows the size of the boundedinterval to be reduced, by choosing the most compact set of points which satisfies the constraint f (a) > f (x) < f (b).
The interval is reduced until it encloses the true minimum to a desired tolerance. This provides a best estimate of thelocation of the minimum and a rigorous error estimate.
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Several bracketing algorithms are available within a single framework. The user provides a high-level driver for thealgorithm, and the library provides the individual functions necessary for each of the steps. There are three main phasesof the iteration. The steps are,• initialize minimizer state, s, for algorithm T• update s using the iteration T• test s for convergence, and repeat iteration if necessary
The state for the minimizers is held in a gsl_min_fminimizer struct. The updating procedure uses only function
evaluations (not derivatives).

37.2 Caveats
Note that minimization functions can only search for one minimum at a time. When there are several minima in the
search area, the first minimum to be found will be returned; however it is difficult to predict which of the minima thiswill be. In most cases, no error will be reported if you try to find a minimum in an area where there is more than one.
With all minimization algorithms it can be difficult to determine the location of the minimum to full numerical precision.
The behavior of the function in the region of the minimum x∗ can be approximated by a Taylor expansion,1
y = f (x∗ ) + f 00 (x∗ )(x − x∗ )22
and the second term of this expansion can be lost
√ when added to the first term at finite precision. This magnifies theerror in locating x∗ , making it proportional to  (where  is the relative accuracy of the floating point numbers). For
functions with higher order minima, such as x4 , the magnification of the error is correspondingly worse. The best thatcan be achieved is to converge to the limit of numerical accuracy in the function values, rather than the location of theminimum itself.

37.3 Initializing the Minimizer
type gsl_min_fminimizer
This is a workspace for minimizing functions.
gsl_min_fminimizer *gsl_min_fminimizer_alloc(const gsl_min_fminimizer_type *T)This function returns a pointer to a newly allocated instance of a minimizer of type T. For example, the following
code creates an instance of a golden section minimizer:
const gsl_min_fminimizer_type * T = gsl_min_fminimizer_goldensection;
gsl_min_fminimizer * s = gsl_min_fminimizer_alloc (T);If there is insufficient memory to create the minimizer then the function returns a null pointer and the error
handler is invoked with an error code of GSL_ENOMEM.
int gsl_min_fminimizer_set(gsl_min_fminimizer *s, gsl_function *f, double x_minimum, double x_lower,
double x_upper)
This function sets, or resets, an existing minimizer s to use the function f and the initial search interval [x_lower,
x_upper], with a guess for the location of the minimum x_minimum.
If the interval given does not contain a minimum, then the function returns an error code of GSL_EINVAL.

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int gsl_min_fminimizer_set_with_values(gsl_min_fminimizer *s, gsl_function *f, double x_minimum, double
f_minimum, double x_lower, double f_lower, double x_upper,
double f_upper)
This function is equivalent to gsl_min_fminimizer_set() but uses the values f_minimum, f_lower and
f_upper instead of computing f(x_minimum), f(x_lower) and f(x_upper).
void gsl_min_fminimizer_free(gsl_min_fminimizer *s)
This function frees all the memory associated with the minimizer s.
const char *gsl_min_fminimizer_name(const gsl_min_fminimizer *s)
This function returns a pointer to the name of the minimizer. For example:
printf ("s is a '%s' minimizer\n", gsl_min_fminimizer_name (s));
would print something like s is a 'brent' minimizer.

37.4 Providing the function to minimize
You must provide a continuous function of one variable for the minimizers to operate on. In order to allow for generalparameters the functions are defined by a gsl_function data type (Providing the function to solve).

37.5 Iteration
The following functions drive the iteration of each algorithm. Each function performs one iteration to update the stateof any minimizer of the corresponding type. The same functions work for all minimizers so that different methods canbe substituted at runtime without modifications to the code.
int gsl_min_fminimizer_iterate(gsl_min_fminimizer *s)
This function performs a single iteration of the minimizer s. If the iteration encounters an unexpected problem
then an error code will be returned,
GSL_EBADFUNC
the iteration encountered a singular point where the function evaluated to Inf or NaN.
GSL_FAILURE
the algorithm could not improve the current best approximation or bounding interval.
The minimizer maintains a current best estimate of the position of the minimum at all times, and the current intervalbounding the minimum. This information can be accessed with the following auxiliary functions,double gsl_min_fminimizer_x_minimum(const gsl_min_fminimizer *s)
This function returns the current estimate of the position of the minimum for the minimizer s.
double gsl_min_fminimizer_x_upper(const gsl_min_fminimizer *s)
double gsl_min_fminimizer_x_lower(const gsl_min_fminimizer *s)
These functions return the current upper and lower bound of the interval for the minimizer s.
double gsl_min_fminimizer_f_minimum(const gsl_min_fminimizer *s)
double gsl_min_fminimizer_f_upper(const gsl_min_fminimizer *s)

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double gsl_min_fminimizer_f_lower(const gsl_min_fminimizer *s)
These functions return the value of the function at the current estimate of the minimum and at the upper andlower bounds of the interval for the minimizer s.

37.6 Stopping Parameters
A minimization procedure should stop when one of the following conditions is true:• A minimum has been found to within the user-specified precision.
• A user-specified maximum number of iterations has been reached.
• An error has occurred.
The handling of these conditions is under user control. The function below allows the user to test the precision of thecurrent result.
int gsl_min_test_interval(double x_lower, double x_upper, double epsabs, double epsrel)This function tests for the convergence of the interval [x_lower, x_upper] with absolute error epsabs andrelative error epsrel. The test returns GSL_SUCCESS if the following condition is achieved,|a − b| < epsabs + epsrel min(|a|, |b|)when the interval x = [a, b] does not include the origin. If the interval includes the origin then min(|a|, |b|) is
replaced by zero (which is the minimum value of |x| over the interval). This ensures that the relative error isaccurately estimated for minima close to the origin.
This condition on the interval also implies that any estimate of the minimum xm in the interval satisfies the same
condition with respect to the true minimum x∗m ,|xm − x∗m | < epsabs + epsrel x∗massuming that the true minimum x∗m is contained within the interval.

37.7 Minimization Algorithms
The minimization algorithms described in this section require an initial interval which is guaranteed to contain a minimum—if a and b are the endpoints of the interval and x is an estimate of the minimum then f (a) > f (x) < f (b).
This ensures that the function has at least one minimum somewhere in the interval. If a valid initial interval is usedthen these algorithm cannot fail, provided the function is well-behaved.
type gsl_min_fminimizer_type
gsl_min_fminimizer_type *gsl_min_fminimizer_goldensection
The golden section algorithm is the simplest method of bracketing the minimum of a function. It is theslowest algorithm provided by the library, with linear convergence.
On each iteration, the algorithm first compares the subintervals from the endpoints to √the current minimum.
The larger subinterval is divided in a golden section (using the famous ratio (3 − 5)/2 ≈ 0.3819660
and the value of the function at this new point is calculated. The new value is used with the constraint
f (a0 ) > f (x0 ) < f (b0 ) to a select new interval containing the minimum, by discarding the least useful point.
This procedure can be continued indefinitely until the interval is sufficiently small. Choosing the golden
section as the bisection ratio can be shown to provide the fastest convergence for this type of algorithm.

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gsl_min_fminimizer_type *gsl_min_fminimizer_brent
The Brent minimization algorithm combines a parabolic interpolation with the golden section algorithm.
This produces a fast algorithm which is still robust.
The outline of the algorithm can be summarized as follows: on each iteration Brent’s method approximates
the function using an interpolating parabola through three existing points. The minimum of the parabola istaken as a guess for the minimum. If it lies within the bounds of the current interval then the interpolating
point is accepted, and used to generate a smaller interval. If the interpolating point is not accepted then thealgorithm falls back to an ordinary golden section step. The full details of Brent’s method include some
additional checks to improve convergence.
gsl_min_fminimizer_type *gsl_min_fminimizer_quad_goldenThis is a variant of Brent’s algorithm which uses the safeguarded step-length algorithm of Gill and Murray.

37.8 Examples
The following program uses the Brent algorithm to find the minimum of the function f (x) = cos(x) + 1, which occurs
at x = π. The starting interval is (0, 6), with an initial guess for the minimum of 2.
#include <stdio.h>#include <gsl/gsl_errno.h>#include <gsl/gsl_math.h>#include <gsl/gsl_min.h>double fn1 (double x, void * params){(void)(params); /* avoid unused parameter warning */return cos(x) + 1.0;}
int
main (void){
int status;
int iter = 0, max_iter = 100;
const gsl_min_fminimizer_type *T;
gsl_min_fminimizer *s;
double m = 2.0, m_expected = M_PI;
double a = 0.0, b = 6.0;
gsl_function F;
F.function = &fn1;
F.params = 0;
T = gsl_min_fminimizer_brent;s = gsl_min_fminimizer_alloc (T);gsl_min_fminimizer_set (s, &F, m, a, b);
printf ("using %s method\n",gsl_min_fminimizer_name (s));
printf ("%5s [%9s, %9s] %9s %10s %9s\n","iter", "lower", "upper", "min",(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)"err", "err(est)");printf ("%5d [%.7f , %.7f ] %.7f %+.7f %.7f \n",iter, a, b,
m, m - m_expected, b - a);do{iter++;
status = gsl_min_fminimizer_iterate (s);m = gsl_min_fminimizer_x_minimum (s);a = gsl_min_fminimizer_x_lower (s);b = gsl_min_fminimizer_x_upper (s);status= gsl_min_test_interval (a, b, 0.001, 0.0);
if (status == GSL_SUCCESS)
printf ("Converged:\n");printf ("%5d [%.7f , %.7f ] ""%.7f %+.7f %.7f \n",iter, a, b,
m, m - m_expected, b - a);}while (status == GSL_CONTINUE && iter < max_iter);
gsl_min_fminimizer_free (s);return status;}
Here are the results of the minimization procedure.
using brent method
iter [
lower,
upper]minerr err(est)
0 [0.0000000, 6.0000000] 2.0000000 -1.1415927 6.0000000
1 [2.0000000, 6.0000000] 3.5278640 +0.3862713 4.0000000
2 [2.0000000, 3.5278640] 3.1748217 +0.0332290 1.5278640
3 [2.0000000, 3.1748217] 3.1264576 -0.0151351 1.1748217
4 [3.1264576, 3.1748217] 3.1414743 -0.0001183 0.0483641
5 [3.1414743, 3.1748217] 3.1415930 +0.0000004 0.0333474
Converged:
6 [3.1414743, 3.1415930] 3.1415927 +0.0000000 0.0001187

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37.9 References and Further Reading
Further information on Brent’s algorithm is available in the following book,• Richard Brent, Algorithms for minimization without derivatives, Prentice-Hall (1973), republished by Dover inpaperback (2002), ISBN 0-486-41998-3.

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CHAPTERTHIRTYEIGHTMULTIDIMENSIONAL ROOT-FINDING

This chapter describes functions for multidimensional root-finding (solving nonlinear systems with n equations in n
unknowns). The library provides low level components for a variety of iterative solvers and convergence tests. Thesecan be combined by the user to achieve the desired solution, with full access to the intermediate steps of the iteration.
Each class of methods uses the same framework, so that you can switch between solvers at runtime without needing
to recompile your program. Each instance of a solver keeps track of its own state, allowing the solvers to be used in
multi-threaded programs. The solvers are based on the original Fortran library MINPACK.
The header file gsl_multiroots.h contains prototypes for the multidimensional root finding functions and related
declarations.

38.1 Overview
The problem of multidimensional root finding requires the simultaneous solution of n equations, fi , in n variables, xi ,fi (x1 , . . . , xn ) = 0

for i = 1 . . . n.

In general there are no bracketing methods available for n dimensional systems, and no way of knowing whether any
solutions exist. All algorithms proceed from an initial guess using a variant of the Newton iteration,x → x0 = x − J −1 f (x)where x, f are vector quantities and J is the Jacobian matrix Jij = ∂fi /∂xj . Additional strategies can be used
to enlarge the region of convergence. These include requiring a decrease in the norm |f | on each step proposed by
Newton’s method, or taking steepest-descent steps in the direction of the negative gradient of |f |.
Several root-finding algorithms are available within a single framework. The user provides a high-level driver for thealgorithms, and the library provides the individual functions necessary for each of the steps. There are three main
phases of the iteration. The steps are,• initialize solver state, s, for algorithm T• update s using the iteration T• test s for convergence, and repeat iteration if necessary
The evaluation of the Jacobian matrix can be problematic, either because programming the derivatives is intractable or
because computation of the n2 terms of the matrix becomes too expensive. For these reasons the algorithms provided
by the library are divided into two classes according to whether the derivatives are available or not.
The state for solvers with an analytic Jacobian matrix is held in a gsl_multiroot_fdfsolver struct. The updatingprocedure requires both the function and its derivatives to be supplied by the user.
The state for solvers which do not use an analytic Jacobian matrix is held in a gsl_multiroot_fsolver struct. The
updating procedure uses only function evaluations (not derivatives). The algorithms estimate the matrix J or J −1 byapproximate methods.
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38.2 Initializing the Solver
The following functions initialize a multidimensional solver, either with or without derivatives. The solver itself depends only on the dimension of the problem and the algorithm and can be reused for different problems.
type gsl_multiroot_fsolverThis is a workspace for multidimensional root-finding without derivatives.
type gsl_multiroot_fdfsolverThis is a workspace for multidimensional root-finding with derivatives.
gsl_multiroot_fsolver *gsl_multiroot_fsolver_alloc(const gsl_multiroot_fsolver_type *T, size_t n)This function returns a pointer to a newly allocated instance of a solver of type T for a system of n dimensions. For
example, the following code creates an instance of a hybrid solver, to solve a 3-dimensional system of equations:const gsl_multiroot_fsolver_type * T = gsl_multiroot_fsolver_hybrid;
gsl_multiroot_fsolver * s = gsl_multiroot_fsolver_alloc (T, 3);
If there is insufficient memory to create the solver then the function returns a null pointer and the error handler
is invoked with an error code of GSL_ENOMEM.
gsl_multiroot_fdfsolver *gsl_multiroot_fdfsolver_alloc(const gsl_multiroot_fdfsolver_type *T, size_t n)This function returns a pointer to a newly allocated instance of a derivative solver of type T for a system of ndimensions. For example, the following code creates an instance of a Newton-Raphson solver, for a 2-dimensional
system of equations:const gsl_multiroot_fdfsolver_type * T = gsl_multiroot_fdfsolver_newton;
gsl_multiroot_fdfsolver * s = gsl_multiroot_fdfsolver_alloc (T, 2);
If there is insufficient memory to create the solver then the function returns a null pointer and the error handler
is invoked with an error code of GSL_ENOMEM.
int gsl_multiroot_fsolver_set(gsl_multiroot_fsolver *s, gsl_multiroot_function *f, const gsl_vector *x)
int gsl_multiroot_fdfsolver_set(gsl_multiroot_fdfsolver *s, gsl_multiroot_function_fdf *fdf, const gsl_vector*x)
These functions set, or reset, an existing solver s to use the function f or function and derivative fdf , and theinitial guess x. Note that the initial position is copied from x, this argument is not modified by subsequent
iterations.
void gsl_multiroot_fsolver_free(gsl_multiroot_fsolver *s)
void gsl_multiroot_fdfsolver_free(gsl_multiroot_fdfsolver *s)
These functions free all the memory associated with the solver s.
const char *gsl_multiroot_fsolver_name(const gsl_multiroot_fsolver *s)
const char *gsl_multiroot_fdfsolver_name(const gsl_multiroot_fdfsolver *s)
These functions return a pointer to the name of the solver. For example:
printf ("s is a '%s' solver\n", gsl_multiroot_fdfsolver_name (s));
would print something like s is a 'newton' solver.

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38.3 Providing the function to solve
You must provide n functions of n variables for the root finders to operate on. In order to allow for general parameters
the functions are defined by the following data types:
type gsl_multiroot_function
This data type defines a general system of functions with parameters.
int (* f) (const gsl_vector * x, void * params, gsl_vector * f)this function should store the vector result f (x, params) in f for argument x and parameters params,
returning an appropriate error code if the function cannot be computed.
size_t n
the dimension of the system, i.e. the number of components of the vectors x and f.
void * params
a pointer to the parameters of the function.
Here is an example using Powell’s test function,
f1 (x) = Ax0 x1 − 1
f2 (x) = exp(−x0 ) + exp(−x1 ) − (1 + 1/A)
with A = 104 . The following code defines a gsl_multiroot_function system F which you could pass to a solver:
struct powell_params { double A; };int
powell (gsl_vector * x, void * p, gsl_vector * f) {struct powell_params * params= (struct powell_params *)p;
const double A = (params->A);
const double x0 = gsl_vector_get(x,0);
const double x1 = gsl_vector_get(x,1);
gsl_vector_set (f, 0, A * x0 * x1 - 1);
gsl_vector_set (f, 1, (exp(-x0) + exp(-x1)- (1.0 + 1.0/A)));
return GSL_SUCCESS}
gsl_multiroot_function F;
struct powell_params params = { 10000.0 };F.f = &powell;
F.n = 2;F.params = &params;
type gsl_multiroot_function_fdf
This data type defines a general system of functions with parameters and the corresponding Jacobian matrix ofderivatives,int (* f) (const gsl_vector * x, void * params, gsl_vector * f)this function should store the vector result f (x, params) in f for argument x and parameters params,
returning an appropriate error code if the function cannot be computed.
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int (* df) (const gsl_vector * x, void * params, gsl_matrix * J)
this function should store the n-by-n matrix result
Jij = ∂fi (x, params)/∂xj
in J for argument x and parameters params, returning an appropriate error code if the function cannot
be computed.
int (* fdf) (const gsl_vector * x, void * params, gsl_vector * f, gsl_matrix * J)This function should set the values of the f and J as above, for arguments x and parameters params.
This function provides an optimization of the separate functions for f (x) and J(x)—it is always fasterto compute the function and its derivative at the same time.
size_t n
the dimension of the system, i.e. the number of components of the vectors x and f.
void * params
a pointer to the parameters of the function.
The example of Powell’s test function defined above can be extended to include analytic derivatives using the following
code:
int
powell_df (gsl_vector * x, void * p, gsl_matrix * J){
struct powell_params * params= (struct powell_params *)p;
const double A = (params->A);
const double x0 = gsl_vector_get(x,0);
const double x1 = gsl_vector_get(x,1);
gsl_matrix_set (J, 0, 0, A * x1);
gsl_matrix_set (J, 0, 1, A * x0);
gsl_matrix_set (J, 1, 0, -exp(-x0));
gsl_matrix_set (J, 1, 1, -exp(-x1));
return GSL_SUCCESS}
int
powell_fdf (gsl_vector * x, void * p,
gsl_matrix * f, gsl_matrix * J) {struct powell_params * params= (struct powell_params *)p;
const double A = (params->A);
const double x0 = gsl_vector_get(x,0);
const double x1 = gsl_vector_get(x,1);
const double u0 = exp(-x0);
const double u1 = exp(-x1);
gsl_vector_set (f, 0, A * x0 * x1 - 1);
gsl_vector_set (f, 1, u0 + u1 - (1 + 1/A));
gsl_matrix_set (J, 0, 0, A * x1);
gsl_matrix_set (J, 0, 1, A * x0);
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GNU Scientific Library, Release 2.7(continued from previous page)gsl_matrix_set (J, 1, 0, -u0);
gsl_matrix_set (J, 1, 1, -u1);
return GSL_SUCCESS}
gsl_multiroot_function_fdf FDF;
FDF.f = &powell_f;
FDF.df = &powell_df;
FDF.fdf = &powell_fdf;
FDF.n = 2;
FDF.params = 0;Note that the function powell_fdf is able to reuse existing terms from the function when calculating the Jacobian,thus saving time.

38.4 Iteration
The following functions drive the iteration of each algorithm. Each function performs one iteration to update the stateof any solver of the corresponding type. The same functions work for all solvers so that different methods can besubstituted at runtime without modifications to the code.
int gsl_multiroot_fsolver_iterate(gsl_multiroot_fsolver *s)
int gsl_multiroot_fdfsolver_iterate(gsl_multiroot_fdfsolver *s)
These functions perform a single iteration of the solver s. If the iteration encounters an unexpected problem thenan error code will be returned,
GSL_EBADFUNC
the iteration encountered a singular point where the function or its derivative evaluated to Inf or NaN.
GSL_ENOPROG
the iteration is not making any progress, preventing the algorithm from continuing.
The solver maintains a current best estimate of the root s->x and its function value s->f at all times. This information
can be accessed with the following auxiliary functions,gsl_vector *gsl_multiroot_fsolver_root(const gsl_multiroot_fsolver *s)
gsl_vector *gsl_multiroot_fdfsolver_root(const gsl_multiroot_fdfsolver *s)
These functions return the current estimate of the root for the solver s, given by s->x.
gsl_vector *gsl_multiroot_fsolver_f(const gsl_multiroot_fsolver *s)
gsl_vector *gsl_multiroot_fdfsolver_f(const gsl_multiroot_fdfsolver *s)
These functions return the function value f (x) at the current estimate of the root for the solver s, given by s->f.
gsl_vector *gsl_multiroot_fsolver_dx(const gsl_multiroot_fsolver *s)
gsl_vector *gsl_multiroot_fdfsolver_dx(const gsl_multiroot_fdfsolver *s)
These functions return the last step dx taken by the solver s, given by s->dx.

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38.5 Search Stopping Parameters
A root finding procedure should stop when one of the following conditions is true:• A multidimensional root has been found to within the user-specified precision.
• A user-specified maximum number of iterations has been reached.
• An error has occurred.
The handling of these conditions is under user control. The functions below allow the user to test the precision of thecurrent result in several standard ways.
int gsl_multiroot_test_delta(const gsl_vector *dx, const gsl_vector *x, double epsabs, double epsrel)This function tests for the convergence of the sequence by comparing the last step dx with the absolute error
epsabs and relative error epsrel to the current position x. The test returns GSL_SUCCESS if the following
condition is achieved,|dxi | < epsabs + epsrel |xi |for each component of x and returns GSL_CONTINUE otherwise.
int gsl_multiroot_test_residual(const gsl_vector *f, double epsabs)
This function tests the residual value f against the absolute error bound epsabs. The test returns GSL_SUCCESS
if the following condition is achieved,X|fi | < epsabsiand returns GSL_CONTINUE otherwise. This criterion is suitable for situations where the precise location of theroot, x, is unimportant provided a value can be found where the residual is small enough.

38.6 Algorithms using Derivatives
The root finding algorithms described in this section make use of both the function and its derivative. They require
an initial guess for the location of the root, but there is no absolute guarantee of convergence—the function must be
suitable for this technique and the initial guess must be sufficiently close to the root for it to work. When the conditions
are satisfied then convergence is quadratic.
type gsl_multiroot_fdfsolver_type
The following are available algorithms for minimizing functions using derivatives.
gsl_multiroot_fdfsolver_type *gsl_multiroot_fdfsolver_hybridsj
This is a modified version of Powell’s Hybrid method as implemented in the HYBRJ algorithm in MINPACK. Minpack was written by Jorge J. Moré, Burton S. Garbow and Kenneth E. Hillstrom. The Hybridalgorithm retains the fast convergence of Newton’s method but will also reduce the residual when Newton’s
method is unreliable.
The algorithm uses a generalized trust region to keep each step under control. In order to be accepted a
proposed new position x0 must satisfy the condition |D(x0 − x)| < δ, where D is a diagonal scaling matrixand δ is the size of the trust region. The components of D are computed internally, using the column
norms of the Jacobian to estimate the sensitivity of the residual to each component of x. This improves the
behavior of the algorithm for badly scaled functions.
On each iteration the algorithm first determines the standard Newton step by solving the system Jdx = −f .
If this step falls inside the trust region it is used as a trial step in the next stage. If not, the algorithm uses

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the linear combination of the Newton and gradient directions which is predicted to minimize the norm of
the function while staying inside the trust region,dx = −αJ −1 f (x) − β∇|f (x)|2This combination of Newton and gradient directions is referred to as a dogleg step.
The proposed step is now tested by evaluating the function at the resulting point, x0 . If the step reduces
the norm of the function sufficiently then it is accepted and size of the trust region is increased. If the
proposed step fails to improve the solution then the size of the trust region is decreased and another trial
step is computed.
The speed of the algorithm is increased by computing the changes to the Jacobian approximately, using arank-1 update. If two successive attempts fail to reduce the residual then the full Jacobian is recomputed.
The algorithm also monitors the progress of the solution and returns an error if several steps fail to makeany improvement,
GSL_ENOPROG
the iteration is not making any progress, preventing the algorithm from continuing.
GSL_ENOPROGJ
re-evaluations of the Jacobian indicate that the iteration is not making any progress, preventing
the algorithm from continuing.
gsl_multiroot_fdfsolver_type *gsl_multiroot_fdfsolver_hybridj
This algorithm is an unscaled version of HYBRIDSJ. The steps are controlled by a spherical trust region|x0 − x| < δ, instead of a generalized region. This can be useful if the generalized region estimated by
HYBRIDSJ is inappropriate.
gsl_multiroot_fdfsolver_type *gsl_multiroot_fdfsolver_newton
Newton’s Method is the standard root-polishing algorithm. The algorithm begins with an initial guess forthe location of the solution. On each iteration a linear approximation to the function F is used to estimate thestep which will zero all the components of the residual. The iteration is defined by the following sequence,x → x0 = x − J −1 f (x)where the Jacobian matrix J is computed from the derivative functions provided by f. The step dx is
obtained by solving the linear system,
Jdx = −f (x)using LU decomposition. If the Jacobian matrix is singular, an error code of GSL_EDOM is returned.
gsl_multiroot_fdfsolver_type *gsl_multiroot_fdfsolver_gnewton
This is a modified version of Newton’s method which attempts to improve global convergence by requiring
every step to reduce the Euclidean norm of the residual, |f (x)|. If the Newton step leads to an increase in
the norm then a reduced step of relative size,√t = ( 1 + 6r − 1)/(3r)is proposed, with r being the ratio of norms |f (x0 )|2 /|f (x)|2 . This procedure is repeated until a suitablestep size is found.

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38.7 Algorithms without Derivatives
The algorithms described in this section do not require any derivative information to be supplied by the user. Any
derivatives needed are approximated by finite differences. Note that if the finite-differencing step size chosen by theseroutines is inappropriate, an explicit user-supplied numerical derivative can always be used with the algorithms described in the previous section.
type gsl_multiroot_fsolver_type
The following are available algorithms for minimizing functions without derivatives.
gsl_multiroot_fsolver_type *gsl_multiroot_fsolver_hybrids
This is a version of the Hybrid algorithm which replaces calls to the Jacobian function by its finite difference
approximation. The finite difference approximation is computed using gsl_multiroots_fdjac() with arelative step size of GSL_SQRT_DBL_EPSILON. Note that this step size will not be suitable for all problems.
gsl_multiroot_fsolver_type *gsl_multiroot_fsolver_hybridThis is a finite difference version of the Hybrid algorithm without internal scaling.
gsl_multiroot_fsolver_type *gsl_multiroot_fsolver_dnewton
The discrete Newton algorithm is the simplest method of solving a multidimensional system. It uses the
Newton iterationx → x − J −1 f (x)where the Jacobian matrix J is approximated by taking finite differences of the function f. The approximation scheme used by this implementation is,
Jij = (fi (x + δj ) − fi (x))/δj√
where δj is a step of size |xj | with  being the machine precision ( ≈ 2.22 × 10−16 ). The order ofconvergence of Newton’s algorithm is quadratic, but the finite differences require n2 function evaluations
on each iteration. The algorithm may become unstable if the finite differences are not a good approximation
to the true derivatives.
gsl_multiroot_fsolver_type *gsl_multiroot_fsolver_broyden
The Broyden algorithm is a version of the discrete Newton algorithm which attempts to avoids the expensive
update of the Jacobian matrix on each iteration. The changes to the Jacobian are also approximated, usinga rank-1 update,J −1 → J −1 − (J −1 df − dx)dxT J −1 /dxT J −1 df
where the vectors dx and df are the changes in x and f . On the first iteration the inverse Jacobian isestimated using finite differences, as in the discrete Newton algorithm.
This approximation gives a fast update but is unreliable if the changes are not small, and the estimate of theinverse Jacobian becomes worse as time passes. The algorithm has a tendency to become unstable unlessit starts close to the root. The Jacobian is refreshed if this instability is detected (consult the source fordetails).
This algorithm is included only for demonstration purposes, and is not recommended for serious use.

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38.8 Examples
The multidimensional solvers are used in a similar way to the one-dimensional root finding algorithms. This firstexample demonstrates the HYBRIDS scaled-hybrid algorithm, which does not require derivatives. The program solves
the Rosenbrock system of equations,f1 (x, y) = a(1 − x)
f2 (x, y) = b(y − x2 )with a = 1, b = 10. The solution of this system lies at (x, y) = (1, 1) in a narrow valley.
The first stage of the program is to define the system of equations:#include <stdlib.h>#include <stdio.h>#include <gsl/gsl_vector.h>#include <gsl/gsl_multiroots.h>
struct rparams{
double a;double b;};
introsenbrock_f (const gsl_vector * x, void *params,
gsl_vector * f){double a = ((struct rparams *) params)->a;
double b = ((struct rparams *) params)->b;
const double x0 = gsl_vector_get (x, 0);const double x1 = gsl_vector_get (x, 1);const double y0 = a * (1 - x0);
const double y1 = b * (x1 - x0 * x0);
gsl_vector_set (f, 0, y0);
gsl_vector_set (f, 1, y1);
return GSL_SUCCESS;}
The main program begins by creating the function object f, with the arguments (x,y) and parameters (a,b). The
solver s is initialized to use this function, with the gsl_multiroot_fsolver_hybrids method:int
main (void){
const gsl_multiroot_fsolver_type *T;
gsl_multiroot_fsolver *s;
int status;
size_t i, iter = 0;
(continues on next page)

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const size_t n = 2;
struct rparams p = {1.0, 10.0};
gsl_multiroot_function f = {&rosenbrock_f, n, &p};double x_init[2] = {-10.0, -5.0};
gsl_vector *x = gsl_vector_alloc (n);gsl_vector_set (x, 0, x_init[0]);gsl_vector_set (x, 1, x_init[1]);T = gsl_multiroot_fsolver_hybrids;s = gsl_multiroot_fsolver_alloc (T, 2);
gsl_multiroot_fsolver_set (s, &f, x);
print_state (iter, s);do{iter++;
status = gsl_multiroot_fsolver_iterate (s);print_state (iter, s);if (status)
break;/* check if solver is stuck */status =gsl_multiroot_test_residual (s->f, 1e-7);}
while (status == GSL_CONTINUE && iter < 1000);
printf ("status = %s\n", gsl_strerror (status));gsl_multiroot_fsolver_free (s);gsl_vector_free (x);return 0;}
Note that it is important to check the return status of each solver step, in case the algorithm becomes stuck. If an error
condition is detected, indicating that the algorithm cannot proceed, then the error can be reported to the user, a newstarting point chosen or a different algorithm used.
The intermediate state of the solution is displayed by the following function. The solver state contains the vector s->x
which is the current position, and the vector s->f with corresponding function values:
int
print_state (size_t iter, gsl_multiroot_fsolver * s){printf ("iter = %3u x = % .3f % .3f ""f(x) = % .3e % .3e\n",iter,
gsl_vector_get (s->x, 0),(continues on next page)464

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GNU Scientific Library, Release 2.7(continued from previous page)gsl_vector_get (s->x, 1),
gsl_vector_get (s->f, 0),
gsl_vector_get (s->f, 1));}
Here are the results of running the program. The algorithm is started at (−10, −5) far from the solution. Since thesolution is hidden in a narrow valley the earliest steps follow the gradient of the function downhill, in an attempt toreduce the large value of the residual. Once the root has been approximately located, on iteration 8, the Newton behavior
takes over and convergence is very rapid:
iter = 0 x = -10.000
iter = 1 x = -10.000
iter = 2 x = -3.976
iter = 3 x = -3.976
iter = 4 x = -3.976
iter = 5 x = -1.274
iter = 6 x = -1.274
iter = 7 x =0.249
iter = 8 x =0.249
iter = 9 x =1.000
iter = 10 x =1.000
iter = 11 x =1.000
status = success-5.000-5.00024.82724.82724.827-5.680-5.680
0.298
0.298
0.878
0.989
1.000

f(x) = 1.100e+01 -1.050e+03
f(x) = 1.100e+01 -1.050e+03
f(x) = 4.976e+00 9.020e+01
f(x) = 4.976e+00 9.020e+01
f(x) = 4.976e+00 9.020e+01
f(x) = 2.274e+00 -7.302e+01
f(x) = 2.274e+00 -7.302e+01
f(x) = 7.511e-01 2.359e+00
f(x) = 7.511e-01 2.359e+00
f(x) = 1.268e-10 -1.218e+00
f(x) = 1.124e-11 -1.080e-01
f(x) = 0.000e+00 0.000e+00

Note that the algorithm does not update the location on every iteration. Some iterations are used to adjust the trust-region
parameter, after trying a step which was found to be divergent, or to recompute the Jacobian, when poor convergence
behavior is detected.
The next example program adds derivative information, in order to accelerate the solution. There are two derivative
functions rosenbrock_df and rosenbrock_fdf. The latter computes both the function and its derivative simultaneously. This allows the optimization of any common terms. For simplicity we substitute calls to the separate f and df
functions at this point in the code below:
introsenbrock_df (const gsl_vector * x, void *params,
gsl_matrix * J){const double a = ((struct rparams *) params)->a;
const double b = ((struct rparams *) params)->b;
const double x0 = gsl_vector_get (x, 0);const double df00 = -a;const double df01 = 0;
const double df10 = -2 * bconst double df11 = b;

* x0;

gsl_matrix_set (J, 0, 0, df00);
gsl_matrix_set (J, 0, 1, df01);
gsl_matrix_set (J, 1, 0, df10);
gsl_matrix_set (J, 1, 1, df11);
return GSL_SUCCESS;(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)}int
rosenbrock_fdf (const gsl_vector * x, void *params,
gsl_vector * f, gsl_matrix * J){rosenbrock_f (x, params, f);rosenbrock_df (x, params, J);
return GSL_SUCCESS;}
The main program now makes calls to the corresponding fdfsolver versions of the functions:int
main (void){
const gsl_multiroot_fdfsolver_type *T;
gsl_multiroot_fdfsolver *s;
int status;
size_t i, iter = 0;
const size_t n = 2;
struct rparams p = {1.0, 10.0};
gsl_multiroot_function_fdf f = {&rosenbrock_f,&rosenbrock_df,&rosenbrock_fdf,
n, &p};
double x_init[2] = {-10.0, -5.0};
gsl_vector *x = gsl_vector_alloc (n);gsl_vector_set (x, 0, x_init[0]);gsl_vector_set (x, 1, x_init[1]);T = gsl_multiroot_fdfsolver_gnewton;s = gsl_multiroot_fdfsolver_alloc (T, n);gsl_multiroot_fdfsolver_set (s, &f, x);
print_state (iter, s);do{iter++;
status = gsl_multiroot_fdfsolver_iterate (s);print_state (iter, s);if (status)
break;(continues on next page)

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status = gsl_multiroot_test_residual (s->f, 1e-7);}
while (status == GSL_CONTINUE && iter < 1000);
printf ("status = %s\n", gsl_strerror (status));gsl_multiroot_fdfsolver_free (s);gsl_vector_free (x);return 0;}
The addition of derivative information to the gsl_multiroot_fsolver_hybrids solver does not make any significant
difference to its behavior, since it able to approximate the Jacobian numerically with sufficient accuracy. To illustrate
the behavior of a different derivative solver we switch to gsl_multiroot_fdfsolver_gnewton. This is a traditional
Newton solver with the constraint that it scales back its step if the full step would lead “uphill”. Here is the output for
the gsl_multiroot_fdfsolver_gnewton algorithm:
iter = 0 x = -10.000 -5.000 f(x) = 1.100e+01 -1.050e+03
iter = 1 x = -4.231 -65.317 f(x) = 5.231e+00 -8.321e+02
iter = 2 x =1.000 -26.358 f(x) = -8.882e-16 -2.736e+02
iter = 3 x =1.000
1.000 f(x) = -2.220e-16 -4.441e-15
status = success
The convergence is much more rapid, but takes a wide excursion out to the point (−4.23, −65.3). This could cause thealgorithm to go astray in a realistic application. The hybrid algorithm follows the downhill path to the solution morereliably.

38.9 References and Further Reading
The original version of the Hybrid method is described in the following articles by Powell,• M.J.D. Powell, “A Hybrid Method for Nonlinear Equations” (Chap 6, p 87–114) and “A Fortran Subroutine forSolving systems of Nonlinear Algebraic Equations” (Chap 7, p 115–161), in Numerical Methods for Nonlinear
Algebraic Equations, P. Rabinowitz, editor. Gordon and Breach, 1970.
The following papers are also relevant to the algorithms described in this section,• J.J. Moré, M.Y. Cosnard, “Numerical Solution of Nonlinear Equations”, ACM Transactions on Mathematical
Software, Vol 5, No 1, (1979), p 64–85• C.G. Broyden, “A Class of Methods for Solving Nonlinear Simultaneous Equations”, Mathematics of Computation, Vol 19 (1965), p 577–593• J.J. Moré, B.S. Garbow, K.E. Hillstrom, “Testing Unconstrained Optimization Software”, ACM Transactions onMathematical Software, Vol 7, No 1 (1981), p 17–41

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CHAPTERTHIRTYNINEMULTIDIMENSIONAL MINIMIZATIONThis chapter describes routines for finding minima of arbitrary multidimensional functions. The library provides low
level components for a variety of iterative minimizers and convergence tests. These can be combined by the user
to achieve the desired solution, while providing full access to the intermediate steps of the algorithms. Each class of
methods uses the same framework, so that you can switch between minimizers at runtime without needing to recompile
your program. Each instance of a minimizer keeps track of its own state, allowing the minimizers to be used in multithreaded programs. The minimization algorithms can be used to maximize a function by inverting its sign.
The header file gsl_multimin.h contains prototypes for the minimization functions and related declarations.

39.1 Overview
The problem of multidimensional minimization requires finding a point x such that the scalar function,f (x1 , . . . , xn )takes a value which is lower than at any neighboring point. For smooth functions the gradient g = ∇f vanishes at theminimum. In general there are no bracketing methods available for the minimization of n-dimensional functions. The
algorithms proceed from an initial guess using a search algorithm which attempts to move in a downhill direction.
Algorithms making use of the gradient of the function perform a one-dimensional line minimisation along this direction
until the lowest point is found to a suitable tolerance. The search direction is then updated with local information fromthe function and its derivatives, and the whole process repeated until the true n-dimensional minimum is found.
Algorithms which do not require the gradient of the function use different strategies. For example, the Nelder-Mead
Simplex algorithm maintains n+1 trial parameter vectors as the vertices of a n-dimensional simplex. On each iteration
it tries to improve the worst vertex of the simplex by geometrical transformations. The iterations are continued until
the overall size of the simplex has decreased sufficiently.
Both types of algorithms use a standard framework. The user provides a high-level driver for the algorithms, and thelibrary provides the individual functions necessary for each of the steps. There are three main phases of the iteration.
The steps are,• initialize minimizer state, s, for algorithm T• update s using the iteration T• test s for convergence, and repeat iteration if necessary
Each iteration step consists either of an improvement to the line-minimisation in the current direction or an updateto the search direction itself. The state for the minimizers is held in a gsl_multimin_fdfminimizer struct or a
gsl_multimin_fminimizer struct.

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39.2 Caveats
Note that the minimization algorithms can only search for one local minimum at a time. When there are several local
minima in the search area, the first minimum to be found will be returned; however it is difficult to predict which of theminima this will be. In most cases, no error will be reported if you try to find a local minimum in an area where there
is more than one.
It is also important to note that the minimization algorithms find local minima; there is no way to determine whether aminimum is a global minimum of the function in question.

39.3 Initializing the Multidimensional Minimizer
The following function initializes a multidimensional minimizer. The minimizer itself depends only on the dimensionof the problem and the algorithm and can be reused for different problems.
type gsl_multimin_fdfminimizer
This is a workspace for minimizing functions using derivatives.
type gsl_multimin_fminimizer
This is a workspace for minimizing functions without derivatives.
gsl_multimin_fdfminimizer *gsl_multimin_fdfminimizer_alloc(const gsl_multimin_fdfminimizer_type *T,
size_t n)
gsl_multimin_fminimizer *gsl_multimin_fminimizer_alloc(const gsl_multimin_fminimizer_type *T, size_t n)This function returns a pointer to a newly allocated instance of a minimizer of type T for an n-dimension function.
If there is insufficient memory to create the minimizer then the function returns a null pointer and the error handler
is invoked with an error code of GSL_ENOMEM.
int gsl_multimin_fdfminimizer_set(gsl_multimin_fdfminimizer *s, gsl_multimin_function_fdf *fdf, const
gsl_vector *x, double step_size, double tol)
int gsl_multimin_fminimizer_set(gsl_multimin_fminimizer *s, gsl_multimin_function *f, const gsl_vector *x,
const gsl_vector *step_size)
The function gsl_multimin_fdfminimizer_set() initializes the minimizer s to minimize the function fdf
starting from the initial point x. The size of the first trial step is given by step_size. The accuracy of the line
minimization is specified by tol. The precise meaning of this parameter depends on the method used. Typicallythe line minimization is considered successful if the gradient of the function g is orthogonal to the current search
direction p to a relative accuracy of tol, where p·g < tol|p||g|. A tol value of 0.1 is suitable for most purposes,
since line minimization only needs to be carried out approximately. Note that setting tol to zero will force theuse of “exact” line-searches, which are extremely expensive.
The function gsl_multimin_fminimizer_set() initializes the minimizer s to minimize the function f , starting from the initial point x. The size of the initial trial steps is given in vector step_size. The precise meaningof this parameter depends on the method used.
void gsl_multimin_fdfminimizer_free(gsl_multimin_fdfminimizer *s)
void gsl_multimin_fminimizer_free(gsl_multimin_fminimizer *s)
This function frees all the memory associated with the minimizer s.
const char *gsl_multimin_fdfminimizer_name(const gsl_multimin_fdfminimizer *s)
const char *gsl_multimin_fminimizer_name(const gsl_multimin_fminimizer *s)
This function returns a pointer to the name of the minimizer. For example:
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printf ("s is a '%s' minimizer\n", gsl_multimin_fdfminimizer_name (s));
would print something like s is a 'conjugate_pr' minimizer.

39.4 Providing a function to minimize
You must provide a parametric function of n variables for the minimizers to operate on. You may also need to provide
a routine which calculates the gradient of the function and a third routine which calculates both the function value andthe gradient together. In order to allow for general parameters the functions are defined by the following data types:
type gsl_multimin_function_fdf
This data type defines a general function of n variables with parameters and the corresponding gradient vector
of derivatives,
double (* f) (const gsl_vector * x, void * params)
this function should return the result f (x, params) for argument x and parameters params. If thefunction cannot be computed, an error value of GSL_NAN should be returned.
void (* df) (const gsl_vector * x, void * params, gsl_vector * g)this function should store the n-dimensional gradientgi = ∂f (x, params)/∂xi
in the vector g for argument x and parameters params, returning an appropriate error code if thefunction cannot be computed.
void (* fdf) (const gsl_vector * x, void * params, double * f, gsl_vector * g)This function should set the values of the f and g as above, for arguments x and parameters params.
This function provides an optimization of the separate functions for f (x) and g(x)—it is always fasterto compute the function and its derivative at the same time.
size_t n
the dimension of the system, i.e. the number of components of the vectors x.
void * params
a pointer to the parameters of the function.
type gsl_multimin_function
This data type defines a general function of n variables with parameters,
double (* f) (const gsl_vector * x, void * params)
this function should return the result f (x, params) for argument x and parameters params. If thefunction cannot be computed, an error value of GSL_NAN should be returned.
size_t n
the dimension of the system, i.e. the number of components of the vectors x.
void * params
a pointer to the parameters of the function.
The following example function defines a simple two-dimensional paraboloid with five parameters,

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GNU Scientific Library, Release 2.7/* Paraboloid centered on (p[0],p[1]), withscale factors (p[2],p[3]) and minimum p[4] */doublemy_f (const gsl_vector *v, void *params){
double x, y;
double *p = (double *)params;
x = gsl_vector_get(v, 0);
y = gsl_vector_get(v, 1);
return p[2] * (x - p[0]) * (x - p[0]) +p[3] * (y - p[1]) * (y - p[1]) + p[4];}/* The gradient of f, df = (df/dx, df/dy). */void
my_df (const gsl_vector *v, void *params,
gsl_vector *df){
double x, y;
double *p = (double *)params;
x = gsl_vector_get(v, 0);
y = gsl_vector_get(v, 1);
gsl_vector_set(df, 0, 2.0 * p[2] * (x - p[0]));gsl_vector_set(df, 1, 2.0 * p[3] * (y - p[1]));}/* Compute both f and df together. */void
my_fdf (const gsl_vector *x, void *params,
double *f, gsl_vector *df){
*f = my_f(x, params);
my_df(x, params, df);}
The function can be initialized using the following code:
gsl_multimin_function_fdf my_func;/* Paraboloid center at (1,2), scale factors (10, 20),
minimum value 30 */double p[5] = { 1.0, 2.0, 10.0, 20.0, 30.0 };my_func.n = 2; /* number of function components */my_func.f = &my_f;
my_func.df = &my_df;
my_func.fdf = &my_fdf;
my_func.params = (void *)p;472

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39.5 Iteration
The following function drives the iteration of each algorithm. The function performs one iteration to update the stateof the minimizer. The same function works for all minimizers so that different methods can be substituted at runtime
without modifications to the code.
int gsl_multimin_fdfminimizer_iterate(gsl_multimin_fdfminimizer *s)
int gsl_multimin_fminimizer_iterate(gsl_multimin_fminimizer *s)
These functions perform a single iteration of the minimizer s. If the iteration encounters an unexpected problem
then an error code will be returned. The error code GSL_ENOPROG signifies that the minimizer is unable toimprove on its current estimate, either due to numerical difficulty or because a genuine local minimum has beenreached.
The minimizer maintains a current best estimate of the minimum at all times. This information can be accessed withthe following auxiliary functions,gsl_vector *gsl_multimin_fdfminimizer_x(const gsl_multimin_fdfminimizer *s)
gsl_vector *gsl_multimin_fminimizer_x(const gsl_multimin_fminimizer *s)
double gsl_multimin_fdfminimizer_minimum(const gsl_multimin_fdfminimizer *s)
double gsl_multimin_fminimizer_minimum(const gsl_multimin_fminimizer *s)
gsl_vector *gsl_multimin_fdfminimizer_gradient(const gsl_multimin_fdfminimizer *s)
gsl_vector *gsl_multimin_fdfminimizer_dx(const gsl_multimin_fdfminimizer *s)
double gsl_multimin_fminimizer_size(const gsl_multimin_fminimizer *s)
These functions return the current best estimate of the location of the minimum, the value of the function atthat point, its gradient, the last step increment of the estimate, and minimizer specific characteristic size for theminimizer s.
int gsl_multimin_fdfminimizer_restart(gsl_multimin_fdfminimizer *s)
This function resets the minimizer s to use the current point as a new starting point.

39.6 Stopping Criteria
A minimization procedure should stop when one of the following conditions is true:• A minimum has been found to within the user-specified precision.
• A user-specified maximum number of iterations has been reached.
• An error has occurred.
The handling of these conditions is under user control. The functions below allow the user to test the precision of thecurrent result.
int gsl_multimin_test_gradient(const gsl_vector *g, double epsabs)
This function tests the norm of the gradient g against the absolute tolerance epsabs. The gradient of a multidimensional function goes to zero at a minimum. The test returns GSL_SUCCESS if the following condition is39.5. Iteration

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achieved,|g| < epsabs
and returns GSL_CONTINUE otherwise. A suitable choice of epsabs can be made from the desired accuracy inthe function for small variations in x. The relationship between these quantities is given by δf = g δx.
int gsl_multimin_test_size(const double size, double epsabs)
This function tests the minimizer specific characteristic size (if applicable to the used minimizer) against absolute
tolerance epsabs. The test returns GSL_SUCCESS if the size is smaller than tolerance, otherwise GSL_CONTINUE
is returned.

39.7 Algorithms with Derivatives
There are several minimization methods available. The best choice of algorithm depends on the problem. The algorithms described in this section use the value of the function and its gradient at each evaluation point.
type gsl_multimin_fdfminimizer_type
This type specifies a minimization algorithm using gradients.
gsl_multimin_fdfminimizer_type *gsl_multimin_fdfminimizer_conjugate_fr
This is the Fletcher-Reeves conjugate gradient algorithm. The conjugate gradient algorithm proceeds as a
succession of line minimizations. The sequence of search directions is used to build up an approximation
to the curvature of the function in the neighborhood of the minimum.
An initial search direction p is chosen using the gradient, and line minimization is carried out in that
direction. The accuracy of the line minimization is specified by the parameter tol. The minimum along thisline occurs when the function gradient g and the search direction p are orthogonal. The line minimization
terminates when p · g < tol|p||g|. The search direction is updated using the Fletcher-Reeves formulap0 = g 0 − βp where β = −|g 0 |2 /|g|2 , and the line minimization is then repeated for the new search
direction.
gsl_multimin_fdfminimizer_type *gsl_multimin_fdfminimizer_conjugate_pr
This is the Polak-Ribiere conjugate gradient algorithm. It is similar to the Fletcher-Reeves method, differing
only in the choice of the coefficient β. Both methods work well when the evaluation point is close enoughto the minimum of the objective function that it is well approximated by a quadratic hypersurface.
gsl_multimin_fdfminimizer_type *gsl_multimin_fdfminimizer_vector_bfgs2
gsl_multimin_fdfminimizer_type *gsl_multimin_fdfminimizer_vector_bfgs
These methods use the vector Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. This is a quasiNewton method which builds up an approximation to the second derivatives of the function f using thedifference between successive gradient vectors. By combining the first and second derivatives the algorithm is able to take Newton-type steps towards the function minimum, assuming quadratic behavior inthat region.
The bfgs2 version of this minimizer is the most efficient version available, and is a faithful implementation of the line minimization scheme described in Fletcher’s Practical Methods of Optimization, Algorithms
2.6.2 and 2.6.4. It supersedes the original bfgs routine and requires substantially fewer function and gradient evaluations. The user-supplied tolerance tol corresponds to the parameter σ used by Fletcher. A valueof 0.1 is recommended for typical use (larger values correspond to less accurate line searches).
gsl_multimin_fdfminimizer_type *gsl_multimin_fdfminimizer_steepest_descent
The steepest descent algorithm follows the downhill gradient of the function at each step. When a downhill
step is successful the step-size is increased by a factor of two. If the downhill step leads to a higher function
value then the algorithm backtracks and the step size is decreased using the parameter tol. A suitablevalue of tol for most applications is 0.1. The steepest descent method is inefficient and is included only
for demonstration purposes.
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39.8 Algorithms without Derivatives
The algorithms described in this section use only the value of the function at each evaluation point.
type gsl_multimin_fminimizer_type
This type specifies minimization algorithms which do not use gradients.
gsl_multimin_fminimizer_type *gsl_multimin_fminimizer_nmsimplex2
gsl_multimin_fminimizer_type *gsl_multimin_fminimizer_nmsimplex
These methods use the Simplex algorithm of Nelder and Mead. Starting from the initial vector x = p0 , thealgorithm constructs an additional n vectors pi using the step size vector s = step_size as follows:
p0 = (x0 , x1 , · · · , xn )p1 = (x0 + s0 , x1 , · · · , xn )p2 = (x0 , x1 + s1 , · · · , xn )... = ...
pn = (x0 , x1 , · · · , xn + sn )These vectors form the n + 1 vertices of a simplex in n dimensions. On each iteration the algorithm
uses simple geometrical transformations to update the vector corresponding to the highest function value.
The geometric transformations are reflection, reflection followed by expansion, contraction and multiple
contraction. Using these transformations the simplex moves through the space towards the minimum, where
it contracts itself.
After each iteration, the best vertex is returned. Note, that due to the nature of the algorithm not every step
improves the current best parameter vector. Usually several iterations are required.
The minimizer-specific characteristic size is calculated as the average distance from the geometrical centerof the simplex to all its vertices. This size can be used as a stopping criteria, as the simplex contracts itself
near the minimum. The size is returned by the function gsl_multimin_fminimizer_size().
The gsl_multimin_fminimizer_nmsimplex2 version of this minimiser is a new O(N ) operations implementation of the earlier O(N 2 ) operations gsl_multimin_fminimizer_nmsimplex minimiser. It
uses the same underlying algorithm, but the simplex updates are computed more efficiently for highdimensional problems. In addition, the size of simplex is calculated as the RMS distance of each vertex
from the center rather than the mean distance, allowing a linear update of this quantity on each step. The
memory usage is O(N 2 ) for both algorithms.
gsl_multimin_fminimizer_type *gsl_multimin_fminimizer_nmsimplex2randThis method is a variant of gsl_multimin_fminimizer_nmsimplex2 which initialises the simplex
around the starting point x using a randomly-oriented set of basis vectors instead of the fixed coordinate axes. The final dimensions of the simplex are scaled along the coordinate axes by the vector step_size. The randomisation uses a simple deterministic generator so that repeated calls to
gsl_multimin_fminimizer_set() for a given solver object will vary the orientation in a well-definedway.

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39.9 Examples
This example program finds the minimum of the paraboloid function defined earlier. The location of the minimum isoffset from the origin in x and y, and the function value at the minimum is non-zero. The main program is given below,
it requires the example function given earlier in this chapter.
int
main (void){
size_t iter = 0;
int status;
const gsl_multimin_fdfminimizer_type *T;
gsl_multimin_fdfminimizer *s;/* Position of the minimum (1,2), scale factors
10,20, height 30. */double par[5] = { 1.0, 2.0, 10.0, 20.0, 30.0 };gsl_vector *x;
gsl_multimin_function_fdf my_func;
my_func.n = 2;
my_func.f = my_f;
my_func.df = my_df;
my_func.fdf = my_fdf;
my_func.params = par;/* Starting point, x = (5,7) */x = gsl_vector_alloc (2);gsl_vector_set (x, 0, 5.0);
gsl_vector_set (x, 1, 7.0);
T = gsl_multimin_fdfminimizer_conjugate_fr;
s = gsl_multimin_fdfminimizer_alloc (T, 2);
gsl_multimin_fdfminimizer_set (s, &my_func, x, 0.01, 1e-4);
do{iter++;
status = gsl_multimin_fdfminimizer_iterate (s);if (status)
break;
status = gsl_multimin_test_gradient (s->gradient, 1e-3);
if (status == GSL_SUCCESS)
printf ("Minimum found at:\n");printf ("%5d %.5f %.5f %10.5f \n", iter,
gsl_vector_get (s->x, 0),(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)gsl_vector_get (s->x, 1),s->f);}
while (status == GSL_CONTINUE && iter < 100);
gsl_multimin_fdfminimizer_free (s);gsl_vector_free (x);return 0;}
The initial step-size is chosen as 0.01, a conservative estimate in this case, and the line minimization parameter is set
at 0.0001. The program terminates when the norm of the gradient has been reduced below 0.001. The output of the
program is shown below,xy
1 4.99629 6.99072
2 4.98886 6.97215
3 4.97400 6.93501
4 4.94429 6.86073
5 4.88487 6.71217
6 4.76602 6.41506
7 4.52833 5.82083
8 4.05295 4.63238
9 3.10219 2.25548
10 2.85185 1.62963
11 2.19088 1.76182
12 0.86892 2.02622
Minimum found at:
13 1.00000 2.00000

f
687.84780
683.55456
675.01278
658.10798
625.01340
561.68440
446.46694
261.79422
75.49762
67.03704
45.31640
30.18555
30.00000

Note that the algorithm gradually increases the step size as it successfully moves downhill, as can be seen by plotting
the successive points in Fig. 39.1.
The conjugate gradient algorithm finds the minimum on its second direction because the function is purely quadratic.
Additional iterations would be needed for a more complicated function.
Here is another example using the Nelder-Mead Simplex algorithm to minimize the same example object function, as
above.
int
main(void){
double par[5] = {1.0, 2.0, 10.0, 20.0, 30.0};
const gsl_multimin_fminimizer_type *T =gsl_multimin_fminimizer_nmsimplex2;
gsl_multimin_fminimizer *s = NULL;
gsl_vector *ss, *x;
gsl_multimin_function minex_func;
size_t iter = 0;
(continues on next page)

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Fig. 39.1: Function contours with path taken by minimization algorithm(continued from previous page)

int status;
double size;/* Starting point */x = gsl_vector_alloc (2);gsl_vector_set (x, 0, 5.0);
gsl_vector_set (x, 1, 7.0);/* Set initial step sizes to 1 */ss = gsl_vector_alloc (2);gsl_vector_set_all (ss, 1.0);/* Initialize method and iterate */minex_func.n = 2;
minex_func.f = my_f;minex_func.params = par;s = gsl_multimin_fminimizer_alloc (T, 2);
gsl_multimin_fminimizer_set (s, &minex_func, x, ss);
do{iter++;
status = gsl_multimin_fminimizer_iterate(s);if (status)
break;
size = gsl_multimin_fminimizer_size (s);status = gsl_multimin_test_size (size, 1e-2);(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)if (status == GSL_SUCCESS){
printf ("converged to minimum at\n");}printf ("%5d %10.3e %10.3e f() = %7.3f size = %.3f \n",iter,
gsl_vector_get (s->x, 0),
gsl_vector_get (s->x, 1),s->fval, size);}
while (status == GSL_CONTINUE && iter < 100);
gsl_vector_free(x);gsl_vector_free(ss);gsl_multimin_fminimizer_free (s);return status;}
The minimum search stops when the Simplex size drops to 0.01. The output is shown below.
1 6.500e+00 5.000e+00 f() = 512.500 size = 1.130
2 5.250e+00 4.000e+00 f() = 290.625 size = 1.409
3 5.250e+00 4.000e+00 f() = 290.625 size = 1.409
4 5.500e+00 1.000e+00 f() = 252.500 size = 1.409
5 2.625e+00 3.500e+00 f() = 101.406 size = 1.847
6 2.625e+00 3.500e+00 f() = 101.406 size = 1.847
7 0.000e+00 3.000e+00 f() = 60.000 size = 1.847
8 2.094e+00 1.875e+00 f() = 42.275 size = 1.321
9 2.578e-01 1.906e+00 f() = 35.684 size = 1.069
10 5.879e-01 2.445e+00 f() = 35.664 size = 0.841
11 1.258e+00 2.025e+00 f() = 30.680 size = 0.476
12 1.258e+00 2.025e+00 f() = 30.680 size = 0.367
13 1.093e+00 1.849e+00 f() = 30.539 size = 0.300
14 8.830e-01 2.004e+00 f() = 30.137 size = 0.172
15 8.830e-01 2.004e+00 f() = 30.137 size = 0.126
16 9.582e-01 2.060e+00 f() = 30.090 size = 0.106
17 1.022e+00 2.004e+00 f() = 30.005 size = 0.063
18 1.022e+00 2.004e+00 f() = 30.005 size = 0.043
19 1.022e+00 2.004e+00 f() = 30.005 size = 0.043
20 1.022e+00 2.004e+00 f() = 30.005 size = 0.027
21 1.022e+00 2.004e+00 f() = 30.005 size = 0.022
22 9.920e-01 1.997e+00 f() = 30.001 size = 0.016
23 9.920e-01 1.997e+00 f() = 30.001 size = 0.013
converged to minimum at
24 9.920e-01 1.997e+00 f() = 30.001 size = 0.008
The simplex size first increases, while the simplex moves towards the minimum. After a while the size begins todecrease as the simplex contracts around the minimum.

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39.10 References and Further Reading
The conjugate gradient and BFGS methods are described in detail in the following book,• R. Fletcher, Practical Methods of Optimization (Second Edition) Wiley (1987), ISBN 0471915475.
A brief description of multidimensional minimization algorithms and more recent references can be found in,• C.W. Ueberhuber, Numerical Computation (Volume 2), Chapter 14, Section 4.4 “Minimization Methods”, p.:
325–335, Springer (1997), ISBN 3-540-62057-5.
The simplex algorithm is described in the following paper,• J.A. Nelder and R. Mead, A simplex method for function minimization, Computer Journal vol.: 7 (1965), 308–313.

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CHAPTERFORTYLINEAR LEAST-SQUARES FITTING

This chapter describes routines for performing least squares fits to experimental data using linear combinations offunctions. The data may be weighted or unweighted, i.e. with known or unknown errors. For weighted data thefunctions compute the best fit parameters and their associated covariance matrix. For unweighted data the covariancematrix is estimated from the scatter of the points, giving a variance-covariance matrix.
The functions are divided into separate versions for simple one- or two-parameter regression and multiple-parameter
fits.

40.1 Overview
Least-squares fits are found by minimizing χ2 (chi-squared), the weighted sum of squared residuals over n experimental
datapoints (xi , yi ) for the model Y (c, x),Xχ2 =wi (yi − Y (c, xi ))2i

The p parameters of the model are c = {c0 , c1 , . . . }. The weight factors wi are given by wi = 1/σi2 where σi is theexperimental error on the data-point yi . The errors are assumed to be Gaussian and uncorrelated. For unweighted data
the chi-squared sum is computed without any weight factors.
The fitting routines return the best-fit parameters c and their p × p covariance matrix. The covariance matrix measures
the statistical errors on the best-fit parameters resulting from the errors on the data, σi , and is defined as
Cab = hδca δcb i
where h i denotes an average over the Gaussian error distributions of the underlying datapoints.
The covariance matrix is calculated by error propagation from the data errors σi . The change in a fitted parameter δcacaused by a small change in the data δyi is given by
δca =

X ∂ca
i

∂yi

δyi

allowing the covariance matrix to be written in terms of the errors on the data,Cab =X ∂ca ∂cbi,j∂yi ∂yjhδyi δyj i

For uncorrelated data the fluctuations of the underlying datapoints satisfyhδyi δyj i = σi2 δij

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giving a corresponding parameter covariance matrix ofCab =X 1 ∂ca ∂cbiwi ∂yi ∂yi

When computing the covariance matrix for unweighted data, i.e. data with unknown errors, the weight factors wi inthis sum are replacedPby the single estimate w = 1/σ 2 , where σ 2 is the computed variance of the residuals about the2best-fit model, σ = (yi − Y (c, xi ))2 /(n − p). This is referred to as the variance-covariance matrix.
The standard deviations of the best-fit√ parameters are given by the square root of the corresponding diagonal elementsof the√covariance matrix, σca = Caa . The correlation coefficient of the fit parameters ca and cb is given by ρab =Cab / Caa Cbb .

40.2 Linear regression
The functions in this section are used to fit simple one or two parameter linear regression models. The functions are
declared in the header file gsl_fit.h.

40.2.1 Linear regression with a constant term
The functions described in this section can be used to perform least-squares fits to a straight line model, Y (c, x) =c0 + c1 x.
int gsl_fit_linear(const double *x, const size_t xstride, const double *y, const size_t ystride, size_t n, double*c0, double *c1, double *cov00, double *cov01, double *cov11, double *sumsq)
This function computes the best-fit linear regression coefficients (c0, c1) of the model Y = c0 + c1 X for thedataset (x, y), two vectors of length n with strides xstride and ystride. The errors on y are assumed unknown
so the variance-covariance matrix for the parameters (c0, c1) is estimated from the scatter of the points aroundthe best-fit line and returned via the parameters (cov00, cov01, cov11). The sum of squares of the residuals
from the best-fit line is returned in sumsq. Note: the correlation coefficient of the data can be computed using
gsl_stats_correlation(), it does not depend on the fit.
int gsl_fit_wlinear(const double *x, const size_t xstride, const double *w, const size_t wstride, const double *y,const size_t ystride, size_t n, double *c0, double *c1, double *cov00, double *cov01, double*cov11, double *chisq)
This function computes the best-fit linear regression coefficients (c0, c1) of the model Y = c0 + c1 X for theweighted dataset (x, y), two vectors of length n with strides xstride and ystride. The vector w, of length nand stride wstride, specifies the weight of each datapoint. The weight is the reciprocal of the variance for eachdatapoint in y.
The covariance matrix for the parameters (c0, c1) is computed using the weights and returned via the parameters(cov00, cov01, cov11). The weighted sum of squares of the residuals from the best-fit line, χ2 , is returned inchisq.
int gsl_fit_linear_est(double x, double c0, double c1, double cov00, double cov01, double cov11, double *y,double *y_err)
This function uses the best-fit linear regression coefficients c0, c1 and their covariance cov00, cov01, cov11
to compute the fitted function y and its standard deviation y_err for the model Y = c0 + c1 X at the point x.

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40.2.2 Linear regression without a constant term
The functions described in this section can be used to perform least-squares fits to a straight line model without aconstant term, Y = c1 X.
int gsl_fit_mul(const double *x, const size_t xstride, const double *y, const size_t ystride, size_t n, double *c1,
double *cov11, double *sumsq)
This function computes the best-fit linear regression coefficient c1 of the model Y = c1 X for the datasets (x,y), two vectors of length n with strides xstride and ystride. The errors on y are assumed unknown so thevariance of the parameter c1 is estimated from the scatter of the points around the best-fit line and returned via
the parameter cov11. The sum of squares of the residuals from the best-fit line is returned in sumsq.
int gsl_fit_wmul(const double *x, const size_t xstride, const double *w, const size_t wstride, const double *y,const size_t ystride, size_t n, double *c1, double *cov11, double *sumsq)
This function computes the best-fit linear regression coefficient c1 of the model Y = c1 X for the weighted
datasets (x, y), two vectors of length n with strides xstride and ystride. The vector w, of length n and stride
wstride, specifies the weight of each datapoint. The weight is the reciprocal of the variance for each datapointin y.
The variance of the parameter c1 is computed using the weights and returned via the parameter cov11. The
weighted sum of squares of the residuals from the best-fit line, χ2 , is returned in chisq.
int gsl_fit_mul_est(double x, double c1, double cov11, double *y, double *y_err)
This function uses the best-fit linear regression coefficient c1 and its covariance cov11 to compute the fitted
function y and its standard deviation y_err for the model Y = c1 X at the point x.

40.3 Multi-parameter regression
This section describes routines which perform least squares fits to a linear model by minimizing the cost functionXX
χ2 =wi (yi −Xij cj )2 = ||y − Xc||2Wijwhere y is a vector of n observations, X is an n-by-p matrix of predictor variables, c is a vector of the p unknown
best-fit parameters to be estimated, and ||r||2W = rT W r. The matrix W = diag(w1 , w2 , ..., wn ) defines the weights
or uncertainties of the observation vector.
This formulation can be used for fits to any number of functions and/or variables by preparing the n-by-p matrix X
appropriately. For example, to fit to a p-th order polynomial in x, use the following matrix,
Xij = xji
where the index i runs over the observations and the index j runs from 0 to p − 1.
To fit to a set of p sinusoidal functions with fixed frequencies ω1 , ω2 , . . ., ωp , use,
Xij = sin(ωj xi )To fit to p independent variables x1 , x2 , . . ., xp , use,
Xij = xj (i)where xj (i) is the i-th value of the predictor variable xj .
The solution of the general linear least-squares system requires an additional working space for intermediate results,
such as the singular value decomposition of the matrix X.
These functions are declared in the header file gsl_multifit.h.
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type gsl_multifit_linear_workspace
This workspace contains internal variables for fitting multi-parameter models.
gsl_multifit_linear_workspace *gsl_multifit_linear_alloc(const size_t n, const size_t p)This function allocates a workspace for fitting a model to a maximum of n observations using a maximum of pparameters. The user may later supply a smaller least squares system if desired. The size of the workspace isO(np + p2 ).
void gsl_multifit_linear_free(gsl_multifit_linear_workspace *work)
This function frees the memory associated with the workspace w.
int gsl_multifit_linear_svd(const gsl_matrix *X, gsl_multifit_linear_workspace *work)
This function performs a singular value decomposition of the matrix X and stores the SVD factors internally inwork.
int gsl_multifit_linear_bsvd(const gsl_matrix *X, gsl_multifit_linear_workspace *work)
This function performs a singular value decomposition of the matrix X and stores the SVD factors internally inwork. The matrix X is first balanced by applying column scaling factors to improve the accuracy of the singularvalues.
int gsl_multifit_linear(const gsl_matrix *X, const gsl_vector *y, gsl_vector *c, gsl_matrix *cov, double*chisq, gsl_multifit_linear_workspace *work)
This function computes the best-fit parameters c of the model y = Xc for the observations y and the matrixof predictor variables X, using the preallocated workspace provided in work. The p-by-p variance-covariancematrix of the model parameters cov is set to σ 2 (X T X)−1 , where σ is the standard deviation of the fit residuals.
The sum of squares of the residuals from the best-fit, χ2 , is returned in chisq. If the coefficient of determination
is desired, it can be computed from the expression R2 = 1 − χ2 /T SS, where the total sum of squares (TSS) of
the observations y may be computed from gsl_stats_tss().
The best-fit is found by singular value decomposition of the matrix X using the modified Golub-Reinsch SVD
algorithm, with column scaling to improve the accuracy of the singular values. Any components which havezero singular value (to machine precision) are discarded from the fit.
int gsl_multifit_linear_tsvd(const gsl_matrix *X, const gsl_vector *y, const double tol, gsl_vector *c,
gsl_matrix *cov, double *chisq, size_t *rank, gsl_multifit_linear_workspace*work)
This function computes the best-fit parameters c of the model y = Xc for the observations y and the matrixof predictor variables X, using a truncated SVD expansion. Singular values which satisfy si ≤ tol × s0 are
discarded from the fit, where s0 is the largest singular value. The p-by-p variance-covariance matrix of themodel parameters cov is set to σ 2 (X T X)−1 , where σ is the standard deviation of the fit residuals. The sum ofsquares of the residuals from the best-fit, χ2 , is returned in chisq. The effective rank (number of singular values
used in solution) is returned in rank. If the coefficient of determination is desired, it can be computed from theexpression R2 = 1 − χ2 /T SS, where the total sum of squares (TSS) of the observations y may be computedfrom gsl_stats_tss().
int gsl_multifit_wlinear(const gsl_matrix *X, const gsl_vector *w, const gsl_vector *y, gsl_vector *c,
gsl_matrix *cov, double *chisq, gsl_multifit_linear_workspace *work)
This function computes the best-fit parameters c of the weighted model y = Xc for the observations y withweights w and the matrix of predictor variables X, using the preallocated workspace provided in work. The p-byp covariance matrix of the model parameters cov is computed as (X T W X)−1 . The weighted sum of squares of
the residuals from the best-fit, χ2 , is returned in chisq. If the coefficient of determination is desired, it can becomputed from the expression R2 = 1 − χ2 /W T SS, where the weighted total sum of squares (WTSS) of theobservations y may be computed from gsl_stats_wtss().
int gsl_multifit_wlinear_tsvd(const gsl_matrix *X, const gsl_vector *w, const gsl_vector *y, const double tol,
gsl_vector *c, gsl_matrix *cov, double *chisq, size_t *rank,
gsl_multifit_linear_workspace *work)
This function computes the best-fit parameters c of the weighted model y = Xc for the observations y with

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weights w and the matrix of predictor variables X, using a truncated SVD expansion. Singular values which
satisfy si ≤ tol × s0 are discarded from the fit, where s0 is the largest singular value. The p-by-p covariancematrix of the model parameters cov is computed as (X T W X)−1 . The weighted sum of squares of the residuals
from the best-fit, χ2 , is returned in chisq. The effective rank of the system (number of singular values usedin the solution) is returned in rank. If the coefficient of determination is desired, it can be computed from theexpression R2 = 1 − χ2 /W T SS, where the weighted total sum of squares (WTSS) of the observations y may
be computed from gsl_stats_wtss().
int gsl_multifit_linear_est(const gsl_vector *x, const gsl_vector *c, const gsl_matrix *cov, double *y, double*y_err)This function uses the best-fit multilinear regression coefficients c and their covariance matrix cov to computethe fitted function value y and its standard deviation y_err for the model y = x.c at the point x.
int gsl_multifit_linear_residuals(const gsl_matrix *X, const gsl_vector *y, const gsl_vector *c, gsl_vector*r)This function computes the vector of residuals r = y − Xc for the observations y, coefficients c and matrix of
predictor variables X.
size_t gsl_multifit_linear_rank(const double tol, const gsl_multifit_linear_workspace *work)
This function returns the rank of the matrix X which must first have its singular value decomposition computed.
The rank is computed by counting the number of singular values σj which satisfy σj > tol × σ0 , where σ0 isthe largest singular value.

40.4 Regularized regression
Ordinary weighted least squares models seek a solution vector c which minimizes the residual
χ2 = ||y − Xc||2W
where y is the n-by-1 observation vector, X is the n-by-p design matrix, c is the p-by-1 solution vector, W =diag(w1 , ..., wn ) is the data weighting matrix, and ||r||2W = rT W r. In cases where the least squares matrix X is
ill-conditioned, small perturbations (ie: noise) in the observation vector could lead to widely different solution vectors c. One way of dealing with ill-conditioned matrices is to use a “truncated SVD” in which small singular values,
below some given tolerance, are discarded from the solution. The truncated SVD method is available using the functions gsl_multifit_linear_tsvd() and gsl_multifit_wlinear_tsvd(). Another way to help solve ill-posed
problems is to include a regularization term in the least squares minimization
χ2 = ||y − Xc||2W + λ2 ||Lc||2
for a suitably chosen regularization parameter λ and matrix L. This type of regularization is known as Tikhonov, or
ridge, regression. In some applications, L is chosen as the identity matrix, giving preference to solution vectors c withsmaller norms. Including this regularization term leads to the explicit “normal equations” solution
−1 T
c = X T W X + λ2 LT L
X Wy
which reduces to the ordinary least squares solution when L = 0. In practice, it is often advantageous to transform aregularized least squares system into the form
χ2 = ||ỹ − X̃ c̃||2 + λ2 ||c̃||2This is known as the Tikhonov “standard form” and has the normal equations solution−1c̃ = X̃ T X̃ + λ2 I
X̃ T ỹ40.4. Regularized regression

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For an m-by-p matrix L which is full rank and has m >= p (ie: L is square or has more rows than columns), we cancalculate the “thin” QR decomposition of L, and note that ||Lc|| = ||Rc|| since the Q factor will not change the norm.
Since R is p-by-p, we can then use the transformation
1

X̃ = W 2 XR−1
1ỹ = W 2 yc̃ = Rc
to achieve the standard form. For a rectangular matrix L with m < p, a more sophisticated approachis needed (see Hansen 1998, chapter 2.3). In practice, the normal equations solution above is not desirable due to numerical instabilities, and so the system is solved using the singular value decomposition of
the matrix X̃. The matrix L is often chosen as the identity matrix, or as a first or second finite difference operator, to ensure a smoothly varying coefficient vector c, or as a diagonal matrix to selectively
damp each model parameter differently. If L 6= I, the user must first convert the least squares problem to standard form using gsl_multifit_linear_stdform1() or gsl_multifit_linear_stdform2(), solve
the system, and then backtransform the solution vector to recover the solution of the original problem (seegsl_multifit_linear_genform1() and gsl_multifit_linear_genform2()).
In many regularization problems, care must be taken when choosing the regularization parameter λ. Since both theresidual norm ||y − Xc|| and solution norm ||Lc|| are being minimized, the parameter λ represents a tradeoff between
minimizing either the residuals or the solution vector. A common tool for visualizing the comprimise between theminimization of these two quantities is known as the L-curve. The L-curve is a log-log plot of the residual norm||y − Xc|| on the horizontal axis and the solution norm ||Lc|| on the vertical axis. This curve nearly always as an L
shaped appearance, with a distinct corner separating the horizontal and vertical sections of the curve. The regularization
parameter corresponding to this corner is often chosen as the optimal value. GSL provides routines to calculate theL-curve for all relevant regularization parameters as well as locating the corner.
Another method of choosing the regularization parameter is known as Generalized Cross Validation (GCV). Thismethod is based on the idea that if an arbitrary element yi is left out of the right hand side, the resulting regularized
solution should predict this element accurately. This leads to choosing the parameter λ which minimizes the GCVfunctionG(λ) =||y − Xcλ ||2
Tr(In − XXλI )2where XλI is the matrix which relates the solution cλ to the right hand side y, ie: cλ = XλI y. GSL provides routines tocompute the GCV curve and its minimum.
For most applications, the steps required to solve a regularized least squares problem are as follows:
1. Construct the least squares system (X, y, W , L)
2. Transform the system to standard form (X̃, ỹ). This step can be skipped if L = Ip and W = In .
3. Calculate the SVD of X̃.
4. Determine an appropriate regularization parameter λ (using for example L-curve or GCV analysis).
5. Solve the standard form system using the chosen λ and the SVD of X̃.
6. Backtransform the standard form solution c̃ to recover the original solution vector c.
int gsl_multifit_linear_stdform1(const gsl_vector *L, const gsl_matrix *X, const gsl_vector *y, gsl_matrix*Xs, gsl_vector *ys, gsl_multifit_linear_workspace *work)
int gsl_multifit_linear_wstdform1(const gsl_vector *L, const gsl_matrix *X, const gsl_vector *w, const
gsl_vector *y, gsl_matrix *Xs, gsl_vector *ys,
gsl_multifit_linear_workspace *work)
These functions define a regularization matrix L = diag(l0 , l1 , ..., lp−1 ). The diagonal matrix element li isprovided by the i-th element of the input vector L. The n-by-p least squares matrix X and vector y of length n are
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then converted to standard form as described above and the parameters (X̃, ỹ) are stored in Xs and ys on output.
Xs and ys have the same dimensions as X and y. Optional data weights may be supplied in the vector w of lengthn. In order to apply this transformation, L−1 must exist and so none of the li may be zero. After the standard
form system has been solved, use gsl_multifit_linear_genform1() to recover the original solution vector.
It is allowed to have X = Xs and y = ys for an in-place transform. In order to perform a weighted regularized fit
with L = I, the user may call gsl_multifit_linear_applyW() to convert to standard form.
int gsl_multifit_linear_L_decomp(gsl_matrix *L, gsl_vector *tau)
This function factors the m-by-p regularization matrix L into a form needed for the later transformation to standard form. L may have any number of rows m. If m ≥ p the QR decomposition of L is computed and storedin L on output. If m < p, the QR decomposition of LT is computed and stored in L on output. On output, the Householder scalars are stored in the vector tau of size M IN (m, p). These outputs will be used by
gsl_multifit_linear_wstdform2() to complete the transformation to standard form.
int gsl_multifit_linear_stdform2(const gsl_matrix *LQR, const gsl_vector *Ltau, const gsl_matrix *X, const
gsl_vector *y, gsl_matrix *Xs, gsl_vector *ys, gsl_matrix *M,
gsl_multifit_linear_workspace *work)
int gsl_multifit_linear_wstdform2(const gsl_matrix *LQR, const gsl_vector *Ltau, const gsl_matrix *X,
const gsl_vector *w, const gsl_vector *y, gsl_matrix *Xs, gsl_vector *ys,
gsl_matrix *M, gsl_multifit_linear_workspace *work)
These functions convert the least squares system (X, y, W, L) to standard form (X̃, ỹ) which are stored in Xs andys respectively. The m-by-p regularization matrix L is specified by the inputs LQR and Ltau, which are outputs
from gsl_multifit_linear_L_decomp(). The dimensions of the standard form parameters (X̃, ỹ) depend
on whether m is larger or less than p. For m ≥ p, Xs is n-by-p, ys is n-by-1, and M is not used. For m < p, Xs is(n−p+m)-by-m, ys is (n−p+m)-by-1, and M is additional n-by-p workspace, which is required to recover theoriginal solution vector after the system has been solved (see gsl_multifit_linear_genform2()). Optional
data weights may be supplied in the vector w of length n, where W = diag(w).
int gsl_multifit_linear_solve(const double lambda, const gsl_matrix *Xs, const gsl_vector *ys, gsl_vector*cs, double *rnorm, double *snorm, gsl_multifit_linear_workspace *work)
This function computes the regularized best-fit parameters c̃ which minimize the cost function χ2 = ||ỹ− X̃ c̃||2 +λ2 ||c̃||2 which is in standard form. The least squares system must therefore be converted to standard form prior
to calling this function. The observation vector ỹ is provided in ys and the matrix of predictor variables X̃ in
Xs. The solution vector c̃ is returned in cs, which has length min(m, p). The SVD of Xs must be computed prior
to calling this function, using gsl_multifit_linear_svd(). The regularization parameter λ is provided inlambda. The residual norm ||ỹ − X̃ c̃|| = ||y − Xc||W is returned in rnorm. The solution norm ||c̃|| = ||Lc||
is returned in snorm.
int gsl_multifit_linear_genform1(const gsl_vector *L, const gsl_vector *cs, gsl_vector *c,
gsl_multifit_linear_workspace *work)
After a regularized system has been solved with L = diag(0 , 1 , ..., p−1 ), this function backtransforms the standard form solution vector cs to recover the solution vector of the original problem c. The diagonal matrix
elements li are provided in the vector L. It is allowed to have c = cs for an in-place transform.
int gsl_multifit_linear_genform2(const gsl_matrix *LQR, const gsl_vector *Ltau, const gsl_matrix *X, const
gsl_vector *y, const gsl_vector *cs, const gsl_matrix *M, gsl_vector *c,
gsl_multifit_linear_workspace *work)
int gsl_multifit_linear_wgenform2(const gsl_matrix *LQR, const gsl_vector *Ltau, const gsl_matrix *X,
const gsl_vector *w, const gsl_vector *y, const gsl_vector *cs, const
gsl_matrix *M, gsl_vector *c, gsl_multifit_linear_workspace *work)
After a regularized system has been solved with a general rectangular matrix L, specified by (LQR, Ltau), thisfunction backtransforms the standard form solution cs to recover the solution vector of the original problem,
which is stored in c, of length p. The original least squares matrix and observation vector are provided in X

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and y respectively. M is the matrix computed by gsl_multifit_linear_stdform2(). For weighted fits, theweight vector w must also be supplied.
int gsl_multifit_linear_applyW(const gsl_matrix *X, const gsl_vector *w, const gsl_vector *y, gsl_matrix*WX, gsl_vector *Wy)
For weighted least squares systems with L = I, this function may be used to convert the system to standard form
by applying the weight matrix W = diag(w) to the least squares matrix X and observation vector y. On output,
WX is equal to W 1/2 X and Wy is equal to W 1/2 y. It is allowed for WX = X and Wy = y for an in-place transform.
int gsl_multifit_linear_lcurve(const gsl_vector *y, gsl_vector *reg_param, gsl_vector *rho, gsl_vector *eta,
gsl_multifit_linear_workspace *work)
This function computes the L-curve for a least squares system using the right hand side vector y and the SVD
decomposition of the least squares matrix X, which must be provided to gsl_multifit_linear_svd() prior
to calling this function. The output vectors reg_param, rho, and eta must all be the same size, and will containthe regularization parameters λi , residual norms ||y − Xci ||, and solution norms ||Lci || which compose the Lcurve, where ci is the regularized solution vector corresponding to λi . The user may determine the number ofpoints on the L-curve by adjusting the size of these input arrays. The regularization parameters λi are estimatedfrom the singular values of X, and chosen to represent the most relevant portion of the L-curve.
int gsl_multifit_linear_lcurvature(const gsl_vector *y, const gsl_vector *reg_param, const gsl_vector *rho,
const gsl_vector *eta, gsl_vector *kappa, gsl_multifit_linear_workspace*work)
This function computes the curvature of the L-curve as a function of the regularization parameter λ, using the right hand side vector y, the vector of regularization parameters, reg_param, vector of residual
norms, rho, and vector of solution norms, eta. The arrays reg_param, rho, and eta can be computed by
gsl_multifit_linear_lcurve(). The curvature is defined as
κ(λ) =

ρ̂0 η̂ 00 − ρ̂00 η̂ 0
3((ρ̂0 )2 + (η̂ 0 )2 ) 2

where ρ̂(λ) = log ||y − Xcλ || and η̂(λ) = log ||Lcλ ||. The curvature values are stored in kappa on output.
int gsl_multifit_linear_lcorner(const gsl_vector *rho, const gsl_vector *eta, size_t *idx)
This function attempts to locate the corner of the L-curve (||y − Xc||, ||Lc||) defined by the rho and eta input
arrays respectively. The corner is defined as the point of maximum curvature of the L-curve in log-log scale.
The rho and eta arrays can be outputs of gsl_multifit_linear_lcurve(). The algorithm used simply
fits a circle to 3 consecutive points on the L-curve and uses the circle’s radius to determine the curvature at the
middle point. Therefore, the input array sizes must be ≥ 3. With more points provided for the L-curve, a better
estimate of the curvature can be obtained. The array index corresponding to maximum curvature (ie: the corner)is returned in idx. If the input arrays contain colinear points, this function could fail and return GSL_EINVAL.
int gsl_multifit_linear_lcorner2(const gsl_vector *reg_param, const gsl_vector *eta, size_t *idx)
This function attempts to locate the corner of an alternate L-curve (λ2 , ||Lc||2 ) studied by Rezghi and Hosseini,2009. This alternate L-curve can provide better estimates of the regularization parameter for smooth solution
vectors. The regularization parameters λ and solution norms ||Lc|| are provided in the reg_param and eta input
arrays respectively. The corner is defined as the point of maximum curvature of this alternate L-curve in linear
scale. The reg_param and eta arrays can be outputs of gsl_multifit_linear_lcurve(). The algorithm
used simply fits a circle to 3 consecutive points on the L-curve and uses the circle’s radius to determine thecurvature at the middle point. Therefore, the input array sizes must be ≥ 3. With more points provided for the Lcurve, a better estimate of the curvature can be obtained. The array index corresponding to maximum curvature(ie: the corner) is returned in idx. If the input arrays contain colinear points, this function could fail and return
GSL_EINVAL.
int gsl_multifit_linear_gcv_init(const gsl_vector *y, gsl_vector *reg_param, gsl_vector *UTy, double*delta0, gsl_multifit_linear_workspace *work)
This function performs some initialization in preparation for computing the GCV curve and its minimum. The
right hand side vector is provided in y. On output, reg_param is set to a vector of regularization parameters

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in decreasing order and may be of any size. The vector UTy of size p is set to U T y. The parameter delta0 isneeded for subsequent steps of the GCV calculation.
int gsl_multifit_linear_gcv_curve(const gsl_vector *reg_param, const gsl_vector *UTy, const double delta0,
gsl_vector *G, gsl_multifit_linear_workspace *work)
This funtion calculates the GCV curve G(λ) and stores it in G on output, which must be the same size as
reg_param. The inputs reg_param, UTy and delta0 are computed in gsl_multifit_linear_gcv_init().
int gsl_multifit_linear_gcv_min(const gsl_vector *reg_param, const gsl_vector *UTy, const gsl_vector *G,
const double delta0, double *lambda, gsl_multifit_linear_workspace *work)
This function computes the value of the regularization parameter which minimizes the GCV curve G(λ) andstores it in lambda. The input G is calculated by gsl_multifit_linear_gcv_curve() and the inputs
reg_param, UTy and delta0 are computed by gsl_multifit_linear_gcv_init().
double gsl_multifit_linear_gcv_calc(const double lambda, const gsl_vector *UTy, const double delta0,
gsl_multifit_linear_workspace *work)
This function returns the value of the GCV curve G(λ) corresponding to the input lambda.
int gsl_multifit_linear_gcv(const gsl_vector *y, gsl_vector *reg_param, gsl_vector *G, double *lambda,
double *G_lambda, gsl_multifit_linear_workspace *work)
This function combines the steps gcv_init, gcv_curve, and gcv_min defined above into a single function.
The input y is the right hand side vector. On output, reg_param and G, which must be the same size, are set tovectors of λ and G(λ) values respectively. The output lambda is set to the optimal value of λ which minimizes
the GCV curve. The minimum value of the GCV curve is returned in G_lambda.
int gsl_multifit_linear_Lk(const size_t p, const size_t k, gsl_matrix *L)This function computes the discrete approximation to the derivative operator Lk of order k on a regular grid of
p points and stores it in L. The dimensions of L are (p − k)-by-p.
int gsl_multifit_linear_Lsobolev(const size_t p, const size_t kmax, const gsl_vector *alpha, gsl_matrix *L,
gsl_multifit_linear_workspace *work)Thisfunctioncomputestheregularization
matrix L corresponding to the weighted Sobolov norm ||Lc||2 =P 2
2k αk ||Lk c|| where Lk approximates the derivative operator of order k. This regularization norm can be
useful in applications where it is necessary to smooth several derivatives of the solution. p is the number ofmodel parameters, kmax is the highest derivative to include in the summation above, and alpha is the vector ofweights of size kmax + 1, where alpha[k] = αk is the weight assigned to the derivative of order k. The output
matrix L is size p-by-p and upper triangular.
double gsl_multifit_linear_rcond(const gsl_multifit_linear_workspace *work)
This function returns the reciprocal condition number of the least squares matrix X, defined as the ratio of thesmallest and largest singular values, rcond = σmin /σmax . The routine gsl_multifit_linear_svd() mustfirst be called to compute the SVD of X.

40.5 Robust linear regression
Ordinary least squares (OLS) models are often heavily influenced by the presence of outliers. Outliers are data points
which do not follow the general trend of the other observations, although there is strictly no precise definition of an
outlier. Robust linear regression refers to regression algorithms which are robust to outliers. The most common typeof robust regression is M-estimation. The general M-estimator minimizes the objective functionXX
ρ(ei ) =ρ(yi − Y (c, xi ))iiwhere ei = yi − Y (c, xi ) is the residual of the ith data point, and ρ(ei ) is a function which should have the following
properties:• ρ(e) ≥ 0
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GNU Scientific Library, Release 2.7• ρ(0) = 0• ρ(−e) = ρ(e)• ρ(e1 ) > ρ(e2 ) for |e1 | > |e2 |The special case of ordinary least squares is given by ρ(ei ) = e2i . Letting ψ = ρ0 be the derivative of ρ, differentiating
the objective function with respect to the coefficients c and setting the partial derivatives to zero produces the systemof equationsXψ(ei )Xi = 0
iwhere Xi is a vector containing row i of the design matrix X. Next, we define a weight function w(e) = ψ(e)/e, andlet wi = w(ei ):X
wi ei Xi = 0
iPThis system of equations is equivalent to solving a weighted ordinary least squares problem, minimizing χ2 = i wi e2i .
The weights however, depend on the residuals ei , which depend on the coefficients c, which depend on the weights.
Therefore, an iterative solution is used, called Iteratively Reweighted Least Squares (IRLS).
1. Compute initial estimates of the coefficients c(0) using ordinary least squares√(k)
2. For iteration k, form the residuals ei = (yi − Xi c(k−1) )/(tσ (k) 1 − hi ), where t is a tuning constant depending on the choice of ψ, and hi are the statistical leverages (diagonal elements of the matrix X(X T X)−1 X T ).
Including t and hi in the residual calculation has been shown to improve the convergence of the method. The
residual standard deviation is approximated as σ (k) = M AD/0.6745, where MAD is the Median-AbsoluteDeviation of the n − p largest residuals from the previous iteration.
(k)

3. Compute new weights wi

(k)

(k)

= ψ(ei )/ei .
(k)

4. Compute new coefficients c(k) by solving the weighted least squares problem with weights wi .
5. Steps 2 through 4 are iterated until the coefficients converge or until some maximum iteration limit is reached.
Coefficients are tested for convergence using the critera:

(k)

|ci

(k−1)

− ci

(k)

(k−1)

| ≤  × max(|ci |, |ci|)

for all 0 ≤ i < p where  is a small tolerance factor.
The key to this method lies in selecting the function ψ(ei ) to assign smaller weights to large residuals, and larger
weights to smaller residuals. As the iteration proceeds, outliers are assigned smaller and smaller weights, eventually
having very little or no effect on the fitted model.
type gsl_multifit_robust_workspace
This workspace is used for robust least squares fitting.
gsl_multifit_robust_workspace *gsl_multifit_robust_alloc(const gsl_multifit_robust_type *T, const size_t n,
const size_t p)This function allocates a workspace for fitting a model to n observations using p parameters. The size of theworkspace is O(np + p2 ). The type T specifies the function ψ and can be selected from the following choices.
type gsl_multifit_robust_type
gsl_multifit_robust_type *gsl_multifit_robust_default
This specifies the gsl_multifit_robust_bisquare type (see below) and is a good general purpose
choice for robust regression.
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gsl_multifit_robust_type *gsl_multifit_robust_bisquareThis is Tukey’s biweight (bisquare) function and is a good general purpose choice for robust regression.
The weight function is given by(1 − e2 )2 , |e| ≤ 1
w(e) =
0,|e| > 1and the default tuning constant is t = 4.685.
gsl_multifit_robust_type *gsl_multifit_robust_cauchy
This is Cauchy’s function, also known as the Lorentzian function. This function does not guarantee
a unique solution, meaning different choices of the coefficient vector c could minimize the objectivefunction. Therefore this option should be used with care. The weight function is given byw(e) =

1
1 + e2and the default tuning constant is t = 2.385.
gsl_multifit_robust_type *gsl_multifit_robust_fair
This is the fair ρ function, which guarantees a unique solution and has continuous derivatives to threeorders. The weight function is given byw(e) =

1
1 + |e|and the default tuning constant is t = 1.400.
gsl_multifit_robust_type *gsl_multifit_robust_huber
This specifies Huber’s ρ function, which is a parabola in the vicinity of zero and increases linearly for agiven threshold |e| > t. This function is also considered an excellent general purpose robust estimator,
however, occasional difficulties can be encountered due to the discontinuous first derivative of the ψfunction. The weight function is given by
1, |e| ≤ 1
w(e) =
1|e| , |e| > 1
and the default tuning constant is t = 1.345.
gsl_multifit_robust_type *gsl_multifit_robust_ols
This specifies the ordinary least squares solution, which can be useful for quickly checking the difference between the various robust and OLS solutions. The weight function is given byw(e) = 1
and the default tuning constant is t = 1.
gsl_multifit_robust_type *gsl_multifit_robust_welschThis specifies the Welsch function which can perform well in cases where the residuals have an exponential distribution. The weight function is given byw(e) = exp (−e2 )and the default tuning constant is t = 2.985.
void gsl_multifit_robust_free(gsl_multifit_robust_workspace *w)
This function frees the memory associated with the workspace w.
const char *gsl_multifit_robust_name(const gsl_multifit_robust_workspace *w)
This function returns the name of the robust type T specified to gsl_multifit_robust_alloc().
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int gsl_multifit_robust_tune(const double tune, gsl_multifit_robust_workspace *w)
This function sets the tuning constant t used to adjust the residuals at each iteration to tune. Decreasing the tuning
constant increases the downweight assigned to large residuals, while increasing the tuning constant decreases thedownweight assigned to large residuals.
int gsl_multifit_robust_maxiter(const size_t maxiter, gsl_multifit_robust_workspace *w)
This function sets the maximum number of iterations in the iteratively reweighted least squares algorithm tomaxiter. By default, this value is set to 100 by gsl_multifit_robust_alloc().
int gsl_multifit_robust_weights(const gsl_vector *r, gsl_vector *wts, gsl_multifit_robust_workspace *w)
This function assigns weights to the vector wts using the residual vector r and previously specified weighting
function. The output weights are given by wtsi = w(ri /(tσ)), where the weighting functions w are detailedin gsl_multifit_robust_alloc(). σ is an estimate of the residual standard deviation based on the MedianAbsolute-Deviation and t is the tuning constant. This function is useful if the user wishes to implement theirown robust regression rather than using the supplied gsl_multifit_robust() routine below.
int gsl_multifit_robust(const gsl_matrix *X, const gsl_vector *y, gsl_vector *c, gsl_matrix *cov,
gsl_multifit_robust_workspace *w)
This function computes the best-fit parameters c of the model y = Xc for the observations y and the matrixof predictor variables X, attemping to reduce the influence of outliers using the algorithm outlined above. The
p-by-p variance-covariance matrix of the model parameters cov is estimated as σ 2 (X T X)−1 , where σ is anapproximation of the residual standard deviation using the theory of robust regression. Special care must be
taken when estimating σ and other statistics such as R2 , and so these are computed internally and are available
by calling the function gsl_multifit_robust_statistics().
If the coefficients do not converge within the maximum iteration limit, the function returns GSL_EMAXITER. In
this case, the current estimates of the coefficients and covariance matrix are returned in c and cov and the internal
fit statistics are computed with these estimates.
int gsl_multifit_robust_est(const gsl_vector *x, const gsl_vector *c, const gsl_matrix *cov, double *y, double*y_err)This function uses the best-fit robust regression coefficients c and their covariance matrix cov to compute thefitted function value y and its standard deviation y_err for the model y = x · c at the point x.
int gsl_multifit_robust_residuals(const gsl_matrix *X, const gsl_vector *y, const gsl_vector *c, gsl_vector*r, gsl_multifit_robust_workspace *w)−(Xc)ifor the observations y, coefficients c
This function computes the vector of studentized residuals ri = yσi√
1−hi
and matrix of predictor variables X. The routine gsl_multifit_robust() must first be called to compute thestatisical leverages hi of the matrix X and residual standard deviation estimate σ.
gsl_multifit_robust_stats gsl_multifit_robust_statistics(const gsl_multifit_robust_workspace *w)
This function returns a structure containing relevant statistics from a robust regression. The function
gsl_multifit_robust() must be called first to perform the regression and calculate these statistics. The
returned gsl_multifit_robust_stats structure contains the following fields.
type gsl_multifit_robust_stats
double sigma_ols
This contains the standard deviation of the residuals as computed from ordinary least squares(OLS).
double sigma_mad
This contains an estimate of the standard deviation of the final residuals using the MedianAbsolute-Deviation statistic
double sigma_rob
This contains an estimate of the standard deviation of the final residuals from the theory of robust
regression (see Street et al, 1988).

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double sigma
This contains an estimate of the standard deviation of the final residuals by attemping to reconcile
sigma_rob and sigma_ols in a reasonable way.
double Rsq
This contains the R2 coefficient of determination statistic using the estimate sigma.
double adj_Rsq
This contains the adjusted R2 coefficient of determination statistic using the estimate sigma.
double rmse
This contains the root mean squared error of the final residuals
double sse
This contains the residual sum of squares taking into account the robust covariance matrix.
size_t dof
This contains the number of degrees of freedom n − psize_t numit
Upon successful convergence, this contains the number of iterations performed
gsl_vector * weights
This contains the final weight vector of length n
gsl_vector * rThis contains the final residual vector of length n, r = y − Xc

40.6 Large dense linear systems
This module is concerned with solving large dense least squares systems Xc = y where the n-by-p matrix X hasn >> p (ie: many more rows than columns). This type of matrix is called a “tall skinny” matrix, and for some
applications, it may not be possible to fit the entire matrix in memory at once to use the standard SVD approach.
Therefore, the algorithms in this module are designed to allow the user to construct smaller blocks of the matrix X andaccumulate those blocks into the larger system one at a time. The algorithms in this module never need to store theentire matrix X in memory. The large linear least squares routines support data weights and Tikhonov regularization,
and are designed to minimize the residual
χ2 = ||y − Xc||2W + λ2 ||Lc||2
where y is the n-by-1 observation vector, X is the n-by-p design matrix, c is the p-by-1 solution vector, W =diag(w1 , ..., wn ) is the data weighting matrix, L is an m-by-p regularization matrix, λ is a regularization parameter, and ||r||2W = rT W r. In the discussion which follows, we will assume that the system has been converted into
Tikhonov standard form,
χ2 = ||ỹ − X̃ c̃||2 + λ2 ||c̃||2and we will drop the tilde characters from the various parameters. For a discussion of the transformation to standard
form, see Regularized regression.

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The basic idea is to partition the matrix X and observation vector y as




y1
X1
 y2 
 X2  X3  c =  y3  .. 
 .. 
 . 
 . 
yk
Xk
into k blocks, where each block (Xi , yi ) may have any number of rows, but each Xi has p columns. The sections
below describe the methods available for solving this partitioned system. The functions are declared in the header file
gsl_multilarge.h.

40.6.1 Normal Equations Approach
The normal equations approach to the large linear least squares problem described above is popular due to its speed
and simplicity. Since the normal equations solution to the problem is given by
c = X T X + λ2 I

−1

XT y

only the p-by-p matrix X T X and p-by-1 vector X T y need to be stored. Using the partition scheme described above,these are given byX
XT X =XiT Xi
i
T

X y=

X

XiT yii

Since the matrix X T X is symmetric, only half of it needs to be calculated. Once all of the blocks (Xi , yi ) have been
accumulated into the final X T X and X T y, the system can be solved with a Cholesky factorization of the X T X matrix.
The X T X matrix is first transformed via a diagonal scaling transformation to attempt to reduce its condition number
as much as possible to recover a more accurate solution vector. The normal equations approach is the fastest method forsolving the large least squares problem, and is accurate for well-conditioned matrices X. However, for ill-conditioned
matrices, as is often the case for large systems, this method can suffer from numerical instabilities (see Trefethen andBau, 1997). The number of operations for this method is O(np2 + 31 p3 ).

40.6.2 Tall Skinny QR (TSQR) Approach
An algorithm which has better numerical stability for ill-conditioned problems is known as the Tall Skinny QR (TSQR)
method. This method is based on computing the thin QR decomposition of the least squares matrix X = QR, where
Q is an n-by-p matrix with orthogonal columns, and R is a p-by-p upper triangular matrix. Once these factors arecalculated, the residual becomesχ2 = ||QT y − Rc||2 + λ2 ||c||2
which can be written as the matrix equation


R
λI




c=

QT y
0



The matrix on the left hand side is now a much smaller 2p-by-p matrix which can be solved with a standard SVD
approach. The Q matrix is just as large as the original matrix X, however it does not need to be explicitly constructed.
The TSQR algorithm computes only the p-by-p matrix R and the p-by-1 vector QT y, and updates these quantities as494

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new blocks are added to the system. Each time a new block of rows (Xi , yi ) is added, the algorithm performs a QR
decomposition of the matrix


Ri−1
Xi
where Ri−1 is the upper triangular R factor for the matrix

X1
 ..
 .





Xi−1
This QR decomposition is done efficiently taking into account the sparse structure of Ri−1 . See Demmel et al, 2008
for more details on how this is accomplished. The number of operations for this method is O(2np2 − 23 p3 ).

40.6.3 Large Dense Linear Systems Solution Steps
The typical steps required to solve large regularized linear least squares problems are as follows:
1. Choose the regularization matrix L.
2. Construct a block of rows of the least squares matrix, right hand side vector, and weight vector (Xi , yi , wi ).
3. Transform the block to standard form (X̃i , y˜i ). This step can be skipped if L = I and W = I.
4. Accumulate the standard form block (X̃i , y˜i ) into the system.
5. Repeat steps 2-4 until the entire matrix and right hand side vector have been accumulated.
6. Determine an appropriate regularization parameter λ (using for example L-curve analysis).
7. Solve the standard form system using the chosen λ.
8. Backtransform the standard form solution c̃ to recover the original solution vector c.

40.6.4 Large Dense Linear Least Squares Routines
type gsl_multilarge_linear_workspace
This workspace contains parameters for solving large linear least squares problems.
gsl_multilarge_linear_workspace *gsl_multilarge_linear_alloc(const gsl_multilarge_linear_type *T, const
size_t p)
This function allocates a workspace for solving large linear least squares systems. The least squares matrix X
has p columns, but may have any number of rows.
type gsl_multilarge_linear_type
The parameter T specifies the method to be used for solving the large least squares system and may be
selected from the following choices
gsl_multilarge_linear_type *gsl_multilarge_linear_normal
This specifies the normal equations approach for solving the least squares system. This method issuitable in cases where performance is critical and it is known that the least squares matrix X is wellconditioned. The size of this workspace is O(p2 ).
gsl_multilarge_linear_type *gsl_multilarge_linear_tsqr
This specifies the sequential Tall Skinny QR (TSQR) approach for solving the least squares system.
This method is a good general purpose choice for large systems, but requires about twice as many
operations as the normal equations method for n >> p. The size of this workspace is O(p2 ).

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void gsl_multilarge_linear_free(gsl_multilarge_linear_workspace *w)
This function frees the memory associated with the workspace w.
const char *gsl_multilarge_linear_name(gsl_multilarge_linear_workspace *w)
This function returns a string pointer to the name of the multilarge solver.
int gsl_multilarge_linear_reset(gsl_multilarge_linear_workspace *w)
This function resets the workspace w so it can begin to accumulate a new least squares system.
int gsl_multilarge_linear_stdform1(const gsl_vector *L, const gsl_matrix *X, const gsl_vector *y, gsl_matrix*Xs, gsl_vector *ys, gsl_multilarge_linear_workspace *work)
int gsl_multilarge_linear_wstdform1(const gsl_vector *L, const gsl_matrix *X, const gsl_vector *w, const
gsl_vector *y, gsl_matrix *Xs, gsl_vector *ys,
gsl_multilarge_linear_workspace *work)
These functions define a regularization matrix L = diag(l0 , l1 , ..., lp−1 ). The diagonal matrix element li is provided by the i-th element of the input vector L. The block (X, y) is converted to standard form and the parameters(X̃, ỹ) are stored in Xs and ys on output. Xs and ys have the same dimensions as X and y. Optional data weights
may be supplied in the vector w. In order to apply this transformation, L−1 must exist and so none of the li
may be zero. After the standard form system has been solved, use gsl_multilarge_linear_genform1() torecover the original solution vector. It is allowed to have X = Xs and y = ys for an in-place transform.
int gsl_multilarge_linear_L_decomp(gsl_matrix *L, gsl_vector *tau)
This function calculates the QR decomposition of the m-by-p regularization matrix L. L must have m ≥ p.
On output, the Householder scalars are stored in the vector tau of size p. These outputs will be used by
gsl_multilarge_linear_wstdform2() to complete the transformation to standard form.
int gsl_multilarge_linear_stdform2(const gsl_matrix *LQR, const gsl_vector *Ltau, const gsl_matrix *X,
const gsl_vector *y, gsl_matrix *Xs, gsl_vector *ys,
gsl_multilarge_linear_workspace *work)
int gsl_multilarge_linear_wstdform2(const gsl_matrix *LQR, const gsl_vector *Ltau, const gsl_matrix *X,
const gsl_vector *w, const gsl_vector *y, gsl_matrix *Xs, gsl_vector*ys, gsl_multilarge_linear_workspace *work)
These functions convert a block of rows (X, y, w) to standard form (X̃, ỹ) which are stored in Xs and ys
respectively. X, y, and w must all have the same number of rows. The m-by-p regularization matrix L
is specified by the inputs LQR and Ltau, which are outputs from gsl_multilarge_linear_L_decomp().
Xs and ys have the same dimensions as X and y. After the standard form system has been solved, use
gsl_multilarge_linear_genform2() to recover the original solution vector. Optional data weights may
be supplied in the vector w, where W = diag(w).
int gsl_multilarge_linear_accumulate(gsl_matrix *X, gsl_vector *y, gsl_multilarge_linear_workspace *w)
This function accumulates the standard form block (X, y) into the current least squares system. X and y havethe same number of rows, which can be arbitrary. X must have p columns. For the TSQR method, X and y aredestroyed on output. For the normal equations method, they are both unchanged.
int gsl_multilarge_linear_solve(const double lambda, gsl_vector *c, double *rnorm, double *snorm,
gsl_multilarge_linear_workspace *w)
After all blocks (Xi , yi ) have been accumulated into the large least squares system, this function will computethe solution vector which is stored in c on output. The regularization parameter λ is provided in lambda. On
output, rnorm contains the residual norm ||y − Xc||W and snorm contains the solution norm ||Lc||.
int gsl_multilarge_linear_genform1(const gsl_vector *L, const gsl_vector *cs, gsl_vector *c,
gsl_multilarge_linear_workspace *work)
After a regularized system has been solved with L = diag(0 , 1 , ..., p−1 ), this function backtransforms the standard form solution vector cs to recover the solution vector of the original problem c. The diagonal matrix
elements li are provided in the vector L. It is allowed to have c = cs for an in-place transform.

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int gsl_multilarge_linear_genform2(const gsl_matrix *LQR, const gsl_vector *Ltau, const gsl_vector *cs,
gsl_vector *c, gsl_multilarge_linear_workspace *work)
After a regularized system has been solved with a regularization matrix L, specified by (LQR, Ltau), this function
backtransforms the standard form solution cs to recover the solution vector of the original problem, which isstored in c, of length p.
int gsl_multilarge_linear_lcurve(gsl_vector *reg_param, gsl_vector *rho, gsl_vector *eta,
gsl_multilarge_linear_workspace *work)
This function computes the L-curve for a large least squares system after it has been fully accumulated into theworkspace work. The output vectors reg_param, rho, and eta must all be the same size, and will contain theregularization parameters λi , residual norms ||y − Xci ||, and solution norms ||Lci || which compose the L-curve,
where ci is the regularized solution vector corresponding to λi . The user may determine the number of points
on the L-curve by adjusting the size of these input arrays. For the TSQR method, the regularization parameters
λi are estimated from the singular values of the triangular R factor. For the normal equations method, they are
estimated from the eigenvalues of the X T X matrix.
const gsl_matrix *gsl_multilarge_linear_matrix_ptr(const gsl_multilarge_linear_workspace *work)
For the normal equations method, this function returns a pointer to the X T X matrix. For the TSQR method,
this function returns a pointer to the upper triangular R matrix.
const gsl_vector *gsl_multilarge_linear_rhs_ptr(const gsl_multilarge_linear_workspace *work)
For the normal equations method, this function returns a pointer to the X T y right hand side vector. For the
TSQR method, this function returns a pointer to the QT y right hand side vector.
int gsl_multilarge_linear_rcond(double *rcond, gsl_multilarge_linear_workspace *work)
This function computes the reciprocal condition number, stored in rcond, of the least squares matrix after it has
been accumulated into the workspace work. For the TSQR algorithm, this is accomplished by calculating the
SVD of the R factor, which has the same singular values as the matrix X. For the normal equations method, thisis done by computing the eigenvalues of X T X, which could be inaccurate for ill-conditioned matrices X.

40.7 Troubleshooting
When using models based on polynomials, care should be taken when constructing the design matrix X. If the x values
are large, then the matrix X could be ill-conditioned since its columns are powers of x, leading to unstable least-squares
solutions. In this case it can often help to center and scale the x values using the mean and standard deviation:x0 =x − µ(x)σ(x)

and then construct the X matrix using the transformed values x0 .

40.8 Examples
The example programs in this section demonstrate the various linear regression methods.

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40.8.1 Simple Linear Regression Example
The following program computes a least squares straight-line fit to a simple dataset, and outputs the best-fit line and its
associated one standard-deviation error bars.
#include <stdio.h>#include <gsl/gsl_fit.h>
intmain (void){
int i, n = 4;
double x[4] = { 1970, 1980, 1990, 2000 };double y[4] = {
12,
11,
14,
13 };
double w[4] = { 0.1, 0.2, 0.3, 0.4 };double c0, c1, cov00, cov01, cov11, chisq;
gsl_fit_wlinear (x, 1, w, 1, y, 1, n,&c0, &c1, &cov00, &cov01, &cov11,&chisq);printf ("# best fit: Y = %g + %g X\n", c0, c1);
printf ("# covariance matrix:\n");printf ("# [ %g, %g\n#%g, %g]\n",cov00, cov01, cov01, cov11);
printf ("# chisq = %g\n", chisq);for (i = 0; i < n; i++)printf ("data: %g %g %g\n",x[i], y[i], 1/sqrt(w[i]));printf ("\n");for (i = -30; i < 130; i++){double xf = x[0] + (i/100.0) * (x[n-1] - x[0]);double yf, yf_err;
gsl_fit_linear_est (xf,c0, c1,
cov00, cov01, cov11,&yf, &yf_err);
printf ("fit: %g %g\n", xf, yf);
printf ("hi : %g %g\n", xf, yf + yf_err);
printf ("lo : %g %g\n", xf, yf - yf_err);}
return 0;}
The following commands extract the data from the output of the program and display it using the GNU plotutils “graph”
utility:

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GNU Scientific Library, Release 2.7$ ./demo > tmp$ more tmp# best fit: Y = -106.6 + 0.06 X# covariance matrix:# [ 39602, -19.9#-19.9, 0.01]# chisq = 0.8$ for n in data fit hi lo ;dogrep "^$n" tmp | cut -d: -f2 > $n ;done
$ graph -T X -X x -Y y -y 0 20 -m 0 -S 2 -Ie data
-S 0 -I a -m 1 fit -m 2 hi -m 2 lo
The result is shown in Fig. 40.1.

40.8.2 Multi-parameter Linear Regression Example
The following program performs a quadratic fit y = c0 + c1 x + c2 x2 to a weighted dataset using the generalised linear
fitting function gsl_multifit_wlinear(). The model matrix X for a quadratic fit is given by,
1 x0 x20
 1 x1 x21 X= 1 x2 x22 ... ... ...
where the column of ones corresponds to the constant term c0 . The two remaining columns corresponds to the terms
c1 x and c2 x2 .
The program reads n lines of data in the format (x, y, err) where err is the error (standard deviation) in the value y.
#include <stdio.h>#include <gsl/gsl_multifit.h>
intmain (int argc, char **argv){
int i, n;double xi, yi, ei, chisq;
gsl_matrix *X, *cov;
gsl_vector *y, *w, *c;
if (argc != 2){
fprintf (stderr,"usage: fit n < data\n");exit (-1);}
n = atoi (argv[1]);X = gsl_matrix_alloc (n, 3);
y = gsl_vector_alloc (n);(continues on next page)

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Fig. 40.1: Straight line fit with 1-σ error bars

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GNU Scientific Library, Release 2.7(continued from previous page)w = gsl_vector_alloc (n);c = gsl_vector_alloc (3);cov = gsl_matrix_alloc (3, 3);
for (i = 0; i < n; i++){int count = fscanf (stdin, "%lg %lg %lg",&xi, &yi, &ei);
if (count != 3){
fprintf (stderr, "error reading file\n");exit (-1);}
printf ("%g %g +/- %g\n", xi, yi, ei);
gsl_matrix_set (X, i, 0, 1.0);
gsl_matrix_set (X, i, 1, xi);
gsl_matrix_set (X, i, 2, xi*xi);
gsl_vector_set (y, i, yi);gsl_vector_set (w, i, 1.0/(ei*ei));}{
gsl_multifit_linear_workspace * work
= gsl_multifit_linear_alloc (n, 3);
gsl_multifit_wlinear (X, w, y, c, cov,
&chisq, work);
gsl_multifit_linear_free (work);}#define C(i) (gsl_vector_get(c,(i)))#define COV(i,j) (gsl_matrix_get(cov,(i),(j))){
printf ("# best fit: Y = %g + %g X + %g X^2\n",C(0), C(1), C(2));
printf ("# covariance matrix:\n");printf ("[ %+.5e, %+.5e, %+.5e \n",COV(0,0), COV(0,1), COV(0,2));
printf (" %+.5e, %+.5e, %+.5e \n",COV(1,0), COV(1,1), COV(1,2));
printf (" %+.5e, %+.5e, %+.5e ]\n",
COV(2,0), COV(2,1), COV(2,2));
printf ("# chisq = %g\n", chisq);}
gsl_matrix_free (X);(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)gsl_vector_free (y);gsl_vector_free (w);gsl_vector_free (c);gsl_matrix_free (cov);
return 0;}
A suitable set of data for fitting can be generated using the following program. It outputs a set of points with gaussian
errors from the curve y = ex in the region 0 < x < 2.
#include <stdio.h>#include <math.h>#include <gsl/gsl_randist.h>intmain (void){
double x;
const gsl_rng_type * T;
gsl_rng * r;
gsl_rng_env_setup ();T = gsl_rng_default;r = gsl_rng_alloc (T);for (x = 0.1; x < 2; x+= 0.1){
double y0 = exp (x);double sigma = 0.1 * y0;
double dy = gsl_ran_gaussian (r, sigma);
printf ("%g %g %g\n", x, y0 + dy, sigma);}
gsl_rng_free(r);return 0;}
The data can be prepared by running the resulting executable program:$ GSL_RNG_TYPE=mt19937_1999 ./generate > exp.dat$ more exp.dat
0.1 0.97935 0.110517
0.2 1.3359 0.12214
0.3 1.52573 0.134986
0.4 1.60318 0.149182
0.5 1.81731 0.164872
0.6 1.92475 0.182212
....

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To fit the data use the previous program, with the number of data points given as the first argument. In this case there
are 19 data points:$ ./fit 19 < exp.dat
0.1 0.97935 +/- 0.110517
0.2 1.3359 +/- 0.12214...
# best fit: Y = 1.02318 + 0.956201 X + 0.876796 X^2# covariance matrix:[ +1.25612e-02, -3.64387e-02, +1.94389e-02-3.64387e-02, +1.42339e-01, -8.48761e-02+1.94389e-02, -8.48761e-02, +5.60243e-02 ]# chisq = 23.0987
The parameters of the quadratic fit match the coefficients of the expansion of ex , taking into account the errors on theparameters and the O(x3 ) difference between the exponential and quadratic functions for the larger values of x. The
errors on the parameters are given by the square-root of the corresponding diagonal elements of the covariance matrix.
The chi-squared per degree of freedom is 1.4, indicating a reasonable fit to the data.
Fig. 40.2 shows the resulting fit.

Fig. 40.2: Weighted fit example with error bars

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40.8.3 Regularized Linear Regression Example 1
The next program demonstrates the difference between ordinary and regularized least squares when the design matrix isnear-singular. In this program, we generate two random normally distributed variables u and v, with v = u + noise sothat u and v are nearly colinear. We then set a third dependent variable y = u + v + noise and solve for the coefficients
c1 , c2 of the model Y (c1 , c2 ) = c1 u + c2 v. Since u ≈ v, the design matrix X is nearly singular, leading to unstableordinary least squares solutions.
Here is the program output:
matrix condition number = 1.025113e+04=== Unregularized fit ===best fit: y = -43.6588 u + 45.6636 v
residual norm = 31.6248
solution norm = 63.1764
chisq/dof = 1.00213=== Regularized fit (L-curve) ===optimal lambda: 4.51103
best fit: y = 1.00113 u + 1.0032 v
residual norm = 31.6547
solution norm = 1.41728
chisq/dof = 1.04499=== Regularized fit (GCV) ===optimal lambda: 0.0232029
best fit: y = -19.8367 u + 21.8417 v
residual norm = 31.6332
solution norm = 29.5051
chisq/dof = 1.00314
We see that the ordinary least squares solution is completely wrong, while the L-curve regularized method with the
optimal λ = 4.51103 finds the correct solution c1 ≈ c2 ≈ 1. The GCV regularized method finds a regularization
parameter λ = 0.0232029 which is too small to give an accurate solution, although it performs better than OLS. The
L-curve and its computed corner, as well as the GCV curve and its minimum are plotted in Fig. 40.3.
The program is given below.
#include <gsl/gsl_math.h>#include <gsl/gsl_vector.h>#include <gsl/gsl_matrix.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>#include <gsl/gsl_multifit.h>intmain(){
const size_t n = 1000; /* number of observations */const size_t p = 2;/* number of model parameters */size_t i;
gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
gsl_matrix *X = gsl_matrix_alloc(n, p);
gsl_vector *y = gsl_vector_alloc(n);504(continues on next page)

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Fig. 40.3: L-curve and GCV curve for example program.
(continued from previous page)for (i = 0; i < n; ++i){/* generate first random variable u */double ui = 5.0 * gsl_ran_gaussian(r, 1.0);/* set v = u + noise */double vi = ui + gsl_ran_gaussian(r, 0.001);/* set y = u + v + noise */double yi = ui + vi + gsl_ran_gaussian(r, 1.0);/* since u =~ v, the matrix X is ill-conditioned */gsl_matrix_set(X, i, 0, ui);
gsl_matrix_set(X, i, 1, vi);/* rhs vector */gsl_vector_set(y, i, yi);}{,→const size_t npoints = 200;/* number of points on L-curve and GCV␣
curve */
gsl_multifit_linear_workspace *w =gsl_multifit_linear_alloc(n, p);
gsl_vector *c = gsl_vector_alloc(p);/* OLS solution */gsl_vector *c_lcurve = gsl_vector_alloc(p);/* regularized solution (L-curve) */gsl_vector *c_gcv = gsl_vector_alloc(p);/* regularized solution (GCV) */gsl_vector *reg_param = gsl_vector_alloc(npoints);(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)gsl_vector *rho = gsl_vector_alloc(npoints);
gsl_vector *eta = gsl_vector_alloc(npoints);
gsl_vector *G = gsl_vector_alloc(npoints);
double lambda_l;,→curve) */double lambda_gcv;,→(GCV) */double G_gcv;
size_t reg_idx;
double rcond;double chisq, rnorm, snorm;/* residual norms *//* solution norms *//* GCV function values *//* optimal regularization parameter (L/* optimal regularization parameter␣/* G(lambda_gcv) *//* index of optimal lambda *//* reciprocal condition number of X *//* compute SVD of X */gsl_multifit_linear_svd(X, w);
rcond = gsl_multifit_linear_rcond(w);fprintf(stderr, "matrix condition number = %e\n\n", 1.0 / rcond);/* unregularized (standard) least squares fit, lambda = 0 */gsl_multifit_linear_solve(0.0, X, y, c, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0);
fprintf(stderr, "=== Unregularized fit ===\n");fprintf(stderr, "best fit: y = %g u + %g v\n",gsl_vector_get(c, 0), gsl_vector_get(c, 1));
fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));/* calculate L-curve and find its corner */gsl_multifit_linear_lcurve(y, reg_param, rho, eta, w);
gsl_multifit_linear_lcorner(rho, eta, &reg_idx);/* store optimal regularization parameter */lambda_l = gsl_vector_get(reg_param, reg_idx);/* regularize with lambda_l */gsl_multifit_linear_solve(lambda_l, X, y, c_lcurve, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0) + pow(lambda_l * snorm, 2.0);
fprintf(stderr, "\n=== Regularized fit (L-curve) ===\n");fprintf(stderr, "optimal lambda: %g\n", lambda_l);fprintf(stderr, "best fit: y = %g u + %g v\n",gsl_vector_get(c_lcurve, 0), gsl_vector_get(c_lcurve, 1));
fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));/* calculate GCV curve and find its minimum */gsl_multifit_linear_gcv(y, reg_param, G, &lambda_gcv, &G_gcv, w);/* regularize with lambda_gcv */(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)gsl_multifit_linear_solve(lambda_gcv, X, y, c_gcv, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0) + pow(lambda_gcv * snorm, 2.0);
fprintf(stderr, "\n=== Regularized fit (GCV) ===\n");fprintf(stderr, "optimal lambda: %g\n", lambda_gcv);
fprintf(stderr, "best fit: y = %g u + %g v\n",gsl_vector_get(c_gcv, 0), gsl_vector_get(c_gcv, 1));
fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));/* output L-curve and GCV curve */for (i = 0; i < npoints; ++i){
printf("%e %e %e %e\n",gsl_vector_get(reg_param, i),
gsl_vector_get(rho, i),
gsl_vector_get(eta, i),
gsl_vector_get(G, i));}/* output L-curve corner point */printf("\n\n%f %f \n",gsl_vector_get(rho, reg_idx),
gsl_vector_get(eta, reg_idx));/* output GCV curve corner minimum */printf("\n\n%e %e\n",lambda_gcv,
G_gcv);
gsl_multifit_linear_free(w);gsl_vector_free(c);gsl_vector_free(c_lcurve);
gsl_vector_free(reg_param);
gsl_vector_free(rho);
gsl_vector_free(eta);
gsl_vector_free(G);}gsl_rng_free(r);gsl_matrix_free(X);gsl_vector_free(y);return 0;}

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40.8.4 Regularized Linear Regression Example 2
The following example program minimizes the cost function||y − Xc||2 + λ2 ||x||2
where X is the 10-by-8 Hilbert matrix whose entries are given by
Xij =

1
i+j−1

and the right hand side vector is given by y = [1, −1, 1, −1, 1, −1, 1, −1, 1, −1]T . Solutions are computed for λ = 0(unregularized) as well as for optimal parameters λ chosen by analyzing the L-curve and GCV curve.
Here is the program output:
matrix condition number = 3.565872e+09=== Unregularized fit ===residual norm = 2.15376
solution norm = 2.92217e+09
chisq/dof = 2.31934=== Regularized fit (L-curve) ===optimal lambda: 7.11407e-07
residual norm = 2.60386
solution norm = 424507
chisq/dof = 3.43565=== Regularized fit (GCV) ===optimal lambda: 1.72278
residual norm = 3.1375
solution norm = 0.139357
chisq/dof = 4.95076
Here we see the unregularized solution results in a large solution norm due to the ill-conditioned matrix. The L-curve
solution finds a small value of λ = 7.11e − 7 which still results in a badly conditioned system and a large solution
norm. The GCV method finds a parameter λ = 1.72 which results in a well-conditioned system and small solutionnorm.
The L-curve and its computed corner, as well as the GCV curve and its minimum are plotted in Fig. 40.4.
The program is given below.
#include <gsl/gsl_math.h>#include <gsl/gsl_vector.h>#include <gsl/gsl_matrix.h>#include <gsl/gsl_multifit.h>#include <gsl/gsl_blas.h>static int
hilbert_matrix(gsl_matrix * m){const size_t N = m->size1;
const size_t M = m->size2;
size_t i, j;(continues on next page)

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Fig. 40.4: L-curve and GCV curve for example program.
(continued from previous page)for (i = 0; i < N; i++){for (j = 0; j < M; j++){gsl_matrix_set(m, i, j, 1.0/(i+j+1.0));}}
return GSL_SUCCESS;}
int
main(){
const size_t n = 10; /* number of observations */const size_t p = 8; /* number of model parameters */size_t i;
gsl_matrix *X = gsl_matrix_alloc(n, p);
gsl_vector *y = gsl_vector_alloc(n);/* construct Hilbert matrix and rhs vector */hilbert_matrix(X);{double val = 1.0;
for (i = 0; i < n; ++i){
gsl_vector_set(y, i, val);(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)val *= -1.0;}}{
const size_t npoints = 200;/* number of points on L-curve and GCV␣
curve */
gsl_multifit_linear_workspace *w =gsl_multifit_linear_alloc(n, p);
gsl_vector *c = gsl_vector_alloc(p);/* OLS solution */gsl_vector *c_lcurve = gsl_vector_alloc(p);/* regularized solution (L-curve) */gsl_vector *c_gcv = gsl_vector_alloc(p);/* regularized solution (GCV) */gsl_vector *reg_param = gsl_vector_alloc(npoints);
gsl_vector *rho = gsl_vector_alloc(npoints); /* residual norms */gsl_vector *eta = gsl_vector_alloc(npoints); /* solution norms */gsl_vector *G = gsl_vector_alloc(npoints);/* GCV function values */double lambda_l;/* optimal regularization parameter (L,→curve) */double lambda_gcv;/* optimal regularization parameter␣
,→(GCV) */double G_gcv;/* G(lambda_gcv) */size_t reg_idx;/* index of optimal lambda */double rcond;/* reciprocal condition number of X */double chisq, rnorm, snorm;,→/* compute SVD of X */gsl_multifit_linear_svd(X, w);
rcond = gsl_multifit_linear_rcond(w);fprintf(stderr, "matrix condition number = %e\n", 1.0 / rcond);/* unregularized (standard) least squares fit, lambda = 0 */gsl_multifit_linear_solve(0.0, X, y, c, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0);
fprintf(stderr, "\n=== Unregularized fit ===\n");fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));/* calculate L-curve and find its corner */gsl_multifit_linear_lcurve(y, reg_param, rho, eta, w);
gsl_multifit_linear_lcorner(rho, eta, &reg_idx);/* store optimal regularization parameter */lambda_l = gsl_vector_get(reg_param, reg_idx);/* regularize with lambda_l */gsl_multifit_linear_solve(lambda_l, X, y, c_lcurve, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0) + pow(lambda_l * snorm, 2.0);
fprintf(stderr, "\n=== Regularized fit (L-curve) ===\n");(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)fprintf(stderr, "optimal lambda: %g\n", lambda_l);fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));/* calculate GCV curve and find its minimum */gsl_multifit_linear_gcv(y, reg_param, G, &lambda_gcv, &G_gcv, w);/* regularize with lambda_gcv */gsl_multifit_linear_solve(lambda_gcv, X, y, c_gcv, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0) + pow(lambda_gcv * snorm, 2.0);
fprintf(stderr, "\n=== Regularized fit (GCV) ===\n");fprintf(stderr, "optimal lambda: %g\n", lambda_gcv);
fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));/* output L-curve and GCV curve */for (i = 0; i < npoints; ++i){
printf("%e %e %e %e\n",gsl_vector_get(reg_param, i),
gsl_vector_get(rho, i),
gsl_vector_get(eta, i),
gsl_vector_get(G, i));}/* output L-curve corner point */printf("\n\n%f %f \n",gsl_vector_get(rho, reg_idx),
gsl_vector_get(eta, reg_idx));/* output GCV curve corner minimum */printf("\n\n%e %e\n",lambda_gcv,
G_gcv);
gsl_multifit_linear_free(w);gsl_vector_free(c);gsl_vector_free(c_lcurve);
gsl_vector_free(reg_param);
gsl_vector_free(rho);
gsl_vector_free(eta);
gsl_vector_free(G);}gsl_matrix_free(X);gsl_vector_free(y);return 0;}

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40.8.5 Robust Linear Regression Example
The next program demonstrates the advantage of robust least squares on a dataset with outliers. The program generates
linear (x, y) data pairs on the line y = 1.45x + 3.88, adds some random noise, and inserts 3 outliers into the dataset.
Both the robust and ordinary least squares (OLS) coefficients are computed for comparison.
#include <stdio.h>#include <gsl/gsl_multifit.h>#include <gsl/gsl_randist.h>
int
dofit(const gsl_multifit_robust_type *T,
const gsl_matrix *X, const gsl_vector *y,
gsl_vector *c, gsl_matrix *cov){
int s;
gsl_multifit_robust_workspace * work
= gsl_multifit_robust_alloc (T, X->size1, X->size2);s = gsl_multifit_robust (X, y, c, cov, work);
gsl_multifit_robust_free (work);
return s;}
int
main (int argc, char **argv){
size_t i;size_t n;
const size_t p = 2; /* linear fit */gsl_matrix *X, *cov;
gsl_vector *x, *y, *c, *c_ols;
const double a = 1.45; /* slope */const double b = 3.88; /* intercept */gsl_rng *r;
if (argc != 2){
fprintf (stderr,"usage: robfit n\n");exit (-1);}
n = atoi (argv[1]);X = gsl_matrix_alloc (n, p);
x = gsl_vector_alloc (n);y = gsl_vector_alloc (n);c = gsl_vector_alloc (p);c_ols = gsl_vector_alloc (p);cov = gsl_matrix_alloc (p, p);(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)r = gsl_rng_alloc(gsl_rng_default);/* generate linear dataset */for (i = 0; i < n - 3; i++){double dx = 10.0 / (n - 1.0);
double ei = gsl_rng_uniform(r);double xi = -5.0 + i * dx;
double yi = a * xi + b;
gsl_vector_set (x, i, xi);
gsl_vector_set (y, i, yi + ei);}/* add a few outliers */gsl_vector_set(x, n - 3, 4.7);
gsl_vector_set(y, n - 3, -8.3);
gsl_vector_set(x, n - 2, 3.5);
gsl_vector_set(y, n - 2, -6.7);
gsl_vector_set(x, n - 1, 4.1);
gsl_vector_set(y, n - 1, -6.0);/* construct design matrix X for linear fit */for (i = 0; i < n; ++i){
double xi = gsl_vector_get(x, i);
gsl_matrix_set (X, i, 0, 1.0);
gsl_matrix_set (X, i, 1, xi);}/* perform robust and OLS fit */dofit(gsl_multifit_robust_ols, X, y, c_ols, cov);
dofit(gsl_multifit_robust_bisquare, X, y, c, cov);/* output data and model */for (i = 0; i < n; ++i){
double xi = gsl_vector_get(x, i);double yi = gsl_vector_get(y, i);
gsl_vector_view v = gsl_matrix_row(X, i);double y_ols, y_rob, y_err;
gsl_multifit_robust_est(&v.vector, c, cov, &y_rob, &y_err);
gsl_multifit_robust_est(&v.vector, c_ols, cov, &y_ols, &y_err);
printf("%g %g %g %g\n", xi, yi, y_rob, y_ols);}#define C(i) (gsl_vector_get(c,(i)))(continues on next page)

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513

GNU Scientific Library, Release 2.7(continued from previous page)#define COV(i,j) (gsl_matrix_get(cov,(i),(j))){
printf ("# best fit: Y = %g + %g X\n",C(0), C(1));
printf ("# covariance matrix:\n");printf ("# [ %+.5e, %+.5e\n",
COV(0,0), COV(0,1));
printf ("#%+.5e, %+.5e\n",
COV(1,0), COV(1,1));}
gsl_matrix_free (X);
gsl_vector_free (x);gsl_vector_free (y);gsl_vector_free (c);gsl_vector_free (c_ols);
gsl_matrix_free (cov);
gsl_rng_free(r);return 0;}
The output from the program is shown in Fig. 40.5.

40.8.6 Large Dense Linear Regression Example
The following program demostrates the large dense linear least squares solvers. This example is adapted from Trefethen
and Bau, and fits the function f (t) = exp (sin3 (10t)) on the interval [0, 1] with a degree 15 polynomial. The program
generates n = 50000 equally spaced points ti on this interval, calculates the function value and adds random noise todetermine the observation value yi . The entries of the least squares matrix are Xij = tji , representing a polynomial
fit. The matrix is highly ill-conditioned, with a condition number of about 2.4 · 1011 . The program accumulates thematrix into the least squares system in 5 blocks, each with 10000 rows. This way the full matrix X is never stored
in memory. We solve the system with both the normal equations and TSQR methods. The results are shown in Fig.
40.6. In the top left plot, the TSQR solution provides a reasonable agreement to the exact solution, while the normal
equations method fails completely since the Cholesky factorization fails due to the ill-conditioning of the matrix. In thebottom left plot, we show the L-curve calculated from TSQR, which exhibits multiple corners. In the top right panel,
we plot a regularized solution using λ = 10−5 . The TSQR and normal solutions now agree, however they are unable toprovide a good fit due to the damping. This indicates that for some ill-conditioned problems, regularizing the normal
equations does not improve the solution. This is further illustrated in the bottom right panel, where we plot the L-curve
calculated from the normal equations. The curve agrees with the TSQR curve for larger damping parameters, but forsmall λ, the normal equations approach cannot provide accurate solution vectors leading to numerical inaccuracies in
the left portion of the curve.
#include <gsl/gsl_math.h>#include <gsl/gsl_vector.h>#include <gsl/gsl_matrix.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>#include <gsl/gsl_multifit.h>(continues on next page)

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GNU Scientific Library, Release 2.7

Fig. 40.5: Linear fit to dataset with outliers.

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Fig. 40.6: Top left: unregularized solutions; top right: regularized solutions; bottom left: L-curve for TSQR method;bottom right: L-curve from normal equations method.

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GNU Scientific Library, Release 2.7(continued from previous page)#include <gsl/gsl_multilarge.h>#include <gsl/gsl_blas.h>#include <gsl/gsl_errno.h>/* function to be fitted */doublefunc(const double t){
double x = sin(10.0 * t);return exp(x*x*x);}/* construct a row of the least squares matrix */intbuild_row(const double t, gsl_vector *row){
const size_t p = row->size;
double Xj = 1.0;
size_t j;for (j = 0; j < p; ++j){
gsl_vector_set(row, j, Xj);
Xj *= t;}return 0;}
int
solve_system(const int print_data, const gsl_multilarge_linear_type * T,const double lambda, const size_t n, const size_t p,
gsl_vector * c){const size_t nblock = 5;/* number of blocks to accumulate */const size_t nrows = n / nblock; /* number of rows per block */gsl_multilarge_linear_workspace * w =gsl_multilarge_linear_alloc(T, p);
gsl_matrix *X = gsl_matrix_alloc(nrows, p);
gsl_vector *y = gsl_vector_alloc(nrows);
gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
const size_t nlcurve = 200;
gsl_vector *reg_param = gsl_vector_alloc(nlcurve);
gsl_vector *rho = gsl_vector_calloc(nlcurve);
gsl_vector *eta = gsl_vector_calloc(nlcurve);size_t rowidx = 0;
double rnorm, snorm, rcond;double t = 0.0;
double dt = 1.0 / (n - 1.0);
while (rowidx < n){(continues on next page)

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517

GNU Scientific Library, Release 2.7(continued from previous page)size_t nleft = n - rowidx;/* number of rows left to accumulate */size_t nr = GSL_MIN(nrows, nleft); /* number of rows in this block */gsl_matrix_view Xv = gsl_matrix_submatrix(X, 0, 0, nr, p);
gsl_vector_view yv = gsl_vector_subvector(y, 0, nr);size_t i;/* build (X,y) block with 'nr' rows */for (i = 0; i < nr; ++i){
gsl_vector_view row = gsl_matrix_row(&Xv.matrix, i);double fi = func(t);double ei = gsl_ran_gaussian (r, 0.1 * fi); /* noise */double yi = fi + ei;/* construct this row of LS matrix */build_row(t, &row.vector);/* set right hand side value with added noise */gsl_vector_set(&yv.vector, i, yi);if (print_data && (i % 100 == 0))
printf("%f %f \n", t, yi);t += dt;}/* accumulate (X,y) block into LS system */gsl_multilarge_linear_accumulate(&Xv.matrix, &yv.vector, w);
rowidx += nr;}
if (print_data)
printf("\n\n");/* compute L-curve */gsl_multilarge_linear_lcurve(reg_param, rho, eta, w);/* solve large LS system and store solution in c */gsl_multilarge_linear_solve(lambda, c, &rnorm, &snorm, w);/* compute reciprocal condition number */gsl_multilarge_linear_rcond(&rcond, w);
fprintf(stderr, "=== Method %s ===\n", gsl_multilarge_linear_name(w));
fprintf(stderr, "condition number = %e\n", 1.0 / rcond);
fprintf(stderr, "residual norm
= %e\n", rnorm);
fprintf(stderr, "solution norm
= %e\n", snorm);/* output L-curve */{size_t i;(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)for (i = 0; i < nlcurve; ++i){
printf("%.12e %.12e %.12e\n",gsl_vector_get(reg_param, i),
gsl_vector_get(rho, i),
gsl_vector_get(eta, i));}printf("\n\n");}gsl_matrix_free(X);gsl_vector_free(y);gsl_multilarge_linear_free(w);gsl_rng_free(r);gsl_vector_free(reg_param);
gsl_vector_free(rho);
gsl_vector_free(eta);
return 0;}
int
main(int argc, char *argv[]){
const size_t n = 50000;/* number of observations */const size_t p = 16;/* polynomial order + 1 */double lambda = 0.0;/* regularization parameter */gsl_vector *c_tsqr = gsl_vector_calloc(p);gsl_vector *c_normal = gsl_vector_calloc(p);if (argc > 1)
lambda = atof(argv[1]);/* turn off error handler so normal equations method won't abort */gsl_set_error_handler_off();/* solve system with TSQR method */solve_system(1, gsl_multilarge_linear_tsqr, lambda, n, p, c_tsqr);/* solve system with Normal equations method */solve_system(0, gsl_multilarge_linear_normal, lambda, n, p, c_normal);/* output solutions */{gsl_vector *v = gsl_vector_alloc(p);double t;for (t = 0.0; t <= 1.0; t += 0.01){
double f_exact = func(t);double f_tsqr, f_normal;(continues on next page)

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519

GNU Scientific Library, Release 2.7(continued from previous page)build_row(t, v);
gsl_blas_ddot(v, c_tsqr, &f_tsqr);
gsl_blas_ddot(v, c_normal, &f_normal);printf("%f %e %e %e\n", t, f_exact, f_tsqr, f_normal);}
gsl_vector_free(v);}gsl_vector_free(c_tsqr);
gsl_vector_free(c_normal);return 0;}40.9 References and Further Reading
A summary of formulas and techniques for least squares fitting can be found in the “Statistics” chapter of the Annual
Review of Particle Physics prepared by the Particle Data Group,• Review of Particle Properties, R.M. Barnett et al., Physical Review D54, 1 (1996) http://pdg.lbl.gov
The Review of Particle Physics is available online at the website given above.
The tests used to prepare these routines are based on the NIST Statistical Reference Datasets. The datasets and their
documentation are available from NIST at the following website,
http://www.nist.gov/itl/div898/strd/index.html
More information on Tikhonov regularization can be found in• Hansen, P. C. (1998), Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion.
SIAM Monogr. on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics• M. Rezghi and S. M. Hosseini (2009), A new variant of L-curve for Tikhonov regularization, Journal of Computational and Applied Mathematics, Volume 231, Issue 2, pages 914-924.
The GSL implementation of robust linear regression closely follows the publications• DuMouchel, W. and F. O’Brien (1989), “Integrating a robust option into a multiple regression computing environment,” Computer Science and Statistics: Proceedings of the 21st Symposium on the Interface, American
Statistical Association• Street, J.O., R.J. Carroll, and D. Ruppert (1988), “A note on computing robust regression estimates via iteratively
reweighted least squares,” The American Statistician, v. 42, pp. 152-154.
More information about the normal equations and TSQR approach for solving large linear least squares systems can befound in the publications• Trefethen, L. N. and Bau, D. (1997), “Numerical Linear Algebra”, SIAM.
• Demmel, J., Grigori, L., Hoemmen, M. F., and Langou, J. “Communication-optimal parallel and sequential QR
and LU factorizations”, UCB Technical Report No. UCB/EECS-2008-89, 2008.

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CHAPTERFORTYONENONLINEAR LEAST-SQUARES FITTING

This chapter describes functions for multidimensional nonlinear least-squares fitting. There are generally two classes ofalgorithms for solving nonlinear least squares problems, which fall under line search methods and trust region methods.
GSL currently implements only trust region methods and provides the user with full access to intermediate steps of theiteration. The user also has the ability to tune a number of parameters which affect low-level aspects of the algorithm
which can help to accelerate convergence for the specific problem at hand. GSL provides two separate interfaces fornonlinear least squares fitting. The first is designed for small to moderate sized problems, and the second is designed
for very large problems, which may or may not have significant sparse structure.
The header file gsl_multifit_nlinear.h contains prototypes for the multidimensional nonlinear fitting functions
and related declarations relating to the small to moderate sized systems.
The header file gsl_multilarge_nlinear.h contains prototypes for the multidimensional nonlinear fitting functions
and related declarations relating to large systems.

41.1 Overview
The problem of multidimensional nonlinear least-squares fitting requires the minimization of the squared residuals ofn functions, fi , in p parameters, xi ,1||f (x)||22
n1X=
fi (x1 , . . . , xp )2
2 i=1

Φ(x) =In trust region methods, the objective (or cost) function Φ(x) is approximated by a model function mk (δ) in the vicinityof some point xk . The model function is often simply a second order Taylor series expansion around the point xk , ie:1
Φ(xk + δ) ≈ mk (δ) = Φ(xk ) + gkT δ + δ T Bk δ
2where gk = ∇Φ(xk ) = J T f is the gradient vector at the point xk , Bk = ∇2 Φ(xk ) is the Hessian matrix at xk , orsome approximation to it, and J is the n-by-p Jacobian matrix
Jij = ∂fi /∂xj
In order to find the next step δ, we minimize the model function mk (δ), but search for solutions only within a region
where we trust that mk (δ) is a good approximation to the objective function Φ(xk + δ). In other words, we seek asolution of the trust region subproblem (TRS)
1minp mk (δ) = Φ(xk ) + gkT δ + δ T Bk δ,
δ∈R
2

s.t.

||Dk δ|| ≤ ∆k521

GNU Scientific Library, Release 2.7

where ∆k > 0 is the trust region radius and Dk is a scaling matrix. If Dk = I, then the trust region is a ball of radius∆k centered at xk . In some applications, the parameter vector x may have widely different scales. For example, one
parameter might be a temperature on the order of 103 K, while another might be a length on the order of 10−6 m. In
such cases, a spherical trust region may not be the best choice, since if Φ changes rapidly along directions with one
scale, and more slowly along directions with a different scale, the model function mk may be a poor approximation toΦ along the rapidly changing directions. In such problems, it may be best to use an elliptical trust region, by setting
Dk to a diagonal matrix whose entries are designed so that the scaled step Dk δ has entries of approximately the sameorder of magnitude.
The trust region subproblem above normally amounts to solving a linear least squares system (or multiple systems) for
the step δ. Once δ is computed, it is checked whether or not it reduces the objective function Φ(x). A useful statistic
for this is to look at the ratioρk =Φ(xk ) − Φ(xk + δk )mk (0) − mk (δk )where the numerator is the actual reduction of the objective function due to the step δk , and the denominator is thepredicted reduction due to the model mk . If ρk is negative, it means that the step δk increased the objective function
and so it is rejected. If ρk is positive, then we have found a step which reduced the objective function and it is accepted.
Furthermore, if ρk is close to 1, then this indicates that the model function is a good approximation to the objectivefunction in the trust region, and so on the next iteration the trust region is enlarged in order to take more ambitious
steps. When a step is rejected, the trust region is made smaller and the TRS is solved again. An outline for the generaltrust region method used by GSL can now be given.
Trust Region Algorithm
1. Initialize: given x0 , construct m0 (δ), D0 and ∆0 > 0
2. For k = 0, 1, 2, . . .
a. If converged, then stopb. Solve TRS for trial step δkc. Evaluate trial step by computing ρk1). if step is accepted, set xk+1 = xk + δk and increase radius, ∆k+1 = α∆k2). if step is rejected, set xk+1 = xk and decrease radius, ∆k+1 = Δβk ; goto 2(b)d. Construct mk+1 (δ) and Dk+1
GSL offers the user a number of different algorithms for solving the trust region subproblem in 2(b), as well as different choices of scaling matrices Dk and different methods of updating the trust region radius ∆k . Therefore, whilereasonable default methods are provided, the user has a lot of control to fine-tune the various steps of the algorithm fortheir specific problem.

41.2 Solving the Trust Region Subproblem (TRS)
Below we describe the methods available for solving the trust region subproblem. The methods available provide
either exact or approximate solutions to the trust region subproblem. In all algorithms below, the Hessian matrix Bk
is approximated as Bk ≈ JkT Jk , where Jk = J(xk ). In all methods, the solution of the TRS involves solving a linear
least squares system involving the Jacobian matrix. For small to moderate sized problems (gsl_multifit_nlinear
interface), this is accomplished by factoring the full Jacobian matrix, which is provided by the user, with the Cholesky,
QR, or SVD decompositions. For large systems (gsl_multilarge_nlinear interface), the user has two choices.
One is to solve the system iteratively, without needing to store the full Jacobian matrix in memory. With this method,
the user must provide a routine to calculate the matrix-vector products Ju or J T u for a given vector u. This iterative
method is particularly useful for systems where the Jacobian has sparse structure, since forming matrix-vector products

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can be done cheaply. The second option for large systems involves forming the normal equations matrix J T J and thenfactoring it using a Cholesky decomposition. The normal equations matrix is p-by-p, typically much smaller than thefull n-by-p Jacobian, and can usually be stored in memory even if the full Jacobian matrix cannot. This option is usefulfor large, dense systems, or if the iterative method has difficulty converging.

41.2.1 Levenberg-Marquardt
There is a theorem which states that if δk is a solution to the trust region subproblem given above, then there existsµk ≥ 0 such that
Bk + µk DkT Dk δk = −gk
with µk (∆k − ||Dk δk ||) = 0. This forms the basis of the Levenberg-Marquardt algorithm, which controls the trust
region size by adjusting the parameter µk rather than the radius ∆k directly. For each radius ∆k , there is a uniqueparameter µk which solves the TRS, and they have an inverse relationship, so that large values of µk correspond tosmaller trust regions, while small values of µk correspond to larger trust regions.
With the approximation Bk ≈ JkT Jk , on each iteration, in order to calculate the step δk , the following linear leastsquares problem is solved:




fk
J
√ k
δk = −µk Dk
0
If the step δk is accepted, then µk is decreased on the next iteration in order to take a larger step, otherwise it is increased
to take a smaller step. The Levenberg-Marquardt algorithm provides an exact solution of the trust region subproblem,
but typically has a higher computational cost per iteration than the approximate methods discussed below, since it may
need to solve the least squares system above several times for different values of µk .

41.2.2 Levenberg-Marquardt with Geodesic Acceleration
This method applies a so-called geodesic acceleration correction to the standard Levenberg-Marquardt step δk(Transtrum et al, 2011). By interpreting δk as a first order step along a geodesic in the model parameter space (ie:a velocity δk = vk ), the geodesic acceleration ak is a second order correction along the geodesic which is determinedby solving the linear least squares system




Jk
fvv (xk )√ak = −
µk Dk
0
where
fvv is the second directional derivative of the residual vector in the velocity direction v, fvv (x) = Dv2 f =P
10αβ vα vβ ∂α ∂β f (x), where α and β are summed over the p parameters. The new total step is then δk = vk + 2 ak .
The second order correction ak can be calculated with a modest additional cost, and has been shown to dramaticallyreduce the number of iterations (and expensive Jacobian evaluations) required to reach convergence on a variety ofdifferent problems. In order to utilize the geodesic acceleration, the user must supply a function which provides thesecond directional derivative vector fvv (x), or alternatively the library can use a finite difference method to estimate
this vector with one additional function evaluation of f (x+hv) where h is a tunable step size (see the h_fvv parameter
description).

41.2. Solving the Trust Region Subproblem (TRS)523

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41.2.3 Dogleg
This is Powell’s dogleg method, which finds an approximate solution to the trust region subproblem, by restricting
its search to a piecewise linear “dogleg” path, composed of the origin, the Cauchy point which represents the model
minimizer along the steepest descent direction, and the Gauss-Newton point, which is the overall minimizer of theunconstrained model. The Gauss-Newton step is calculated by solving
Jk δgn = −fk
which is the main computational task for each iteration, but only needs to be performed once per iteration. If the GaussNewton point is inside the trust region, it is selected as the step. If it is outside, the method then calculates the Cauchy
point, which is located along the gradient direction. If the Cauchy point is also outside the trust region, the method
assumes that it is still far from the minimum and so proceeds along the gradient direction, truncating the step at thetrust region boundary. If the Cauchy point is inside the trust region, with the Gauss-Newton point outside, the method
uses a dogleg step, which is a linear combination of the gradient direction and the Gauss-Newton direction, stoppingat the trust region boundary.

41.2.4 Double Dogleg
This method is an improvement over the classical dogleg algorithm, which attempts to include information about theGauss-Newton step while the iteration is still far from the minimum. When the Cauchy point is inside the trust region
and the Gauss-Newton point is outside, the method computes a scaled Gauss-Newton point and then takes a dogleg step
between the Cauchy point and the scaled Gauss-Newton point. The scaling is calculated to ensure that the reduction in
the model mk is about the same as the reduction provided by the Cauchy point.

41.2.5 Two Dimensional Subspace
The dogleg methods restrict the search for the TRS solution to a 1D curve defined by the Cauchy and Gauss-Newton
points. An improvement to this is to search for a solution using the full two dimensional subspace spanned by the
Cauchy and Gauss-Newton directions. The dogleg path is of course inside this subspace, and so this method solves the
TRS at least as accurately as the dogleg methods. Since this method searches a larger subspace for a solution, it canconverge more quickly than dogleg on some problems. Because the subspace is only two dimensional, this method isvery efficient and the main computation per iteration is to determine the Gauss-Newton point.

41.2.6 Steihaug-Toint Conjugate Gradient
One difficulty of the dogleg methods is calculating the Gauss-Newton step when the Jacobian matrix is singular. The
Steihaug-Toint method also computes a generalized dogleg step, but avoids solving for the Gauss-Newton step directly, instead using an iterative conjugate gradient algorithm. This method performs well at points where the Jacobian
is singular, and is also suitable for large-scale problems where factoring the Jacobian matrix could be prohibitively
expensive.

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41.3 Weighted Nonlinear Least-Squares
Weighted nonlinear least-squares fitting minimizes the function1||f ||2W2n
1X=
wi fi (x1 , . . . , xp )2
2 i=1

Φ(x) =

where W = diag(w1 , w2 , ..., wn ) is the weighting matrix, and ||f ||2W = f T W f . The weights wi are commonly
1
defined as wi = 1/σi2 , where σi is the error in the i-th measurement. A simple change of variables f˜ = W 2 f yields
Φ(x) = 12 ||f˜||2 , which is in the same form as the unweighted case. The user can either perform this transform directlyon their function residuals and Jacobian, or use the gsl_multifit_nlinear_winit() interface which automatically
performs the correct scaling. To manually perform this transformation, the residuals and Jacobian should be modified
according to√fif˜i = wi fi =
σi
√ ∂fi
1 ∂fi
J˜ij = wi=∂xj
σi ∂xjFor large systems, the user must perform their own weighting.

41.4 Tunable Parameters
The user can tune nearly all aspects of the iteration at allocation time. For the gsl_multifit_nlinear interface, theuser may modify the gsl_multifit_nlinear_parameters structure, which is defined as follows:
type gsl_multifit_nlinear_parameters
typedef struct{const gsl_multifit_nlinear_trs *trs;
const gsl_multifit_nlinear_scale *scale;const gsl_multifit_nlinear_solver *solver;
gsl_multifit_nlinear_fdtype fdtype;
double factor_up;,→*/
double factor_down;,→*/
double avmax;
double h_df;,→Jacobian */double h_fvv;,→fvv */
} gsl_multifit_nlinear_parameters;/* trust region subproblem method *//* scaling method *//* solver method *//* finite difference method *//* factor for increasing trust radius␣
/* factor for decreasing trust radius␣/* max allowed |a|/|v| *//* step size for finite difference␣
/* step size for finite difference␣For the gsl_multilarge_nlinear interface, the user may modify the gsl_multilarge_nlinear_parameters
structure, which is defined as follows:
type gsl_multilarge_nlinear_parameters41.3. Weighted Nonlinear Least-Squares

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typedef struct{const gsl_multilarge_nlinear_trs *trs;/* trust region subproblem method */const gsl_multilarge_nlinear_scale *scale;/* scaling method */const gsl_multilarge_nlinear_solver *solver; /* solver method */gsl_multilarge_nlinear_fdtype fdtype;/* finite difference method */double factor_up;/* factor for increasing trust␣,→radius */double factor_down;/* factor for decreasing trust␣,→radius */double avmax;/* max allowed |a|/|v| */double h_df;/* step size for finite difference␣,→Jacobian */double h_fvv;/* step size for finite difference␣,→fvv */
size_t max_iter;/* maximum iterations for trs method␣,→*/
double tol;/* tolerance for solving trs */} gsl_multilarge_nlinear_parameters;
Each of these parameters is discussed in further detail below.
type gsl_multifit_nlinear_trs
type gsl_multilarge_nlinear_trs
The parameter trs determines the method used to solve the trust region subproblem, and may be selected fromthe following choices,
gsl_multifit_nlinear_trs *gsl_multifit_nlinear_trs_lm
gsl_multilarge_nlinear_trs *gsl_multilarge_nlinear_trs_lm
This selects the Levenberg-Marquardt algorithm.
gsl_multifit_nlinear_trs *gsl_multifit_nlinear_trs_lmaccel
gsl_multilarge_nlinear_trs *gsl_multilarge_nlinear_trs_lmaccel
This selects the Levenberg-Marquardt algorithm with geodesic acceleration.
gsl_multifit_nlinear_trs *gsl_multifit_nlinear_trs_dogleg
gsl_multilarge_nlinear_trs *gsl_multilarge_nlinear_trs_dogleg
This selects the dogleg algorithm.
gsl_multifit_nlinear_trs *gsl_multifit_nlinear_trs_ddogleg
gsl_multilarge_nlinear_trs *gsl_multilarge_nlinear_trs_ddogleg
This selects the double dogleg algorithm.
gsl_multifit_nlinear_trs *gsl_multifit_nlinear_trs_subspace2D
gsl_multilarge_nlinear_trs *gsl_multilarge_nlinear_trs_subspace2D
This selects the 2D subspace algorithm.
gsl_multilarge_nlinear_trs *gsl_multilarge_nlinear_trs_cgst
This selects the Steihaug-Toint conjugate gradient algorithm. This method is available only for large systems.
type gsl_multifit_nlinear_scale
type gsl_multilarge_nlinear_scale
The parameter scale determines the diagonal scaling matrix D and may be selected from the following choices,

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gsl_multifit_nlinear_scale *gsl_multifit_nlinear_scale_more
gsl_multilarge_nlinear_scale *gsl_multilarge_nlinear_scale_more
This damping strategy was suggested by Moré, and corresponds to DT D = max(diag(J T J)), in otherwords the maximum elements of diag(J T J) encountered thus far in the iteration. This choice of D makes
the problem scale-invariant, so that if the model parameters xi are each scaled by an arbitrary constant,x̃i = ai xi , then the sequence of iterates produced by the algorithm would be unchanged. This method canwork very well in cases where the model parameters have widely different scales (ie: if some parameters
are measured in nanometers, while others are measured in degrees Kelvin). This strategy has been proveneffective on a large class of problems and so it is the library default, but it may not be the best choice for all
problems.
gsl_multifit_nlinear_scale *gsl_multifit_nlinear_scale_levenberg
gsl_multilarge_nlinear_scale *gsl_multilarge_nlinear_scale_levenberg
This damping strategy was originally suggested by Levenberg, and corresponds to DT D = I. This method
has also proven effective on a large class of problems, but is not scale-invariant. However, some authors(e.g. Transtrum and Sethna 2012) argue that this choice is better for problems which are susceptible toparameter evaporation (ie: parameters go to infinity)
gsl_multifit_nlinear_scale *gsl_multifit_nlinear_scale_marquardt
gsl_multilarge_nlinear_scale *gsl_multilarge_nlinear_scale_marquardt
This damping strategy was suggested by Marquardt, and corresponds to DT D = diag(J T J). This method
is scale-invariant, but it is generally considered inferior to both the Levenberg and Moré strategies, though
may work well on certain classes of problems.
type gsl_multifit_nlinear_solver
type gsl_multilarge_nlinear_solver
Solving the trust region subproblem on each iteration almost always requires the solution of the following linear
least squares system




f
J
√
δ=−
µD
0
The solver parameter determines how the system is solved and can be selected from the following choices:
gsl_multifit_nlinear_solver *gsl_multifit_nlinear_solver_qr
This method solves the system using a rank revealing QR decomposition of the Jacobian J. This method
will produce reliable solutions in cases where the Jacobian is rank deficient or near-singular but does require
about twice as many operations as the Cholesky method discussed below.
gsl_multifit_nlinear_solver *gsl_multifit_nlinear_solver_cholesky
gsl_multilarge_nlinear_solver *gsl_multilarge_nlinear_solver_choleskyThis method solves the alternate normal equations problem
J T J + µDT D δ = −J T f
by using a Cholesky decomposition of the matrix J T J + µDT D. This method is faster than the QR
approach, however it is susceptible to numerical instabilities if the Jacobian matrix is rank deficient or
near-singular. In these cases, an attempt is made to reduce the condition number of the matrix using Jacobi
preconditioning, but for highly ill-conditioned problems the QR approach is better. If it is known that theJacobian matrix is well conditioned, this method is accurate and will perform faster than the QR approach.
gsl_multifit_nlinear_solver *gsl_multifit_nlinear_solver_mcholesky
gsl_multilarge_nlinear_solver *gsl_multilarge_nlinear_solver_mcholeskyThis method solves the alternate normal equations problem
J T J + µDT D δ = −J T f
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by using a modified Cholesky decomposition of the matrix J T J + µDT D. This is more suitable for thedogleg methods where the parameter µ = 0, and the matrix J T J may be ill-conditioned or indefinite
causing the standard Cholesky decomposition to fail. This method is based on Level 2 BLAS and is thus
slower than the standard Cholesky decomposition, which is based on Level 3 BLAS.
gsl_multifit_nlinear_solver *gsl_multifit_nlinear_solver_svd
This method solves the system using a singular value decomposition of the Jacobian J. This method willproduce the most reliable solutions for ill-conditioned Jacobians but is also the slowest solver method.
type gsl_multifit_nlinear_fdtype
The parameter fdtype specifies whether to use forward or centered differences when approximating the Jacobian. This is only used when an analytic Jacobian is not provided to the solver. This parameter may be set to oneof the following choices.
GSL_MULTIFIT_NLINEAR_FWDIFF
This specifies a forward finite difference to approximate the Jacobian matrix. The Jacobian matrix will becalculated as
Jij =1(fi (x + ∆j ej ) − fi (x))∆jwhere ∆j = h|xj | and ej is the standard j-th Cartesian unit basis vector so that x + ∆j ej represents asmall (forward) perturbation of the j-th parameter by an amount ∆j . The perturbation ∆j is proportional
to the current value |xj | which helps to calculate an accurate Jacobian when the various parameters havedifferent scale sizes. The value of h is specified by the h_df parameter. The accuracy of this method isO(h), and evaluating this matrix requires an additional p function evaluations.
GSL_MULTIFIT_NLINEAR_CTRDIFF
This specifies a centered finite difference to approximate the Jacobian matrix. The Jacobian matrix will becalculated as


1
1
1
Jij =
fi (x + ∆j ej ) − fi (x − ∆j ej )∆j2
2
See above for a description of ∆j . The accuracy of this method is O(h2 ), but evaluating this matrix requires
an additional 2p function evaluations.
double factor_up
When a step is accepted, the trust region radius will be increased by this factor. The default value is 3.
double factor_down
When a step is rejected, the trust region radius will be decreased by this factor. The default value is 2.
double avmax
When using geodesic acceleration to solve a nonlinear least squares problem, an important parameter to monitor is theratio of the acceleration term to the velocity term,
||a||
||v||
If this ratio is small, it means the acceleration correction is contributing very little to the step. This could be because
the problem is not “nonlinear” enough to benefit from the acceleration. If the ratio is large (> 1) it means that theacceleration is larger than the velocity, which shouldn’t happen since the step represents a truncated series and so thesecond order term a should be smaller than the first order term v to guarantee convergence. Therefore any steps witha ratio larger than the parameter avmax are rejected. avmax is set to 0.75 by default. For problems which experience
difficulty converging, this threshold could be lowered.
double h_df

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This parameter specifies the step size for approximating the Jacobian matrix with finite differences. It is set todefault, where  is GSL_DBL_EPSILON.

√

 by

double h_fvv
When using geodesic acceleration, the user must either supply a function to calculate fvv (x) or the library can estimate
this second directional derivative using a finite difference method. When using finite differences, the library must
calculate f (x + hv) where h represents a small step in the velocity direction. The parameter h_fvv defines this step
size and is set to 0.02 by default.

41.5 Initializing the Solver
type gsl_multifit_nlinear_type
This structure specifies the type of algorithm which will be used to solve a nonlinear least squares problem. It
may be selected from the following choices,
gsl_multifit_nlinear_type *gsl_multifit_nlinear_trust
This specifies a trust region method. It is currently the only implemented nonlinear least squares method.
gsl_multifit_nlinear_workspace *gsl_multifit_nlinear_alloc(const gsl_multifit_nlinear_type *T, const
gsl_multifit_nlinear_parameters *params, const
size_t n, const size_t p)
gsl_multilarge_nlinear_workspace *gsl_multilarge_nlinear_alloc(const gsl_multilarge_nlinear_type *T,
const gsl_multilarge_nlinear_parameters*params, const size_t n, const size_t p)
These functions return a pointer to a newly allocated instance of a derivative solver of type T for n observations and p parameters. The params input specifies a tunable set of parameters which will affect important details in each iteration of the trust region subproblem algorithm. It is recommended tostart with the suggested default parameters (see gsl_multifit_nlinear_default_parameters() and
gsl_multilarge_nlinear_default_parameters()) and then tune the parameters once the code is working correctly. See Tunable Parameters. for descriptions of the various parameters. For example, the following
code creates an instance of a Levenberg-Marquardt solver for 100 data points and 3 parameters, using suggested
defaults:
const gsl_multifit_nlinear_type * T = gsl_multifit_nlinear_trust;
gsl_multifit_nlinear_parameters params = gsl_multifit_nlinear_default_parameters();gsl_multifit_nlinear_workspace * w = gsl_multifit_nlinear_alloc (T, &params, 100,␣
,→3);
The number of observations n must be greater than or equal to parameters p.
If there is insufficient memory to create the solver then the function returns a null pointer and the error handler
is invoked with an error code of GSL_ENOMEM.
gsl_multifit_nlinear_parameters gsl_multifit_nlinear_default_parameters(void)
gsl_multilarge_nlinear_parameters gsl_multilarge_nlinear_default_parameters(void)
These functions return a set of recommended default parameters for use in solving nonlinear least squares problems. The user can tune each parameter to improve the performance on their particular problem, see Tunable
Parameters.
int gsl_multifit_nlinear_init(const gsl_vector *x, gsl_multifit_nlinear_fdf *fdf,
gsl_multifit_nlinear_workspace *w)

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int gsl_multifit_nlinear_winit(const gsl_vector *x, const gsl_vector *wts, gsl_multifit_nlinear_fdf *fdf,
gsl_multifit_nlinear_workspace *w)
int gsl_multilarge_nlinear_init(const gsl_vector *x, gsl_multilarge_nlinear_fdf *fdf,
gsl_multilarge_nlinear_workspace *w)
These functions initialize, or reinitialize, an existing workspace w to use the system fdf and the initial guess x.
See Providing the Function to be Minimized for a description of the fdf structure.
Optionally, a weight vector wts can be given to perform a weighted nonlinear regression. Here, the weighting
matrix is W = diag(w1 , w2 , ..., wn ).
void gsl_multifit_nlinear_free(gsl_multifit_nlinear_workspace *w)
void gsl_multilarge_nlinear_free(gsl_multilarge_nlinear_workspace *w)
These functions free all the memory associated with the workspace w.
const char *gsl_multifit_nlinear_name(const gsl_multifit_nlinear_workspace *w)
const char *gsl_multilarge_nlinear_name(const gsl_multilarge_nlinear_workspace *w)
These functions return a pointer to the name of the solver. For example:
printf ("w is a '%s' solver\n", gsl_multifit_nlinear_name (w));
would print something like w is a 'trust-region' solver.
const char *gsl_multifit_nlinear_trs_name(const gsl_multifit_nlinear_workspace *w)
const char *gsl_multilarge_nlinear_trs_name(const gsl_multilarge_nlinear_workspace *w)
These functions return a pointer to the name of the trust region subproblem method. For example:
printf ("w is a '%s' solver\n", gsl_multifit_nlinear_trs_name (w));
would print something like w is a 'levenberg-marquardt' solver.

41.6 Providing the Function to be Minimized
The user must provide n functions of p variables for the minimization algorithm to operate on. In order to allow for
arbitrary parameters the functions are defined by the following data types:
type gsl_multifit_nlinear_fdf
This data type defines a general system of functions with arbitrary parameters, the corresponding Jacobian matrixof derivatives, and optionally the second directional derivative of the functions for geodesic acceleration.
int (* f) (const gsl_vector * x, void * params, gsl_vector * f)This function should store the n components of the vector f (x) in f for argument x and arbitrary
parameters params, returning an appropriate error code if the function cannot be computed.
int (* df) (const gsl_vector * x, void * params, gsl_matrix * J)This function should store the n-by-p matrix result
Jij = ∂fi (x)/∂xj
in J for argument x and arbitrary parameters params, returning an appropriate error code if the matrix cannot be computed. If an analytic Jacobian is unavailable, or too expensive to compute, this

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function pointer may be set to NULL, in which case the Jacobian will be internally computed using
finite difference approximations of the function f.
int (* fvv) (const gsl_vector * x, const gsl_vector * v, void * params, gsl_vector *fvv)
When geodesic
P acceleration is enabled, this function should store the n components of the vector
fvv (x) = αβ vα vβ ∂x∂α ∂x∂ β f (x), representing second directional derivatives of the function to beminimized, into the output fvv. The parameter vector is provided in x and the velocity vector isprovided in v, both of which have p components. The arbitrary parameters are given in params. If
analytic expressions for fvv (x) are unavailable or too difficult to compute, this function pointer may be
set to NULL, in which case fvv (x) will be computed internally using a finite difference approximation.
size_t n
the number of functions, i.e. the number of components of the vector f.
size_t p
the number of independent variables, i.e. the number of components of the vector x.
void * params
a pointer to the arbitrary parameters of the function.
size_t nevalfThis does not need to be set by the user. It counts the number of function evaluations and is initialized
by the _init function.
size_t nevaldf
This does not need to be set by the user. It counts the number of Jacobian evaluations and is initialized
by the _init function.
size_t nevalfvv
This does not need to be set by the user. It counts the number of fvv (x) evaluations and is initialized
by the _init function.
type gsl_multilarge_nlinear_fdf
This data type defines a general system of functions with arbitrary parameters, a function to compute Ju or J T u
for a given vector u, the normal equations matrix J T J, and optionally the second directional derivative of thefunctions for geodesic acceleration.
int (* f) (const gsl_vector * x, void * params, gsl_vector * f)This function should store the n components of the vector f (x) in f for argument x and arbitrary
parameters params, returning an appropriate error code if the function cannot be computed.
int (* df) (CBLAS_TRANSPOSE_t TransJ, const gsl_vector * x, const gsl_vector * u,void * params, gsl_vector * v, gsl_matrix * JTJ)
If TransJ is equal to CblasNoTrans, then this function should compute the matrix-vector productJu and store the result in v. If TransJ is equal to CblasTrans, then this function should computethe matrix-vector product J T u and store the result in v. Additionally, the normal equations matrix
J T J should be stored in the lower half of JTJ. The input matrix JTJ could be set to NULL, for example by iterative methods which do not require this matrix, so the user should check for this prior toconstructing the matrix. The input params contains the arbitrary parameters.
int (* fvv) (const gsl_vector * x, const gsl_vector * v, void * params, gsl_vector *fvv)

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When geodesic
P acceleration is enabled, this function should store the n components of the vector
fvv (x) = αβ vα vβ ∂x∂α ∂x∂ β f (x), representing second directional derivatives of the function to beminimized, into the output fvv. The parameter vector is provided in x and the velocity vector isprovided in v, both of which have p components. The arbitrary parameters are given in params. If
analytic expressions for fvv (x) are unavailable or too difficult to compute, this function pointer may be
set to NULL, in which case fvv (x) will be computed internally using a finite difference approximation.
size_t n
the number of functions, i.e. the number of components of the vector f.
size_t p
the number of independent variables, i.e. the number of components of the vector x.
void * params
a pointer to the arbitrary parameters of the function.
size_t nevalfThis does not need to be set by the user. It counts the number of function evaluations and is initialized
by the _init function.
size_t nevaldfu
This does not need to be set by the user. It counts the number of Jacobian matrix-vector evaluations(Ju or J T u) and is initialized by the _init function.
size_t nevaldf2
This does not need to be set by the user. It counts the number of J T J evaluations and is initialized
by the _init function.
size_t nevalfvv
This does not need to be set by the user. It counts the number of fvv (x) evaluations and is initialized
by the _init function.
Note that when fitting a non-linear model against experimental data, the data is passed to the functions above using theparams argument and the trial best-fit parameters through the x argument.

41.7 Iteration
The following functions drive the iteration of each algorithm. Each function performs one iteration of the trust region
method and updates the state of the solver.
int gsl_multifit_nlinear_iterate(gsl_multifit_nlinear_workspace *w)
int gsl_multilarge_nlinear_iterate(gsl_multilarge_nlinear_workspace *w)
These functions perform a single iteration of the solver w. If the iteration encounters an unexpected problem thenan error code will be returned. The solver workspace maintains a current estimate of the best-fit parameters at
all times.
The solver workspace w contains the following entries, which can be used to track the progress of the solution:
gsl_vector * x
The current position, length p.
gsl_vector * f

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The function residual vector at the current position f (x), length n.
gsl_matrix * J
The Jacobian matrix at the current position J(x), size n-by-p (only for gsl_multifit_nlinear interface).
gsl_vector * dx
The difference between the current position and the previous position, i.e. the last step δ, taken as a vector,
length p.
These quantities can be accessed with the following functions,gsl_vector *gsl_multifit_nlinear_position(const gsl_multifit_nlinear_workspace *w)
gsl_vector *gsl_multilarge_nlinear_position(const gsl_multilarge_nlinear_workspace *w)
These functions return the current position x (i.e. best-fit parameters) of the solver w.
gsl_vector *gsl_multifit_nlinear_residual(const gsl_multifit_nlinear_workspace *w)
gsl_vector *gsl_multilarge_nlinear_residual(const gsl_multilarge_nlinear_workspace *w)
These functions return the current√ residual vector f (x) of the solver w. For weighted systems, the residual vector
includes the weighting factor W .
gsl_matrix *gsl_multifit_nlinear_jac(const gsl_multifit_nlinear_workspace *w)
This function returns a pointer to the n-by-p Jacobian matrix for the current iteration of the solver w. This function
is available only for the gsl_multifit_nlinear interface.
size_t gsl_multifit_nlinear_niter(const gsl_multifit_nlinear_workspace *w)
size_t gsl_multilarge_nlinear_niter(const gsl_multilarge_nlinear_workspace *w)
These functions return the number of iterations performed so far. The iteration counter is updated on each callto the _iterate functions above, and reset to 0 in the _init functions.
int gsl_multifit_nlinear_rcond(double *rcond, const gsl_multifit_nlinear_workspace *w)
int gsl_multilarge_nlinear_rcond(double *rcond, const gsl_multilarge_nlinear_workspace *w)
This function estimates the reciprocal condition number of the Jacobian matrix at the current position x andstores it in rcond. The computed value is only an estimate to give the user a guideline as to the conditioning of
their particular problem. Its calculation is based on which factorization method is used (Cholesky, QR, or SVD).
• For the Cholesky solver, the matrix J T J is factored at each iteration. Therefore this function will estimate
the 1-norm condition number rcond2 = 1/(||J T J||1 · ||(J T J)−1 ||1 )• For the QR solver, J is factored as J = QR at each iteration. For simplicity, this function calculates the
1-norm conditioning of only the R factor, rcond = 1/(||R||1 · ||R−1 ||1 ). This can be computed efficiently
since R is upper triangular.
• For the SVD solver, in order to efficiently solve the trust region subproblem, the matrix which is factored is
JD−1 , instead of J itself. The resulting singular values are used to provide the 2-norm reciprocal condition
number, as rcond = σmin /σmax . Note that when using Moré scaling, D 6= I and the resulting rcond
estimate may be significantly different from the true rcond of J itself.
double gsl_multifit_nlinear_avratio(const gsl_multifit_nlinear_workspace *w)
double gsl_multilarge_nlinear_avratio(const gsl_multilarge_nlinear_workspace *w)
This function returns the current ratio |a|/|v| of the acceleration correction term to the velocity stepterm.
The acceleration term is computed only by the gsl_multifit_nlinear_trs_lmaccel andgsl_multilarge_nlinear_trs_lmaccel methods, so this ratio will be zero for other TRS methods.
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41.8 Testing for Convergence
A minimization procedure should stop when one of the following conditions is true:• A minimum has been found to within the user-specified precision.
• A user-specified maximum number of iterations has been reached.
• An error has occurred.
The handling of these conditions is under user control. The functions below allow the user to test the current estimateof the best-fit parameters in several standard ways.
int gsl_multifit_nlinear_test(const double xtol, const double gtol, const double ftol, int *info, const
gsl_multifit_nlinear_workspace *w)
int gsl_multilarge_nlinear_test(const double xtol, const double gtol, const double ftol, int *info, const
gsl_multilarge_nlinear_workspace *w)
These functions test for convergence of the minimization method using the following criteria:• Testing for a small step size relative to the current parameter vector|δi | ≤ xtol(|xi | + xtol)
for each 0 <= i < p. Each element of the step vector δ is tested individually in case the different parameters
have widely different scales. Adding xtol to |xi | helps the test avoid breaking down in situations where
the true solution value xi = 0. If this test succeeds, info is set to 1 and the function returns GSL_SUCCESS.
A general guideline for selecting the step tolerance is to choose xtol = 10−d where d is the number ofaccurate decimal digits desired in the solution x. See Dennis and Schnabel for more information.
• Testing for a small gradient (g = ∇Φ(x) = J T f ) indicating a local function minimum:
max |gi × max(xi , 1)| ≤ gtol × max(Φ(x), 1)i

This expression tests whether the ratio (∇Φ)i xi /Φ is small. Testing this scaled gradient is a better than∇Φ alone since it is a dimensionless quantity and so independent of the scale of the problem. The max
arguments help ensure the test doesn’t break down in regions where xi or Φ(x) are close to 0. If this test
succeeds, info is set to 2 and the function returns GSL_SUCCESS.
A general guideline for choosing the gradient tolerance is to set gtol = GSL_DBL_EPSILON^(1/3). See
Dennis and Schnabel for more information.
If none of the tests succeed, info is set to 0 and the function returns GSL_CONTINUE, indicating further iterations
are required.

41.9 High Level Driver
These routines provide a high level wrapper that combines the iteration and convergence testing for easy use.
int gsl_multifit_nlinear_driver(const size_t maxiter, const double xtol, const double gtol, const double ftol,
void (*callback)(const size_t iter, void *params, const
gsl_multifit_linear_workspace *w), void *callback_params, int *info,
gsl_multifit_nlinear_workspace *w)

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int gsl_multilarge_nlinear_driver(const size_t maxiter, const double xtol, const double gtol, const double
ftol, void (*callback)(const size_t iter, void *params, const
gsl_multilarge_linear_workspace *w), void *callback_params, int *info,
gsl_multilarge_nlinear_workspace *w)
These functions iterate the nonlinear least squares solver w for a maximum of maxiter iterations. After each
iteration, the system is tested for convergence with the error tolerances xtol, gtol and ftol. Additionally, theuser may supply a callback function callback which is called after each iteration, so that the user may saveor print relevant quantities for each iteration. The parameter callback_params is passed to the callback
function. The parameters callback and callback_params may be set to NULL to disable this feature. Upon
successful convergence, the function returns GSL_SUCCESS and sets info to the reason for convergence (seegsl_multifit_nlinear_test()). If the function has not converged after maxiter iterations, GSL_EMAXITER
is returned. In rare cases, during an iteration the algorithm may be unable to find a new acceptable step δ to take.
In this case, GSL_ENOPROG is returned indicating no further progress can be made. If your problem is having
difficulty converging, see Troubleshooting for further guidance.

41.10 Covariance matrix of best fit parameters
int gsl_multifit_nlinear_covar(const gsl_matrix *J, const double epsrel, gsl_matrix *covar)
int gsl_multilarge_nlinear_covar(gsl_matrix *covar, gsl_multilarge_nlinear_workspace *w)
This function computes the covariance matrix of best-fit parameters using the Jacobian matrix J and stores it incovar. The parameter epsrel is used to remove linear-dependent columns when J is rank deficient.
The covariance matrix is given by,
C = (J T J)−1
or in the weighted case,C = (J T W J)−1
and is computed using the factored form of the Jacobian (Cholesky, QR, or SVD). Any columns of R which
satisfy|Rkk | ≤ epsrel|R11 |are considered linearly-dependent and are excluded from the covariance matrix (the corresponding rows andcolumns of the covariance matrix are set to zero).
If the minimisation uses the weighted least-squares function fi = (Y (x, ti ) − yi )/σi then the covariance matrix
above gives the statistical error on the best-fit parameters resulting from the Gaussian errors σi on the underlying
data yi . This can be verified from the relation δf = Jδc and the fact that the fluctuations in f from the data yi
are normalised by σi and so satisfyhδf δf T i = I
For an unweighted least-squares function fi = (Y (x, tPi ) − yi ) the covariance matrix above should be multipliedby the variance of the residuals about the best-fit σ 2 = (yi −Y (x, ti ))2 /(n−p) to give the variance-covariancematrix σ 2 C. This estimates the statistical error on the best-fit parameters from the scatter of the underlying data.
For more information about covariance matrices see Linear Least-Squares Overview.

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41.11 Troubleshooting
When developing a code to solve a nonlinear least squares problem, here are a few considerations to keep in mind.
1. The most common difficulty is the accurate implementation of the Jacobian matrix. If the analytic Jacobian is
not properly provided to the solver, this can hinder and many times prevent convergence of the method. Whendeveloping a new nonlinear least squares code, it often helps to compare the program output with the internallycomputed finite difference Jacobian and the user supplied analytic Jacobian. If there is a large difference incoefficients, it is likely the analytic Jacobian is incorrectly implemented.
2. If your code is having difficulty converging, the next thing to check is the starting point provided to the solver.
The methods of this chapter are local methods, meaning if you provide a starting point far away from the true
minimum, the method may converge to a local minimum or not converge at all. Sometimes it is possible tosolve a linearized approximation to the nonlinear problem, and use the linear solution as the starting point to thenonlinear problem.
3. If the various parameters of the coefficient vector x vary widely in magnitude, then the problem is said to bebadly scaled. The methods of this chapter do attempt to automatically rescale the elements of x to have roughly
the same order of magnitude, but in extreme cases this could still cause problems for convergence. In these cases
it is recommended for the user to scale their parameter vector x so that each parameter spans roughly the samerange, say [−1, 1]. The solution vector can be backscaled to recover the original units of the problem.

41.12 Examples
The following example programs demonstrate the nonlinear least squares fitting capabilities.

41.12.1 Exponential Fitting Example
The following example program fits a weighted exponential model with background to experimental data, Y =A exp(−λt) + b. The first part of the program sets up the functions expb_f() and expb_df() to calculate the modeland its Jacobian. The appropriate fitting function is given by,fi = (A exp(−λti ) + b) − yiwhere we have chosen ti = iT /(N − 1), where N is the number of data points fitted, so that ti ∈ [0, T ]. The Jacobianmatrix J is the derivative of these functions with respect to the three parameters (A, λ, b). It is given by,
Jij =

∂fi
∂xj

where x0 = A, x1 = λ and x2 = b. The i-th row of the Jacobian is therefore
Ji· =exp(−λti ) −ti A exp(−λti )1The main part of the program sets up a Levenberg-Marquardt solver and some simulated random data. The data uses
the known parameters (5.0,1.5,1.0) combined with Gaussian noise (standard deviation = 0.1) with a maximum timeT = 3 and N = 100 timesteps. The initial guess for the parameters is chosen as (1.0, 1.0, 0.0). The iteration terminates
when the relative change in x is smaller than 10−8 , or when the magnitude of the gradient falls below 10−8 . Here are
the results of running the program:
iter
iter
iter

0: A = 1.0000, lambda = 1.0000, b = 0.0000, cond(J) =1: A = 4.5109, lambda = 2.5258, b = 1.0704, cond(J) =2: A = 4.8565, lambda = 1.7442, b = 1.1669, cond(J) =inf, |f(x)| = 88.4448
26.2686, |f(x)| = 24.0646
23.7470, |f(x)| = 11.9797(continues on next page)

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iter 3: A = 4.9356, lambda = 1.5713, b = 1.0767, cond(J) =iter 4: A = 4.8678, lambda = 1.4838, b = 1.0252, cond(J) =iter 5: A = 4.8118, lambda = 1.4481, b = 1.0076, cond(J) =iter 6: A = 4.7983, lambda = 1.4404, b = 1.0041, cond(J) =iter 7: A = 4.7967, lambda = 1.4395, b = 1.0037, cond(J) =iter 8: A = 4.7965, lambda = 1.4394, b = 1.0037, cond(J) =iter 9: A = 4.7965, lambda = 1.4394, b = 1.0037, cond(J) =iter 10: A = 4.7965, lambda = 1.4394, b = 1.0037, cond(J) =iter 11: A = 4.7965, lambda = 1.4394, b = 1.0037, cond(J) =summary from method 'trust-region/levenberg-marquardt'
number of iterations: 11function evaluations: 16
Jacobian evaluations: 12
reason for stopping: small gradientinitial |f(x)| = 88.444756
final|f(x)| = 10.477801
chisq/dof = 1.1318
A= 4.79653 +/- 0.18704
lambda = 1.43937 +/- 0.07390
b= 1.00368 +/- 0.03473
status = success17.5849, |f(x)| = 10.7355
16.3428, |f(x)| = 10.5000
15.7925, |f(x)| = 10.4786
15.5840, |f(x)| = 10.4778
15.5396, |f(x)| = 10.4778
15.5344, |f(x)| = 10.4778
15.5339, |f(x)| = 10.4778
15.5339, |f(x)| = 10.4778
15.5339, |f(x)| = 10.4778

The approximate values of the parameters are found correctly, and the chi-squared value indicates a good fit (the chisquared per degree of freedom is approximately 1). In this case the errors on the parameters can be estimated fromthe square roots of the diagonal elements of the covariance matrix. If the chi-squared value shows a poor fit (i.e.
χ2 /(n − p)  1 then the error estimates obtained
from the covariance matrix will be too small. In the examplep
program the error estimates are multiplied by χ2 /(n − p) in this case, a common way of increasing the errors for apoor fit. Note that a poor fit will result from the use of an inappropriate model, and the scaled error estimates may thenbe outside the range of validity for Gaussian errors.
Additionally, we see that the condition number of J(x) stays reasonably small throughout the iteration. This indicates
we could safely switch to the Cholesky solver for speed improvement, although this particular system is too small toreally benefit.
Fig. 41.1 shows the fitted curve with the original data.
#include <stdlib.h>#include <stdio.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>#include <gsl/gsl_matrix.h>#include <gsl/gsl_vector.h>#include <gsl/gsl_blas.h>#include <gsl/gsl_multifit_nlinear.h>#define N#define TMAX

100
(3.0)

/* number of data points to fit *//* time variable in [0,TMAX] */struct data {size_t n;double * t;double * y;};(continues on next page)

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Fig. 41.1: Exponential fitted curve with data(continued from previous page)int
expb_f (const gsl_vector * x, void *data,
gsl_vector * f){size_t n = ((struct data *)data)->n;double *t = ((struct data *)data)->t;
double *y = ((struct data *)data)->y;double A = gsl_vector_get (x, 0);
double lambda = gsl_vector_get (x, 1);
double b = gsl_vector_get (x, 2);
size_t i;for (i = 0; i < n; i++){/* Model Yi = A * exp(-lambda * t_i) + b */double Yi = A * exp (-lambda * t[i]) + b;
gsl_vector_set (f, i, Yi - y[i]);}return GSL_SUCCESS;}
int
expb_df (const gsl_vector * x, void *data,
gsl_matrix * J){
size_t n = ((struct data *)data)->n;(continues on next page)

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double *t = ((struct data *)data)->t;
double A = gsl_vector_get (x, 0);
double lambda = gsl_vector_get (x, 1);
size_t i;for (i = 0; i < n; i++){/* Jacobian matrix J(i,j) = dfi / dxj, *//* where fi = (Yi - yi)/sigma[i],*//*Yi = A * exp(-lambda * t_i) + b *//* and the xj are the parameters (A,lambda,b) */double e = exp(-lambda * t[i]);gsl_matrix_set (J, i, 0, e);
gsl_matrix_set (J, i, 1, -t[i] * A * e);
gsl_matrix_set (J, i, 2, 1.0);}
return GSL_SUCCESS;}
void
callback(const size_t iter, void *params,
const gsl_multifit_nlinear_workspace *w){
gsl_vector *f = gsl_multifit_nlinear_residual(w);gsl_vector *x = gsl_multifit_nlinear_position(w);double rcond;/* compute reciprocal condition number of J(x) */gsl_multifit_nlinear_rcond(&rcond, w);
fprintf(stderr, "iter %2zu: A = %.4f, lambda = %.4f, b = %.4f, cond(J) = %8.4f, |f(x)|␣= %.4f\n",iter,
gsl_vector_get(x, 0),
gsl_vector_get(x, 1),
gsl_vector_get(x, 2),
1.0 / rcond,
gsl_blas_dnrm2(f));

,→

}
int
main (void){
const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
gsl_multifit_nlinear_workspace *w;
gsl_multifit_nlinear_fdf fdf;
gsl_multifit_nlinear_parameters fdf_params =gsl_multifit_nlinear_default_parameters();
const size_t n = N;(continues on next page)

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const size_t p = 3;
gsl_vector *f;
gsl_matrix *J;
gsl_matrix *covar = gsl_matrix_alloc (p, p);double t[N], y[N], weights[N];struct data d = { n, t, y };double x_init[3] = { 1.0, 1.0, 0.0 }; /* starting values */gsl_vector_view x = gsl_vector_view_array (x_init, p);
gsl_vector_view wts = gsl_vector_view_array(weights, n);gsl_rng * r;
double chisq, chisq0;
int status, info;
size_t i;const double xtol = 1e-8;
const double gtol = 1e-8;
const double ftol = 0.0;
gsl_rng_env_setup();r = gsl_rng_alloc(gsl_rng_default);/* define the function to be minimized */fdf.f = expb_f;
fdf.df = expb_df;/* set to NULL for finite-difference Jacobian */fdf.fvv = NULL;/* not using geodesic acceleration */fdf.n = n;fdf.p = p;
fdf.params = &d;/* this is the data to be fitted */for (i = 0; i < n; i++){double ti = i * TMAX / (n - 1.0);
double yi = 1.0 + 5 * exp (-1.5 * ti);double si = 0.1 * yi;double dy = gsl_ran_gaussian(r, si);t[i] = ti;y[i] = yi + dy;
weights[i] = 1.0 / (si * si);printf ("data: %g %g %g\n", ti, y[i], si);};/* allocate workspace with default parameters */w = gsl_multifit_nlinear_alloc (T, &fdf_params, n, p);/* initialize solver with starting point and weights */gsl_multifit_nlinear_winit (&x.vector, &wts.vector, &fdf, w);/* compute initial cost function */f = gsl_multifit_nlinear_residual(w);(continues on next page)

540

Chapter 41. Nonlinear Least-Squares Fitting

GNU Scientific Library, Release 2.7(continued from previous page)gsl_blas_ddot(f, f, &chisq0);/* solve the system with a maximum of 100 iterations */status = gsl_multifit_nlinear_driver(100, xtol, gtol, ftol,
callback, NULL, &info, w);/* compute covariance of best fit parameters */J = gsl_multifit_nlinear_jac(w);gsl_multifit_nlinear_covar (J, 0.0, covar);/* compute final cost */gsl_blas_ddot(f, f, &chisq);#define FIT(i) gsl_vector_get(w->x, i)#define ERR(i) sqrt(gsl_matrix_get(covar,i,i))
fprintf(stderr, "summary from method '%s/%s'\n",gsl_multifit_nlinear_name(w),gsl_multifit_nlinear_trs_name(w));
fprintf(stderr, "number of iterations: %zu\n",gsl_multifit_nlinear_niter(w));
fprintf(stderr, "function evaluations: %zu\n", fdf.nevalf);
fprintf(stderr, "Jacobian evaluations: %zu\n", fdf.nevaldf);
fprintf(stderr, "reason for stopping: %s\n",(info == 1) ? "small step size" : "small gradient");
fprintf(stderr, "initial |f(x)| = %f\n", sqrt(chisq0));
fprintf(stderr, "final|f(x)| = %f\n", sqrt(chisq));{
double dof = n - p;double c = GSL_MAX_DBL(1, sqrt(chisq / dof));fprintf(stderr, "chisq/dof = %g\n", chisq / dof);
fprintf (stderr, "A= %.5f +/- %.5f\n", FIT(0), c*ERR(0));
fprintf (stderr, "lambda = %.5f +/- %.5f\n", FIT(1), c*ERR(1));
fprintf (stderr, "b= %.5f +/- %.5f\n", FIT(2), c*ERR(2));}
fprintf (stderr, "status = %s\n", gsl_strerror (status));gsl_multifit_nlinear_free (w);gsl_matrix_free (covar);
gsl_rng_free (r);return 0;}41.12. Examples

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41.12.2 Geodesic Acceleration Example 1
The following example program minimizes a modified Rosenbrock function, which is characterized by a narrow canyon
with steep walls. The starting point is selected high on the canyon wall, so the solver must first find the canyon bottom
and then navigate to the minimum. The problem is solved both with and without using geodesic acceleration forcomparison. The cost function is given by
1 2(f + f22 )2 1

f1 = 100 x2 − x21

Φ(x) =

f2 = 1 − x1
The Jacobian matrix is
J=

∂f1
∂x1
∂f2
∂x1

∂f1
∂x2
∂f2
∂x2

!


=

−200x1
−1

100
0



In order to use geodesic acceleration,
the user must provide the second directional derivative of each residual in thePvelocity direction, Dv2 fi = αβ vα vβ ∂α ∂β fi . The velocity vector v is provided by the solver. For this example, thesederivatives are

 

f1
−200v12
fvv = Dv2=
f2
0
The solution of this minimization problem is
x∗ =



1
1



Φ(x∗ ) = 0
The program output is shown below:=== Solving system without acceleration ===
NITER
= 53
NFEV
= 56
NJEV
= 54
NAEV
= 0
initial cost = 2.250225000000e+04
final cost
= 6.674986031430e-18
final x= (9.999999974165e-01, 9.999999948328e-01)
final cond(J) = 6.000096055094e+02=== Solving system with acceleration ===
NITER
= 15
NFEV
= 17
NJEV
= 16
NAEV
= 16
initial cost = 2.250225000000e+04
final cost
= 7.518932873279e-19
final x= (9.999999991329e-01, 9.999999982657e-01)
final cond(J) = 6.000097233278e+02
We can see that enabling geodesic acceleration requires less than a third of the number of Jacobian evaluations in orderto locate the minimum. The path taken by both methods is shown in Fig. 41.2. The contours show the cost function
Φ(x1 , x2 ). We see that both methods quickly find the canyon bottom, but the geodesic acceleration method navigatesalong the bottom to the solution with significantly fewer iterations.
The program is given below.
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Fig. 41.2: Paths taken by solver for Rosenbrock function

41.12. Examples543

GNU Scientific Library, Release 2.7#include <stdlib.h>#include <stdio.h>#include <gsl/gsl_vector.h>#include <gsl/gsl_matrix.h>#include <gsl/gsl_blas.h>#include <gsl/gsl_multifit_nlinear.h>int
func_f (const gsl_vector * x, void *params, gsl_vector * f){double x1 = gsl_vector_get(x, 0);
double x2 = gsl_vector_get(x, 1);
gsl_vector_set(f, 0, 100.0 * (x2 - x1*x1));
gsl_vector_set(f, 1, 1.0 - x1);
return GSL_SUCCESS;}
int
func_df (const gsl_vector * x, void *params, gsl_matrix * J){
double x1 = gsl_vector_get(x, 0);
gsl_matrix_set(J, 0, 0, -200.0*x1);
gsl_matrix_set(J, 0, 1, 100.0);
gsl_matrix_set(J, 1, 0, -1.0);
gsl_matrix_set(J, 1, 1, 0.0);
return GSL_SUCCESS;}
int
func_fvv (const gsl_vector * x, const gsl_vector * v,
void *params, gsl_vector * fvv){
double v1 = gsl_vector_get(v, 0);
gsl_vector_set(fvv, 0, -200.0 * v1 * v1);
gsl_vector_set(fvv, 1, 0.0);
return GSL_SUCCESS;}
void
callback(const size_t iter, void *params,
const gsl_multifit_nlinear_workspace *w){
gsl_vector * x = gsl_multifit_nlinear_position(w);/* print out current location */printf("%f %f \n",gsl_vector_get(x, 0),544(continues on next page)

Chapter 41. Nonlinear Least-Squares Fitting

GNU Scientific Library, Release 2.7(continued from previous page)gsl_vector_get(x, 1));}void
solve_system(gsl_vector *x0, gsl_multifit_nlinear_fdf *fdf,
gsl_multifit_nlinear_parameters *params){
const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;const size_t max_iter = 200;
const double xtol = 1.0e-8;
const double gtol = 1.0e-8;
const double ftol = 1.0e-8;
const size_t n = fdf->n;
const size_t p = fdf->p;
gsl_multifit_nlinear_workspace *work =gsl_multifit_nlinear_alloc(T, params, n, p);
gsl_vector * f = gsl_multifit_nlinear_residual(work);
gsl_vector * x = gsl_multifit_nlinear_position(work);
int info;
double chisq0, chisq, rcond;/* initialize solver */gsl_multifit_nlinear_init(x0, fdf, work);/* store initial cost */gsl_blas_ddot(f, f, &chisq0);/* iterate until convergence */gsl_multifit_nlinear_driver(max_iter, xtol, gtol, ftol,
callback, NULL, &info, work);/* store final cost */gsl_blas_ddot(f, f, &chisq);/* store cond(J(x)) */gsl_multifit_nlinear_rcond(&rcond, work);/* print summary */fprintf(stderr, "NITER= %zu\n", gsl_multifit_nlinear_niter(work));
fprintf(stderr, "NFEV= %zu\n", fdf->nevalf);
fprintf(stderr, "NJEV= %zu\n", fdf->nevaldf);
fprintf(stderr, "NAEV= %zu\n", fdf->nevalfvv);
fprintf(stderr, "initial cost = %.12e\n", chisq0);
fprintf(stderr, "final cost
= %.12e\n", chisq);
fprintf(stderr, "final x= (%.12e, %.12e)\n",gsl_vector_get(x, 0), gsl_vector_get(x, 1));
fprintf(stderr, "final cond(J) = %.12e\n", 1.0 / rcond);
printf("\n\n");gsl_multifit_nlinear_free(work);(continues on next page)

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545

GNU Scientific Library, Release 2.7(continued from previous page)}int
main (void){
const size_t n = 2;
const size_t p = 2;
gsl_vector *f = gsl_vector_alloc(n);gsl_vector *x = gsl_vector_alloc(p);gsl_multifit_nlinear_fdf fdf;
gsl_multifit_nlinear_parameters fdf_params =gsl_multifit_nlinear_default_parameters();/* print map of Phi(x1, x2) */{double x1, x2, chisq;
double *f1 = gsl_vector_ptr(f, 0);
double *f2 = gsl_vector_ptr(f, 1);for (x1 = -1.2; x1 < 1.3; x1 += 0.1){
for (x2 = -0.5; x2 < 2.1; x2 += 0.1){
gsl_vector_set(x, 0, x1);
gsl_vector_set(x, 1, x2);
func_f(x, NULL, f);chisq = (*f1) * (*f1) + (*f2) * (*f2);
printf("%f %f %f \n", x1, x2, chisq);}
printf("\n");}printf("\n\n");}/* define function to be minimized */fdf.f = func_f;
fdf.df = func_df;
fdf.fvv = func_fvv;
fdf.n = n;fdf.p = p;
fdf.params = NULL;/* starting point */gsl_vector_set(x, 0, -0.5);
gsl_vector_set(x, 1, 1.75);
fprintf(stderr, "=== Solving system without acceleration ===\n");fdf_params.trs = gsl_multifit_nlinear_trs_lm;
solve_system(x, &fdf, &fdf_params);
fprintf(stderr, "=== Solving system with acceleration ===\n");(continues on next page)

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fdf_params.trs = gsl_multifit_nlinear_trs_lmaccel;
solve_system(x, &fdf, &fdf_params);
gsl_vector_free(f);gsl_vector_free(x);return 0;}41.12.3 Geodesic Acceleration Example 2
The following example fits a set of data to a Gaussian model using the Levenberg-Marquardt method with geodesic
acceleration. The cost function is
1X 2
f
Φ(x) =
2 i i
fi = yi − Y (a, b, c; ti )where yi is the measured data point at time ti , and the model is specified by
"

2 #
1 t−b
Y (a, b, c; t) = a exp −2c
The parameters a, b, c represent the amplitude, mean, and width of the Gaussian respectively. The program below
generates the yi data on [0, 1] using the values a = 5, b = 0.4, c = 0.15 and adding random noise. The i-th row of theJacobian isJi,: =

∂fi
∂a

∂fi
∂b

∂fi
∂c



=

−ei

− ac zi ei− ac zi2 ei



where
zi =

ti − bc

1ei = exp − zi22In order toPuse geodesic acceleration, we need the second directional derivative of the residuals in the velocity direction,
Dv2 fi = αβ vα vβ ∂α ∂β fi , where v is provided by the solver. To compute this, it is helpful to make a table of all second
derivatives of the residuals fi with respect to each combination of model parameters. This table is
∂
∂a
∂
∂a
∂
∂b
∂
∂c

0
a
c2

∂
∂b
− zci ei 
1 − zi2 ei

a
c2 zi
a 2
c2 zi

∂
∂c
zi2
− c ei
2 − zi2  ei
3 − zi2 ei

The lower half of the table is omitted since it is symmetric. Then, the second directional derivative of fi is
Dv2 fi = va2 ∂a2 fi + 2va vb ∂a ∂b fi + 2va vc ∂a ∂c fi + vb2 ∂b2 fi + 2vb vc ∂b ∂c fi + vc2 ∂c2 fi
The factors of 2 come from the symmetry of the mixed second partial derivatives. The iteration is started using theinitial guess a = 1, b = 0, c = 1. The program output is shown below:

41.12. Examples

547

GNU Scientific Library, Release 2.7

iter 0: a = 1.0000, b = 0.0000, c = 1.0000, |a|/|v| = 0.0000 cond(J) =,→= 35.4785
iter 1: a = 1.5708, b = 0.5321, c = 0.5219, |a|/|v| = 0.3093 cond(J) =,→= 31.1042
iter 2: a = 1.7387, b = 0.4040, c = 0.4568, |a|/|v| = 0.1199 cond(J) =,→= 28.7217
iter 3: a = 2.2340, b = 0.3829, c = 0.3053, |a|/|v| = 0.3308 cond(J) =,→= 23.8074
iter 4: a = 3.2275, b = 0.3952, c = 0.2243, |a|/|v| = 0.2784 cond(J) =,→= 15.6003
iter 5: a = 4.3347, b = 0.3974, c = 0.1752, |a|/|v| = 0.2029 cond(J) =,→= 7.5908
iter 6: a = 4.9352, b = 0.3992, c = 0.1536, |a|/|v| = 0.1001 cond(J) =,→= 4.8402
iter 7: a = 5.0716, b = 0.3994, c = 0.1498, |a|/|v| = 0.0166 cond(J) =,→= 4.7103
iter 8: a = 5.0828, b = 0.3994, c = 0.1495, |a|/|v| = 0.0012 cond(J) =,→= 4.7095
iter 9: a = 5.0831, b = 0.3994, c = 0.1495, |a|/|v| = 0.0000 cond(J) =,→= 4.7095
iter 10: a = 5.0831, b = 0.3994, c = 0.1495, |a|/|v| = 0.0000 cond(J) =,→= 4.7095
iter 11: a = 5.0831, b = 0.3994, c = 0.1495, |a|/|v| = 0.0000 cond(J) =,→= 4.7095
NITER
= 11
NFEV
= 18
NJEV
= 12
NAEV
= 17
initial cost = 1.258724737288e+03
final cost
= 2.217977560180e+01
final x= (5.083101559156e+00, 3.994484109594e-01, 1.494898e-01)
final cond(J) = 3.669757713403e+01

inf, |f(x)|␣
29.0443, |f(x)|␣
3.5256, |f(x)|␣
4.5121, |f(x)|␣
8.6499, |f(x)|␣
15.1732, |f(x)|␣
26.6621, |f(x)|␣
34.6922, |f(x)|␣
36.5422, |f(x)|␣
36.6929, |f(x)|␣
36.6975, |f(x)|␣
36.6976, |f(x)|␣We see the method converges after 11 iterations. For comparison the standard Levenberg-Marquardt method requires
26 iterations and so the Gaussian fitting problem benefits substantially from the geodesic acceleration correction. The
column marked |a|/|v| above shows the ratio of the acceleration term to the velocity term as the iteration progresses.
Larger values of this ratio indicate that the geodesic acceleration correction term is contributing substantial information
to the solver relative to the standard LM velocity step.
The data and fitted model are shown in Fig. 41.3.
The program is given below.
#include <stdlib.h>#include <stdio.h>#include <gsl/gsl_vector.h>#include <gsl/gsl_matrix.h>#include <gsl/gsl_blas.h>#include <gsl/gsl_multifit_nlinear.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>
struct data{(continues on next page)

548

Chapter 41. Nonlinear Least-Squares Fitting

GNU Scientific Library, Release 2.7

Fig. 41.3: Gaussian model fitted to data

41.12. Examples

549

GNU Scientific Library, Release 2.7(continued from previous page)

double *t;
double *y;size_t n;};/* model function: a * exp( -1/2 * [ (t - b) / c ]^2 ) */doublegaussian(const double a, const double b, const double c, const double t){const double z = (t - b) / c;return (a * exp(-0.5 * z * z));}int
func_f (const gsl_vector * x, void *params, gsl_vector * f){struct data *d = (struct data *) params;
double a = gsl_vector_get(x, 0);
double b = gsl_vector_get(x, 1);
double c = gsl_vector_get(x, 2);
size_t i;for (i = 0; i < d->n; ++i){
double ti = d->t[i];double yi = d->y[i];double y = gaussian(a, b, c, ti);
gsl_vector_set(f, i, yi - y);}return GSL_SUCCESS;}
int
func_df (const gsl_vector * x, void *params, gsl_matrix * J){
struct data *d = (struct data *) params;
double a = gsl_vector_get(x, 0);
double b = gsl_vector_get(x, 1);
double c = gsl_vector_get(x, 2);
size_t i;for (i = 0; i < d->n; ++i){
double ti = d->t[i];double zi = (ti - b) / c;double ei = exp(-0.5 * zi * zi);
gsl_matrix_set(J, i, 0, -ei);
gsl_matrix_set(J, i, 1, -(a / c) * ei * zi);
gsl_matrix_set(J, i, 2, -(a / c) * ei * zi * zi);(continues on next page)

550

Chapter 41. Nonlinear Least-Squares Fitting

GNU Scientific Library, Release 2.7(continued from previous page)}
return GSL_SUCCESS;}
int
func_fvv (const gsl_vector * x, const gsl_vector * v,
void *params, gsl_vector * fvv){
struct data *d = (struct data *) params;
double a = gsl_vector_get(x, 0);
double b = gsl_vector_get(x, 1);
double c = gsl_vector_get(x, 2);double va = gsl_vector_get(v, 0);
double vb = gsl_vector_get(v, 1);
double vc = gsl_vector_get(v, 2);
size_t i;for (i = 0; i < d->n; ++i){
double ti = d->t[i];double zi = (ti - b) / c;double ei = exp(-0.5 * zi * zi);
double Dab = -zi * ei / c;double Dac = -zi * zi * ei / c;double Dbb = a * ei / (c * c) * (1.0 - zi*zi);
double Dbc = a * zi * ei / (c * c) * (2.0 - zi*zi);
double Dcc = a * zi * zi * ei / (c * c) * (3.0 - zi*zi);
double sum;
sum = 2.0 * va * vb * Dab +2.0 * va * vc * Dac +vb * vb * Dbb +2.0 * vb * vc * Dbc +vc * vc * Dcc;
gsl_vector_set(fvv, i, sum);}
return GSL_SUCCESS;}
void
callback(const size_t iter, void *params,
const gsl_multifit_nlinear_workspace *w){
gsl_vector *f = gsl_multifit_nlinear_residual(w);gsl_vector *x = gsl_multifit_nlinear_position(w);double avratio = gsl_multifit_nlinear_avratio(w);double rcond;(void) params; /* not used */(continues on next page)

41.12. Examples

551

GNU Scientific Library, Release 2.7(continued from previous page)/* compute reciprocal condition number of J(x) */gsl_multifit_nlinear_rcond(&rcond, w);
fprintf(stderr, "iter %2zu: a = %.4f , b = %.4f , c = %.4f , |a|/|v| = %.4f cond(J) = %8.
4f , |f(x)| = %.4f \n",iter,
gsl_vector_get(x, 0),
gsl_vector_get(x, 1),
gsl_vector_get(x, 2),
avratio,
1.0 / rcond,
gsl_blas_dnrm2(f));

,→

}
void
solve_system(gsl_vector *x, gsl_multifit_nlinear_fdf *fdf,
gsl_multifit_nlinear_parameters *params){
const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;const size_t max_iter = 200;
const double xtol = 1.0e-8;
const double gtol = 1.0e-8;
const double ftol = 1.0e-8;
const size_t n = fdf->n;
const size_t p = fdf->p;
gsl_multifit_nlinear_workspace *work =gsl_multifit_nlinear_alloc(T, params, n, p);
gsl_vector * f = gsl_multifit_nlinear_residual(work);
gsl_vector * y = gsl_multifit_nlinear_position(work);
int info;
double chisq0, chisq, rcond;/* initialize solver */gsl_multifit_nlinear_init(x, fdf, work);/* store initial cost */gsl_blas_ddot(f, f, &chisq0);/* iterate until convergence */gsl_multifit_nlinear_driver(max_iter, xtol, gtol, ftol,
callback, NULL, &info, work);/* store final cost */gsl_blas_ddot(f, f, &chisq);/* store cond(J(x)) */gsl_multifit_nlinear_rcond(&rcond, work);
gsl_vector_memcpy(x, y);/* print summary */(continues on next page)

552

Chapter 41. Nonlinear Least-Squares Fitting

GNU Scientific Library, Release 2.7(continued from previous page)fprintf(stderr, "NITER= %zu\n", gsl_multifit_nlinear_niter(work));
fprintf(stderr, "NFEV= %zu\n", fdf->nevalf);
fprintf(stderr, "NJEV= %zu\n", fdf->nevaldf);
fprintf(stderr, "NAEV= %zu\n", fdf->nevalfvv);
fprintf(stderr, "initial cost = %.12e\n", chisq0);
fprintf(stderr, "final cost
= %.12e\n", chisq);
fprintf(stderr, "final x= (%.12e, %.12e, %12e)\n",gsl_vector_get(x, 0), gsl_vector_get(x, 1), gsl_vector_get(x, 2));
fprintf(stderr, "final cond(J) = %.12e\n", 1.0 / rcond);
gsl_multifit_nlinear_free(work);}
intmain (void){
const size_t n = 300; /* number of data points to fit */const size_t p = 3;/* number of model parameters */const double a = 5.0; /* amplitude */const double b = 0.4; /* center */const double c = 0.15; /* width */const gsl_rng_type * T = gsl_rng_default;
gsl_vector *f = gsl_vector_alloc(n);gsl_vector *x = gsl_vector_alloc(p);gsl_multifit_nlinear_fdf fdf;
gsl_multifit_nlinear_parameters fdf_params =gsl_multifit_nlinear_default_parameters();
struct data fit_data;
gsl_rng * r;
size_t i;
gsl_rng_env_setup ();r = gsl_rng_alloc (T);fit_data.t = malloc(n * sizeof(double));fit_data.y = malloc(n * sizeof(double));fit_data.n = n;/* generate synthetic data with noise */for (i = 0; i < n; ++i){
double t = (double)i / (double) n;double y0 = gaussian(a, b, c, t);double dy = gsl_ran_gaussian (r, 0.1 * y0);
fit_data.t[i] = t;
fit_data.y[i] = y0 + dy;}/* define function to be minimized */fdf.f = func_f;(continues on next page)

41.12. Examples

553

GNU Scientific Library, Release 2.7(continued from previous page)

fdf.df = func_df;
fdf.fvv = func_fvv;
fdf.n = n;fdf.p = p;
fdf.params = &fit_data;/* starting point */gsl_vector_set(x, 0, 1.0);
gsl_vector_set(x, 1, 0.0);
gsl_vector_set(x, 2, 1.0);
fdf_params.trs = gsl_multifit_nlinear_trs_lmaccel;
solve_system(x, &fdf, &fdf_params);/* print data and model */{double A = gsl_vector_get(x, 0);
double B = gsl_vector_get(x, 1);
double C = gsl_vector_get(x, 2);
for (i = 0; i < n; ++i){
double ti = fit_data.t[i];double yi = fit_data.y[i];double fi = gaussian(A, B, C, ti);printf("%f %f %f \n", ti, yi, fi);}}
gsl_vector_free(f);gsl_vector_free(x);gsl_rng_free(r);return 0;}41.12.4 Comparing TRS Methods Example
The following program compares all available nonlinear least squares trust-region subproblem (TRS) methods on theBranin function, a common optimization test problem. The cost function is
1 2(f + f22 )2 1
f1 = x2 + a1 x21 + a2 x1 + a3
√ p
f2 = a4 1 + (1 − a5 ) cos x1

Φ(x) =

5.1
51
with a1 = − 4π
2 , a2 = π , a3 = −6, a4 = 10, a5 = 8π . There are three minima of this function in the range(x1 , x2 ) ∈ [−5, 15] × [−5, 15]. The program below uses the starting point (x1 , x2 ) = (6, 14.5) and calculates thesolution with all available nonlinear least squares TRS methods. The program output is shown below:

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Chapter 41. Nonlinear Least-Squares Fitting

GNU Scientific Library, Release 2.7

MethodNITER,→Final xlevenberg-marquardt
20,→3.14e+00, 1.23e+01)
levenberg-marquardt+accel 27,→14e+00, 2.27e+00)
dogleg23,→14e+00, 2.28e+00)
double-dogleg
24,→14e+00, 2.27e+00)
2D-subspace
23,→14e+00, 2.27e+00)

NFEV

NJEV

Initial Cost

Final cost

Final cond(J)␣

27

21

1.9874e+02

3.9789e-01

6.1399e+07

(-

36

28

1.9874e+02

3.9789e-01

1.4465e+07

(3.

64

23

1.9874e+02

3.9789e-01

5.0692e+08

(3.

69

24

1.9874e+02

3.9789e-01

3.4879e+07

(3.

54

24

1.9874e+02

3.9789e-01

2.5142e+07(3.

The first row of output above corresponds to standard Levenberg-Marquardt, while the second row includes geodesic
acceleration. We see that the standard LM method converges to the minimum at (−π, 12.275) and also uses the least
number of iterations and Jacobian evaluations. All other methods converge to the minimum (π, 2.275) and perform
similarly in terms of number of Jacobian evaluations. We see that J is fairly ill-conditioned at both minima, indicatingthat the QR (or SVD) solver is the best choice for this problem. Since there are only two parameters in this optimization
problem, we can easily visualize the paths taken by each method, which are shown in Fig. 41.4. The figure shows
contours of the cost function Φ(x1 , x2 ) which exhibits three global minima in the range [−5, 15] × [−5, 15]. The paths
taken by each solver are shown as colored lines.

Fig. 41.4: Paths taken for different TRS methods for the Branin function
The program is given below.
#include <stdlib.h>#include <stdio.h>#include <gsl/gsl_vector.h>(continues on next page)

41.12. Examples

555

GNU Scientific Library, Release 2.7(continued from previous page)#include <gsl/gsl_matrix.h>#include <gsl/gsl_blas.h>#include <gsl/gsl_multifit_nlinear.h>/* parameters to model */struct model_params{
double a1;
double a2;
double a3;
double a4;
double a5;};/* Branin function */int
func_f (const gsl_vector * x, void *params, gsl_vector * f){struct model_params *par = (struct model_params *) params;
double x1 = gsl_vector_get(x, 0);
double x2 = gsl_vector_get(x, 1);
double f1 = x2 + par->a1 * x1 * x1 + par->a2 * x1 + par->a3;
double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * cos(x1));
gsl_vector_set(f, 0, f1);
gsl_vector_set(f, 1, f2);
return GSL_SUCCESS;}
int
func_df (const gsl_vector * x, void *params, gsl_matrix * J){
struct model_params *par = (struct model_params *) params;
double x1 = gsl_vector_get(x, 0);
double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * cos(x1));
gsl_matrix_set(J, 0, 0, 2.0 * par->a1 * x1 + par->a2);gsl_matrix_set(J, 0, 1, 1.0);
gsl_matrix_set(J, 1, 0, -0.5 * par->a4 / f2 * (1.0 - par->a5) * sin(x1));
gsl_matrix_set(J, 1, 1, 0.0);
return GSL_SUCCESS;}
int
func_fvv (const gsl_vector * x, const gsl_vector * v,
void *params, gsl_vector * fvv){
struct model_params *par = (struct model_params *) params;
double x1 = gsl_vector_get(x, 0);(continues on next page)

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Chapter 41. Nonlinear Least-Squares Fitting

GNU Scientific Library, Release 2.7(continued from previous page)

double v1 = gsl_vector_get(v, 0);
double c = cos(x1);double s = sin(x1);double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * c);double t = 0.5 * par->a4 * (1.0 - par->a5) / f2;
gsl_vector_set(fvv, 0, 2.0 * par->a1 * v1 * v1);
gsl_vector_set(fvv, 1, -t * (c + s*s/f2) * v1 * v1);
return GSL_SUCCESS;}
void
callback(const size_t iter, void *params,
const gsl_multifit_nlinear_workspace *w){
gsl_vector * x = gsl_multifit_nlinear_position(w);double x1 = gsl_vector_get(x, 0);
double x2 = gsl_vector_get(x, 1);/* print out current location */printf("%f %f \n", x1, x2);}
void
solve_system(gsl_vector *x0, gsl_multifit_nlinear_fdf *fdf,
gsl_multifit_nlinear_parameters *params){
const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;const size_t max_iter = 200;
const double xtol = 1.0e-8;
const double gtol = 1.0e-8;
const double ftol = 1.0e-8;
const size_t n = fdf->n;
const size_t p = fdf->p;
gsl_multifit_nlinear_workspace *work =gsl_multifit_nlinear_alloc(T, params, n, p);
gsl_vector * f = gsl_multifit_nlinear_residual(work);
gsl_vector * x = gsl_multifit_nlinear_position(work);
int info;
double chisq0, chisq, rcond;
printf("# %s/%s\n",gsl_multifit_nlinear_name(work),
gsl_multifit_nlinear_trs_name(work));/* initialize solver */gsl_multifit_nlinear_init(x0, fdf, work);/* store initial cost */gsl_blas_ddot(f, f, &chisq0);(continues on next page)

41.12. Examples

557

GNU Scientific Library, Release 2.7(continued from previous page)/* iterate until convergence */gsl_multifit_nlinear_driver(max_iter, xtol, gtol, ftol,
callback, NULL, &info, work);/* store final cost */gsl_blas_ddot(f, f, &chisq);/* store cond(J(x)) */gsl_multifit_nlinear_rcond(&rcond, work);/* print summary */fprintf(stderr, "%-25s %-6zu %-5zu %-5zu %-13.4e %-12.4e %-13.4e (%.2e, %.2e)\n",gsl_multifit_nlinear_trs_name(work),
gsl_multifit_nlinear_niter(work),
fdf->nevalf,
fdf->nevaldf,
chisq0,chisq,
1.0 / rcond,gsl_vector_get(x, 0),
gsl_vector_get(x, 1));
printf("\n\n");gsl_multifit_nlinear_free(work);}
intmain (void){
const size_t n = 2;
const size_t p = 2;
gsl_vector *f = gsl_vector_alloc(n);gsl_vector *x = gsl_vector_alloc(p);gsl_multifit_nlinear_fdf fdf;
gsl_multifit_nlinear_parameters fdf_params =gsl_multifit_nlinear_default_parameters();
struct model_params params;
params.a1 = -5.1 / (4.0 * M_PI * M_PI);
params.a2 = 5.0 / M_PI;
params.a3 = -6.0;
params.a4 = 10.0;
params.a5 = 1.0 / (8.0 * M_PI);/* print map of Phi(x1, x2) */{double x1, x2, chisq;
for (x1 = -5.0; x1 < 15.0; x1 += 0.1){
for (x2 = -5.0; x2 < 15.0; x2 += 0.1)
(continues on next page)

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Chapter 41. Nonlinear Least-Squares Fitting

GNU Scientific Library, Release 2.7(continued from previous page){gsl_vector_set(x, 0, x1);
gsl_vector_set(x, 1, x2);
func_f(x, &params, f);
gsl_blas_ddot(f, f, &chisq);
printf("%f %f %f \n", x1, x2, chisq);}
printf("\n");}printf("\n\n");}/* define function to be minimized */fdf.f = func_f;
fdf.df = func_df;
fdf.fvv = func_fvv;
fdf.n = n;fdf.p = p;
fdf.params = &params;/* starting point */gsl_vector_set(x, 0, 6.0);
gsl_vector_set(x, 1, 14.5);
fprintf(stderr, "%-25s %-6s %-5s %-5s %-13s %-12s %-13s %-15s\n","Method", "NITER", "NFEV", "NJEV", "Initial Cost","Final cost", "Final cond(J)", "Final x");fdf_params.trs = gsl_multifit_nlinear_trs_lm;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multifit_nlinear_trs_lmaccel;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multifit_nlinear_trs_dogleg;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multifit_nlinear_trs_ddogleg;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multifit_nlinear_trs_subspace2D;
solve_system(x, &fdf, &fdf_params);
gsl_vector_free(f);gsl_vector_free(x);return 0;}41.12. Examples

559

GNU Scientific Library, Release 2.7

41.12.5 Large Nonlinear Least Squares Example
The following program illustrates the large nonlinear least squares solvers on a system with significant sparse structurein the Jacobian. The cost function is
p+1

1X 2
f
2 i=1 i√fi = α(xi − 1),1
fp+1 = ||x||2 −4

Φ(x) =

1≤i≤pwith α = 10−5 . The residual fp+1 imposes a constraint on the p parameters x, to ensure that ||x||2 ≈ 14 . The(p + 1)-by-p Jacobian for this system is
 √

αIp
J(x) =
2xT
and the normal equations matrix is
J T J = αIp + 4xxT
Finally, the second directional derivative of f for the geodesic acceleration method is02
fvv = Dv f =2||v||2
Since the upper p-by-p block of J is diagonal, this sparse structure should be exploited in the nonlinear solver. For
comparison, the following program solves the system for p = 2000 using the dense direct Cholesky solver based on thenormal equations matrix J T J, as well as the iterative Steihaug-Toint solver, based on sparse matrix-vector products
Ju and J T u. The program output is shown below:
MethodNITER NFEV NJUEV NJTJEV NAEV Init Cost Final cost cond(J)␣
,→Final |x|^2 Time (s)levenberg-marquardt
25
31
26
26
0
7.1218e+18 1.9555e-02 447.50 2.
,→5044e-01
46.28
levenberg-marquardt+accel 22
23
45
23
22
7.1218e+18 1.9555e-02 447.64 2.
,→5044e-01
33.92
dogleg
37
87
36
36
0
7.1218e+18 1.9555e-02 447.59 2.
,→5044e-01
56.05
double-dogleg
35
88
34
34
0
7.1218e+18 1.9555e-02 447.62 2.
,→5044e-01
52.65
2D-subspace
37
88
36
36
0
7.1218e+18 1.9555e-02 447.71 2.
,→5044e-01
59.75
steihaug-toint
35
88
345
0
0
7.1218e+18 1.9555e-02 inf
2.
,→5044e-01
0.09
The first five rows use methods based on factoring the dense J T J matrix while the last row uses the iterative SteihaugToint method. While the number of Jacobian matrix-vector products (NJUEV) is less for the dense methods, the added
time to construct and factor the J T J matrix (NJTJEV) results in a much larger runtime than the iterative method (seelast column).
The program is given below.
#include <stdlib.h>#include <stdio.h>(continues on next page)

560

Chapter 41. Nonlinear Least-Squares Fitting

GNU Scientific Library, Release 2.7(continued from previous page)#include <sys/time.h>#include <gsl/gsl_vector.h>#include <gsl/gsl_matrix.h>#include <gsl/gsl_blas.h>#include <gsl/gsl_multilarge_nlinear.h>#include <gsl/gsl_spblas.h>#include <gsl/gsl_spmatrix.h>/* parameters for functions */struct model_params{
double alpha;gsl_spmatrix *J;};/* penalty function */int
penalty_f (const gsl_vector * x, void *params, gsl_vector * f){struct model_params *par = (struct model_params *) params;
const double sqrt_alpha = sqrt(par->alpha);
const size_t p = x->size;
size_t i;
double sum = 0.0;
for (i = 0; i < p; ++i){
double xi = gsl_vector_get(x, i);
gsl_vector_set(f, i, sqrt_alpha*(xi - 1.0));
sum += xi * xi;}
gsl_vector_set(f, p, sum - 0.25);
return GSL_SUCCESS;}
int
penalty_df (CBLAS_TRANSPOSE_t TransJ, const gsl_vector * x,const gsl_vector * u, void * params, gsl_vector * v,
gsl_matrix * JTJ){
struct model_params *par = (struct model_params *) params;
const size_t p = x->size;
size_t j;/* store 2*x in last row of J */for (j = 0; j < p; ++j){
double xj = gsl_vector_get(x, j);(continues on next page)

41.12. Examples

561

GNU Scientific Library, Release 2.7(continued from previous page)gsl_spmatrix_set(par->J, p, j, 2.0 * xj);}/* compute v = op(J) u */if (v)
gsl_spblas_dgemv(TransJ, 1.0, par->J, u, 0.0, v);
if (JTJ){
gsl_vector_view diag = gsl_matrix_diagonal(JTJ);/* compute J^T J = [ alpha*I_p + 4 x x^T ] */gsl_matrix_set_zero(JTJ);/* store 4 x x^T in lower half of JTJ */gsl_blas_dsyr(CblasLower, 4.0, x, JTJ);/* add alpha to diag(JTJ) */gsl_vector_add_constant(&diag.vector, par->alpha);}return GSL_SUCCESS;}
int
penalty_fvv (const gsl_vector * x, const gsl_vector * v,
void *params, gsl_vector * fvv){
const size_t p = x->size;
double normv = gsl_blas_dnrm2(v);gsl_vector_set_zero(fvv);
gsl_vector_set(fvv, p, 2.0 * normv * normv);(void)params; /* avoid unused parameter warning */return GSL_SUCCESS;}
void
solve_system(const gsl_vector *x0, gsl_multilarge_nlinear_fdf *fdf,
gsl_multilarge_nlinear_parameters *params){
const gsl_multilarge_nlinear_type *T = gsl_multilarge_nlinear_trust;const size_t max_iter = 200;
const double xtol = 1.0e-8;
const double gtol = 1.0e-8;
const double ftol = 1.0e-8;
const size_t n = fdf->n;
const size_t p = fdf->p;
gsl_multilarge_nlinear_workspace *work =gsl_multilarge_nlinear_alloc(T, params, n, p);(continues on next page)

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Chapter 41. Nonlinear Least-Squares Fitting

GNU Scientific Library, Release 2.7(continued from previous page)gsl_vector * f = gsl_multilarge_nlinear_residual(work);
gsl_vector * x = gsl_multilarge_nlinear_position(work);
int info;
double chisq0, chisq, rcond, xsq;
struct timeval tv0, tv1;
gettimeofday(&tv0, NULL);/* initialize solver */gsl_multilarge_nlinear_init(x0, fdf, work);/* store initial cost */gsl_blas_ddot(f, f, &chisq0);/* iterate until convergence */gsl_multilarge_nlinear_driver(max_iter, xtol, gtol, ftol,
NULL, NULL, &info, work);
gettimeofday(&tv1, NULL);/* store final cost */gsl_blas_ddot(f, f, &chisq);/* compute final ||x||^2 */gsl_blas_ddot(x, x, &xsq);/* store cond(J(x)) */gsl_multilarge_nlinear_rcond(&rcond, work);/* print summary */fprintf(stderr, "%-25s %-5zu %-4zu %-5zu %-6zu %-4zu %-10.4e %-10.4e %-7.2f %-11.4e %.
,→2f \n",gsl_multilarge_nlinear_trs_name(work),
gsl_multilarge_nlinear_niter(work),
fdf->nevalf,
fdf->nevaldfu,
fdf->nevaldf2,
fdf->nevalfvv,
chisq0,chisq,
1.0 / rcond,
xsq,(tv1.tv_sec - tv0.tv_sec) + 1.0e-6 * (tv1.tv_usec - tv0.tv_usec));
gsl_multilarge_nlinear_free(work);}
intmain (void){
const size_t p = 2000;
const size_t n = p + 1;
(continues on next page)

41.12. Examples

563

GNU Scientific Library, Release 2.7(continued from previous page)gsl_vector *f = gsl_vector_alloc(n);gsl_vector *x = gsl_vector_alloc(p);/* allocate sparse Jacobian matrix with 2*p non-zero elements in triplet format */gsl_spmatrix *J = gsl_spmatrix_alloc_nzmax(n, p, 2 * p, GSL_SPMATRIX_TRIPLET);
gsl_multilarge_nlinear_fdf fdf;
gsl_multilarge_nlinear_parameters fdf_params =gsl_multilarge_nlinear_default_parameters();
struct model_params params;
size_t i;params.alpha = 1.0e-5;
params.J = J;/* define function to be minimized */fdf.f = penalty_f;
fdf.df = penalty_df;
fdf.fvv = penalty_fvv;
fdf.n = n;fdf.p = p;
fdf.params = &params;for (i = 0; i < p; ++i){/* starting point */gsl_vector_set(x, i, i + 1.0);/* store sqrt(alpha)*I_p in upper p-by-p block of J */gsl_spmatrix_set(J, i, i, sqrt(params.alpha));}fprintf(stderr, "%-25s %-4s %-4s %-5s %-6s %-4s %-10s %-10s %-7s %-11s %-10s\n","Method", "NITER", "NFEV", "NJUEV", "NJTJEV", "NAEV", "Init Cost","Final cost", "cond(J)", "Final |x|^2", "Time (s)");fdf_params.scale = gsl_multilarge_nlinear_scale_levenberg;
fdf_params.trs = gsl_multilarge_nlinear_trs_lm;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multilarge_nlinear_trs_lmaccel;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multilarge_nlinear_trs_dogleg;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multilarge_nlinear_trs_ddogleg;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multilarge_nlinear_trs_subspace2D;
solve_system(x, &fdf, &fdf_params);(continues on next page)

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Chapter 41. Nonlinear Least-Squares Fitting

GNU Scientific Library, Release 2.7(continued from previous page)

fdf_params.trs = gsl_multilarge_nlinear_trs_cgst;
solve_system(x, &fdf, &fdf_params);
gsl_vector_free(f);gsl_vector_free(x);gsl_spmatrix_free(J);return 0;}41.13 References and Further Reading
The following publications are relevant to the algorithms described in this section,• J.J. Moré, The Levenberg-Marquardt Algorithm: Implementation and Theory, Lecture Notes in Mathematics,
v630 (1978), ed G. Watson.
• H. B. Nielsen, “Damping Parameter in Marquardt’s Method”, IMM Department of Mathematical Modeling,
DTU, Tech. Report IMM-REP-1999-05 (1999).
• K. Madsen and H. B. Nielsen, “Introduction to Optimization and Data Fitting”, IMM Department of Mathematical Modeling, DTU, 2010.
• J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations,
SIAM, 1996.
• M. K. Transtrum, B. B. Machta, and J. P. Sethna, Geometry of nonlinear least squares with applications to sloppy
models and optimization, Phys. Rev. E 83, 036701, 2011.
• M. K. Transtrum and J. P. Sethna, Improvements to the Levenberg-Marquardt algorithm for nonlinear leastsquares minimization, arXiv:1201.5885, 2012.
• J.J. Moré, B.S. Garbow, K.E. Hillstrom, “Testing Unconstrained Optimization Software”, ACM Transactions onMathematical Software, Vol 7, No 1 (1981), p 17–41.
• H. B. Nielsen, “UCTP Test Problems for Unconstrained Optimization”, IMM Department of Mathematical Modeling, DTU, Tech. Report IMM-REP-2000-17 (2000).

41.13. References and Further Reading565

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Chapter 41. Nonlinear Least-Squares Fitting

CHAPTERFORTYTWOBASIS SPLINESThis chapter describes functions for the computation of smoothing basis splines (B-splines). A smoothing spline differsfrom an interpolating spline in that the resulting curve is not required to pass through each data point. For information
about interpolating splines, see Interpolation.
The header file gsl_bspline.h contains the prototypes for the bspline functions and related declarations.

42.1 Overview
Basis splines are commonly used to fit smooth curves to data sets. They are defined on some interval [a, b] which is
sub-divided into smaller intervals by defining a non-decreasing sequence of knots. Between any two adjacent knots, a
B-spline of order k is a polynomial of degree k − 1, and so B-splines defined over the whole interval [a, b] are called
piecewise polynomials. In order to generate a smooth spline spanning the full interval, the different polynomial pieces
are often required to match one or more derivatives at the knots.
B-splines of order k are basis functions for spline functions of the same order, so that all possible spline functions onsome interval [a, b] can be expressed as linear combinations of B-splines,f (x; t) =n
X

ci Bi,k (x; t),

a≤x≤b

i=1

Here, the n coefficients ci are known as control points and are often determined from a least squares fit. The Bi,k (x; t)
are the basis splines of order k, and depend on a knot vector t, which is a non-decreasing sequence of n + k knots
t = {t1 , t2 , . . . , tn+k }On each knot interval ti ≤ x < ti+1 , the B-splines Bi,k (x; t) are polynomials of degree k − 1. The n basis splines oforder k are defined by
1, ti ≤ x < ti+1
Bi,1 (x) =0,
else
Bi,k (x) =(x − ti )(ti+k − x)Bi,k−1 (x) +Bi+1,k−1 (x)(ti+k−1 − ti )(ti+k − ti+1 )for i = 1, . . . , n. The common case of cubic B-splines is given by k = 4. The above recurrence relation can beevaluated in a numerically stable way by the de Boor algorithm.
In what follows, we will drop the t and simply refer to the spline functions as f (x) and Bi,k (x) for simplicity, with theunderstanding that each set of spline basis functions depends on the chosen knot vector t. We will also occasionallydrop the order k and use Bi (x) and Bi,k (x) interchangeably.

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GNU Scientific Library, Release 2.7

42.1.1 Vector Splines
The spline above can be generalized to vector control points,
f (x) =nX

ci Bi,k (x)i=1where the ci may have some length m.

42.1.2 Knot vector
In practice, one will often identify locations inside the interval [a, b] where it makes sense to end one polynomial pieceand begin another. These locations are called interior knots or breakpoints. It may be desirable to increase the densityof knots in regions with rapid changes, and decrease the density when the underlying data is changing slowly. For some
applications it may be simply best to distribute the knots with equal spacing on the interval, which are called uniform
knots.
The full knot vector will look liket = {t1 , . . . , tk−1 , tk , . . . , tn+1 , tn+2 , . . . , tn+k }
{z
} |
{z
} |
{z
}
|
n−k+2 interior knots

k−1

k−1

The n − k + 2 interior knots are in [a, b]. In order to have a full basis of B-splines, we require an additional 2(k − 1)exterior knots, {t1 , . . . , tk−1 } and {tn+2 , . . . , tn+k }. These knots must satisfy
t1 ≤ · · · ≤ tk−1 = ab = tn+2 ≤ · · · ≤ tn+k
but can otherwise be arbitrary.

42.2 Allocation of B-splines
type gsl_bspline_workspace
The computation of B-spline functions requires a preallocated workspace.
gsl_bspline_workspace *gsl_bspline_alloc(const size_t k, const size_t nbreak)
gsl_bspline_workspace *gsl_bspline_alloc_ncontrol(const size_t k, const size_t ncontrol)
These functions allocate a workspace for computing B-splines of order k. The number of breakpoints is given
by nbreak. Alternatively, the number of control points n may be specified in ncontrol. The relationship isnbreak = n − k + 2. Cubic B-splines are specified by k = 4. The size of the workspace is O(2k 2 + 5k + nbreak).
void gsl_bspline_free(gsl_bspline_workspace *w)
This function frees the memory associated with the workspace w.

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42.3 Initializing the knots vector
int gsl_bspline_init_augment(const gsl_vector *tau, gsl_bspline_workspace *w)
This function initializes the knot vector t using the non-decreasing interior knot vector tau of length n − k + 2
by choosing the exterior knots to be equal to the first and last interior knot. The total knot vector ist = {τ1 , . . . , τ1 , τ1 , τ2 , . . . , τn−k+2 , τn−k+2 , . . . , τn−k+2 }| {z }|{z}
k−1 copies

k−1 copies

where τi is specified by the i-th element of tau.
This function does not verify that the tau vector is non-decreasing, so the user must ensure this.
int gsl_bspline_init_uniform(const double a, const double b, gsl_bspline_workspace *w)
This function constructs a knot vector using uniformly spaced interior knots on [a, b] and sets the exterior knots
to the interval endpoints. The total knot vector ist = {a, . . . , a , τ1 , τ2 , . . . , τn−k+2 , b, . . . , b }| {z }| {z }
k−1 copies

k−1 copies

withτi = a +i−1(b − a),n−k+1

i = 1, · · · , n − k + 2

int gsl_bspline_init_periodic(const double a, const double b, gsl_bspline_workspace *w)
This function constructs a knot vector suitable for defining a periodic B-spline on [a, b] with a period P = b − a.
There are several ways to choose a periodic knot vector. Here, we adopt the approach of Dierckx, 1982, andrequireti = ti+n−k+1 − P, i = 1, . . . , k − 1
tn−k+1+i = ti + P, i = k + 1, . . . , 2k − 1
We further define a uniform spacing between each consecutive knot pair. The knot values are given by
ti =

i−k
(b − a) + a,n−k+1

i = 1, . . . , n + k

int gsl_bspline_init_interp(const gsl_vector *x, gsl_bspline_workspace *w)
This function calculates a knot vector suitable for interpolating data at the n abscissa values provided in x, whichmust be in ascending order. The knot vector for interpolation must satisfy the Schoenberg-Whitney conditions,ti < xi < ti+k ,This function selects the knots according to x1 Pi−1
1
ti =x
 k−1 j=i−k+1 jxn

i = 1, . . . , ni = 1, . . . , k
i = k + 1, . . . , n
i = n + 1, . . . , n + kwhich satisfy the Schoenberg-Whitney conditions, although this may not be the optimal knot vector for a given
interpolation problem.
int gsl_bspline_init(const gsl_vector *t, gsl_bspline_workspace *w)
This function sets the knot vector equal to the user-supplied vector t which must have length n + k and be
non-decreasing.

42.3. Initializing the knots vector

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42.4 Properties of B-splines
size_t gsl_bspline_ncontrol(const gsl_bspline_workspace *w)
This function returns the number of B-spline control points given by n = nbreak + k − 2.
size_t gsl_bspline_order(const gsl_bspline_workspace *w)
This function returns the B-spline order k.
size_t gsl_bspline_nbreak(const gsl_bspline_workspace *w)
This function returns the number of breakpoints (interior knots) defined for the B-spline.

42.5 Evaluation of B-splines
int gsl_bspline_calc(const double x, const gsl_vector *c, double *result, gsl_bspline_workspace *w)
This function evaluates the B-splinef (x) =nX

ci Bi (x)i=1

at the point x given the n control points c. The result f (x) is stored in result. If x lies outside the knot interval,a Taylor series approximation about the closest knot endpoint is used to extrapolate the spline.
int gsl_bspline_vector_calc(const double x, const gsl_matrix *c, gsl_vector *result, gsl_bspline_workspace*w)
This function evaluates the vector valued B-splinef (x) =nX

ci Bi (x)i=1

at the point x given the n control vectors ci . The vector ci is stored in the i-th column of c. The result f (x) is
stored in result. The parameters c and result must have the same number of rows. If x lies outside the knot
interval, a Taylor series approximation about the closest knot endpoint is used to extrapolate the spline.
int gsl_bspline_basis(const double x, gsl_vector *B, size_t *istart, gsl_bspline_workspace *w)
This function evaluates all potentially nonzero B-spline basis functions at the position x and stores them in thevector B of length k, so that the i-th element is B(istart+i) (x) for i = 0, 1, . . . , k − 1. The output istartis guaranteed to be in [0, n − k]. By returning only the nonzero basis functions, this function allows quantities
involving linear combinations of the Bi (x) to be computed without unnecessary terms (such linear combinations
occur, for example, when evaluating an interpolated function).

42.6 Evaluation of B-spline derivatives
int gsl_bspline_calc_deriv(const double x, const gsl_vector *c, const size_t nderiv, double *result,
gsl_bspline_workspace *w)
This function evaluates the B-spline derivative
n
X
dj
dj
f
(x)
=
ci j Bi (x)
j
dx
dx
i=1

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Chapter 42. Basis Splines

GNU Scientific Library, Release 2.7

at the point x given the n control points c. The derivative order j is specified by nderiv. The result dj f (x)/dxj
is stored in result. If x lies outside the knot interval, a Taylor series approximation about the closest knot
endpoint is used to extrapolate the spline.
int gsl_bspline_vector_calc_deriv(const double x, const gsl_matrix *c, const size_t nderiv, gsl_vector *result,
gsl_bspline_workspace *w)
This function evaluates the vector valued B-spline derivative
n
X
dj
dj
f
(x)
=
Bi (x)
c
i
dxj
dxj
i=1

at the point x given the n control vectors ci . The vector ci is stored in the i-th column of c. The result dj f (x)/dxj
is stored in result. The parameters c and result must have the same number of rows. The derivative order jis specified in nderiv. If x lies outside the knot interval, a Taylor series approximation about the closest knot
endpoint is used to extrapolate the spline.
int gsl_bspline_basis_deriv(const double x, const size_t nderiv, gsl_matrix *dB, size_t *istart,
gsl_bspline_workspace *w)
This function evaluates all potentially nonzero B-spline basis function derivatives of orders 0 through nderiv(inclusive) at the position x and stores them in the matrix dB. The (i, j)-th element of dB is dj B(istart+i) (x)/dxj .
The last row of dB contains dj Bistart+k−1 (x)/dxj . The matrix dB must be of size k by at least nderiv + 1. The
output istart is guaranteed to be in [0, n − k]. Note that function evaluations are included as the zeroth orderderivatives in dB. By returning only the nonzero basis functions, this function allows quantities involving linear
combinations of the Bi (x) and their derivatives to be computed without unnecessary terms.

42.7 Evaluation of B-spline integrals
int gsl_bspline_calc_integ(const double a, const double b, const gsl_vector *c, double *result,
gsl_bspline_workspace *w)
This function computes the integral
Z b
f (x)dx =
a

n
X
i=1

Z b
ci

Bi (x)dxausing the coefficients stored in c and the integration limits (a, b). The integral value is stored in result on
output.
int gsl_bspline_basis_integ(const double a, const double b, gsl_vector *y, gsl_bspline_workspace *w)
This function computes the integral from a to b of each basis function Bi (x),
Z b
yi =

Bi (x)dx
astoring the results in the vector y, of length ncontrol. The algorithm uses Gauss-Legendre quadrature on eachknot interval.

42.7. Evaluation of B-spline integrals571

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42.8 Least Squares Fitting with B-splines
A common application is to fit a B-spline to a set of m data points (xi , yi ) without requiring the spline to pass througheach point, but rather the spline fits the data in a least square sense. In this case, the number of control points can bemuch less than m and the result is a smooth spline which minimizes the cost function,
χ2 =

m
X

2

wi (yi − f (xi )) = ||y − Xc||2W

i=1

Pn

where f (x) = j=1 cj Bj,k (x) is the B-spline defined above, n is the number of control points for the spline, W =diag(w), and Xij = Bj,k (xi ) is the least squares design matrix. The parameters wi are optional weights which
can be assigned to each data point (xi , yi ). Because each basis spline Bj,k (x) has local support (Bj,k (x) = 0 forx ∈/ [tj , . . . , tj+k ]), the least squares design matrix X has only k nonzero entries per row. Therefore, the normal
equations matrix X T W X is symmetric and banded, with lower bandwidth k − 1. The GSL routines below use abanded Cholesky factorization to solve the normal equations system for the unknown coefficient vector,
c = XT W X

−1

XT W yThe normal equations approach is often avoided in least squares problems, since if X is badly conditioned, then X T W X
is much worse conditioned, which can lead to loss of accuracy in the solution vector. However, a theorem of de Boor[de Boor (2001), XI(4)] states that the condition number of the B-spline basis is independent of the knot vector t andbounded, depending only on k. Numerical evidence has shown that for reasonably small k the normal equations matrix
is well conditioned, and a banded Cholesky factorization can be used safely.
int gsl_bspline_lssolve(const gsl_vector *x, const gsl_vector *y, gsl_vector *c, double *chisq,
gsl_bspline_workspace *w)
int gsl_bspline_wlssolve(const gsl_vector *x, const gsl_vector *y, const gsl_vector *wts, gsl_vector *c, double*chisq, gsl_bspline_workspace *w)
These functions fit a B-spline to the input data (x, y) which must be the same length m. Optional weights may
be given in the wts vector. The banded normal equations matrix is assembled and factored with a Cholesky
factorization, and then solved, storing the coefficients in the vector c, which has length n. The cost function χ2
is output in chisq. If the normal equations matrix is singular, the error code GSL_EDOM is returned. This could
occur, for example, if there is a large gap in the data vectors so that no data points constrain a particular Bi,k (x)basis function.
On output, w->XTX contains the Cholesky factor of the symmetric banded normal equations matrix, and this canbe passed directly to the functions gsl_bspline_covariance() and gsl_bspline_rcond().
int gsl_bspline_lsnormal(const gsl_vector *x, const gsl_vector *y, const gsl_vector *wts, gsl_vector *XTy,
gsl_matrix *XTX, gsl_bspline_workspace *w)
This function builds the normal equations matrix X T W X and right hand side vector X T W y, which are stored
in XTX and XTy on output. The diagonal weight matrix W is specified by the input vector wts, which may be
set to NULL, in which case W = I. The inputs x, y, and wts represent the data and weights to be fitted in aleast-squares sense, and they all have length m. While the full normal equations matrix has dimension n-by-n, it
has a significant number of zeros, and so the output matrix XTX is stored in banded symmetric format. Therefore,
its dimensions are n-by-k. The vector XTy has length n. The output matrix XTX can be passed directly into abanded Cholesky factorization routine, such as gsl_linalg_cholesky_band_decomp().
This function is normally called by gsl_bspline_wlssolve() but is available for users with more specialized
applications.
int gsl_bspline_lsnormalm(const gsl_vector *x, const gsl_matrix *Y, const gsl_vector *wts, gsl_matrix *XTY,
gsl_matrix *XTX, gsl_bspline_workspace *w)
This function builds the normal equations matrix X T W X and right hand side vectors X T W Y , which are stored
in XTX and XTY on output. The diagonal weight matrix W is specified by the input vector wts, which may be
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set to NULL, in which case W = I. The input Y is a matrix of the right hand side vectors, of size m-by-nrhs.
The inputs x and wts represent the data and weights to be fitted in a least-squares sense, and they have length m.
While the full normal equations matrix has dimension n-by-n, it has a significant number of zeros, and so the
output matrix XTX is stored in banded symmetric format. Therefore, its dimensions are n-by-k. The matrix XTY
has size n-by-nrhs. The output matrix XTX can be passed directly into a banded Cholesky factorization routine,such as gsl_linalg_cholesky_band_decomp().
int gsl_bspline_residuals(const gsl_vector *x, const gsl_vector *y, const gsl_vector *c, gsl_vector *r,
gsl_bspline_workspace *w)
This function computes the residual vector ri = yi − f (xi ) by evaluating the spline f (x) using the coefficientsc.
int gsl_bspline_covariance(const gsl_matrix *cholesky, gsl_matrix *cov, gsl_bspline_workspace *w)
−1This function computes the n-by-n covariance matrix X T W X
using the banded symmetric Cholesky
decomposition stored in cholesky, which has dimensions n-by-k. The output is stored in cov. The function gsl_linalg_cholesky_band_decomp() must be called on the normal equations matrix to compute theCholesky factor prior to calling this function.
int gsl_bspline_err(const double x, const size_t nderiv, const gsl_matrix *cov, double *err,
gsl_bspline_workspace *w)
This function computes the standard deviation error of a spline (or its derivative),f (q) (x) =

n
X

(q)

cj Bj (x)j=1

at the point x using the covariance matrix cov calculated previously with gsl_bspline_covariance(). The
derivative order is specified by nderiv. The error is stored in err on output, which is defined as,qδf = B(q) (x)T CB(q) (x)


(q)
(q)
where B(q) (x) = B1 (x), . . . , Bn (x) is the vector of B-spline values evaluated at x, and C is the covariancematrix.
int gsl_bspline_rcond(const gsl_matrix *XTX, double *rcond, gsl_bspline_workspace *w)
This function estimates the reciprocal condition number (using the 1-norm) of the normal equations matrix X T W X, using its Cholesky decomposition, which must be computed previously by calling gsl_linalg_cholesky_band_decomp(). The reciprocal condition number estimate, defined as
1/(||X T W X||1 · ||(X T W X)−1 ||1 ), is stored in rcond on output.

42.8.1 Periodic Spline Fitting
Some applications require fitting a spline to a dataset which has an underlying periodicity associated with it. In thesecases it is desirable to construct a spline which exhibits the same periodicity. A spline f (x) is called periodic on [a, b]
if it satisfies the conditions
f (j) (a) = f (j) (b),j = 0, . . . , k − 2

These conditions can be satisfied with the following constraints on the knot vector and the control points (Dierckx,1982),
ti = ti+n−k+1 − P,
tn−k+1+i = ti + P,
ci = cn−k+1+i ,42.8. Least Squares Fitting with B-splines

i = 1, . . . , k − 1

i = k + 1, . . . , 2k − 1
i = 1, . . . , k − 1

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where P = b − a is the period. The constraint on the control points ci means that there are only n − k + 1 independent
control points, and then the remaining k − 1 are determined. The independent control points c∗ = {c1 , . . . , cn−k+1 }can be determined by solving the following least squares problem (Dierckx, 1982):
2||y − X ∗ c∗ ||W
min
∗
c

∗
∗
= Bj,k(xi ) and
where Xij
∗
Bj,k
(x) =



Bj,k (x) + Bn−k+j+1,k (x),Bj,k (x),j = 1, . . . , k − 1
j = k, . . . , n − k + 1

int gsl_bspline_plssolve(const gsl_vector *x, const gsl_vector *y, gsl_vector *c, double *chisq,
gsl_bspline_workspace *w)
int gsl_bspline_pwlssolve(const gsl_vector *x, const gsl_vector *y, const gsl_vector *wts, gsl_vector *c,double *chisq, gsl_bspline_workspace *w)
These functions fit a periodic B-spline to the input data (x, y) which must be the same length m. Optional weights
may be given in the wts vector. The input vector x must be sorted in ascending order, in order to efficiently
compute the QR decomposition of the least squares system. The algorithm computes the QR decomposition of
the least squares matrix, one row at a time using Givens transformations, taking advantage of sparse structure
due to the local properties of the B-spline basis. On output, the coefficients are stored in the vector c, which has
length n. The cost function χ2 = ||y − Xc||2W is output in chisq.
int gsl_bspline_plsqr(const gsl_vector *x, const gsl_vector *y, const gsl_vector *wts, gsl_matrix *R, gsl_vector*QTy, double *rnorm, gsl_bspline_workspace *w)
This function calculates the R factor and QT y vector for a periodic spline fitting problem on the input data (x,y) with optional weights wts. The input vector x must be sorted in ascending order. On output, the (n − k + 1)by-(n − k + 1) R factor is stored in R, while the first n − k + 1 elements of the vector QT y are stored in QTy.
The residual norm ||y − X ∗ c∗ ||W is stored in rnorm.

42.9 Gram Matrix
Let B(x) = (B1 (x), B2 (x), . . . , Bn (x))T be the n-vector whose elements are the B-spline basis functions evaluatedat x. Then the n-by-n outer product matrix of the B-spline basis at x is defined as
A(x) = B(x) ⊗ B(x) = B(x)B(x)T ,Aij (x) = Bi (x)Bj (x)This can be generalized to higher order derivatives,
 q
 q

d
d
(q)
Aij (x) =Bi (x)Bj (x)
dxq
dxq
Then, the n-by-n Gram matrix of the B-spline basis is defined as
(q)
Gij =

Z b
a

(q)
Aij (x)dx =

Z b
a

 q

dq
d
Bi (x)Bj (x) dx
dxq
dxq

where the integral is usually taken over the full support of the B-splines, i.e. from min(t) to max(t). However it canbe computed over any interval [a, b].
The B-spline outer product and Gram matrices are symmetric and banded with lower bandwidth k − 1. These matrices
are positive semi-definite, and in the case of q = 0, A(0) and G(0) are positive definite. These matrices have applicationsin least-squares fitting.
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int gsl_bspline_oprod(const size_t nderiv, const double x, gsl_matrix *A, gsl_bspline_workspace *w)
This function calculates the outer product matrix A(q) (x) at the location x, where the derivative order q is specified by nderiv. The matrix is stored in A in symmetric banded format, and so A has dimensions n-by-k. Since
the B-spline functions have local support, many elements of the outer product matrix could be zero.
int gsl_bspline_gram(const size_t nderiv, gsl_matrix *G, gsl_bspline_workspace *w)(q)This function calculates the positive semi-definite Gram matrix Gij , where the derivative order q is specified bynderiv. The matrix is stored in G in symmetric banded format, and so G has dimensions n-by-k. The integrals
are computed over the full support of the B-spline basis.
The algorithm uses Gauss-Legendre quadrature on each knot interval. See algorithm 5.22 of L. Schumaker,“Spline Functions: Basic Theory”.
int gsl_bspline_gram_interval(const double a, const double b, const size_t nderiv, gsl_matrix *G,
gsl_bspline_workspace *w)(q)This function calculates the positive semi-definite Gram matrix Gij , where the derivative order q is specified bynderiv. The matrix is stored in G in symmetric banded format, and so G has dimensions n-by-k. The integrals
are computed over the interval [a, b].
The algorithm uses Gauss-Legendre quadrature on each knot interval. See algorithm 5.22 of L. Schumaker,“Spline Functions: Basic Theory”.

42.10 Interpolation with B-splines
Given a set of n data points, (xi , yi ), we can define an interpolating spline with n control points which passes througheach data point,f (xi ) =nX

cj Bj (xi ) = yi ,i = 1, . . . , n

j=1

This is a square system of equations which can be solved by inverting the n-by-n collocation matrix, whose elements
are Bj (xi ). In order for the collocation matrix to be invertible, the xi and spline knots must satisfy the SchoenbergWhitney conditions, which areti < xi < ti+k ,i = 1, . . . , n

If these conditions are met, then the collocation matrix is banded, with both the upper and lower bandwidths equal tok − 1. This property leads to savings in storage and computation when solving interpolation problems. To construct aknot vector which satisfies these conditions, see the function gsl_bspline_init_interp().
int gsl_bspline_col_interp(const gsl_vector *x, gsl_matrix *XB, gsl_bspline_workspace *w)
This function constructs the banded collocation matrix to interpolate the n abscissa values provided in x. The
values in x must be sorted in ascending order, although this is not checked explicitly by the function. The
collocation matrix is stored in XB on output in banded storage format suitable for use with the banded LU routines.
Therefore, XB must have dimension n-by-3(k − 1) + 1.

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42.11 Projection onto the B-spline Basis
A function f (x) can be projected onto the subspace V = span(Bi ) spanned by the B-splines as follows,Xf (x) ≈ projV (f ) =cj Bj (x)j

If f (x) is a piecewise polynomial of degree less than k, then f (x) will lie in V and f (x) = projV (f ). The expansion
coefficients cj can be calculated as follows,Xf (x)Bi (x) =cj Bj (x)Bi (x)jZ
f (x)Bi (x)dx =
{z
}
|

X

yi

j

Z
cj

Bi (x)Bj (x)dx
{z
}
|
Gij

Gc = ywhere G is the B-spline Gram Matrix. The above equation can be solved using a Cholesky factorization of G.
int gsl_bspline_proj_rhs(const gsl_function *F, gsl_vector R*y, gsl_bspline_workspace *w)
This function computes the right hand side vector yi = f (x)Bi (x)dx for projecting a function F onto theB-spline basis. The output is stored in y which must have size ncontrol. The integrals are computed using
Gauss-Legendre quadrature, which will produce exact results if f (x) is a piecewise polynomial of degree lessthan k.

42.12 Greville abscissae
The Greville abscissae are defined to be the mean location of k − 1 consecutive knots in the knot vector for each basis
spline function of order k. With the first and last knots in the knot vector excluded, there are n Greville abscissae
for any given B-spline basis. These values are often used in B-spline collocation applications and may also be called
Marsden-Schoenberg points.
double gsl_bspline_greville_abscissa(const size_t i, gsl_bspline_workspace *w)
Returns the location of the i-th Greville abscissa for the given B-spline basis. For the ill-defined case whenk = 1, the implementation chooses to return breakpoint interval midpoints.

42.13 Examples
42.13.1 Example 1: Basis Splines and Uniform Knots
The following program computes and outputs linear, quadratic, cubic, and quartic basis splines using uniform knots on
the interval [0, 1]. The knot vector ist = 0, . . . , 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, . . . , 1
| {z }| {z }kk

where the first and last knot are repeated k times, where k is the spline order. The output is shown in Fig. 42.1.
This figure makes it clear that each B-spline has local support and spans k + 1 knots. The B-splines near the interval
endpoints appear to span less knots, but note that the endpoint knots have multiplicities of k.
The program is given below.
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Fig. 42.1: Linear, quadratic, cubic, and quartic basis splines defined on the interval [0, 1] with uniform knots. Blackcircles denote the location of the knots.

#include <stdio.h>#include <stdlib.h>#include <math.h>#include <gsl/gsl_math.h>#include <gsl/gsl_bspline.h>#include <gsl/gsl_math.h>#include <gsl/gsl_vector.h>void
print_basis(FILE *fp_knots, FILE *fp_spline, const size_t order, const size_t nbreak){
gsl_bspline_workspace *w = gsl_bspline_alloc(order, nbreak);const size_t p = gsl_bspline_ncontrol(w);const size_t n = 300;
const double a = 0.0;
const double b = 1.0;
const double dx = (b - a) / (n - 1.0);
gsl_vector *B = gsl_vector_alloc(p);size_t i, j;/* use uniform breakpoints on [a, b] */gsl_bspline_init_uniform(a, b, w);
gsl_vector_fprintf(fp_knots, w->knots, "%f ");42.13. Examples(continues on next page)577

GNU Scientific Library, Release 2.7(continued from previous page)for (i = 0; i < n; ++i){
double xi = i * dx;
gsl_bspline_eval_basis(xi, B, w);
fprintf(fp_spline, "%f ", xi);for (j = 0; j < p; ++j)
fprintf(fp_spline, "%f ", gsl_vector_get(B, j));fprintf(fp_spline, "\n");}fprintf(fp_knots, "\n\n");fprintf(fp_spline, "\n\n");gsl_vector_free(B);gsl_bspline_free(w);}intmain (void){
const size_t nbreak = 11; /* number of breakpoints */FILE *fp_knots = fopen("bspline1_knots.txt", "w");
FILE *fp_spline = fopen("bspline1.txt", "w");print_basis(fp_knots, fp_spline, 2, nbreak); /* linear splines */print_basis(fp_knots, fp_spline, 3, nbreak); /* quadratic splines */print_basis(fp_knots, fp_spline, 4, nbreak); /* cubic splines */print_basis(fp_knots, fp_spline, 5, nbreak); /* quartic splines */fclose(fp_knots);
fclose(fp_spline);
return 0;}42.13.2 Example 2: Derivatives of Basis Splines
The following program computes and outputs cubic B-splines and their derivatives using 6 breakpoints and uniform
knots on the interval [0, 1]. All derivatives up to order 3 are computed. The output is shown in Fig. 42.2.
The program is given below.
#include <stdio.h>#include <stdlib.h>#include <math.h>(continues on next page)

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Fig. 42.2: Cubic B-splines and derivatives up to order 3. Black circles denote the location of the knots.
(continued from previous page)#include <gsl/gsl_math.h>#include <gsl/gsl_matrix.h>#include <gsl/gsl_bspline.h>intmain (void){
const size_t nbreak = 6;
const size_t spline_order = 4;
gsl_bspline_workspace *w = gsl_bspline_alloc(spline_order, nbreak);const size_t p = gsl_bspline_ncontrol(w);const size_t n = 300;
const double a = 0.0;
const double b = 1.0;
const double dx = (b - a) / (n - 1.0);
gsl_matrix *dB = gsl_matrix_alloc(p, spline_order);size_t i, j, k;/* uniform breakpoints on [a, b] */gsl_bspline_init_uniform(a, b, w);/* output knot vector */gsl_vector_fprintf(stdout, w->knots, "%f ");printf("\n\n");(continues on next page)42.13. Examples

579

GNU Scientific Library, Release 2.7(continued from previous page)for (i = 0; i < spline_order; ++i){for (j = 0; j < n; ++j){
double xj = j * dx;
gsl_bspline_eval_deriv_basis(xj, i, dB, w);
printf("%f ", xj);for (k = 0; k < p; ++k)
printf("%f ", gsl_matrix_get(dB, k, i));printf("\n");}printf("\n\n");}gsl_matrix_free(dB);
gsl_bspline_free(w);return 0;}42.13.3 Example 3: Least squares and breakpoints
The following program computes a linear least squares fit to data using cubic B-splines with uniform breakpoints. The
data is generated from the curve y(x) = cos x exp (−x/10) with Gaussian noise added. Splines are fitted with 10 and
40 breakpoints for comparison.
The program output is shown below:$ ./a.out > bspline_lsbreak.txt
40 breakpoints: chisq/dof = 1.008999e+00
10 breakpoints: chisq/dof = 1.014761e+00
The data and fitted models are shown in Fig. 42.3.
The program is given below.
#include <stdio.h>#include <stdlib.h>#include <math.h>#include <gsl/gsl_math.h>#include <gsl/gsl_bspline.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>#include <gsl/gsl_statistics.h>
int(continues on next page)

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Fig. 42.3: Cubic B-spline least squares fit

42.13. Examples

581

GNU Scientific Library, Release 2.7(continued from previous page)main (void){
const size_t n = 500;/* number of data points to fit */const size_t k = 4;/* spline order */const double a = 0.0;/* data interval [a,b] */const double b = 15.0;
const double sigma = 0.2; /* noise */gsl_bspline_workspace *work1 = gsl_bspline_alloc(k, 40);/* 40 breakpoints */gsl_bspline_workspace *work2 = gsl_bspline_alloc(k, 10);/* 10 breakpoints */const size_t p1 = gsl_bspline_ncontrol(work1);/* number of control points␣
,→*/
const size_t p2 = gsl_bspline_ncontrol(work2);/* number of control points␣
,→*/
const size_t dof1 = n - p1;/* degrees of freedom */const size_t dof2 = n - p2;/* degrees of freedom */gsl_vector *c1 = gsl_vector_alloc(p1);gsl_vector *c2 = gsl_vector_alloc(p2);gsl_vector *x = gsl_vector_alloc(n);gsl_vector *y = gsl_vector_alloc(n);gsl_vector *w = gsl_vector_alloc(n);size_t i;
gsl_rng *r;
double chisq1, chisq2;
gsl_rng_env_setup();r = gsl_rng_alloc(gsl_rng_default);/* this is the data to be fitted */for (i = 0; i < n; ++i){
double xi = (b - a) / (n - 1.0) * i + a;
double yi = cos(xi) * exp(-0.1 * xi);
double dyi = gsl_ran_gaussian(r, sigma);
yi += dyi;
gsl_vector_set(x, i, xi);
gsl_vector_set(y, i, yi);gsl_vector_set(w, i, 1.0 / (sigma * sigma));printf("%f %f \n", xi, yi);}/* use uniform breakpoints on [a, b] */gsl_bspline_init_uniform(a, b, work1);
gsl_bspline_init_uniform(a, b, work2);/* solve least squares problem */gsl_bspline_wlssolve(x, y, w, c1, &chisq1, work1);
gsl_bspline_wlssolve(x, y, w, c2, &chisq2, work2);
fprintf(stderr, "40 breakpoints: chisq/dof = %e\n", chisq1 / dof1);(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)fprintf(stderr, "10 breakpoints: chisq/dof = %e\n", chisq2 / dof2);
printf("\n\n");/* output the spline curves */{double xi;
for (xi = a; xi <= b; xi += 0.1){
double result1, result2;
gsl_bspline_calc(xi, c1, &result1, work1);
gsl_bspline_calc(xi, c2, &result2, work2);
printf("%f %f %f \n", xi, result1, result2);}}
gsl_rng_free(r);gsl_vector_free(x);gsl_vector_free(y);gsl_vector_free(w);gsl_vector_free(c1);gsl_vector_free(c2);gsl_bspline_free(work1);
gsl_bspline_free(work2);
return 0;}42.13.4 Example 4: Least squares and spline order
The following program computes least squares fits to the same dataset in the previous example, using 10 uniform
breakpoints and computing splines of order 1, 2, 3, 4, 5.
The data and fitted models are shown in Fig. 42.4.
The program is given below.
#include <stdio.h>#include <stdlib.h>#include <math.h>#include <gsl/gsl_math.h>#include <gsl/gsl_bspline.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>#include <gsl/gsl_statistics.h>
intmain (void){(continues on next page)42.13. Examples

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Fig. 42.4: B-spline least squares fit with varying spline orders

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const size_t n = 500;/* number of data points to fit */const double a = 0.0;/* data interval [a,b] */const double b = 15.0;
const size_t max_spline_order = 5; /* maximum spline order */const size_t nbreak = 10;/* number of breakpoints */const double sigma = 0.2;/* noise */gsl_bspline_workspace **work = malloc(max_spline_order * sizeof(gsl_bspline_workspace␣,→*));
gsl_vector **c = malloc(max_spline_order * sizeof(gsl_vector *));
gsl_vector *x = gsl_vector_alloc(n);gsl_vector *y = gsl_vector_alloc(n);gsl_vector *w = gsl_vector_alloc(n);size_t i;
gsl_rng *r;
double chisq;
gsl_rng_env_setup();r = gsl_rng_alloc(gsl_rng_default);/* this is the data to be fitted */for (i = 0; i < n; ++i){
double xi = (b - a) / (n - 1.0) * i + a;
double yi = cos(xi) * exp(-0.1 * xi);
double dyi = gsl_ran_gaussian(r, sigma);
yi += dyi;
gsl_vector_set(x, i, xi);
gsl_vector_set(y, i, yi);gsl_vector_set(w, i, 1.0 / (sigma * sigma));printf("%f %f \n", xi, yi);}printf("\n\n");for (i = 0; i < max_spline_order; ++i){/* allocate workspace for this spline order */work[i] = gsl_bspline_alloc(i + 1, nbreak);c[i] = gsl_vector_alloc(gsl_bspline_ncontrol(work[i]));/* use uniform breakpoints on [a, b] */gsl_bspline_init_uniform(a, b, work[i]);/* solve least squares problem */gsl_bspline_wlssolve(x, y, w, c[i], &chisq, work[i]);}/* output the spline curves */{(continues on next page)42.13. Examples

585

GNU Scientific Library, Release 2.7(continued from previous page)

double xi;
for (xi = a; xi <= b; xi += 0.1){
printf("%f ", xi);for (i = 0; i < max_spline_order; ++i){
double result;
gsl_bspline_calc(xi, c[i], &result, work[i]);printf("%f ", result);}
printf("\n");}}for (i = 0; i < max_spline_order; ++i){
gsl_vector_free(c[i]);gsl_bspline_free(work[i]);}gsl_rng_free(r);gsl_vector_free(x);gsl_vector_free(y);gsl_vector_free(w);free(work);
free(c);return 0;}42.13.5 Example 5: Least squares, regularization and the Gram matrix
The following example program fits a cubic B-spline to data generated from the Gaussiang(x) = e−x2

on the interval [−1.5, 1.5] with noise added. Two data gaps are deliberately introduced on [−1.1, −0.7] and [0.1, 0.55],
which makes the problem ill-posed. Fig. 42.5 shows that the ordinary least squares solution exhibits large errors in thedata gap regions due to the ill-conditioned least squares matrix.
One way to correct the situation is to minimize the data misfit but also minimize the curvature of the spline averaged

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over the full interval. We can define the average curvature using the second derivative of the spline as follows:
curvature ≈d2
f (x) dx
dx2

Z

d2 X
ci Bi (x) dx
dx2 i=1

=

=

2

Z

2

n

Z X
n

2

ci Bi00 (x)

dx

i=1

Z
=

2

cT B 00 (x) dx

Z

cT B 00 (x)B 00T (x)cdxZB 00 (x)B 00T (x)dx c= cT== cT G(2) cwhere G(2) is the Gram matrix of second derivatives of the B-splines. Therefore, our regularized least squares problem
ismin ||y − Xc||2W + λ2 cT G(2) ccwhere the regularization parameter λ2 represents a tradeoff between minimizing the data misfit and minimizing thespline curvature. The solution of this least-squares problem is−1
c = X T W X + λ2 G(2)XT W yThe example program below solves this problem without regularization (λ2 = 0) and with regularization (λ2 = 0.1) forcomparison. We also plot the 1σ confidence intervals for both solutions using the diagonal elements of the covariance−1matrix Cov = X T W X + λ2 G(2).
Fig. 42.5 shows the result of regularizing the spline over the full interval, using the command./a.out > data.txt
Fig. 42.6 shows the result of regularizing the spline over the smaller interval [0, 0.6], correcting only the second gap,
using the command./a.out 0 0.6 > data.txt
The program is given below.
#include <stdio.h>#include <stdlib.h>#include <math.h>#include <gsl/gsl_math.h>#include <gsl/gsl_bspline.h>#include <gsl/gsl_matrix.h>#include <gsl/gsl_vector.h>#include <gsl/gsl_linalg.h>#include <gsl/gsl_rng.h>(continues on next page)42.13. Examples

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GNU Scientific Library, Release 2.7

Fig. 42.5: B-spline least squares fit with exact (green), unregularized (red with 1σ confidence intervals), and fully
regularized (blue with 1σ confidence intervals) solutions.

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Fig. 42.6: B-spline least squares fit with exact (green), unregularized (red with 1σ confidence intervals), and partiallyregularized (blue with 1σ confidence intervals) solutions.

42.13. Examples

589

GNU Scientific Library, Release 2.7(continued from previous page)#include <gsl/gsl_randist.h>intmain (int argc, char * argv[]){
const size_t n = 500;,→fit */
const size_t k = 4;
const double a = -1.5;
const double b = 1.5;
const double sigma = 0.02;
const double lambda_sq = 0.1;,→*/
gsl_bspline_workspace *work = gsl_bspline_alloc(k, 25);
const size_t p = gsl_bspline_ncontrol(work);,→*/
gsl_vector *c = gsl_vector_alloc(p);,→unregularized model */gsl_vector *creg = gsl_vector_alloc(p);,→regularized model */gsl_vector *x = gsl_vector_alloc(n);gsl_vector *y = gsl_vector_alloc(n);gsl_vector *w = gsl_vector_alloc(n);gsl_matrix *XTX = gsl_matrix_alloc(p, k);
gsl_vector *XTy = gsl_vector_alloc(p);gsl_matrix *cov = gsl_matrix_alloc(p, p);
gsl_matrix *cov_reg = gsl_matrix_alloc(p, p);,→regularized model */gsl_matrix *G = gsl_matrix_alloc(p, k);double reg_a = -1.0;,→[reg_a,reg_b] */double reg_b = -1.0;
int reg_interval = 0;,→smaller interval */size_t i;
gsl_rng *r;/* number of data points to␣/* spline order *//* data interval [a,b] *//* noise *//* regularization parameter␣/* 25 breakpoints *//* number of control points␣
/* control points for␣/* control points for␣/* x data *//* y data *//* data weights *//* normal equations matrix *//* normal equations rhs *//* covariance matrix *//* covariance matrix for␣/* regularization matrix *//* regularization interval␣

/* apply regularization on␣

if (argc > 2){
reg_a = atof(argv[1]);reg_b = atof(argv[2]);reg_interval = 1;}
gsl_rng_env_setup();r = gsl_rng_alloc(gsl_rng_default);/* this is the data to be fitted */i = 0;
while (i < n){double ui = gsl_rng_uniform(r);(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)

double xi = (b - a) * ui + a;
double yi, dyi;/* data gaps between [-1.1,-0.7] and [0.1,0.5] */if ((xi >= -1.1 && xi <= -0.7) || (xi >= 0.1 && xi <= 0.55))
continue;dyi = gsl_ran_gaussian(r, sigma);
yi = exp(-xi*xi) + dyi;
gsl_vector_set(x, i, xi);
gsl_vector_set(y, i, yi);gsl_vector_set(w, i, 1.0 / (sigma * sigma));printf("%f %f \n", xi, yi);++i;}
printf("\n\n");/* use uniform breakpoints on [a, b] */gsl_bspline_init_uniform(a, b, work);
gsl_bspline_lsnormal(x, y, w, XTy, XTX, work); /* form normal equations matrix */gsl_linalg_cholesky_band_decomp(XTX);/* banded Cholesky decomposition */gsl_linalg_cholesky_band_solve(XTX, XTy, c);/* solve for unregularized solution */gsl_bspline_covariance(XTX, cov, work);/* compute covariance matrix *//* compute regularization matrix */if (reg_interval)
gsl_bspline_gram_interval(reg_a, reg_b, 2, G, work);else
gsl_bspline_gram(2, G, work);/* multiply by lambda^2 */gsl_matrix_scale(G, lambda_sq);
gsl_bspline_lsnormal(x, y, w, XTy, XTX, work); /* form normal equations matrix */gsl_matrix_add(XTX, G);/* add regularization term */gsl_linalg_cholesky_band_decomp(XTX);/* banded Cholesky decomposition */gsl_linalg_cholesky_band_solve(XTX, XTy, creg); /* solve for regularized solution */gsl_bspline_covariance(XTX, cov_reg, work);/* compute covariance matrix *//* output the spline curves */{double xi;
for (xi = a; xi <= b; xi += 0.01){
double result_unreg, result_reg;
double err_unreg, err_reg;(continues on next page)42.13. Examples

591

GNU Scientific Library, Release 2.7(continued from previous page)/* compute unregularized spline value and error at xi */gsl_bspline_calc(xi, c, &result_unreg, work);
gsl_bspline_err(xi, 0, cov, &err_unreg, work);/* compute regularized spline value and error at xi */gsl_bspline_calc(xi, creg, &result_reg, work);
gsl_bspline_err(xi, 0, cov_reg, &err_reg, work);
printf("%f %e %e %e %e %e\n",xi,
exp(-xi*xi),
result_unreg,
err_unreg,
result_reg,
err_reg);}}
gsl_rng_free(r);gsl_vector_free(x);gsl_vector_free(y);gsl_vector_free(w);gsl_vector_free(c);gsl_vector_free(creg);
gsl_bspline_free(work);
gsl_matrix_free(XTX);
gsl_vector_free(XTy);
gsl_matrix_free(cov);
gsl_matrix_free(cov_reg);
gsl_matrix_free(G);return 0;}42.13.6 Example 6: Least squares and endpoint regularization
Data fitting with splines can often produce undesired effects at the spline endpoints, where they are less constrained bydata. As an example, we fit order k = 10 splines to the Runge function,g(x) =

1
1 + 25x2

with 20 uniform breakpoints on the interval [a, b] = [−1, 1]. Fig. 42.7 shows that the fitted spline (red) exhibits a sharp
upward feature at the left endpoint. This is due to the lack of data constraining this part of the spline, and would likely
result in errors if the user wanted to extrapolate the spline outside the fit interval. One method to correct this is to apply
regularization to the spline at the endpoints by minimizing one or more of its derivatives. In this example, we minimize

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the first derivative of the spline at both endpoints, |f 0 (a)|2 and |f 0 (b)|2 . Note that
d
f (x)
dx

2

n

d X
=
ci Bi (x)dx i=1= cT B 0 (x)22= cT B 0 (x)B 0T (x)c
= cT A(1) (x)cwhere A(1) is the outer product matrix of first derivatives. Therefore, our regularized least squares problem ismin ||y − Xc||2W + λ2 cT A(1) (a)c + λ2 cT A(1) (b)ccwhere we have chosen to use the same regularization parameter λ2 to regularize both ends of the spline. In the example
program below, we compute the unregularized solution (λ2 = 0) and regularized solution (λ2 = 10). The program
outputs the first derivative of the spline at both endpoints:
unregularized endpoint deriv 1: [ -1.081170e+01,
regularized endpoint deriv 1: [ -2.735857e-02,-2.963725e+00]-5.371758e-03]

showing that the regularization has forced the first derivative smaller at both endpoints. The results are shown in Fig.
42.7.

Fig. 42.7: Runge function B-spline least squares fit with exact (green), unregularized (red) and regularized (blue)
solutions. The zoomed inset views show the spline endpoints with 1σ confidence intervals.
The program is given below.
42.13. Examples

593

GNU Scientific Library, Release 2.7#include <stdio.h>#include <stdlib.h>#include <math.h>#include <gsl/gsl_math.h>#include <gsl/gsl_bspline.h>#include <gsl/gsl_matrix.h>#include <gsl/gsl_vector.h>#include <gsl/gsl_linalg.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>doublef(double x){
return (1.0 / (1.0 + 25.0*x*x));}int
main (void){
const size_t n = 500;,→fit */
const size_t k = 10;
const double a = -1.0;
const double b = 1.0;
const double sigma = 0.03;
const double lambda_sq = 10.0;,→*/
const size_t nderiv = 1;,→regularize at endpoints */gsl_bspline_workspace *work = gsl_bspline_alloc(k, 20);
const size_t p = gsl_bspline_ncontrol(work);,→*/
gsl_vector *c = gsl_vector_alloc(p);,→unregularized model */gsl_vector *creg = gsl_vector_alloc(p);,→regularized model */gsl_vector *x = gsl_vector_alloc(n);gsl_vector *y = gsl_vector_alloc(n);gsl_vector *w = gsl_vector_alloc(n);gsl_matrix *XTX = gsl_matrix_alloc(p, k);
gsl_vector *XTy = gsl_vector_alloc(p);gsl_matrix *cov = gsl_matrix_alloc(p, p);
gsl_matrix *cov_reg = gsl_matrix_alloc(p, p);,→regularized model */gsl_matrix *A1 = gsl_matrix_alloc(p, k);,→matrix */gsl_matrix *A2 = gsl_matrix_alloc(p, k);,→matrix */size_t i;
gsl_rng *r;/* number of data points to␣/* spline order *//* data interval [a,b] *//* noise *//* regularization parameter␣/* derivative order to␣/* 20 breakpoints *//* number of control points␣
/* control points for␣/* control points for␣/* x data *//* y data *//* data weights *//* normal equations matrix *//* normal equations rhs *//* covariance matrix *//* covariance matrix for␣/* left regularization␣
/* right regularization␣

(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)gsl_rng_env_setup();r = gsl_rng_alloc(gsl_rng_default);/* this is the data to be fitted */i = 0;
while (i < n){double ui = gsl_rng_uniform(r);double xi = (b - a) * ui + a;
double yi, dyi;
dyi = gsl_ran_gaussian(r, sigma);
yi = f(xi) + dyi;
gsl_vector_set(x, i, xi);
gsl_vector_set(y, i, yi);gsl_vector_set(w, i, 1.0 / (sigma * sigma));printf("%f %f \n", xi, yi);++i;}
printf("\n\n");/* use uniform breakpoints on [a, b] */gsl_bspline_init_uniform(a, b, work);
gsl_bspline_lsnormal(x, y, w, XTy, XTX, work); /* form normal equations matrix */gsl_linalg_cholesky_band_decomp(XTX);/* banded Cholesky decomposition */gsl_linalg_cholesky_band_solve(XTX, XTy, c);/* solve for unregularized solution */gsl_bspline_covariance(XTX, cov, work);/* compute covariance matrix *//* compute regularization matrices */gsl_bspline_oprod(nderiv, a, A1, work);
gsl_bspline_oprod(nderiv, b, A2, work);/* multiply by lambda^2 */gsl_matrix_scale(A1, lambda_sq);
gsl_matrix_scale(A2, lambda_sq);
gsl_bspline_lsnormal(x, y, w, XTy, XTX, work); /* form normal equations matrix */gsl_matrix_add(XTX, A1);/* add regularization terms */gsl_matrix_add(XTX, A2);
gsl_linalg_cholesky_band_decomp(XTX);/* banded Cholesky decomposition */gsl_linalg_cholesky_band_solve(XTX, XTy, creg); /* solve for regularized solution */gsl_bspline_covariance(XTX, cov_reg, work);/* compute covariance matrix *//* output the spline curves */{double xi;(continues on next page)42.13. Examples

595

GNU Scientific Library, Release 2.7(continued from previous page)for (xi = a; xi <= b; xi += 0.001){
double result_unreg, result_reg;
double err_unreg, err_reg;/* compute unregularized spline value and error at xi */gsl_bspline_calc(xi, c, &result_unreg, work);
gsl_bspline_err(xi, 0, cov, &err_unreg, work);/* compute regularized spline value and error at xi */gsl_bspline_calc(xi, creg, &result_reg, work);
gsl_bspline_err(xi, 0, cov_reg, &err_reg, work);
printf("%f %e %e %e %e %e\n",xi,
f(xi),
result_unreg,
err_unreg,
result_reg,
err_reg);}}{
double result0, result1;
gsl_bspline_calc_deriv(a, c, nderiv, &result0, work);
gsl_bspline_calc_deriv(b, c, nderiv, &result1, work);
fprintf(stderr, "unregularized endpoint deriv %zu: [%14.6e, %14.6e]\n", nderiv,␣
,→result0, result1);
gsl_bspline_calc_deriv(a, creg, nderiv, &result0, work);
gsl_bspline_calc_deriv(b, creg, nderiv, &result1, work);
fprintf(stderr, " regularized endpoint deriv %zu: [%14.6e, %14.6e]\n", nderiv,␣
,→result0, result1);}
gsl_rng_free(r);gsl_vector_free(x);gsl_vector_free(y);gsl_vector_free(w);gsl_vector_free(c);gsl_vector_free(creg);
gsl_bspline_free(work);
gsl_matrix_free(XTX);
gsl_vector_free(XTy);
gsl_matrix_free(cov);
gsl_matrix_free(cov_reg);
gsl_matrix_free(A1);
gsl_matrix_free(A2);
return 0;(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)}42.13.7 Example 7: Least squares and periodic splines
The following example program fits B-spline curves to a dataset which is periodic. Both a non-periodic and periodic
spline are fitted to the data for comparison. The data is derived from the function on [0, 2π],f (x) = sin x − cos 2x
with Gaussian noise added. The program fits order k = 6 splines to the dataset.

Fig. 42.8: Data (black) with non-periodic spline fit (red) and periodic spline fit (green).
The program additionally computes the derivatives up to order 5 at the endpoints x = 0 and x = 2π:=== Non-periodic spline endpoint derivatives ===deriv 0: [ -8.697939e-01, -8.328553e-01]
deriv 1: [ -2.423132e+00,
4.277439e+00]
deriv 2: [ 3.904362e+01,
3.710592e+01]
deriv 3: [ -2.096142e+02,
2.036125e+02]
deriv 4: [ 7.240121e+02,
7.384888e+02]
deriv 5: [ -1.230036e+03,
1.294264e+03]=== Periodic spline endpoint derivatives ===deriv 0: [ -1.022798e+00, -1.022798e+00](continues on next page)42.13. Examples

597

GNU Scientific Library, Release 2.7(continued from previous page)deriv 1: [ 1.041910e+00,
deriv 2: [ 4.466384e+00,
deriv 3: [ -1.356654e+00,
deriv 4: [ -2.751544e+01,
deriv 5: [ 5.055419e+01,

1.041910e+00]
4.466384e+00]-1.356654e+00]-2.751544e+01]
2.674261e+01]

We can see that the periodic fit matches derivative values up to order 4, with the 5-th derivative being non-continuous.
The program is given below.
#include <stdio.h>#include <stdlib.h>#include <math.h>#include <gsl/gsl_math.h>#include <gsl/gsl_bspline.h>#include <gsl/gsl_rng.h>#include <gsl/gsl_randist.h>#include <gsl/gsl_statistics.h>
intmain (void){
const size_t n = 500;/* number of data points to fit */const double a = 0.0;/* data interval [a,b] */const double b = 2.0 * M_PI;
const size_t spline_order = 6;/* spline order */const size_t ncontrol = 15;/* number of control points */const double sigma = 0.2;/* noise */gsl_bspline_workspace *w = gsl_bspline_alloc_ncontrol(spline_order, ncontrol);
gsl_bspline_workspace *wper = gsl_bspline_alloc_ncontrol(spline_order, ncontrol);
gsl_vector *c = gsl_vector_alloc(ncontrol);/* non-periodic coefficients */gsl_vector *cper = gsl_vector_alloc(ncontrol); /* periodic coefficients */gsl_vector *x = gsl_vector_alloc(n);gsl_vector *y = gsl_vector_alloc(n);gsl_vector *wts = gsl_vector_alloc(n);size_t i;
gsl_rng *r;
double chisq;
gsl_rng_env_setup();r = gsl_rng_alloc(gsl_rng_default);/* this is the data to be fitted */for (i = 0; i < n; ++i){
double xi = (b - a) / (n - 1.0) * i + a;
double yi = sin(xi) - cos(2.0 * xi);
double dyi = gsl_ran_gaussian(r, sigma);
yi += dyi;
gsl_vector_set(x, i, xi);
(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)gsl_vector_set(y, i, yi);gsl_vector_set(wts, i, 1.0 / (sigma * sigma));printf("%f %f \n", xi, yi);}printf("\n\n");/* use uniform non-periodic knots on [a, b] */gsl_bspline_init_uniform(a, b, w);/* solve least squares problem for non-periodic spline */gsl_bspline_wlssolve(x, y, wts, c, &chisq, w);/* use periodic knots on [a, b] */gsl_bspline_init_periodic(a, b, wper);/* solve least squares problem for periodic spline */gsl_bspline_pwlssolve(x, y, wts, cper, &chisq, wper);/* output the spline curves */{double xi;
for (xi = a; xi <= b; xi += 0.01){
double result, result_per;gsl_bspline_calc(xi, c, &result, w);
gsl_bspline_calc(xi, cper, &result_per, wper);printf("%f %f %f \n", xi, result, result_per);}}fprintf(stderr, "=== Non-periodic spline endpoint derivatives ===\n");for (i = 0; i < spline_order; ++i){
double result0, result1;
gsl_bspline_calc_deriv(a, c, i, &result0, w);
gsl_bspline_calc_deriv(b, c, i, &result1, w);
fprintf(stderr, "deriv %zu: [%14.6e, %14.6e]\n", i, result0, result1);}
fprintf(stderr, "=== Periodic spline endpoint derivatives ===\n");for (i = 0; i < spline_order; ++i){
double result0, result1;
gsl_bspline_calc_deriv(a, cper, i, &result0, wper);(continues on next page)42.13. Examples

599

GNU Scientific Library, Release 2.7(continued from previous page)gsl_bspline_calc_deriv(b, cper, i, &result1, wper);fprintf(stderr, "deriv %zu: [%14.6e, %14.6e]\n", i, result0, result1);}
gsl_vector_free(c);gsl_vector_free(cper);gsl_bspline_free(w);gsl_bspline_free(wper);gsl_rng_free(r);gsl_vector_free(x);gsl_vector_free(y);gsl_vector_free(wts);
return 0;}42.13.8 Example 8: Projecting onto the B-spline basis
The following example program projects the function
f (x) = 3x3 − 2x2 − 7x
onto a cubic B-spline basis with uniform knots on the interval [−2, 2]. Because the function is cubic, it can be exactly
decomposed in the B-spline basis.
The program is given below.
#include <stdio.h>#include <stdlib.h>#include <math.h>#include <gsl/gsl_math.h>#include <gsl/gsl_bspline.h>#include <gsl/gsl_matrix.h>#include <gsl/gsl_vector.h>#include <gsl/gsl_linalg.h>/* function to be projected */doublef(double x, void * params){(void) params;
return (x * (-7.0 + x*(-2.0 + 3.0*x)));}int
main (void){
const size_t k = 4;
const double a = -2.0;
const double b = 2.0;/* spline order *//* spline interval [a,b] */(continues on next page)

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Fig. 42.9: Exact function (red) and spline projection (green). Small offset deliberately added to green curve.

42.13. Examples

601

GNU Scientific Library, Release 2.7(continued from previous page)gsl_bspline_workspace *work = gsl_bspline_alloc(k, 10);
const size_t n = gsl_bspline_ncontrol(work);,→*/
gsl_vector *c = gsl_vector_alloc(n);,→*/gsl_vector *y = gsl_vector_alloc(n);gsl_matrix *G = gsl_matrix_alloc(n, k);
gsl_function F;/* 10 breakpoints *//* number of control points␣
/* control points for spline␣/* rhs vector *//* Gram matrix */F.function = f;F.params = NULL;/* use uniform breakpoints on [a, b] */gsl_bspline_init_uniform(a, b, work);/* compute Gram matrix */gsl_bspline_gram(0, G, work);/* construct rhs vector */gsl_bspline_proj_rhs(&F, y, work);/* solve system */gsl_linalg_cholesky_band_decomp(G);gsl_linalg_cholesky_band_solve(G, y, c);/* output the result */{double x;for (x = a; x <= b; x += 0.01){
double result;
gsl_bspline_calc(x, c, &result, work);
printf("%f %f %f \n", x, GSL_FN_EVAL(&F, x), result);}}
gsl_vector_free(y);gsl_vector_free(c);gsl_matrix_free(G);gsl_bspline_free(work);
return 0;}602

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GNU Scientific Library, Release 2.7

42.13.9 Example 9: Interpolation
The following example program computes an interpolating cubic B-spline for a set of data. It constructs the collocation
matrix for the dataset and factors it with a banded LU decomposition. The results are shown in Fig. 42.10.

Fig. 42.10: Data points are shown in black and interpolated spline in light blue.
The program is given below.
#include <stdio.h>#include <stdlib.h>#include <math.h>#include <gsl/gsl_math.h>#include <gsl/gsl_bspline.h>#include <gsl/gsl_matrix.h>#include <gsl/gsl_vector.h>#include <gsl/gsl_linalg.h>
int
main (void){
const size_t n = 9;/* number of data to interpolate */const double x_data[] = { 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 };const double y_data[] = { 3.0, 2.9, 2.5, 1.0, 0.9, 0.8, 0.5, 0.2, 0.1 };gsl_vector_const_view xv = gsl_vector_const_view_array(x_data, n);(continues on next page)42.13. Examples

603

GNU Scientific Library, Release 2.7(continued from previous page)gsl_vector_const_view yv = gsl_vector_const_view_array(y_data, n);const size_t k = 4;/* spline order */gsl_vector *c = gsl_vector_alloc(n);/* control points for spline */gsl_bspline_workspace *work = gsl_bspline_alloc_ncontrol(k, n);gsl_matrix * XB = gsl_matrix_alloc(n, 3*(k-1) + 1); /* banded collocation matrix */gsl_vector_uint * ipiv = gsl_vector_uint_alloc(n);double t;size_t i;/* initialize knots for interpolation */gsl_bspline_init_interp(&xv.vector, work);/* compute collocation matrix for interpolation */gsl_bspline_col_interp(&xv.vector, XB, work);/* solve linear system with banded LU */gsl_linalg_LU_band_decomp(n, k - 1, k - 1, XB, ipiv);
gsl_linalg_LU_band_solve(k - 1, k - 1, XB, ipiv, &yv.vector, c);/* output the data */for (i = 0; i < n; ++i){
double xi = x_data[i];double yi = y_data[i];printf("%f %f \n", xi, yi);}printf("\n\n");/* output the spline */for (t = x_data[0]; t <= x_data[n-1]; t += 0.005){
double result;
gsl_bspline_calc(t, c, &result, work);
printf("%f %f \n", t, result);}
gsl_vector_free(c);gsl_bspline_free(work);
gsl_vector_uint_free(ipiv);
gsl_matrix_free(XB);
return 0;}604

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GNU Scientific Library, Release 2.7

42.14 References and Further Reading
Further information on the algorithms described in this section can be found in the following references,• C. de Boor, A Practical Guide to Splines (2001), Springer-Verlag, ISBN 0-387-95366-3.
• L. L. Schumaker, Spline Functions: Basic Theory, 3rd edition, Cambridge University Press, 2007.
• P. Dierckx, Algorithms for smoothing data with periodic and parametric splines, Computer Graphics and Image
Processing, 20, 1982.
•

P. Dierckx, Curve and surface fitting with splines, Oxford University Press, 1995.

Further information of Greville abscissae and B-spline collocation can be found in the following paper,• Richard W. Johnson, Higher order B-spline collocation at the Greville abscissae. Applied Numerical Mathematics. vol.: 52, 2005, 63–75.
A large collection of B-spline routines is available in the PPPACK library available at http://www.netlib.org/pppack,which is also part of SLATEC.

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CHAPTERFORTYTHREESPARSE MATRICESThis chapter describes functions for the construction and manipulation of sparse matrices, matrices which are populated
primarily with zeros and contain only a few non-zero elements. Sparse matrices often appear in the solution of partial
differential equations. It is beneficial to use specialized data structures and algorithms for storing and working withsparse matrices, since dense matrix algorithms and structures can be prohibitively slow and use huge amounts ofmemory when applied to sparse matrices.
The header file gsl_spmatrix.h contains the prototypes for the sparse matrix functions and related declarations.

43.1 Data types
All the functions are available for each of the standard data-types. The versions for double have the prefix
gsl_spmatrix, Similarly the versions for single-precision float arrays have the prefix gsl_spmatrix_float. The
full list of available types is given below,
Prefixgsl_spmatrix
gsl_spmatrix_float
gsl_spmatrix_long_double
gsl_spmatrix_int
gsl_spmatrix_uint
gsl_spmatrix_long
gsl_spmatrix_ulong
gsl_spmatrix_short
gsl_spmatrix_ushort
gsl_spmatrix_char
gsl_spmatrix_uchar
gsl_spmatrix_complex
gsl_spmatrix_complex_float
gsl_spmatrix_complex_long_doubleTypedoublefloat
long double
int
unsigned int
longunsigned longshort
unsigned shortchar
unsigned charcomplex double
complex float
complex long double

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43.2 Sparse Matrix Storage Formats
GSL currently supports three storage formats for sparse matrices: the coordinate (COO) representation, compressedsparse column (CSC) and compressed sparse row (CSR) formats. These are discussed in more detail below. In order toillustrate the different storage formats, the following sections will reference this M -by-N sparse matrix, with M = 4
and N = 5:

9 0 0 0 −3
4 7 0 0 0 
0 8 −1 8 0 
4 0 5 6 0
The number of non-zero elements in the matrix, also abbreviated as nnz is equal to 10 in this case.

43.2.1 Coordinate Storage (COO)
The coordinate storage format, also known as triplet format, stores triplets (i, j, x) for each non-zero element of thematrix. This notation means that the (i, j) element of the matrix A is Aij = x. The matrix is stored using three arraysof the same length, representing the row indices, column indices, and matrix data. For the reference matrix above, one
possible storage format is:
data
row
col

9
0
0

7
1
1

4
1
0

8
2
1

-3
0
4

-1
2
2

8
2
3

5
3
2

6
3
3

4
3
0

Note that this representation is not unique - the coordinate triplets may appear in any ordering and would still represent
the same sparse matrix. The length of the three arrays is equal to the number of non-zero elements in the matrix,
nnz, which in this case is 10. The coordinate format is extremely convenient for sparse matrix assembly, the processof adding new elements, or changing existing elements, in a sparse matrix. However, it is generally not suitable forthe efficient implementation of matrix-matrix products, or matrix factorization algorithms. For these applications it isbetter to use one of the compressed formats discussed below.
In order to faciliate efficient sparse matrix assembly, GSL stores the coordinate data in a balanced binary search tree,
specifically an AVL tree, in addition to the three arrays discussed above. This allows GSL to efficiently determine
whether an entry (i, j) already exists in the matrix, and to replace an existing matrix entry with a new value, without
needing to search unsorted arrays.

43.2.2 Compressed Sparse Column (CSC)
Compressed sparse column storage stores each column of non-zero values in the sparse matrix in a continuous memory
block, keeping pointers to the beginning of each column in that memory block, and storing the row indices of each
non-zero element. For the reference matrix above, these arrays look likedatarow
col_ptr

9
0
0

4
1
3

4
3
5

7
1
7

8
2
9

-1
2
10

5
3

8
2

6
3

-3
0

The data and row arrays are of length nnz and are the same as the COO storage format. The col_ptr array has length
N + 1, and col_ptr[j] gives the index in data of the start of column j. Therefore, the j-th column of the matrix
is stored in data[col_ptr[j]], data[col_ptr[j] + 1], . . . , data[col_ptr[j+1] - 1]. The last element of
col_ptr is nnz.

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43.2.3 Compressed Sparse Row (CSR)
Compressed row storage stores each row of non-zero values in a continuous memory block, keeping pointers to thebeginning of each row in the block and storing the column indices of each non-zero element. For the reference matrix
above, these arrays look likedata
colrow_ptr

9
0
0

-3
4
2

4
0
4

7
1
7

8
1
10

-1
2

8
3

4
0

5
2

6
3

The data and col arrays are of length nnz and are the same as the COO storage format. The row_ptr array has length
M + 1, and row_ptr[i] gives the index in data of the start of row i. Therefore, the i-th row of the matrix is stored
in data[row_ptr[i]], data[row_ptr[i] + 1], . . . , data[row_ptr[i+1] - 1]. The last element of row_ptr
is nnz.

43.3 Overview
These routines provide support for constructing and manipulating sparse matrices in GSL, using an API similar to
gsl_matrix. The basic structure is called gsl_spmatrix.
type gsl_spmatrix
This structure is defined as:
typedef struct{
size_t size1;
size_t size2;
int *i;double *data;
int *p;size_t nzmax;
size_t nz;[ ... variables for binary tree and memory management ... ]size_t sptype;} gsl_spmatrix;
This defines a size1-by-size2 sparse matrix. The number of non-zero elements currently in the matrix is given
by nz. For the triplet representation, i, p, and data are arrays of size nz which contain the row indices, column
indices, and element value, respectively. So if data[k] = A(i, j), then i = i[k] and j = p[k].
For compressed column storage, i and data are arrays of size nz containing the row indices and element values,
identical to the triplet case. p is an array of size size2 + 1 where p[j] points to the index in data of the startof column j. Thus, if data[k] = A(i, j), then i = i[k] and p[j] <= k < p[j + 1].
For compressed row storage, i and data are arrays of size nz containing the column indices and element values,
identical to the triplet case. p is an array of size size1 + 1 where p[i] points to the index in data of the startof row i. Thus, if data[k] = A(i, j), then j = i[k] and p[i] <= k < p[i + 1].
There are additional variables in the gsl_spmatrix structure related to binary tree storage and memory management. The GSL implementation of sparse matrices uses balanced AVL trees to sort matrix elements in the triplet
representation. This speeds up element searches and duplicate detection during the matrix assembly process.
The gsl_spmatrix structure also contains additional workspace variables needed for various operations likeconverting from triplet to compressed storage. sptype indicates the type of storage format being used (COO,
CSC or CSR).
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The compressed storage format defined above makes it very simple to interface with sophisticated external linear
solver libraries which accept compressed storage input. The user can simply pass the arrays i, p, and data as
the inputs to external libraries.

43.4 Allocation
The functions for allocating memory for a sparse matrix follow the style of malloc() and free(). They also performtheir own error checking. If there is insufficient memory available to allocate a matrix then the functions call the GSL
error handler with an error code of GSL_ENOMEM in addition to returning a null pointer.
gsl_spmatrix *gsl_spmatrix_alloc(const size_t n1, const size_t n2)This function allocates a sparse matrix of size n1-by-n2 and initializes it to all zeros. If the size of the matrix
is not known at allocation time, both n1 and n2 may be set to 1, and they will automatically grow as elements
are added to the matrix. This function sets the matrix to the triplet representation, which is the easiest foradding and accessing matrix elements. This function tries to make a reasonable guess for the number of nonzero elements (nzmax) which will be added to the matrix by assuming a sparse density of 10%. The function
gsl_spmatrix_alloc_nzmax() can be used if this number is known more accurately. The workspace is ofsize O(nzmax).
gsl_spmatrix *gsl_spmatrix_alloc_nzmax(const size_t n1, const size_t n2, const size_t nzmax, const size_t
sptype)This function allocates a sparse matrix of size n1-by-n2 and initializes it to all zeros. If the size of the matrix is
not known at allocation time, both n1 and n2 may be set to 1, and they will automatically grow as elements areadded to the matrix. The parameter nzmax specifies the maximum number of non-zero elements which will be
added to the matrix. It does not need to be precisely known in advance, since storage space will automaticallygrow using gsl_spmatrix_realloc() if nzmax is not large enough. Accurate knowledge of this parameter
reduces the number of reallocation calls required. The parameter sptype specifies the storage format of thesparse matrix. Possible values are
GSL_SPMATRIX_COO
This flag specifies coordinate (triplet) storage.
GSL_SPMATRIX_CSC
This flag specifies compressed sparse column storage.
GSL_SPMATRIX_CSRThis flag specifies compressed sparse row storage.
The allocated gsl_spmatrix structure is of size O(nzmax).
int gsl_spmatrix_realloc(const size_t nzmax, gsl_spmatrix *m)This function reallocates the storage space for m to accomodate nzmax non-zero elements. It is typically calledinternally by gsl_spmatrix_set() if the user wants to add more elements to the sparse matrix than the previously specified nzmax.
Input matrix formats supported: COO, CSC, CSR
void gsl_spmatrix_free(gsl_spmatrix *m)This function frees the memory associated with the sparse matrix m.
Input matrix formats supported: COO, CSC, CSR

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43.5 Accessing Matrix Elements
double gsl_spmatrix_get(const gsl_spmatrix *m, const size_t i, const size_t j)
This function returns element (i, j) of the matrix m.
Input matrix formats supported: COO, CSC, CSR
int gsl_spmatrix_set(gsl_spmatrix *m, const size_t i, const size_t j, const double x)
This function sets element (i, j) of the matrix m to the value x.
Input matrix formats supported: COO
double *gsl_spmatrix_ptr(gsl_spmatrix *m, const size_t i, const size_t j)
This function returns a pointer to the (i, j) element of the matrix m. If the (i, j) element is not explicitly stored
in the matrix, a null pointer is returned.
Input matrix formats supported: COO, CSC, CSR

43.6 Initializing Matrix Elements
Since the sparse matrix format only stores the non-zero elements, it is automatically initialized to zero upon allocation.
The function gsl_spmatrix_set_zero() may be used to re-initialize a matrix to zero after elements have been added
to it.
int gsl_spmatrix_set_zero(gsl_spmatrix *m)This function sets (or resets) all the elements of the matrix m to zero. For CSC and CSR matrices, the cost of
this operation is O(1). For COO matrices, the binary tree structure must be dismantled, so the cost is O(nz).
Input matrix formats supported: COO, CSC, CSR

43.7 Reading and Writing Matrices
int gsl_spmatrix_fwrite(FILE *stream, const gsl_spmatrix *m)This function writes the elements of the matrix m to the stream stream in binary format. The return value is 0
for success and GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native
binary format it may not be portable between different architectures.
Input matrix formats supported: COO, CSC, CSR
int gsl_spmatrix_fread(FILE *stream, gsl_spmatrix *m)This function reads into the matrix m from the open stream stream in binary format. The matrix m must be
preallocated with the correct storage format, dimensions and have a sufficiently large nzmax in order to read in
all matrix elements, otherwise GSL_EBADLEN is returned. The return value is 0 for success and GSL_EFAILED if
there was a problem reading from the file. The data is assumed to have been written in the native binary format
on the same architecture.
Input matrix formats supported: COO, CSC, CSR
int gsl_spmatrix_fprintf(FILE *stream, const gsl_spmatrix *m, const char *format)
This function writes the elements of the matrix m line-by-line to the stream stream using the format specifier
format, which should be one of the %g, %e or %f formats for floating point numbers. The function returns 0 for
success and GSL_EFAILED if there was a problem writing to the file. The input matrix m may be in any storage
format, and the output file will be written in MatrixMarket format.
Input matrix formats supported: COO, CSC, CSR

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gsl_spmatrix *gsl_spmatrix_fscanf(FILE *stream)
This function reads sparse matrix data in the MatrixMarket format from the stream stream and stores it in a
newly allocated matrix which is returned in COO format. The function returns 0 for success and GSL_EFAILED if
there was a problem reading from the file. The user should free the returned matrix when it is no longer needed.

43.8 Copying Matrices
int gsl_spmatrix_memcpy(gsl_spmatrix *dest, const gsl_spmatrix *src)
This function copies the elements of the sparse matrix src into dest. The two matrices must have the same
dimensions and be in the same storage format.
Input matrix formats supported: COO, CSC, CSR

43.9 Exchanging Rows and Columns
int gsl_spmatrix_transpose_memcpy(gsl_spmatrix *dest, const gsl_spmatrix *src)
This function copies the transpose of the sparse matrix src into dest. The dimensions of dest must match thetranspose of the matrix src. Also, both matrices must use the same sparse storage format.
Input matrix formats supported: COO, CSC, CSR
int gsl_spmatrix_transpose(gsl_spmatrix *m)This function replaces the matrix m by its transpose, but changes the storage format for compressed matrix inputs.
Since compressed column storage is the transpose of compressed row storage, this function simply converts a
CSC matrix to CSR and vice versa. This is the most efficient way to transpose a compressed storage matrix, but
the user should note that the storage format of their compressed matrix will change on output. For COO matrix
inputs, the output matrix is also in COO storage.
Input matrix formats supported: COO, CSC, CSR

43.10 Matrix Operations
int gsl_spmatrix_scale(gsl_spmatrix *m, const double x)
This function scales all elements of the matrix m by the constant factor x. The result m(i, j) ← xm(i, j) isstored in m.
Input matrix formats supported: COO, CSC, CSR
int gsl_spmatrix_scale_columns(gsl_spmatrix *A, const gsl_vector *x)
This function scales the columns of the M -by-N sparse matrix A by the elements of the vector x, of length N .
The j-th column of A is multiplied by x[j]. This is equivalent to forming
A → AX
where X = diag(x).
Input matrix formats supported: COO, CSC, CSR
int gsl_spmatrix_scale_rows(gsl_spmatrix *A, const gsl_vector *x)
This function scales the rows of the M -by-N sparse matrix A by the elements of the vector x, of length M . The
i-th row of A is multiplied by x[i]. This is equivalent to forming
A → XA
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where X = diag(x).
Input matrix formats supported: COO, CSC, CSR
int gsl_spmatrix_add(gsl_spmatrix *c, const gsl_spmatrix *a, const gsl_spmatrix *b)
This function computes the sum c = a + b. The three matrices must have the same dimensions.
Input matrix formats supported: CSC, CSR
int gsl_spmatrix_dense_add(gsl_matrix *a, const gsl_spmatrix *b)
This function adds the elements of the sparse matrix b to the elements of the dense matrix a. The result a(i, j) ←a(i, j) + b(i, j) is stored in a and b remains unchanged. The two matrices must have the same dimensions.
Input matrix formats supported: COO, CSC, CSR
int gsl_spmatrix_dense_sub(gsl_matrix *a, const gsl_spmatrix *b)
This function subtracts the elements of the sparse matrix b from the elements of the dense matrix a. The resulta(i, j) ← a(i, j) − b(i, j) is stored in a and b remains unchanged. The two matrices must have the same
dimensions.
Input matrix formats supported: COO, CSC, CSR

43.11 Matrix Properties
const char *gsl_spmatrix_type(const gsl_spmatrix *m)This function returns a string describing the sparse storage format of the matrix m. For example:
printf ("matrix is '%s' format.\n", gsl_spmatrix_type (m));
would print something like:matrix is 'CSR' format.
Input matrix formats supported: COO, CSC, CSR
size_t gsl_spmatrix_nnz(const gsl_spmatrix *m)This function returns the number of non-zero elements in m.
Input matrix formats supported: COO, CSC, CSR
int gsl_spmatrix_equal(const gsl_spmatrix *a, const gsl_spmatrix *b)
This function returns 1 if the matrices a and b are equal (by comparison of element values) and 0 otherwise. The
matrices a and b must be in the same sparse storage format for comparison.
Input matrix formats supported: COO, CSC, CSR
double gsl_spmatrix_norm1(const gsl_spmatrix *A)
This function returns the 1-norm of the m-by-n matrix A, defined as the maximum column sum,||A||1 = max1≤j≤nm
X|Aij |

i=1

Input matrix formats supported: COO, CSC, CSR

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43.12 Finding Maximum and Minimum Elements
int gsl_spmatrix_minmax(const gsl_spmatrix *m, double *min_out, double *max_out)
This function returns the minimum and maximum elements of the matrix m, storing them in min_out andmax_out, and searching only the non-zero values.
Input matrix formats supported: COO, CSC, CSR
int gsl_spmatrix_min_index(const gsl_spmatrix *m, size_t *imin, size_t *jmin)This function returns the indices of the minimum value in the matrix m, searching only the non-zero values, andstoring them in imin and jmin. When there are several equal minimum elements then the first element found isreturned.
Input matrix formats supported: COO, CSC, CSR

43.13 Compressed Format
These routines calculate a compressed matrix from a coordinate representation.
int gsl_spmatrix_csc(gsl_spmatrix *dest, const gsl_spmatrix *src)
This function creates a sparse matrix in compressed sparse column format from the input sparse matrix src
which must be in COO format. The compressed matrix is stored in dest.
Input matrix formats supported: COO
int gsl_spmatrix_csr(gsl_spmatrix *dest, const gsl_spmatrix *src)
This function creates a sparse matrix in compressed sparse row format from the input sparse matrix src whichmust be in COO format. The compressed matrix is stored in dest.
Input matrix formats supported: COO
gsl_spmatrix *gsl_spmatrix_compress(const gsl_spmatrix *src, const int sptype)This function allocates a new sparse matrix, and stores src into it using the format specified by sptype. The
input sptype can be one of GSL_SPMATRIX_COO, GSL_SPMATRIX_CSC, or GSL_SPMATRIX_CSR. A pointer tothe newly allocated matrix is returned, and must be freed by the caller when no longer needed.

43.14 Conversion Between Sparse and Dense Matrices
The gsl_spmatrix structure can be converted into the dense gsl_matrix format and vice versa with the following
routines.
int gsl_spmatrix_d2sp(gsl_spmatrix *S, const gsl_matrix *A)
This function converts the dense matrix A into sparse COO format and stores the result in S.
Input matrix formats supported: COO
int gsl_spmatrix_sp2d(gsl_matrix *A, const gsl_spmatrix *S)
This function converts the sparse matrix S into a dense matrix and stores the result in A.
Input matrix formats supported: COO, CSC, CSR

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43.15 Examples
The following example program builds a 5-by-4 sparse matrix and prints it in coordinate, compressed column, andcompressed row format. The matrix which is constructed is
0
0 3.1 4.6
 1
0 7.2 0 


 0
0
0
0 


 2.1 2.9 0 8.5 
4.1 0
0
0
The output of the program is:
printing all matrix elements:
A(0,0) = 0
A(0,1) = 0
A(0,2) = 3.1
A(0,3) = 4.6
A(1,0) = 1.
.
.
A(4,0) = 4.1
A(4,1) = 0
A(4,2) = 0
A(4,3) = 0
matrix in triplet format (i,j,Aij):(0, 2, 3.1)
(0, 3, 4.6)
(1, 0, 1.0)
(1, 2, 7.2)
(3, 0, 2.1)
(3, 1, 2.9)
(3, 3, 8.5)
(4, 0, 4.1)
matrix in compressed column format:
i = [ 1, 3, 4, 3, 0, 1, 0, 3, ]p = [ 0, 3, 4, 6, 8, ]d = [ 1, 2.1, 4.1, 2.9, 3.1, 7.2, 4.6, 8.5, ]matrix in compressed row format:
i = [ 2, 3, 0, 2, 0, 1, 3, 0, ]p = [ 0, 2, 4, 4, 7, 8, ]d = [ 3.1, 4.6, 1, 7.2, 2.1, 2.9, 8.5, 4.1, ]We see in the compressed column output, the data array stores each column contiguously, the array i stores the row
index of the corresponding data element, and the array p stores the index of the start of each column in the data array.
Similarly, for the compressed row output, the data array stores each row contiguously, the array i stores the column
index of the corresponding data element, and the p array stores the index of the start of each row in the data array.
#include <stdio.h>#include <stdlib.h>#include <gsl/gsl_spmatrix.h>(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)intmain(){
gsl_spmatrix *A = gsl_spmatrix_alloc(5, 4); /* triplet format */gsl_spmatrix *B, *C;
size_t i, j;/* build the sparse matrix */gsl_spmatrix_set(A, 0, 2, 3.1);
gsl_spmatrix_set(A, 0, 3, 4.6);
gsl_spmatrix_set(A, 1, 0, 1.0);
gsl_spmatrix_set(A, 1, 2, 7.2);
gsl_spmatrix_set(A, 3, 0, 2.1);
gsl_spmatrix_set(A, 3, 1, 2.9);
gsl_spmatrix_set(A, 3, 3, 8.5);
gsl_spmatrix_set(A, 4, 0, 4.1);
printf("printing all matrix elements:\n");for (i = 0; i < 5; ++i)for (j = 0; j < 4; ++j)
printf("A(%zu,%zu) = %g\n", i, j,
gsl_spmatrix_get(A, i, j));/* print out elements in triplet format */printf("matrix in triplet format (i,j,Aij):\n");gsl_spmatrix_fprintf(stdout, A, "%.1f ");/* convert to compressed column format */B = gsl_spmatrix_ccs(A);printf("matrix in compressed column format:\n");printf("i = [ ");for (i = 0; i < B->nz; ++i)
printf("%d, ", B->i[i]);printf("]\n");printf("p = [ ");for (i = 0; i < B->size2 + 1; ++i)
printf("%d, ", B->p[i]);printf("]\n");printf("d = [ ");for (i = 0; i < B->nz; ++i)
printf("%g, ", B->data[i]);printf("]\n");/* convert to compressed row format */C = gsl_spmatrix_crs(A);printf("matrix in compressed row format:\n");printf("i = [ ");for (i = 0; i < C->nz; ++i)(continues on next page)

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printf("%d, ", C->i[i]);printf("]\n");printf("p = [ ");for (i = 0; i < C->size1 + 1; ++i)
printf("%d, ", C->p[i]);printf("]\n");printf("d = [ ");for (i = 0; i < C->nz; ++i)
printf("%g, ", C->data[i]);printf("]\n");gsl_spmatrix_free(A);gsl_spmatrix_free(B);gsl_spmatrix_free(C);return 0;}43.16 References and Further Reading
The algorithms used by these functions are described in the following sources,• Davis, T. A., Direct Methods for Sparse Linear Systems, SIAM, 2006.
• CSparse software library, https://www.cise.ufl.edu/research/sparse/CSparse

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CHAPTERFORTYFOURSPARSE BLAS SUPPORTThe Sparse Basic Linear Algebra Subprograms (BLAS) define a set of fundamental operations on vectors and sparse
matrices which can be used to create optimized higher-level linear algebra functionality. GSL supports a limited numberof BLAS operations for sparse matrices.
The header file gsl_spblas.h contains the prototypes for the sparse BLAS functions and related declarations.

44.1 Sparse BLAS operations
int gsl_spblas_dgemv(const CBLAS_TRANSPOSE_t TransA, const double alpha, const gsl_spmatrix *A, const
gsl_vector *x, const double beta, gsl_vector *y)This function computes the matrix-vector product and sum y ← αop(A)x + βy, where op(A) = A, AT forTransA = CblasNoTrans, CblasTrans. In-place computations are not supported, so x and y must be distinct
vectors. The matrix A may be in triplet or compressed format.
int gsl_spblas_dgemm(const double alpha, const gsl_spmatrix *A, const gsl_spmatrix *B, gsl_spmatrix *C)
This function computes the sparse matrix-matrix product C = αAB. The matrices must be in compressed
format.

44.2 References and Further Reading
The algorithms used by these functions are described in the following sources:• Davis, T. A., Direct Methods for Sparse Linear Systems, SIAM, 2006.
• CSparse software library, https://www.cise.ufl.edu/research/sparse/CSparse619

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CHAPTERFORTYFIVESPARSE LINEAR ALGEBRAThis chapter describes functions for solving sparse linear systems of equations. The library provides linear algebra
routines which operate directly on the gsl_spmatrix and gsl_vector objects.
The functions described in this chapter are declared in the header file gsl_splinalg.h.

45.1 Overview
This chapter is primarily concerned with the solution of the linear system
Ax = bwhere A is a general square n-by-n non-singular sparse matrix, x is an unknown n-by-1 vector, and b is a given n-by-1
right hand side vector. There exist many methods for solving such sparse linear systems, which broadly fall into either
direct or iterative categories. Direct methods include LU and QR decompositions, while iterative methods start with an
initial guess for the vector x and update the guess through iteration until convergence. GSL does not currently provide
any direct sparse solvers.

45.2 Sparse Iterative Solvers
45.2.1 Overview
Many practical iterative methods of solving large n-by-n sparse linear systems involve projecting an approximate solution for x onto a subspace of Rn . If we define a m-dimensional subspace K as the subspace of approximations tothe solution x, then m constraints must be imposed to determine the next approximation. These m constraints define
another m-dimensional subspace denoted by L. The subspace dimension m is typically chosen to be much smaller
than n in order to reduce the computational effort needed to generate the next approximate solution vector. The many
iterative algorithms which exist differ mainly in their choice of K and L.

45.2.2 Types of Sparse Iterative Solvers
The sparse linear algebra library provides the following types of iterative solvers:
type gsl_splinalg_itersolve_type621

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gsl_splinalg_itersolve_type *gsl_splinalg_itersolve_gmresThis specifies the Generalized Minimum Residual Method (GMRES). This is a projection method using
K = Km and L = AKm where Km is the m-th Krylov subspace
Km = span r0 , Ar0 , ..., Am−1 r0
and r0 = b − Ax0 is the residual vector of the initial guess x0 . If m is set equal to n, then the Krylov
subspace is Rn and GMRES will provide the exact solution x. However, the goal is for the method toarrive at a very good approximation to x using a much smaller subspace Km . By default, the GMRES
method selects m = M IN (n, 10) but the user may specify a different value for m. The GMRES storage
requirements grow as O(n(m + 1)) and the number of flops grow as O(4m2 n − 4m3 /3).
In the below function gsl_splinalg_itersolve_iterate(), one GMRES iteration is defined as projecting the approximate solution vector onto each Krylov subspace K1 , ..., Km , and so m can be kept smaller
by “restarting” the method and calling gsl_splinalg_itersolve_iterate() multiple times, providing
the updated approximation x to each new call. If the method is not adequately converging, the user may try
increasing the parameter m.
GMRES is considered a robust general purpose iterative solver, however there are cases where the method
stagnates if the matrix is not positive-definite and fails to reduce the residual until the very last projection
onto the subspace Kn = Rn . In these cases, preconditioning the linear system can help, but GSL does not
currently provide any preconditioners.

45.2.3 Iterating the Sparse Linear System
The following functions are provided to allocate storage for the sparse linear solvers and iterate the system to a solution.
gsl_splinalg_itersolve *gsl_splinalg_itersolve_alloc(const gsl_splinalg_itersolve_type *T, const size_t n,
const size_t m)
This function allocates a workspace for the iterative solution of n-by-n sparse matrix systems. The iterativesolver type is specified by T. The argument m specifies the size of the solution candidate subspace Km . The
dimension m may be set to 0 in which case a reasonable default value is used.
void gsl_splinalg_itersolve_free(gsl_splinalg_itersolve *w)
This function frees the memory associated with the workspace w.
const char *gsl_splinalg_itersolve_name(const gsl_splinalg_itersolve *w)
This function returns a string pointer to the name of the solver.
int gsl_splinalg_itersolve_iterate(const gsl_spmatrix *A, const gsl_vector *b, const double tol, gsl_vector*x, gsl_splinalg_itersolve *w)
This function performs one iteration of the iterative method for the sparse linear system specfied by the matrix
A, right hand side vector b and solution vector x. On input, x must be set to an initial guess for the solution.
On output, x is updated to give the current solution estimate. The parameter tol specifies the relative tolerance
between the residual norm and norm of b in order to check for convergence. When the following condition issatisfied:||Ax − b|| ≤ tol × ||b||the method has converged, the function returns GSL_SUCCESS and the final solution is provided in x. Otherwise,
the function returns GSL_CONTINUE to signal that more iterations are required. Here, ||·|| represents the Euclidean
norm. The input matrix A may be in triplet or compressed format.
double gsl_splinalg_itersolve_normr(const gsl_splinalg_itersolve *w)
This function returns the current residual norm ||r|| = ||Ax − b||, which is updated after each call to
gsl_splinalg_itersolve_iterate().
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45.3 Examples
This example program demonstrates the sparse linear algebra routines on the solution of a simple 1D Poisson equation
on [0, 1]:d2 u(x)= f (x) = −π 2 sin (πx)
dx2with boundary conditions u(0) = u(1) = 0. The analytic solution of this simple problem is u(x) = sin πx. We willsolve this problem by finite differencing the left hand side to give1
(ui+1 − 2ui + ui−1 ) = fih2Defining a grid of N points, h = 1/(N − 1). In the finite difference equation above, u0 = uN −1 = 0 are known fromthe boundary conditions, so we will only put the equations for i = 1, ..., N − 2 into the matrix system. The resulting(N − 2) × (N − 2) matrix equation is


 

u1
−2 1
0
0 ... 0f1

 1 −2 1
 

0 ... 0 
  u2   f2 





1 −2 1 . . . 0   u3   f3 
1  0


=

..  
..
.
..
..
..
..
..

 

h2  ..
.
.
.
.
.
.
.


 

 0 . . . . . . 1 −2 1   uN −3   fN −3 
0 . . . . . . . . . 1 −2
uN −2
fN −2
An example program which constructs and solves this system is given below. The system is solved using the iterative
GMRES solver. Here is the output of the program:iter 0 residual = 4.297275996844e-11
Converged
showing that the method converged in a single iteration. The calculated solution is shown in Fig. 45.1.
#include <stdio.h>#include <stdlib.h>#include <math.h>#include <gsl/gsl_math.h>#include <gsl/gsl_vector.h>#include <gsl/gsl_spmatrix.h>#include <gsl/gsl_splinalg.h>
int
main(){
const size_t N = 100;/* number of grid points */const size_t n = N - 2;/* subtract 2 to exclude boundaries */const double h = 1.0 / (N - 1.0);/* grid spacing */gsl_spmatrix *A = gsl_spmatrix_alloc(n ,n); /* triplet format */gsl_spmatrix *C;/* compressed format */gsl_vector *f = gsl_vector_alloc(n);/* right hand side vector */gsl_vector *u = gsl_vector_alloc(n);/* solution vector */size_t i;(continues on next page)

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Fig. 45.1: Solution of PDE

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GNU Scientific Library, Release 2.7(continued from previous page)/* construct the sparse matrix for the finite difference equation *//* construct first row */gsl_spmatrix_set(A, 0, 0, -2.0);
gsl_spmatrix_set(A, 0, 1, 1.0);/* construct rows [1:n-2] */for (i = 1; i < n - 1; ++i){
gsl_spmatrix_set(A, i, i + 1, 1.0);
gsl_spmatrix_set(A, i, i, -2.0);
gsl_spmatrix_set(A, i, i - 1, 1.0);}/* construct last row */gsl_spmatrix_set(A, n - 1, n - 1, -2.0);
gsl_spmatrix_set(A, n - 1, n - 2, 1.0);/* scale by h^2 */gsl_spmatrix_scale(A, 1.0 / (h * h));/* construct right hand side vector */for (i = 0; i < n; ++i){
double xi = (i + 1) * h;double fi = -M_PI * M_PI * sin(M_PI * xi);
gsl_vector_set(f, i, fi);}/* convert to compressed column format */C = gsl_spmatrix_ccs(A);/* now solve the system with the GMRES iterative solver */{const double tol = 1.0e-6; /* solution relative tolerance */const size_t max_iter = 10; /* maximum iterations */const gsl_splinalg_itersolve_type *T = gsl_splinalg_itersolve_gmres;
gsl_splinalg_itersolve *work =gsl_splinalg_itersolve_alloc(T, n, 0);
size_t iter = 0;
double residual;
int status;/* initial guess u = 0 */gsl_vector_set_zero(u);/* solve the system A u = f */do{status = gsl_splinalg_itersolve_iterate(C, f, tol, u, work);/* print out residual norm ||A*u - f|| */(continues on next page)

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residual = gsl_splinalg_itersolve_normr(work);
fprintf(stderr, "iter %zu residual = %.12e\n", iter, residual);if (status == GSL_SUCCESS)
fprintf(stderr, "Converged\n");}while (status == GSL_CONTINUE && ++iter < max_iter);/* output solution */for (i = 0; i < n; ++i){
double xi = (i + 1) * h;double u_exact = sin(M_PI * xi);
double u_gsl = gsl_vector_get(u, i);printf("%f %.12e %.12e\n", xi, u_gsl, u_exact);}
gsl_splinalg_itersolve_free(work);}
gsl_spmatrix_free(A);gsl_spmatrix_free(C);gsl_vector_free(f);gsl_vector_free(u);return 0;} /* main() */45.4 References and Further Reading
The implementation of the GMRES iterative solver closely follows the publications• H. F. Walker, Implementation of the GMRES method using Householder transformations, SIAM J. Sci. Stat.
Comput. 9(1), 1988.
• Y. Saad, Iterative methods for sparse linear systems, 2nd edition, SIAM, 2003.

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CHAPTERFORTYSIXPHYSICAL CONSTANTS

This chapter describes macros for the values of physical constants, such as the speed of light, c, and gravitational
constant, G. The values are available in different unit systems, including the standard MKSA system (meters, kilograms,
seconds, amperes) and the CGSM system (centimeters, grams, seconds, gauss), which is commonly used in Astronomy.
The definitions of constants in the MKSA system are available in the file gsl_const_mksa.h. The constants in the
CGSM system are defined in gsl_const_cgsm.h. Dimensionless constants, such as the fine structure constant, which
are pure numbers are defined in gsl_const_num.h.
The full list of constants is described briefly below. Consult the header files themselves for the values of the constants
used in the library.

46.1 Fundamental Constants
GSL_CONST_MKSA_SPEED_OF_LIGHT
The speed of light in vacuum, c.
GSL_CONST_MKSA_VACUUM_PERMEABILITY
The permeability of free space, µ0 . This constant is defined in the MKSA system only.
GSL_CONST_MKSA_VACUUM_PERMITTIVITY
The permittivity of free space, 0 . This constant is defined in the MKSA system only.
GSL_CONST_MKSA_PLANCKS_CONSTANT_H
Planck’s constant, h.
GSL_CONST_MKSA_PLANCKS_CONSTANT_HBAR
Planck’s constant divided by 2π, ~.
GSL_CONST_NUM_AVOGADRO
Avogadro’s number, Na .
GSL_CONST_MKSA_FARADAY
The molar charge of 1 Faraday.
GSL_CONST_MKSA_BOLTZMANNThe Boltzmann constant, k.
GSL_CONST_MKSA_MOLAR_GAS
The molar gas constant, R0 .
GSL_CONST_MKSA_STANDARD_GAS_VOLUME
The standard gas volume, V0 .
GSL_CONST_MKSA_STEFAN_BOLTZMANN_CONSTANT
The Stefan-Boltzmann radiation constant, σ.

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GSL_CONST_MKSA_GAUSSThe magnetic field of 1 Gauss.

46.2 Astronomy and Astrophysics
GSL_CONST_MKSA_ASTRONOMICAL_UNIT
The length of 1 astronomical unit (mean earth-sun distance), au.
GSL_CONST_MKSA_GRAVITATIONAL_CONSTANT
The gravitational constant, G.
GSL_CONST_MKSA_LIGHT_YEAR
The distance of 1 light-year, ly.
GSL_CONST_MKSA_PARSEC
The distance of 1 parsec, pc.
GSL_CONST_MKSA_GRAV_ACCELThe standard gravitational acceleration on Earth, g.
GSL_CONST_MKSA_SOLAR_MASS
The mass of the Sun.

46.3 Atomic and Nuclear Physics
GSL_CONST_MKSA_ELECTRON_CHARGE
The charge of the electron, e.
GSL_CONST_MKSA_ELECTRON_VOLTThe energy of 1 electron volt, eV .
GSL_CONST_MKSA_UNIFIED_ATOMIC_MASS
The unified atomic mass, amu.
GSL_CONST_MKSA_MASS_ELECTRON
The mass of the electron, me .
GSL_CONST_MKSA_MASS_MUON
The mass of the muon, mµ .
GSL_CONST_MKSA_MASS_PROTON
The mass of the proton, mp .
GSL_CONST_MKSA_MASS_NEUTRON
The mass of the neutron, mn .
GSL_CONST_NUM_FINE_STRUCTURE
The electromagnetic fine structure constant α.
GSL_CONST_MKSA_RYDBERG
The Rydberg constant, Ry, in units of energy. This is related to the Rydberg inverse wavelength R∞ by Ry =hcR∞ .
GSL_CONST_MKSA_BOHR_RADIUS
The Bohr radius, a0 .
GSL_CONST_MKSA_ANGSTROM
The length of 1 angstrom.
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GSL_CONST_MKSA_BARN
The area of 1 barn.
GSL_CONST_MKSA_BOHR_MAGNETON
The Bohr Magneton, µB .
GSL_CONST_MKSA_NUCLEAR_MAGNETON
The Nuclear Magneton, µN .
GSL_CONST_MKSA_ELECTRON_MAGNETIC_MOMENTThe absolute value of the magnetic moment of the electron, µe . The physical magnetic moment of the electron
is negative.
GSL_CONST_MKSA_PROTON_MAGNETIC_MOMENTThe magnetic moment of the proton, µp .
GSL_CONST_MKSA_THOMSON_CROSS_SECTION
The Thomson cross section, σT .
GSL_CONST_MKSA_DEBYE
The electric dipole moment of 1 Debye, D.

46.4 Measurement of Time
GSL_CONST_MKSA_MINUTE
The number of seconds in 1 minute.
GSL_CONST_MKSA_HOUR
The number of seconds in 1 hour.
GSL_CONST_MKSA_DAY
The number of seconds in 1 day.
GSL_CONST_MKSA_WEEKThe number of seconds in 1 week.

46.5 Imperial Units
GSL_CONST_MKSA_INCH
The length of 1 inch.
GSL_CONST_MKSA_FOOT
The length of 1 foot.
GSL_CONST_MKSA_YARD
The length of 1 yard.
GSL_CONST_MKSA_MILE
The length of 1 mile.
GSL_CONST_MKSA_MIL
The length of 1 mil (1/1000th of an inch).

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46.6 Speed and Nautical Units
GSL_CONST_MKSA_KILOMETERS_PER_HOUR
The speed of 1 kilometer per hour.
GSL_CONST_MKSA_MILES_PER_HOUR
The speed of 1 mile per hour.
GSL_CONST_MKSA_NAUTICAL_MILE
The length of 1 nautical mile.
GSL_CONST_MKSA_FATHOM
The length of 1 fathom.
GSL_CONST_MKSA_KNOTThe speed of 1 knot.

46.7 Printers Units
GSL_CONST_MKSA_POINT
The length of 1 printer’s point (1/72 inch).
GSL_CONST_MKSA_TEXPOINT
The length of 1 TeX point (1/72.27 inch).

46.8 Volume, Area and Length
GSL_CONST_MKSA_MICRON
The length of 1 micron.
GSL_CONST_MKSA_HECTARE
The area of 1 hectare.
GSL_CONST_MKSA_ACRE
The area of 1 acre.
GSL_CONST_MKSA_LITER
The volume of 1 liter.
GSL_CONST_MKSA_US_GALLON
The volume of 1 US gallon.
GSL_CONST_MKSA_CANADIAN_GALLON
The volume of 1 Canadian gallon.
GSL_CONST_MKSA_UK_GALLON
The volume of 1 UK gallon.
GSL_CONST_MKSA_QUART
The volume of 1 quart.
GSL_CONST_MKSA_PINT
The volume of 1 pint.

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46.9 Mass and Weight
GSL_CONST_MKSA_POUND_MASS
The mass of 1 pound.
GSL_CONST_MKSA_OUNCE_MASS
The mass of 1 ounce.
GSL_CONST_MKSA_TON
The mass of 1 ton.
GSL_CONST_MKSA_METRIC_TON
The mass of 1 metric ton (1000 kg).
GSL_CONST_MKSA_UK_TON
The mass of 1 UK ton.
GSL_CONST_MKSA_TROY_OUNCEThe mass of 1 troy ounce.
GSL_CONST_MKSA_CARATThe mass of 1 carat.
GSL_CONST_MKSA_GRAM_FORCEThe force of 1 gram weight.
GSL_CONST_MKSA_POUND_FORCEThe force of 1 pound weight.
GSL_CONST_MKSA_KILOPOUND_FORCEThe force of 1 kilopound weight.
GSL_CONST_MKSA_POUNDAL
The force of 1 poundal.

46.10 Thermal Energy and Power
GSL_CONST_MKSA_CALORIEThe energy of 1 calorie.
GSL_CONST_MKSA_BTU
The energy of 1 British Thermal Unit, btu.
GSL_CONST_MKSA_THERM
The energy of 1 Therm.
GSL_CONST_MKSA_HORSEPOWER
The power of 1 horsepower.

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46.11 Pressure
GSL_CONST_MKSA_BAR
The pressure of 1 bar.
GSL_CONST_MKSA_STD_ATMOSPHERE
The pressure of 1 standard atmosphere.
GSL_CONST_MKSA_TORR
The pressure of 1 torr.
GSL_CONST_MKSA_METER_OF_MERCURY
The pressure of 1 meter of mercury.
GSL_CONST_MKSA_INCH_OF_MERCURY
The pressure of 1 inch of mercury.
GSL_CONST_MKSA_INCH_OF_WATER
The pressure of 1 inch of water.
GSL_CONST_MKSA_PSI
The pressure of 1 pound per square inch.

46.12 Viscosity
GSL_CONST_MKSA_POISE
The dynamic viscosity of 1 poise.
GSL_CONST_MKSA_STOKESThe kinematic viscosity of 1 stokes.

46.13 Light and Illumination
GSL_CONST_MKSA_STILBThe luminance of 1 stilb.
GSL_CONST_MKSA_LUMEN
The luminous flux of 1 lumen.
GSL_CONST_MKSA_LUXThe illuminance of 1 lux.
GSL_CONST_MKSA_PHOT
The illuminance of 1 phot.
GSL_CONST_MKSA_FOOTCANDLE
The illuminance of 1 footcandle.
GSL_CONST_MKSA_LAMBERTThe luminance of 1 lambert.
GSL_CONST_MKSA_FOOTLAMBERTThe luminance of 1 footlambert.

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46.14 Radioactivity
GSL_CONST_MKSA_CURIE
The activity of 1 curie.
GSL_CONST_MKSA_ROENTGEN
The exposure of 1 roentgen.
GSL_CONST_MKSA_RAD
The absorbed dose of 1 rad.

46.15 Force and Energy
GSL_CONST_MKSA_NEWTONThe SI unit of force, 1 Newton.
GSL_CONST_MKSA_DYNE
The force of 1 Dyne = 10−5 Newton.
GSL_CONST_MKSA_JOULEThe SI unit of energy, 1 Joule.
GSL_CONST_MKSA_ERG
The energy 1 erg = 10−7 Joule.

46.16 Prefixes
These constants are dimensionless scaling factors.
GSL_CONST_NUM_YOTTA
1024
GSL_CONST_NUM_ZETTA
1021
GSL_CONST_NUM_EXA
1018
GSL_CONST_NUM_PETA
1015
GSL_CONST_NUM_TERA
1012
GSL_CONST_NUM_GIGA109
GSL_CONST_NUM_MEGA106
GSL_CONST_NUM_KILO
103
GSL_CONST_NUM_MILLI
10−3

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GSL_CONST_NUM_MICRO
10−6
GSL_CONST_NUM_NANO
10−9
GSL_CONST_NUM_PICO
10−12
GSL_CONST_NUM_FEMTO
10−15
GSL_CONST_NUM_ATTO
10−18
GSL_CONST_NUM_ZEPTO
10−21
GSL_CONST_NUM_YOCTO
10−24

46.17 Examples
The following program demonstrates the use of the physical constants in a calculation. In this case, the goal is tocalculate the range of light-travel times from Earth to Mars.
The required data is the average distance of each planet from the Sun in astronomical units (the eccentricities andinclinations of the orbits will be neglected for the purposes of this calculation). The average radius of the orbit of Marsis 1.52 astronomical units, and for the orbit of Earth it is 1 astronomical unit (by definition). These values are combinedwith the MKSA values of the constants for the speed of light and the length of an astronomical unit to produce a result
for the shortest and longest light-travel times in seconds. The figures are converted into minutes before being displayed.
#include <stdio.h>#include <gsl/gsl_const_mksa.h>
intmain (void){
double c = GSL_CONST_MKSA_SPEED_OF_LIGHT;
double au = GSL_CONST_MKSA_ASTRONOMICAL_UNIT;
double minutes = GSL_CONST_MKSA_MINUTE;/* distance stored in meters */double r_earth = 1.00 * au;
double r_mars = 1.52 * au;
double t_min, t_max;t_min = (r_mars - r_earth) / c;t_max = (r_mars + r_earth) / c;printf ("light travel time from Earth to Mars:\n");printf ("minimum = %.1f minutes\n", t_min / minutes);
printf ("maximum = %.1f minutes\n", t_max / minutes);(continues on next page)

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return 0;}
Here is the output from the program,light travel time from Earth to Mars:
minimum = 4.3 minutes
maximum = 21.0 minutes

46.18 References and Further Reading
The authoritative sources for physical constants are the 2006 CODATA recommended values, published in the article
below. Further information on the values of physical constants is also available from the NIST website.
• P.J. Mohr, B.N. Taylor, D.B. Newell, “CODATA Recommended Values of the Fundamental Physical Constants:
2006”, Reviews of Modern Physics, 80(2), pp. 633–730 (2008).
• http://www.physics.nist.gov/cuu/Constants/index.html• http://physics.nist.gov/Pubs/SP811/appenB9.html

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CHAPTERFORTYSEVEN

IEEE FLOATING-POINT ARITHMETIC

This chapter describes functions for examining the representation of floating point numbers and controlling the floating point environment of your program. The functions described in this chapter are declared in the header file
gsl_ieee_utils.h.

47.1 Representation of floating point numbers
The IEEE Standard for Binary Floating-Point Arithmetic defines binary formats for single and double precision numbers. Each number is composed of three parts: a sign bit (s), an exponent (E) and a fraction (f ). The numerical valueof the combination (s, E, f ) is given by the following formula,(−1)s (1 · f f f f f . . . )2E
The sign bit is either zero or one. The exponent ranges from a minimum value Emin to a maximum value Emax
depending on the precision. The exponent is converted to an unsigned number e, known as the biased exponent, forstorage by adding a bias parameter,e = E + bias
The sequence f f f f f... represents the digits of the binary fraction f . The binary digits are stored in normalized form,
by adjusting the exponent to give a leading digit of 1. Since the leading digit is always 1 for normalized numbers it isassumed implicitly and does not have to be stored. Numbers smaller than 2Emin are be stored in denormalized form
with a leading zero,(−1)s (0 · f f f f f . . . )2EminThis allows gradual underflow down to 2Emin −p for p bits of precision. A zero is encoded with the special exponentof 2Emin −1 and infinities with the exponent of 2Emax +1 .
The format for single precision numbers uses 32 bits divided in the following way:
seeeeeeeefffffffffffffffffffffffs = sign bit, 1 bite = exponent, 8 bits (E_min=-126, E_max=127, bias=127)
f = fraction, 23 bits
The format for double precision numbers uses 64 bits divided in the following way:637

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seeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffffffs = sign bit, 1 bite = exponent, 11 bits
f = fraction, 52 bits

(E_min=-1022, E_max=1023, bias=1023)

It is often useful to be able to investigate the behavior of a calculation at the bit-level and the library provides functions
for printing the IEEE representations in a human-readable form.
void gsl_ieee_fprintf_float(FILE *stream, const float *x)void gsl_ieee_fprintf_double(FILE *stream, const double *x)
These functions output a formatted version of the IEEE floating-point number pointed to by x to the stream
stream. A pointer is used to pass the number indirectly, to avoid any undesired promotion from float to
double. The output takes one of the following forms,
NaN
the Not-a-Number symbol
Inf, -Inf
positive or negative infinity
1.fffff...*2^E, -1.fffff...*2^E
a normalized floating point number
0.fffff...*2^E, -0.fffff...*2^E
a denormalized floating point number
0, -0
positive or negative zero
The output can be used directly in GNU Emacs Calc mode by preceding it with 2# to indicate binary.
void gsl_ieee_printf_float(const float *x)void gsl_ieee_printf_double(const double *x)
These functions output a formatted version of the IEEE floating-point number pointed to by x to the stream
stdout.
The following program demonstrates the use of the functions by printing the single and double precision representationsof the fraction 1/3. For comparison the representation of the value promoted from single to double precision is also
printed.
#include <stdio.h>#include <gsl/gsl_ieee_utils.h>
int
main (void){
float f = 1.0/3.0;
double d = 1.0/3.0;
double fd = f; /* promote from float to double */(continues on next page)

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printf (" f="); gsl_ieee_printf_float(&f);
printf ("\n");printf ("fd="); gsl_ieee_printf_double(&fd);
printf ("\n");printf (" d="); gsl_ieee_printf_double(&d);
printf ("\n");return 0;}
The binary representation of 1/3 is 0.01010101.... The output below shows that the IEEE format normalizes thisfraction to give a leading digit of 1:
f= 1.01010101010101010101011*2^-2
fd= 1.0101010101010101010101100000000000000000000000000000*2^-2
d= 1.0101010101010101010101010101010101010101010101010101*2^-2
The output also shows that a single-precision number is promoted to double-precision by adding zeros in the binary
representation.

47.2 Setting up your IEEE environment
The IEEE standard defines several modes for controlling the behavior of floating point operations. These modes specify
the important properties of computer arithmetic: the direction used for rounding (e.g. whether numbers should berounded up, down or to the nearest number), the rounding precision and how the program should handle arithmetic
exceptions, such as division by zero.
Many of these features can now be controlled via standard functions such as fpsetround(), which should be
used whenever they are available. Unfortunately in the past there has been no universal API for controlling their
behavior—each system has had its own low-level way of accessing them. To help you write portable programs
GSL allows you to specify modes in a platform-independent way using the environment variable GSL_IEEE_MODE.
The library then takes care of all the necessary machine-specific initializations for you when you call the function
gsl_ieee_env_setup().
GSL_IEEE_MODE
Environment variable which specifies IEEE mode.
void gsl_ieee_env_setup()This function reads the environment variable GSL_IEEE_MODE and attempts to set up the corresponding specified
IEEE modes. The environment variable should be a list of keywords, separated by commas, like this:GSL_IEEE_MODE = "keyword, keyword, ..."
where keyword is one of the following mode-names:
single-precision
double-precisionextended-precisionround-to-nearest
round-down(continues on next page)

47.2. Setting up your IEEE environment639

GNU Scientific Library, Release 2.7(continued from previous page)round-upround-to-zeromask-allmask-invalid
mask-denormalizedmask-division-by-zeromask-overflow
mask-underflow
trap-inexact
trap-common
If GSL_IEEE_MODE is empty or undefined then the function returns immediately and no attempt is made tochange the system’s IEEE mode. When the modes from GSL_IEEE_MODE are turned on the function prints a
short message showing the new settings to remind you that the results of the program will be affected.
If the requested modes are not supported by the platform being used then the function calls the error handler andreturns an error code of GSL_EUNSUP.
When options are specified using this method, the resulting mode is based on a default setting of the highest available precision (double precision or extended precision, depending on the platform) in round-to-nearest
mode, with all exceptions enabled apart from the INEXACT exception. The INEXACT exception is generated
whenever rounding occurs, so it must generally be disabled in typical scientific calculations. All other floatingpoint exceptions are enabled by default, including underflows and the use of denormalized numbers, for safety.
They can be disabled with the individual mask- settings or together using mask-all.
The following adjusted combination of modes is convenient for many purposes:
GSL_IEEE_MODE="double-precision,"\"mask-underflow,"\"mask-denormalized"
This choice ignores any errors relating to small numbers (either denormalized, or underflowing to zero) but traps
overflows, division by zero and invalid operations.
Note that on the x86 series of processors this function sets both the original x87 mode and the newer MXCSR
mode, which controls SSE floating-point operations. The SSE floating-point units do not have a precision-control
bit, and always work in double-precision. The single-precision and extended-precision keywords have no effectin this case.
To demonstrate the effects of different rounding modes consider the following program which computes e, the base ofnatural logarithms, by summing a rapidly-decreasing series,
1
1
1
1
1
+ + + + + ...
0! 1! 2! 3! 4!
= 2.71828182846...

e=#include <stdio.h>#include <gsl/gsl_math.h>#include <gsl/gsl_ieee_utils.h>
int
main (void){
double x = 1, oldsum = 0, sum = 0;
(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)

int i = 0;
gsl_ieee_env_setup (); /* read GSL_IEEE_MODE */
do
{
i++;
oldsum = sum;sum += x;x = x / i;printf ("i=%2d sum=%.18f error=%g\n",i, sum, sum - M_E);
if (i > 30)
break;}
while (sum != oldsum);
return 0;}
Here are the results of running the program in round-to-nearest mode. This is the IEEE default so it isn’t reallynecessary to specify it here:$ GSL_IEEE_MODE="round-to-nearest" ./a.out
i= 1 sum=1.000000000000000000 error=-1.71828
i= 2 sum=2.000000000000000000 error=-0.718282....
i=18 sum=2.718281828459045535 error=4.44089e-16
i=19 sum=2.718281828459045535 error=4.44089e-16
After nineteen terms the sum converges to within 4 × 10−16 of the correct value. If we now change the rounding modeto round-down the final result is less accurate:$ GSL_IEEE_MODE="round-down" ./a.out
i= 1 sum=1.000000000000000000 error=-1.71828....
i=19 sum=2.718281828459041094 error=-3.9968e-15
The result is about 4 × 10−15 below the correct value, an order of magnitude worse than the result obtained in theround-to-nearest mode.
If we change to rounding mode to round-up then the final result is higher than the correct value (when we add eachterm to the sum the final result is always rounded up, which increases the sum by at least one tick until the added term
underflows to zero). To avoid this problem we would need to use a safer converge criterion, such as while (fabs(sum- oldsum) > epsilon), with a suitably chosen value of epsilon.
Finally we can see the effect of computing the sum using single-precision rounding, in the default round-to-nearest
mode. In this case the program thinks it is still using double precision numbers but the CPU rounds the result of eachfloating point operation to single-precision accuracy. This simulates the effect of writing the program using singleprecision float variables instead of double variables. The iteration stops after about half the number of iterations
and the final result is much less accurate:

47.2. Setting up your IEEE environment641

GNU Scientific Library, Release 2.7$ GSL_IEEE_MODE="single-precision" ./a.out....
i=12 sum=2.718281984329223633 error=1.5587e-07
with an error of O(10−7 ), which corresponds to single precision accuracy (about 1 part in 107 ). Continuing theiterations further does not decrease the error because all the subsequent results are rounded to the same value.

47.3 References and Further Reading
The reference for the IEEE standard is,• ANSI/IEEE Std 754-1985, IEEE Standard for Binary Floating-Point Arithmetic.
A more pedagogical introduction to the standard can be found in the following paper,• David Goldberg: What Every Computer Scientist Should Know About Floating-Point Arithmetic. ACM Computing Surveys, Vol.: 23, No.: 1 (March 1991), pages 5–48.
• Corrigendum: ACM Computing Surveys, Vol.: 23, No.: 3 (September 1991), page 413. and see also the sections
by B. A. Wichmann and Charles B. Dunham in Surveyor’s Forum: “What Every Computer Scientist Should
Know About Floating-Point Arithmetic”. ACM Computing Surveys, Vol.: 24, No.: 3 (September 1992), page319.
A detailed textbook on IEEE arithmetic and its practical use is available from SIAM Press,• Michael L. Overton, Numerical Computing with IEEE Floating Point Arithmetic, SIAM Press, ISBN0898715717.

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Chapter 47. IEEE floating-point arithmetic

CHAPTERFORTYEIGHTDEBUGGING NUMERICAL PROGRAMSThis chapter describes some tips and tricks for debugging numerical programs which use GSL.

48.1 Using gdb
Any errors reported by the library are passed to the function gsl_error(). By running your programs under gdb andsetting a breakpoint in this function you can automatically catch any library errors. You can add a breakpoint for every
session by putting:break gsl_error
into your .gdbinit file in the directory where your program is started.
If the breakpoint catches an error then you can use a backtrace (bt) to see the call-tree, and the arguments which
possibly caused the error. By moving up into the calling function you can investigate the values of variables at thatpoint. Here is an example from the program fft/test_trap, which contains the following line:
status = gsl_fft_complex_wavetable_alloc (0, &complex_wavetable);
The function gsl_fft_complex_wavetable_alloc() takes the length of an FFT as its first argument. When thisline is executed an error will be generated because the length of an FFT is not allowed to be zero.
To debug this problem we start gdb, using the file .gdbinit to define a breakpoint in gsl_error():$ gdb test_trap
GDB is free software and you are welcome to distribute copies
of it under certain conditions; type "show copying" to see
the conditions. There is absolutely no warranty for GDB;
type "show warranty" for details. GDB 4.16 (i586-debian-linux),
Copyright 1996 Free Software Foundation, Inc.
Breakpoint 1 at 0x8050b1e: file error.c, line 14.
When we run the program this breakpoint catches the error and shows the reason for it:(gdb) runStarting program: test_trap
Breakpoint 1, gsl_error (reason=0x8052b0d"length n must be positive integer",(continues on next page)643

GNU Scientific Library, Release 2.7(continued from previous page)14

file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
at error.c:14
if (gsl_error_handler)

The first argument of gsl_error() is always a string describing the error. Now we can look at the backtrace to see
what caused the problem:(gdb) bt
#0 gsl_error (reason=0x8052b0d"length n must be positive integer",file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
at error.c:14#1 0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,wavetable=0xbffff778) at c_init.c:108#2 0x8048a00 in main (argc=1, argv=0xbffff9bc)
at test_trap.c:94#3 0x80488be in ___crt_dummy__ ()We can see that the error was generated in the function gsl_fft_complex_wavetable_alloc() when it was calledwith an argument of n = 0. The original call came from line 94 in the file test_trap.c.
By moving up to the level of the original call we can find the line that caused the error:(gdb) up
#1 0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,wavetable=0xbffff778) at c_init.c:108
108
GSL_ERROR ("length n must be positive integer", GSL_EDOM);(gdb) up
#2 0x8048a00 in main (argc=1, argv=0xbffff9bc)
at test_trap.c:94
94
status = gsl_fft_complex_wavetable_alloc (0,&complex_wavetable);Thus we have found the line that caused the problem. From this point we could also print out the values of othervariables such as complex_wavetable.

48.2 Examining floating point registers
The contents of floating point registers can be examined using the command info float (on supported platforms):(gdb) info float
st0: 0xc4018b895aa17a945000 Valid Normal -7.838871e+308
st1: 0x3ff9ea3f50e4d7275000 Valid Normal 0.0285946
st2: 0x3fe790c64ce27dad4800 Valid Normal 6.7415931e-08
st3: 0x3ffaa3ef0df6607d7800 Spec Normal 0.0400229
st4: 0x3c028000000000000000 Valid Normal 4.4501477e-308
st5: 0x3ffef5412c22219d9000 Zero Normal 0.9580257
st6: 0x3fff8000000000000000 Valid Normal 1
st7: 0xc4028b65a1f6d243c800 Valid Normal -1.566206e+309
fctrl: 0x0272 53 bit; NEAR; mask DENOR UNDER LOS;
fstat: 0xb9ba flags 0001; top 7; excep DENOR OVERF UNDER LOS(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)

ftag: 0x3fff
fip: 0x08048b5c
fcs: 0x051a0023
fopoff: 0x08086820
fopsel: 0x002b
Individual registers can be examined using the variables $reg, where reg is the register name:(gdb) p $st1
$1 = 0.02859464454261210347719

48.3 Handling floating point exceptions
It is possible to stop the program whenever a SIGFPE floating point exception occurs. This can be useful for finding thecause of an unexpected infinity or NaN. The current handler settings can be shown with the command info signal
SIGFPE:(gdb) info signal SIGFPE
Signal Stop Print Pass to program Description
SIGFPE Yes
Yes
Yes
Arithmetic exception
Unless the program uses a signal handler the default setting should be changed so that SIGFPE is not passed to theprogram, as this would cause it to exit. The command handle SIGFPE stop nopass prevents this:(gdb) handle SIGFPE stop nopass
Signal Stop Print Pass to program Description
SIGFPE Yes
Yes
NoArithmetic exception
Depending on the platform it may be necessary to instruct the kernel to generate signals for floating point exceptions.
For programs using GSL this can be achieved using the GSL_IEEE_MODE environment variable in conjunction with thefunction gsl_ieee_env_setup() as described in IEEE floating-point arithmetic:(gdb) set env GSL_IEEE_MODE=double-precision

48.4 GCC warning options for numerical programs
Writing reliable numerical programs in C requires great care. The following GCC warning options are recommended
when compiling numerical programs:
gcc -ansi -pedantic -Werror -Wall -W-Wmissing-prototypes -Wstrict-prototypes-Wconversion -Wshadow -Wpointer-arith-Wcast-qual -Wcast-align-Wwrite-strings -Wnested-externs-fshort-enums -fno-common -Dinline= -g -O2
For details of each option consult the manual Using and Porting GCC. The following table gives a brief explanation ofwhat types of errors these options catch.
-ansi -pedantic
48.3. Handling floating point exceptions

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Use ANSI C, and reject any non-ANSI extensions. These flags help in writing portable programs that willcompile on other systems.
-WerrorConsider warnings to be errors, so that compilation stops. This prevents warnings from scrolling off the topof the screen and being lost. You won’t be able to compile the program until it is completely warning-free.
-WallThis turns on a set of warnings for common programming problems. You need -Wall, but it is not enoughon its own.
-O2
Turn on optimization. The warnings for uninitialized variables in -Wall rely on the optimizer to analyze
the code. If there is no optimization then these warnings aren’t generated.
-W
This turns on some extra warnings not included in -Wall, such as missing return values and comparisons
between signed and unsigned integers.
-Wmissing-prototypes -Wstrict-prototypes
Warn if there are any missing or inconsistent prototypes. Without prototypes it is harder to detect problems
with incorrect arguments.
-Wconversion
The main use of this option is to warn about conversions from signed to unsigned integers. For example,
unsigned int x = -1. If you need to perform such a conversion you can use an explicit cast.
-Wshadow
This warns whenever a local variable shadows another local variable. If two variables have the same namethen it is a potential source of confusion.
-Wpointer-arith -Wcast-qual -Wcast-alignThese options warn if you try to do pointer arithmetic for types which don’t have a size, such as void,
if you remove a const cast from a pointer, or if you cast a pointer to a type which has a different size,
causing an invalid alignment.
-Wwrite-strings
This option gives string constants a const qualifier so that it will be a compile-time error to attempt tooverwrite them.
-fshort-enums
This option makes the type of enum as short as possible. Normally this makes an enum different froman int. Consequently any attempts to assign a pointer-to-int to a pointer-to-enum will generate a castalignment warning.
-fno-common
This option prevents global variables being simultaneously defined in different object files (you get an error
at link time). Such a variable should be defined in one file and referred to in other files with an extern
declaration.
-Wnested-externs
This warns if an extern declaration is encountered within a function.
-Dinline=

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GNU Scientific Library, Release 2.7

The inline keyword is not part of ANSI C. Thus if you want to use -ansi with a program which uses
inline functions you can use this preprocessor definition to remove the inline keywords.
-g
It always makes sense to put debugging symbols in the executable so that you can debug it using gdb. The
only effect of debugging symbols is to increase the size of the file, and you can use the strip command
to remove them later if necessary.

48.5 References and Further Reading
The following books are essential reading for anyone writing and debugging numerical programs with gcc and gdb.
• R.M. Stallman, Using and Porting GNU CC, Free Software Foundation, ISBN 1882114388• R.M. Stallman, R.H. Pesch, Debugging with GDB: The GNU Source-Level Debugger, Free Software Foundation,
ISBN 1882114779
For a tutorial introduction to the GNU C Compiler and related programs, see• B.J. Gough, http://www.network-theory.co.uk/gcc/intro/,’ An Introduction to GCC, Network Theory Ltd, ISBN
0954161793

48.5. References and Further Reading647

GNU Scientific Library, Release 2.7648

Chapter 48. Debugging Numerical Programs

CHAPTERFORTYNINECONTRIBUTORS TO GSL(See the AUTHORS file in the distribution for up-to-date information.)
Mark Galassi
Conceived GSL (with James Theiler) and wrote the design document. Wrote the simulated annealing
package and the relevant chapter in the manual.
James Theiler
Conceived GSL (with Mark Galassi). Wrote the random number generators and the relevant chapter inthis manual.
Jim Davies
Wrote the statistical routines and the relevant chapter in this manual.
Brian Gough
FFTs, numerical integration, random number generators and distributions, root finding, minimization andfitting, polynomial solvers, complex numbers, physical constants, permutations, vector and matrix functions, histograms, statistics, ieee-utils, revised CBLAS Level 2 & 3, matrix decompositions, eigensystems,
cumulative distribution functions, testing, documentation and releases.
Reid Priedhorsky
Wrote and documented the initial version of the root finding routines while at Los Alamos National Laboratory, Mathematical Modeling and Analysis Group.
Gerard Jungman
Special Functions, Series acceleration, ODEs, BLAS, Linear Algebra, Eigensystems, Hankel Transforms.
Patrick Alken
Implementation of nonsymmetric and generalized eigensystems, B-splines, linear and nonlinear leastsquares, matrix decompositions, associated Legendre functions, running statistics, sparse matrices, andsparse linear algebra.
Mike Booth
Wrote the Monte Carlo library.
Jorma Olavi Tähtinen
Wrote the initial complex arithmetic functions.
Thomas Walter
Wrote the initial heapsort routines and Cholesky decomposition.
Fabrice Rossi

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Multidimensional minimization.
Carlo Perassi
Implementation of the random number generators in Knuth’s Seminumerical Algorithms, 3rd Ed.
Szymon Jaroszewicz
Wrote the routines for generating combinations.
Nicolas Darnis
Wrote the cyclic functions and the initial functions for canonical permutations.
Jason H. Stover
Wrote the major cumulative distribution functions.
Ivo Alxneit
Wrote the routines for wavelet transforms.
Tuomo Keskitalo
Improved the implementation of the ODE solvers and wrote the ode-initval2 routines.
Lowell Johnson
Implementation of the Mathieu functions.
Rhys Ulerich
Wrote the multiset routines.
Pavel Holoborodko
Wrote the fixed order Gauss-Legendre quadrature routines.
Pedro Gonnet
Wrote the CQUAD integration routines.
Thanks to Nigel Lowry for help in proofreading the manual.
The non-symmetric eigensystems routines contain code based on the LAPACK linear algebra library. LAPACK is
distributed under the following license:
Copyright (c) 1992-2006 The University of Tennessee.

All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions aremet:* Redistributions of source code must retain the above copyrightnotice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyrightnotice, this list of conditions and the following disclaimer listed
in this license in the documentation and/or other materials
provided with the distribution.
* Neither the name of the copyright holders nor the names of its
contributors may be used to endorse or promote products derived fromthis software without specific prior written permission.
(continues on next page)

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GNU Scientific Library, Release 2.7(continued from previous page)

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHTOWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

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CHAPTERFIFTY

AUTOCONF MACROS

For applications using autoconf the standard macro AC_CHECK_LIB can be used to link with GSL automatically froma configure script. The library itself depends on the presence of a CBLAS and math library as well, so these must also
be located before linking with the main libgsl file. The following commands should be placed in the configure.ac
file to perform these tests:
AC_CHECK_LIB([m],[cos])
AC_CHECK_LIB([gslcblas],[cblas_dgemm])
AC_CHECK_LIB([gsl],[gsl_blas_dgemm])
It is important to check for libm and libgslcblas before libgsl, otherwise the tests will fail. Assuming the libraries
are found the output during the configure stage looks like this:checking for cos in -lm... yeschecking for cblas_dgemm in -lgslcblas... yeschecking for gsl_blas_dgemm in -lgsl... yes
If the library is found then the tests will define the macros HAVE_LIBGSL, HAVE_LIBGSLCBLAS, HAVE_LIBM and add
the options -lgsl -lgslcblas -lm to the variable LIBS.
The tests above will find any version of the library. They are suitable for general use, where the versions of the functions
are not important. An alternative macro is available in the file gsl.m4 to test for a specific version of the library. To
use this macro simply add the following line to your configure.in file instead of the tests above:
AX_PATH_GSL(GSL_VERSION,[action-if-found],[action-if-not-found])
The argument GSL_VERSION should be the two or three digit major.minor or major.minor.micro version numberof the release you require. A suitable choice for action-if-not-found is:
AC_MSG_ERROR(could not find required version of GSL)
Then you can add the variables GSL_LIBS and GSL_CFLAGS to your Makefile.am files to obtain the correct compiler flags. GSL_LIBS is equal to the output of the gsl-config --libs command and GSL_CFLAGS is equal to
gsl-config --cflags command. For example:
libfoo_la_LDFLAGS = -lfoo $(GSL_LIBS) -lgslcblas
Note that the macro AX_PATH_GSL needs to use the C compiler so it should appear in the configure.in file before
the macro AC_LANG_CPLUSPLUS for programs that use C++.
To test for inline the following test should be placed in your configure.in file:653

GNU Scientific Library, Release 2.7

AC_C_INLINE
if test "$ac_cv_c_inline" != no ; then
AC_DEFINE(HAVE_INLINE,1)
AC_SUBST(HAVE_INLINE)fi
and the macro will then be defined in the compilation flags or by including the file config.h before any library headers.
The following autoconf test will check for extern inline:
dnl Check for "extern inline", using a modified version
dnl of the test for AC_C_INLINE from acspecific.mt
dnl
AC_CACHE_CHECK([for extern inline], ac_cv_c_extern_inline,[ac_cv_c_extern_inline=no
AC_TRY_COMPILE([extern $ac_cv_c_inline double foo(double x);
extern $ac_cv_c_inline double foo(double x) { return x+1.0; };double foo (double x) { return x + 1.0; };],
[ foo(1.0) ],[ac_cv_c_extern_inline="yes"])])
if test "$ac_cv_c_extern_inline" != no ; then
AC_DEFINE(HAVE_INLINE,1)
AC_SUBST(HAVE_INLINE)fi
The substitution of portability functions can be made automatically if you use autoconf. For example, to test whether
the BSD function hypot() is available you can include the following line in the configure file configure.in for yourapplication:
AC_CHECK_FUNCS(hypot)
and place the following macro definitions in the file config.h.in:/* Substitute gsl_hypot for missing system hypot */#ifndef HAVE_HYPOT#define hypot gsl_hypot#endif
The application source files can then use the include command #include <config.h> to substitute gsl_hypot()
for each occurrence of hypot() when hypot() is not available.

654

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CHAPTERFIFTYONEGSL CBLAS LIBRARY

The prototypes for the low-level CBLAS functions are declared in the file gsl_cblas.h. For the definition of thefunctions consult the documentation available from Netlib (see BLAS References and Further Reading).

51.1 Level 1
float cblas_sdsdot(const int N, const float alpha, const float *x, const int incx, const float *y, const int incy)
double cblas_dsdot(const int N, const float *x, const int incx, const float *y, const int incy)
float cblas_sdot(const int N, const float *x, const int incx, const float *y, const int incy)
double cblas_ddot(const int N, const double *x, const int incx, const double *y, const int incy)
void cblas_cdotu_sub(const int N, const void *x, const int incx, const void *y, const int incy, void *dotu)
void cblas_cdotc_sub(const int N, const void *x, const int incx, const void *y, const int incy, void *dotc)
void cblas_zdotu_sub(const int N, const void *x, const int incx, const void *y, const int incy, void *dotu)
void cblas_zdotc_sub(const int N, const void *x, const int incx, const void *y, const int incy, void *dotc)
float cblas_snrm2(const int N, const float *x, const int incx)
float cblas_sasum(const int N, const float *x, const int incx)
double cblas_dnrm2(const int N, const double *x, const int incx)
double cblas_dasum(const int N, const double *x, const int incx)
float cblas_scnrm2(const int N, const void *x, const int incx)
float cblas_scasum(const int N, const void *x, const int incx)655

GNU Scientific Library, Release 2.7

double cblas_dznrm2(const int N, const void *x, const int incx)
double cblas_dzasum(const int N, const void *x, const int incx)
CBLAS_INDEX cblas_isamax(const int N, const float *x, const int incx)
CBLAS_INDEX cblas_idamax(const int N, const double *x, const int incx)
CBLAS_INDEX cblas_icamax(const int N, const void *x, const int incx)
CBLAS_INDEX cblas_izamax(const int N, const void *x, const int incx)
void cblas_sswap(const int N, float *x, const int incx, float *y, const int incy)
void cblas_scopy(const int N, const float *x, const int incx, float *y, const int incy)
void cblas_saxpy(const int N, const float alpha, const float *x, const int incx, float *y, const int incy)
void cblas_dswap(const int N, double *x, const int incx, double *y, const int incy)
void cblas_dcopy(const int N, const double *x, const int incx, double *y, const int incy)
void cblas_daxpy(const int N, const double alpha, const double *x, const int incx, double *y, const int incy)
void cblas_cswap(const int N, void *x, const int incx, void *y, const int incy)
void cblas_ccopy(const int N, const void *x, const int incx, void *y, const int incy)
void cblas_caxpy(const int N, const void *alpha, const void *x, const int incx, void *y, const int incy)
void cblas_zswap(const int N, void *x, const int incx, void *y, const int incy)
void cblas_zcopy(const int N, const void *x, const int incx, void *y, const int incy)
void cblas_zaxpy(const int N, const void *alpha, const void *x, const int incx, void *y, const int incy)
void cblas_srotg(float *a, float *b, float *c, float *s)void cblas_srotmg(float *d1, float *d2, float *b1, const float b2, float *P)
void cblas_srot(const int N, float *x, const int incx, float *y, const int incy, const float c, const float s)

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void cblas_srotm(const int N, float *x, const int incx, float *y, const int incy, const float *P)
void cblas_drotg(double *a, double *b, double *c, double *s)void cblas_drotmg(double *d1, double *d2, double *b1, const double b2, double *P)
void cblas_drot(const int N, double *x, const int incx, double *y, const int incy, const double c, const double s)void cblas_drotm(const int N, double *x, const int incx, double *y, const int incy, const double *P)
void cblas_sscal(const int N, const float alpha, float *x, const int incx)
void cblas_dscal(const int N, const double alpha, double *x, const int incx)
void cblas_cscal(const int N, const void *alpha, void *x, const int incx)
void cblas_zscal(const int N, const void *alpha, void *x, const int incx)
void cblas_csscal(const int N, const float alpha, void *x, const int incx)
void cblas_zdscal(const int N, const double alpha, void *x, const int incx)

51.2 Level 2
void cblas_sgemv(const enum CBLAS_ORDER order, const enum CBLAS_TRANSPOSE TransA, const int M,
const int N, const float alpha, const float *A, const int lda, const float *x, const int incx, const
float beta, float *y, const int incy)
void cblas_sgbmv(const enum CBLAS_ORDER order, const enum CBLAS_TRANSPOSE TransA, const int M,
const int N, const int KL, const int KU, const float alpha, const float *A, const int lda, const float*x, const int incx, const float beta, float *y, const int incy)
void cblas_strmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const float *A,
const int lda, float *x, const int incx)
void cblas_stbmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const int K, const
float *A, const int lda, float *x, const int incx)
void cblas_stpmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const float *Ap,float *x, const int incx)

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void cblas_strsv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const float *A,
const int lda, float *x, const int incx)
void cblas_stbsv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const int K, const
float *A, const int lda, float *x, const int incx)
void cblas_stpsv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const float *Ap,float *x, const int incx)
void cblas_dgemv(const enum CBLAS_ORDER order, const enum CBLAS_TRANSPOSE TransA, const int M,
const int N, const double alpha, const double *A, const int lda, const double *x, const int incx,
const double beta, double *y, const int incy)
void cblas_dgbmv(const enum CBLAS_ORDER order, const enum CBLAS_TRANSPOSE TransA, const int M,
const int N, const int KL, const int KU, const double alpha, const double *A, const int lda, const
double *x, const int incx, const double beta, double *y, const int incy)
void cblas_dtrmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const double *A,
const int lda, double *x, const int incx)
void cblas_dtbmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const int K, const
double *A, const int lda, double *x, const int incx)
void cblas_dtpmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const double*Ap, double *x, const int incx)
void cblas_dtrsv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const double *A,
const int lda, double *x, const int incx)
void cblas_dtbsv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const int K, const
double *A, const int lda, double *x, const int incx)
void cblas_dtpsv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const double*Ap, double *x, const int incx)
void cblas_cgemv(const enum CBLAS_ORDER order, const enum CBLAS_TRANSPOSE TransA, const int M,
const int N, const void *alpha, const void *A, const int lda, const void *x, const int incx, const
void *beta, void *y, const int incy)

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void cblas_cgbmv(const enum CBLAS_ORDER order, const enum CBLAS_TRANSPOSE TransA, const int M,
const int N, const int KL, const int KU, const void *alpha, const void *A, const int lda, const
void *x, const int incx, const void *beta, void *y, const int incy)
void cblas_ctrmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const void *A,
const int lda, void *x, const int incx)
void cblas_ctbmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const int K, const
void *A, const int lda, void *x, const int incx)
void cblas_ctpmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const void *Ap,void *x, const int incx)
void cblas_ctrsv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const void *A,
const int lda, void *x, const int incx)
void cblas_ctbsv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const int K, const
void *A, const int lda, void *x, const int incx)
void cblas_ctpsv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const void *Ap,void *x, const int incx)
void cblas_zgemv(const enum CBLAS_ORDER order, const enum CBLAS_TRANSPOSE TransA, const int M,
const int N, const void *alpha, const void *A, const int lda, const void *x, const int incx, const
void *beta, void *y, const int incy)
void cblas_zgbmv(const enum CBLAS_ORDER order, const enum CBLAS_TRANSPOSE TransA, const int M,
const int N, const int KL, const int KU, const void *alpha, const void *A, const int lda, const
void *x, const int incx, const void *beta, void *y, const int incy)
void cblas_ztrmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const void *A,
const int lda, void *x, const int incx)
void cblas_ztbmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const int K, const
void *A, const int lda, void *x, const int incx)
void cblas_ztpmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const void *Ap,void *x, const int incx)

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void cblas_ztrsv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const void *A,
const int lda, void *x, const int incx)
void cblas_ztbsv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const int K, const
void *A, const int lda, void *x, const int incx)
void cblas_ztpsv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const void *Ap,void *x, const int incx)
void cblas_ssymv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const floatalpha, const float *A, const int lda, const float *x, const int incx, const float beta, float *y, const
int incy)
void cblas_ssbmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const int K,
const float alpha, const float *A, const int lda, const float *x, const int incx, const float beta, float*y, const int incy)
void cblas_sspmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const floatalpha, const float *Ap, const float *x, const int incx, const float beta, float *y, const int incy)
void cblas_sger(const enum CBLAS_ORDER order, const int M, const int N, const float alpha, const float *x,
const int incx, const float *y, const int incy, float *A, const int lda)
void cblas_ssyr(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const floatalpha, const float *x, const int incx, float *A, const int lda)
void cblas_sspr(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const floatalpha, const float *x, const int incx, float *Ap)void cblas_ssyr2(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const floatalpha, const float *x, const int incx, const float *y, const int incy, float *A, const int lda)
void cblas_sspr2(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const floatalpha, const float *x, const int incx, const float *y, const int incy, float *A)
void cblas_dsymv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const doublealpha, const double *A, const int lda, const double *x, const int incx, const double beta, double*y, const int incy)
void cblas_dsbmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const int K,
const double alpha, const double *A, const int lda, const double *x, const int incx, const double
beta, double *y, const int incy)

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void cblas_dspmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const doublealpha, const double *Ap, const double *x, const int incx, const double beta, double *y, const int
incy)
void cblas_dger(const enum CBLAS_ORDER order, const int M, const int N, const double alpha, const double *x,
const int incx, const double *y, const int incy, double *A, const int lda)
void cblas_dsyr(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const doublealpha, const double *x, const int incx, double *A, const int lda)
void cblas_dspr(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const doublealpha, const double *x, const int incx, double *Ap)void cblas_dsyr2(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const doublealpha, const double *x, const int incx, const double *y, const int incy, double *A, const int lda)
void cblas_dspr2(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const doublealpha, const double *x, const int incx, const double *y, const int incy, double *A)
void cblas_chemv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const void*alpha, const void *A, const int lda, const void *x, const int incx, const void *beta, void *y, const
int incy)
void cblas_chbmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const int K,
const void *alpha, const void *A, const int lda, const void *x, const int incx, const void *beta,
void *y, const int incy)
void cblas_chpmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const void*alpha, const void *Ap, const void *x, const int incx, const void *beta, void *y, const int incy)
void cblas_cgeru(const enum CBLAS_ORDER order, const int M, const int N, const void *alpha, const void *x,
const int incx, const void *y, const int incy, void *A, const int lda)
void cblas_cgerc(const enum CBLAS_ORDER order, const int M, const int N, const void *alpha, const void *x,
const int incx, const void *y, const int incy, void *A, const int lda)
void cblas_cher(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const floatalpha, const void *x, const int incx, void *A, const int lda)
void cblas_chpr(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const floatalpha, const void *x, const int incx, void *A)
void cblas_cher2(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const void*alpha, const void *x, const int incx, const void *y, const int incy, void *A, const int lda)

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void cblas_chpr2(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const void*alpha, const void *x, const int incx, const void *y, const int incy, void *Ap)void cblas_zhemv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const void*alpha, const void *A, const int lda, const void *x, const int incx, const void *beta, void *y, const
int incy)
void cblas_zhbmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const int K,
const void *alpha, const void *A, const int lda, const void *x, const int incx, const void *beta,
void *y, const int incy)
void cblas_zhpmv(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const void*alpha, const void *Ap, const void *x, const int incx, const void *beta, void *y, const int incy)
void cblas_zgeru(const enum CBLAS_ORDER order, const int M, const int N, const void *alpha, const void *x,
const int incx, const void *y, const int incy, void *A, const int lda)
void cblas_zgerc(const enum CBLAS_ORDER order, const int M, const int N, const void *alpha, const void *x,
const int incx, const void *y, const int incy, void *A, const int lda)
void cblas_zher(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const doublealpha, const void *x, const int incx, void *A, const int lda)
void cblas_zhpr(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const doublealpha, const void *x, const int incx, void *A)
void cblas_zher2(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const void*alpha, const void *x, const int incx, const void *y, const int incy, void *A, const int lda)
void cblas_zhpr2(const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const void*alpha, const void *x, const int incx, const void *y, const int incy, void *Ap)51.3 Level 3
void cblas_sgemm(const enum CBLAS_ORDER Order, const enum CBLAS_TRANSPOSE TransA, const enum
CBLAS_TRANSPOSE TransB, const int M, const int N, const int K, const float alpha, const
float *A, const int lda, const float *B, const int ldb, const float beta, float *C, const int ldc)
void cblas_ssymm(const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum
CBLAS_UPLO Uplo, const int M, const int N, const float alpha, const float *A, const int lda,
const float *B, const int ldb, const float beta, float *C, const int ldc)
void cblas_ssyrk(const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE Trans, const int N, const int K, const float alpha, const float *A, const int
lda, const float beta, float *C, const int ldc)

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void cblas_ssyr2k(const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE Trans, const int N, const int K, const float alpha, const float *A, const
int lda, const float *B, const int ldb, const float beta, float *C, const int ldc)
void cblas_strmm(const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum
CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG
Diag, const int M, const int N, const float alpha, const float *A, const int lda, float *B, const int
ldb)
void cblas_strsm(const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum
CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG
Diag, const int M, const int N, const float alpha, const float *A, const int lda, float *B, const int
ldb)
void cblas_dgemm(const enum CBLAS_ORDER Order, const enum CBLAS_TRANSPOSE TransA, const enum
CBLAS_TRANSPOSE TransB, const int M, const int N, const int K, const double alpha, const
double *A, const int lda, const double *B, const int ldb, const double beta, double *C, const int
ldc)
void cblas_dsymm(const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum
CBLAS_UPLO Uplo, const int M, const int N, const double alpha, const double *A, const int
lda, const double *B, const int ldb, const double beta, double *C, const int ldc)
void cblas_dsyrk(const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE Trans, const int N, const int K, const double alpha, const double *A,
const int lda, const double beta, double *C, const int ldc)
void cblas_dsyr2k(const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE Trans, const int N, const int K, const double alpha, const double *A,
const int lda, const double *B, const int ldb, const double beta, double *C, const int ldc)
void cblas_dtrmm(const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum
CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG
Diag, const int M, const int N, const double alpha, const double *A, const int lda, double *B,
const int ldb)
void cblas_dtrsm(const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum
CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG
Diag, const int M, const int N, const double alpha, const double *A, const int lda, double *B,
const int ldb)
void cblas_cgemm(const enum CBLAS_ORDER Order, const enum CBLAS_TRANSPOSE TransA, const enum
CBLAS_TRANSPOSE TransB, const int M, const int N, const int K, const void *alpha, const
void *A, const int lda, const void *B, const int ldb, const void *beta, void *C, const int ldc)
void cblas_csymm(const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum
CBLAS_UPLO Uplo, const int M, const int N, const void *alpha, const void *A, const int lda,
const void *B, const int ldb, const void *beta, void *C, const int ldc)

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void cblas_csyrk(const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE Trans, const int N, const int K, const void *alpha, const void *A, const
int lda, const void *beta, void *C, const int ldc)
void cblas_csyr2k(const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE Trans, const int N, const int K, const void *alpha, const void *A, const
int lda, const void *B, const int ldb, const void *beta, void *C, const int ldc)
void cblas_ctrmm(const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum
CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG
Diag, const int M, const int N, const void *alpha, const void *A, const int lda, void *B, const int
ldb)
void cblas_ctrsm(const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum
CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG
Diag, const int M, const int N, const void *alpha, const void *A, const int lda, void *B, const int
ldb)
void cblas_zgemm(const enum CBLAS_ORDER Order, const enum CBLAS_TRANSPOSE TransA, const enum
CBLAS_TRANSPOSE TransB, const int M, const int N, const int K, const void *alpha, const
void *A, const int lda, const void *B, const int ldb, const void *beta, void *C, const int ldc)
void cblas_zsymm(const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum
CBLAS_UPLO Uplo, const int M, const int N, const void *alpha, const void *A, const int lda,
const void *B, const int ldb, const void *beta, void *C, const int ldc)
void cblas_zsyrk(const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE Trans, const int N, const int K, const void *alpha, const void *A, const
int lda, const void *beta, void *C, const int ldc)
void cblas_zsyr2k(const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE Trans, const int N, const int K, const void *alpha, const void *A, const
int lda, const void *B, const int ldb, const void *beta, void *C, const int ldc)
void cblas_ztrmm(const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum
CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG
Diag, const int M, const int N, const void *alpha, const void *A, const int lda, void *B, const int
ldb)
void cblas_ztrsm(const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum
CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG
Diag, const int M, const int N, const void *alpha, const void *A, const int lda, void *B, const int
ldb)
void cblas_chemm(const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum
CBLAS_UPLO Uplo, const int M, const int N, const void *alpha, const void *A, const int lda,
const void *B, const int ldb, const void *beta, void *C, const int ldc)

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void cblas_cherk(const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE Trans, const int N, const int K, const float alpha, const void *A, const int
lda, const float beta, void *C, const int ldc)
void cblas_cher2k(const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE Trans, const int N, const int K, const void *alpha, const void *A, const
int lda, const void *B, const int ldb, const float beta, void *C, const int ldc)
void cblas_zhemm(const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum
CBLAS_UPLO Uplo, const int M, const int N, const void *alpha, const void *A, const int lda,
const void *B, const int ldb, const void *beta, void *C, const int ldc)
void cblas_zherk(const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE Trans, const int N, const int K, const double alpha, const void *A, const
int lda, const double beta, void *C, const int ldc)
void cblas_zher2k(const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum
CBLAS_TRANSPOSE Trans, const int N, const int K, const void *alpha, const void *A, const
int lda, const void *B, const int ldb, const double beta, void *C, const int ldc)
void cblas_xerbla(int p, const char *rout, const char *form, ...)

51.4 Examples
The following program computes the product of two matrices using the Level-3 BLAS function SGEMM,






1011 1012
367.76 368.12
0.11 0.12 0.13 1021 1022  =674.06 674.72
0.21 0.22 0.23
1031 1032
The matrices are stored in row major order but could be stored in column major order if the first argument of the callto cblas_sgemm() was changed to CblasColMajor.
#include <stdio.h>#include <gsl/gsl_cblas.h>intmain (void){
int lda = 3;
float A[] = { 0.11, 0.12, 0.13,
0.21, 0.22, 0.23 };int ldb = 2;
float B[] = { 1011, 1012,
1021, 1022,
1031, 1032 };(continues on next page)51.4. Examples

665

GNU Scientific Library, Release 2.7(continued from previous page)

int ldc = 2;
float C[] = { 0.00, 0.00,
0.00, 0.00 };/* Compute C = A B */cblas_sgemm (CblasRowMajor,
CblasNoTrans, CblasNoTrans, 2, 2, 3,
1.0, A, lda, B, ldb, 0.0, C, ldc);
printf ("[ %g, %g\n", C[0], C[1]);printf (" %g, %g ]\n", C[2], C[3]);return 0;}
To compile the program use the following command line:$ gcc -Wall demo.c -lgslcblas
There is no need to link with the main library -lgsl in this case as the CBLAS library is an independent unit. Here is
the output from the program,[ 367.76, 368.12
674.06, 674.72 ]666

Chapter 51. GSL CBLAS LibraryCHAPTERFIFTYTWO

GNU GENERAL PUBLIC LICENSE

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state the exclusion of warranty; and each file should have at least
the "copyright" line and a pointer to where the full notice is found.
<one line to give the program's name and a brief idea of what it does.>
Copyright (C) <year> <name of author>
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public Licensealong with this program. If not, see <http://www.gnu.org/licenses/>.
Also add information on how to contact you by electronic and paper mail.
If the program does terminal interaction, make it output a short
notice like this when it starts in an interactive mode:<program> Copyright (C) <year> <name of author>
This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
This is free software, and you are welcome to redistribute it
under certain conditions; type `show c' for details.
The hypothetical commands `show w' and `show c' should show the appropriate
parts of the General Public License. Of course, your program's commands
might be different; for a GUI interface, you would use an "about box".
You should also get your employer (if you work as a programmer) or school,(continues on next page)679

GNU Scientific Library, Release 2.7(continued from previous page)

if any, to sign a "copyright disclaimer" for the program, if necessary.
For more information on this, and how to apply and follow the GNU GPL, see<http://www.gnu.org/licenses/>.
The GNU General Public License does not permit incorporating your program
into proprietary programs. If your program is a subroutine library, you
may consider it more useful to permit linking proprietary applications withthe library. If this is what you want to do, use the GNU Lesser General
Public License instead of this License. But first, please read
<http://www.gnu.org/philosophy/why-not-lgpl.html>.

680

Chapter 52. GNU General Public LicenseCHAPTERFIFTYTHREE

GNU FREE DOCUMENTATION LICENSE

GNU Free Documentation License
Version 1.3, 3 November 2008

Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc.
<http://fsf.org/>
Everyone is permitted to copy and distribute verbatim copiesof this license document, but changing it is not allowed.
0. PREAMBLE
The purpose of this License is to make a manual, textbook, or otherfunctional and useful document "free" in the sense of freedom: toassure everyone the effective freedom to copy and redistribute it,with or without modifying it, either commercially or noncommercially.
Secondarily, this License preserves for the author and publisher a way
to get credit for their work, while not being considered responsible
for modifications made by others.
This License is a kind of "copyleft", which means that derivative
works of the document must themselves be free in the same sense. It
complements the GNU General Public License, which is a copyleftlicense designed for free software.
We have designed this License in order to use it for manuals for free
software, because free software needs free documentation: a freeprogram should come with manuals providing the same freedoms that thesoftware does. But this License is not limited to software manuals;
it can be used for any textual work, regardless of subject matter orwhether it is published as a printed book. We recommend this License
principally for works whose purpose is instruction or reference.

1. APPLICABILITY AND DEFINITIONSThis License applies to any manual or other work, in any medium, that
contains a notice placed by the copyright holder saying it can be
distributed under the terms of this License. Such a notice grants aworld-wide, royalty-free license, unlimited in duration, to use that(continues on next page)681

GNU Scientific Library, Release 2.7(continued from previous page)work under the conditions stated herein. The "Document", below,
refers to any such manual or work. Any member of the public is alicensee, and is addressed as "you". You accept the license if you
copy, modify or distribute the work in a way requiring permission
under copyright law.
A "Modified Version" of the Document means any work containing theDocument or a portion of it, either copied verbatim, or withmodifications and/or translated into another language.
A "Secondary Section" is a named appendix or a front-matter section of
the Document that deals exclusively with the relationship of the
publishers or authors of the Document to the Document's overall
subject (or to related matters) and contains nothing that could fall
directly within that overall subject. (Thus, if the Document is inpart a textbook of mathematics, a Secondary Section may not explain
any mathematics.) The relationship could be a matter of historicalconnection with the subject or with related matters, or of legal,commercial, philosophical, ethical or political position regardingthem.
The "Invariant Sections" are certain Secondary Sections whose titles
are designated, as being those of Invariant Sections, in the noticethat says that the Document is released under this License. If asection does not fit the above definition of Secondary then it is notallowed to be designated as Invariant. The Document may contain zero
Invariant Sections. If the Document does not identify any Invariant
Sections then there are none.
The "Cover Texts" are certain short passages of text that are listed,
as Front-Cover Texts or Back-Cover Texts, in the notice that says thatthe Document is released under this License. A Front-Cover Text may
be at most 5 words, and a Back-Cover Text may be at most 25 words.
A "Transparent" copy of the Document means a machine-readable copy,
represented in a format whose specification is available to thegeneral public, that is suitable for revising the document
straightforwardly with generic text editors or (for images composed ofpixels) generic paint programs or (for drawings) some widely available
drawing editor, and that is suitable for input to text formatters orfor automatic translation to a variety of formats suitable for input
to text formatters. A copy made in an otherwise Transparent fileformat whose markup, or absence of markup, has been arranged to thwart
or discourage subsequent modification by readers is not Transparent.
An image format is not Transparent if used for any substantial amountof text. A copy that is not "Transparent" is called "Opaque".
Examples of suitable formats for Transparent copies include plain
ASCII without markup, Texinfo input format, LaTeX input format, SGMLor XML using a publicly available DTD, and standard-conforming simple
HTML, PostScript or PDF designed for human modification. Examples oftransparent image formats include PNG, XCF and JPG. Opaque formats(continues on next page)682

Chapter 53. GNU Free Documentation LicenseGNU Scientific Library, Release 2.7(continued from previous page)

include proprietary formats that can be read and edited only by
proprietary word processors, SGML or XML for which the DTD and/or
processing tools are not generally available, and themachine-generated HTML, PostScript or PDF produced by some word
processors for output purposes only.
The "Title Page" means, for a printed book, the title page itself,plus such following pages as are needed to hold, legibly, the material
this License requires to appear in the title page. For works informats which do not have any title page as such, "Title Page" means
the text near the most prominent appearance of the work's title,
preceding the beginning of the body of the text.
The "publisher" means any person or entity that distributes copies of
the Document to the public.
A section "Entitled XYZ" means a named subunit of the Document whose
title either is precisely XYZ or contains XYZ in parentheses following
text that translates XYZ in another language. (Here XYZ stands for a
specific section name mentioned below, such as "Acknowledgements","Dedications", "Endorsements", or "History".) To "Preserve the Title"of such a section when you modify the Document means that it remains asection "Entitled XYZ" according to this definition.
The Document may include Warranty Disclaimers next to the notice whichstates that this License applies to the Document. These Warranty
Disclaimers are considered to be included by reference in thisLicense, but only as regards disclaiming warranties: any other
implication that these Warranty Disclaimers may have is void and has
no effect on the meaning of this License.
2. VERBATIM COPYING
You may copy and distribute the Document in any medium, either
commercially or noncommercially, provided that this License, thecopyright notices, and the license notice saying this License applies
to the Document are reproduced in all copies, and that you add noother conditions whatsoever to those of this License. You may not use
technical measures to obstruct or control the reading or further
copying of the copies you make or distribute. However, you may accept
compensation in exchange for copies. If you distribute a large enoughnumber of copies you must also follow the conditions in section 3.
You may also lend copies, under the same conditions stated above, andyou may publicly display copies.

3. COPYING IN QUANTITY
If you publish printed copies (or copies in media that commonly haveprinted covers) of the Document, numbering more than 100, and theDocument's license notice requires Cover Texts, you must enclose the(continues on next page)683

GNU Scientific Library, Release 2.7(continued from previous page)

copies in covers that carry, clearly and legibly, all these CoverTexts: Front-Cover Texts on the front cover, and Back-Cover Texts on
the back cover. Both covers must also clearly and legibly identify
you as the publisher of these copies. The front cover must present
the full title with all words of the title equally prominent andvisible. You may add other material on the covers in addition.
Copying with changes limited to the covers, as long as they preserve
the title of the Document and satisfy these conditions, can be treatedas verbatim copying in other respects.
If the required texts for either cover are too voluminous to fit
legibly, you should put the first ones listed (as many as fit
reasonably) on the actual cover, and continue the rest onto adjacentpages.
If you publish or distribute Opaque copies of the Document numbering
more than 100, you must either include a machine-readable Transparent
copy along with each Opaque copy, or state in or with each Opaque copy
a computer-network location from which the general network-using
public has access to download using public-standard network protocols
a complete Transparent copy of the Document, free of added material.
If you use the latter option, you must take reasonably prudent steps,
when you begin distribution of Opaque copies in quantity, to ensurethat this Transparent copy will remain thus accessible at the statedlocation until at least one year after the last time you distribute an
Opaque copy (directly or through your agents or retailers) of that
edition to the public.
It is requested, but not required, that you contact the authors of theDocument well before redistributing any large number of copies, togive them a chance to provide you with an updated version of theDocument.

4. MODIFICATIONS
You may copy and distribute a Modified Version of the Document under
the conditions of sections 2 and 3 above, provided that you release
the Modified Version under precisely this License, with the Modified
Version filling the role of the Document, thus licensing distribution
and modification of the Modified Version to whoever possesses a copyof it. In addition, you must do these things in the Modified Version:
A. Use in the Title Page (and on the covers, if any) a title distinct
from that of the Document, and from those of previous versions(which should, if there were any, be listed in the History sectionof the Document). You may use the same title as a previous version
if the original publisher of that version gives permission.
B. List on the Title Page, as authors, one or more persons or entities
responsible for authorship of the modifications in the Modified
Version, together with at least five of the principal authors of theDocument (all of its principal authors, if it has fewer than five),(continues on next page)

684

Chapter 53. GNU Free Documentation LicenseGNU Scientific Library, Release 2.7(continued from previous page)

unless they release you from this requirement.
C. State on the Title page the name of the publisher of the
Modified Version, as the publisher.
D. Preserve all the copyright notices of the Document.
E. Add an appropriate copyright notice for your modifications
adjacent to the other copyright notices.
F. Include, immediately after the copyright notices, a license noticegiving the public permission to use the Modified Version under theterms of this License, in the form shown in the Addendum below.
G. Preserve in that license notice the full lists of Invariant Sections
and required Cover Texts given in the Document's license notice.
H. Include an unaltered copy of this License.
I. Preserve the section Entitled "History", Preserve its Title, and add
to it an item stating at least the title, year, new authors, andpublisher of the Modified Version as given on the Title Page. If
there is no section Entitled "History" in the Document, create one
stating the title, year, authors, and publisher of the Document asgiven on its Title Page, then add an item describing the Modified
Version as stated in the previous sentence.
J. Preserve the network location, if any, given in the Document for
public access to a Transparent copy of the Document, and likewise
the network locations given in the Document for previous versions
it was based on. These may be placed in the "History" section.
You may omit a network location for a work that was published atleast four years before the Document itself, or if the original
publisher of the version it refers to gives permission.
K. For any section Entitled "Acknowledgements" or "Dedications",
Preserve the Title of the section, and preserve in the section all
the substance and tone of each of the contributor acknowledgements
and/or dedications given therein.
L. Preserve all the Invariant Sections of the Document,
unaltered in their text and in their titles. Section numbers
or the equivalent are not considered part of the section titles.
M. Delete any section Entitled "Endorsements". Such a section
may not be included in the Modified Version.
N. Do not retitle any existing section to be Entitled "Endorsements"
or to conflict in title with any Invariant Section.
O. Preserve any Warranty Disclaimers.
If the Modified Version includes new front-matter sections orappendices that qualify as Secondary Sections and contain no material
copied from the Document, you may at your option designate some or allof these sections as invariant. To do this, add their titles to thelist of Invariant Sections in the Modified Version's license notice.
These titles must be distinct from any other section titles.
You may add a section Entitled "Endorsements", provided it contains
nothing but endorsements of your Modified Version by various
parties--for example, statements of peer review or that the text has
been approved by an organization as the authoritative definition of astandard.
(continues on next page)685

GNU Scientific Library, Release 2.7(continued from previous page)You may add a passage of up to five words as a Front-Cover Text, and apassage of up to 25 words as a Back-Cover Text, to the end of the listof Cover Texts in the Modified Version. Only one passage ofFront-Cover Text and one of Back-Cover Text may be added by (orthrough arrangements made by) any one entity. If the Document alreadyincludes a cover text for the same cover, previously added by you or
by arrangement made by the same entity you are acting on behalf of,you may not add another; but you may replace the old one, on explicit
permission from the previous publisher that added the old one.
The author(s) and publisher(s) of the Document do not by this License
give permission to use their names for publicity for or to assert or
imply endorsement of any Modified Version.

5. COMBINING DOCUMENTS
You may combine the Document with other documents released under thisLicense, under the terms defined in section 4 above for modified
versions, provided that you include in the combination all of theInvariant Sections of all of the original documents, unmodified, andlist them all as Invariant Sections of your combined work in its
license notice, and that you preserve all their Warranty Disclaimers.
The combined work need only contain one copy of this License, andmultiple identical Invariant Sections may be replaced with a single
copy. If there are multiple Invariant Sections with the same name but
different contents, make the title of each such section unique byadding at the end of it, in parentheses, the name of the original
author or publisher of that section if known, or else a unique number.
Make the same adjustment to the section titles in the list ofInvariant Sections in the license notice of the combined work.
In the combination, you must combine any sections Entitled "History"
in the various original documents, forming one section Entitled"History"; likewise combine any sections Entitled "Acknowledgements",
and any sections Entitled "Dedications". You must delete all sections
Entitled "Endorsements".

6. COLLECTIONS OF DOCUMENTS
You may make a collection consisting of the Document and otherdocuments released under this License, and replace the individual
copies of this License in the various documents with a single copythat is included in the collection, provided that you follow the rulesof this License for verbatim copying of each of the documents in allother respects.
You may extract a single document from such a collection, anddistribute it individually under this License, provided you insert a
copy of this License into the extracted document, and follow this(continues on next page)

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License in all other respects regarding verbatim copying of that
document.

7. AGGREGATION WITH INDEPENDENT WORKS
A compilation of the Document or its derivatives with other separate
and independent documents or works, in or on a volume of a storage or
distribution medium, is called an "aggregate" if the copyright
resulting from the compilation is not used to limit the legal rightsof the compilation's users beyond what the individual works permit.
When the Document is included in an aggregate, this License does not
apply to the other works in the aggregate which are not themselves
derivative works of the Document.
If the Cover Text requirement of section 3 is applicable to thesecopies of the Document, then if the Document is less than one half ofthe entire aggregate, the Document's Cover Texts may be placed on
covers that bracket the Document within the aggregate, or theelectronic equivalent of covers if the Document is in electronic form.
Otherwise they must appear on printed covers that bracket the wholeaggregate.

8. TRANSLATION
Translation is considered a kind of modification, so you may
distribute translations of the Document under the terms of section 4.
Replacing Invariant Sections with translations requires special
permission from their copyright holders, but you may include
translations of some or all Invariant Sections in addition to theoriginal versions of these Invariant Sections. You may include atranslation of this License, and all the license notices in theDocument, and any Warranty Disclaimers, provided that you also include
the original English version of this License and the original versionsof those notices and disclaimers. In case of a disagreement between
the translation and the original version of this License or a noticeor disclaimer, the original version will prevail.
If a section in the Document is Entitled "Acknowledgements","Dedications", or "History", the requirement (section 4) to Preserve
its Title (section 1) will typically require changing the actual
title.

9. TERMINATION
You may not copy, modify, sublicense, or distribute the Document
except as expressly provided under this License. Any attempt
otherwise to copy, modify, sublicense, or distribute it is void, andwill automatically terminate your rights under this License.
(continues on next page)687

GNU Scientific Library, Release 2.7(continued from previous page)

However, if you cease all violation of this License, then your licensefrom a particular copyright holder is reinstated (a) provisionally,
unless and until the copyright holder explicitly and finallyterminates your license, and (b) permanently, if the copyright holder
fails to notify you of the violation by some reasonable means prior to
60 days after the cessation.
Moreover, your license from a particular copyright holder isreinstated permanently if the copyright holder notifies you of theviolation by some reasonable means, this is the first time you havereceived notice of violation of this License (for any work) from thatcopyright holder, and you cure the violation prior to 30 days after
your receipt of the notice.
Termination of your rights under this section does not terminate thelicenses of parties who have received copies or rights from you under
this License. If your rights have been terminated and not permanently
reinstated, receipt of a copy of some or all of the same material does
not give you any rights to use it.

10. FUTURE REVISIONS OF THIS LICENSE
The Free Software Foundation may publish new, revised versions of the
GNU Free Documentation License from time to time. Such new versions
will be similar in spirit to the present version, but may differ indetail to address new problems or concerns. See
http://www.gnu.org/copyleft/.
Each version of the License is given a distinguishing version number.
If the Document specifies that a particular numbered version of thisLicense "or any later version" applies to it, you have the option offollowing the terms and conditions either of that specified version orof any later version that has been published (not as a draft) by theFree Software Foundation. If the Document does not specify a version
number of this License, you may choose any version ever published (notas a draft) by the Free Software Foundation. If the Document
specifies that a proxy can decide which future versions of thisLicense can be used, that proxy's public statement of acceptance of aversion permanently authorizes you to choose that version for theDocument.
11. RELICENSING"Massive Multiauthor Collaboration Site" (or "MMC Site") means any
World Wide Web server that publishes copyrightable works and also
provides prominent facilities for anybody to edit those works. A
public wiki that anybody can edit is an example of such a server. A"Massive Multiauthor Collaboration" (or "MMC") contained in the sitemeans any set of copyrightable works thus published on the MMC site.
"CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0(continues on next page)

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Chapter 53. GNU Free Documentation LicenseGNU Scientific Library, Release 2.7(continued from previous page)

license published by Creative Commons Corporation, a not-for-profit
corporation with a principal place of business in San Francisco,California, as well as future copyleft versions of that license
published by that same organization.
"Incorporate" means to publish or republish a Document, in whole or inpart, as part of another Document.
An MMC is "eligible for relicensing" if it is licensed under thisLicense, and if all works that were first published under this License
somewhere other than this MMC, and subsequently incorporated in whole orin part into the MMC, (1) had no cover texts or invariant sections, and(2) were thus incorporated prior to November 1, 2008.
The operator of an MMC Site may republish an MMC contained in the site
under CC-BY-SA on the same site at any time before August 1, 2009,
provided the MMC is eligible for relicensing.

ADDENDUM: How to use this License for your documents
To use this License in a document you have written, include a copy of
the License in the document and put the following copyright andlicense notices just after the title page:
Copyright (c) YEAR YOUR NAME.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3
or any later version published by the Free Software Foundation;with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
A copy of the license is included in the section entitled "GNUFree Documentation License".
If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts,
replace the "with...Texts." line with this:with the Invariant Sections being LIST THEIR TITLES, with the
Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST.
If you have Invariant Sections without Cover Texts, or some othercombination of the three, merge those two alternatives to suit thesituation.
If your document contains nontrivial examples of program code, we
recommend releasing these examples in parallel under your choice of
free software license, such as the GNU General Public License,
to permit their use in free software.
• genindex689

GNU Scientific Library, Release 2.7

690

Chapter 53. GNU Free Documentation License

INDEX

Symbols$, shell prompt, 3
2D histograms, 341
2D random direction vector, 265
3D random direction vector, 265
3-j symbols, 47
6-j symbols, 47
9-j symbols, 47

Aacceleration of series, 418
acosh, 18
Adams method, 385
Adaptive step-size control, differential
equations, 386
Ai(x), 37
Airy functions, 37
Akima splines, 396
aliasing of arrays, 11
alternative optimized functions, 9
AMAX, Level-1 BLAS, 131
Angular Mathieu Functions, 75
angular reduction, 79
ANSI C, use of, 4Apell symbol, see Pochhammer symbol, 60
approximate comparison of floating point
numbers, 21
arctangent integral, 56
argument of complex number, 25
arithmetic exceptions, 639
asinh, 18
astronomical constants, 628
ASUM, Level-1 BLAS, 131
atanh, 18
atomic physics, constants, 628
autoconf, using with GSL, 651
AXPY, Level-1 BLAS, 132

B
B-spline wavelets, 424
Bader and Deuflhard, Bulirsch-Stoer method.,385

balancing matrices, 167
banded Cholesky Decomposition, 166
banded general matrices, 164banded LDLT decomposition, 167
banded LU Decomposition, 165
banded matrices, 163
banded symmetric matrices, 164
Basic Linear Algebra Subroutines (BLAS), 127,654
basis splines, allocating, 568
basis splines, B-splines, 565
basis splines, breakpoints, 568
basis splines, derivatives, 570
basis splines, evaluation, 570
basis splines, examples, 576
basis splines, Gram matrix, 574
basis splines, Greville abscissae, 576
basis splines, integration, 571
basis splines, interpolation, 575
basis splines, knots, 568
basis splines, least squares, 571
basis splines, Marsden-Schoenberg points, 576
basis splines, overview, 567
basis splines, properties, 569
BDF method, 386
Bernoulli trial, random variates, 272
Bessel functions, 38
Bessel Functions, Fractional Order, 43
best-fit parameters, covariance, 535
Beta distribution, 262
Beta function, 61
Beta function, incomplete normalized, 61
BFGS algorithm, minimization, 474
Bi(x), 37
bias, IEEE format, 637
bicubic interpolation, 404
bidiagonalization of real matrices, 160
bilinear interpolation, 404
binning data, 333
Binomial random variates, 273
biorthogonal wavelets, 424
bisection algorithm for finding roots, 438

691

GNU Scientific Library, Release 2.7

Bivariate Gaussian distribution, 241, 242
BLAS, 127
BLAS, Low-level C interface, 654
BLAS, sparse, 617
blocks, 83
bounds checking, extension to GCC, 86
breakpoints, 643
Brent's method for finding minima, 450
Brent's method for finding roots, 439
Broyden algorithm for multidimensionalroots, 462
BSD random number generator, 223
bug-gsl, 2
bug-gsl mailing list, 2bugs, 2
Bulirsch-Stoer method, 385

C
C extensions, compatible use of, 4C++, compatibility, 11
C99, inline keyword, 7
Carlson forms of Elliptic integrals, 50
Cash-Karp, Runge-Kutta method, 385
Cauchy distribution, 246
Cauchy principal value, by numericalquadrature, 203
CBLAS, 127
CBLAS, Low-level interface, 654
cblas_caxpy (C function), 656
cblas_ccopy (C function), 656
cblas_cdotc_sub (C function), 655
cblas_cdotu_sub (C function), 655
cblas_cgbmv (C function), 658
cblas_cgemm (C function), 663
cblas_cgemv (C function), 658
cblas_cgerc (C function), 661
cblas_cgeru (C function), 661
cblas_chbmv (C function), 661
cblas_chemm (C function), 664
cblas_chemv (C function), 661
cblas_cher (C function), 661
cblas_cher2 (C function), 661
cblas_cher2k (C function), 665
cblas_cherk (C function), 664
cblas_chpmv (C function), 661
cblas_chpr (C function), 661
cblas_chpr2 (C function), 661
cblas_cscal (C function), 657
cblas_csscal (C function), 657
cblas_cswap (C function), 656
cblas_csymm (C function), 663
cblas_csyr2k (C function), 664
cblas_csyrk (C function), 663
cblas_ctbmv (C function), 659
692

cblas_ctbsv (C function), 659
cblas_ctpmv (C function), 659
cblas_ctpsv (C function), 659
cblas_ctrmm (C function), 664
cblas_ctrmv (C function), 659
cblas_ctrsm (C function), 664
cblas_ctrsv (C function), 659
cblas_dasum (C function), 655
cblas_daxpy (C function), 656
cblas_dcopy (C function), 656
cblas_ddot (C function), 655
cblas_dgbmv (C function), 658
cblas_dgemm (C function), 663
cblas_dgemv (C function), 658
cblas_dger (C function), 661
cblas_dnrm2 (C function), 655
cblas_drot (C function), 657
cblas_drotg (C function), 657
cblas_drotm (C function), 657
cblas_drotmg (C function), 657
cblas_dsbmv (C function), 660
cblas_dscal (C function), 657
cblas_dsdot (C function), 655
cblas_dspmv (C function), 660
cblas_dspr (C function), 661
cblas_dspr2 (C function), 661
cblas_dswap (C function), 656
cblas_dsymm (C function), 663
cblas_dsymv (C function), 660
cblas_dsyr (C function), 661
cblas_dsyr2 (C function), 661
cblas_dsyr2k (C function), 663
cblas_dsyrk (C function), 663
cblas_dtbmv (C function), 658
cblas_dtbsv (C function), 658
cblas_dtpmv (C function), 658
cblas_dtpsv (C function), 658
cblas_dtrmm (C function), 663
cblas_dtrmv (C function), 658
cblas_dtrsm (C function), 663
cblas_dtrsv (C function), 658
cblas_dzasum (C function), 656
cblas_dznrm2 (C function), 655
cblas_icamax (C function), 656
cblas_idamax (C function), 656
cblas_isamax (C function), 656
cblas_izamax (C function), 656
cblas_sasum (C function), 655
cblas_saxpy (C function), 656
cblas_scasum (C function), 655
cblas_scnrm2 (C function), 655
cblas_scopy (C function), 656
cblas_sdot (C function), 655
cblas_sdsdot (C function), 655
IndexGNU Scientific Library, Release 2.7

cblas_sgbmv (C function), 657
cblas_sgemm (C function), 662
cblas_sgemv (C function), 657
cblas_sger (C function), 660
cblas_snrm2 (C function), 655
cblas_srot (C function), 656
cblas_srotg (C function), 656
cblas_srotm (C function), 656
cblas_srotmg (C function), 656
cblas_ssbmv (C function), 660
cblas_sscal (C function), 657
cblas_sspmv (C function), 660
cblas_sspr (C function), 660
cblas_sspr2 (C function), 660
cblas_sswap (C function), 656
cblas_ssymm (C function), 662
cblas_ssymv (C function), 660
cblas_ssyr (C function), 660
cblas_ssyr2 (C function), 660
cblas_ssyr2k (C function), 662
cblas_ssyrk (C function), 662
cblas_stbmv (C function), 657
cblas_stbsv (C function), 658
cblas_stpmv (C function), 657
cblas_stpsv (C function), 658
cblas_strmm (C function), 663
cblas_strmv (C function), 657
cblas_strsm (C function), 663
cblas_strsv (C function), 657
cblas_xerbla (C function), 665
cblas_zaxpy (C function), 656
cblas_zcopy (C function), 656
cblas_zdotc_sub (C function), 655
cblas_zdotu_sub (C function), 655
cblas_zdscal (C function), 657
cblas_zgbmv (C function), 659
cblas_zgemm (C function), 664
cblas_zgemv (C function), 659
cblas_zgerc (C function), 662
cblas_zgeru (C function), 662
cblas_zhbmv (C function), 662
cblas_zhemm (C function), 665
cblas_zhemv (C function), 662
cblas_zher (C function), 662
cblas_zher2 (C function), 662
cblas_zher2k (C function), 665
cblas_zherk (C function), 665
cblas_zhpmv (C function), 662
cblas_zhpr (C function), 662
cblas_zhpr2 (C function), 662
cblas_zscal (C function), 657
cblas_zswap (C function), 656
cblas_zsymm (C function), 664
cblas_zsyr2k (C function), 664
Indexcblas_zsyrk (C function), 664
cblas_ztbmv (C function), 659
cblas_ztbsv (C function), 660
cblas_ztpmv (C function), 659
cblas_ztpsv (C function), 660
cblas_ztrmm (C function), 664
cblas_ztrmv (C function), 659
cblas_ztrsm (C function), 664
cblas_ztrsv (C function), 659
CDFs, cumulative distribution functions, 233ce(q,x), Mathieu function, 75
Chebyshev series, 413
checking combination for validity, 114
checking multiset for validity, 118
checking permutation for validity, 108
Chi(x), 55
Chi-squared distribution, 256
Cholesky decomposition, 154
Cholesky decomposition, banded, 166
Cholesky decomposition, modified, 157
Cholesky decomposition, pivoted, 156
Cholesky decomposition, square root free, 157
Ci(x), 56
Clausen functions, 45
Clenshaw-Curtis quadrature, 200
CMRG, combined multiple recursive random
number generator, 221
code reuse in applications, 11
combinations, 112
combinatorial factor C(m, 59
combinatorial optimization, 366
comparison functions, definition, 123
compatibility, 4compiling programs, include paths, 5compiling programs, library paths, 6complementary incomplete Gamma function, 60
complete Fermi-Dirac integrals, 57
complete orthogonal decomposition, 151
complex arithmetic, 25complex cosine function, special functions,78
Complex Gamma function, 59
complex hermitian matrix, eigensystem, 172
complex log sine function, special
functions, 78
complex numbers, 21
complex sinc function, special functions, 78
complex sine function, special functions, 78
confluent hypergeometric function, 67
confluent hypergeometric functions, 65
conical functions, 68
Conjugate gradient algorithm, minimization,
474
conjugate of complex number, 25
693

GNU Scientific Library, Release 2.7

constant matrix, 95
constants, fundamental, 627
constants, mathematical (defined as macros), 17
constants, physical, 626
constants, prefixes, 633
contacting the GSL developers, 3
conventions, used in manual, 3convergence, accelerating a series, 418
conversion of units, 626
cooling schedule, 367
COPY, Level-1 BLAS, 131
correlation, of two datasets, 290
cosine function, special functions, 78
cosine of complex number, 26
cost function, 366
Coulomb wave functions, 45
coupling coefficients, 47
covariance matrix, from linear regression,
482
covariance matrix, linear fits, 481
covariance matrix, nonlinear fits, 535
covariance, of two datasets, 289
cquad, doubly-adaptive integration, 206
CRAY random number generator, RANF, 224
cubic equation, solving, 31
cubic splines, 396
cumulative distribution functions (CDFs), 233
Cylindrical Bessel Functions, 38

D
Daubechies wavelets, 423
Dawson function, 48
DAXPY, Level-1 BLAS, 132
debugging numerical programs, 643
Debye functions, 48
denormalized form, IEEE format, 637
deprecated functions, 11derivatives, calculating numerically, 409
determinant of a matrix, by LU
decomposition, 142
Deuflhard and Bader, Bulirsch-Stoer method.,
385
DFTs, see FFT, 182
diagonal, of a matrix, 99
differential equations, initial value
problems, 381
differentiation of functions, numeric, 409
digamma function, 76
dilogarithm, 49
direction vector, random 2D, 265
direction vector, random 3D, 265
direction vector, random N-dimensional, 265
Dirichlet distribution, 269
discontinuities, in ODE systems, 388
694

Discrete Fourier Transforms, see FFT, 182
discrete Hankel transforms, 429
Discrete Newton algorithm for
multidimensional roots, 462
Discrete random numbers, 270
Discrete random numbers, preprocessing, 270
divided differences, polynomials, 29
division by zero, IEEE exceptions, 639
Dogleg algorithm, 523
Dogleg algorithm, double, 524
dollar sign $, shell prompt, 3
DOT, Level-1 BLAS, 130
double Dogleg algorithm, 524
double factorial, 59
double precision, IEEE format, 637
downloading GSL, 2
DWT initialization, 423
DWT, mathematical definition, 423
DWT, one dimensional, 424
DWT, see wavelet transforms, 422
DWT, two dimensional, 425Ee, defined as a macro, 17
E1(x), 55
E2(x), 55
Ei(x), 55
eigenvalues and eigenvectors, 170
elementary functions, 16elementary operations, 49
elliptic functions (Jacobi), 52
elliptic integrals, 50
energy function, 366
energy, units of, 631
erf(x), 52
erfc(x), 52
Erlang distribution, 252error codes, 14error codes, reserved, 13
error function, 52
error handlers, 14error handling, 12error handling macros, 15
estimated standard deviation, 285
estimated variance, 285
estimation, location, 293
estimation, scale, 294
Eta Function, 81
euclidean distance function, hypot, 18
euclidean distance function, hypot3, 18
Euler's constant, defined as a macro, 17
evaluation of polynomials, 29
evaluation of polynomials, in divided
difference form, 29
Index

GNU Scientific Library, Release 2.7

examples, conventions used in, 3
exceptions, C++, 11
exceptions, floating point, 645
exceptions, IEEE arithmetic, 639
exchanging permutation elements, 108exp, 53
expm1, 18
exponent, IEEE format, 637
Exponential distribution, 243
exponential function, 53
exponential integrals, 55
Exponential power distribution, 245
exponential, difference from 1 computed
accurately, 18
exponentiation of complex number, 26
extern inline, 7

Fourier integrals, numerical, 205
Fourier Transforms, see FFT, 182
Fractional Order Bessel Functions, 43
free software, explanation of, 1
frexp, 19
functions, numerical differentiation, 409
fundamental constants, 627

G

Gamma distribution, 252
gamma functions, 58
Gastwirth estimator, 294
Gauss-Kronrod quadrature, 200
Gaussian distribution, 237
Gaussian distribution, bivariate, 241, 242
Gaussian Tail distribution, 239
gcc extensions, range-checking, 86
F
gcc warning options, 645
F-distribution, 258
gdb, 643
factorial, 59
Gegenbauer functions, 61
factorization of matrices, 139
GEMM, Level-3 BLAS, 135
false position algorithm for finding roots,
GEMV, Level-2 BLAS, 133
439
general polynomial equations, solving, 31
Fast Fourier Transforms, see FFT, 182
generalized eigensystems, 176
Fehlberg method, differential equations, 385
generalized hermitian definite eigensystems,
Fermi-Dirac function, 57
175
FFT, 182
generalized symmetric eigensystems, 174
FFT mathematical definition, 183
Geometric random variates, 277, 278
FFT of complex data, mixed-radix algorithm,
GER, Level-2 BLAS, 134
186
GERC, Level-2 BLAS, 134
FFT of complex data, radix-2 algorithm, 185
GERU, Level-2 BLAS, 134
FFT of real data, 191
Givens rotation, 160
FFT of real data, mixed-radix algorithm, 193
Givens Rotation, BLAS, 132
FFT of real data, radix-2 algorithm, 191
Givens Rotation, Modified, BLAS, 132
FFT, complex data, 184
gmres, 621
finding minima, 445
GNU General Public License, 1finding roots, 432
golden section algorithm for finding minima,
finding zeros, 432
450
fits, multi-parameter linear, 483
gsl_acosh (C function), 18
fitting, 480
gsl_asinh (C function), 18
fitting, using Chebyshev polynomials, 413
gsl_atanh (C function), 18
Fj(x), Fermi-Dirac integral, 57
gsl_blas_caxpy (C function), 132
Fj(x,b), incomplete Fermi-Dirac integral, 58 gsl_blas_ccopy (C function), 131
flat distribution, 254
gsl_blas_cdotc (C function), 130
Fletcher-Reeves conjugate gradientgsl_blas_cdotu (C function), 130
algorithm, minimization, 474
gsl_blas_cgemm (C function), 135
floating point exceptions, 645
gsl_blas_cgemv (C function), 133
floating point numbers, approximate
gsl_blas_cgerc (C function), 134
comparison, 21
gsl_blas_cgeru (C function), 134
floating point registers, 644
gsl_blas_chemm (C function), 136
force and energy, 633
gsl_blas_chemv (C function), 134
Fortran range checking, equivalent in gcc, 86 gsl_blas_cher (C function), 134
Four-tap Generalized Feedback Shiftgsl_blas_cher2 (C function), 135
Register, 222
gsl_blas_cher2k (C function), 137
Index

695

GNU Scientific Library, Release 2.7

gsl_blas_cherk (C function), 137
gsl_blas_cscal (C function), 132
gsl_blas_csscal (C function), 132
gsl_blas_cswap (C function), 131
gsl_blas_csymm (C function), 135
gsl_blas_csyr2k (C function), 137
gsl_blas_csyrk (C function), 137
gsl_blas_ctrmm (C function), 136
gsl_blas_ctrmv (C function), 133
gsl_blas_ctrsm (C function), 136
gsl_blas_ctrsv (C function), 133
gsl_blas_dasum (C function), 131
gsl_blas_daxpy (C function), 132
gsl_blas_dcopy (C function), 131
gsl_blas_ddot (C function), 130
gsl_blas_dgemm (C function), 135
gsl_blas_dgemv (C function), 133
gsl_blas_dger (C function), 134
gsl_blas_dnrm2 (C function), 131
gsl_blas_drot (C function), 132
gsl_blas_drotg (C function), 132
gsl_blas_drotm (C function), 132
gsl_blas_drotmg (C function), 132
gsl_blas_dscal (C function), 132
gsl_blas_dsdot (C function), 130
gsl_blas_dswap (C function), 131
gsl_blas_dsymm (C function), 135
gsl_blas_dsymv (C function), 133
gsl_blas_dsyr (C function), 134
gsl_blas_dsyr2 (C function), 134
gsl_blas_dsyr2k (C function), 137
gsl_blas_dsyrk (C function), 137
gsl_blas_dtrmm (C function), 136
gsl_blas_dtrmv (C function), 133
gsl_blas_dtrsm (C function), 136
gsl_blas_dtrsv (C function), 133
gsl_blas_dzasum (C function), 131
gsl_blas_dznrm2 (C function), 131
gsl_blas_icamax (C function), 131
gsl_blas_idamax (C function), 131
gsl_blas_isamax (C function), 131
gsl_blas_izamax (C function), 131
gsl_blas_sasum (C function), 131
gsl_blas_saxpy (C function), 132
gsl_blas_scasum (C function), 131
gsl_blas_scnrm2 (C function), 131
gsl_blas_scopy (C function), 131
gsl_blas_sdot (C function), 130
gsl_blas_sdsdot (C function), 130
gsl_blas_sgemm (C function), 135
gsl_blas_sgemv (C function), 133
gsl_blas_sger (C function), 134
gsl_blas_snrm2 (C function), 131
gsl_blas_srot (C function), 132696

gsl_blas_srotg (C function), 132
gsl_blas_srotm (C function), 132
gsl_blas_srotmg (C function), 132
gsl_blas_sscal (C function), 132
gsl_blas_sswap (C function), 131
gsl_blas_ssymm (C function), 135
gsl_blas_ssymv (C function), 133
gsl_blas_ssyr (C function), 134
gsl_blas_ssyr2 (C function), 134
gsl_blas_ssyr2k (C function), 137
gsl_blas_ssyrk (C function), 137
gsl_blas_strmm (C function), 136
gsl_blas_strmv (C function), 133
gsl_blas_strsm (C function), 136
gsl_blas_strsv (C function), 133
gsl_blas_zaxpy (C function), 132
gsl_blas_zcopy (C function), 131
gsl_blas_zdotc (C function), 130
gsl_blas_zdotu (C function), 130
gsl_blas_zdscal (C function), 132
gsl_blas_zgemm (C function), 135
gsl_blas_zgemv (C function), 133
gsl_blas_zgerc (C function), 134
gsl_blas_zgeru (C function), 134
gsl_blas_zhemm (C function), 136
gsl_blas_zhemv (C function), 134
gsl_blas_zher (C function), 134
gsl_blas_zher2 (C function), 135
gsl_blas_zher2k (C function), 137
gsl_blas_zherk (C function), 137
gsl_blas_zscal (C function), 132
gsl_blas_zswap (C function), 131
gsl_blas_zsymm (C function), 135
gsl_blas_zsyr2k (C function), 137
gsl_blas_zsyrk (C function), 137
gsl_blas_ztrmm (C function), 136
gsl_blas_ztrmv (C function), 133
gsl_blas_ztrsm (C function), 136
gsl_blas_ztrsv (C function), 133
gsl_block (C type), 84
gsl_block_alloc (C function), 84
gsl_block_calloc (C function), 84
gsl_block_fprintf (C function), 84
gsl_block_fread (C function), 84
gsl_block_free (C function), 84
gsl_block_fscanf (C function), 85
gsl_block_fwrite (C function), 84
gsl_bspline_alloc (C function), 568
gsl_bspline_alloc_ncontrol (C function), 568
gsl_bspline_basis (C function), 570
gsl_bspline_basis_deriv (C function), 571
gsl_bspline_basis_integ (C function), 571
gsl_bspline_calc (C function), 570
gsl_bspline_calc_deriv (C function), 570
Index

GNU Scientific Library, Release 2.7

gsl_bspline_calc_integ (C function), 571
gsl_bspline_col_interp (C function), 575
gsl_bspline_covariance (C function), 573
gsl_bspline_err (C function), 573
gsl_bspline_free (C function), 568
gsl_bspline_gram (C function), 575
gsl_bspline_gram_interval (C function), 575
gsl_bspline_greville_abscissa (C function), 576
gsl_bspline_init (C function), 569
gsl_bspline_init_augment (C function), 569
gsl_bspline_init_interp (C function), 569
gsl_bspline_init_periodic (C function), 569
gsl_bspline_init_uniform (C function), 569
gsl_bspline_lsnormal (C function), 572
gsl_bspline_lsnormalm (C function), 572
gsl_bspline_lssolve (C function), 572
gsl_bspline_nbreak (C function), 570
gsl_bspline_ncontrol (C function), 570
gsl_bspline_oprod (C function), 574
gsl_bspline_order (C function), 570
gsl_bspline_plsqr (C function), 574
gsl_bspline_plssolve (C function), 574
gsl_bspline_proj_rhs (C function), 576
gsl_bspline_pwlssolve (C function), 574
gsl_bspline_rcond (C function), 573
gsl_bspline_residuals (C function), 573
gsl_bspline_vector_calc (C function), 570
gsl_bspline_vector_calc_deriv (C function), 571
gsl_bspline_wlssolve (C function), 572
gsl_bspline_workspace (C type), 568
GSL_C99_INLINE, 7
GSL_C99_INLINE (C macro), 86
gsl_cdf_beta_P (C function), 262
gsl_cdf_beta_Pinv (C function), 262
gsl_cdf_beta_Q (C function), 262
gsl_cdf_beta_Qinv (C function), 262
gsl_cdf_binomial_P (C function), 273
gsl_cdf_binomial_Q (C function), 273
gsl_cdf_cauchy_P (C function), 246
gsl_cdf_cauchy_Pinv (C function), 246
gsl_cdf_cauchy_Q (C function), 246
gsl_cdf_cauchy_Qinv (C function), 246
gsl_cdf_chisq_P (C function), 256
gsl_cdf_chisq_Pinv (C function), 256
gsl_cdf_chisq_Q (C function), 256
gsl_cdf_chisq_Qinv (C function), 256
gsl_cdf_exponential_P (C function), 243
gsl_cdf_exponential_Pinv (C function), 243
gsl_cdf_exponential_Q (C function), 243
gsl_cdf_exponential_Qinv (C function), 243
gsl_cdf_exppow_P (C function), 245
gsl_cdf_exppow_Q (C function), 245
gsl_cdf_fdist_P (C function), 258
gsl_cdf_fdist_Pinv (C function), 258
Index

gsl_cdf_fdist_Q (C function), 258
gsl_cdf_fdist_Qinv (C function), 258
gsl_cdf_flat_P (C function), 254
gsl_cdf_flat_Pinv (C function), 254
gsl_cdf_flat_Q (C function), 254
gsl_cdf_flat_Qinv (C function), 254
gsl_cdf_gamma_P (C function), 252
gsl_cdf_gamma_Pinv (C function), 252
gsl_cdf_gamma_Q (C function), 252
gsl_cdf_gamma_Qinv (C function), 252
gsl_cdf_gaussian_P (C function), 238
gsl_cdf_gaussian_Pinv (C function), 238
gsl_cdf_gaussian_Q (C function), 238
gsl_cdf_gaussian_Qinv (C function), 238
gsl_cdf_geometric_P (C function), 277
gsl_cdf_geometric_Q (C function), 277
gsl_cdf_gumbel1_P (C function), 267
gsl_cdf_gumbel1_Pinv (C function), 267
gsl_cdf_gumbel1_Q (C function), 267
gsl_cdf_gumbel1_Qinv (C function), 267
gsl_cdf_gumbel2_P (C function), 268
gsl_cdf_gumbel2_Pinv (C function), 268
gsl_cdf_gumbel2_Q (C function), 268
gsl_cdf_gumbel2_Qinv (C function), 268
gsl_cdf_hypergeometric_P (C function), 278
gsl_cdf_hypergeometric_Q (C function), 278
gsl_cdf_laplace_P (C function), 244
gsl_cdf_laplace_Pinv (C function), 244
gsl_cdf_laplace_Q (C function), 244
gsl_cdf_laplace_Qinv (C function), 244
gsl_cdf_logistic_P (C function), 263
gsl_cdf_logistic_Pinv (C function), 263
gsl_cdf_logistic_Q (C function), 263
gsl_cdf_logistic_Qinv (C function), 263
gsl_cdf_lognormal_P (C function), 255
gsl_cdf_lognormal_Pinv (C function), 255
gsl_cdf_lognormal_Q (C function), 255
gsl_cdf_lognormal_Qinv (C function), 255
gsl_cdf_negative_binomial_P (C function), 275
gsl_cdf_negative_binomial_Q (C function), 275
gsl_cdf_pareto_P (C function), 264
gsl_cdf_pareto_Pinv (C function), 264
gsl_cdf_pareto_Q (C function), 264
gsl_cdf_pareto_Qinv (C function), 264
gsl_cdf_pascal_P (C function), 276
gsl_cdf_pascal_Q (C function), 276
gsl_cdf_poisson_P (C function), 271
gsl_cdf_poisson_Q (C function), 271
gsl_cdf_rayleigh_P (C function), 247
gsl_cdf_rayleigh_Pinv (C function), 247
gsl_cdf_rayleigh_Q (C function), 247
gsl_cdf_rayleigh_Qinv (C function), 247
gsl_cdf_tdist_P (C function), 260
gsl_cdf_tdist_Pinv (C function), 260
697

GNU Scientific Library, Release 2.7

gsl_cdf_tdist_Q (C function), 260
gsl_cdf_tdist_Qinv (C function), 260
gsl_cdf_ugaussian_P (C function), 238
gsl_cdf_ugaussian_Pinv (C function), 238
gsl_cdf_ugaussian_Q (C function), 238
gsl_cdf_ugaussian_Qinv (C function), 238
gsl_cdf_weibull_P (C function), 266
gsl_cdf_weibull_Pinv (C function), 266
gsl_cdf_weibull_Q (C function), 266
gsl_cdf_weibull_Qinv (C function), 266
gsl_cheb_alloc (C function), 415
gsl_cheb_calc_deriv (C function), 416
gsl_cheb_calc_integ (C function), 416
gsl_cheb_coeffs (C function), 416
gsl_cheb_eval (C function), 416
gsl_cheb_eval_err (C function), 416
gsl_cheb_eval_n (C function), 416
gsl_cheb_eval_n_err (C function), 416
gsl_cheb_free (C function), 415
gsl_cheb_init (C function), 415
gsl_cheb_order (C function), 416
gsl_cheb_series (C type), 415
gsl_cheb_size (C function), 416
gsl_check_range (C var), 87
gsl_combination (C type), 113
gsl_combination_alloc (C function), 113
gsl_combination_calloc (C function), 113
gsl_combination_data (C function), 114
gsl_combination_fprintf (C function), 115
gsl_combination_fread (C function), 115
gsl_combination_free (C function), 113
gsl_combination_fscanf (C function), 115
gsl_combination_fwrite (C function), 115
gsl_combination_get (C function), 114
gsl_combination_init_first (C function), 113
gsl_combination_init_last (C function), 113
gsl_combination_k (C function), 114
gsl_combination_memcpy (C function), 114
gsl_combination_n (C function), 114
gsl_combination_next (C function), 114
gsl_combination_prev (C function), 114
gsl_combination_valid (C function), 114
gsl_complex, 23
gsl_complex_abs (C function), 25
gsl_complex_abs2 (C function), 25
gsl_complex_add (C function), 25
gsl_complex_add_imag (C function), 25
gsl_complex_add_real (C function), 25
gsl_complex_arccos (C function), 27
gsl_complex_arccos_real (C function), 27
gsl_complex_arccosh (C function), 28
gsl_complex_arccosh_real (C function), 28
gsl_complex_arccot (C function), 27
gsl_complex_arccoth (C function), 28
698

gsl_complex_arccsc (C function), 27
gsl_complex_arccsc_real (C function), 27
gsl_complex_arccsch (C function), 28
gsl_complex_arcsec (C function), 27
gsl_complex_arcsec_real (C function), 27
gsl_complex_arcsech (C function), 28
gsl_complex_arcsin (C function), 27
gsl_complex_arcsin_real (C function), 27
gsl_complex_arcsinh (C function), 28
gsl_complex_arctan (C function), 27
gsl_complex_arctanh (C function), 28
gsl_complex_arctanh_real (C function), 28
gsl_complex_arg (C function), 25
gsl_complex_conjugate (C function), 25
gsl_complex_cos (C function), 26
gsl_complex_cosh (C function), 27
gsl_complex_cot (C function), 26
gsl_complex_coth (C function), 27
gsl_complex_csc (C function), 26
gsl_complex_csch (C function), 27
gsl_complex_div (C function), 25
gsl_complex_div_imag (C function), 25
gsl_complex_div_real (C function), 25
gsl_complex_exp (C function), 26
gsl_complex_inverse (C function), 25
gsl_complex_log (C function), 26
gsl_complex_log10 (C function), 26
gsl_complex_log_b (C function), 26
gsl_complex_logabs (C function), 25
gsl_complex_mul (C function), 25
gsl_complex_mul_imag (C function), 25
gsl_complex_mul_real (C function), 25
gsl_complex_negative (C function), 25
gsl_complex_polar (C function), 24
gsl_complex_poly_complex_eval (C function), 29
gsl_complex_pow (C function), 26
gsl_complex_pow_real (C function), 26
gsl_complex_rect (C function), 24
gsl_complex_sec (C function), 26
gsl_complex_sech (C function), 27
gsl_complex_sin (C function), 26
gsl_complex_sinh (C function), 27
gsl_complex_sqrt (C function), 26
gsl_complex_sqrt_real (C function), 26
gsl_complex_sub (C function), 25
gsl_complex_sub_imag (C function), 25
gsl_complex_sub_real (C function), 25
gsl_complex_tan (C function), 26
gsl_complex_tanh (C function), 27
GSL_CONST_MKSA_ACRE (C macro), 630
GSL_CONST_MKSA_ANGSTROM (C macro), 628
GSL_CONST_MKSA_ASTRONOMICAL_UNIT (C macro), 628
GSL_CONST_MKSA_BAR (C macro), 632
GSL_CONST_MKSA_BARN (C macro), 628
IndexGNU Scientific Library, Release 2.7

GSL_CONST_MKSA_BOHR_MAGNETON (C macro), 629
GSL_CONST_MKSA_BOHR_RADIUS (C macro), 628
GSL_CONST_MKSA_BOLTZMANN (C macro), 627
GSL_CONST_MKSA_BTU (C macro), 631
GSL_CONST_MKSA_CALORIE (C macro), 631
GSL_CONST_MKSA_CANADIAN_GALLON (C macro), 630
GSL_CONST_MKSA_CARAT (C macro), 631
GSL_CONST_MKSA_CURIE (C macro), 633
GSL_CONST_MKSA_DAY (C macro), 629
GSL_CONST_MKSA_DEBYE (C macro), 629
GSL_CONST_MKSA_DYNE (C macro), 633
GSL_CONST_MKSA_ELECTRON_CHARGE (C macro), 628
GSL_CONST_MKSA_ELECTRON_MAGNETIC_MOMENT(C
macro), 629
GSL_CONST_MKSA_ELECTRON_VOLT (C macro), 628
GSL_CONST_MKSA_ERG (C macro), 633
GSL_CONST_MKSA_FARADAY (C macro), 627
GSL_CONST_MKSA_FATHOM (C macro), 630
GSL_CONST_MKSA_FOOT (C macro), 629
GSL_CONST_MKSA_FOOTCANDLE (C macro), 632
GSL_CONST_MKSA_FOOTLAMBERT (C macro), 632
GSL_CONST_MKSA_GAUSS (C macro), 627
GSL_CONST_MKSA_GRAM_FORCE (C macro), 631
GSL_CONST_MKSA_GRAV_ACCEL (C macro), 628
GSL_CONST_MKSA_GRAVITATIONAL_CONSTANT(Cmacro), 628
GSL_CONST_MKSA_HECTARE (C macro), 630
GSL_CONST_MKSA_HORSEPOWER (C macro), 631
GSL_CONST_MKSA_HOUR (C macro), 629
GSL_CONST_MKSA_INCH (C macro), 629
GSL_CONST_MKSA_INCH_OF_MERCURY (C macro), 632
GSL_CONST_MKSA_INCH_OF_WATER (C macro), 632
GSL_CONST_MKSA_JOULE (C macro), 633
GSL_CONST_MKSA_KILOMETERS_PER_HOUR (C macro),
630
GSL_CONST_MKSA_KILOPOUND_FORCE (C macro), 631
GSL_CONST_MKSA_KNOT (C macro), 630
GSL_CONST_MKSA_LAMBERT (C macro), 632
GSL_CONST_MKSA_LIGHT_YEAR (C macro), 628
GSL_CONST_MKSA_LITER (C macro), 630
GSL_CONST_MKSA_LUMEN (C macro), 632
GSL_CONST_MKSA_LUX (C macro), 632
GSL_CONST_MKSA_MASS_ELECTRON (C macro), 628
GSL_CONST_MKSA_MASS_MUON (C macro), 628
GSL_CONST_MKSA_MASS_NEUTRON (C macro), 628
GSL_CONST_MKSA_MASS_PROTON (C macro), 628
GSL_CONST_MKSA_METER_OF_MERCURY (C macro), 632
GSL_CONST_MKSA_METRIC_TON (C macro), 631
GSL_CONST_MKSA_MICRON (C macro), 630
GSL_CONST_MKSA_MIL (C macro), 629
GSL_CONST_MKSA_MILE (C macro), 629
GSL_CONST_MKSA_MILES_PER_HOUR (C macro), 630
GSL_CONST_MKSA_MINUTE (C macro), 629
GSL_CONST_MKSA_MOLAR_GAS (C macro), 627
Index

GSL_CONST_MKSA_NAUTICAL_MILE (C macro), 630
GSL_CONST_MKSA_NEWTON (C macro), 633
GSL_CONST_MKSA_NUCLEAR_MAGNETON (C macro), 629
GSL_CONST_MKSA_OUNCE_MASS (C macro), 631
GSL_CONST_MKSA_PARSEC (C macro), 628
GSL_CONST_MKSA_PHOT (C macro), 632
GSL_CONST_MKSA_PINT (C macro), 630
GSL_CONST_MKSA_PLANCKS_CONSTANT_H (C macro),
627
GSL_CONST_MKSA_PLANCKS_CONSTANT_HBAR(Cmacro), 627
GSL_CONST_MKSA_POINT (C macro), 630
GSL_CONST_MKSA_POISE (C macro), 632
GSL_CONST_MKSA_POUND_FORCE (C macro), 631
GSL_CONST_MKSA_POUND_MASS (C macro), 631
GSL_CONST_MKSA_POUNDAL (C macro), 631
GSL_CONST_MKSA_PROTON_MAGNETIC_MOMENT(C
macro), 629
GSL_CONST_MKSA_PSI (C macro), 632
GSL_CONST_MKSA_QUART (C macro), 630
GSL_CONST_MKSA_RAD (C macro), 633
GSL_CONST_MKSA_ROENTGEN (C macro), 633
GSL_CONST_MKSA_RYDBERG (C macro), 628
GSL_CONST_MKSA_SOLAR_MASS (C macro), 628
GSL_CONST_MKSA_SPEED_OF_LIGHT (C macro), 627
GSL_CONST_MKSA_STANDARD_GAS_VOLUME (C macro),
627
GSL_CONST_MKSA_STD_ATMOSPHERE (C macro), 632
GSL_CONST_MKSA_STEFAN_BOLTZMANN_CONSTANT (Cmacro), 627
GSL_CONST_MKSA_STILB (C macro), 632
GSL_CONST_MKSA_STOKES (C macro), 632
GSL_CONST_MKSA_TEXPOINT (C macro), 630
GSL_CONST_MKSA_THERM (C macro), 631
GSL_CONST_MKSA_THOMSON_CROSS_SECTION(Cmacro), 629
GSL_CONST_MKSA_TON (C macro), 631
GSL_CONST_MKSA_TORR (C macro), 632
GSL_CONST_MKSA_TROY_OUNCE (C macro), 631
GSL_CONST_MKSA_UK_GALLON (C macro), 630
GSL_CONST_MKSA_UK_TON (C macro), 631
GSL_CONST_MKSA_UNIFIED_ATOMIC_MASS (C macro),
628
GSL_CONST_MKSA_US_GALLON (C macro), 630
GSL_CONST_MKSA_VACUUM_PERMEABILITY (C macro),
627
GSL_CONST_MKSA_VACUUM_PERMITTIVITY (C macro),
627
GSL_CONST_MKSA_WEEK (C macro), 629
GSL_CONST_MKSA_YARD (C macro), 629
GSL_CONST_NUM_ATTO (C macro), 634
GSL_CONST_NUM_AVOGADRO (C macro), 627
GSL_CONST_NUM_EXA (C macro), 633
GSL_CONST_NUM_FEMTO (C macro), 634
699

GNU Scientific Library, Release 2.7

GSL_CONST_NUM_FINE_STRUCTURE (C macro), 628
GSL_CONST_NUM_GIGA (C macro), 633
GSL_CONST_NUM_KILO (C macro), 633
GSL_CONST_NUM_MEGA (C macro), 633
GSL_CONST_NUM_MICRO (C macro), 633
GSL_CONST_NUM_MILLI (C macro), 633
GSL_CONST_NUM_NANO (C macro), 634
GSL_CONST_NUM_PETA (C macro), 633
GSL_CONST_NUM_PICO (C macro), 634
GSL_CONST_NUM_TERA (C macro), 633
GSL_CONST_NUM_YOCTO (C macro), 634
GSL_CONST_NUM_YOTTA (C macro), 633
GSL_CONST_NUM_ZEPTO (C macro), 634
GSL_CONST_NUM_ZETTA (C macro), 633
gsl_deriv_backward (C function), 411
gsl_deriv_central (C function), 411
gsl_deriv_forward (C function), 411
gsl_dht (C type), 432
gsl_dht_alloc (C function), 432
gsl_dht_apply (C function), 432
gsl_dht_free (C function), 432
gsl_dht_init (C function), 432
gsl_dht_k_sample (C function), 432
gsl_dht_new (C function), 432
gsl_dht_x_sample (C function), 432
GSL_EDOM (C var), 14
gsl_eigen_gen (C function), 177
gsl_eigen_gen_alloc (C function), 176
gsl_eigen_gen_free (C function), 176
gsl_eigen_gen_params (C function), 176
gsl_eigen_gen_QZ (C function), 177
gsl_eigen_gen_workspace (C type), 176
gsl_eigen_genherm (C function), 175
gsl_eigen_genherm_alloc (C function), 175
gsl_eigen_genherm_free (C function), 175
gsl_eigen_genherm_workspace (C type), 175
gsl_eigen_genhermv (C function), 175
gsl_eigen_genhermv_alloc (C function), 175
gsl_eigen_genhermv_free (C function), 175
gsl_eigen_genhermv_sort (C function), 178
gsl_eigen_genhermv_workspace (C type), 175
gsl_eigen_gensymm (C function), 174
gsl_eigen_gensymm_alloc (C function), 174
gsl_eigen_gensymm_free (C function), 174
gsl_eigen_gensymm_workspace (C type), 174
gsl_eigen_gensymmv (C function), 174
gsl_eigen_gensymmv_alloc (C function), 174
gsl_eigen_gensymmv_free (C function), 174
gsl_eigen_gensymmv_sort (C function), 178
gsl_eigen_gensymmv_workspace (C type), 174
gsl_eigen_genv (C function), 177
gsl_eigen_genv_alloc (C function), 177
gsl_eigen_genv_free (C function), 177
gsl_eigen_genv_QZ (C function), 177
700

gsl_eigen_genv_sort (C function), 178
gsl_eigen_genv_workspace (C type), 177
gsl_eigen_herm (C function), 172
gsl_eigen_herm_alloc (C function), 172
gsl_eigen_herm_free (C function), 172
gsl_eigen_herm_workspace (C type), 172
gsl_eigen_hermv (C function), 172
gsl_eigen_hermv_alloc (C function), 172
gsl_eigen_hermv_free (C function), 172
gsl_eigen_hermv_sort (C function), 178
gsl_eigen_hermv_workspace (C type), 172
gsl_eigen_nonsymm (C function), 173
gsl_eigen_nonsymm_alloc (C function), 173
gsl_eigen_nonsymm_free (C function), 173
gsl_eigen_nonsymm_params (C function), 173
gsl_eigen_nonsymm_workspace (C type), 172
gsl_eigen_nonsymm_Z (C function), 173
gsl_eigen_nonsymmv (C function), 174
gsl_eigen_nonsymmv_alloc (C function), 173
gsl_eigen_nonsymmv_free (C function), 173
gsl_eigen_nonsymmv_params (C function), 173
gsl_eigen_nonsymmv_sort (C function), 178
gsl_eigen_nonsymmv_workspace (C type), 173
gsl_eigen_nonsymmv_Z (C function), 174
gsl_eigen_symm (C function), 171
gsl_eigen_symm_alloc (C function), 171
gsl_eigen_symm_free (C function), 171
gsl_eigen_symm_workspace (C type), 171
gsl_eigen_symmv (C function), 171
gsl_eigen_symmv_alloc (C function), 171
gsl_eigen_symmv_free (C function), 171
gsl_eigen_symmv_sort (C function), 177
gsl_eigen_symmv_sort.gsl_eigen_sort_t(Ctype), 177
gsl_eigen_symmv_workspace (C type), 171
GSL_EINVAL (C var), 14
GSL_ENOMEM (C var), 14
GSL_ERANGE (C var), 14
GSL_ERROR (C macro), 15
gsl_error_handler_t (C type), 14
GSL_ERROR_VAL (C macro), 16
gsl_expm1 (C function), 18
gsl_fcmp (C function), 21
gsl_fft_complex_backward (C function), 189
gsl_fft_complex_forward (C function), 189
gsl_fft_complex_inverse (C function), 189
gsl_fft_complex_radix2_backward (C function),185
gsl_fft_complex_radix2_dif_backward (C function), 185
gsl_fft_complex_radix2_dif_forward (C function), 185
gsl_fft_complex_radix2_dif_inverse (C function), 185
Index

GNU Scientific Library, Release 2.7

gsl_fft_complex_radix2_dif_transform (C function), 185
gsl_fft_complex_radix2_forward (C function), 185
gsl_fft_complex_radix2_inverse (C function), 185
gsl_fft_complex_radix2_transform (C function),185
gsl_fft_complex_transform (C function), 189
gsl_fft_complex_wavetable (C type), 188
gsl_fft_complex_wavetable_alloc (C function),188
gsl_fft_complex_wavetable_free (C function), 188
gsl_fft_complex_workspace (C type), 189
gsl_fft_complex_workspace_alloc (C function),189
gsl_fft_complex_workspace_free (C function), 189
gsl_fft_halfcomplex_radix2_backward (C function), 192
gsl_fft_halfcomplex_radix2_inverse (C function), 192
gsl_fft_halfcomplex_radix2_unpack (C function),
192
gsl_fft_halfcomplex_transform (C function), 194
gsl_fft_halfcomplex_unpack (C function), 195
gsl_fft_halfcomplex_wavetable (C type), 194
gsl_fft_halfcomplex_wavetable_alloc (C function), 194
gsl_fft_halfcomplex_wavetable_free (C function), 194
gsl_fft_real_radix2_transform (C function), 192
gsl_fft_real_transform (C function), 194
gsl_fft_real_unpack (C function), 195
gsl_fft_real_wavetable (C type), 194
gsl_fft_real_wavetable_alloc (C function), 194
gsl_fft_real_wavetable_free (C function), 194
gsl_fft_real_workspace (C type), 194
gsl_fft_real_workspace_alloc (C function), 194
gsl_fft_real_workspace_free (C function), 194
gsl_filter_end_t (C type), 319
gsl_filter_end_t.GSL_FILTER_END_PADVALUE (Cmacro), 319
gsl_filter_end_t.GSL_FILTER_END_PADZERO(C
macro), 319
gsl_filter_end_t.GSL_FILTER_END_TRUNCATE (Cmacro), 319
gsl_filter_gaussian (C function), 320
gsl_filter_gaussian_alloc (C function), 320
gsl_filter_gaussian_free (C function), 320
gsl_filter_gaussian_kernel (C function), 320
gsl_filter_impulse (C function), 323
gsl_filter_impulse_alloc (C function), 323
gsl_filter_impulse_free (C function), 323
gsl_filter_median (C function), 321
gsl_filter_median_alloc (C function), 321
gsl_filter_median_free (C function), 321
Index

gsl_filter_rmedian (C function), 321
gsl_filter_rmedian_alloc (C function), 321
gsl_filter_rmedian_free (C function), 321
gsl_filter_scale_t (C type), 322
gsl_filter_scale_t.GSL_FILTER_SCALE_IQR(Cmacro), 322
gsl_filter_scale_t.GSL_FILTER_SCALE_MAD(C
macro), 322
gsl_filter_scale_t.GSL_FILTER_SCALE_QN(Cmacro), 322
gsl_filter_scale_t.GSL_FILTER_SCALE_SN(C
macro), 322
gsl_finite (C function), 18
gsl_fit_linear (C function), 482
gsl_fit_linear_est (C function), 482
gsl_fit_mul (C function), 483
gsl_fit_mul_est (C function), 483
gsl_fit_wlinear (C function), 482
gsl_fit_wmul (C function), 483
gsl_frexp (C function), 19
gsl_function (C type), 435
gsl_function_fdf (C type), 436
gsl_heapsort (C function), 123
gsl_heapsort.gsl_comparison_fn_t (C type), 123
gsl_heapsort_index (C function), 124
gsl_histogram (C type), 335
gsl_histogram2d (C type), 342
gsl_histogram2d_accumulate (C function), 344
gsl_histogram2d_add (C function), 346
gsl_histogram2d_alloc (C function), 343
gsl_histogram2d_clone (C function), 343
gsl_histogram2d_cov (C function), 345
gsl_histogram2d_div (C function), 346
gsl_histogram2d_equal_bins_p (C function), 346
gsl_histogram2d_find (C function), 345
gsl_histogram2d_fprintf (C function), 346
gsl_histogram2d_fread (C function), 346
gsl_histogram2d_free (C function), 343
gsl_histogram2d_fscanf (C function), 347
gsl_histogram2d_fwrite (C function), 346
gsl_histogram2d_get (C function), 344
gsl_histogram2d_get_xrange (C function), 344
gsl_histogram2d_get_yrange (C function), 344
gsl_histogram2d_increment (C function), 344
gsl_histogram2d_max_bin (C function), 345
gsl_histogram2d_max_val (C function), 345
gsl_histogram2d_memcpy (C function), 343
gsl_histogram2d_min_bin (C function), 345
gsl_histogram2d_min_val (C function), 345
gsl_histogram2d_mul (C function), 346
gsl_histogram2d_nx (C function), 344
gsl_histogram2d_ny (C function), 344
gsl_histogram2d_pdf (C type), 347
gsl_histogram2d_pdf_alloc (C function), 348
701

GNU Scientific Library, Release 2.7

gsl_histogram2d_pdf_free (C function), 348
gsl_histogram2d_pdf_init (C function), 348
gsl_histogram2d_pdf_sample (C function), 348
gsl_histogram2d_reset (C function), 344
gsl_histogram2d_scale (C function), 346
gsl_histogram2d_set_ranges (C function), 343
gsl_histogram2d_set_ranges_uniform (C function), 343
gsl_histogram2d_shift (C function), 346
gsl_histogram2d_sub (C function), 346
gsl_histogram2d_sum (C function), 345
gsl_histogram2d_xmax (C function), 344
gsl_histogram2d_xmean (C function), 345
gsl_histogram2d_xmin (C function), 344
gsl_histogram2d_xsigma (C function), 345
gsl_histogram2d_ymax (C function), 344
gsl_histogram2d_ymean (C function), 345
gsl_histogram2d_ymin (C function), 344
gsl_histogram2d_ysigma (C function), 345
gsl_histogram_accumulate (C function), 337
gsl_histogram_add (C function), 338
gsl_histogram_alloc (C function), 336
gsl_histogram_bins (C function), 337
gsl_histogram_clone (C function), 337
gsl_histogram_div (C function), 339
gsl_histogram_equal_bins_p (C function), 338
gsl_histogram_find (C function), 338
gsl_histogram_fprintf (C function), 339
gsl_histogram_fread (C function), 339
gsl_histogram_free (C function), 336
gsl_histogram_fscanf (C function), 339
gsl_histogram_fwrite (C function), 339
gsl_histogram_get (C function), 337
gsl_histogram_get_range (C function), 337
gsl_histogram_increment (C function), 337
gsl_histogram_max (C function), 337
gsl_histogram_max_bin (C function), 338
gsl_histogram_max_val (C function), 338
gsl_histogram_mean (C function), 338
gsl_histogram_memcpy (C function), 337
gsl_histogram_min (C function), 337
gsl_histogram_min_bin (C function), 338
gsl_histogram_min_val (C function), 338
gsl_histogram_mul (C function), 338
gsl_histogram_pdf (C type), 340
gsl_histogram_pdf_alloc (C function), 340
gsl_histogram_pdf_free (C function), 340
gsl_histogram_pdf_init (C function), 340
gsl_histogram_pdf_sample (C function), 340
gsl_histogram_reset (C function), 337
gsl_histogram_scale (C function), 339
gsl_histogram_set_ranges (C function), 336
gsl_histogram_set_ranges_uniform (C function),336
702

gsl_histogram_shift (C function), 339
gsl_histogram_sigma (C function), 338
gsl_histogram_sub (C function), 338
gsl_histogram_sum (C function), 338
gsl_hypot (C function), 18
gsl_hypot3 (C function), 18
gsl_ieee_env_setup (C function), 639
gsl_ieee_fprintf_double (C function), 638
gsl_ieee_fprintf_float (C function), 638
GSL_IEEE_MODE (C macro), 639
gsl_ieee_printf_double (C function), 638
gsl_ieee_printf_float (C function), 638
GSL_IMAG (C macro), 24
gsl_integration_cquad (C function), 206
gsl_integration_cquad_workspace, 206
gsl_integration_cquad_workspace_alloc (C function), 206
gsl_integration_cquad_workspace_free (C function), 206
gsl_integration_fixed (C function), 209
gsl_integration_fixed_alloc (C function), 208
gsl_integration_fixed_alloc.gsl_integration_fixed_type(C type), 209
gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl(C var), 209
gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl(C var), 209
gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl(C var), 209
gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl(C var), 209
gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl(C var), 209
gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl(C var), 209
gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl(C var), 209
gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl(C var), 209
gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl(C var), 209
gsl_integration_fixed_free (C function), 209
gsl_integration_fixed_n (C function), 209
gsl_integration_fixed_nodes (C function), 209
gsl_integration_fixed_weights (C function), 209
gsl_integration_fixed_workspace (C type), 208
gsl_integration_glfixed (C function), 207
gsl_integration_glfixed_point (C function), 207
gsl_integration_glfixed_table, 207, 208
gsl_integration_glfixed_table_alloc (C function), 207
gsl_integration_glfixed_table_free (C function), 208
gsl_integration_qag (C function), 201
Index

GNU Scientific Library, Release 2.7

gsl_integration_qagi (C function), 202
gsl_interp2d_get (C function), 404
gsl_integration_qagil (C function), 203
gsl_interp2d_idx (C function), 404
gsl_integration_qagiu (C function), 202
gsl_interp2d_init (C function), 403
gsl_integration_qagp (C function), 202
gsl_interp2d_min_size (C function), 404
gsl_integration_qags (C function), 202
gsl_interp2d_name (C function), 404
gsl_integration_qawc (C function), 203
gsl_interp2d_set (C function), 404
gsl_integration_qawf (C function), 205
gsl_interp2d_type (C type), 404
gsl_integration_qawo (C function), 205
gsl_interp2d_type.gsl_interp2d_bicubic(C
gsl_integration_qawo_table, 204var), 404
gsl_integration_qawo_table_alloc (C function), gsl_interp2d_type.gsl_interp2d_bilinear(C204var), 404
gsl_integration_qawo_table_alloc.GSL_INTEG_COSINE
gsl_interp2d_type_min_size (C function), 404(C macro), 204
gsl_interp_accel (C type), 397
gsl_integration_qawo_table_alloc.GSL_INTEG_SINE
gsl_interp_accel_alloc (C function), 397(C macro), 204
gsl_interp_accel_find (C function), 397
gsl_integration_qawo_table_free (C function), gsl_interp_accel_free (C function), 397
205
gsl_interp_accel_reset (C function), 397
gsl_integration_qawo_table_set (C function), 205 gsl_interp_alloc (C function), 395
gsl_integration_qawo_table_set_length (C func- gsl_interp_bsearch (C function), 397tion), 205
gsl_interp_eval (C function), 397
gsl_integration_qaws (C function), 204
gsl_interp_eval_deriv (C function), 397
gsl_integration_qaws_table (C type), 203
gsl_interp_eval_deriv2 (C function), 397
gsl_integration_qaws_table_alloc (C function), gsl_interp_eval_deriv2_e (C function), 397
203
gsl_interp_eval_deriv_e (C function), 397
gsl_integration_qaws_table_free (C function), gsl_interp_eval_e (C function), 397
204
gsl_interp_eval_integ (C function), 398
gsl_integration_qaws_table_set (C function), 204 gsl_interp_eval_integ_e (C function), 398
gsl_integration_qng (C function), 201
gsl_interp_free (C function), 395
gsl_integration_romberg (C function), 207
gsl_interp_init (C function), 395
gsl_integration_romberg_alloc (C function), 207
gsl_interp_min_size (C function), 396
gsl_integration_romberg_free (C function), 207
gsl_interp_name (C function), 396
gsl_integration_workspace, 201
gsl_interp_type (C type), 396
gsl_integration_workspace (C type), 201
gsl_interp_type.gsl_interp_akima (C var), 396
gsl_integration_workspace_alloc (C function), gsl_interp_type.gsl_interp_akima_periodic (C201var), 396
gsl_integration_workspace_free (C function), 201 gsl_interp_type.gsl_interp_cspline (C var), 396
gsl_interp (C type), 395
gsl_interp_type.gsl_interp_cspline_periodic
gsl_interp2d (C type), 403(C var), 396
gsl_interp2d_alloc (C function), 403
gsl_interp_type.gsl_interp_linear (C var), 396
gsl_interp2d_eval (C function), 405
gsl_interp_type.gsl_interp_polynomial (C var),
gsl_interp2d_eval_deriv_x (C function), 405
396
gsl_interp2d_eval_deriv_x_e (C function), 405
gsl_interp_type.gsl_interp_steffen (C var), 396
gsl_interp2d_eval_deriv_xx (C function), 405
gsl_interp_type_min_size (C function), 396
gsl_interp2d_eval_deriv_xx_e (C function), 405
GSL_IS_EVEN (C macro), 20
gsl_interp2d_eval_deriv_xy (C function), 406
GSL_IS_ODD (C macro), 20
gsl_interp2d_eval_deriv_xy_e (C function), 406
gsl_isinf (C function), 18
gsl_interp2d_eval_deriv_y (C function), 405
gsl_isnan (C function), 18
gsl_interp2d_eval_deriv_y_e (C function), 405
gsl_ldexp (C function), 19
gsl_interp2d_eval_deriv_yy (C function), 405
gsl_linalg_balance_matrix (C function), 168
gsl_interp2d_eval_deriv_yy_e (C function), 405
gsl_linalg_bidiag_decomp (C function), 160
gsl_interp2d_eval_e (C function), 405
gsl_linalg_bidiag_unpack (C function), 160
gsl_interp2d_eval_extrap (C function), 405
gsl_linalg_bidiag_unpack2 (C function), 160
gsl_interp2d_eval_extrap_e (C function), 405
gsl_linalg_bidiag_unpack_B (C function), 160
gsl_interp2d_free (C function), 404
Index

703

GNU Scientific Library, Release 2.7

gsl_linalg_cholesky_band_decomp (C function),166
gsl_linalg_cholesky_band_invert (C function),166
gsl_linalg_cholesky_band_rcond (C function), 167
gsl_linalg_cholesky_band_scale (C function), 167
gsl_linalg_cholesky_band_scale_apply (C function), 167
gsl_linalg_cholesky_band_solve (C function), 166
gsl_linalg_cholesky_band_solvem (C function),166
gsl_linalg_cholesky_band_svx (C function), 166
gsl_linalg_cholesky_band_svxm (C function), 166
gsl_linalg_cholesky_band_unpack (C function),166
gsl_linalg_cholesky_decomp (C function), 154
gsl_linalg_cholesky_decomp1 (C function), 154
gsl_linalg_cholesky_decomp2 (C function), 155
gsl_linalg_cholesky_invert (C function), 155
gsl_linalg_cholesky_rcond (C function), 155
gsl_linalg_cholesky_scale (C function), 155
gsl_linalg_cholesky_scale_apply (C function),155
gsl_linalg_cholesky_solve (C function), 155
gsl_linalg_cholesky_solve2 (C function), 155
gsl_linalg_cholesky_svx (C function), 155
gsl_linalg_cholesky_svx2 (C function), 155
gsl_linalg_COD_decomp (C function), 152
gsl_linalg_COD_decomp_e (C function), 152
gsl_linalg_COD_lssolve (C function), 152
gsl_linalg_COD_lssolve2 (C function), 153
gsl_linalg_COD_matZ (C function), 153
gsl_linalg_COD_unpack (C function), 153
gsl_linalg_complex_cholesky_decomp (C function), 154
gsl_linalg_complex_cholesky_invert (C function), 155
gsl_linalg_complex_cholesky_solve (C function),155
gsl_linalg_complex_cholesky_svx (C function),155
gsl_linalg_complex_householder_hm (C function),161
gsl_linalg_complex_householder_hv (C function),162
gsl_linalg_complex_householder_mh (C function),162
gsl_linalg_complex_householder_transform (Cfunction), 161
gsl_linalg_complex_LU_decomp (C function), 141
gsl_linalg_complex_LU_det (C function), 142
gsl_linalg_complex_LU_invert (C function), 142
gsl_linalg_complex_LU_invx (C function), 142
gsl_linalg_complex_LU_lndet (C function), 142704

gsl_linalg_complex_LU_refine (C function), 142
gsl_linalg_complex_LU_sgndet (C function), 142
gsl_linalg_complex_LU_solve (C function), 141
gsl_linalg_complex_LU_svx (C function), 141
gsl_linalg_complex_QR_decomp (C function), 145
gsl_linalg_complex_QR_decomp_r (C function), 143
gsl_linalg_complex_QR_lssolve (C function), 145
gsl_linalg_complex_QR_lssolve_r (C function),144
gsl_linalg_complex_QR_QHvec (C function), 145
gsl_linalg_complex_QR_QHvec_r (C function), 144
gsl_linalg_complex_QR_Qvec (C function), 145
gsl_linalg_complex_QR_solve (C function), 145
gsl_linalg_complex_QR_solve_r (C function), 143
gsl_linalg_complex_QR_svx (C function), 145
gsl_linalg_complex_QR_unpack_r (C function), 144
gsl_linalg_complex_tri_invert (C function), 163
gsl_linalg_complex_tri_LHL (C function), 163
gsl_linalg_complex_tri_UL (C function), 163
gsl_linalg_givens (C function), 161
gsl_linalg_givens_gv (C function), 161
gsl_linalg_hermtd_decomp (C function), 159
gsl_linalg_hermtd_unpack (C function), 159
gsl_linalg_hermtd_unpack_T (C function), 159
gsl_linalg_hessenberg_decomp (C function), 159
gsl_linalg_hessenberg_set_zero (C function), 160
gsl_linalg_hessenberg_unpack (C function), 159
gsl_linalg_hessenberg_unpack_accum (C function), 159
gsl_linalg_hesstri_decomp (C function), 160
gsl_linalg_HH_solve (C function), 162
gsl_linalg_HH_svx (C function), 162
gsl_linalg_householder_hm (C function), 161
gsl_linalg_householder_hv (C function), 162
gsl_linalg_householder_mh (C function), 162
gsl_linalg_householder_transform (C function),161
gsl_linalg_ldlt_band_decomp (C function), 167
gsl_linalg_ldlt_band_rcond (C function), 167
gsl_linalg_ldlt_band_solve (C function), 167
gsl_linalg_ldlt_band_svx (C function), 167
gsl_linalg_ldlt_band_unpack (C function), 167
gsl_linalg_ldlt_decomp (C function), 158
gsl_linalg_ldlt_rcond (C function), 158
gsl_linalg_ldlt_solve (C function), 158
gsl_linalg_ldlt_svx (C function), 158
gsl_linalg_LQ_decomp (C function), 151
gsl_linalg_LQ_lssolve (C function), 151
gsl_linalg_LQ_QTvec (C function), 151
gsl_linalg_LQ_unpack (C function), 151
gsl_linalg_LU_band_decomp (C function), 165
gsl_linalg_LU_band_solve (C function), 165
gsl_linalg_LU_band_svx (C function), 166
gsl_linalg_LU_band_unpack (C function), 166
IndexGNU Scientific Library, Release 2.7

gsl_linalg_LU_decomp (C function), 141
gsl_linalg_LU_det (C function), 142
gsl_linalg_LU_invert (C function), 142
gsl_linalg_LU_invx (C function), 142
gsl_linalg_LU_lndet (C function), 142
gsl_linalg_LU_refine (C function), 142
gsl_linalg_LU_sgndet (C function), 142
gsl_linalg_LU_solve (C function), 141
gsl_linalg_LU_svx (C function), 141
gsl_linalg_mcholesky_decomp (C function), 157
gsl_linalg_mcholesky_rcond (C function), 157
gsl_linalg_mcholesky_solve (C function), 157
gsl_linalg_mcholesky_svx (C function), 157
gsl_linalg_pcholesky_decomp (C function), 156
gsl_linalg_pcholesky_decomp2 (C function), 156
gsl_linalg_pcholesky_invert (C function), 157
gsl_linalg_pcholesky_rcond (C function), 157
gsl_linalg_pcholesky_solve (C function), 156
gsl_linalg_pcholesky_solve2 (C function), 156
gsl_linalg_pcholesky_svx (C function), 156
gsl_linalg_pcholesky_svx2 (C function), 156
gsl_linalg_QL_decomp (C function), 151
gsl_linalg_QL_unpack (C function), 151
gsl_linalg_QR_decomp (C function), 145
gsl_linalg_QR_decomp_r (C function), 143
gsl_linalg_QR_lssolve (C function), 145
gsl_linalg_QR_lssolve_r (C function), 144
gsl_linalg_QR_QRsolve (C function), 146
gsl_linalg_QR_QTmat (C function), 146
gsl_linalg_QR_QTmat_r (C function), 144
gsl_linalg_QR_QTvec (C function), 145
gsl_linalg_QR_QTvec_r (C function), 144
gsl_linalg_QR_Qvec (C function), 145
gsl_linalg_QR_rcond (C function), 144
gsl_linalg_QR_Rsolve (C function), 146
gsl_linalg_QR_Rsvx (C function), 146
gsl_linalg_QR_solve (C function), 145
gsl_linalg_QR_solve_r (C function), 143
gsl_linalg_QR_svx (C function), 145
gsl_linalg_QR_UD_decomp (C function), 148
gsl_linalg_QR_UD_lssolve (C function), 148
gsl_linalg_QR_unpack (C function), 146
gsl_linalg_QR_unpack_r (C function), 144
gsl_linalg_QR_update (C function), 146
gsl_linalg_QR_UR_decomp (C function), 146
gsl_linalg_QR_UU_decomp (C function), 147
gsl_linalg_QR_UU_lssolve (C function), 147
gsl_linalg_QR_UU_QTec (C function), 147
gsl_linalg_QR_UZ_decomp (C function), 148
gsl_linalg_QRPT_decomp (C function), 149
gsl_linalg_QRPT_decomp2 (C function), 149
gsl_linalg_QRPT_lssolve (C function), 149
gsl_linalg_QRPT_lssolve2 (C function), 150
gsl_linalg_QRPT_QRsolve (C function), 150
Index

gsl_linalg_QRPT_rank (C function), 150
gsl_linalg_QRPT_rcond (C function), 150
gsl_linalg_QRPT_Rsolve (C function), 150
gsl_linalg_QRPT_Rsvx (C function), 150
gsl_linalg_QRPT_solve (C function), 149
gsl_linalg_QRPT_svx (C function), 149
gsl_linalg_QRPT_update (C function), 150
gsl_linalg_R_solve (C function), 146
gsl_linalg_R_svx (C function), 146
gsl_linalg_solve_cyc_tridiag (C function), 163
gsl_linalg_solve_symm_cyc_tridiag (C function),163
gsl_linalg_solve_symm_tridiag (C function), 162
gsl_linalg_solve_tridiag (C function), 162
gsl_linalg_SV_decomp (C function), 153
gsl_linalg_SV_decomp_jacobi (C function), 154
gsl_linalg_SV_decomp_mod (C function), 154
gsl_linalg_SV_leverage (C function), 154
gsl_linalg_SV_solve (C function), 154
gsl_linalg_symmtd_decomp (C function), 158
gsl_linalg_symmtd_unpack (C function), 158
gsl_linalg_symmtd_unpack_T (C function), 159
gsl_linalg_tri_invert (C function), 163
gsl_linalg_tri_LTL (C function), 163
gsl_linalg_tri_rcond (C function), 163
gsl_linalg_tri_UL (C function), 163
gsl_log1p (C function), 18
gsl_matrix (C type), 94
gsl_matrix_add (C function), 101
gsl_matrix_add_constant (C function), 101
gsl_matrix_alloc (C function), 95
gsl_matrix_calloc (C function), 95
gsl_matrix_column (C function), 98
gsl_matrix_complex_conjtrans_memcpy (C function), 100
gsl_matrix_const_column (C function), 98
gsl_matrix_const_diagonal (C function), 99
gsl_matrix_const_ptr (C function), 95
gsl_matrix_const_row (C function), 98
gsl_matrix_const_subcolumn (C function), 99
gsl_matrix_const_subdiagonal (C function), 99
gsl_matrix_const_submatrix (C function), 96
gsl_matrix_const_subrow (C function), 99
gsl_matrix_const_superdiagonal (C function), 99
gsl_matrix_const_view (C type), 96
gsl_matrix_const_view_array (C function), 97
gsl_matrix_const_view_array_with_tda (C function), 97
gsl_matrix_const_view_vector (C function), 98
gsl_matrix_const_view_vector_with_tda (C function), 98
gsl_matrix_diagonal (C function), 99
gsl_matrix_div_elements (C function), 101
gsl_matrix_equal (C function), 102705

GNU Scientific Library, Release 2.7

gsl_matrix_fprintf (C function), 96
gsl_matrix_fread (C function), 96
gsl_matrix_free (C function), 95
gsl_matrix_fscanf (C function), 96
gsl_matrix_fwrite (C function), 96
gsl_matrix_get (C function), 95
gsl_matrix_get_col (C function), 100
gsl_matrix_get_row (C function), 100
gsl_matrix_isneg (C function), 102
gsl_matrix_isnonneg (C function), 102
gsl_matrix_isnull (C function), 102
gsl_matrix_ispos (C function), 102
gsl_matrix_max (C function), 102
gsl_matrix_max_index (C function), 102
gsl_matrix_memcpy (C function), 100
gsl_matrix_min (C function), 102
gsl_matrix_min_index (C function), 102
gsl_matrix_minmax (C function), 102
gsl_matrix_minmax_index (C function), 102
gsl_matrix_mul_elements (C function), 101
gsl_matrix_norm1 (C function), 102
gsl_matrix_ptr (C function), 95
gsl_matrix_row (C function), 98
gsl_matrix_scale (C function), 101
gsl_matrix_scale_columns (C function), 101
gsl_matrix_scale_rows (C function), 101
gsl_matrix_set (C function), 95
gsl_matrix_set_all (C function), 96
gsl_matrix_set_col (C function), 100
gsl_matrix_set_identity (C function), 96
gsl_matrix_set_row (C function), 100
gsl_matrix_set_zero (C function), 96
gsl_matrix_sub (C function), 101
gsl_matrix_subcolumn (C function), 99
gsl_matrix_subdiagonal (C function), 99
gsl_matrix_submatrix (C function), 96
gsl_matrix_subrow (C function), 99
gsl_matrix_superdiagonal (C function), 99
gsl_matrix_swap (C function), 100
gsl_matrix_swap_columns (C function), 100
gsl_matrix_swap_rowcol (C function), 100
gsl_matrix_swap_rows (C function), 100
gsl_matrix_transpose (C function), 100
gsl_matrix_transpose_memcpy (C function), 100
gsl_matrix_view (C type), 96
gsl_matrix_view_array (C function), 97
gsl_matrix_view_array_with_tda (C function), 97
gsl_matrix_view_vector (C function), 98
gsl_matrix_view_vector_with_tda (C function), 98
GSL_MAX (C macro), 20
GSL_MAX_DBL (C function), 20
GSL_MAX_INT (C function), 20
GSL_MAX_LDBL (C function), 20
GSL_MIN (C macro), 20
706

GSL_MIN_DBL (C function), 20
gsl_min_fminimizer (C type), 448
gsl_min_fminimizer_alloc (C function), 448
gsl_min_fminimizer_f_lower (C function), 449
gsl_min_fminimizer_f_minimum (C function), 449
gsl_min_fminimizer_f_upper (C function), 449
gsl_min_fminimizer_free (C function), 449
gsl_min_fminimizer_iterate (C function), 449
gsl_min_fminimizer_name (C function), 449
gsl_min_fminimizer_set (C function), 448
gsl_min_fminimizer_set_with_values (C function), 448
gsl_min_fminimizer_type (C type), 450
gsl_min_fminimizer_type.gsl_min_fminimizer_brent(C var), 450
gsl_min_fminimizer_type.gsl_min_fminimizer_goldensection(C var), 450
gsl_min_fminimizer_type.gsl_min_fminimizer_quad_golden(C var), 451
gsl_min_fminimizer_x_lower (C function), 449
gsl_min_fminimizer_x_minimum (C function), 449
gsl_min_fminimizer_x_upper (C function), 449
GSL_MIN_INT (C function), 20
GSL_MIN_LDBL (C function), 20
gsl_min_test_interval (C function), 450
gsl_mode_t (C type), 36
gsl_mode_t.GSL_PREC_APPROX (C macro), 36
gsl_mode_t.GSL_PREC_DOUBLE (C macro), 36
gsl_mode_t.GSL_PREC_SINGLE (C macro), 36
gsl_monte_function (C type), 357
gsl_monte_miser_alloc (C function), 359
gsl_monte_miser_free (C function), 360
gsl_monte_miser_init (C function), 359
gsl_monte_miser_integrate (C function), 360
gsl_monte_miser_params (C type), 360
gsl_monte_miser_params.alpha (C var), 360
gsl_monte_miser_params.dither (C var), 360
gsl_monte_miser_params.estimate_frac (C var),360
gsl_monte_miser_params.min_calls (C var), 360
gsl_monte_miser_params.min_calls_per_bisection(C var), 360
gsl_monte_miser_params_get (C function), 360
gsl_monte_miser_params_set (C function), 360
gsl_monte_miser_state (C type), 359
gsl_monte_plain_alloc (C function), 358
gsl_monte_plain_free (C function), 359
gsl_monte_plain_init (C function), 359
gsl_monte_plain_integrate (C function), 359
gsl_monte_plain_state (C type), 358
gsl_monte_vegas_alloc (C function), 361
gsl_monte_vegas_chisq (C function), 362
gsl_monte_vegas_free (C function), 361
gsl_monte_vegas_init (C function), 361
Index

GNU Scientific Library, Release 2.7

gsl_monte_vegas_integrate (C function), 361
gsl_monte_vegas_params (C type), 362
gsl_monte_vegas_params.alpha (C var), 362
gsl_monte_vegas_params.iterations (C var), 362
gsl_monte_vegas_params.mode (C var), 363
gsl_monte_vegas_params.ostream (C var), 363
gsl_monte_vegas_params.stage (C var), 362
gsl_monte_vegas_params.verbose (C var), 363
gsl_monte_vegas_params_get (C function), 362
gsl_monte_vegas_params_set (C function), 362
gsl_monte_vegas_runval (C function), 362
gsl_monte_vegas_state (C type), 361
gsl_movstat_accum (C type), 310
gsl_movstat_accum.delete (C member), 311
gsl_movstat_accum.get (C member), 311
gsl_movstat_accum.init (C member), 311
gsl_movstat_accum.insert (C member), 311
gsl_movstat_accum.size (C member), 311
gsl_movstat_accum_max (C var), 311
gsl_movstat_accum_mean (C var), 311
gsl_movstat_accum_median (C var), 311
gsl_movstat_accum_min (C var), 311
gsl_movstat_accum_minmax (C var), 311
gsl_movstat_accum_Qn (C var), 311
gsl_movstat_accum_qqr (C var), 311
gsl_movstat_accum_sd (C var), 311
gsl_movstat_accum_Sn (C var), 311
gsl_movstat_accum_sum (C var), 311
gsl_movstat_accum_variance (C var), 311
gsl_movstat_alloc (C function), 306
gsl_movstat_alloc2 (C function), 306
gsl_movstat_apply (C function), 310
gsl_movstat_end_t (C type), 305
gsl_movstat_end_t.GSL_MOVSTAT_END_PADVALUE(C macro), 305
gsl_movstat_end_t.GSL_MOVSTAT_END_PADZERO (Cmacro), 305
gsl_movstat_end_t.GSL_MOVSTAT_END_TRUNCATE(C macro), 306
gsl_movstat_fill (C function), 310
gsl_movstat_free (C function), 306
gsl_movstat_function (C type), 310
gsl_movstat_function.function (C member), 310
gsl_movstat_function.params (C member), 310
gsl_movstat_mad (C function), 308
gsl_movstat_mad0 (C function), 308
gsl_movstat_max (C function), 307
gsl_movstat_mean (C function), 306
gsl_movstat_median (C function), 308
gsl_movstat_min (C function), 307
gsl_movstat_minmax (C function), 307
gsl_movstat_Qn (C function), 309
gsl_movstat_qqr (C function), 309
gsl_movstat_sd (C function), 307
Index

gsl_movstat_Sn (C function), 309
gsl_movstat_sum (C function), 308
gsl_movstat_variance (C function), 307
gsl_movstat_workspace (C type), 306
gsl_multifit_linear (C function), 484
gsl_multifit_linear_alloc (C function), 484
gsl_multifit_linear_applyW (C function), 488
gsl_multifit_linear_bsvd (C function), 484
gsl_multifit_linear_est (C function), 485
gsl_multifit_linear_free (C function), 484
gsl_multifit_linear_gcv (C function), 489
gsl_multifit_linear_gcv_calc (C function), 489
gsl_multifit_linear_gcv_curve (C function), 489
gsl_multifit_linear_gcv_init (C function), 488
gsl_multifit_linear_gcv_min (C function), 489
gsl_multifit_linear_genform1 (C function), 487
gsl_multifit_linear_genform2 (C function), 487
gsl_multifit_linear_L_decomp (C function), 487
gsl_multifit_linear_lcorner (C function), 488
gsl_multifit_linear_lcorner2 (C function), 488
gsl_multifit_linear_lcurvature (C function), 488
gsl_multifit_linear_lcurve (C function), 488
gsl_multifit_linear_Lk (C function), 489
gsl_multifit_linear_Lsobolev (C function), 489
gsl_multifit_linear_rank (C function), 485
gsl_multifit_linear_rcond (C function), 489
gsl_multifit_linear_residuals (C function), 485
gsl_multifit_linear_solve (C function), 487
gsl_multifit_linear_stdform1 (C function), 486
gsl_multifit_linear_stdform2 (C function), 487
gsl_multifit_linear_svd (C function), 484
gsl_multifit_linear_tsvd (C function), 484
gsl_multifit_linear_wgenform2 (C function), 487
gsl_multifit_linear_workspace (C type), 483
gsl_multifit_linear_wstdform1 (C function), 486
gsl_multifit_linear_wstdform2 (C function), 487
gsl_multifit_nlinear_alloc (C function), 529
gsl_multifit_nlinear_avratio (C function), 533
gsl_multifit_nlinear_covar (C function), 535
gsl_multifit_nlinear_default_parameters(Cfunction), 529
gsl_multifit_nlinear_driver (C function), 534
gsl_multifit_nlinear_fdf (C type), 530
gsl_multifit_nlinear_fdtype (C type), 528
gsl_multifit_nlinear_fdtype.GSL_MULTIFIT_NLINEAR_CTRDIFF(C macro), 528
gsl_multifit_nlinear_fdtype.GSL_MULTIFIT_NLINEAR_FWDIFF(C macro), 528
gsl_multifit_nlinear_free (C function), 530
gsl_multifit_nlinear_init (C function), 529
gsl_multifit_nlinear_iterate (C function), 532
gsl_multifit_nlinear_jac (C function), 533
gsl_multifit_nlinear_name (C function), 530
gsl_multifit_nlinear_niter (C function), 533
707

GNU Scientific Library, Release 2.7

gsl_multifit_nlinear_parameters (C type), 525
gsl_multilarge_linear_genform1 (C function), 496
gsl_multifit_nlinear_position (C function), 533
gsl_multilarge_linear_genform2 (C function), 496
gsl_multifit_nlinear_rcond (C function), 533
gsl_multilarge_linear_L_decomp (C function), 496
gsl_multifit_nlinear_residual (C function), 533
gsl_multilarge_linear_lcurve (C function), 497
gsl_multifit_nlinear_scale (C type), 526
gsl_multilarge_linear_matrix_ptr (C function),
gsl_multifit_nlinear_solver (C type), 527497
gsl_multifit_nlinear_test (C function), 534
gsl_multilarge_linear_name (C function), 496
gsl_multifit_nlinear_trs (C type), 526
gsl_multilarge_linear_rcond (C function), 497
gsl_multifit_nlinear_trs_name (C function), 530
gsl_multilarge_linear_reset (C function), 496
gsl_multifit_nlinear_type (C type), 529
gsl_multilarge_linear_rhs_ptr (C function), 497
gsl_multifit_nlinear_type.gsl_multifit_nlinear_trust
gsl_multilarge_linear_solve (C function), 496(C var), 529
gsl_multilarge_linear_stdform1 (C function), 496
gsl_multifit_nlinear_winit (C function), 529
gsl_multilarge_linear_stdform2 (C function), 496
gsl_multifit_robust (C function), 492
gsl_multilarge_linear_workspace (C type), 495
gsl_multifit_robust_alloc (C function), 490
gsl_multilarge_linear_wstdform1 (C function),
gsl_multifit_robust_alloc.gsl_multifit_robust_type496(C type), 490
gsl_multilarge_linear_wstdform2 (C function),
gsl_multifit_robust_alloc.gsl_multifit_robust_type.gsl_multifit_robust_bisquare
496(C var), 490
gsl_multilarge_nlinear_alloc (C function), 529
gsl_multifit_robust_alloc.gsl_multifit_robust_type.gsl_multifit_robust_cauchy
gsl_multilarge_nlinear_avratio (C function), 533(C var), 491
gsl_multilarge_nlinear_covar (C function), 535
gsl_multifit_robust_alloc.gsl_multifit_robust_type.gsl_multifit_robust_default
gsl_multilarge_nlinear_default_parameters (C(C var), 490
function), 529
gsl_multifit_robust_alloc.gsl_multifit_robust_type.gsl_multifit_robust_fair
gsl_multilarge_nlinear_driver (C function), 534(C var), 491
gsl_multilarge_nlinear_fdf (C type), 531
gsl_multifit_robust_alloc.gsl_multifit_robust_type.gsl_multifit_robust_huber
gsl_multilarge_nlinear_free (C function), 530(C var), 491
gsl_multilarge_nlinear_init (C function), 529
gsl_multifit_robust_alloc.gsl_multifit_robust_type.gsl_multifit_robust_ols
gsl_multilarge_nlinear_iterate (C function), 532(C var), 491
gsl_multilarge_nlinear_name (C function), 530
gsl_multifit_robust_alloc.gsl_multifit_robust_type.gsl_multifit_robust_welsch
gsl_multilarge_nlinear_niter (C function), 533(C var), 491
gsl_multilarge_nlinear_parameters (C type), 525
gsl_multifit_robust_est (C function), 492
gsl_multilarge_nlinear_position (C function),
gsl_multifit_robust_free (C function), 491533
gsl_multifit_robust_maxiter (C function), 492
gsl_multilarge_nlinear_rcond (C function), 533
gsl_multifit_robust_name (C function), 491
gsl_multilarge_nlinear_residual (C function),
gsl_multifit_robust_residuals (C function), 492
533
gsl_multifit_robust_statistics (C function), 492 gsl_multilarge_nlinear_scale (C type), 526
gsl_multifit_robust_statistics.gsl_multifit_robust_stats
gsl_multilarge_nlinear_scale.gsl_multifit_nlinear_scale_le(C type), 492(C var), 527
gsl_multifit_robust_tune (C function), 491
gsl_multilarge_nlinear_scale.gsl_multifit_nlinear_scale_ma
gsl_multifit_robust_weights (C function), 492(C var), 527
gsl_multifit_robust_workspace (C type), 490
gsl_multilarge_nlinear_scale.gsl_multifit_nlinear_scale_mo
gsl_multifit_wlinear (C function), 484(C var), 526
gsl_multifit_wlinear_tsvd (C function), 484
gsl_multilarge_nlinear_scale.gsl_multilarge_nlinear_scale_
gsl_multilarge_linear_accumulate (C function),(C var), 527496
gsl_multilarge_nlinear_scale.gsl_multilarge_nlinear_scale_
gsl_multilarge_linear_alloc (C function), 495(C var), 527
gsl_multilarge_linear_alloc.gsl_multilarge_linear_type
gsl_multilarge_nlinear_scale.gsl_multilarge_nlinear_scale_(C type), 495(C var), 526
gsl_multilarge_linear_alloc.gsl_multilarge_linear_type.gsl_multilarge_linear_normal
gsl_multilarge_nlinear_solver (C type), 527(C var), 495
gsl_multilarge_nlinear_solver.gsl_multifit_nlinear_solver_
gsl_multilarge_linear_alloc.gsl_multilarge_linear_type.gsl_multilarge_linear_tsqr(C var), 527(C var), 495
gsl_multilarge_nlinear_solver.gsl_multifit_nlinear_solver_
gsl_multilarge_linear_free (C function), 495(C var), 527
708

Index

GNU Scientific Library, Release 2.7

gsl_multilarge_nlinear_solver.gsl_multifit_nlinear_solver_qr
gsl_multimin_fdfminimizer_type.gsl_multimin_fdfminimizer_s(C var), 527(C var), 474
gsl_multilarge_nlinear_solver.gsl_multifit_nlinear_solver_svd
gsl_multimin_fdfminimizer_type.gsl_multimin_fdfminimizer_v(C var), 528(C var), 474
gsl_multilarge_nlinear_solver.gsl_multilarge_nlinear_solver_cholesky
gsl_multimin_fdfminimizer_type.gsl_multimin_fdfminimizer_v(C var), 527(C var), 474
gsl_multilarge_nlinear_solver.gsl_multilarge_nlinear_solver_mcholesky
gsl_multimin_fdfminimizer_x (C function), 473(C var), 527
gsl_multimin_fminimizer (C type), 470
gsl_multilarge_nlinear_test (C function), 534
gsl_multimin_fminimizer_alloc (C function), 470
gsl_multilarge_nlinear_trs (C type), 526
gsl_multimin_fminimizer_free (C function), 470
gsl_multilarge_nlinear_trs.gsl_multifit_nlinear_trs_ddogleg
gsl_multimin_fminimizer_iterate (C function),(C var), 526473
gsl_multilarge_nlinear_trs.gsl_multifit_nlinear_trs_dogleg
gsl_multimin_fminimizer_minimum (C function),(C var), 526473
gsl_multilarge_nlinear_trs.gsl_multifit_nlinear_trs_lm
gsl_multimin_fminimizer_name (C function), 470(C var), 526
gsl_multimin_fminimizer_set (C function), 470
gsl_multilarge_nlinear_trs.gsl_multifit_nlinear_trs_lmaccel
gsl_multimin_fminimizer_size (C function), 473(C var), 526
gsl_multimin_fminimizer_type (C type), 475
gsl_multilarge_nlinear_trs.gsl_multifit_nlinear_trs_subspace2D
gsl_multimin_fminimizer_type.gsl_multimin_fminimizer_nmsim(C var), 526(C var), 475
gsl_multilarge_nlinear_trs.gsl_multilarge_nlinear_trs_cgst
gsl_multimin_fminimizer_type.gsl_multimin_fminimizer_nmsim(C var), 526(C var), 475
gsl_multilarge_nlinear_trs.gsl_multilarge_nlinear_trs_ddogleg
gsl_multimin_fminimizer_type.gsl_multimin_fminimizer_nmsim(C var), 526(C var), 475
gsl_multilarge_nlinear_trs.gsl_multilarge_nlinear_trs_dogleg
gsl_multimin_fminimizer_x (C function), 473(C var), 526
gsl_multimin_function (C type), 471
gsl_multilarge_nlinear_trs.gsl_multilarge_nlinear_trs_lm
gsl_multimin_function_fdf (C type), 471(C var), 526
gsl_multimin_test_gradient (C function), 473
gsl_multilarge_nlinear_trs.gsl_multilarge_nlinear_trs_lmaccel
gsl_multimin_test_size (C function), 474(C var), 526
gsl_multiroot_fdfsolver (C type), 456
gsl_multilarge_nlinear_trs.gsl_multilarge_nlinear_trs_subspace2D
gsl_multiroot_fdfsolver_alloc (C function), 456(C var), 526
gsl_multiroot_fdfsolver_dx (C function), 459
gsl_multilarge_nlinear_trs_name (C function), gsl_multiroot_fdfsolver_f (C function), 459
530
gsl_multiroot_fdfsolver_free (C function), 456
gsl_multimin_fdfminimizer (C type), 470
gsl_multiroot_fdfsolver_iterate (C function),
gsl_multimin_fdfminimizer_alloc (C function),459
470
gsl_multiroot_fdfsolver_name (C function), 456
gsl_multimin_fdfminimizer_dx (C function), 473
gsl_multiroot_fdfsolver_root (C function), 459
gsl_multimin_fdfminimizer_free (C function), 470 gsl_multiroot_fdfsolver_set (C function), 456
gsl_multimin_fdfminimizer_gradient (C func- gsl_multiroot_fdfsolver_type (C type), 460tion), 473
gsl_multiroot_fdfsolver_type.gsl_multiroot_fdfsolver_gnewt
gsl_multimin_fdfminimizer_iterate (C function),(C var), 461473
gsl_multiroot_fdfsolver_type.gsl_multiroot_fdfsolver_hybri
gsl_multimin_fdfminimizer_minimum (C function),(C var), 461473
gsl_multiroot_fdfsolver_type.gsl_multiroot_fdfsolver_hybri
gsl_multimin_fdfminimizer_name (C function), 470(C var), 460
gsl_multimin_fdfminimizer_restart (C function), gsl_multiroot_fdfsolver_type.gsl_multiroot_fdfsolver_newto
473(C var), 461
gsl_multimin_fdfminimizer_set (C function), 470
gsl_multiroot_fsolver (C type), 456
gsl_multimin_fdfminimizer_type (C type), 474
gsl_multiroot_fsolver_alloc (C function), 456
gsl_multimin_fdfminimizer_type.gsl_multimin_fdfminimizer_conjugate_fr
gsl_multiroot_fsolver_dx (C function), 459(C var), 474
gsl_multiroot_fsolver_f (C function), 459
gsl_multimin_fdfminimizer_type.gsl_multimin_fdfminimizer_conjugate_pr
gsl_multiroot_fsolver_free (C function), 456(C var), 474
gsl_multiroot_fsolver_iterate (C function), 459
Index

709

GNU Scientific Library, Release 2.7

gsl_multiroot_fsolver_name (C function), 456
gsl_odeiv2_control_set_driver (C function), 387
gsl_multiroot_fsolver_root (C function), 459
gsl_odeiv2_control_standard_new (C function),
gsl_multiroot_fsolver_set (C function), 456386
gsl_multiroot_fsolver_type (C type), 462
gsl_odeiv2_control_type (C type), 386
gsl_multiroot_fsolver_type.gsl_multiroot_fsolver_broyden
gsl_odeiv2_control_y_new (C function), 386(C var), 462
gsl_odeiv2_control_yp_new (C function), 386
gsl_multiroot_fsolver_type.gsl_multiroot_fsolver_dnewton
gsl_odeiv2_driver_alloc_scaled_new (C func(C var), 462tion), 389
gsl_multiroot_fsolver_type.gsl_multiroot_fsolver_hybrid
gsl_odeiv2_driver_alloc_standard_new (C func(C var), 462tion), 389
gsl_multiroot_fsolver_type.gsl_multiroot_fsolver_hybrids
gsl_odeiv2_driver_alloc_y_new (C function), 389(C var), 462
gsl_odeiv2_driver_alloc_yp_new (C function), 389
gsl_multiroot_function (C type), 457
gsl_odeiv2_driver_apply (C function), 389
gsl_multiroot_function_fdf (C type), 457
gsl_odeiv2_driver_apply_fixed_step (C funcgsl_multiroot_test_delta (C function), 460tion), 389
gsl_multiroot_test_residual (C function), 460
gsl_odeiv2_driver_free (C function), 390
gsl_multiset (C type), 117
gsl_odeiv2_driver_reset (C function), 389
gsl_multiset_alloc (C function), 117
gsl_odeiv2_driver_reset_hstart (C function), 389
gsl_multiset_calloc (C function), 117
gsl_odeiv2_driver_set_hmax (C function), 389
gsl_multiset_data (C function), 118
gsl_odeiv2_driver_set_hmin (C function), 389
gsl_multiset_fprintf (C function), 119
gsl_odeiv2_driver_set_nmax (C function), 389
gsl_multiset_fread (C function), 119
gsl_odeiv2_evolve (C type), 388
gsl_multiset_free (C function), 117
gsl_odeiv2_evolve_alloc (C function), 388
gsl_multiset_fscanf (C function), 119
gsl_odeiv2_evolve_apply (C function), 388
gsl_multiset_fwrite (C function), 119
gsl_odeiv2_evolve_apply_fixed_step (C funcgsl_multiset_get (C function), 118tion), 388
gsl_multiset_init_first (C function), 117
gsl_odeiv2_evolve_free (C function), 388
gsl_multiset_init_last (C function), 117
gsl_odeiv2_evolve_reset (C function), 388
gsl_multiset_k (C function), 118
gsl_odeiv2_evolve_set_driver (C function), 388
gsl_multiset_memcpy (C function), 118
gsl_odeiv2_step (C type), 384
gsl_multiset_n (C function), 118
gsl_odeiv2_step_alloc (C function), 384
gsl_multiset_next (C function), 118
gsl_odeiv2_step_apply (C function), 384
gsl_multiset_prev (C function), 118
gsl_odeiv2_step_free (C function), 384
gsl_multiset_valid (C function), 118
gsl_odeiv2_step_name (C function), 384
GSL_NAN (C macro), 18
gsl_odeiv2_step_order (C function), 384
GSL_NEGINF (C macro), 18
gsl_odeiv2_step_reset (C function), 384
gsl_ntuple (C type), 351
gsl_odeiv2_step_set_driver (C function), 384
gsl_ntuple_bookdata (C function), 352
gsl_odeiv2_step_type (C type), 385
gsl_ntuple_close (C function), 352
gsl_odeiv2_step_type.gsl_odeiv2_step_bsimp
gsl_ntuple_create (C function), 351(C var), 385
gsl_ntuple_open (C function), 352
gsl_odeiv2_step_type.gsl_odeiv2_step_msadams
gsl_ntuple_project (C function), 353(C var), 385
gsl_ntuple_read (C function), 352
gsl_odeiv2_step_type.gsl_odeiv2_step_msbdf
gsl_ntuple_select_fn (C type), 352(C var), 386
gsl_ntuple_value_fn (C type), 352
gsl_odeiv2_step_type.gsl_odeiv2_step_rk1imp
gsl_ntuple_write (C function), 352(C var), 385
gsl_odeiv2_control (C type), 386
gsl_odeiv2_step_type.gsl_odeiv2_step_rk2 (Cgsl_odeiv2_control_alloc (C function), 387var), 385
gsl_odeiv2_control_errlevel (C function), 387
gsl_odeiv2_step_type.gsl_odeiv2_step_rk2imp
gsl_odeiv2_control_free (C function), 387(C var), 385
gsl_odeiv2_control_hadjust (C function), 387
gsl_odeiv2_step_type.gsl_odeiv2_step_rk4 (Cgsl_odeiv2_control_init (C function), 387var), 385
gsl_odeiv2_control_name (C function), 387
gsl_odeiv2_step_type.gsl_odeiv2_step_rk4imp
gsl_odeiv2_control_scaled_new (C function), 387(C var), 385
710

Index

GNU Scientific Library, Release 2.7

gsl_odeiv2_step_type.gsl_odeiv2_step_rk8pd(C var), 385
gsl_odeiv2_step_type.gsl_odeiv2_step_rkck (Cvar), 385
gsl_odeiv2_step_type.gsl_odeiv2_step_rkf45(C var), 385
gsl_odeiv2_system (C type), 383
gsl_permutation (C type), 107
gsl_permutation_alloc (C function), 107
gsl_permutation_calloc (C function), 107
gsl_permutation_canonical_cycles (C function),110
gsl_permutation_canonical_to_linear (C function), 110
gsl_permutation_data (C function), 108
gsl_permutation_fprintf (C function), 109
gsl_permutation_fread (C function), 109
gsl_permutation_free (C function), 107
gsl_permutation_fscanf (C function), 109
gsl_permutation_fwrite (C function), 109
gsl_permutation_get (C function), 108
gsl_permutation_init (C function), 107
gsl_permutation_inverse (C function), 108
gsl_permutation_inversions (C function), 110
gsl_permutation_linear_cycles (C function), 110
gsl_permutation_linear_to_canonical (C function), 110
gsl_permutation_memcpy (C function), 108
gsl_permutation_mul (C function), 109
gsl_permutation_next (C function), 108
gsl_permutation_prev (C function), 108
gsl_permutation_reverse (C function), 108
gsl_permutation_size (C function), 108
gsl_permutation_swap (C function), 108
gsl_permutation_valid (C function), 108
gsl_permute (C function), 109
gsl_permute_inverse (C function), 109
gsl_permute_matrix (C function), 109
gsl_permute_vector (C function), 109
gsl_permute_vector_inverse (C function), 109
gsl_poly_complex_eval (C function), 29
gsl_poly_complex_solve (C function), 31
gsl_poly_complex_solve_cubic (C function), 31
gsl_poly_complex_solve_quadratic (C function),30
gsl_poly_complex_workspace (C type), 31
gsl_poly_complex_workspace_alloc (C function),31
gsl_poly_complex_workspace_free (C function), 31
gsl_poly_dd_eval (C function), 30
gsl_poly_dd_hermite_init (C function), 30
gsl_poly_dd_init (C function), 30
gsl_poly_dd_taylor (C function), 30
gsl_poly_eval (C function), 29
Index

gsl_poly_eval_derivs (C function), 29
gsl_poly_solve_cubic (C function), 31
gsl_poly_solve_quadratic (C function), 30
GSL_POSINF (C macro), 18
gsl_pow_2 (C function), 19
gsl_pow_3 (C function), 19
gsl_pow_4 (C function), 19
gsl_pow_5 (C function), 19
gsl_pow_6 (C function), 19
gsl_pow_7 (C function), 19
gsl_pow_8 (C function), 19
gsl_pow_9 (C function), 19
gsl_pow_int (C function), 19
gsl_pow_uint (C function), 19
gsl_qrng (C type), 231
gsl_qrng_alloc (C function), 231
gsl_qrng_clone (C function), 232
gsl_qrng_free (C function), 231
gsl_qrng_get (C function), 231
gsl_qrng_init (C function), 231
gsl_qrng_memcpy (C function), 232
gsl_qrng_name (C function), 232
gsl_qrng_size (C function), 232
gsl_qrng_state (C function), 232
gsl_qrng_type (C type), 232
gsl_qrng_type.gsl_qrng_halton (C var), 232
gsl_qrng_type.gsl_qrng_niederreiter_2 (C var),
232
gsl_qrng_type.gsl_qrng_reversehalton (C var),
232
gsl_qrng_type.gsl_qrng_sobol (C var), 232
gsl_ran_bernoulli (C function), 272
gsl_ran_bernoulli_pdf (C function), 272
gsl_ran_beta (C function), 262
gsl_ran_beta_pdf (C function), 262
gsl_ran_binomial (C function), 273
gsl_ran_binomial_pdf (C function), 273
gsl_ran_bivariate_gaussian (C function), 241
gsl_ran_bivariate_gaussian_pdf (C function), 241
gsl_ran_cauchy (C function), 246
gsl_ran_cauchy_pdf (C function), 246
gsl_ran_chisq (C function), 256
gsl_ran_chisq_pdf (C function), 256
gsl_ran_choose (C function), 281
gsl_ran_dir_2d (C function), 265
gsl_ran_dir_2d_trig_method (C function), 265
gsl_ran_dir_3d (C function), 265
gsl_ran_dir_nd (C function), 265
gsl_ran_dirichlet (C function), 269
gsl_ran_dirichlet_lnpdf (C function), 269
gsl_ran_dirichlet_pdf (C function), 269
gsl_ran_discrete (C function), 270
gsl_ran_discrete_free (C function), 270
gsl_ran_discrete_pdf (C function), 270
711

GNU Scientific Library, Release 2.7

gsl_ran_discrete_preproc (C function), 270
gsl_ran_discrete_t (C type), 270
gsl_ran_exponential (C function), 243
gsl_ran_exponential_pdf (C function), 243
gsl_ran_exppow (C function), 245
gsl_ran_exppow_pdf (C function), 245
gsl_ran_fdist (C function), 258
gsl_ran_fdist_pdf (C function), 258
gsl_ran_flat (C function), 254
gsl_ran_flat_pdf (C function), 254
gsl_ran_gamma (C function), 252
gsl_ran_gamma_knuth (C function), 252
gsl_ran_gamma_pdf (C function), 252
gsl_ran_gaussian (C function), 237
gsl_ran_gaussian_pdf (C function), 237
gsl_ran_gaussian_ratio_method (C function), 237
gsl_ran_gaussian_tail (C function), 239
gsl_ran_gaussian_tail_pdf (C function), 239
gsl_ran_gaussian_ziggurat (C function), 237
gsl_ran_geometric (C function), 277
gsl_ran_geometric_pdf (C function), 277
gsl_ran_gumbel1 (C function), 267
gsl_ran_gumbel1_pdf (C function), 267
gsl_ran_gumbel2 (C function), 268
gsl_ran_gumbel2_pdf (C function), 268
gsl_ran_hypergeometric (C function), 278
gsl_ran_hypergeometric_pdf (C function), 278
gsl_ran_landau (C function), 249
gsl_ran_landau_pdf (C function), 249
gsl_ran_laplace (C function), 244
gsl_ran_laplace_pdf (C function), 244
gsl_ran_levy (C function), 250
gsl_ran_levy_skew (C function), 251
gsl_ran_logarithmic (C function), 279
gsl_ran_logarithmic_pdf (C function), 279
gsl_ran_logistic (C function), 263
gsl_ran_logistic_pdf (C function), 263
gsl_ran_lognormal (C function), 255
gsl_ran_lognormal_pdf (C function), 255
gsl_ran_multinomial (C function), 274
gsl_ran_multinomial_lnpdf (C function), 274
gsl_ran_multinomial_pdf (C function), 274
gsl_ran_multivariate_gaussian (C function), 242
gsl_ran_multivariate_gaussian_log_pdf (C function), 242
gsl_ran_multivariate_gaussian_mean (C function), 242
gsl_ran_multivariate_gaussian_pdf (C function),242
gsl_ran_multivariate_gaussian_vcov (C function), 242
gsl_ran_negative_binomial (C function), 275
gsl_ran_negative_binomial_pdf (C function), 275
gsl_ran_pareto (C function), 264712

gsl_ran_pareto_pdf (C function), 264
gsl_ran_pascal (C function), 276
gsl_ran_pascal_pdf (C function), 276
gsl_ran_poisson (C function), 271
gsl_ran_poisson_pdf (C function), 271
gsl_ran_rayleigh (C function), 247
gsl_ran_rayleigh_pdf (C function), 247
gsl_ran_rayleigh_tail (C function), 248
gsl_ran_rayleigh_tail_pdf (C function), 248
gsl_ran_sample (C function), 281
gsl_ran_shuffle (C function), 281
gsl_ran_tdist (C function), 260
gsl_ran_tdist_pdf (C function), 260
gsl_ran_ugaussian (C function), 237
gsl_ran_ugaussian_pdf (C function), 237
gsl_ran_ugaussian_ratio_method (C function), 237
gsl_ran_ugaussian_tail (C function), 239
gsl_ran_ugaussian_tail_pdf (C function), 239
gsl_ran_weibull (C function), 266
gsl_ran_weibull_pdf (C function), 266
gsl_ran_wishart (C function), 280
gsl_ran_wishart_log_pdf (C function), 280
gsl_ran_wishart_pdf (C function), 280
GSL_RANGE_CHECK_OFF (C macro), 86
GSL_REAL (C macro), 24
gsl_rng (C type), 215
gsl_rng_alloc (C function), 216
gsl_rng_borosh13 (C var), 226
gsl_rng_clone (C function), 219
gsl_rng_cmrg (C var), 221
gsl_rng_coveyou (C var), 226
gsl_rng_default (C var), 218
gsl_rng_default_seed (C var), 218
gsl_rng_env_setup (C function), 218
gsl_rng_fishman18 (C var), 226
gsl_rng_fishman20 (C var), 226
gsl_rng_fishman2x (C var), 226
gsl_rng_fread (C function), 220
gsl_rng_free (C function), 216
gsl_rng_fwrite (C function), 220
gsl_rng_get (C function), 216
gsl_rng_gfsr4 (C var), 222
gsl_rng_knuthran (C var), 226
gsl_rng_knuthran2 (C var), 226
gsl_rng_knuthran2002 (C var), 226
gsl_rng_lecuyer21 (C var), 226
gsl_rng_max (C function), 217
gsl_rng_memcpy (C function), 219
gsl_rng_min (C function), 217
gsl_rng_minstd (C var), 225
gsl_rng_mrg (C var), 221
gsl_rng_mt19937 (C var), 220
gsl_rng_name (C function), 217
gsl_rng_r250 (C var), 224
Index

GNU Scientific Library, Release 2.7

gsl_rng_rand (C var), 223
gsl_root_fsolver_root (C function), 437
gsl_rng_rand48 (C var), 223
gsl_root_fsolver_set (C function), 434
gsl_rng_random_bsd (C var), 223
gsl_root_fsolver_type (C type), 438
gsl_rng_random_glibc2 (C var), 223
gsl_root_fsolver_type.gsl_root_fsolver_bisection
gsl_rng_random_libc5 (C var), 223(C var), 438
gsl_rng_randu (C var), 225
gsl_root_fsolver_type.gsl_root_fsolver_brent
gsl_rng_ranf (C var), 224(C var), 439
gsl_rng_ranlux (C var), 221
gsl_root_fsolver_type.gsl_root_fsolver_falsepos
gsl_rng_ranlux389 (C var), 221(C var), 439
gsl_rng_ranlxd1 (C var), 221
gsl_root_fsolver_x_lower (C function), 437
gsl_rng_ranlxd2 (C var), 221
gsl_root_fsolver_x_upper (C function), 437
gsl_rng_ranlxs0 (C var), 220
gsl_root_test_delta (C function), 438
gsl_rng_ranlxs1 (C var), 220
gsl_root_test_interval (C function), 438
gsl_rng_ranlxs2 (C var), 220
gsl_root_test_residual (C function), 438
gsl_rng_ranmar (C var), 224
gsl_rstat_add (C function), 299
GSL_RNG_SEED (C macro), 218
gsl_rstat_alloc (C function), 299
gsl_rng_set (C function), 216
gsl_rstat_free (C function), 299
gsl_rng_size (C function), 217
gsl_rstat_kurtosis (C function), 300
gsl_rng_slatec (C var), 225
gsl_rstat_max (C function), 300
gsl_rng_state (C function), 217
gsl_rstat_mean (C function), 300
gsl_rng_taus (C var), 222
gsl_rstat_median (C function), 300
gsl_rng_taus2 (C var), 222
gsl_rstat_min (C function), 300
gsl_rng_transputer (C var), 225
gsl_rstat_n (C function), 299
gsl_rng_tt800 (C var), 224
gsl_rstat_quantile_add (C function), 301
GSL_RNG_TYPE (C macro), 218
gsl_rstat_quantile_alloc (C function), 301
gsl_rng_type (C type), 215
gsl_rstat_quantile_free (C function), 301
gsl_rng_types_setup (C function), 217
gsl_rstat_quantile_get (C function), 301
gsl_rng_uni (C var), 225
gsl_rstat_quantile_reset (C function), 301
gsl_rng_uni32 (C var), 225
gsl_rstat_quantile_workspace (C type), 301
gsl_rng_uniform (C function), 216
gsl_rstat_reset (C function), 299
gsl_rng_uniform_int (C function), 217
gsl_rstat_rms (C function), 300
gsl_rng_uniform_pos (C function), 217
gsl_rstat_sd (C function), 300
gsl_rng_vax (C var), 225
gsl_rstat_sd_mean (C function), 300
gsl_rng_waterman14 (C var), 226
gsl_rstat_skew (C function), 300
gsl_rng_zuf (C var), 225
gsl_rstat_variance (C function), 300
gsl_root_fdfsolver (C type), 434
gsl_rstat_workspace (C type), 299
gsl_root_fdfsolver_alloc (C function), 434
GSL_SET_COMPLEX (C macro), 24
gsl_root_fdfsolver_free (C function), 434
gsl_set_error_handler (C function), 15
gsl_root_fdfsolver_iterate (C function), 437
gsl_set_error_handler_off (C function), 15
gsl_root_fdfsolver_name (C function), 435
gsl_sf_airy_Ai (C function), 37
gsl_root_fdfsolver_root (C function), 437
gsl_sf_airy_Ai_deriv (C function), 37
gsl_root_fdfsolver_set (C function), 434
gsl_sf_airy_Ai_deriv_e (C function), 37
gsl_root_fdfsolver_type (C type), 439
gsl_sf_airy_Ai_deriv_scaled (C function), 37
gsl_root_fdfsolver_type.gsl_root_fdfsolver_newton
gsl_sf_airy_Ai_deriv_scaled_e (C function), 37(C var), 439
gsl_sf_airy_Ai_e (C function), 37
gsl_root_fdfsolver_type.gsl_root_fdfsolver_secant
gsl_sf_airy_Ai_scaled (C function), 37(C var), 439
gsl_sf_airy_Ai_scaled_e (C function), 37
gsl_root_fdfsolver_type.gsl_root_fdfsolver_steffenson
gsl_sf_airy_Bi (C function), 37(C var), 440
gsl_sf_airy_Bi_deriv (C function), 37
gsl_root_fsolver (C type), 434
gsl_sf_airy_Bi_deriv_e (C function), 37
gsl_root_fsolver_alloc (C function), 434
gsl_sf_airy_Bi_deriv_scaled (C function), 38
gsl_root_fsolver_free (C function), 434
gsl_sf_airy_Bi_deriv_scaled_e (C function), 38
gsl_root_fsolver_iterate (C function), 437
gsl_sf_airy_Bi_e (C function), 37
gsl_root_fsolver_name (C function), 435
gsl_sf_airy_Bi_scaled (C function), 37
Index

713

GNU Scientific Library, Release 2.7

gsl_sf_airy_Bi_scaled_e (C function), 37
gsl_sf_airy_zero_Ai (C function), 38
gsl_sf_airy_zero_Ai_deriv (C function), 38
gsl_sf_airy_zero_Ai_deriv_e (C function), 38
gsl_sf_airy_zero_Ai_e (C function), 38
gsl_sf_airy_zero_Bi (C function), 38
gsl_sf_airy_zero_Bi_deriv (C function), 38
gsl_sf_airy_zero_Bi_deriv_e (C function), 38
gsl_sf_airy_zero_Bi_e (C function), 38
gsl_sf_angle_restrict_pos (C function), 79
gsl_sf_angle_restrict_pos_e (C function), 79
gsl_sf_angle_restrict_symm (C function), 79
gsl_sf_angle_restrict_symm_e (C function), 79
gsl_sf_atanint (C function), 56
gsl_sf_atanint_e (C function), 56
gsl_sf_bessel_I0 (C function), 39
gsl_sf_bessel_I0_e (C function), 39
gsl_sf_bessel_I0_scaled (C function), 39
gsl_sf_bessel_i0_scaled (C function), 42
gsl_sf_bessel_I0_scaled_e (C function), 39
gsl_sf_bessel_i0_scaled_e (C function), 42
gsl_sf_bessel_I1 (C function), 39
gsl_sf_bessel_I1_e (C function), 39
gsl_sf_bessel_I1_scaled (C function), 40
gsl_sf_bessel_i1_scaled (C function), 42
gsl_sf_bessel_I1_scaled_e (C function), 40
gsl_sf_bessel_i1_scaled_e (C function), 42
gsl_sf_bessel_i2_scaled (C function), 42
gsl_sf_bessel_i2_scaled_e (C function), 42
gsl_sf_bessel_il_scaled (C function), 42
gsl_sf_bessel_il_scaled_array (C function), 42
gsl_sf_bessel_il_scaled_e (C function), 42
gsl_sf_bessel_In (C function), 39
gsl_sf_bessel_In_array (C function), 39
gsl_sf_bessel_In_e (C function), 39
gsl_sf_bessel_In_scaled (C function), 40
gsl_sf_bessel_In_scaled_array (C function), 40
gsl_sf_bessel_In_scaled_e (C function), 40
gsl_sf_bessel_Inu (C function), 44
gsl_sf_bessel_Inu_e (C function), 44
gsl_sf_bessel_Inu_scaled (C function), 44
gsl_sf_bessel_Inu_scaled_e (C function), 44
gsl_sf_bessel_J0 (C function), 38
gsl_sf_bessel_j0 (C function), 41
gsl_sf_bessel_J0_e (C function), 38
gsl_sf_bessel_j0_e (C function), 41
gsl_sf_bessel_J1 (C function), 38
gsl_sf_bessel_j1 (C function), 41
gsl_sf_bessel_J1_e (C function), 38
gsl_sf_bessel_j1_e (C function), 41
gsl_sf_bessel_j2 (C function), 41
gsl_sf_bessel_j2_e (C function), 41
gsl_sf_bessel_jl (C function), 41
gsl_sf_bessel_jl_array (C function), 41714

gsl_sf_bessel_jl_e (C function), 41
gsl_sf_bessel_jl_steed_array (C function), 41
gsl_sf_bessel_Jn (C function), 38
gsl_sf_bessel_Jn_array (C function), 39
gsl_sf_bessel_Jn_e (C function), 38
gsl_sf_bessel_Jnu (C function), 43
gsl_sf_bessel_Jnu_e (C function), 43
gsl_sf_bessel_K0 (C function), 40
gsl_sf_bessel_K0_e (C function), 40
gsl_sf_bessel_K0_scaled (C function), 40
gsl_sf_bessel_k0_scaled (C function), 43
gsl_sf_bessel_K0_scaled_e (C function), 40
gsl_sf_bessel_k0_scaled_e (C function), 43
gsl_sf_bessel_K1 (C function), 40
gsl_sf_bessel_K1_e (C function), 40
gsl_sf_bessel_K1_scaled (C function), 40
gsl_sf_bessel_k1_scaled (C function), 43
gsl_sf_bessel_K1_scaled_e (C function), 40
gsl_sf_bessel_k1_scaled_e (C function), 43
gsl_sf_bessel_k2_scaled (C function), 43
gsl_sf_bessel_k2_scaled_e (C function), 43
gsl_sf_bessel_kl_scaled (C function), 43
gsl_sf_bessel_kl_scaled_array (C function), 43
gsl_sf_bessel_kl_scaled_e (C function), 43
gsl_sf_bessel_Kn (C function), 40
gsl_sf_bessel_Kn_array (C function), 40
gsl_sf_bessel_Kn_e (C function), 40
gsl_sf_bessel_Kn_scaled (C function), 40
gsl_sf_bessel_Kn_scaled_array (C function), 41
gsl_sf_bessel_Kn_scaled_e (C function), 40
gsl_sf_bessel_Knu (C function), 44
gsl_sf_bessel_Knu_e (C function), 44
gsl_sf_bessel_Knu_scaled (C function), 44
gsl_sf_bessel_Knu_scaled_e (C function), 44
gsl_sf_bessel_lnKnu (C function), 44
gsl_sf_bessel_lnKnu_e (C function), 44
gsl_sf_bessel_sequence_Jnu_e (C function), 43
gsl_sf_bessel_Y0 (C function), 39
gsl_sf_bessel_y0 (C function), 42
gsl_sf_bessel_Y0_e (C function), 39
gsl_sf_bessel_y0_e (C function), 42
gsl_sf_bessel_Y1 (C function), 39
gsl_sf_bessel_y1 (C function), 42
gsl_sf_bessel_Y1_e (C function), 39
gsl_sf_bessel_y1_e (C function), 42
gsl_sf_bessel_y2 (C function), 42
gsl_sf_bessel_y2_e (C function), 42
gsl_sf_bessel_yl (C function), 42
gsl_sf_bessel_yl_array (C function), 42
gsl_sf_bessel_yl_e (C function), 42
gsl_sf_bessel_Yn (C function), 39
gsl_sf_bessel_Yn_array (C function), 39
gsl_sf_bessel_Yn_e (C function), 39
gsl_sf_bessel_Ynu (C function), 44
Index

GNU Scientific Library, Release 2.7

gsl_sf_bessel_Ynu_e (C function), 44
gsl_sf_bessel_zero_J0 (C function), 44
gsl_sf_bessel_zero_J0_e (C function), 44
gsl_sf_bessel_zero_J1 (C function), 44
gsl_sf_bessel_zero_J1_e (C function), 44
gsl_sf_bessel_zero_Jnu (C function), 44
gsl_sf_bessel_zero_Jnu_e (C function), 44
gsl_sf_beta (C function), 61
gsl_sf_beta_e (C function), 61
gsl_sf_beta_inc (C function), 61
gsl_sf_beta_inc_e (C function), 61
gsl_sf_Chi (C function), 56
gsl_sf_Chi_e (C function), 56
gsl_sf_choose (C function), 59
gsl_sf_choose_e (C function), 59
gsl_sf_Ci (C function), 56
gsl_sf_Ci_e (C function), 56
gsl_sf_clausen (C function), 45
gsl_sf_clausen_e (C function), 45
gsl_sf_complex_cos_e (C function), 78
gsl_sf_complex_dilog_e (C function), 49
gsl_sf_complex_log_e (C function), 73
gsl_sf_complex_logsin_e (C function), 78
gsl_sf_complex_sin_e (C function), 78
gsl_sf_conicalP_0 (C function), 72
gsl_sf_conicalP_0_e (C function), 72
gsl_sf_conicalP_1 (C function), 72
gsl_sf_conicalP_1_e (C function), 72
gsl_sf_conicalP_cyl_reg (C function), 72
gsl_sf_conicalP_cyl_reg_e (C function), 72
gsl_sf_conicalP_half (C function), 72
gsl_sf_conicalP_half_e (C function), 72
gsl_sf_conicalP_mhalf (C function), 72
gsl_sf_conicalP_mhalf_e (C function), 72
gsl_sf_conicalP_sph_reg (C function), 72
gsl_sf_conicalP_sph_reg_e (C function), 72
gsl_sf_cos (C function), 78
gsl_sf_cos_e (C function), 78
gsl_sf_cos_err_e (C function), 79
gsl_sf_coulomb_CL_array (C function), 46
gsl_sf_coulomb_CL_e (C function), 46
gsl_sf_coulomb_wave_F_array (C function), 46
gsl_sf_coulomb_wave_FG_array (C function), 46
gsl_sf_coulomb_wave_FG_e (C function), 46
gsl_sf_coulomb_wave_FGp_array (C function), 46
gsl_sf_coulomb_wave_sphF_array (C function), 46
gsl_sf_coupling_3j (C function), 47
gsl_sf_coupling_3j_e (C function), 47
gsl_sf_coupling_6j (C function), 47
gsl_sf_coupling_6j_e (C function), 47
gsl_sf_coupling_9j (C function), 47
gsl_sf_coupling_9j_e (C function), 47
gsl_sf_dawson (C function), 48
gsl_sf_dawson_e (C function), 48
Index

gsl_sf_debye_1 (C function), 48
gsl_sf_debye_1_e (C function), 48
gsl_sf_debye_2 (C function), 48
gsl_sf_debye_2_e (C function), 48
gsl_sf_debye_3 (C function), 48
gsl_sf_debye_3_e (C function), 48
gsl_sf_debye_4 (C function), 48
gsl_sf_debye_4_e (C function), 48
gsl_sf_debye_5 (C function), 48
gsl_sf_debye_5_e (C function), 48
gsl_sf_debye_6 (C function), 48
gsl_sf_debye_6_e (C function), 48
gsl_sf_dilog (C function), 49
gsl_sf_dilog_e (C function), 49
gsl_sf_doublefact (C function), 59
gsl_sf_doublefact_e (C function), 59
gsl_sf_ellint_D (C function), 51
gsl_sf_ellint_D_e (C function), 51
gsl_sf_ellint_E (C function), 51
gsl_sf_ellint_E_e (C function), 51
gsl_sf_ellint_Ecomp (C function), 50
gsl_sf_ellint_Ecomp_e (C function), 50
gsl_sf_ellint_F (C function), 51
gsl_sf_ellint_F_e (C function), 51
gsl_sf_ellint_Kcomp (C function), 50
gsl_sf_ellint_Kcomp_e (C function), 50
gsl_sf_ellint_P (C function), 51
gsl_sf_ellint_P_e (C function), 51
gsl_sf_ellint_Pcomp (C function), 50
gsl_sf_ellint_Pcomp_e (C function), 50
gsl_sf_ellint_RC (C function), 51
gsl_sf_ellint_RC_e (C function), 51
gsl_sf_ellint_RD (C function), 51
gsl_sf_ellint_RD_e (C function), 51
gsl_sf_ellint_RF (C function), 51
gsl_sf_ellint_RF_e (C function), 51
gsl_sf_ellint_RJ (C function), 52
gsl_sf_ellint_RJ_e (C function), 52
gsl_sf_elljac_e (C function), 52
gsl_sf_erf (C function), 52
gsl_sf_erf_e (C function), 52
gsl_sf_erf_Q (C function), 53
gsl_sf_erf_Q_e (C function), 53
gsl_sf_erf_Z (C function), 53
gsl_sf_erf_Z_e (C function), 53
gsl_sf_erfc (C function), 52
gsl_sf_erfc_e (C function), 52
gsl_sf_eta (C function), 81
gsl_sf_eta_e (C function), 81
gsl_sf_eta_int (C function), 81
gsl_sf_eta_int_e (C function), 81
gsl_sf_exp (C function), 53
gsl_sf_exp_e (C function), 53
gsl_sf_exp_e10_e (C function), 53
715

GNU Scientific Library, Release 2.7

gsl_sf_exp_err_e (C function), 54
gsl_sf_exp_err_e10_e (C function), 54
gsl_sf_exp_mult (C function), 53
gsl_sf_exp_mult_e (C function), 53
gsl_sf_exp_mult_e10_e (C function), 53
gsl_sf_exp_mult_err_e (C function), 54
gsl_sf_exp_mult_err_e10_e (C function), 54
gsl_sf_expint_3 (C function), 56
gsl_sf_expint_3_e (C function), 56
gsl_sf_expint_E1 (C function), 55
gsl_sf_expint_E1_e (C function), 55
gsl_sf_expint_E2 (C function), 55
gsl_sf_expint_E2_e (C function), 55
gsl_sf_expint_Ei (C function), 55
gsl_sf_expint_Ei_e (C function), 55
gsl_sf_expint_En (C function), 55
gsl_sf_expint_En_e (C function), 55
gsl_sf_expm1 (C function), 54
gsl_sf_expm1_e (C function), 54
gsl_sf_exprel (C function), 54
gsl_sf_exprel_2 (C function), 54
gsl_sf_exprel_2_e (C function), 54
gsl_sf_exprel_e (C function), 54
gsl_sf_exprel_n (C function), 54
gsl_sf_exprel_n_e (C function), 54
gsl_sf_fact (C function), 59
gsl_sf_fact_e (C function), 59
gsl_sf_fermi_dirac_0 (C function), 57
gsl_sf_fermi_dirac_0_e (C function), 57
gsl_sf_fermi_dirac_1 (C function), 57
gsl_sf_fermi_dirac_1_e (C function), 57
gsl_sf_fermi_dirac_2 (C function), 57
gsl_sf_fermi_dirac_2_e (C function), 57
gsl_sf_fermi_dirac_3half (C function), 57
gsl_sf_fermi_dirac_3half_e (C function), 57
gsl_sf_fermi_dirac_half (C function), 57
gsl_sf_fermi_dirac_half_e (C function), 57
gsl_sf_fermi_dirac_inc_0 (C function), 58
gsl_sf_fermi_dirac_inc_0_e (C function), 58
gsl_sf_fermi_dirac_int (C function), 57
gsl_sf_fermi_dirac_int_e (C function), 57
gsl_sf_fermi_dirac_m1 (C function), 57
gsl_sf_fermi_dirac_m1_e (C function), 57
gsl_sf_fermi_dirac_mhalf (C function), 57
gsl_sf_fermi_dirac_mhalf_e (C function), 57
gsl_sf_gamma (C function), 58
gsl_sf_gamma_e (C function), 58
gsl_sf_gamma_inc (C function), 60
gsl_sf_gamma_inc_e (C function), 60
gsl_sf_gamma_inc_P (C function), 60
gsl_sf_gamma_inc_P_e (C function), 60
gsl_sf_gamma_inc_Q (C function), 60
gsl_sf_gamma_inc_Q_e (C function), 60
gsl_sf_gammainv (C function), 59716

gsl_sf_gammainv_e (C function), 59
gsl_sf_gammastar (C function), 58
gsl_sf_gammastar_e (C function), 58
gsl_sf_gegenpoly_1 (C function), 61
gsl_sf_gegenpoly_1_e (C function), 61
gsl_sf_gegenpoly_2 (C function), 61
gsl_sf_gegenpoly_2_e (C function), 61
gsl_sf_gegenpoly_3 (C function), 61
gsl_sf_gegenpoly_3_e (C function), 61
gsl_sf_gegenpoly_array (C function), 62
gsl_sf_gegenpoly_n (C function), 62
gsl_sf_gegenpoly_n_e (C function), 62
gsl_sf_hazard (C function), 53
gsl_sf_hazard_e (C function), 53
gsl_sf_hermite (C function), 62
gsl_sf_hermite_array (C function), 62
gsl_sf_hermite_array_deriv (C function), 63
gsl_sf_hermite_deriv (C function), 63
gsl_sf_hermite_deriv_array (C function), 63
gsl_sf_hermite_deriv_e (C function), 63
gsl_sf_hermite_e (C function), 62
gsl_sf_hermite_func (C function), 64
gsl_sf_hermite_func_array (C function), 64
gsl_sf_hermite_func_der (C function), 64
gsl_sf_hermite_func_der_e (C function), 64
gsl_sf_hermite_func_e (C function), 64
gsl_sf_hermite_func_fast (C function), 64
gsl_sf_hermite_func_fast_e (C function), 64
gsl_sf_hermite_func_series (C function), 64
gsl_sf_hermite_func_series_e (C function), 64
gsl_sf_hermite_func_zero (C function), 65
gsl_sf_hermite_func_zero_e (C function), 65
gsl_sf_hermite_prob (C function), 63
gsl_sf_hermite_prob_array (C function), 63
gsl_sf_hermite_prob_array_deriv (C function), 63
gsl_sf_hermite_prob_deriv (C function), 63
gsl_sf_hermite_prob_deriv_array (C function), 63
gsl_sf_hermite_prob_deriv_e (C function), 63
gsl_sf_hermite_prob_e (C function), 63
gsl_sf_hermite_prob_series (C function), 63
gsl_sf_hermite_prob_series_e (C function), 63
gsl_sf_hermite_prob_zero (C function), 65
gsl_sf_hermite_prob_zero_e (C function), 65
gsl_sf_hermite_series (C function), 62
gsl_sf_hermite_series_e (C function), 62
gsl_sf_hermite_zero (C function), 65
gsl_sf_hermite_zero_e (C function), 65
gsl_sf_hydrogenicR (C function), 45
gsl_sf_hydrogenicR_1 (C function), 45
gsl_sf_hydrogenicR_1_e (C function), 45
gsl_sf_hydrogenicR_e (C function), 45
gsl_sf_hyperg_0F1 (C function), 65
gsl_sf_hyperg_0F1_e (C function), 65
gsl_sf_hyperg_1F1 (C function), 65
Index

GNU Scientific Library, Release 2.7

gsl_sf_hyperg_1F1_e (C function), 65
gsl_sf_hyperg_1F1_int (C function), 65
gsl_sf_hyperg_1F1_int_e (C function), 65
gsl_sf_hyperg_2F0 (C function), 66
gsl_sf_hyperg_2F0_e (C function), 66
gsl_sf_hyperg_2F1 (C function), 66
gsl_sf_hyperg_2F1_conj (C function), 66
gsl_sf_hyperg_2F1_conj_e (C function), 66
gsl_sf_hyperg_2F1_conj_renorm (C function), 66
gsl_sf_hyperg_2F1_conj_renorm_e (C function), 66
gsl_sf_hyperg_2F1_e (C function), 66
gsl_sf_hyperg_2F1_renorm (C function), 66
gsl_sf_hyperg_2F1_renorm_e (C function), 66
gsl_sf_hyperg_U (C function), 66
gsl_sf_hyperg_U_e (C function), 66
gsl_sf_hyperg_U_e10_e (C function), 66
gsl_sf_hyperg_U_int (C function), 65
gsl_sf_hyperg_U_int_e (C function), 65
gsl_sf_hyperg_U_int_e10_e (C function), 66
gsl_sf_hypot (C function), 78
gsl_sf_hypot_e (C function), 78
gsl_sf_hzeta (C function), 80
gsl_sf_hzeta_e (C function), 80
gsl_sf_laguerre_1 (C function), 67
gsl_sf_laguerre_1_e (C function), 67
gsl_sf_laguerre_2 (C function), 67
gsl_sf_laguerre_2_e (C function), 67
gsl_sf_laguerre_3 (C function), 67
gsl_sf_laguerre_3_e (C function), 67
gsl_sf_laguerre_n (C function), 67
gsl_sf_laguerre_n_e (C function), 67
gsl_sf_lambert_W0 (C function), 67
gsl_sf_lambert_W0_e (C function), 67
gsl_sf_lambert_Wm1 (C function), 68
gsl_sf_lambert_Wm1_e (C function), 68
gsl_sf_legendre_array (C function), 70
gsl_sf_legendre_array_e (C function), 70
gsl_sf_legendre_array_index (C function), 71
gsl_sf_legendre_array_n (C function), 71
gsl_sf_legendre_array_size (C function), 72
gsl_sf_legendre_deriv2_alt_array (C function),71
gsl_sf_legendre_deriv2_alt_array_e (C function), 71
gsl_sf_legendre_deriv2_array (C function), 70
gsl_sf_legendre_deriv2_array_e (C function), 70
gsl_sf_legendre_deriv_alt_array (C function), 70
gsl_sf_legendre_deriv_alt_array_e (C function),70
gsl_sf_legendre_deriv_array (C function), 70
gsl_sf_legendre_deriv_array_e (C function), 70
gsl_sf_legendre_H3d (C function), 73
gsl_sf_legendre_H3d_0 (C function), 72
gsl_sf_legendre_H3d_0_e (C function), 72
Index

gsl_sf_legendre_H3d_1 (C function), 73
gsl_sf_legendre_H3d_1_e (C function), 73
gsl_sf_legendre_H3d_array (C function), 73
gsl_sf_legendre_H3d_e (C function), 73
gsl_sf_legendre_nlm (C function), 71
gsl_sf_legendre_P1 (C function), 68
gsl_sf_legendre_P1_e (C function), 68
gsl_sf_legendre_P2 (C function), 68
gsl_sf_legendre_P2_e (C function), 68
gsl_sf_legendre_P3 (C function), 68
gsl_sf_legendre_P3_e (C function), 68
gsl_sf_legendre_Pl (C function), 68
gsl_sf_legendre_Pl_array (C function), 68
gsl_sf_legendre_Pl_deriv_array (C function), 68
gsl_sf_legendre_Pl_e (C function), 68
gsl_sf_legendre_Plm (C function), 71
gsl_sf_legendre_Plm_array (C function), 71
gsl_sf_legendre_Plm_deriv_array (C function), 71
gsl_sf_legendre_Plm_e (C function), 71
gsl_sf_legendre_Q0 (C function), 68
gsl_sf_legendre_Q0_e (C function), 68
gsl_sf_legendre_Q1 (C function), 68
gsl_sf_legendre_Q1_e (C function), 68
gsl_sf_legendre_Ql (C function), 68
gsl_sf_legendre_Ql_e (C function), 68
gsl_sf_legendre_sphPlm (C function), 71
gsl_sf_legendre_sphPlm_array (C function), 71
gsl_sf_legendre_sphPlm_deriv_array (C function), 71
gsl_sf_legendre_sphPlm_e (C function), 71
gsl_sf_legendre_t (C type), 69
gsl_sf_lnbeta (C function), 61
gsl_sf_lnbeta_e (C function), 61
gsl_sf_lnchoose (C function), 60
gsl_sf_lnchoose_e (C function), 60
gsl_sf_lncosh (C function), 78
gsl_sf_lncosh_e (C function), 78
gsl_sf_lndoublefact (C function), 59
gsl_sf_lndoublefact_e (C function), 59
gsl_sf_lnfact (C function), 59
gsl_sf_lnfact_e (C function), 59
gsl_sf_lngamma (C function), 58
gsl_sf_lngamma_complex_e (C function), 59
gsl_sf_lngamma_e (C function), 58
gsl_sf_lngamma_sgn_e (C function), 58
gsl_sf_lnpoch (C function), 60
gsl_sf_lnpoch_e (C function), 60
gsl_sf_lnpoch_sgn_e (C function), 60
gsl_sf_lnsinh (C function), 78
gsl_sf_lnsinh_e (C function), 78
gsl_sf_log (C function), 73
gsl_sf_log_1plusx (C function), 73
gsl_sf_log_1plusx_e (C function), 73
gsl_sf_log_1plusx_mx (C function), 73
717

GNU Scientific Library, Release 2.7

gsl_sf_log_1plusx_mx_e (C function), 73
gsl_sf_log_abs (C function), 73
gsl_sf_log_abs_e (C function), 73
gsl_sf_log_e (C function), 73
gsl_sf_log_erfc (C function), 52
gsl_sf_log_erfc_e (C function), 52
gsl_sf_mathieu_a (C function), 74
gsl_sf_mathieu_a_array (C function), 74
gsl_sf_mathieu_a_e (C function), 74
gsl_sf_mathieu_alloc (C function), 74
gsl_sf_mathieu_b (C function), 74
gsl_sf_mathieu_b_array (C function), 74
gsl_sf_mathieu_b_e (C function), 74
gsl_sf_mathieu_ce (C function), 75
gsl_sf_mathieu_ce_array (C function), 75
gsl_sf_mathieu_ce_e (C function), 75
gsl_sf_mathieu_free (C function), 74
gsl_sf_mathieu_Mc (C function), 75
gsl_sf_mathieu_Mc_array (C function), 75
gsl_sf_mathieu_Mc_e (C function), 75
gsl_sf_mathieu_Ms (C function), 75
gsl_sf_mathieu_Ms_array (C function), 75
gsl_sf_mathieu_Ms_e (C function), 75
gsl_sf_mathieu_se (C function), 75
gsl_sf_mathieu_se_array (C function), 75
gsl_sf_mathieu_se_e (C function), 75
gsl_sf_mathieu_workspace (C type), 74
gsl_sf_multiply (C function), 49
gsl_sf_multiply_e (C function), 49
gsl_sf_multiply_err_e (C function), 49
gsl_sf_poch (C function), 60
gsl_sf_poch_e (C function), 60
gsl_sf_pochrel (C function), 60
gsl_sf_pochrel_e (C function), 60
gsl_sf_polar_to_rect (C function), 79
gsl_sf_pow_int (C function), 75
gsl_sf_pow_int_e (C function), 75
gsl_sf_psi (C function), 76
gsl_sf_psi_1 (C function), 76
gsl_sf_psi_1_e (C function), 76
gsl_sf_psi_1_int (C function), 76
gsl_sf_psi_1_int_e (C function), 76
gsl_sf_psi_1piy (C function), 76
gsl_sf_psi_1piy_e (C function), 76
gsl_sf_psi_e (C function), 76
gsl_sf_psi_int (C function), 76
gsl_sf_psi_int_e (C function), 76
gsl_sf_psi_n (C function), 77
gsl_sf_psi_n_e (C function), 77
gsl_sf_rect_to_polar (C function), 79
gsl_sf_result (C type), 36
gsl_sf_result_e10 (C type), 36
gsl_sf_Shi (C function), 55
gsl_sf_Shi_e (C function), 55
718

gsl_sf_Si (C function), 56
gsl_sf_Si_e (C function), 56
gsl_sf_sin (C function), 78
gsl_sf_sin_e (C function), 78
gsl_sf_sin_err_e (C function), 79
gsl_sf_sinc (C function), 78
gsl_sf_sinc_e (C function), 78
gsl_sf_synchrotron_1 (C function), 77
gsl_sf_synchrotron_1_e (C function), 77
gsl_sf_synchrotron_2 (C function), 77
gsl_sf_synchrotron_2_e (C function), 77
gsl_sf_taylorcoeff (C function), 60
gsl_sf_taylorcoeff_e (C function), 60
gsl_sf_transport_2 (C function), 77
gsl_sf_transport_2_e (C function), 77
gsl_sf_transport_3 (C function), 77
gsl_sf_transport_3_e (C function), 77
gsl_sf_transport_4 (C function), 77
gsl_sf_transport_4_e (C function), 77
gsl_sf_transport_5 (C function), 77
gsl_sf_transport_5_e (C function), 77
gsl_sf_zeta (C function), 80
gsl_sf_zeta_e (C function), 80
gsl_sf_zeta_int (C function), 80
gsl_sf_zeta_int_e (C function), 80
gsl_sf_zetam1 (C function), 80
gsl_sf_zetam1_e (C function), 80
gsl_sf_zetam1_int (C function), 80
gsl_sf_zetam1_int_e (C function), 80
GSL_SIGN (C macro), 20
gsl_siman_copy_construct_t (C type), 368
gsl_siman_copy_t (C type), 368
gsl_siman_destroy_t (C type), 368
gsl_siman_Efunc_t (C type), 368
gsl_siman_metric_t (C type), 368
gsl_siman_params_t (C type), 369
gsl_siman_print_t (C type), 368
gsl_siman_solve (C function), 367
gsl_siman_step_t (C type), 368
gsl_sort (C function), 124
gsl_sort2 (C function), 124
gsl_sort_index (C function), 124
gsl_sort_largest (C function), 125
gsl_sort_largest_index (C function), 125
gsl_sort_smallest (C function), 125
gsl_sort_smallest_index (C function), 125
gsl_sort_vector (C function), 124
gsl_sort_vector2 (C function), 124
gsl_sort_vector_index (C function), 124
gsl_sort_vector_largest (C function), 125
gsl_sort_vector_largest_index (C function), 125
gsl_sort_vector_smallest (C function), 125
gsl_sort_vector_smallest_index (C function), 125
gsl_spblas_dgemm (C function), 619
IndexGNU Scientific Library, Release 2.7

gsl_spblas_dgemv (C function), 619
gsl_spmatrix_alloc_nzmax.GSL_SPMATRIX_CSR (Cgsl_splinalg_itersolve_alloc (C function), 622
macro), 610
gsl_splinalg_itersolve_free (C function), 622
gsl_spmatrix_compress (C function), 614
gsl_splinalg_itersolve_iterate (C function), 622 gsl_spmatrix_csc (C function), 614
gsl_splinalg_itersolve_name (C function), 622
gsl_spmatrix_csr (C function), 614
gsl_splinalg_itersolve_normr (C function), 622
gsl_spmatrix_d2sp (C function), 614
gsl_splinalg_itersolve_type (C type), 621
gsl_spmatrix_dense_add (C function), 613
gsl_splinalg_itersolve_type.gsl_splinalg_itersolve_gmres
gsl_spmatrix_dense_sub (C function), 613(C var), 621
gsl_spmatrix_equal (C function), 613
gsl_spline (C type), 398
gsl_spmatrix_fprintf (C function), 611
gsl_spline2d (C type), 406
gsl_spmatrix_fread (C function), 611
gsl_spline2d_alloc (C function), 406
gsl_spmatrix_free (C function), 610
gsl_spline2d_eval (C function), 406
gsl_spmatrix_fscanf (C function), 611
gsl_spline2d_eval_deriv_x (C function), 407
gsl_spmatrix_fwrite (C function), 611
gsl_spline2d_eval_deriv_x_e (C function), 407
gsl_spmatrix_get (C function), 611
gsl_spline2d_eval_deriv_xx (C function), 407
gsl_spmatrix_memcpy (C function), 612
gsl_spline2d_eval_deriv_xx_e (C function), 407
gsl_spmatrix_min_index (C function), 614
gsl_spline2d_eval_deriv_xy (C function), 407
gsl_spmatrix_minmax (C function), 614
gsl_spline2d_eval_deriv_xy_e (C function), 407
gsl_spmatrix_nnz (C function), 613
gsl_spline2d_eval_deriv_y (C function), 407
gsl_spmatrix_norm1 (C function), 613
gsl_spline2d_eval_deriv_y_e (C function), 407
gsl_spmatrix_ptr (C function), 611
gsl_spline2d_eval_deriv_yy (C function), 407
gsl_spmatrix_realloc (C function), 610
gsl_spline2d_eval_deriv_yy_e (C function), 407
gsl_spmatrix_scale (C function), 612
gsl_spline2d_eval_e (C function), 406
gsl_spmatrix_scale_columns (C function), 612
gsl_spline2d_eval_extrap (C function), 406
gsl_spmatrix_scale_rows (C function), 612
gsl_spline2d_eval_extrap_e (C function), 406
gsl_spmatrix_set (C function), 611
gsl_spline2d_free (C function), 406
gsl_spmatrix_set_zero (C function), 611
gsl_spline2d_get (C function), 407
gsl_spmatrix_sp2d (C function), 614
gsl_spline2d_init (C function), 406
gsl_spmatrix_transpose (C function), 612
gsl_spline2d_min_size (C function), 406
gsl_spmatrix_transpose_memcpy (C function), 612
gsl_spline2d_name (C function), 406
gsl_spmatrix_type (C function), 613
gsl_spline2d_set (C function), 407
gsl_stats_absdev (C function), 288
gsl_spline_alloc (C function), 398
gsl_stats_absdev_m (C function), 288
gsl_spline_eval (C function), 398
gsl_stats_correlation (C function), 290
gsl_spline_eval_deriv (C function), 398
gsl_stats_covariance (C function), 290
gsl_spline_eval_deriv2 (C function), 398
gsl_stats_covariance_m (C function), 290
gsl_spline_eval_deriv2_e (C function), 398
gsl_stats_gastwirth_from_sorted_data (C funcgsl_spline_eval_deriv_e (C function), 398tion), 294
gsl_spline_eval_e (C function), 398
gsl_stats_kurtosis (C function), 289
gsl_spline_eval_integ (C function), 398
gsl_stats_kurtosis_m_sd (C function), 289
gsl_spline_eval_integ_e (C function), 398
gsl_stats_lag1_autocorrelation (C function), 289
gsl_spline_free (C function), 398
gsl_stats_lag1_autocorrelation_m (C function),
gsl_spline_init (C function), 398289
gsl_spline_min_size (C function), 398
gsl_stats_mad (C function), 295
gsl_spline_name (C function), 398
gsl_stats_mad0 (C function), 294
gsl_spmatrix (C type), 609
gsl_stats_max (C function), 292
gsl_spmatrix_add (C function), 613
gsl_stats_max_index (C function), 292
gsl_spmatrix_alloc (C function), 610
gsl_stats_mean (C function), 287
gsl_spmatrix_alloc_nzmax (C function), 610
gsl_stats_median (C function), 293
gsl_spmatrix_alloc_nzmax.GSL_SPMATRIX_COO (C gsl_stats_median_from_sorted_data (C function),
macro), 610
293
gsl_spmatrix_alloc_nzmax.GSL_SPMATRIX_CSC (C gsl_stats_min (C function), 292
macro), 610
gsl_stats_min_index (C function), 292
gsl_stats_minmax (C function), 292
Index

719

GNU Scientific Library, Release 2.7

gsl_stats_minmax_index (C function), 292
gsl_stats_Qn0_from_sorted_data (C function), 295
gsl_stats_Qn_from_sorted_data (C function), 295
gsl_stats_quantile_from_sorted_data (C function), 293
gsl_stats_sd (C function), 288
gsl_stats_sd_m (C function), 288
gsl_stats_sd_with_fixed_mean (C function), 288
gsl_stats_select (C function), 293
gsl_stats_skew (C function), 289
gsl_stats_skew_m_sd (C function), 289
gsl_stats_Sn0_from_sorted_data (C function), 295
gsl_stats_Sn_from_sorted_data (C function), 295
gsl_stats_spearman (C function), 290
gsl_stats_trmean_from_sorted_data (C function),294
gsl_stats_tss (C function), 288
gsl_stats_tss_m (C function), 288
gsl_stats_variance (C function), 287
gsl_stats_variance_m (C function), 287
gsl_stats_variance_with_fixed_mean (C function), 288
gsl_stats_wabsdev (C function), 291
gsl_stats_wabsdev_m (C function), 291
gsl_stats_wkurtosis (C function), 292
gsl_stats_wkurtosis_m_sd (C function), 292
gsl_stats_wmean (C function), 290
gsl_stats_wsd (C function), 291
gsl_stats_wsd_m (C function), 291
gsl_stats_wsd_with_fixed_mean (C function), 291
gsl_stats_wskew (C function), 291
gsl_stats_wskew_m_sd (C function), 292
gsl_stats_wtss (C function), 291
gsl_stats_wtss_m (C function), 291
gsl_stats_wvariance (C function), 290
gsl_stats_wvariance_m (C function), 291
gsl_stats_wvariance_with_fixed_mean (C function), 291
gsl_strerror (C function), 14
gsl_sum_levin_u_accel (C function), 419
gsl_sum_levin_u_alloc (C function), 419
gsl_sum_levin_u_free (C function), 419
gsl_sum_levin_u_workspace (C type), 419
gsl_sum_levin_utrunc_accel (C function), 420
gsl_sum_levin_utrunc_alloc (C function), 420
gsl_sum_levin_utrunc_free (C function), 420
gsl_sum_levin_utrunc_workspace (C type), 420
gsl_vector (C type), 85
gsl_vector_add (C function), 91
gsl_vector_add_constant (C function), 91
gsl_vector_alloc (C function), 86
gsl_vector_axpby (C function), 91
gsl_vector_calloc (C function), 86
gsl_vector_complex_conj_memcpy (C function), 90
720

gsl_vector_complex_const_imag (C function), 89
gsl_vector_complex_const_real (C function), 89
gsl_vector_complex_imag (C function), 89
gsl_vector_complex_real (C function), 89
gsl_vector_const_ptr (C function), 87
gsl_vector_const_subvector (C function), 88
gsl_vector_const_subvector_with_stride(Cfunction), 88
gsl_vector_const_view (C type), 88
gsl_vector_const_view_array (C function), 89
gsl_vector_const_view_array_with_stride(Cfunction), 89
gsl_vector_div (C function), 91
gsl_vector_equal (C function), 92
gsl_vector_fprintf (C function), 87
gsl_vector_fread (C function), 87
gsl_vector_free (C function), 86
gsl_vector_fscanf (C function), 88
gsl_vector_fwrite (C function), 87
gsl_vector_get (C function), 87
gsl_vector_isneg (C function), 92
gsl_vector_isnonneg (C function), 92
gsl_vector_isnull (C function), 92
gsl_vector_ispos (C function), 92
gsl_vector_max (C function), 91
gsl_vector_max_index (C function), 91
gsl_vector_memcpy (C function), 90
gsl_vector_min (C function), 91
gsl_vector_min_index (C function), 91
gsl_vector_minmax (C function), 91
gsl_vector_minmax_index (C function), 91
gsl_vector_mul (C function), 91
gsl_vector_ptr (C function), 87
gsl_vector_reverse (C function), 90
gsl_vector_scale (C function), 91
gsl_vector_set (C function), 87
gsl_vector_set_all (C function), 87
gsl_vector_set_basis (C function), 87
gsl_vector_set_zero (C function), 87
gsl_vector_sub (C function), 91
gsl_vector_subvector (C function), 88
gsl_vector_subvector_with_stride (C function),88
gsl_vector_sum (C function), 91
gsl_vector_swap (C function), 90
gsl_vector_swap_elements (C function), 90
gsl_vector_view (C type), 88
gsl_vector_view_array (C function), 89
gsl_vector_view_array_with_stride (C function),89
gsl_wavelet (C type), 423
gsl_wavelet2d_nstransform (C function), 426
gsl_wavelet2d_nstransform_forward (C function),
426
Index

GNU Scientific Library, Release 2.7

gsl_wavelet2d_nstransform_inverse (C function), HER2K, Level-3 BLAS, 137
426
HERK, Level-3 BLAS, 137
gsl_wavelet2d_nstransform_matrix (C function), Hermite functions, 63426
Hermite functions, derivatives, 64
gsl_wavelet2d_nstransform_matrix_forward (C Hermite functions, zeros, 65
function), 426
Hermite polynomials, 62
gsl_wavelet2d_nstransform_matrix_inverse (C Hermite polynomials, derivatives, 63
function), 426
Hermite polynomials, zeros, 65
gsl_wavelet2d_transform (C function), 425
hermitian matrix, complex, eigensystem, 172
gsl_wavelet2d_transform_forward (C function), Hessenberg decomposition, 159
425
Hessenberg triangular decomposition, 160
gsl_wavelet2d_transform_inverse (C function), histogram statistics, 338
425
histogram, from ntuple, 353
gsl_wavelet2d_transform_matrix (C function), 425 histograms, 333
gsl_wavelet2d_transform_matrix_forward(C histograms, random sampling from, 340
function), 425
Householder linear solver, 162
gsl_wavelet2d_transform_matrix_inverse(C Householder matrix, 161
function), 425
Householder transformation, 161
gsl_wavelet_alloc (C function), 423
how to report, 2
gsl_wavelet_free (C function), 424
Hurwitz Zeta Function, 80
gsl_wavelet_name (C function), 424
HYBRID algorithm, unscaled without
gsl_wavelet_transform (C function), 424
derivatives, 462
gsl_wavelet_transform_forward (C function), 424
HYBRID algorithms for nonlinear systems, 460
gsl_wavelet_transform_inverse (C function), 424
HYBRIDJ algorithm, 461
gsl_wavelet_type (C type), 423
HYBRIDS algorithm, scaled without
gsl_wavelet_type.gsl_wavelet_bspline (C var),derivatives, 462
424
HYBRIDSJ algorithm, 460
gsl_wavelet_type.gsl_wavelet_bspline_centered hydrogen atom, 45(C var), 424
hyperbolic cosine, inverse, 18
gsl_wavelet_type.gsl_wavelet_daubechies(C hyperbolic functions, complex numbers, 27var), 423
hyperbolic integrals, 55
gsl_wavelet_type.gsl_wavelet_daubechies_centeredhyperbolic sine, inverse, 18(C var), 423
hyperbolic space, 68
gsl_wavelet_type.gsl_wavelet_haar (C var), 424 hyperbolic tangent, inverse, 18
gsl_wavelet_type.gsl_wavelet_haar_centeredhypergeometric functions, 65(C var), 424
hypergeometric random variates, 278
gsl_wavelet_workspace (C type), 424
hypot, 18
gsl_wavelet_workspace_alloc (C function), 424
hypot function, special functions, 78
gsl_wavelet_workspace_free (C function), 424
I
Gumbel distribution (Type 1), 267
Gumbel distribution (Type 2), 268
I(x), Bessel Functions, 39i(x), Bessel Functions, 42Hidentity matrix, 95
identity permutation, 107
Haar wavelets, 424
IEEE exceptions, 639
Hankel transforms, discrete, 429
IEEE floating point, 635
HAVE_INLINE, 7IEEE format for floating point numbers, 637
hazard function, normal distribution, 53
IEEE infinity, defined as a macro, 17
HBOOK, 356
IEEE NaN, defined as a macro, 18
header files, including, 5illumination, units of, 632
heapsort, 121
imperial units, 629
HEMM, Level-3 BLAS, 136
Implicit Euler method, 385
HEMV, Level-2 BLAS, 134
Implicit Runge-Kutta method, 385
HER, Level-2 BLAS, 134
importance sampling, VEGAS, 361
HER2, Level-2 BLAS, 135
Index

721

GNU Scientific Library, Release 2.7

including GSL header files, 5
incomplete Beta function, normalized, 61
incomplete Fermi-Dirac integral, 58
incomplete Gamma function, 60
indirect sorting, 124
indirect sorting, of vector elements, 124
infinity, defined as a macro, 17
infinity, IEEE format, 637
info-gsl mailing list, 2
initial value problems, differential
equations, 381
initializing matrices, 95
initializing vectors, 87
inline functions, 7
integer powers, 75
integrals, exponential, 55
integration, numerical (quadrature), 198interpolating quadrature, 208interpolation, 394
interpolation, using Chebyshev polynomials,
413
inverse complex trigonometric functions, 26
inverse cumulative distribution functions,233
inverse hyperbolic cosine, 18
inverse hyperbolic functions, complex
numbers, 27
inverse hyperbolic sine, 18
inverse hyperbolic tangent, 18
inverse of a matrix, by LU decomposition, 142
inverting a permutation, 108
Irregular Cylindrical Bessel Functions, 39
Irregular Modified Bessel Functions,
Fractional Order, 44
Irregular Modified Cylindrical Bessel
Functions, 40
Irregular Modified Spherical Bessel
Functions, 43
Irregular Spherical Bessel Functions, 42
iterating through combinations, 114
iterating through multisets, 118
iterating through permutations, 108
iterative refinement of solutions in linear
systems, 142JJ(x), Bessel Functions, 38j(x), Bessel Functions, 41
Jacobi elliptic functions, 52
Jacobi orthogonalization, 154
Jacobian matrix, ODEs, 383
Jacobian matrix, root finding, 455

722

KK(x), Bessel Functions, 40k(x), Bessel Functions, 43knots, basis splines, 568
kurtosis, 288

L
Laguerre functions, 67
Lambert function, 67
Landau distribution, 249
LAPACK, 182
Laplace distribution, 244
large dense linear least squares, 493
large linear least squares, normal
equations, 494
large linear least squares, routines, 495
large linear least squares, steps, 495
large linear least squares, TSQR, 494
LD_LIBRARY_PATH, 6
ldexp, 19
LDL decomposition, 157
LDLT decomposition, 157
LDLT decomposition, banded, 167
leading dimension, matrices, 94
least squares fit, 480
least squares troubleshooting, 497
least squares, covariance of best-fit
parameters, 535
least squares, nonlinear, 520
least squares, regularized, 485
least squares, robust, 489
Legendre forms of elliptic integrals, 50
Legendre functions, 68
Legendre polynomials, 68
length, computed accurately using hypot, 18
length, computed accurately using hypot3, 18
Levenberg-Marquardt algorithm, 523
Levenberg-Marquardt algorithm, geodesic
acceleration, 523
Levin u-transform, 418
Levy distribution, 250
Levy distribution, skew, 251
libraries, linking with, 6
libraries, shared, 6
license of GSL, 1
light, units of, 632
linear algebra, 139
linear algebra, BLAS, 127
linear algebra, sparse, 619
linear interpolation, 396
linear least squares, large, 493
linear regression, 482
linear systems, 141
linear systems, refinement of solutions, 142
IndexGNU Scientific Library, Release 2.7

linking with GSL libraries, 6
location estimation, 293
log1p, 18
logarithm and related functions, 73
logarithm of Beta function, 61
logarithm of combinatorial factor C(m, 60
logarithm of complex number, 26
logarithm of cosh function, special
functions, 78
logarithm of double factorial, 59
logarithm of factorial, 59
logarithm of Gamma function, 58
logarithm of Pochhammer symbol, 60
logarithm of sinh function, 78
logarithm of the determinant of a matrix, 142
logarithm, computed accurately near 1, 18
Logarithmic random variates, 279
Logistic distribution, 263
Lognormal distribution, 255
low discrepancy sequences, 229
Low-level CBLAS, 654
LQ decomposition, 150
LU decomposition, 141
LU decomposition, banded, 165

max, 285
maximal phase, Daubechies wavelets, 423
maximization, see minimization, 445
maximum of two numbers, 20
maximum value, from histogram, 338
mean, 285
mean value, from histogram, 338
mean, trimmed, 294
mean, truncated, 294
median absolute deviation, 294
Mills' ratio, inverse, 53
min, 285
minimization, BFGS algorithm, 474
minimization, caveats, 448
minimization, conjugate gradient algorithm,
474
minimization, multidimensional, 467
minimization, one-dimensional, 445
minimization, overview, 447
minimization, Polak-Ribiere algorithm, 474
minimization, providing a function tominimize, 449
minimization, simplex algorithm, 475
minimization, steepest descent algorithm, 474
minimization, stopping parameters, 450
Mminimum finding, Brent's method, 450
minimum finding, golden section algorithm,
macros for mathematical constants, 17450
magnitude of complex number, 25
minimum of two numbers, 20
mailing list, 2
minimum value, from histogram, 338
mailing list archives, 3
MINPACK, minimization algorithms, 460
mailing list for GSL announcements, 2
MISCFUN, 82
mantissa, IEEE format, 637
MISER monte carlo integration, 359
mass, units of, 630
mathematical constants, defined as macros, 17 Mixed-radix FFT, complex data, 186
Mixed-radix FFT, real data, 193
mathematical functions, elementary, 16
Modified Bessel Functions, Fractional Order,
Mathieu Function Characteristic Values, 74
44
Mathieu functions, 74
Modified Cholesky Decomposition, 157
matrices, 94
Modified Clenshaw-Curtis quadrature, 200
matrices, banded, 163
Modified Cylindrical Bessel Functions, 39
matrices, initializing, 95
Modified Givens Rotation, BLAS, 132
matrices, range-checking, 95
Modified Newton's method for nonlinear
matrices, sparse, 605
systems, 461
matrix determinant, 142
Modified Spherical Bessel Functions, 42
matrix diagonal, 99
Monte Carlo integration, 356
matrix factorization, 139
moving maximum, 307
matrix inverse, 142
moving mean, 306
matrix square root, Cholesky decomposition,
moving median, 308
154
moving median absolute deviation, 308
matrix subdiagonal, 99
moving minimum, 307
matrix superdiagonal, 99
moving quantile range, 309
matrix, constant, 95
moving standard deviation, 306
matrix, identity, 95
moving sum, 307
matrix, operations, 127
moving variance, 306
matrix, zero, 95
Index

723

GNU Scientific Library, Release 2.7

moving window accumulators, 309
moving window statistics, 304
moving window, allocation, 306
MRG, multiple recursive random number
generator, 221
MT19937 random number generator, 220
multi-parameter regression, 483
multidimensional integration, 356
multidimensional root finding, Broyden
algorithm, 462
multidimensional root finding, overview, 455
multidimensional root finding, providing afunction to solve, 456
Multimin, caveats, 469
Multinomial distribution, 274
multiplication, 49
multisets, 116
multistep methods, ODEs, 385

Nn), 59, 60
N-dimensional random direction vector, 265
NaN, defined as a macro, 18
nautical units, 629
Negative Binomial distribution, randomvariates, 275
Nelder-Mead simplex algorithm forminimization, 475
Newton algorithm, discrete, 462
Newton algorithm, globally convergent, 461
Newton's method for finding roots, 439
Newton's method for systems of nonlinear
equations, 461
Niederreiter sequence, 229
NIST Statistical Reference Datasets, 520
non-normalized incomplete Gamma function, 60
nonlinear equation, solutions of, 432
nonlinear fitting, stopping parameters,
convergence, 533
nonlinear functions, minimization, 445
nonlinear least squares, 520
nonlinear least squares, dogleg, 523
nonlinear least squares, double dogleg, 524
nonlinear least squares,
levenberg-marquardt, 523
nonlinear least squares,
levenberg-marquardt, geodesic
acceleration, 523
nonlinear least squares, overview, 521
nonlinear systems of equations, solution of,453
nonsymmetric matrix, real, eigensystem, 172
Nordsieck form, 385
normalized form, IEEE format, 637
724

normalized incomplete Beta function, 61
Not-a-number, defined as a macro, 18
NRM2, Level-1 BLAS, 131
ntuples, 349
nuclear physics, constants, 628
numerical constants, defined as macros, 17
numerical derivatives, 409
numerical integration (quadrature), 198O
obtaining GSL, 2
ODEs, initial value problems, 381
online statistics, 298, 304
optimization, combinatorial, 366
optimization, see minimization, 445
optimized functions, alternatives, 9ordering, matrix elements, 94
ordinary differential equations, initialvalue problem, 381
oscillatory functions, numericalintegration of, 204
overflow, IEEE exceptions, 639

P
Pareto distribution, 264
PAW, 356
permutations, 105
physical constants, 626
physical dimension, matrices, 94
pi, defined as a macro, 17
Pivoted Cholesky Decomposition, 156
plain Monte Carlo, 358
Pochhammer symbol, 60
Poisson random numbers, 271
Polak-Ribiere algorithm, minimization, 474
polar form of complex numbers, 23
polar to rectangular conversion, 79
polygamma functions, 76
polynomial evaluation, 29
polynomial interpolation, 396
polynomials, roots of, 28
power function, 75
power of complex number, 26
power, units of, 631
precision, IEEE arithmetic, 639
predictor-corrector method, ODEs, 385
prefixes, 633
pressure, 631
Prince-Dormand, Runge-Kutta method, 385
printers units, 630
probability distribution, from histogram, 340
probability distributions, from histograms,
339
projection of ntuples, 353
Index

GNU Scientific Library, Release 2.7

psi function, 76

Q
QAG quadrature algorithm, 201
QAGI quadrature algorithm, 202
QAGP quadrature algorithm, 202
QAGS quadrature algorithm, 202
QAWC quadrature algorithm, 203
QAWF quadrature algorithm, 205
QAWO quadrature algorithm, 204
QAWS quadrature algorithm, 203
QL decomposition, 151
Qn statistic, 295
QNG quadrature algorithm, 201
QR decomposition, 142
QR decomposition with column pivoting, 148
QUADPACK, 198
quadratic equation, solving, 30
quadrature, 198
quadrature, fixed point, 208
quadrature, interpolating, 208
quantile functions, 233
quasi-random sequences, 229

R
R250 shift-register random number generator,
224
Racah coefficients, 47
Radial Mathieu Functions, 75
radioactivity, 632
Radix-2 FFT for real data, 191
Radix-2 FFT, complex data, 185
rand, BSD random number generator, 223
rand48 random number generator, 223
random number distributions, 233
random number generators, 213
random sampling from histograms, 340
RANDU random number generator, 225
RANF random number generator, 224
range, 285
range-checking for matrices, 95
range-checking for vectors, 86
RANLUX random number generator, 221
RANLXD random number generator, 221
RANLXS random number generator, 220
RANMAR random number generator, 224, 225
Rayleigh distribution, 247
Rayleigh Tail distribution, 248
real nonsymmetric matrix, eigensystem, 172
real symmetric matrix, eigensystem, 171
Reciprocal Gamma function, 59
rectangular to polar conversion, 79
recursive stratified sampling, MISER, 359
reduction of angular variables, 79
Index

refinement of solutions in linear systems,
142
regression, least squares, 480
regression, ridge, 485
regression, robust, 489
regression, Tikhonov, 485
Regular Bessel Functions, Fractional Order,
43
Regular Bessel Functions, Zeros of, 44
Regular Cylindrical Bessel Functions, 38
Regular Modified Bessel Functions,
Fractional Order, 44
Regular Modified Cylindrical Bessel
Functions, 39
Regular Modified Spherical Bessel Functions,42
Regular Spherical Bessel Functions, 41
Regulated Gamma function, 58
relative Pochhammer symbol, 60
reporting bugs in GSL, 2
representations of complex numbers, 23
resampling from histograms, 339
residual, in nonlinear systems of equations,460
reversing a permutation, 108
ridge regression, 485
Riemann Zeta Function, 80
RK2, Runge-Kutta method, 385
RK4, Runge-Kutta method, 385
RKF45, Runge-Kutta-Fehlberg method, 385
robust location estimators, 293
robust regression, 489
robust scale estimators, 294rolling maximum, 307
rolling mean, 306
rolling median, 308
rolling median absolute deviation, 308
rolling minimum, 307
rolling quantile range, 309
rolling standard deviation, 306
rolling sum, 307
rolling variance, 306
rolling window accumulators, 309
root finding, 432
root finding, bisection algorithm, 438
root finding, Brent's method, 439
root finding, caveats, 433
root finding, false position algorithm, 439
root finding, initial guess, 437
root finding, Newton's method, 439
root finding, overview, 433
root finding, providing a function to solve,
435
root finding, search bounds, 437
725

GNU Scientific Library, Release 2.7

root finding, secant method, 439
root finding, Steffenson's method, 440
root finding, stopping parameters, 437, 459
roots, 432
ROTG, Level-1 BLAS, 132
rounding mode, 639
Runge-Kutta Cash-Karp method, 385
Runge-Kutta methods, ordinary differential
equations, 385
Runge-Kutta Prince-Dormand method, 385
running statistics, 298

sorting vector elements, 124
source code, reuse in applications, 11sparse BLAS, 617
sparse BLAS, references, 619
sparse linear algebra, 619
sparse linear algebra, examples, 622
sparse linear algebra, iterative solvers, 621
sparse linear algebra, overview, 621
sparse linear algebra, references, 626
sparse matrices, 605
sparse matrices, accessing elements, 610
sparse matrices, allocation, 610
Ssparse matrices, BLAS operations, 619
safe comparison of floating point numbers, 21 sparse matrices, compressed column storage,608
safeguarded step-length algorithm, 451
sparse matrices, compressed row storage, 608
sampling from histograms, 339, 340
sparse matrices, compressed sparse column,
SAXPY, Level-1 BLAS, 132
608
SCAL, Level-1 BLAS, 132
sparse matrices, compressed sparse row, 608
scale estimation, 294
sparse matrices, compression, 614
schedule, cooling, 367
sparse matrices, conversion, 614se(q,x), Mathieu function, 75
sparse matrices, coordinate format, 608
secant method for finding roots, 439
sparse matrices, copying, 612
selection function, ntuples, 352
sparse matrices, data types, 607
series, acceleration, 418
sparse matrices, examples, 614
shared libraries, 6
sparse matrices, exchanging rows andshell prompt, 3columns, 612
Shi(x), 55
sparse matrices, initializing elements, 611
shift-register random number generator, 224
sparse matrices, iterative solvers, 621
Si(x), 56
sparse matrices, min/max elements, 613
sign bit, IEEE format, 637
sparse matrices, operations, 612
sign of the determinant of a matrix, 142
sparse matrices, overview, 609
simplex algorithm, minimization, 475
sparse matrices, properties, 613
simulated annealing, 366
sparse matrices, reading, 611
sin, of complex number, 26
sparse matrices, references, 617
sine function, special functions, 78
sparse matrices, storage formats, 607
single precision, IEEE format, 637
sparse matrices, triplet format, 608
singular functions, numerical integration
sparse matrices, writing, 611
of, 203sparse, iterative solvers, 621
singular points, specifying positions in
special functions, 33, 78
quadrature, 202
Spherical Bessel Functions, 41
singular value decomposition, 153
spherical harmonics, 68
Skew Levy distribution, 251
spherical random variates, 2D, 265
skewness, 288
spherical random variates, 3D, 265
slope, see numerical derivative, 409
spherical random variates, N-dimensional, 265
Sn statistic, 295
spline, 394
Sobol sequence, 229
splines, basis, 565
solution of, 141
squareroot of a matrix, Cholesky
solution of linear system by Householder
decomposition, 154transformations, 162
squarerootof complex number, 26
solution of linear systems, Ax=b, 139
standard
deviation,
285
solving a nonlinear equation, 432
standard
deviation,from histogram, 338
solving nonlinear systems of equations, 453
standards
conformance,
ANSI C, 4
sorting, 121
Statistical
Reference
Datasets(StRD), 520
sorting eigenvalues and eigenvectors, 177
726

Index

GNU Scientific Library, Release 2.7

statistics, 285
statistics, from histogram, 338
statistics, moving window, 304
steepest descent algorithm, minimization, 474
Steffenson's method for finding roots, 440
stratified sampling in Monte Carlointegration, 356
stride, of vector index, 85
Student t-distribution, 260
subdiagonal, of a matrix, 99
summation, acceleration, 418
superdiagonal, matrix, 99
SVD, 153
SWAP, Level-1 BLAS, 131
swapping permutation elements, 108
SYMM, Level-3 BLAS, 135
symmetric matrices, banded, 164symmetric matrix, real, eigensystem, 171
SYMV, Level-2 BLAS, 133
synchrotron functions, 77
SYR, Level-2 BLAS, 134
SYR2, Level-2 BLAS, 134
SYR2K, Level-3 BLAS, 137
SYRK, Level-3 BLAS, 137
systems of equations, nonlinear, 453Tt-distribution, 260
t-test, 285
tangent of complex number, 26
Tausworthe random number generator, 222
Taylor coefficients, computation of, 60
testing combination for validity, 114
testing multiset for validity, 118
testing permutation for validity, 108
thermal energy, units of, 631
Tikhonov regression, 485
time units, 629
trailing dimension, matrices, 94transformation, Householder, 161transforms, Hankel, 429
transforms, wavelet, 422
transport functions, 77
traveling salesman problem, 371
triangular systems, 163
tridiagonal decomposition, 158, 159
tridiagonal systems, 162
trigonometric functions, 78
trigonometric functions of complex numbers,
26
trigonometric integrals, 56
trimmed mean, 294
TRMM, Level-3 BLAS, 136
TRMV, Level-2 BLAS, 133
Index

TRSM, Level-3 BLAS, 136
TRSV, Level-2 BLAS, 133
truncated mean, 294
TSP, 371
TT800 random number generator, 224
two dimensional Gaussian distribution,
241,
242
two dimensional histograms, 341
two-sided exponential distribution, 244
Type 1 Gumbel distribution, random variates,
267
Type 2 Gumbel distribution, 268

Uu-transform for series, 418
underflow, IEEE exceptions, 639
uniform distribution, 254
units of, 631–633
units, conversion of, 626
units, imperial, 629
Unix random number generators, rand, 223
Unix random number generators, rand48, 223
unnormalized incomplete Gamma function, 60
unweighted linear fits, 480

Vvalue function, ntuples, 352
Van der Pol oscillator, example, 390
variance, 285
variance, from histogram, 338
variance-covariance matrix, linear fits, 482
VAX random number generator, 225
vector, operations, 127
vector, sorting elements of, 124
vectors, 85
vectors, initializing, 87
vectors, range-checking, 86
VEGAS Monte Carlo integration, 361
viscosity, 632
volume units, 630

W
W function, 67
warning options, 645
warranty (none), 2
wavelet transforms, 422
website, developer information, 3
Weibull distribution, 266
weight, units of, 630
weighted linear fits, 480
Wigner coefficients, 47
Wishart random variates, 280

727

GNU Scientific Library, Release 2.7

Y
Y(x), Bessel Functions, 39y(x), Bessel Functions, 42Zzero finding, 432
zero matrix, 95
zero, IEEE format, 637
Zeros of Regular Bessel Functions, 44
Zeta functions, 79
Ziggurat method, 237

728

Index