标题：数值算法的精确性与稳定性
图书信息
出版社: 清华大学出版社; 第1版 (2011年2月1日)
外文书名: Accuracy and Stability of Numerical Algorithms 2nd edition
丛书名: 国际著名数学图书
平装: 680页
正文语种: 英语
开本: 16
ISBN: 9787302244936, 7302244936
条形码: 9787302244936
尺寸: 24.4 x 17.4 x 3.4 cm
重量: 939 g
作者简介
作者：（美国）汉安（Nicholas J.Higham）
Nicholas J. Higham is Richardson Professor of Applied Mathematics at the University of Manchester, England. He is the author of more than 80 publications and is a member of the editorial boards of Foundations of Computational Mathematics, the IMA Journal of Numerical Analysis, Linear Algebra and Its Applications, and the SIAM Journal on Matrix Analysis and Applications.
内容简介
《数值算法的精确性与稳定性(第2版)(影印版)》内容简介：accuracy and stability of numerical algorithms gives a thorough, up-to-date treatment of the behavior of numerical algorithms in finite precision arithmetic. it combines algorithmic derivations, perturbation theory, and rounding error analysis, all enlivened by historical perspective and informative quotations. 
this second edition expands and updates the coverage of the first edition (1996) and includes numerous improvements to the original material. two new chapters treat symmetric indefinite systems and skew-symmetric systems, and nonlinear systems and newton's method. twelve new sections include coverage of additional error bounds for gaussian elimination, rank revealing lu factorizations, weighted and constrained least squares problems, and the fused multiply-add operation found on some modern computer architectures. although not designed specifically as a textbook, this new edition is a suitable reference for an advanced course. it can also be used by instructors at all levels as a supplementary text from which to draw examples, historical perspective, statements of results, and exercises.
媒体评论
"This definitive source on the accuracy and stability of numerical algorithms is quite a bargain and a worthwhile addition to the library of any statistician heavily involved in computing." 
——Robert L. Strawderman, Journal of the American Statistical Association, March 1999. 
"This text may become the new 'Bible' about accuracy and stability for the solution of system of linear equations. It covers 688 pages carefully collected, investigated, and written.. One will find that this book is a very suitable and comprehensive reference for research in numerical linear algebra, software usage and development, and for numerical linear algebra courses." 
—— N. Kockler, Zentrablatt for Mathematik, Band 847/96. 
"Nick Higham has assembled an enormous amount of important and useful material in a coherent, readable form. His book belongs on the shelf of anyone who has more than a casual interest in rounding error and matrix computation." 
—— G.W. Stewart, SIAM Review, March 1997.
目录
list of figures 
list of tables 
preface to second edition 
preface to first edition 
about the dedication 
1 principles of finite precision computation 
1.1 notation and background 
1.2 relative error and significant digits 
1.3 sources of errors 
1.4 precision versus accuracy 
1.5 backward and forward errors 
1.6 conditioning 
1.7 cancellation 
1.8 solving a quadratic equation 
1.9 computing the sample variance 
1.10 solving linear equations 
1.10.1 gepp versus cramer's rule 
1.11 accumulation of rounding errors 
1.12 instability without cancellation 
1.12.1 the need for pivoting 
1.12.2 an innocuous calculation? 
1.12.3 an infinite sum 
1.13 increasing the precision 
1.14 cancellation of rounding errors 
1.14.1 computing (ex - 1)ix 
1.14.2 qr factorization 
1.15 rounding errors can be beneficial 
1.16 stability of an algorithm depends on the problem 
1.17 rounding errors are not random 
1.18 designing stable algorithms 
1.19 misconceptions 
1.20 rounding errors in numerical analysis 
1.21 notes and references 
problems 
2 floating point arithmetic 
2.1 floating point number system 
2.2 model of arithmetic 
2.3 ieee arithmetic 
2.4 aberrant arithmetics 
2.5 exact subtraction 
2.6 fused multiply-add operation 
2.7 choice of base and distribution of numbers 
2.8 statistical distribution of rounding errors 
2.9 alternative number systems 
2.10 elementary functions 
2.11 accuracy tests 
2.12 notes and references 
problems 
3 basics 
3.1 inner and outer products 
3.2 the purpose of rounding error analysis 
3.3 running error analysis 
3.4 notation for error analysis 
3.5 matrix multiplication 
3.6 complex arithmetic 
3.7 miscellany 
3.8 error analysis demystified 
3.9 other approaches 
3.10 notes and references 
problems 
4 summation 
4.1 summation methods 
4.2 error analysis 
4.3 compensated summation 
4.4 other summation methods 
4.5 statistical estimates of accuracy 
4.6 choice of method 
4.7 notes and references 
problems 
5 polynomials 
5.1 hornet's method 
5.2 evaluating derivatives 
5.3 the newton form and polynomial interpolation 
5.4 matrix polynomials 
5.5 notes and references 
problems 
6 norms 
6.1 vector norms 
6.2 matrix norms 
6.3 the matrix p-norm 
6.4 singular value decomposition 
6.5 notes and references 
problems 
7 perturbation theory for linear systems 
7.1 normwise analysis 
7.2 componentwise analysis 
7.3 scaling to minimize the condition number 
7.4 the matrix inverse 
7.5 extensions 
7.6 numerical stability 
7.7 practical error bounds 
7.8 perturbation theory by calculus 
7.9 notes and references 
problems 
8 triangular systems 
8.1 backward error analysis 
8.2 forward error analysis 
8.3 bounds for the inverse 
8.4 a parallel fan-in algorithm 
8.5 notes and references 
8.5.1 lapack 
problems 
9 lu factorization and linear equations 
9.1 gaussian elimination and pivoting strategies 
9.2 lu factorization 
9.3 error analysis 
9.4 the growth factor 
9.5 diagonally dominant and banded matrices 
9.6 tridiagonal matrices 
9.7 more error bounds 
9.8 scaling and choice of pivoting strategy 
9.9 variants of ganssian elimination 
9.10 a posteriori stability tests 
9.11 sensitivity of the lu factorization 
9.12 rank-revealing lu factorizations 
9.13 historical perspective 
9.14 notes and references 
9.14.1 lapack 
problems 
10 cholesky factorization 
10.1 symmetric positive definite matrices 
10.1.1 error analysis 
10.2 sensitivity of the cholesky factorization 
10.3 positive semidefinite matrices 
10.3.1 perturbation theory 
10.3.2 error analysis 
10.4 matrices with positive definite symmetric part 
10.5 notes and references 
10.5.1 lapack 
problems 
11 symmetric indefinite and skew-symmetric systems 
11.1 block ldlt factorization for symmetric matrices 
11.1.1 complete pivoting 
11.1.2 partial pivoting 
11.1.3 rook pivoting 
11.1.4 tridiagonal matrices 
11.2 aasen's method 
11.2.1 aasen's method versus block ldlt factorization 
11.3 block ldlt factorization for skew-symmetric matrices 
11.4 notes and references 
11.4.1 lapack 
problems 
12 iterative refinement 
12.1 behaviour of the forward error 
12.2 iterative refinement implies stability 
12.3 notes and references 
12.3.1 lapack 
problems 
13 block lu factorization 
13.1 block versus partitioned lu factorization 
13.2 error analysis of partitioned lu factorization 
13.3 error analysis of block lu factorization 
13.3.1 block diagonal dominance 
13.3.2 symmetric positive definite matrices 
13.4 notes and references 
13.4.1 lapack 
problems 
14 matrix inversion 
14.1 use and abuse of the matrix inverse 
14.2 inverting a triangular matrix 
14.2.1 unblocked methods 
14.2.2 block methods 
14.3 inverting a full matrix by lu factorization 
14.3.1 method a 
14.3.2 method b 
14.3.3 method c 
14.3.4 method d 
14.3.5 summary 
14.4 gauss-jordan elimination 
14.5 parallel inversion methods 
14.6 the determinant 
14.6.1 hyman's method 
14.7 notes and references 
14.7.1 lapack 
problems 
15 condition number estimation 
15.1 how to estimate componentwise condition numbers 
15.2 the p-norm power method 
15.3 lapack 1-norm estimator 
15.4 block 1-norm estimator 
15.5 other condition estimators 
15.6 condition numbers of tridiagonal matrices 
15.7 notes and references 
15.7.1 lapack 
problems 
16 the sylvester equation 
16.1 solving the sylvester equation 
16.2 backward error 
16.2.1 the lyapunov equation 
16.3 perturbation result 
16.4 practical error bounds 
16.5 extensions 
16.6 notes and references 
16.6.1 lapack 
problems 
17 stationary iterative methods 
17.1 survey of error analysis 
17.2 forward error analysis 
17.2.1 jacobi's method 
17.2.2 successive overrelaxation 
17.3 backward error analysis 
17.4 singular systems 
17.4.1 theoretical background 
17.4.2 forward error analysis 
17.5 stopping an iterative method 
17.6 notes and references 
problems 
18 matrix powers 
18.1 matrix powers in exact arithmetic 
18.2 bounds for finite precision arithmetic 
18.3 application to stationary iteration 
18.4 notes and references 
problems 
19 qr factorization 
19.1 householder transformations 
19.2 qr factorization 
19.3 error analysis of householder computations 
19.4 pivoting and row-wise stability 
19.5 aggregated householder transformations 
19.6 givens rotations 
19.7 iterative refinement 
19.8 gram-schmidt orthogonalization 
19.9 sensitivity of the qr factorization 
19.10 notes and references 
19.10.1 lapack 
problems 
20 the least squares problem 
20.1 perturbation theory 
20.2 solution by qr factorization 
20.3 solution by the modified gram-schmidt method 
20.4 the normal equations 
20.5 iterative refinement 
20.6 the seminormal equations 
20.7 backward error 
20.8 weighted least squares problems 
20.9 the equality constrained least squares problem 
20.9.1 perturbation theory 
20.9.2 methods 
20.10 proof of wedin's theorem 
20.11 notes and references 
20.11.1 lapack 
problems 
21 underdetermined systems 
21.1 solution methods 
21.2 perturbation theory and backward error 
21.3 error analysis 
21.4 notes and references 
21.4.1 lapack 
problems 
22 vandermonde systems 
22.1 matrix inversion 
22.2 primal and dual systems 
22.3 stability 
22.3.1 forward error 
22.3.2 residual 
22.3.3 dealing with instability 
22.4 notes and references 
problems 
23 fast matrix multiplication 
23.1 methods 
23.2 error analysis 
23.2.1 winograd's method 
23.2.2 strassen's method 
23.2.3 bilinear noncommutative algorithms 
23.2.4 the 3m method 
23.3 notes and references 
problems 
24 the fast fourier transform and applications 
24.1 the fast fourier transform 
24.2 circulant linear systems 
24.3 notes and references 
problems 
25 nonlinear systems and newton's method 
25.1 newton's method 
25.2 error analysis 
25.3 special cases and experiments 
25.4 conditioning 
25.5 stopping an iterative method 
25.6 notes and references 
problems 
26 automatic error analysis 
26.1 exploiting direct search optimization 
26.2 direct search methods 
26.3 examples of direct search 
26.3.1 condition estimation 
26.3.2 fast matrix inversion 
26.3.3 roots of a cubic 
26.4 interval analysis 
26.5 other work 
26.6 notes and references 
problems 
27 software issues in floating point arithmetic 
27.1 exploiting ieee arithmetic 
27.2 subtleties of floating point arithmetic 
27.3 cray peculiarities 
27.4 compilers 
27.5 determining properties of floating point arithme 
27.6 testing a floating point arithmetic 
27.7 portability 
27.7.1 arithmetic parameters 
27.7.2 2 x 2 problems in lapack 
27.7.3 numerical constants 
27.7.4 models of floating point arithmetic 
27.8 avoiding underflow and overflow 
27.9 multiple precision arithmetic 
27.10 extended and mixed precision blas 
27.11 patriot missile software problem 
27.12 notes and references 
problems 
28 a gallery of test matrices 
28.1 the hilbert and cauchy matrices 
28.2 random matrices 
28.3 "randsvd" matrices 
28.4 the pascal matrix 
28.5 tridiagonal toeplitz matrices 
28.6 companion matrices 
28.7 notes and references 
28.7.1 lapack 
problems 
a solutions to problems 
b acquiring software 
b.1 internet 
b.2 netlib 
b.3 matlab 
b.4 nag library and nagware f95 compiler 
c program libraries 
c.1 basic linear algebra subprograms 
c.2 eispack 
c.3 linpack 
c.4 lapack 
c.4.1 structure of lapack 
d the matrix computation toolbox 
bibliography 
name index 
subject index