bluuebunny/crossref_metadata_2025 huggingface dataset:
A comparison of algorithms for rational 𝑙_{∞} approximation
Authors: Lee C. M. , Roberts F. D. K.
Published: June 2010

Results are reported of a numerical study to compare eight algorithms for obtaining rational

l
∞

{l_\infty }

approximations. The algorithms investigated are Loeb’s algorithm, the linear inequality algorithm, the Osborne-Watson algorithm, the differential correction algorithms I, II and III, the Remes algorithm and Maehly’s algorithm. The results of the study indicate that the Remes algorithm and the differential correction algorithm III are the most satisfactory methods to use in practice.
Bounds on iterated coerror functions and their ratios
Authors: Amos D. E.
Published: June 2010

Upper and lower bounds on

y
n

=

i
n

erfc

(
x
)

{y_n} = {i^n}\;{\operatorname {erfc}}(x)

and

r
n

=

y
n

/

y

n
−
1

,
n
≧
1
,
−
∞
>
x
>
∞

{r_n} = {y_n}/{y_{n - 1}}, n \geqq 1, - \infty > x > \infty

, are established in terms of elementary functions. Numerical procedures for refining these bounds are presented so that

r
n

{r_n}

and

y
k

,
k
=
0
,
1
,
…
,
n

{y_k},k = 0,1, \ldots ,n

, can be computed to a specified accuracy. Some relations establishing bounds on

r

n

′

r’_{n}

and

r

n

r_{n}

are also derived.
An error analysis of a method for solving matrix equations
Authors: Paige C. C.
Published: June 2010

Let

B
=
[
L

0
]
Q

B = [L\;0]Q

be a decomposition of the m by n matrix B of rank m such that L is lower triangular and Q is orthonormal. It is possible to solve

B
x
=
b

Bx = b

, using L but not Q, in the following manner: solve

L
y
=
b

Ly = b

, solve

L
T

w
=
y

{L^T}w = y

, and form

x
=

B
T

w

x = {B^T}w

. It is shown that the numerical stability of this method is comparable to that of the method which uses Q. This is important for some methods used in mathematical programming where B is very large and sparse and Q is discarded to save storage.
Multi-dimensional extension of the Chebyshev polynomials
Authors: Hays Richard O.
Published: June 2010

Two families of polynomials are introduced which satisfy multi-dimensional (or multi-indiced) recursion relationships. These polynomials are developed from the Chebyshev polynomials. Also two additional polynomials are presented which satisfy a special two-dimensional recursion relationship.
Computation of Hermite polynomials
Authors: Eisenhart Laurance C. , Trapp George E.
Published: June 2010

Projection methods are commonly used to approximate solutions of ordinary and partial differential equations. A basis of the subspace under consideration is needed to apply the projection method. This paper discusses methods of obtaining a basis for piecewise polynomial Hermite subspaces. A simple recursive procedure is derived for generating piecewise Hermite polynomials. These polynomials are then used to obtain approximate solutions of differential equations.
A note on Dirichlet characters
Authors: Hudson Richard H.
Published: June 2010

Denoting by

r
(
k
,
m
,
p
)

r(k,m,p)

the first occurrence of m consecutive kth power residues of a prime

p
≡
1

(
mod

k
)

p \equiv 1 \pmod k

, we show that

r
(
k
,
m
,
p
)
>
c
log
⁡
p

r(k,m,p) > c\log p

for infinitely many p (c is an absolute constant) provided that k is even and

m
≧
3

m \geqq 3

.
The calculation of best linear one-sided 𝐿_{𝑝} approximations
Authors: Watson G. A.
Published: June 2010

The calculation of linear one-sided approximations is considered, using the discrete

L
p

{L_p}

norm. For

p
=
1

p = 1

and

p
=
∞

p = \infty

, this gives rise to a linear programming problem, and for

1
>
p
>
∞

1 > p > \infty

, to a convex programming problem. Numerical results are presented, including some applications to the approximate numerical solution of ordinary differential equations, with error bounds.
On the numerical computation of parabolic problems for preceding times
Authors: Buzbee B. L. , Carasso Alfred
Published: June 2010

We develop and analyze a general procedure for computing selfadjoint parabolic problems backwards in time, given an a priori bound on the solutions. The method is applicable to mixed problems with variable coefficients which may depend on time. We obtain error bounds which are naturally related to certain convexity inequalities in parabolic equations. In the time-dependent case, our difference scheme discerns three classes of problems. In the most severe case, we recover a convexity result of Agmon and Nirenberg. We illustrate the method with a numerical experiment.
Mesh refinements for parabolic equations of second order
Authors: Venit Stewart
Published: June 2010

Given certain implicit difference approximations to

u
t

=
a
(
x
)

u

x
x

+
b
(
x
)

u
x

+
c
(
x
)
u

{u_t} = a(x){u_{xx}} + b(x){u_x} + c(x)u

in the region

−
∞
>
x
>
∞
,
t
≧
0

- \infty > x > \infty ,t \geqq 0

, with a finer x-mesh width in the left half-plane than in the right, we consider the stability in the maximum norm of these schemes using several different interface conditions (at

x
=
0

x = 0

). In order to obtain our results, we first prove a stability theorem for certain simple second-order parabolic initial boundary systems on an evenly spaced mesh in the right half-plane alone. By a standard procedure, the first problem is converted into the second one, and solved in this manner.
Inequalities for modified Bessel functions
Authors: Nȧsell Ingemar
Published: June 2010

A sequence of sharp versions of the inequality

I

v
+
1

(
x
)
>

I
v

(
x
)
,
v
>
−

1
2

,
x
>
0

{I_{v + 1}}(x) > {I_v}(x),v > - \frac {1}{2},x > 0

, is established.
Evaluation of a constant associated with a parking problem
Authors: Lal M. , Gillard P.
Published: June 2010

A constant associated with a random space filling problem is computed to 19D. This is achieved by numerically solving an integral-difference equation.
Quadrature formulas for semi-infinite integrals
Authors: Kumar Ravindra , Jain M. K.
Published: June 2010

Polynomials orthogonal on

[
0
,
∞
]

[0,\infty ]

with regard to the weight function

w
(
x
)
=

x
α

(
1
+
x

)

−
β

w(x) = {x^\alpha }{(1 + x)^{ - \beta }}

are obtained, recurrence relations are found and the differential equation, which is satisfied by them, is given. Formulas for weights and abscissas in the corresponding quadrature formula are given.
On the modification of 𝐿𝐷𝐿^{𝑇} factorizations
Authors: Fletcher R. , Powell M. J. D.
Published: June 2010

A number of methods are compared for modifying an

L
D

L
T

LD{L^T}

factorization of a positive definite matrix A when a matrix of rank one is added to A. Both the efficiency and error propagation characteristics are discussed, and it is shown that a suitable method must be chosen with care. An error analysis of some of the algorithms is given which has novel features. A worked example which also illustrates error growth is presented. Extensions to the methods are desribed which enable them to be used advantageously in representing and updating positive semidefinite matrices.
A compact prime listing
Authors: Weintraub Sol
Published: June 2010

A very compact listing of the primes to 100000 is given and applications to computer storage of primes is indicated.
The character table of an eight-dimensional orthogonal group
Authors: Hunt David C.
Published: June 2010

This paper describes the calculation of the character table of the 8-dimensional orthogonal group of maximal index over the field with 3 elements. The group is of interest as it is a subgroup of relatively small index in the sporadic simple group

M
(
23
)

M(23)

defined by B. Fischer [2]. The group also has an outer automorphism group of order 24, isomorphic to the symmetric group on 4 symbols.
Semidiscrete least-squares methods for second order parabolic problems with nonhomogenous data
Authors: King J. Thomas
Published: June 2010

Recently, Bramble and Thomée proposed semidiscrete least-squares methods for the heat equation. In this paper we extend these methods to variable coefficient parabolic operators with nonhomogeneous equations and boundary conditions.
Semianalytic numerical studies of turning points arising in stiff boundary value problems
Authors: Miranker W. L. , Morreeuw J. P.
Published: June 2010

A numerical algorithm for solving stiff boundary value problems with turning points is presented. The stiff systems are characterized as singularly perturbed differential equations. The numerical method is derived by appropriately discretizing the boundary layer and connection theory for such systems. Numerical results demonstrate the effectiveness of the method. In many cases the calculation proceeds with mesh increments which are orders of magnitude larger than those used by other known methods.
On the distribution of pseudo-random numbers generated by the linear congruential method. II
Authors: Niederreiter Harald
Published: June 2010

The discrepancy of a sequence of pseudo-random numbers generated by the linear congruential method is estimated for parts of the period which are somewhat larger than the square root of the modulus. Applications to numerical integration are mentioned.
On factorisation, with a suggested new approach
Authors: Miller J. C. P.
Published: June 2010

This paper gives a brief survey of methods based mainly on Fermat’s Theorem, for testing and establishing primality of large integers. It gives an extension of the Fermat-Lucas-Lehmer Theorems which allows us to establish primality, or to factorise composites, in cases where the Carmichael

λ
\lambda

-exponent is known (or a multiple or submultiple of it, by a moderate factor). The main part of the paper is concerned with describing a method for determining the

λ
\lambda

-exponent in cases where the Fermat test is not satisfied. This method is a variation of A. E. Western’s method for finding indices and primitive roots, based on congruences

N
=
a
+
b

N = a + b

, where N is the number whose exponent is required, and both a and b are

A
k

{A_k}

-numbers, that is, having no factor larger than

p
k

{p_k}

, the kth prime. The most onerous problem lies in the finding of a sufficient number of congruences (at least k) and in the choice of a suitable value of k. The determination of the approximate number of

A
k

{A_k}

-splittings available is considered, to allow an estimate of the amount of labour (human or electronic) needed to be made. The final suggestion, rather inconclusive, is that the method has possibilities worth exploring further and may be as economical, after development, as existing methods, and possibly more so when N is large.
A generating function for triangular partitions
Authors: Carlitz L. , Scoville Richard
Published: June 2010

LetTk(n){T_k}(n)denote the number of solutions in nonnegative integersai{a_i}, of the equation\[n=∑i=1k∑j=1k−i+1aijn = \sum \limits _{i = 1}^k {\sum \limits _{j = 1}^{k - i + 1} {{a_{ij}}} }\]where theaij{a_{ij}}satisfy the inequalitiesaij⩾ai+1,j,aij⩾ai+1,j−1{a_{ij}} \geqslant {a_{i + 1,j}},{a_{ij}} \geqslant {a_{i + 1,j - 1}}. We show that\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts upper T Subscript k Baseline left-parenthesis n right-parenthesis x Superscript n Baseline equals left-parenthesis 1 minus x right-parenthesis Superscript negative k Baseline left-parenthesis 1 minus x cubed right-parenthesis Superscript negative k plus 1 Baseline left-parenthesis 1 minus x Superscript 5 Baseline right-parenthesis Superscript negative k plus 2 Baseline midline-horizontal-ellipsis left-parenthesis 1 minus x Superscript 2 k minus 1 Baseline right-parenthesis Superscript negative 1 Baseline period">∑n=1∞Tk(n)xn=(1−x)−k(1−x3)−k+1(1−x5)−k+2⋯(1−x2k−1)−1.\sum \limits _{n = 1}^\infty {{T_k}(n){x^n} = {{(1 - x)}^{ - k}}{{(1 - {x^3})}^{ - k + 1}}{{(1 - {x^5})}^{ - k + 2}} \cdots {{(1 - {x^{2k - 1}})}^{ - 1}}.}\]
Cubature formulas of degree nine for symmetric planar regions
Authors: Piessens Robert , Haegemans Ann
Published: June 2010

A method of constructing 19-point cubature formulas with degree of exactness 9 is given for two-dimensional regions and weight functions which are symmetric in each variable. For some regions, e.g., the square and the circle, these formulas can be reduced to 18-point formulas.
A numerical comparison of integral equations of the first and second kind for conformal mapping
Authors: Hayes John K. , Kahaner David K. , Kellner Richard G.
Published: June 2010

Two methods for computing numerical conformal mappings are compared. The first, due to Symm, uses a Fredholm integral equation of the first kind while the other, due to Lichtenstein, uses a Fredholm integral equation of the second kind. The two methods are tested on ellipses with different ratios of major to minor axes. The method based on the integral equation of the second kind is superior if the ratio is less than or equal to 2.5. The opposite is true if the ratio is greater than or equal to 10. Similar results are obtained for other regions.
Optimal 𝐿_{∞} error estimates for Galerkin approximations to solutions of two-point boundary value problems
Authors: Douglas Jim , Dupont Todd , Wahlbin Lars
Published: June 2010

A priori error estimates in the maximum norm are derived for Galerkin approximations to solutions of two-point boundary value problems. The class of Galerkin spaces considered includes almost all (quasiuniform) piecewise-polynomial spaces that are used in practice. The estimates are optimal in the sense that no better rate of approximation is possible in general in the spaces employed.
Numerical solution for eigenvalues and eigenfunctions of a Hermitian kernel and an error estimate
Authors: Rakotch E.
Published: June 2010

New error estimates for eigenvalues of symmetric integral equations are obtained. These estimates are applicable to a more general class of integration methods and, in many cases, are better than those of Wielandt. For every eigenvalue, a numerical solution for the corresponding eigenfunction is also obtained. Whenever the exact eigenvalue happens to be simple, an error estimate for the corresponding eigenfunction is also derived.
A lower bound on the angles of triangles constructed by bisecting the longest side
Authors: Rosenberg Ivo G. , Stenger Frank
Published: June 2010

Let

Δ

A
1

A
2

A
3

\Delta {A^1}{A^2}{A^3}

be a triangle with vertices at

A
1

,

A
2

{A^1},{A^2}

and

A
3

{A^3}

. The process of "bisecting

Δ

A
1

A
2

A
3

\Delta {A^1}{A^2}{A^3}

" is defined as follows. We first locate the longest edge,

A
i

A

i
+
1

{A^i}{A^{i + 1}}

of

Δ

A
1

A
2

A
3

\Delta {A^1}{A^2}{A^3}

where

A

i
+
3

=

A
i

{A^{i + 3}} = {A^i}

, set

D
=
(

A
i

+

A

i
+
1

)

/

2

D = ({A^i} + {A^{i + 1}})/2

, and then define two new triangles,

Δ

A
i

D

A

i
+
2

\Delta {A^i}D{A^{i + 2}}

and

Δ
D

A

i
+
1

A

i
+
2

\Delta D{A^{i + 1}}{A^{i + 2}}

. Let

Δ

00

{\Delta _{00}}

be a given triangle, with smallest interior angle

α
>
0

\alpha > 0

. Bisect

Δ

00

{\Delta _{00}}

into two new triangles,

Δ

1
i

,
i
=
1
,
2

{\Delta _{1i}},i = 1,2

. Next, bisect each triangle

Δ

1
i

{\Delta _{1i}}

, to form four new triangles

Δ

2
i

,
i
=
1
,
2
,
3
,
4

{\Delta _{2i}},i = 1,2,3,4

, and so on, to form an infinite sequence T of triangles. It is shown that if

Δ
∈
T

\Delta \in T

, and

θ
\theta

is any interior angle of

Δ
\Delta

, then

θ
⩾
α

/

2

\theta \geqslant \alpha /2

.
Not every number is the sum or difference of two prime powers
Authors: Cohen Fred , Selfridge J. L.
Published: June 2010

Every odd number less than 262144 is the sum or difference of a power of two and a prime. An interesting example is

113921
=
p
−

2

141

113921 = p - {2^{141}}

. Using covering congruences, we exhibit a 26-digit odd number which is neither the sum nor difference of a power of two and a prime. The method is then modified to exhibit an arithmetic progression of numbers which are not the sum or difference of two prime powers.
How to calculate shortest vectors in a lattice
Authors: Dieter U.
Published: June 2010

A method for calculating vectors of smallest norm in a given lattice is outlined. The norm is defined by means of a convex, compact, and symmetric subset of the given space. The main tool is the systematic use of the dual lattice. The method generalizes an algorithm presented by Coveyou and MacPherson, and improved by Knuth, for the determination of vectors of smallest Euclidean norm.
Bifurcation in difference approximations to two-point boundary value problems
Authors: Weiss Richard
Published: June 2010

Numerical methods for bifurcation problems of the form

(

∗

)

L
y
=
λ
f
(
y
)
,

B
y
=
0
,

\begin{equation}\tag {$\ast $} Ly = \lambda f(y),\quad By = 0,\end{equation}

where

f
(
0
)
=
0

f(0) = 0

and

f
′

(
0
)
≠
0

f’(0) \ne 0

, are considered. Here y is a scalar function,

λ
\lambda

is a real scalar, L is a linear differential operator and

B
y
=
0

By = 0

represents some linear homogeneous two-point boundary conditions. Under certain assumptions, it is shown that if

(
∗
)

(\ast )

is replaced by an appropriate difference scheme, then there exists a unique branch of nontrivial solutions of the discrete problem in a neighborhood of a branch of nontrivial solutions of

(
∗
)

(\ast )

bifurcating from the trivial solution and that the discrete branch converges to the continuous one. Error estimates are derived and an illustrative numerical example is included.
The structure of certain triple systems
Authors: Robinson Raphael M.
Published: June 2010

For each prime power

q
≡
7

(
mod

12
)

q \equiv 7 \pmod {12}

, there is a triple system of order q whose automorphism group is transitive on unordered pairs. The object of this paper is to study these systems. This is done by analyzing how pairs of elements are linked. The linkage of a and b consists of a triple (a, b, c) and of some cycles in which adjacent pairs of elements form triples alternately with a and with b. Because of the transitivity, the lengths of the cycles will be independent of the choice of a and b. Using a computer, the linkage between two elements was determined for each

q
>
1000

q > 1000

. Some curious facts concerning the lengths of the cycles were uncovered; for example, the number of cycles of length greater than 4 is even. The systems of prime order

p
>
1000

p > 1000

were found to have no proper subsystems of order greater than 3. In the remaining case,

q
=
343

q = 343

, there are subsystems of orders 7 and 49, and all subsystems of the same order are isomorphic. For no q with

7
>
q
>
1000

7 > q > 1000

is the automorphism group doubly transitive. Finally, some general results are proved. The cycles of lengths 4 and 6 are determined. Using this result, it is shown that there can be no subsystem of order 7 or 9, except for the subsystems of order 7 when q is a power of 7. Hence, by a theorem of Marshall Hall, the automorphism group cannot be doubly transitive, except possibly when q is a power of 7. (Added August 1974. In a postscript, it is shown that the automorphism group is not doubly transitive in this case either.)
Convergence of the Fraser-Hart algorithm for rational Chebyshev approximation
Authors: Dunham Charles B.
Published: June 2010

The Fraser-Hart variant of the Remez algorithm is used to determine the best rational Chebyshev approximation to a continuous function on an interval. A necessary and sufficient condition for the matrix of the associated linear system to be nonsingular at the solution to the approximation problem is given. It is shown that the Fraser-Hart method may fail even if started arbitrarily close to the solution of the approximation problem. Use of the secant method in place of the Fraser-Hart iteration is also considered.
Methods for computing and modifying the 𝐿𝐷𝑉 factors of a matrix
Authors: Gill Philip E. , Murray Walter , Saunders Michael A.
Published: June 2010

Methods are given for computing the LDV factorization of a matrix B and modifying the factorization when columns of B are added or deleted. The methods may be viewed as a means for updating the orthogonal (LQ) factorization of B without the use of square roots. It is also shown how these techniques lead to two numerically stable methods for updating the Cholesky factorization of a matrix following the addition or subtraction, respectively, of a matrix of rank one. The first method turns out to be one given recently by Fletcher and Powell; the second method has not appeared before.
Free boundaries and finite elements in one dimension
Authors: Hager William W. , Strang Gilbert
Published: June 2010

Two problems in control theory, one with state constraints and the other with control constraints, have been approximated by the finite element method. This discretization has been applied to both the primal and the dual formulation, in order to make a number of observations and comparisons: 1. The rate of convergence as the grid interval h is decreased, for polynomial elements of different degrees. 2. The presence or absence of a boundary layer in the error, concentrated at the "contact points" where the constraints change between binding and nonbinding. 3. The advantages of simpler constraints in the dual formulation, and the disadvantages of replacing strict convexity by ordinary convexity. 4. The numerical efficiency of each possible variation in achieving an approximate solution of reasonable accuracy. We concluded that in our model problems, linear elements and the dual method provide the most efficient combination.
A comparison of global methods for linear two-point boundary value problems
Authors: Russell R. D. , Varah J. M.
Published: June 2010

Recently there has been a great deal of interest in numerical methods of a global nature for boundary value problems. In this paper we discuss and compare these global methods from a computational point of view, for the case of a single linear two-point boundary value problem.
Maximum-norm interior estimates for Ritz-Galerkin methods
Authors: Bramble James H. , Nitsche Joachim A. , Schatz Alfred H.
Published: June 2010

In this paper we obtain, by simple means, interior maximum-norm estimates for a class of Ritz-Galerkin methods used for approximating solutions of second order elliptic boundary value problems in

R

N

{{\mathbf {R}}^N}

. The estimates are proved when the approximating subspaces are any of a large class of piecewise polynomial subspaces which we assume here to be defined on a uniform mesh on the interior domain. Optimal rates of convergence are obtained.
A finite element method for first order hyperbolic equations
Authors: Baker Garth A.
Published: June 2010

A class of finite element methods is proposed for first order hyperbolic equations. The expository example chosen is of a single equation in one space dimension with constant coefficients. Optimal

L
2

{L^2}

error estimates are derived for both approximations continuous in the time variable and an approximation scheme discrete in time.
Calculation and applications of Epstein zeta functions
Authors: Shanks Daniel
Published: June 2010

Rapidly convergent series are given for computing Epstein zeta functions at integer arguments. From these one may rapidly and accurately compute Dirichlet L functions and Dedekind zeta functions for quadratic and cubic fields of any negative discriminant. Tables of such functions computed in this way are described and numerous applications are given, including the evaluation of very slowly convergent products such as those that give constants of Landau and of Hardy-Littlewood.
On existence criteria and approximation procedures for integral equations of the first kind
Authors: Groetsch C. W.
Published: June 2010

The existence of solutions of Fredholm integral equations of the first kind is characterized in terms of the convergence properties of a general approximation procedure based on a spectral analysis of the integral operator. Applications are given to some iterative and regularization methods. In particular, some results of Diaz and Metcalf are generalized.
A computational technique for determining the class number of a pure cubic field
Authors: Barrucand Pierre , Williams H. C. , Baniuk L.
Published: June 2010

Two different computational techniques for determining the class number of a pure cubic field are discussed. These techniques were implemented on an IBM/370-158 computer, and the class number for each pure cubic field

Q
(

D

1

/

3

)

Q({D^{1/3}})

for

D
=
2
,
3
,
…
,
9999

D = 2,3, \ldots ,9999

was obtained. Several tables are presented which summarize the results of these computations. Some theorems concerning the class group structure of pure cubic fields are also given. The paper closes with some conjectures which were inspired by the computer results.
A nearest point algorithm for convex polyhedral cones and applications to positive linear approximation
Authors: Wilhelmsen Don R.
Published: June 2010

Suppose K is a convex polyhedral cone in

E
n

{E_n}

and is defined in terms of some generating set

{

e
1

,

e
2

,
…
,

e
N

}

\{ {e_1},{e_2}, \ldots ,{e_N}\}

. A procedure is devised so that, given any point

q
∈

E
n

q \in {E_n}

, the nearest point p in K to q can be found as a positive linear sum of

N
∗

⩽
n

{N^\ast } \leqslant n

points from the generating set. The procedure requires at most finitely many linear steps. The algorithm is then applied to find a positive representation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L f equals sigma-summation Underscript i equals 1 Overscript upper N Superscript asterisk Baseline Endscripts lamda Subscript i Baseline f left-parenthesis x Subscript i Baseline right-parenthesis comma lamda Subscript i Baseline greater-than 0 comma f element-of normal upper Phi comma">

L
f
=

∑

i
=
1

N
∗

λ
i

f
(

x
i

)
,

λ
i

>
0
,
f
∈
Φ
,

Lf = \sum \limits _{i = 1}^{{N^\ast }} {{\lambda _i}f({x_i}),} \quad {\lambda _i} > 0,f \in \Phi ,

\]
for a positive linear functional L acting on a suitable finite-dimensional function space

Φ
\Phi

.
A stability analysis for perturbed nonlinear iterative methods
Authors: Boggs Paul T. , Dennis J. E.
Published: June 2010

This paper applies the asymptotic stability theory for ordinary differential equations to Gavurin’s continuous analogue of several well-known nonlinear iterative methods. In particular, a general theory is developed which extends the Ortega-Rheinboldt concept of consistency to include the widely used finite-difference approximations to the gradient as well as the finite-difference approximations to the Jacobian in Newton’s method. The theory is also shown to be applicable to the Levenberg-Marquardt and finite-difference Levenberg-Marquardt methods.
A quadratically convergent iteration method for computing zeros of operators satisfying autonomous differential equations
Authors: Rall L. B.
Published: June 2010

If the Fréchet derivative P’ of the operator P in a Banach space X is Lipschitz continuous, satisfies an autonomous differential equation

P
′

(
x
)
=
f
(
P
(
x
)
)

P’(x) = f(P(x))

, and

f
(
0
)

f(0)

has the bounded inverse

Γ
\Gamma

, then the iteration process
\[

x

n
+
1

=

x
n

−
Γ
P
(

x
n

)
,

n
=
0
,
1
,
2
,
…
,

{x_{n + 1}} = {x_n} - \Gamma P({x_n}),\quad n = 0,1,2, \ldots ,

\]
is shown to be locally quadratically convergent to solutions

x
=

x
∗

x = {x^\ast }

of the equation

P
(
x
)
=
0

P(x) = 0

. If f is Lipschitz continuous and

Γ
\Gamma

exists, then the global existence of

x
∗

{x^\ast }

is shown to follow if

P
(
x
)

P(x)

is uniformly bounded by a sufficiently small constant. The replacement of the uniform boundedness of P by Lipschitz continuity gives a semilocal theorem for the existence of

x
∗

{x^\ast }

and the quadratic convergence of the sequence

{

x
n

}

\{ {x_n}\}

to

x
∗

{x^\ast }

.
Explicit criteria for quintic residuacity
Authors: Williams Kenneth S.
Published: June 2010

Let p be a

prime

≡
1

(
mod

5
)

{\text {prime}} \equiv 1 \pmod 5

. Necessary and sufficient conditions are determined for the prime

q
(
q
⩽
19
)

q(q \leqslant 19)

to be a quintic residue of p. The results for

q
⩽
7

q \leqslant 7

are known, the rest are new.
Dyadotropic polynomials
Authors: Cohn Harvey
Published: June 2010

Polynomials which tend to represent powers of two arise in connection with certain problems of class field theory of dihedral biquadratic fields. The availability of independent units is an immediate consequence for an infinitude of parametrized cases. An exhaustive search for such types of polynomials is made by use of computer.
A computational technique for evaluating 𝐿(1,𝜒) and the class number of a real quadratic field
Authors: Williams H. C. , Broere J.
Published: June 2010

A description is given of a method for estimating

L
(
1
,
χ
)

L(1,\chi )

to sufficient accuracy to determine the class number of a real quadratic field. This algorithm was implemented on an IBM/370-158 computer and the class number, regulator, and value of

L
(
1
,
χ
)

L(1,\chi )

were obtained for each real quadratic field

Q
(

D

)

(
D
=
2
,
3
,
…
,
149999
)

Q(\sqrt D )\;(D = 2,3, \ldots ,149999)

. Several tables, summarizing various results of these computations, are also presented.
Accelerating convergence of eigenfunction expansions
Authors: Shaw J. K. , Johnson L. W. , Riess R. D.
Published: June 2010

A general procedure is presented for accelerating the convergence of eigenfunction expansions associated with selfadjoint boundary-value problems. The results obtained reduce, in special cases, to certain well-known methods of acceleration, including the Lanczos representation. The generality of the procedure allows the user to take advantage of structural properties of the expanded function.
Extensions of the Mehler-Weisner and other results for the Hermite function
Authors: Cohen M. E.
Published: June 2010

The purpose of this paper is to present expansions which generalize some well-known formulae for the Hermite function. Among these are the Weisner [20] extension of Mehler’s [17] bilinear relation, some recent results of Carlitz [4], and the Bateman [2] addition theorem. A bilateral generating function involving the product of the Hermite and ultraspherical polynomials is given. Finally, some general polynomial expansion theorems are derived.
A rational approximation to Weierstrass’ 𝑃-function
Authors: Eckhardt Ulrich
Published: June 2010

A rational approximation to Weierstrass’

℘
\wp

-function in the equianharmonic case for unit period parallelogram is given. With a third-degree numerator polynomial and a fourth-degree denominator polynomial the maximal error for

|

z

|

>
1

/

3

|z| > 1/\sqrt 3

becomes

3
⋅

10

−
14

3 \cdot {10^{ - 14}}

. The approximation of

℘
(
z
)

\wp (z)

is then used to calculate a rational approximation to

℘
′

(
z
)

\wp ’(z)

together with an error bound.
Reorthogonalization and stable algorithms for updating the Gram-Schmidt 𝑄𝑅 factorization
Authors: Daniel J. W. , Gragg W. B. , Kaufman L. , Stewart G. W.
Published: June 2010

Numerically stable algorithms are given for updating the Gram-Schmidt QR factorization of an

m
×
n

m \times n

matrix

A

(
m
⩾
n
)

A\;(m \geqslant n)

when A is modified by a matrix of rank one, or when a row or column is inserted or deleted. The algorithms require

O
(
m
n
)

O(mn)

operations per update, and are based on the use of elementary two-by-two reflection matrices and the Gram-Schmidt process with reorthogonalization. An error analysis of the reorthogonalization process provides rigorous justification for the corresponding ALGOL procedures.
Euclid’s algorithm in the cyclotomic field 𝑄(𝜁₁₆)
Authors: Ojala T.
Published: June 2010

Let

ζ

16

{\zeta _{16}}

denote a primitive 16th root of unity. It is proved that

Z

[

ζ

16

]

{\mathbf {Z}}[{\zeta _{16}}]

is Euclidean for the norm map.
Osculatory and hyperosculatory cubature formulas
Authors: Salzer Herbert E.
Published: June 2010

Osculatory and hyperosculatory cubature formulas for a rectangular region, employing the function with either first, or first and second partial derivatives, were obtained together with dominant remainder terms involving higher derivatives at one point, providing exact accuracy through the fifth (seventh) degree without (with) remainders.
On a posteriori error estimates
Authors: Miel George
Published: June 2010

Consider a sequence

{

x
n

}

n
=
0

∞

\{ {x_n}\} _{n = 0}^\infty

in a normed space X converging to some

x
∗

∈
X

{x^\ast } \in X

. It is shown that the sequence satisfies a condition of the type
\[

‖

x
∗

−

x
n

‖

⩽
α

‖

x
n

−

x

n
−
1

‖

\left \| {{x^\ast } - {x_n}} \right \| \leqslant \alpha \left \| {{x_n} - {x_{n - 1}}} \right \|

\]
for some constant

α
\alpha

and every

n
⩾
1

n \geqslant 1

, if the associated null sequence

{

e
n

}

n
=
0

∞

,

e
n

=

x
∗

−

x
n

\{ {e_n}\} _{n = 0}^\infty ,{e_n} = {x^\ast } - {x_n}

, is uniformly decreasing in norm or if it is alternating with respect to any ordering whose cone of positive elements is acute.
Multistep methods using higher derivatives and damping at infinity
Authors: Jeltsch Rolf
Published: June 2010

Linear multistep methods using higher derivatives are discussed. The order of damping at infinity which measures the stability behavior of a k-step method for large h is introduced, A-stable methods with positive damping order are most suitable for stiff problems. A method for computing the damping order is given. Necessary and sufficient conditions for A-stability, A

A
(
α
)

A(\alpha )

-stability and stiff stability are presented. A new A-stable two-step method of order 4 with damping order 1 is found and numerical results are given.
The orders of the known simple groups as far as one trillion
Authors: Cato Craig
Published: June 2010

In this paper an update of L. E. Dickson’s 1901 list of the orders of the known simple groups up to one billion is given. On the microfiche supplement is a complete list of all of the orders of the known simple groups as far as one trillion together with the Lie notation for these groups.
Inverse iteration on defective matrices
Authors: Chen Nai Fu
Published: June 2010

Very often, inverse iteration is used with shifts to accelerate convergence to an eigenvector. In this paper, it is shown that, if the eigenvalue belongs to a nonlinear elementary divisor, the vector sequences may diverge even when the shift sequences converge to the eigenvalue. The local behavior is discussed through a

2
×
2

2 \times 2

example, and a sufficient condition for the convergence of the vector sequence is given.
An error estimate for the truncation method for the solution of parabolic obstacle variational inequalities
Authors: Berger Alan E. , Falk Richard S.
Published: June 2010

A rate of convergence is obtained for a truncation method for the numerical solution of a class of parabolic variational inequalities.
Numbers generated by the reciprocal of 𝑒^{𝑥}-𝑥-1
Authors: Howard F. T.
Published: June 2010

In this paper we examine the polynomialsAn(z){A_n}(z)and the rational numbersAn=An(0){A_n} = {A_n}(0)defined by means of\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Superscript x z Baseline x squared left-parenthesis e Superscript x Baseline minus x minus 1 right-parenthesis Superscript negative 1 Baseline equals 2 sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts upper A Subscript n Baseline left-parenthesis z right-parenthesis x Superscript n Baseline slash n factorial period">exzx2(ex−x−1)−1=2∑n=0∞An(z)xn/n!.{e^{xz}}{x^2}{({e^x} - x - 1)^{ - 1}} = 2\sum \limits _{n = 0}^\infty {{A_n}(z){x^n}/n!} .\]We prove that the numbersAn{A_n}are related to the Stirling numbers and associated Stirling numbers of the second kind, and we show that this relationship appears to be a logical extension of a similar relationship involving Bernoulli and Stirling numbers. Other similarities betweenAn{A_n}and the Bernoulli numbers are pointed out. We also reexamine and extend previous results concerningAn{A_n}andAn(z){A_n}(z). In particular, it has been conjectured thatAn{A_n}has the same sign as−cos⁡nθ- \cos n\theta, wherereiθr{e^{i\theta }}is the zero ofex−x−1{e^x} - x - 1with smallest absolute value. We verify this for1⩽n⩽143291 \leqslant n \leqslant 14329and show that if the conjecture is not true forAn{A_n}, then|cos⁡nθ|>10−(n−1)/5|\cos n\theta | > {10^{ - (n - 1)/5}}. We also show thatAn(z){A_n}(z)has no integer roots, and in the interval[0,1][0,1],An(z){A_n}(z)has either two or three real roots.
Modifying singular values: existence of solutions to sytems of nonlinear equations having a possibly singular Jacobian matrix
Authors: Gay David
Published: June 2010

We show that if a certain nondegeneracy assumption holds, it is possible to guarantee the existence of a solution to a system of nonlinear equations

f
(
x
)
=
0

f(x) = 0

whose Jacobian matrix

J
(
x
)

J(x)

exists but may be singular. The main idea is to modify small singular values of

J
(
x
)

J(x)

in such a way that the modified Jacobian matrix

J
^

(
x
)

\hat J(x)

has a continuous pseudoinverse

J
^

+

(
x
)

{\hat J^ + }(x)

and that a solution

x
∗

{x^\ast }

of

f
(
x
)
=
0

f(x) = 0

may be found by determining an asymptote of the solution to the initial value problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x left-parenthesis 0 right-parenthesis equals x 0 comma x prime left-parenthesis t right-parenthesis equals minus ModifyingAbove upper J With caret Superscript plus Baseline left-parenthesis x right-parenthesis f left-parenthesis x right-parenthesis">

x
(
0
)
=

x
0

,
x
′
(
t
)
=
−

J
^

+

(
x
)
f
(
x
)

x(0) = {x_0},x\prime (t) = - {\hat J^ + }(x)f(x)

. We briefly discuss practical (algorithmic) implications of this result. Although the nondegeneracy assumption may fail for many systems of interest (indeed, if the assumption holds and

J
(

x
∗

)

J({x^\ast })

is nonsingular, then

x
∗

{x^\ast }

is unique), algorithms using

J
^

+

(
x
)

{\hat J^ + }(x)

may enjoy a larger region of convergence than those that require (an approximation to)

J

−
1

(
x
)

{J^{ - 1}}(x)

.
Comparison of algorithms for multivariate rational approximation
Authors: Henry Jackson N.
Published: June 2010

Let F be a continuous real-valued function defined on the unit square

[
−
1
,
1
]
×
[
−
1
,
1
]

[ - 1,1] \times [ - 1,1]

. When developing the rational product approximation to F, a certain type of discontinuity may arise. We develop a variation of a known technique to overcome this discontinuity so that the approximation can be programmed. Rational product approximations to F have been computed using both the second algorithm of Remez and the differential correction algorithm. A discussion of the differences in errors and computing time for each of these algorithms is presented and compared with the surface fit approximation also obtained using the differential correction algorithm.
On congruences related to the first case of Fermat’s last theorem
Authors: Johnson Wells
Published: June 2010

Solutions to the congruences

(
1
+
a

)

p
n

≡
1
+

a

p
n

(
mod

p

n
+
2

)

{(1 + a)^{{p^n}}} \equiv 1 + {a^{{p^n}}}\pmod {p^{n + 2}}

and

(
1
+
s

)
p

≡
1
+

s
p

(
mod

p
n

)

{(1 + s)^p} \equiv 1 + {s^p}\pmod {p^n}

are discussed. Congruences of this type arise in the study of the first case of Fermat’s Last Theorem. Solutions to these congruences always exist for primes

p
≡
1

(
mod

6
)

p \equiv 1\;\pmod 6

. They are derived from the existence of a primitive cube root of unity

(
mod

p
)

\pmod p

. Constructive techniques for finding numerical examples are presented. The results are obtained by examining the p-adic expansions of the p-adic

(
p
−
1
)

(p - 1)

st roots of unity.
Uniqueness of Padé approximants from series of orthogonal polynomials
Authors: Sidi Avram
Published: June 2010

It is proved that whenever a nonlinear Padé approximant, derived from a series of orthogonal polynomials, exists, it is unique.
The numerical solution of boundary value problems for stiff differential equations
Authors: Flaherty Joseph E. , O’Malley R. E.
Published: June 2010

The numerical solution of boundary value problems for certain stiff ordinary differential equations is studied. The methods developed use singular perturbation theory to construct approximate numerical solutions which are valid asymptotically; hence, they have the desirable feature of becoming more accurate as the equations become stiffer. Several numerical examples are presented which demonstrate the effectiveness of these methods.
A double shooting scheme for certain unstable and singular boundary value problems
Authors: Bayliss Alvin
Published: June 2010

A scheme is presented to obtain the unique bounded solution for an exponentially unstable linear system. The scheme consists of choosing random data at large initial values and integrating forwards and backwards until accurate regular boundary values are obtained. Proofs of convergence are given for the case that the homogeneous equation has an exponential dichotomy. Applications to other types of problems are discussed and numerical results are presented.
Computational experiments and techniques for the penalty method with extrapolation
Authors: King J. Thomas , Serbin Steven M.
Published: June 2010

In this note we present results of a computational investigation of the extrapolated penalty method for approximate solution of elliptic boundary value problems. We investigate the effects of extrapolation and present an iterative technique for solving the extra linear algebraic systems necessary to perform the process. We indicate how convergence of the iterative procedure may be accelerated when boundary weights are appropriately selected. We consider the Euclidean relative error in the iterative procedure and the effect of conditioning. We develop a bound for the difference between an extrapolate obtained assuming exact solution of all linear systems and the corresponding quantity computed by a terminated iterative procedure.
Determining the fundamental unit of a pure cubic field given any unit
Authors: Jeans N. S. , Hendy M. D.
Published: June 2010

A number of algorithms which have been used to derive fundamental units for pure cubic fields suffer from the lack of absolute certainty that the units obtained are fundamental. We present here an algorithm which will correct this deficiency. Briefly, if

η
\eta

is any nontrivial unit of a pure cubic field, then for some positive integer

N
,

η

1

/

N

N, {\eta ^{1/N}}

will be a fundamental unit. Our method determines which of the real numbers

η

1

/

N

{\eta ^{1/N}}

are integers of the field and, subsequently, will determine the coefficients of the fundamental unit. We illustrate the process with several numerical examples.
A scalar expression for matrices with symplectic involution
Authors: Rowen Louis Halle , Schild Uri
Published: June 2010

Various algebraic reductions are made to facilitate computer verification of the following result: If x and y are

8
×
8

8 \times 8

matrices such that [x, y] is regular,

tr
⁡
(
x
)
=
0

\operatorname {tr} (x) = 0

, and, with respect to the canonical symplectic involution, x is symmetric and y is antisymmetric, then the element

(
x
+
[
x
,
y
]
x

[
x
,
y

]

−
1

)
2

{(x + [x,y]x{[x,y]^{ - 1}})^2}

satisfies a minimal equation of degree

⩽
2

\leqslant 2

.
Interpolation by convex quadratic splines
Authors: McAllister David F. , Roulier John A.
Published: June 2010

Algorithms are presented for computing a quadratic spline interpolant with variable knots which preserves the monotonicity and convexity of the data. It is also shown that such a spline may not exist for fixed knots.
Interpolation error estimates for the reduced Hsieh-Clough-Tocher triangle
Authors: Ciarlet Philippe G.
Published: June 2010

We study the unisolvence and interpolation properties of the reduced Hsieh-Clough-Tocher triangle. This finite element of class

C

1

{\mathcal {C}^1}

, which has only nine degrees of freedom, can be used in the numerical approximation of plate problems.
Linear Chebyshev approximation of complex-valued functions
Authors: Barrodale I. , Delves L. M. , Mason J. C.
Published: June 2010

This paper is concerned with Chebyshev approximation by linear functions to complex-valued data. The problem is nonlinear, and we present a convergent algorithm for its solution. We also pose a related linear problem which is simple to solve, and which produces approximations which are near-best in the Chebyshev sense within a factor of

2

\sqrt 2

. Some numerical examples are provided.
Some numerical results using a sparse matrix updating formula in unconstrained optimization
Authors: Toint Ph. L.
Published: June 2010

This paper presents a numerical comparison between algorithms for unconstrained optimization that take account of sparsity in the second derivative matrix of the objective function. Some of the methods included in the comparison use difference approximation schemes to evaluate the second derivative matrix and others use an approximation to it which is updated regularly using the changes in the gradient. These results show what method to use in what circumstances and also suggest interesting future developments.
The construction of rational iterating functions
Authors: Smyth W. F.
Published: June 2010

Suppose integers

n
⩾
1

n \geqslant 1

and

σ
⩾
2

\sigma \geqslant 2

are given, together with n distinct points

z
2

,
…
,

z
n

{z_2}, \ldots ,{z_n}

, in the complex plane. Define

Φ
M

=

Φ
M

(
σ
;

z
1

,
…
,

z
n

)

{\Phi _M} = {\Phi _M}(\sigma ;{z_1}, \ldots ,{z_n})

to be the class of rational functions

ϕ

p
,
q

(
z
)
=

g
p

(
z
)

/

h
q

(
z
)

{\phi _{p,q}}(z) = {g_p}(z)/{h_q}(z)

(where g and h are polynomials of degree

p
⩾
1

p \geqslant 1

and

q
⩾
1

q \geqslant 1

, respectively) such that

(
1
)

p
+
q
+
1
=
M
,
(
2
)

ϕ

(1)\;p + q + 1 = M,(2)\;\phi

when iterated converges with order

σ
\sigma

at each

z
i

,
i
=
1
,
…
,
n

{z_i}, i = 1, \ldots ,n

. Then if

M
>
σ
n
,

Φ
M

M > \sigma n,{\Phi _M}

is null; if

M
=
σ
n

M = \sigma n

Φ
M

{\Phi _M}

contains exactly

σ
n

\sigma n

elements. For every

M
⩾
σ
n

M \geqslant \sigma n

, we show how to construct all the elements of

Φ
M

{\Phi _M}

by expressing, for each choice of p and q which satisfies

p
+
q
+
1
=
M

p + q + 1 = M

, the coefficients of

g
p

{g_p}

and

h
q

{h_q}

in terms of

M
−
σ
n

M - \sigma n

arbitrarily chosen values. In fact, these coefficients are expressed in terms of generalized Newton sums

S
n

j
,
k

=

S
n

j
,
k

(

z
1

,
…
,

z
n

)

S_n^{j,k} = S_n^{j,k}({z_1}, \ldots ,{z_n})

,

1
⩽
j
⩽
n
,
k
⩾
n

1 \leqslant j \leqslant n,k \geqslant n

, which we show may be calculated by recursion from the normal Newton sums

S
n

j
,
n

S_n^{j,n}

. Hence, given a polynomial

f
n

(
z
)

{f_n}(z)

with n distinct (unknown) zeros

z
1

,
…
,

z
n

{z_1}, \ldots ,{z_n}

, we may construct all

ϕ

p
,
q

(
z
)

{\phi _{p,q}}(z)

which converge to the

z
i

{z_i}

with order

σ
\sigma

in the case

σ
=
2

\sigma = 2

, the choice

p
=
n

p = n

,

q
=
n
−
1

q = n - 1

, yields the Newton-Raphson iteration

ϕ

n
,
n
−
1

∈

Φ

2
n

{\phi _{n,n - 1}} \in {\Phi _{2n}}

; the Schröder and König iterations are shown to be elements of

Φ

2
(
2
σ
−
3
)
(
n
−
1
)
+
2

{\Phi _{2(2\sigma - 3)(n - 1) + 2}}

and

Φ

2
(
σ
−
1
)
(
n
−
1
)
+
2

{\Phi _{2(\sigma - 1)(n - 1) + 2}}

, respectively. We show by example that there exist cases in which

ϕ

n
,
n
−
1

{\phi _{n,n - 1}}

has an undesirable property (attractive cycles) not shared by other iterating functions in the same class

Φ

2
n

{\Phi _{2n}}

.