bluuebunny/crossref_metadata_2025 huggingface dataset:
Limits on the number of function evaluations required by certain high-dimensional integration rules of hypercubic symmetry
Authors: Lyness J. N.
Published: June 2010

We consider an

n
n

-dimensional integration rule

R
t

(
n
)

R_t^{(n)}

of degree

2
t
−
1

2t - 1

and of hypercubic symmetry. We derive theorems which set a lower bound in terms of

n
n

and

t
t

on the number of function evaluations such a rule requires. These results apply to spaces of integration which have hypercubic symmetry. In certain cases this bound is very close to the number of points required by a known rule.
Symmetric integration rules for hypercubes. I. Error coefficients
Authors: Lyness J. N.
Published: June 2010

A compact notation is introduced to describe and systematise symmetric integration rules and the Euler-Maclaurin expansion is used to describe their error terms. The application to cytolic rules is discussed especially in relation to the number of function evaluations required. This paper is devoted exclusively to theory, illustrated by well-known results. This theory leads to new powerful integration rules which will be published shortly.
Symmetric integration rules for hypercubes. II. Rule projection and rule extension
Authors: Lyness J. N.
Published: June 2010

A theory is described which facilitates the construction of highdimensional integration rules. It is found that, for large

n
n

, an

n
n

-dimensional integration rule of degree

2
t
+
1

2t + 1

man be constructed requiring a number of function evaluations of order

2
t

n
t

/

t
!

{2^t}{n^t}/t!

. In an example we construct a

15
15

-dimensional rule of degree 9 which requires 52,701 function evaluations. The corresponding number for the product Gaussian is

3
×

10

10

3 \times {10^{10}}

.
Complex zeros of cylinder functions
Authors: Döring Boro
Published: June 2010

All complex zeros of the cylinder functions

Y
n

(
z
)
,

H
n

(
1
)

(
z
)

 for 

n
=
0

{Y_n}(z),{H_n}^{(1)}(z){\text { for }}n = 0

and

1
1

,

|

z

|

>
158

|z| > 158

,

|

arg
⁡
z

|

≦
π

|\arg z| \leqq \pi

and some complex zeros small in modulus of

Y
n

(
z
)

{Y_n}(z)

,

H
n

(
1
)

(
z
)

{H_n}^{(1)}(z)

for

n
=
2

n = 2

,

3
3

,

4
4

,

5
5

,

15
15

and

|

arg
⁡
z

|

≦
π

|\arg z| \leqq \pi

are tabulated accurate to

10
D

10D

. The methods (being partly new) for computing these zeros are described.
On the relative efficiencies of gradient methods
Authors: Greenstadt John
Published: June 2010

A comparison is made among various gradient methods for maximizing a function, based on a characterization by Crockett and Chernoff of the class of these methods. By defining the “efficiency” of a gradient step in a certain way, it becomes easy to compare the efficiencies of different schemes with that of Newton’s method, which can be regarded as a particular gradient scheme. For quadratic functions, it is shown that Newton’s method is the most efficient (a conclusion which may be approximately true for nonquadratic functions). For functions which are not concave (downward), it is shown that the Newton direction may be just the opposite of the most desirable one. A simple way of correcting this is explained.
A class of single-step methods for systems of nonlinear differential equations
Authors: Cooper G. J.
Published: June 2010

The numerical solution of a system of nonlinear differential equations of arbitrary orders is considered. General implicit single-step methods are obtained and some convergence properties studied.
Rational Chebyshev approximations for the exponential integral 𝐸₁(𝑥)
Authors: Cody W. J. , Thacher Henry C.
Published: June 2010

Rational Chebyshev approximations are presented for the exponential integral

E
1

(
x
)

{E_1}(x)

in the intervals

(
0
,
1
]

(0,1]

,

[
1
,
4
]

[1,4]

, and

[
4
,
∞
)

[4,\infty )

with maximal relative errors ranging down to

10

−
21

{10^{ - 21}}

.

25
S

25S

coefficients are also given for a continued-fraction expansion for small

X
X

.
Experiments on error growth associated with some linear least-squares procedures
Authors: Jordan T. L.
Published: June 2010

Some numerical experiments were performed to compare the performance of procedures for solving the linear least-squares problem based on GramSchmidt, Modified Gram-Schmidt, and Householder transformations, as well as the classical method of forming and solving the normal equations. In addition, similar comparisons were made of the first three procedures and a procedure based on Gaussian elimination for solving an

n
×
n

n \times n

system of equations. The results of these experiments suggest that: (1) the Modified Gram-Schmidt procedure is best for the least-squares problem and that the procedure based on Householder transformations performed competitively; (2) all the methods for solving least-squares problems suffer the effects of the condition number of

A

a
m
p
;

T

a
m
p
;
A

\begin {array}{*{20}{c}} A & {^T} & A \\ \end {array}

, although in a different manner for the first three procedures than for the fourth; and (3) the procedure based on Gaussian elimination is the most economical and essentially, the most accurate for solving

n
×
n

n \times n

systems of linear equations. Some effects of pivoting in each of the procedures are included.
Primitive trinomials of high degree
Authors: Rodemich Eugene R. , Rumsey Howard
Published: June 2010

New primitive polynomials

(
mod
2
)

(\bmod 2)

of high degree were found by the methods described here. The polynomials are all trinomials; such trinomials are useful for generating pseudo-random sequences of

0
0

’s and

1
1

’s of long length [1].
On the relations between eigenvalues and eigenvectors for matrices resulting from pre and post multiplication by the transpose
Authors: Law V. J. , Fariss R. H.
Published: June 2010

Relationships between the eigenproblem associated with the matrices

A
T

A

{A^T}A

and

A

A
T

A{A^T}

are derived. If the larger problem is solved first, then the eigenvalues and eigenvectors associated with the smaller problem may easily be computed from the derived relationships.
Numerical solution of the Navier-Stokes equations
Authors: Chorin Alexandre Joel
Published: June 2010

A finite-difference method for solving the time-dependent NavierStokes equations for an incompressible fluid is introduced. This method uses the primitive variables, i.e. the velocities and the pressure, and is equally applicable to problems in two and three space dimensions. Test problems are solved, and an application to a three-dimensional convection problem is presented.
A simple set of test matrices for eigenvalue programs
Authors: Gear C. W.
Published: June 2010

Sets of simple matrices of order

N
N

are given, together with all of their eigenvalues and right eigenvectors, and simple rules for generating their inverses in the nonsingular cases. In general, these matrices are nonsymmetric. They can have sets of double and triple roots. In each of these cases, two of the roots of the doublet or triplet can correspond to a single eigenvector.
The explicit inverses of two commonly occurring matrices
Authors: Kershaw D.
Published: June 2010

Explicit formulae are given for the inverses of certain tridiagonal scalar and block matrices. During an investigation into the convergence properties of natural splines it was found convenient to have the explicit forms for the inverses of certain tridiagonal matrices. Special forms of these matrices arise in other branches of numerical analysis and so it may be useful to record them. The derivation is elementary but complicated and will only be indicated.
Reduction of functions of some partitioned matrices
Authors: Lovass-Nagy Victor , Powers David L.
Published: June 2010

In the numerical analysis of physical problems, there often arise large matrices which exhibit certain kinds of block-symmetry when partitioned appropriately. In this article, the structures of the frequently-occurring hyper-circulant and hyper-Jacobi matrices are examined, and it is shown how the calculation of any analytic function of such matrices may be reduced to the calculation of functions of the submatrices. Examples drawn from current engineering literature are given as well as small illustrative examples.
Computation of isomorphism classes of 𝑝-groups
Authors: James Rodney , Cannon John
Published: June 2010

p
p

-groups may be classified by splitting the groups up into classes having the same commutator relations (isoclinism classes) and then determining the nonisomorphic groups in each class. This paper reduces the problem of determining the isomorphism classes to that of finding the equivalence classes of a set of matrices under some equivalence relation. A computer is used to find the equivalence classes for the first few values of

p
p

, and these are then used as a guide for finding the solution for general

p
p

.
On the series expansion method for computing incomplete elliptic integrals of the first and second kinds
Authors: Van de Vel H.
Published: June 2010

In the present paper an attempt is made to improve the series expansion method for computing the incomplete integrals

F
(
ϕ
,
k
)

F(\phi ,k)

and

E
(
ϕ
,
k
)

E(\phi ,k)

. Therefore the following three pairs of series covering the region

−
1
≦
k
≦
1
,
0
≦
ϕ
>
π

/

2

- 1 \leqq k \leqq 1,0 \leqq \phi > \pi /2

are used: series obtained by a straightforward binomial expansion of the integrands, series valid for

k

′
2

tan
2

ϕ
>
1

{k’^2}{\tan ^2}\phi > 1

, and new series which converge for

ϕ
>
π

/

4

\phi > \pi /4

and for all values of

k
k

. Terms of the last two pairs of series can be generated by means of the same recurrence relations, so that the coding of the whole is not longer than that for similar methods using only two pairs of series. Any degree of accuracy can be obtained. In general the method is a little bit slower than Bulirsch’ calculation procedures which are based on the Landen transformation, but it works more quickly in case of large values of

k
2

{k^2}

and/or

ϕ
\phi

. The new series introduced are also represented in trigonometric form, and the double passage to the limit

k
2

→
1

{k^2} \to 1

,

ϕ
→
π

/

2

\phi \to \pi /2

is discussed.
On the lattice constant for |𝑥³+𝑦³+𝑧³|≦1
Authors: Spohn W. G.
Published: June 2010

There has been no published work on this intractable problem in the Geometry of Numbers since 1946. In 1944 and 1946 L. J. Mordell and H. Davenport gave bounds for the lattice constant in the Journal of the London Mathematical Society. The present attack stems from considering natural lattices with 9 points on the boundary of the region. The points of these lattices which are interior to the region are removed in the most efficient way by applying a convergent linear programming process. Apparently an infinite number of points must be removed in an infinite number of stages. A conjecture is made about the critical lattices for the region and the conjectured value

.948754
.
…

.948754. \ldots

is given for the lattice constant.
Chebyshev approximations for the exponential integral 𝐸𝑖(𝑥)
Authors: Cody W. J. , Thacher Henry C.
Published: June 2010

The computation of the exponential integral

E
i
(
x
)

Ei(x)

,

x
>
0

x > 0

, using rational Chebyshev approximations is discussed. The necessary approximations are presented in well-conditioned forms for the intervals

(
0
,
6
]

(0,6]

,

[
6
,
12
]

[6,12]

,

[
12
,
24
]

[12,24]

and

[
24
,
∞
)

[24,\infty )

. Maximal relative errors are as low as from

8
×

10

−
19

t
o
2
×

10

−
21

8 \times {10^{ - 19}}to2 \times {10^{ - 21}}

. In addition, the value of the zero of

E
i
(
x
)

Ei(x)

is presented to 30 decimal places.
A general algorithm for nonnegative quadrature formulas
Authors: Wilson M. Wayne
Published: June 2010

A general algorithm is presented for determining numerical integration formulas exact for an arbitrary finite set of continuous functions defined on a compact set, involving nonnegative combinations of function values at a finite number of points in the set. Examples are given.
On the solid-packing constant for circles
Authors: Melzak Z. A.
Published: June 2010

A solid packing of a circular disk

U
U

is a sequence of disjoint open circular subdisks

U

1
,

U

2
,

⋯

{U_{1,}}{U_{2,}} \cdots

whose total area equals that of

U
U

. The MergelyanWesler theorem asserts that the sum of radii diverges; here numerical evidence is presented that the sum of ath powers of the radii diverges for every

a
>
1.306951

a > 1.306951

. This is based on inscribing a particular sequence of 19660 disks, fitting a power law for the radii, and relating the exponent of the power law to the above constant.
A note on sums of four cubes
Authors: Lal M. , Russell W. , Blundon W. J.
Published: June 2010

A search for the integral solutions of the Diophantine equation

x
3

+

y
3

+
2

z
3

=
k

{x^3} + {y^3} + 2{z^3} = k

for

|

x

|

|x|

|

y

|

|y|

and

|

z

|

>

10
5

|z| > {10^5}

was made on an I.B.M. 1620 Model 1. These results showed that there are now just 19 values of

k
k

in the range

1
≦
k
≦
999

1 \leqq k \leqq 999

for which no solution is known.
Quadrature methods based on complex function values
Authors: Lyness J. N.
Published: June 2010

A method of numerical quadrature over a finite interval is described. This method is applicable if the integrand is an analytic function, regular within the circle in the complex plane having the integration interval as diameter. The method is iterative in nature and relies on function values at equally spaced points on this circle. It is flexible enough to take into account certain simple nonanalytic singularities in the integrand lying on the interval of integration or its extension. Numerical examples are given which illustrate various advantages and disadvantages of this method when compared with standard quadrature procedures.
On the numerical solution of a differential-difference equation arising in analytic number theory
Authors: van de Lune J. , Wattel E.
Published: June 2010

In the January 1962 issue of this Journal R. Bellman and B. Kotkin published a short paper under the same title as this one (cf. [1]). In that paper Bellman and Kotkin presented some of their results concerning the numerical computation of the continuous function

y
(
x
)

y(x)

, defined by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row y left-parenthesis x right-parenthesis equals 1 left-parenthesis 0 less-than-over-equals x less-than-over-equals 1 right-parenthesis 2nd Row y prime left-parenthesis x right-parenthesis equals minus StartFraction 1 Over x EndFraction dot y left-parenthesis x minus 1 right-parenthesis left-parenthesis x greater-than 1 right-parenthesis EndLayout">

y
(
x
)
=
1
(
0
≦
x
≦
1
)

y
′

(
x
)
=
−

1
x

⋅
y
(
x
−
1
)
(
x
>
1
)

\begin {gathered} y(x) = 1(0 \leqq x \leqq 1) \hfill \\ y’ (x) = - \frac {1} {x} \cdot y(x - 1)(x > 1) \hfill \\ \end {gathered}

\]
Tables of

y
(
x
)

y(x)

were given for

x
=
1
(
0.0625
)
6

x = 1(0.0625)6

and

x
=
6
(
1
)
20

x = 6(1)20

. In the process of extending these tables beyond

x
=
20

x = 20

we discovered that the second table was rather inaccurate for all values of

x
>
9

x > 9

. Bellman and Kotkin found, for example, that

y
(
20
)
=
0.149
⋅

10

−
8

y(20) = 0.149 \cdot {10^{ - 8}}

, whereas the actual value of

y
(
20
)

y(20)

can be shown to be smaller than

10

−
20

{10^{ - 20}}

. Moreover, in view of the method used by Bellman and Kotkin, one may expect that it would be quite time consuming to compute

y
(
x
)

y(x)

for values of

x
x

up to say

x
=
1
,
000

x = 1,000

. In this paper we describe a different method which enables us to compute

y
(
x
)

y(x)

for values of

x
x

up to about “as far as one would like."
Summation of a slowly convergent series arising in antenna study
Authors: Den Chi Fu
Published: June 2010

An equivalent series for the slowly convergent series
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts left-bracket integral Subscript negative pi slash 2 Superscript pi slash 2 Baseline cosine Superscript alpha Baseline theta cosine left-parenthesis n element-of sine theta right-parenthesis right-bracket squared slash n">

∑

n
=
1

∞

[

∫

−
π

/

2

π

/

2

cos

α

θ
cos
⁡
(
n
∈
sin
⁡
θ
)

]

2

/

n

\sum \limits _{n = 1}^\infty {\left [ {\smallint _{ - \pi /2}^{\pi /2}{{\cos }^\alpha }\theta \cos (n \in \sin \theta )} \right ]} {}^2/n

\]
which arises in antenna theory is obtained. The new form is found to consist of two rapidly convergent series for small

∈
\in

.
A method for the approximation of functions defined by formal series expansions in orthogonal polynomials
Authors: Holdeman Jonas T.
Published: June 2010

An algorithm is described for numerically evaluating functions defined by formal (and possibly divergent) series as well as convergent series of orthogonal functions which are, apart from a factor, orthogonal polynomials. When the orthogonal functions are polynomials, the approximations are rational functions. The algorithm is similar in some respects to the method of Padé approximants. A rational approximation involving Tchebychev polynomials due to H. Maehley and described by E. Kogbetliantz [1] is a special case of the algorithm.
Applications of Hilbert transform theory to numerical quadrature
Authors: Smith W. E. , Lyness J. N.
Published: June 2010

Some finite integrals are difficult to evaluate numerically because the integrand has a high peak or contains a rapidly oscillating function as a factor. If the integrand is an analytic function Cauchy’s theorem may be applied to replace the integral by a contour integral, the path being chosen to avoid singularities of the integrand, together with a possible residue contribution. If the integrand has branch singularities in the complex plane close to the interval of integration, the direct application of Cauchy’s theorem is not practical. In this paper we show how the theory of Hilbert transforms may be applied to replace the integrand by a different complex valued function whose real part coincides with the integrand on the real line, but which has no singularities in the upper half plane. Using these transformations, integrands whose difficult behavior arises from a factor whose Hilbert transform is known analytically may be treated by carrying out a contour integral of a different function and taking the real part of the result. It is shown by means of examples that such a procedure may result in significant savings in terms of computational effort.
Extensions and applications of the Householder algorithm for solving linear least squares problems
Authors: Hanson Richard J. , Lawson Charles L.
Published: June 2010

The mathematical and numerical least squares solution of a general linear system of equations is discussed. Perturbation and differentiability theorems for pseudoinverses are given. Computational procedures for calculating least squares solutions using orthonormal transformations, multiplying matrices by a matrix of orthonormal basis vectors for the null-space of a given matrix, sequential processing of data, and processing of block diagonal matrices form a partial list of numerical topics presented.
Convergence estimates for essentially positive type discrete Dirichlet problems
Authors: Bramble J. H. , Hubbard B. E. , Thomée Vidar
Published: June 2010

In this paper we consider a class of difference approximations to the Dirichlet problem for second-order elliptic operators with smooth coefficients. The main result is that if the order of accuracy of the approximate problem is

V

\mathcal {V}

, and

F
F

(the right-hand side) and

f
f

(the boundary values) both belong to

C
λ

{C^\lambda }

for

λ
>

V

\lambda > \mathcal {V}

, then the rate of convergence is

O
(

h
λ

)

O({h^\lambda })

.
On generating infinitely many solutions of the Diophantine equation 𝐴⁶+𝐵⁶+𝐶⁶=𝐷⁶+𝐸⁶+𝐹⁶
Authors: Brudno Simcha
Published: June 2010

A method of generating infinitely many solutions of the Diophantine equation

A
6

+

B
6

+

C
6

=

D
6

+

E
6

+

F
6

{A^6} + {B^6} + {C^6} = {D^6} + {E^6} + {F^6}

is presented. The technique is to reduce the equation to one of fourth degree and to use the known recursive solutions to the fourth-order equations.
The generalized serial test applied to expansions of some irrational square roots in various bases
Authors: Beyer W. A. , Metropolis N. , Neergaard J. R.
Published: June 2010

A brief summary is given of the application of the generalized serial test for randomness to the digits of irrational

√
n

\surd n

in bases

t
t

where

2
≦
n
,
t
≦
15

2 \leqq n,t \leqq 15

. The results are consistent, except for a few aberrations, with the hypothesis of randomness of the digits.
The calculation of multidimensional Hermite polynomials and Gram-Charlier coefficients
Authors: Berkowitz S. , Garner F. J.
Published: June 2010

The paper documents derivations of: (a) a recurrence relation for calculating values of multidimensional Hermite polynomials, (b) a recurrence relation for calculating an approximation to the Gram-Charlier coefficients of the probability density distribution associated with a random process, based on (a), (c) an efficient algorithm to utilize the formulae in (a) and (b).
Best rational starting approximations and improved Newton iteration for the square root
Authors: Ninomiya Ichizo
Published: June 2010

The most important class of the best rational approximations to the square root is obtained analytically by means of elliptic function theory. An improvement of the Newton iteration procedure is proposed.
Chebyshev type quadrature formulas
Authors: Kahaner David K.
Published: June 2010

Quadrature formulas of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript negative 1 Superscript 1 Baseline f left-parenthesis x right-parenthesis d x almost-equals StartFraction 2 Over n EndFraction sigma-summation Underscript i equals 1 Overscript n Endscripts f left-parenthesis x Subscript i Baseline Superscript left-parenthesis n right-parenthesis Baseline right-parenthesis">

∫

−
1

1

f
(
x
)
d
x
≈

2
n

∑

i
=
1

n

f
(

x
i

(
n
)

)

\int _{ - 1}^1 {f(x)dx \approx \frac {2} {n}\sum \limits _{i = 1}^n {f({x_i}^{(n)})} }

\]
are associated with the name of Chebyshev. Various constraints may be posed on the formula to determine the nodes

x
i

(
n
)

{x_i}^{(n)}

. Classically the formula is required to integrate

n
n

th degree polynomials exactly. For

n
=
8

n = 8

and

n
≧
10

n \geqq 10

this leads to some complex nodes. In this note we point out a simple way of determining the nodes so that the formula is exact for polynomials of degree less than

n
n

. For

n
=
8

n = 8

,

10
10

and

11
11

we compare our results with others obtained by minimizing the

l
2

{l^2}

-norm of the deviations of the first

n
+
1

n + 1

monomials from their moments and point out an error in one of these latter calculations.
Some observations on interpolation in higher dimensions
Authors: Guenther R. B. , Roetman E. L.
Published: June 2010

This paper presents a method, based on successive displacement of the coordinates, both for finding suitable interpolation points and for constructing the interpolating polynomial for functions of more than one independent variable.
Numerical differentiation and the solution of multidimensional Vandermonde systems
Authors: Galimberti G. , Pereyra V.
Published: June 2010

We define multidimensional Vandermonde matrices (MV) to be certain submatrices of Kronecker products of standard Vandermonde matrices. These MV matrices appear naturally in multidimensional problems of polynomial interpolation. An explicit algorithm is produced to solve systems of linear equations with MV matrices of coefficients. This is an extension of work of Stenger for the two-dimensional case. Numerical results for three-dimensional numerical differentiation are given.
On the numerical solution of the diffusion equation
Authors: Tødenes Øystein
Published: June 2010

A proof given by C. E. Pearson

[
1
]

[1]

for the asymptotic convergence of the numerical solution of the diffusion equation is discussed, and found insufficient. A new, direct proof is given. A method given by Pearson, for improving the numerical solution when a discontinuity is present in the initial-boundary conditions, is considered in more detail.
Lower bounds for the disk packing constant
Authors: Boyd David W.
Published: June 2010

An osculatory packing of a disk,

U
U

, is an infinite sequence of disjoint disks,

{

U
n

}

\{ {U_n}\}

, contained in

U
U

, chosen so that, for

n
≧
2

n \geqq 2

,

U
n

{U_n}

has the largest possible radius,

r
n

{r_n}

, of all disks fitting in

U
∖
(

U
1

∪
⋯
∪

U

n
−
1

)

U\backslash ({U_1} \cup \cdots \cup {U_{n - 1}})

. The exponent of the packing,

S
S

, is the least upper bound of numbers,

t
t

, such that

∑

r
n
t

=
∞

\sum {r_n^t} = \infty

. Here, we present a number of methods for obtaining lower bounds for

S
S

, based on obtaining weighted averages of the curvatures of the

U
n

{U_n}

. We are able to prove that

S
>
1.28467

S > 1.28467

. We use a number of well-known results from the analytic theory of matrices.
On Hadamard matrices constructible by circulant submatrices
Authors: Yang C. H.
Published: June 2010

Let

V

2
n

{V_{2n}}

be an H-matrix of order 2n constructible by using circulant

n
×
n

n \times n

submatrices. A recursive method has been found to construct

V

4
n

{V_{4n}}

by using circulant

2
n
×
2
n

2n \times 2n

submatrices which are derived from

n
×
n

n \times n

submatrices of a given

V

2
n

{V_{2n}}

. A similar method can be applied to a given

W

4
n

{W_{4n}}

, an H-matrix of Williamson type with odd n, to construct

W

8
n

{W_{8n}}

. All

V

2
n

{V_{2n}}

constructible by the standard type, for

1
≦
n
≦
16

1 \leqq n \leqq 16

, and some

V

2
n

{V_{2n}}

, for

n
≧
20

n \geqq 20

, are listed and classified by this method.
Error analysis of the algorithm for shifting the zeros of a polynomial by synthetic division
Authors: Stewart G. W.
Published: June 2010

An analysis is given of the role of rounding errors in the synthetic division algorithm for computing the coefficients of the polynomial

g
(
z
)
=
f
(
z
+
s
)

g(z) = f(z + s)

from the coefficients of the polynomial f. It is shown that if

|

z
+
s

|

≅

|

z

|

+

|

s

|

|z + s| \cong |z| + |s|

then the value of the computed polynomial

g
∗

(
z
)

{g^\ast }(z)

differs from

g
(
z
)

g(z)

by no more than a bound on the error made in computing

f
(
z
+
s
)

f(z + s)

with rounding error. It may be concluded that well-conditioned zeros of f lying near s will not be seriously disturbed by the shift.
An example of ill-conditioning in the numerical solution of singular perturbation problems
Authors: Dorr Fred W.
Published: June 2010

The use of finite-difference methods is considered for solving a singular perturbation problem for a linear ordinary differential equation with an interior turning point. Computational results demonstrate that such problems can lead to very ill-conditioned matrix equations.
Covering the set of integers by congruence classes of distinct moduli
Authors: Choi S. L. G.
Published: June 2010

A set of congruences is called a covering set if every integer belongs to at least one of the congruences. Erdös has raised the following question: given any number N, does there exist a covering set of distinct moduli such that the least of such moduli is N. This has been answered in the affirmative for N up to 9. The aim of this paper is to show that there exists a covering set of distinct moduli the least of which is 20. Recently, Krukenberg independently and by other methods has also obtained results up through

N
=
18

N = 18

.
Convergent generalized monotone splitting of matrices
Authors: Mangasarian O. L.
Published: June 2010

Let B and T be

n
×
n

n \times n

real matrices and r an n-vector and consider the system

u
=
B
T
u
+
r

u = BTu + r

. A new sufficient condition is given for the existence of a solution and convergence of a monotone process to a solution. The monotone process is a generalization of the Collatz-Schröder procedure.
Convergence of difference methods for initial and boundary value problems with discontinuous data
Authors: Chartres Bruce , Stepleman Robert
Published: June 2010

This paper extends the classical convergence theory for numerical solutions to initial and boundary value problems with continuous data (the right-hand side) to problems with Riemann integrable data. Order of convergence results are also obtained.
The lattice structure of multiplicative congruential pseudo-random vectors
Authors: Beyer W. A. , Roof R. B. , Williamson Dorothy
Published: June 2010

The lattice structure of points in an n-dimensional space produced by an appropriate grouping of pseudo-random numbers obtained from multiplicative congruential generators is discussed. Examples are given for

2
≦
n
≦
6

2 \leqq n \leqq 6

. The work is based on the theory of the reduction of positive quadratic forms in n variables.
On the Gaussian integration of Chebyshev polynomials
Authors: Curtis A. R. , Rabinowitz P.
Published: June 2010

It is shown that as

m
m

tends to infinity, the error in the integration of the Chebyshev polynomial of the first kind,

T

(
4
m
+
2
)
j
±
2
l

(
x
)

{T_{(4m + 2)j \pm 2l}}(x)

, by an

m
m

-point Gauss integration rule approaches

(
−
1

)
j

⋅
2

/

(
4

l
2

−
1
)
,
l
=
0
,
1
,
⋯
,
m
−
1

{( - 1)^j} \cdot 2/(4{l^2} - 1),l = 0,1, \cdots ,m - 1

, and

(
−
1

)
j

⋅
π

/

2
,
l
=
m

{( - 1)^j} \cdot \pi /2,l = m

, for all

j
j

.
Coefficients in series expansions for certain classes of functions
Authors: Tuan P. D. , Elliott David
Published: June 2010

The problem of evaluating or estimating the coefficients for the expansion of a function in a series of classical orthogonal polynomials is examined. By restricting the functions under consideration to the classes of integral transforms and inverse integral transforms, the coefficients may be expressed in alternative forms which often are more amenable to analysis.
Norms on direct sums and tensor products
Authors: Lancaster P. , Farahat H. K.
Published: June 2010

We first consider the construction of a norm on a direct sum of normed linear spaces and call a norm absolute if it depends only on the norms of the component spaces. Several characterizations are given of absolute norms. Absolute norms are then used to construct norms on tensor products of normed linear spaces and on tensor products of operators on normed linear spaces.
An algebraic theory of integration methods
Authors: Butcher J. C.
Published: June 2010

A class of integration methods which includes Runge-Kutta methods, as well as the Picard successive approximation method, is shown to be related to a certain group which can be represented as the family of real-valued functions on the set of rooted trees. For each integration method, a group element is defined corresponding to it and it is shown that the numerical result obtained using the method is characterised by this group element. If two methods are given, then a new method may be defined in such a way that when it is applied to a given initial-value problem the result is the same as for the successive application of the given methods. It is shown that the group element for this new method is the product of the group elements corresponding to the given methods. Various properties of the group and certain of its subgroups are examined. The concept of order is defined as a relationship between group elements.
An integral analogue of Taylor’s series and its use in computing Fourier transforms
Authors: Osler Thomas J.
Published: June 2010

In this paper, an integral analogue of Taylor’s series
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis equals integral Subscript negative normal infinity Superscript normal infinity Baseline upper D Superscript omega Baseline f left-parenthesis z 0 right-parenthesis left-parenthesis z minus z 0 right-parenthesis Superscript omega Baseline slash normal upper Gamma left-parenthesis omega plus 1 right-parenthesis d omega">

f
(
z
)
=

∫

−
∞

∞

D
ω

f
(

z
0

)

(
z
−

z
0

)
ω

/

Γ
(
ω
+
1
)
d
ω

f(z) = \int _{ - \infty }^\infty {{D^\omega }f({z_0})} {(z - {z_0})^\omega }/\Gamma (\omega + 1)d\omega

\]
is discussed.

D
ω

f
(
z
)

{D^\omega }f(z)

is a fractional derivative of order

ω
\omega

. Extensions of this integral are also given, one of which is an integral analogue of Lagrange’s expansion. These integrals are shown to be generalizations of the Fourier integral theorem. Several special cases of these integrals are computed, and a table of Fourier transforms emerges.
An algorithm for computing logarithms and arctangents
Authors: Carlson B. C.
Published: June 2010

An iterative algorithm with fast convergence can be used to compute logarithms, inverse circular functions, or inverse hyperbolic functions according to the choice of initial conditions. Only rational operations and square roots are required. The method consists in adding an auxiliary recurrence relation to Borchardt’s algorithm to speed the convergence.
Best 𝐿_{𝑝} approximation
Authors: Kahng S. W.
Published: June 2010

A new algorithm is presented for the best

L
p

{L_p}

approximation of a continuous function over a discrete set or a finite interval with

2
>
p
>
∞

2 > p > \infty

. Methods to accelerate the convergence of the Rice-Usow extension of Lawson’s algorithm as well as the new algorithm are presented, and the result of a numerical example is given.
Minimum norm differentiation formulas with improved roundoff error bounds
Authors: Kahaner David K.
Published: June 2010

Numerical differentiation formulas of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma Subscript i equals 1 Superscript upper N Baseline w Subscript i Baseline f left-parenthesis x Subscript i Baseline right-parenthesis almost-equals f Superscript left-parenthesis m right-parenthesis Baseline left-parenthesis a right-parenthesis comma alpha less-than-over-equals x Subscript i Baseline less-than-over-equals beta">

Σ

i
=
1

N

w
i

f
(

x
i

)
≈

f

(
m
)

(
a
)
,
α
≦

x
i

≦
β

\Sigma _{i = 1}^N{w_i}f({x_i}) \approx {f^{(m)}}(a),\alpha \leqq {x_i} \leqq \beta

, are considered. The roundoff error of such formulas is bounded by a value proportional to

Σ

i
=
1

N

|

w
i

|

\Sigma _{i = 1}^N|{w_i}|

. We consider formulas that have minimum norm

Σ

i
=
1

N

w
i
2

\Sigma _{i = 1}^Nw_i^2

and converge to

f

(
m
)

(
a
)

{f^{(m)}}(a)

as

β
−
α
→
0

\beta - \alpha \to 0

. The resulting roundoff error bounds can be several orders of magnitude less than corresponding bounds for high order differences.
On Thabit ibn Kurrah’s formula for amicable numbers
Authors: Borho Walter
Published: June 2010

This note describes some methods for deriving explicit formulas for amicable numbers which are similar to Thabit’s famous rule. The search for new amicable pairs by such rules consists merely in primality tests for certain big numbers. Only two new pairs are actually given, demonstrating the usefulness of the results and simultaneously illustrating the quite strange form of pairs obtained by this new method. It is proposed to use a computer in order to get more solutions.
Complex zeros of the modified Bessel function 𝐾_{𝑛}(𝑍)
Authors: Parnes R.
Published: June 2010

The complex zeros of

K
n

(
Z
)

{K_n}(Z)

are computed for integer orders

n
=
2
(
1
)
10

n = 2(1)10

, to

9D

{\text {9D}}

figures, using an iterative interpolation scheme.
Remark on a paper by Huddleston
Authors: Merz Gerhard
Published: June 2010

Using a function-theoretic approach, we obtain, in a quite simple way, linear relations between the values of a function and its first derivatives at

n
n

abscissa points

x
1

{x_1}

. The derivation of these formulae in a recent paper by Huddleston was rather cumbersome. Possible generalizations are indicated.
Note on representing a prime as a sum of two squares
Authors: Brillhart John
Published: June 2010

An improvement is given to the method of Hermite for finding

a
a

and

b
b

in

p
=

a
2

+

b
2

p = {a^2} + {b^2}

, where

p
p

is a

prime

≡
1

(
mod

4
)

{\text {prime}} \equiv 1\pmod 4

.
On the general Hermite cardinal interpolation
Authors: Kreß R.
Published: June 2010

A sequence of interpolation series is given which generalizes Whittaker’s cardinal function to the case of Hermite interpolation. By integrating the interpolation series, a sequence of new quadrature formulae for

∫

−
∞

∞

f
(
x
)
d
x

\int _{ - \infty }^\infty {f(x)dx}

is obtained. Derivative-free remainders are stated for these interpolation and quadrature formulae.
A note on modified optimal linear multistep methods
Authors: Brunner H.
Published: June 2010

Modified optimal linear

k
k

-step methods (whose coefficients depend on the stepsize and on a parameter

L
L

) are used for the numerical integration of systems of nonlinear ordinary differential equations. It is shown that, by choosing

L
L

suitably (depending essentially on the growth parameters of the

k
k

-step method and on the logarithmic norm of the Jacobian of the given system), weak stability does no longer occur, and one of two types of stability (called asymptotical relative and asymptotical absolute stability) may be obtained.
Note on backward recurrence algorithms
Authors: Olver F. W. J. , Sookne D. J.
Published: June 2010

An algorithm is given for the computation of the recessive solution of a secondorder linear difference equation, based upon a combination of algorithms due to J.C.P. Miller and F.W.J. Olver. A special feature is automatic and rigorous control of truncation error. The method is illustrated by application to the well-used example of the Bessel functions

J
r

(
x
)

{J_r}(x)

.
Generalized recursive multivariate interpolation
Authors: McKinney Earl H.
Published: June 2010

A generalized recursive interpolation technique for a set of linear functionals over a set of general univariate basis functions has been previously developed. This paper extends these results to restricted multivariate interpolation over a set of general multivariate basis functions. When the data array is a suitable configuration (e.g., an

n
n

-dimensional simplex), minimal degree multivariate interpolating polynomials are produced by this recursive interpolation scheme. By using product rules, recursive univariate interpolation applied to each variable singly produces multivariate interpolating polynomials (not of minimal degree) when the data are arranged in a hyper-rectangular array. By proper ordering of points in a data array, multivariate polynomial interpolation is accomplished over other arrays such as diamonds and truncated diamonds in two dimensions and their counterparts in

n
n

dimensions.
New approximations to familiar functions
Authors: Dutt J. E. , Lin T. K. , Tao L. C.
Published: June 2010

Using an integral representation of the Hermite polynomial and then Gaussian quadrature, very accurate representations are obtained for

exp
⁡
(
−

z
2

)
,

erf

(
z
)

\exp ( - {z^2}),{\operatorname {erf}}(z)

, and

arcsin
⁡
(
z
)

\arcsin (z)

.
Three thirteens
Authors: Lander L. J.
Published: June 2010

From a computed solution to the Tarry-Escott problem two sets of thirteen integers are obtained having equal sums of odd powers through the thirteenth.
A simple approach to the Perron-Frobenius theory for positive operators on general partially-ordered finite-dimensional linear spaces
Authors: Rheinboldt Werner C. , Vandergraft James S.
Published: June 2010

This paper presents simple proofs of the principal results of the Perron-Frobenius theory for linear mappings on finite-dimensional spaces which are nonnegative relative to a general partial ordering on the space. The principal tool for these proofs is an application of the theory of norms in finite dimensions to the study of order inequalities of the form

A
x
≦
α
x
,
x
≧
0

Ax \leqq \alpha x,x \geqq 0

where

A
≧
0

A \geqq 0

. This approach also permits the derivation of various inclusion and comparison theorems.
Saddle points of the complementary error function
Authors: Fettis Henry E. , Caslin James C. , Cramer Kenneth R.
Published: June 2010

The first one hundred zeros of the derivative of the function

w
(
z
)
=

e

−

z
2

Erfc

(
−
i
z
)

w(z) = {e^{ - {z^2}}}\;{\operatorname {Erfc}}( - iz)

are given, together with an asymptotic formula for estimating the higher zeros.
Calculation of the Ramanujan 𝜏-Dirichlet series
Authors: Spira Robert
Published: June 2010

A method is found for calculating the Ramanujan

τ
\tau

-Dirichlet series

F
(
s
)

F(s)

. An inequality connecting points symmetric with the critical line,

σ
=
6

\sigma = 6

, is proved, and a table is given for

Γ
(
s
)
F
(
s
)

\Gamma (s)F(s)

for

σ
=
6.0
,
6.5
,
t
=
0
(
.25
)

16

\sigma = 6.0,6.5,t = 0(.25)\,16

. Two zeros are found in

0
>
t
≦
16

0 > t \leqq 16

; they appear to be simple and on the critical line.
More about four biquadrates equal one biquadrate
Authors: Rose Kermit , Brudno Simcha
Published: June 2010

A computer-generated table of the first 82 numerical solutions of

A
4

+

B
4

+

C
4

+

D
4

=

E
4

{A^4} + {B^4} + {C^4} + {D^4} = {E^4}

is presented. Some regularities are noted.
Iteration methods for finding all zeros of a polynomial simultaneously
Authors: Aberth Oliver
Published: June 2010

Durand and Kerner independently have proposed a quadratically convergent iteration method for finding all zeros of a polynomial simultaneously. Here, a new derivation of their iteration equation is given, and a second, cubically convergent iteration method is proposed. A relatively simple procedure for choosing the initial approximations is described, which is applicable to either method.
A probabilistic approach to a differential-difference equation arising in analytic number theory
Authors: Chamayou Jean-Marie-François
Published: June 2010

The differential-difference equation\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column t v prime left-parenthesis t right-parenthesis plus v left-parenthesis t minus 1 right-parenthesis equals 0 comma 2nd Column a m p semicolon t greater-than 1 comma 2nd Row 1st Column v left-parenthesis t right-parenthesis equals 0 comma 2nd Column a m p semicolon t greater-than 0 comma 3rd Row 1st Column v left-parenthesis t right-parenthesis equals c o n s t a n t comma 2nd Column a m p semicolon 0 less-than-over-equals t less-than-over-equals 1 comma EndLayout">tv′(t)+v(t−1)=0,amp;t>1,v(t)=0,amp;t>0,v(t)=constant,amp;0≦t≦1,\begin {array}{*{20}{c}} {tv’(t) + v(t - 1) = 0,} \hfill & {t > 1,} \hfill \\ {v(t) = 0,} \hfill & {t > 0,} \hfill \\ {v(t) = {\operatorname {constant}},} \hfill & {0 \leqq t \leqq 1,} \hfill \\ \end {array}\]can be solved by the Monte-Carlo method, for the initial conditionv(t)=e−γ,0≦t≦1v(t) = {e^{ - \gamma }},0 \leqq t \leqq 1, where thev(t)v(t)represent the probability density of a random variable:\[t=limn→∞∑i=1n∏j=1ixj,t = \lim \limits _{n \to \infty } \sum \limits _{i = 1}^n {\prod \limits _{j = 1}^i {{x_j},} }\]where thexj{x_j}are independent and uniformly distributed on (0, 1).
Integration formulas and schemes based on 𝑔-splines
Authors: Andria George D. , Byrne George D. , Hill David R.
Published: June 2010

Numerical integration formulas of interpolatory type are generated by the integration of g-splines. These formulas, which are best in the sense of Sard, are used to construct predictor-corrector and block implicit schemes. The schemes are then compared with Adams-Bashforth-Adams-Moulton and Rosser schemes for a particular set of prototype problems. Moreover, an improved error bound for linear multistep formulas based on g-splines and a comparison of

L
2

{L^2}

norms of Peano kernels for Adams and natural spline formulas are given.
Almost-interpolatory Chebyshev quadrature
Authors: Salkauskas K.
Published: June 2010

The requirement that a Chebyshev quadrature formula have distinct real nodes is not always compatible with the requirement that the degree of precision of an n-point formula be at least equal to n. This condition may be expressed as

‖
d
‖

ν

=
0
,
1
≦
p

{\left \| d \right \|_\nu } = 0,1 \leqq p

, where

d
=
(

d
1

,
⋯
,

d
n

)

d = ({d_1}, \cdots ,{d_n})

with
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d Subscript j Baseline equals StartFraction mu 0 left-parenthesis omega right-parenthesis Over n EndFraction sigma-summation Underscript i equals 1 Overscript n Endscripts x Subscript i Superscript j Baseline minus mu Subscript j Baseline left-parenthesis omega right-parenthesis comma j equals 1 comma 2 comma midline-horizontal-ellipsis comma n comma">

d
j

=

μ
0

(
ω
)

n

∑

i
=
1

n

x
i
j

−

μ
j

(
ω
)
,

j
=
1
,
2
,
⋯
,
n
,

{d_j} = \frac {{{\mu _0}(\omega )}}{n}\sum \limits _{i = 1}^n {x_i^j - {\mu _j}(\omega ),\quad j = 1,2, \cdots ,n,}

\]

μ
j

(
ω
)
,
j
=
0
,
1
,
⋯

{\mu _j}(\omega ),j = 0,1, \cdots

, are the moments of the weight function

ω
\omega

used in the quadrature, and

x
1

,
⋯
,

x
n

{x_1}, \cdots ,{x_n}

are the nodes. In those cases when

‖
d
‖

2

{\left \| d \right \|_2}

does not vanish for a real choice of nodes, it has been proposed that a real minimizer of

‖
d
‖

2

{\left \| d \right \|_2}

be used to supply the nodes. It is shown in this paper that, in such cases, minimizers of

‖
d
‖

p

,
1
≦
p
>
∞

{\left \| d \right \|_p},1 \leqq p > \infty

, always lead to formulae that are degenerate in the sense that the nodes are not all distinct. The results are valid for a large class of weight functions.
Discrete Green’s functions
Authors: McAllister G. T. , Sabotka E. F.
Published: June 2010

Let

G
(
P
;
Q
)

G(P;Q)

be the discrete Green’s function over a discrete h-convex region

Ω
\Omega

of the plane; i.e.,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a left-parenthesis upper P right-parenthesis upper G Subscript x x overbar Baseline left-parenthesis upper P semicolon upper Q right-parenthesis plus c left-parenthesis upper P right-parenthesis upper G Subscript y y overbar Baseline left-parenthesis upper P semicolon upper Q right-parenthesis equals minus delta left-parenthesis upper P semicolon upper Q right-parenthesis slash h squared">

a
(
P
)

G

x

x
¯

(
P
;
Q
)
+
c
(
P
)

G

y

y
¯

(
P
;
Q
)
=
−
δ
(
P
;
Q
)

/

h
2

a(P){G_{x\bar x}}(P;Q) + c(P){G_{y\bar y}}(P;Q) = - \delta (P;Q)/{h^2}

for

P
∈

Ω
h

,
G
(
P
;
Q
)
=
0

P \in {\Omega _h},G(P;Q) = 0

for

P
∈
∂

Ω
h

P \in \partial {\Omega _h}

. Assume that

a
(
P
)

a(P)

and

c
(
P
)

c(P)

are Hölder continuous over

Ω
\Omega

and positive. We show that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper D Superscript left-parenthesis m right-parenthesis Baseline upper G left-parenthesis upper P semicolon upper Q right-parenthesis EndAbsoluteValue less-than-over-equals upper A Subscript m Baseline slash rho Subscript upper P upper Q Superscript m">

|

D

(
m
)

G
(
P
;
Q
)

|

≦

A
m

/

ρ

P

Q

m

|{D^{(m)}}G(P;Q)| \leqq {A_m}/\rho _{P\;Q}^m

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper D overTilde Superscript left-parenthesis m right-parenthesis Baseline upper G left-parenthesis upper P semicolon upper Q right-parenthesis EndAbsoluteValue less-than-over-equals upper B Subscript m Baseline d left-parenthesis upper Q right-parenthesis slash rho Subscript upper P upper Q Superscript m plus 1">

|

D
~

(
m
)

G
(
P
;
Q
)

|

≦

B
m

d
(
Q
)

/

ρ

P

Q

m
+
1

|{\tilde D^{(m)}}G(P;Q)| \leqq {B_m}d(Q)/\rho _{P\;Q}^{m + 1}

, where

D

(
m
)

{D^{(m)}}

is an mth order difference quotient with respect to the components of P or Q, and

D
~

(
m
)

{\tilde D^{(m)}}

denotes an mth order difference quotient only with respect to the components of P.
The finite element method with penalty
Authors: Babuška Ivo
Published: June 2010

An application of the penalty method to the finite element method is analyzed. For a model Poisson equation with homogeneous Dirichlet boundary conditions, a variational principle with penalty is discussed. This principle leads to the solution of the Poisson equation by using functions that do not satisfy the boundary condition. The rate of convergence is discussed.