bluuebunny/crossref_metadata_2025 huggingface dataset:
Convergence properties of the symmetric and unsymmetric successive overrelaxation methods and related methods
Authors: Young David M.
Published: June 2010

The paper is concerned with variants of the successive overrelaxation method (SOR method) for solving the linear system

A
u
=
b

Au = b

. Necessary and sufficient conditions are given for the convergence of the symmetric and unsymmetric SOR methods when

A
A

is symmetric. The modified SOR, symmetric SOR, and unsymmetric SOR methods are also considered for systems of the form

D
1

u
1

−

C
U

u
2

=

b
1

,
−

C
L

u
1

+

D
2

u
2

=

b
2

{D_1}{u_1} - {C_U}{u_2} = {b_1}, - {C_L}{u_1} + {D_2}{u_2} = {b_2}

where

D
1

{D_1}

and

D
2

{D_2}

are square diagonal matrices. Different values of the relaxation factor are used on each set of equations. It is shown that if the matrix corresponding to the Jacobi method of iteration has real eigenvalues and has spectral radius

μ
¯

>
1

\bar \mu > 1

, then the spectral radius of the matrix

G
G

associated with any of the methods is not less than that of the ordinary SOR method with

ω
=
2

(
1
+

(
1
−

μ
¯

2

)

1

/

2

)

−
1

\omega = 2{(1 + {(1 - {\bar \mu ^2})^{1/2}})^{ - 1}}

. Moreover, if the eigenvalues of

G
G

are real then no improvement is possible by the use of semi-iterative methods.
“Best” interpolation, differentiation, and integration approximations on the Hardy space 𝐻²
Authors: Winslow Leon
Published: June 2010

A general formula is developed which gives the "best" approximation for any linear functional on the Hardy space

H
2

{H^2}

. Some "best" approximations are given for interpolation, differentiation, and integration and are compared to polynomial approximations.
Asymptotic properties of minimal integration rules
Authors: Rabinowitz Philip , Richter-Dyn Nira
Published: June 2010

The error of a particular integration rule applied to a Hilbert space of functions analytic within an ellipse containing the interval of integration is a bounded linear functional. Its norm, which depends on the size of the ellipse, has proved useful in estimating the truncation error occurring when the integral of a particular analytic function is approximated using the rule in question. It is thus of interest to study rules which minimize this norm, namely minimal integration rules. The present paper deals with asymptotic properties of such minimal integration rules as the underlying ellipses shrink to the interval of integration.
An implementation of Christoffel’s theorem in the theory of orthogonal polynomials
Authors: Galant David
Published: June 2010

An algorithm for the construction of the polynomials associated with the weight function

w
(
t
)
P
(
t
)

w(t)P(t)

from those associated with

w
(
t
)

w(t)

is given for the case when

P
(
t
)

P(t)

is a polynomial which is nonnegative in the interval of orthogonality. The relation of the algorithm to the LR algorithm is also discussed.
Existence of quadrature formulae with almost equal weights
Authors: Salkauskas K.
Published: June 2010

The condition that an interpolatory quadrature formula on n nodes have degree of precision at least n and positive weights defines a homeomorphism between the sets of admissible nodes and weights of such formulae for each n. This is used to prove that the only formulae having "almost equal" weights are the Chebyshev formulae.
Homomorphisms of knot groups on finite groups
Authors: Riley Robert
Published: June 2010

We describe trial and error computer programs for finding certain homomorphisms of a knot group on a special projective group

L
F
(
2
,
p
)
,
p

LF(2,p),p

prime, and programs to evaluate

H
1

(

M

;

Z

)

{H_1}(\mathfrak {M};{\text {Z}})

where

M

\mathfrak {M}

is a finitely sheeted branched covering space of

S
3

{S^3}

associated with such a homomorphism. These programs have been applied to several collections of examples, in particular to the Kinoshita-Terasaka knots, and we state numerous conjectures based on these experiments.
A new method of evaluation of Howland integrals
Authors: Ling Chih Bing , Lin Jung
Published: June 2010

In this paper, two Howland integrals are evaluated to 25D when the index is an odd integer. Those Howland integrals when the index is an even integer have been evaluated to 18D by Nelson. A new method of evaluation is used.
The triangular decomposition of Hankel matrices
Authors: Phillips James L.
Published: June 2010

An algorithm for determining the triangular decomposition

H
=

R
∗

D
R

H = {R^ \ast }DR

of a Hankel matrix H using

O
(

n
2

)

O({n^2})

operations is derived. The derivation is based on the Lanczos algorithm and the relation between orthogonalization of vectors and the triangular decomposition of moment matrices. The algorithm can be used to compute the three-term recurrence relation for orthogonal polynomials from a moment matrix.
A quadratic formula for finding the root of an equation
Authors: Chambers Ll. G.
Published: June 2010

An extension is made to Reguli Falsi involving a parabolic approximation. Formulae are derived which have convergence exponents 2, 2.414, 2.732, respectively.
Gauss’s ternary form reduction and the 2-Sylow subgroup
Authors: Shanks Daniel
Published: June 2010

An algorithm is developed for determining the 2-Sylow subgroup of the class group of a quadratic field provided the complete factorization of the discriminant d is known. It uses Gauss’s ternary form reduction with some new improvements and is applicable even if d is so large that the class number

h
(
d
)

h(d)

is inaccessible. Examples are given for various d that illustrate a number of special problems.
The square root of 2 to 1,000,000 decimals
Authors: Dutka Jacques
Published: June 2010

The square root of 2 has been calculated to 1,000,000 decimals on a large-scale digital computer and the result has been verified. The calculation was based on a specially developed algorithm for square roots which does not appear to have been used in previous computations of this type.
Some polynomials for complex quadrature
Authors: Kahaner David K.
Published: June 2010

Equal-weight Chebyshev quadrature is not generally used because the nodes become complex for large n. However, interest in these schemes remains because of recent work on minimal norm quadrature as well as schemes for doing real integrals of analytic functions by complex methods. This note presents some properties of these Chebyshev quadratures that may be of interest to other researchers in this area. Proofs are sketched to save space.
Some numerical evidence concerning the uniqueness of the Markov numbers
Authors: Rosen D. , Patterson G. S.
Published: June 2010

A Markov triple is a set of three positive integers satisfying the diophantine equation

(

x
2

+

y
2

+

z
2

=
3
x
y
z
)

({x^2} + {y^2} + {z^2} = 3xyz)

. The maximum of the triple is called a Markov number. Although all Markov triples can be generated from the triple (1,1,1), it is not known whether it is possible to obtain

(
p
,

a
1

,

b
1

)

(p,{a_1},{b_1})

and

(
p
,

a
2

,

b
2

)

(p,{a_2},{b_2})

, where p is the same Markov number for both triples. All Markov numbers not exceeding 30 digits were computed without turning up a duplication, lending some credence to the conjecture that the Markov numbers are unique.
Whittaker’s cardinal function in retrospect
Authors: McNamee J. , Stenger F. , Whitney E. L.
Published: June 2010

This paper exposes properties of the Whittaker cardinal function and illustrates the use of this function as a mathematical tool. The cardinal function is derived using the Paley-Wiener theorem. The cardinal function and the central-difference expansions are linked through their similarities. A bound is obtained on the difference between the cardinal function and the function which it interpolates. Several cardinal functions of a number of special functions are examined. It is shown how the cardinal function provides a link between Fourier series and Fourier transforms, and how the cardinal function may be used to solve integral equations.
Matricial difference schemes for integrating stiff systems of ordinary differential equations
Authors: Miranker W. L.
Published: June 2010

In this paper we give a description and analysis of a class of matricial difference schemes. This class of schemes is based in part on a generalization of the feature of classical numerical methods of being characterized by approximations at a single point in the complex plane. The schemes introduced here are effective for integrating stiff systems.
The fast Fourier transform in a finite field
Authors: Pollard J. M.
Published: June 2010

A transform analogous to the discrete Fourier transform may be defined in a finite field, and may be calculated efficiently by the ’fast Fourier transform’ algorithm. The transform may be applied to the problem of calculating convolutions of long integer sequences by means of integer arithmetic.
Complex zeros of two incomplete Riemann zeta functions
Authors: Kölbig K. S.
Published: June 2010

The computation of the complex zeros of an incomplete Riemann zeta function defined in an earlier paper is extended and new zero trajectories are given. A second incomplete Riemann zeta function is defined and its zero trajectories are investigated numerically as functions of the upper limit

λ
\lambda

of the definition integral. It becomes apparent that there exist three different classes of zero trajectories for this function, distinguished by their behaviour for

λ
→
∞

\lambda \to \infty

.
On evaluation of moments of 𝐾_{𝜈}(𝑡)/𝐼_{𝜈}(𝑡)
Authors: Ling Chih Bing , Lin Jung
Published: June 2010

This paper presents a method of evaluation of the moments of

K
ν

(
t
)

/

I
ν

(
t
)

{K_\nu }(t)/{I_\nu }(t)

. Two pairs of expressions, each consisting of two series, are obtained according to the index being an even or an odd integer. The method is an extension of the method used by Watson. Values are tabulated to 12D for

ν
=
0
(
1
)
2

\nu = 0(1)2

.
Some results for 𝑘!±1 and 2⋅3⋅5⋯𝑝±1
Authors: Borning Alan
Published: June 2010

The numbers

k
!
±
1

k! \pm 1

for

k
=
2
(
1
)
100

k = 2(1)100

, and

2
⋅
3
⋅
5
⋯
p
±
1

2 \cdot 3 \cdot 5 \cdots p \pm 1

for

p
p

prime,

2
≦
p
≦
307

2 \leqq p \leqq 307

, were tested for primality. For

k
=
2
(
1
)
30

k = 2(1)30

, factorizations of

k
!
±
1

k! \pm 1

are given.
Block five diagonal matrices and the fast numerical solution of the biharmonic equation
Authors: Bauer Louis , Reiss Edward L.
Published: June 2010

A factoring and block elimination method for the fast numerical solution of block five diagonal linear algebraic equations is described. Applications of the method are given for the numerical solution of several boundary-value problems involving the biharmonic operator. In particular, 22 eigenvalues and eigenfunctions of the clamped square plate are computed and sketched.
Numerical investigation of certain asymptotic results in the theory of partitions
Authors: Cheema M. S. , Conway W. E.
Published: June 2010

A numerical investigation of some of the asymptotic formulas in partitions is made. Comparisons with actual computed values show that in certain cases only the relative error tends to zero and the errors are significant. Only in the case of

p
k

(
n
)

{p_k}(n)

is it found that only a few terms of the asymptotic series are sufficient to obtain the exact value.
Existence and error estimates for solutions of a discrete analog of nonlinear eigenvalue problems
Authors: Simpson R. B.
Published: June 2010

Finite-difference methods using the five-point discrete Laplacian and suitable boundary modifications for approximating

(
1
)
−
Δ
u
=
λ
f
(
x
,
u
)

(1) - \Delta u = \lambda f(x,u)

in a plane domain

D
,
u
=
0

D,u = 0

on its boundary are considered. It is shown that if (1) has an isolated solution,

u
u

, then the discrete problem has a solution,

U
h

{U_h}

, for which

U
h

−
u
=
O
(

h
2

)

{U_h} - u = O({h^2})

. If the discrete problem has solutions,

U
h

{U_h}

, such that

|

U
h

|

≦
M

|{U_h}| \leqq M

as

h
h

tends to zero, then (1) has a solution,

u
u

, satisfying

|

u

|

≦
M

|u| \leqq M

. Let

λ
∗

{\lambda ^\ast }

be a critical value of

λ
\lambda

so that (1) has positive solutions for

λ
≦

λ
∗

\lambda \leqq \lambda ^\ast

but not for

λ
>

λ
∗

\lambda > {\lambda ^\ast }

, then the discrete problem has an analogous critical value

λ
h
∗

\lambda _h^\ast

and, under suitable conditions,

λ
h
∗

−

λ
∗

=
O
(

h

4

/

3
−
ϵ

)
,
ϵ
>
0

\lambda _h^\ast - {\lambda ^\ast } = O({h^{4/3 - \epsilon }}),\epsilon > 0

. Computed results for the case

f
(
x
,
u
)
=

e
u

f(x,u) = {e^u}

and

D
D

the unit square are given.
Modification methods for inverting matrices and solving systems of linear algebraic equations
Authors: Goldfarb D.
Published: June 2010

Modification methods for inverting matrices and solving systems of linear algebraic equations are developed from Broyden’s rank-one modification formula. Several algorithms are presented that take as few, or nearly as few, arithmetic operations as Gaussian elimination and are well suited for the handling of data. The effect of rounding errors is discussed briefly. Some of these algorithms are essentially equivalent to, or “compact” forms of, such known methods as Sherman and Morrison’s modification method, Hestenes’ biorthogonalization method, Gauss-Jordan elimination, Aitken’s below-the-line elimination method, Purcell’s vector method, and its equivalent, Pietrzykowski’s projection method, and the bordering method. These methods are thus shown to be directly related to each other. Iterative methods and methods for inverting symmetric matrices are also given, as are the results of some computational experiments.
On the zeros of the incomplete gamma function
Authors: Kölbig K. S.
Published: June 2010

Some asymptotic formulae given elsewhere for the zeros of the incomplete gamma function

γ
(
a
,
x
)

\gamma (a,x)

are corrected. A plot of a few of the zero trajectories of the function

γ
(
x
w
,
x
)

\gamma (xw,x)

is given, where

x
x

is a real parameter. Based on theoretical work by Mahler, it is seen that the zero trajectories of

γ
(
x
w
,
x
)

\gamma (xw,x)

lie in a finite region of the complex

w
w

-plane.
Boundary expansions for spline interpolation
Authors: Hoskins W. D.
Published: June 2010

An explicit method is given for deriving the formulae for derivatives of the spline of order

m
+
1

m + 1

at two boundaries

x
=
a
,
x
=
b

x = a,x = b

in terms of known function values and computed mth derivatives of the spline.
A note on the Catalan-Dickson conjecture
Authors: te Riele H. J. J.
Published: June 2010

In this note, the existence of an aliquot sequence with more than 5092 monotonically increasing even terms is proved; the first 125 terms are listed.
Computing the Brouwer degree in 𝑅²
Authors: Erdelsky P. J.
Published: June 2010

A very simple rigorous procedure is derived for computing the Brouwer degree in

R
2

{R^2}

, a generalization of the zero-counting integral

∮

f
′

(
z
)

d
z

/

f
(
z
)

\oint {f’(z)\;dz/f(z)}

, for functions which are Lipschitz continuous on a piecewise linear path of integration, using only computed or observed values of

f
(
z
)

f(z)

, a bound for the error in them, and a bound for the Lipschitz constant. It is used to locate zeros and to test the numerical significance of zeros found by other methods.
An extrapolated Gauss-Seidel iteration for Hessenberg matrices
Authors: Lardy L. J.
Published: June 2010

We show that for certain systems of linear equations with coefficient matrices of Hessenberg form it is possible to use Gaussian elimination to obtain an extrapolated version of the Gauss-Seidel iterative process where the iteration matrix has spectral radius zero. Computational aspects of the procedure are discussed.
Extensions of Forsythe’s method for random sampling from the normal distribution
Authors: Ahrens J. H. , Dieter U.
Published: June 2010

This article is an expansion of G. E. Forsythe’s paper "Von Neumann’s comparison method for random sampling from the normal and other distributions" [5]. It is shown that Forsythe’s method for the normal distribution can be adjusted so that the average number

N
¯

\bar {N}

of uniform deviates required drops to 2.53947 in spite of a shorter program. In a further series of algorithms,

N
¯

\bar {N}

is reduced to values close to 1 at the expense of larger tables. Extensive computational experience is reported which indicates that the new methods compare extremely well with known sampling algorithms for the normal distribution.
Asymptotic expansions for product integration
Authors: de Hoog Frank , Weiss Richard
Published: June 2010

A generalized Euler-Maclaurin sum formula is established for product integration based on piecewise Lagrangian interpolation. The integrands considered may have algebraic or logarithmic singularities. The results are used to obtain accurate convergence rates of numerical methods for Fredholm and Volterra integral equations with singular kernels.
A modified Butcher formula for integration of stiff systems of ordinary differential equations
Authors: Nosrati H.
Published: June 2010

An A-stable one-step integration formula, called the modified Butcher (MB), is presented and is shown to have an order of accuracy

p
=
3

p = 3

, when the differential system is linear, and

p
=
2

p = 2

otherwise. A method for evaluating the local truncation error of the formula is also suggested. Finally, the main features of this formula, vis-a-vis the trapezoidal, are compared.
Higher order accuracy finite difference algorithms for quasi-linear, conservation law hyperbolic systems
Authors: Abarbanel S. , Gottlieb D.
Published: June 2010

An explicit algorithm that yields finite difference schemes of aly desired order of accuracy for solving quasi-linear hyperbolic systems of partial differential equations in several space dimensions is presented. These schemes are shown to be stable under certain conditions. The stability conditions in the one-dimensional case are derived for any order of accuracy. Analytic stability proofs for two and

d

(
d
>
2
)

d\;(d > 2)

space dimensions are also obtained up to and including third order accuracy. A conjecture is submitted for the highest accuracy schemes in the multi-dimensional cases. Numerical examples show that the above schemes have the stipulated accuracy and stability.
Integer vectors with interprimed components
Authors: Shapiro Harold N.
Published: June 2010

Vectors are considered whose components are positive integers. Such a vector is called interprimed if the components all contain exactly the same distinct prime factors. A method is provided for estimating the number of such vectors, all of whose components are less than a given bound. These estimates resolve a conjecture of Erdös and Motzkin.
An algorithm for generating the sphere coordinates in a three-dimensional osculatory packing
Authors: Boyd David W.
Published: June 2010

This paper develops an efficient algorithm which generates the pentaspherical coordinates of the spheres in an osculatory packing of the three-dimensional unit sphere. The algorithm has a tree-like structure and is easily modified so that, given a prescribed bound, it counts the number of spheres in the packing whose curvatures are less than this bound. The algorithm has been used to produce heuristic estimates of the exponent M of the packing, and these indicate that M is approximately 2.42.
A finite element collocation method for quasilinear parabolic equations
Authors: Douglas Jim , Dupont Todd
Published: June 2010

Let the parabolic problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c left-parenthesis x comma t comma u right-parenthesis u Subscript t Baseline equals a left-parenthesis x comma t comma u right-parenthesis u Subscript x x Baseline plus b left-parenthesis x comma t comma u comma u Subscript x Baseline right-parenthesis comma 0 greater-than x greater-than 1 comma 0 greater-than t less-than-over-equals upper T comma u left-parenthesis x comma 0 right-parenthesis equals f left-parenthesis x right-parenthesis comma u left-parenthesis 0 comma t right-parenthesis equals g 0 left-parenthesis t right-parenthesis comma u left-parenthesis 1 comma t right-parenthesis equals g 1 left-parenthesis t right-parenthesis">

c
(
x
,
t
,
u
)

u
t

=
a
(
x
,
t
,
u
)

u

x
x

+
b
(
x
,
t
,
u
,

u
x

)
,
0
>
x
>
1
,
0
>
t
≦
T
,
u
(
x
,
0
)
=
f
(
x
)
,
u
(
0
,
t
)
=

g
0

(
t
)
,
u
(
1
,
t
)
=

g
1

(
t
)

c(x,t,u){u_t} = a(x,t,u){u_{xx}} + b(x,t,u,{u_x}),0 > x > 1,0 > t \leqq T,u(x,0) = f(x),u(0,t) = {g_0}(t),u(1,t) = {g_1}(t)

, be solved approximately by the continuous-time collocation process based on having the differential equation satisfied at Gaussian points

ξ

i
,
1

{\xi _{i,1}}

and

ξ

i
,
2

{\xi _{i,2}}

in subintervals

(

x

i
−
1

,

x
i

)

({x_{i - 1}},{x_i})

for a function

U
:
[
0
,
T
]
→

H

3

U:[0,T] \to {\mathcal {H}_3}

, the class of Hermite piecewise-cubic polynomial functions with knots

0
=

x
0

>

x
1

>
⋯
>

x
n

=
1

0 = {x_0} > {x_1} > \cdots > {x_n} = 1

. It is shown that

u
−
U
=
O
(

h
4

)

u - U = O({h^4})

uniformly in x and t, where

h
=
max
(

x
i

−

x

i
−
1

)

h = \max ({x_i} - {x_{i - 1}})

.
Note on error bounds for numerical integration
Authors: Hetherington J. H.
Published: June 2010

A length functional was defined in a previous publication which associates a length scale r to every function analytic on a given interval. The ratio of r to the mean panel size for numerical integration on that interval was defined as the sketchability. Stronger error bounds are given here for interpolative integration schemes in terms of the sketchability. Since these are not least upper bounds we also present lower limits for the least upper bounds obtained from actual numerical integrations of specific sketchability. We find the gap between the bounds and their lower limits to be small enough to be ignored for most applications.
The order of numerical methods for ordinary differential equations
Authors: Butcher J. C.
Published: June 2010

For a general class of methods, which includes linear multistep and Runge-Kutta methods as special cases, a concept of order relative to a given starting procedure is defined and an order of convergence theorem is proved. The definition is given an algebraic interpretation and illustrated by the derivation of a particular fourth-order method.
Derivatives of Whittaker functions 𝑊_{𝜅,1/2} and 𝑀_{𝜅,1/2} with respect to order 𝜅
Authors: Laurenzi Bernard J.
Published: June 2010

The Whittaker function derivatives

[
∂

W

κ
,
1

/

2

/

∂
κ

]

κ
=
n

{[\partial {W_{\kappa ,1/2}}/\partial \kappa ]_{\kappa = n}}

and

[
∂

M

κ
,
1

/

2

/

∂
κ

]

κ
=
n

{[\partial {M_{\kappa ,1/2}}/\partial \kappa ]_{\kappa = n}}

which arise in calculations involving the hydrogen atom’s generalized Green’s functions are computed.
Error analysis for direct linear integral equation methods
Authors: Phillips James L.
Published: June 2010

An error analysis of projection methods for solving linear integral equations of the second kind is presented. The relationships between several direct methods for solving integral equations are examined. It is shown that the error analysis given is applicable to other methods, including a modified Nyström method and certain degenerate kernel methods.
On the vanishing of the Iwasawa invariant 𝜇_{𝑝} for 𝑝<8000
Authors: Johnson Wells
Published: June 2010

The irregular primes less than 8000 are computed, and it is shown that the Iwasawa invariant

μ
p

=
0

{\mu _p} = 0

for all primes

p
>
8000

p > 8000

.
A catalogue of combinatorial geometries
Authors: Blackburn John E. , Crapo Henry H. , Higgs Denis A.
Published: June 2010

We include in the microfiche section of this issue a catalogue of all the different geometric configurations which may be constructed from n points,

n
≦
8

n \leqq 8

, by specifying their points, lines, planes, . . . , colines, copoints, their flats of each rank. It suffices to list the copoints of each combinatorial geometry G, because the colines of G are the copoints of a geometry earlier in the list, which may be located by deleting one component of the designator of G.
Elliptical membranes with smallest second eigenvalue
Authors: Troesch B. Andreas
Published: June 2010

The elliptic membranes with fixed boundary are determined, for which the second eigenfrequency is a minimum if the area or if the circumference is prescribed. The results are compared with those of some other shapes. A remark is made about the overtones of elliptic membranes.
Limiting precision in differential equation solvers
Authors: Shampine L. F.
Published: June 2010

Machine dependent limits on the step size and local error tolerance are discussed. By taking them into account codes can be made more robust.
Numercial solution of Lundquist equations of magnetohydrodynamics
Authors: Johnston R. L. , Pal S. K.
Published: June 2010

A method of bicharacteristics [3] is used to derive a numerical method for solving multidimensional nonlinear Lundquist equations of magnetohydrodynamics. Actual numerical computations are carried out to solve two specific problems of magnetohydrodynamics—the magnetohydrodynamic initial-pressure problem and a problem of cylindrical waves in a transverse magnetic field due to a thin current-carrying wire perpendicular to the plane of the fluid.
Four large amicable pairs
Authors: te Riele H. J. J.
Published: June 2010

This note gives a report of systematic computer tests of Euler’s rule and several Thabit-ibn-Kurrah-rules, in search of large amicable pairs. The tests have yielded four amicable pairs, which are much larger than the largest amicable pair thus far known.
A new function associated with the prime factors of (ⁿ_{𝑘})
Authors: Ecklund E. F. , Erdös P. , Selfridge J. L.
Published: June 2010

Let

g
(
k
)

g(k)

denote the least integer

>
k
+
1

> k + 1

so that all the prime factors of

(

g
(
k
)

k

)

\left ( {\begin {array}{*{20}{c}} {g(k)} \\ k \\ \end {array} } \right )

are greater than k. The irregular behavior of

g
(
k
)

g(k)

is studied, obtaining the following bounds:

k

1
+
c

>
g
(
k
)
>
exp

(
k
(
1
+
o
(
1
)
)
)
.

{k^{1 + c}} > g(k) > \exp \,(k(1 + o(1))).

Numerical values obtained for

g
(
k
)

g(k)

with

k
≦
52

k \leqq 52

are listed.
On the conditional equivalence of two starting methods for the second algorithm of Remez
Authors: Huddleston R. E.
Published: June 2010

In computing best min-max rational approximations by the second algorithm of Remez (which is an iterative procedure), one must provide a starting approximation. A method proposed by Ralston and one by Werner are shown to be equivalent under reasonable conditions.
On semicardinal quadrature formulae
Authors: Schoenberg I. J. , Silliman S. D.
Published: June 2010

The present paper concerns the semicardinal quadrature formulae introduced in Part III of the reference [3]. These were the limiting forms of Sard’s best quadrature formulae as the number of nodes increases indefinitely. Here we give a new derivation and characterization of these formulae. This derivation uses appropriate generating functions and also allows us to compute the coefficients very accurately.
The application of implicit Runge-Kutta and collection methods to boundary-value problems
Authors: Weiss Richard
Published: June 2010

The solution of a nonlinear system of first order differential equations with nonlinear boundary conditions by implicit Runge-Kutta methods based on interpolatory quadrature formulae is examined. An equivalence between implicit Runge-Kutta and collocation schemes is established. It is shown that the difference equations obtained have a unique solution in a neighbourhood of an isolated solution of the continuous problem, that this solution can be computed by Newton iteration and that it converges to the isolated solution. The order of convergence is equal to the degree of precision of the related quadrature formula plus one. The efficient implementation of the methods is discussed and numerical examples are given.
Modifying pivot elements in Gaussian elimination
Authors: Stewart G. W.
Published: June 2010

The rounding-error analysis of Gaussian elimination shows that the method is stable only when the elements of the matrix do not grow excessively in the course of the reduction. Usually such growth is prevented by interchanging rows and columns of the matrix so that the pivot element is acceptably large. In this paper the alternative of simply altering the pivot element is examined. The alteration, which amounts to a rank one modification of the matrix, is undone at a later stage by means of the well-known formula for the inverse of a modified matrix. The technique should prove useful in applications in which the pivoting strategy has been fixed, say to preserve sparseness in the reduction.
A characterization of superlinear convergence and its application to quasi-Newton methods
Authors: Dennis J. E. , Moré Jorge J.
Published: June 2010

Let F be a mapping from real n-dimensional Euclidean space into itself. Most practical algorithms for finding a zero of F are of the form
\[

x

k
+
1

=

x
k

−

B
k

−
1

F

x
k

,

{x_{k + 1}} = {x_k} - B_k^{ - 1}F{x_k},

\]
where

{

B
k

}

\{ {B_k}\}

is a sequence of nonsingular matrices. The main result of this paper is a characterization theorem for the superlinear convergence to a zero of F of sequences of the above form. This result is then used to give a unified treatment of the results on the superlinear convergence of the Davidon-Fletcher-Powell method obtained by Powell for the case in which exact line searches are used, and by Broyden, Dennis, and Moré for the case without line searches. As a by-product, several results on the asymptotic behavior of the sequence

{

B
k

}

\{ {B_k}\}

are obtained. An interesting aspect of these results is that superlinear convergence is obtained without any consistency conditions; i.e., without requiring that the sequence

{

B
k

}

\{ {B_k}\}

converge to the Jacobian matrix of F at the zero. In fact, a modification of an example due to Powell shows that most of the known quasi-Newton methods are not, in general, consistent. Finally, it is pointed out that the above-mentioned characterization theorem applies to other single and double rank quasi-Newton methods, and that the results of this paper can be used to obtain their superlinear convergence.
Approximation by aliasing with application to “Certaine” stiff differential equations
Authors: Snider Arthur David , Fleming Gary Charles
Published: June 2010

The usual method of finding an accurate trigonometric interpolation for a function with dominant high frequencies requires a large number of calculations. This paper shows how aliasing can be used to achieve a great reduction in the computations in cases when the high frequencies are known beforehand. The technique is applied to stiff differential equations, extending the applicability of the method of Certaine to systems with oscillatory forcing functions.
An inequality about factors of polynomials
Authors: Mignotte M.
Published: June 2010

A sharp inequality is proved about the product of some roots of a polynomial. It is used to bound the height of the factors of a polynomial. Applications are given to the problem of factorization and numerical examples show that these bounds strongly improve the previous ones.
High-order finite-difference methods for Poisson’s equation
Authors: van Linde H. J.
Published: June 2010

Finite-difference approximations to the three boundary value problems for Poisson’s equation are given with discretization errors of

O
(

h
3

)

O({h^3})

for the mixed boundary value problem,

O
(

h
3

|

ln
⁡
h

|

)

O({h^3}|\ln h|)

for the Neumann problem and

O
(

h
4

)

O({h^4})

for the Dirichlet problem, respectively. These error bounds are an improvement upon similar results obtained by Bramble and Hubbard; moreover, all resulting coefficient matrices are of positive type.
Negative integral powers of a bidiagonal matrix
Authors: Chatterjee Gurudas
Published: June 2010

The elements of the inverse of a bidiagonal matrix have been expressed in a convenient form. The higher negative integral powers of the bidiagonal matrix exhibit an interesting property: the (ij)th element of the

(
−
m
)

( - m)

th power is equal to the product of the corresponding element of the inverse by a Wronski polynomial, viz., the complete symmetric function of degree

(
m
−
1
)

(m - 1)

of the diagonal elements,

d
i

,

d

i
+
1

,
…
,

d
j

{d_i},{d_{i + 1}}, \ldots ,{d_j}

, of the inverse matrix.
Generating the nine-point graphs
Authors: Baker H. H. , Dewdney A. K. , Szilard A. L.
Published: June 2010

A program has been written which recently generated all the (unlabelled) nine-point graphs. Written in MACRO-10 assembly language and run on a 165K PDP-10, it generates the complete set of 274,668 graphs in less than six hours. The algorithm on which this program is based is discussed with an emphasis on coding of graphs and various programming techniques designed to save space and time during execution. The methods developed may have applications in other combinatorial generating problems.
Error analysis for polynomial evaluation
Authors: Newbery A. C. R.
Published: June 2010

A floating-point error analysis is given for the evaluation of a real polynomial at a real argument by Horner’s scheme. A computable error bound is derived. It is observed that when a polynomial has coefficients of constant sign or of strictly alternating sign, one cannot expect better accuracy by reformulating the problem in terms of Chebyshev polynomials.
A least squares procedure for the wave equation
Authors: Carasso Alfred
Published: June 2010

We develop and analyze a least squares procedure for approximating the homogeneous Dirichlet problem for the wave equation in a bounded domain

Ω
\Omega

in

R
N

{R^N}

. This procedure is based on the pure implicit scheme for time differencing. Surprisingly, it is the normal derivative of u rather than u itself which must be included in the boundary functional. This normal derivative is an unknown quantity. We show that it may be set equal to zero while retaining the

O
(
k
)

O(k)

accuracy of the pure implicit scheme. The penalty is that one must use smoother trial functions to obtain this accuracy.
An application of approximation theory to an error estimate in linear algebra
Authors: Belford Geneva G. , Kaufman E. H.
Published: June 2010

Recently developed theorems characterizing best uniform approximations from a family with a fixed point are shown to be useful in the estimation of errors in computational techniques for solving linear algebraic equations. Specifically, a gap in the proof of a published theorem is filled in.
𝑆𝑂𝑅-methods for the eigenvalue problem with large sparse matrices
Authors: Ruhe Axel
Published: June 2010

The eigenvalue problem

A
x
=
λ
B
x

Ax = \lambda Bx

, where A and B are large and sparse symmetric matrices, is considered. An iterative algorithm for computing the smallest eigenvalue and its corresponding eigenvector, based on the successive overrelaxation splitting of the matrices, is developed, and its global convergence is proved. An expression for the optimal overrelaxation factor is found in the case where A and B are two-cyclic (property A). Further, it is shown that this SOR algorithm is the first order approximation to the coordinate relaxation algorithm, which implies that the same overrelaxation can be applied to this latter algorithm. Several numerical tests are reported. It is found that the SOR method is more effective than coordinate relaxation. If the separation of the eigenvalues is not too bad, the SOR algorithm has a fast rate of convergence, while, for problems with more severe clustering, the c-g or Lanczos algorithms should be preferred.
The solution of systems of equations using the 𝜖-algorithm, and an application to boundary-value problems
Authors: Brezinski Claude , Rieu Alain C.
Published: June 2010

In this paper, the authors describe the properties of an algorithm to solve systems of nonlinear equations. The algorithm does not use any derivatives. The convergence of this algorithm is quadratic under quite mild conditions. This method can also be used to solve systems of linear equations with infinitely many solutions. The second part of the paper is devoted to the application of this algorithm to the solution of multipoint boundary-value problems for differential equations. A theorem of convergence is proved and various numerical examples are given.
On multiple prime divisors of cyclotomic polynomials
Authors: McDaniel Wayne L.
Published: June 2010

Let q be a prime

>
150

> 150

and

F
n

{F_n}

be the cyclotomic polynomial of order n. All triples (p, n, q) with p an odd prime

>

10
6

> {10^6}

when

q
>
100

q > 100

and

p
>

10
4

p > {10^4}

when

100
>
q
>
150

100 > q > 150

are given for which

F
n

(
q
)

{F_n}(q)

is divisible by

p
t

(
t
>
1
)

{p^t}(t > 1)

.
On Fourier-Toeplitz methods for separable elliptic problems
Authors: Fischer D. , Golub G. , Hald O. , Leiva C. , Widlund O.
Published: June 2010

Some very fast numerical methods have been developed in recent years for the solution of elliptic differential equations which allow for separation of variables. In this paper, a Fourier-Toeplitz method is developed as an alternative to the well-known methods of Hockney and Buneman. It is based on the fast Fourier transform and Toeplitz factorizations. The use of Toeplitz factorizations combined with the Sherman-Morrison formula is also systematically explored for linear systems of algebraic equations with band matrices of Toeplitz, or almost Toeplitz form. Finally, results of numerical experiments are described.
Nonexistence of Chebyshev-type quadratures on infinite intervals
Authors: Gautschi Walter
Published: June 2010

Quadrature rules on semi-infinite and infinite intervals are considered involving weight functions of the Laguerre and Hermite type. It is shown that such quadrature rules cannot have equal coefficients and real nodes unless the algebraic degree of accuracy is severely limited.
On complex quadratic fields wth class-number two
Authors: Stark H. M.
Published: June 2010

Let

d
>
0

d > 0

be the discriminant of a complex quadratic field of class-number

h
(
d
)

h(d)

. In a previous paper the author has effectively shown how to find all d with

h
(
d
)
=
2

h(d) = 2

. In this paper, it is proved that, if

h
(
d
)
=
2

h(d) = 2

, then

|

d

|

⩽
427

|d|\; \leqslant 427

.
Finite element multistep discretizations of parabolic boundary value problems
Authors: Zlámal Miloš
Published: June 2010

The initial-boundary value problem for a linear parabolic equation in an infinite cylinder under the Dirichlet boundary condition is solved by applying the finite element discretization in the space dimension and

A
0

{A_0}

-stable multistep discretizations in time. No restriction on the ratio of the time and space increments is imposed. The methods are analyzed and bounds for the discretization error in the

L
2

{L_2}

-norm are given.
A method of factoring and the factorization of 𝐹₇
Authors: Morrison Michael A. , Brillhart John
Published: June 2010

The continued fraction method for factoring integers, which was introduced by D. H. Lehmer and R. E. Powers, is discussed along with its computer implementation. The power of the method is demonstrated by the factorization of the seventh Fermat number

F
7

{F_7}

and other large numbers of interest.
Some new multistep methods for solving ordinary differential equations
Authors: Gupta G. K. , Wallace C. S.
Published: June 2010

Three sets of linear multistep formulae for solving stiff and nonstiff ordinary differential equations are presented. Two of the sets are based on Adams-Moulton and stiff formulae used by Gear [1969]. A third set of formulae based on least-squares approximation is shown to be stiffly-stable up to order 8 and is suitable for solving stiff differential equations.
On small zeros of Dirichlet 𝐿-functions
Authors: Weinberger Peter J.
Published: June 2010

A method is given for calculating the value of Dirichlet L-functions near the real axis in the critical strip. As an application, some zeros for zeta functions of complex quadratic fields are calculated.
The zeros of regular Coulomb wave functions and of their derivatives
Authors: Ikebe Yasuhiko
Published: June 2010

A simple and efficient numerical method for computing the zeros of regular Coulomb wave functions and of their derivatives is presented. The method is based on the characterization of the zeros of the functions and of their derivatives in terms of eigenvalues of certain compact matrix operators. A similar approach has been reported for the computation of the zeros of Bessel functions and of their derivatives [9], [14].
What drives an aliquot sequence?
Authors: Guy Richard K. , Selfridge J. L.
Published: June 2010

The concept of the "driver" of an aliquot sequence is discussed. It is shown that no driver can be expected to persist indefinitely. A definition of driver is given which leads to just 5 drivers apart from the even perfect numbers.
A modified Galerkin procedure with Hermite cubics for hyperbolic problems
Authors: Wahlbin Lars
Published: June 2010

The Galerkin method, modified to include a term of artificial viscosity type, is applied to model problems for linear and quasilinear hyperbolic systems. Asymptotic error estimates are derived.
Computational investigations of least-squares type methods for the approximate solution of boundary value problems
Authors: Serbin Steven M.
Published: June 2010

Several Galerkin schemes for approximate solution of linear elliptic boundary value problems are studied for such computational aspects as obtainable accuracy, sensitivity to parameters and conditioning of linear systems. Methods studied involve computing subspaces (e.g., splines) whose elements need not satisfy boundary conditions. A Poisson problem study on the square produces computed error reflective of theoretical

L
2

{L_2}

estimates and

L
∞

{L_\infty }

behavior optimal for smooth data but loss according to Sobolev’s lemma for nonsmooth data. Insensitivity to parameters is evidenced. Analogous one-dimensional methods enhance the conditioning study. Studies are included for parallelogram and L-shaped domains.
The distribution of ideal class numbers of real quadratic fields
Authors: Hendy M. D.
Published: June 2010

A table of class numbers of real quadratic number fields

Q
(
√
d
)

Q(\surd d)

with square-free determinant d,

1000
>
d
>
100000

1000 > d > 100000

is examined and several analyses of the distribution of the class numbers, and the number of classes per genus are made. From these, two conjectures on the possible distribution of the class numbers as

d
→
∞

d \to \infty

are made, which are consistent with Gauss’s related conjecture.
On the evaluation of some integrals occurring in scattering problems
Authors: Kadlec J.
Published: June 2010

Some definite integrals which occur in transport problems through a scattering medium are studied. They are expressed in terms of such functions as the exponential integral of the first and second order, the dilogarithm, and a newly introduced and tabulated function.
The uniqueness of the Markoff numbers
Authors: Rosenberger Gerhard
Published: June 2010

A Markoff triple is a set of three positive integers satisfying the diophantine equation

x
2

+

y
2

+

z
2

=
3
x
y
z

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

. The maximum of the three numbers is called a Markoff number. We show: If there are Markoff triples

(

x
1

,

y
1

,
z
)

({x_1},{y_1},z)

and

(

x
2

,

y
2

,
z
)

({x_2},{y_2},z)

with the same Markoff number z, then

x
1

=

x
2

{x_1} = {x_2}

or

x
1

=

y
2

{x_1} = {y_2}

.
Decision problems for HNN groups and vector addition systems
Authors: Anshel Michael
Published: June 2010

Our purpose is to show the equivalence of the conjugacy problem for certain HNN extensions of the infinite cyclic groups and the reachability problem for the class of self-dual vector addition systems. In addition, we extend an endomorphism theorem of the author’s to a homomorphism theorem and indicate a problem related to the isomorphism problem for a class of HNN groups.
Generation of conjugate directions for unconstrained minimization without derivatives
Authors: Nazareth Larry
Published: June 2010

We analyze a technique for unconstrained minimization without derivatives. This stems from two theorems proved by M. J. D. Powell. A particular version, which we consider in detail, is related to the Jacobi process for finding the eigensystem of a symmetric matrix, and the two processes, although different, help to illuminate one another. We study convergence of the search directions to mutual conjugacy, cases when cycling occurs and identify a broad class of ’cyclic patterns’ for which convergence to mutual conjugacy is proven.
Error analysis for a stiff system procedure
Authors: Snider Arthur David
Published: June 2010

The analysis of the local truncation error in a numerical scheme for integrating stiff differential equations, presented in a recent paper by Guderley and Hsu, holds only in special circumstances, but a more general analysis preserves the main conclusions. Several modifications of the scheme are also considered.
Higher order approximations to the boundary conditions for the finite element method
Authors: Blair J. J.
Published: June 2010

We consider here the approximation of essential boundary conditions for the finite element solutions of second order elliptic equations in two dimensions. Nonhomogeneous boundary conditions on curved boundaries are treated. The approach is to use trial functions which interpolate (in a generalized sense) functions satisfying the boundary conditions. The work is directed to showing in what manner this interpolation should be done to achieve the maximum accuracy and computational simplicity. These methods can be used to construct approximations of arbitrary high order of accuracy. Several examples are given.
Class groups of quadratic fields
Authors: Buell Duncan A.
Published: June 2010

The author has computed the class groups of all complex quadratic number fields

Q
(

−
D

)

Q(\sqrt { - D} )

of discriminant

−
D

- D

for

0
>
D
>
4000000

0 > D > 4000000

. In so doing, it was found that the first occurrences of rank three in the 3-Sylow subgroup are

D
=
3321607
=

prime

D = 3321607 = {\text {prime}}

, class group

C
(
3
)
×
C
(
3
)
×
C
(
9.7
)

(
C
(
n
)

C(3) \times C(3) \times C(9.7)\quad (C(n)

a cyclic group of order n), and

D
=
3640387
=
421.8647

D = 3640387 = 421.8647

, class group

C
(
3
)
×
C
(
3
)
×
C
(
9.2
)

C(3) \times C(3) \times C(9.2)

. The author has also found polynomials representing discriminants of 3-rank

⩾
2

\geqslant 2

, and has found 3-rank 3 for

D
=
6562327
=
367.17881
,
8124503
,
10676983
,
193816927

D = 6562327 = 367.17881,8124503,10676983,193816927

, all prime,

390240895
=
5.11.7095289

390240895 = 5.11.7095289

, and

503450951
=

prime

503450951 = {\text {prime}}

. The first five of these were discovered by Diaz y Diaz, using a different method. The author believes, however, that his computation independently establishes the fact that 3321607 and 3640387 are the smallest D with 3-rank 3. The smallest examples of noncyclic 13-, 17-, and 19-Sylow subgroups have been found, and of groups noncyclic in two odd p-Sylow subgroups.

D
=
119191
=

prime

D = 119191 = {\text {prime}}

, class group

C
(
15
)
×
C
(
15
)

C(15) \times C(15)

, had been found by A. O. L. Atkin; the next such D is

2075343
=
3.17.40693

2075343 = 3.17.40693

, class group

C
(
30
)
×
C
(
30
)

C(30) \times C(30)

. Finally,

D
=
3561799
=

prime

D = 3561799 = {\text {prime}}

has class group

C
(
21
)
×
C
(
63
)

C(21) \times C(63)

, the smallest D noncyclic for 3 and 7 together.