bluuebunny/crossref_metadata_2025 huggingface dataset:
A “sinc-Galerkin” method of solution of boundary value problems
Authors: Stenger Frank
Published: June 2010

This paper illustrates the application of a "Sinc-Galerkin" method to the approximate solution of linear and nonlinear second order ordinary differential equations, and to the approximate solution of some linear elliptic and parabolic partial differential equations in the plane. The method is based on approximating functions and their derivatives by use of the Whittaker cardinal function. The DE is reduced to a system of algebraic equations via new accurate explicit approximations of the inner products, the evaluation of which does not require any numerical integration. Usingnfunction evaluations, the error in the final approximation to the solution of the DE isO(e−cn1/2d)O({e^{ - c{n^{1/2d}}}}), wherecis independent ofn, andddenotes the dimension of the region on which the DE is defined. This rate of convergence is optimal in the class ofn-point methods which assume that the solution is analytic in the interior of the interval, and which ignore possible singularities of the solution at the endpoints of the interval.
A new step-size changing technique for multistep methods
Authors: Gupta G. K. , Wallace C. S.
Published: June 2010

The step-size changing technique is an important component of a Variable Step Variable Order algorithm for solving ordinary differential equations using multi-step methods. This paper presents a new technique for changing the step-size and compares its performance to that of the Variable-Step and Fixed-Step Interpolation techniques.
A parallel algorithm for solving general tridiagonal equations
Authors: Swarztrauber Paul N.
Published: June 2010

A parallel algorithm for the solution of the general tridiagonal system is presented. The method is based on an efficient implementation of Cramer’s rule, in which the only divisions are by the determinant of the matrix. Therefore, the algorithm is defined without pivoting for any nonsingular system.

O
(
n
)

O(n)

storage is required for n equations and

O
(
log
⁡
n
)

O(\log n)

operations are required on a parallel computer with n processors.

O
(
n
)

O(n)

operations are required on a sequential computer. Experimental results are presented from both the CDC 7600 and CRAY-1 computers.
Convergence properties of some nonlinear sequence transformations
Authors: Sidi Avram
Published: June 2010

The nonlinear transformations to accelerate the convergence of sequences due to Levin are considered and bounds on the errors are derived. Convergence theorems for oscillatory and some monotone sequences are proved.
Polynomial minimum root separation
Authors: Rump Siegfried M.
Published: June 2010

The minimum root separation of an arbitrary polynomial P is defined as the minimum of the distances between distinct (real or complex) roots of P. Some asymptotically good lower bounds for the root separation of P are given, where P may have multiple zeros. There are applications in the analysis of complexity of algorithms and in the theory of algebraic and transcendental numbers.
The condition of polynomials in power form
Authors: Gautschi Walter
Published: June 2010

A study is made of the numerical condition of the coordinate map

M
n

{M_n}

which associates to each polynomial of degree

⩽
n
−
1

\leqslant n - 1

on the compact interval [a, b] the n-vector of its coefficients with respect to the power basis. It is shown that the condition number

‖

M
n

‖

∞

‖

M
n

−
1

‖

∞

{\left \| {{M_n}} \right \|_\infty }{\left \| {M_n^{ - 1}} \right \|_\infty }

increases at an exponential rate if the interval [a, b] is symmetric or on one side of the origin, the rate of growth being at least equal to

1
+

2

1 + \sqrt 2

. In the more difficult case of an asymmetric interval around the origin we obtain upper bounds for the condition number which also grow exponentially.
The 𝜈-zeros of 𝐽_{-𝜈}(𝑥)
Authors: Conde S. , Kalla S. L.
Published: June 2010

We compute the positive

ν
\nu

-zeros of

J

−
ν

(
x
)

{J_{ - \nu }}(x)

, regarded as a function of

ν
\nu

.

J
ν

(
x
)

{J_\nu }(x)

stands for the Bessel function of first kind of order

ν
\nu

. Some related inequalities are verified and algorithms the computer uses are described briefly.
An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations
Authors: Steihaug Trond , Wolfbrandt Arne
Published: June 2010

A class of linear implicit methods for numerical solution of stiff ODE’s is presented. These require only occasional calculation of the Jacobian matrix while maintaining stability. Especially, an effective second order stable algorithm with automatic stepsize control is designed and tested.
Stability of sequences generated by nonlinear differential systems
Authors: Brown R. Leonard
Published: June 2010

A local stability analysis is given for both the analytic and numerical solutions of the initial value problem for a system of ordinary differential equations. The standard linear stability analysis is reviewed, then it is shown that, using a proper choice of Liapunov function, a connected region of stable initial values of both the analytic solution and of the one-leg k-step numerical solution can be approximated computationally. Correspondence between the one-leg k-step solution and its associated linear k-step solution is shown, and two examples are given.
Two linear programming algorithms for the linear discrete 𝐿₁ norm problem
Authors: Armstrong Ronald D. , Godfrey James P.
Published: June 2010

Computational studies by several authors have indicated that linear programming is currently the most efficient procedure for obtaining

L
1

{L_1}

, norm estimates for a discrete linear problem. However, there are several linear programming algorithms, and the "best" approach may depend on the problem’s structure (e.g., sparsity, triangularity, stability). In this paper we shall compare two published simplex algorithms, one referred to as primal and the other referred to as dual, and show that they are conceptually equivalent.
Finite element collocation methods for first-order systems
Authors: Lesaint P. , Raviart P.-A.
Published: June 2010

Finite element methods and the associate collocation methods are considered for solving first-order hyperbolic systems, positive in the sense of Friedrichs. Applied in the case when the meshes are rectangle, those methods lead for example to the successfully used box scheme for the heat equation or D.S.N. scheme for the neutron transport equation. Generalizations of these methods are described here for nonrectangle meshes and (or) noncylindrical domains; stability results and error estimates are derived.
New Euler-Maclaurin expansions and their application to quadrature over the 𝑠-dimensional simplex
Authors: de Doncker Elise
Published: June 2010

The

μ
\mu

-panel offset trapezodial rule for noninteger values of

μ
\mu

, is introduced in a one-dimensional context. An asymptotic series describing the error functional is derived. The values of

μ
\mu

for which this is an even Euler-Maclaurin expansion are determined, together with the conditions under which it terminates after a finite number of terms. This leads to a new variant of one-dimensional Romberg integration. The theory is then extended to quadrature over the s-dimensional simplex, the basic rules being obtained by an iterated use of one-dimensional rules. The application to Romberg integration is discussed, and it is shown how Romberg integration over the simplex has properties analogous to those for standard one-dimensional Romberg integration and Romberg integration over the hypercube. Using extrapolation, quadrature rules for the s-simplex can be generated, and a set of formulas can be obtained which are the optimum so far discovered in the sense of requiring fewest function values to obtain a specific polynomial degree.
A finite difference scheme for a system of two conservation laws with artificial viscosity
Authors: Hoff David
Published: June 2010

In this paper we analyze an implicit finite difference scheme for the mixed initial-value Dirichlet problem for a system of two conservation laws with artificial viscosity. The system we consider is a model for isentropic flow in one space dimension. First, we show that, under certain conditions on the mesh, the scheme is stable in the sense that it possesses an invariant set (defined by the so-called Riemann invariants). We obtain this result as an extension of the same stability theorem for the Lax-Friedrichs scheme in the inviscid case. Second, we show that the approximants remain bounded and, in fact, decay to the boundary values as

t
→
∞

t \to \infty

. Finally, we obtain two

O
(
Δ

x
2

)

O(\Delta {x^2})

error bounds; the first grows exponentially in time while the second, which requires that the data have small oscillation, is independent of time.
The exact order of convergence for finite difference approximations to ordinary boundary value problems
Authors: Beyn Wolf-Jürgen
Published: June 2010

This paper deals with the problem of determining the exact order of convergence for the finite difference method applied to ordinary boundary value problems when formulas of different orders are used at different points of the grid. Under rather general assumptions, it is shown that the global discretization error is

O
(

h
τ

)

O({h^\tau })

if the local truncation error is

O
(

h
τ

)

O({h^\tau })

on the boundary and at interior grid points, while it is only

O
(

h

τ
−
(
k
−
μ
)

)

O({h^{\tau - (k - \mu )}})

at grid points near the boundary. Here k and

μ
\mu

denote the order of the differential and the boundary operator, respectively.
On bounding 𝐴⁻¹_{∞} for banded 𝐴
Authors: Demko Stephen
Published: June 2010

Upper bounds for

‖

A

−
1

‖

∞

\left \|{A^{ - 1}}\right \|_\infty

in terms of inverses of certain submatrices are obtained for band matrices. An application to a problem in spline theory is made.
On maximal finite irreducible subgroups of 𝐺𝐿(𝑛,𝑍). V. The eight-dimensional case and a complete description of dimensions less than ten
Authors: Plesken Wilhelm , Pohst Michael
Published: June 2010

All maximal finite (absolutely) irreducible subgroups ofGL(8,Z)GL(8,{\mathbf {Z}})are determined up toZ-equivalence. Moreover, we present a full set of representatives of theZ-classes of the maximal finite irreducible subgroups ofGL(n,Z)GL(n,{\mathbf {Z}})forn⩽9n \leqslant 9by listing generators of the groups, the corresponding quadratic forms fixed by these groups, and the shortest vectors of these forms.
Corrigendum to: “What drives an aliquot sequence?” [Math. Comp. 29 (1975), 101–107; MR 52 #5542]
Authors: Guy Richard K. , Selfridge J. L.
Published: June 2010

An aliquot sequence

n
:
k

n:k

,

k
=
0
,
1
,
2
,
…

k = 0,1,2, \ldots

, is defined by

n
:
0
=
n
,

n:0 = n,

,

n
:
k
+
1
=
σ
(
n
:
k
)
−
n
:
k

n:k + 1 = \sigma (n:k) - n:k

, and a driver of an aliquot sequence is a number

2
A

υ

{2^A}\upsilon

with

A
>
0

A > 0

,

υ
\upsilon

odd,

υ

|

2

A
+
1

−
1

\upsilon |{2^{A + 1}} - 1

and

2

A
−
1

|

σ
(
υ
)

{2^{A - 1}}|\sigma (\upsilon )

. Pollard has noted some errors in a proof in [1] that the drivers comprise the even perfect numbers and a finite set. These are now corrected in a revised proof.
A stable quotient-difference algorithm
Authors: Stokes A. N.
Published: June 2010

This paper shows how the arithmetic of the quotient-difference algorithm can be performed using the forward and backward difference tables of each column. This removes the tendency of the algorithm to amplify errors. As an application, 70 continued fraction coefficients are calculated for the modified Bessel function

K
0

(
z
)

{K_0}(z)

in single-precision arithmetic. There is no significant build-up of error.
Issues in nonlinear hyperperfect numbers
Authors: Minoli Daniel
Published: June 2010

Hyperperfect numbers (HP) are a generalization of perfect numbers and as such share remarkably similar properties. In this note we show, among other things, that ifm=p1α1p2α2m = p_1^{{\alpha _1}}p_2^{{\alpha _2}}is 2-HP thenα2=1{\alpha _2} = 1, withp1=3{p_1} = 3,p2=3α1+1−2{p_2} = {3^{{\alpha _1} + 1}} - 2; this is in agreement with the structure of the perfect case (1-HP) stating that such a number is of the formm=p1α1p2m = p_1^{{\alpha _1}}{p_2}withp1=2{p_1} = 2andp2=2α1+1−1{p_2} = {2^{{\alpha _1} + 1}} - 1.
𝑝-divisibility of certain sets of Bernoulli numbers
Authors: Wagstaff Samuel S.
Published: June 2010

Recently, Ullom has proved an upper bound on the number of Bernoulli numbers in certain sets which are divisible by a given prime. We report on a search for such Bernoulli numbers and primes up to 1000000.
Updating quasi-Newton matrices with limited storage
Authors: Nocedal Jorge
Published: June 2010

We study how to use the BFGS quasi-Newton matrices to precondition minimization methods for problems where the storage is critical. We give an update formula which generates matrices using information from the last m iterations, where m is any number supplied by the user. The quasi-Newton matrix is updated at every iteration by dropping the oldest information and replacing it by the newest information. It is shown that the matrices generated have some desirable properties. The resulting algorithms are tested numerically and compared with several well-known methods.
A mean value theorem for linear functionals
Authors: Meek D.
Published: June 2010

When working out the errors in discretization formulas, one usually hopes to obtain a mean value type of error. This occurs if the associated Peano kernel is a function which does not change sign. In this paper an expansion is developed which will express any error in mean value form, even when the associated Peano kernel is a function which changes sign.
On the numerical evaluation of two infinite products
Authors: Allasia G. , Bonardo F.
Published: June 2010

A numerical evaluation of two infinite products of the type

Π

n
=
0

∞

(
1
−
a

q
n

)

\Pi _{n = 0}^\infty (1 - a{q^n})

, which are important in some mathematical fields, is considered. The numerical evaluation is based on a recursive formula of the type

x

n
+
1

=

x
n

f
(

y
n

/

x
n

)

{x_{n + 1}} = {x_n}f({y_n}/{x_n})

,

y

n
+
1

=

x

n
+
1

g
(

y
n

/

x
n

)

{y_{n + 1}} = {x_{n + 1}}g({y_n}/{x_n})

, and it is compared with a series expansion that was previously used for the computation. Two tables of the infinite products are provided with twenty significant figures which check and extend existing data.
Further comments on the computation of modified Bessel function ratios
Authors: Tretter Marietta J. , Walster G. W.
Published: June 2010

A theorem by W. J. Thron is used to enhance and further substantiate empirical results by Gautschi and Slavik on the convergence of continued fractions used to compute ratios of modified Bessel functions.
Integrating ODEs in the complex plane—pole vaulting
Authors: Corliss George F.
Published: June 2010

Most existing algorithms for solving initial value problems in ordinary differential equations implicitly assume that all variables are real. If the real-valued assumption is removed, the solution can be extended by analytic continuation along a path of integration in the complex plane of the independent variable. This path is chosen to avoid singularities which can make the solution difficult or impossible for standard methods. We restrict our attention to Taylor series methods, although other methods can be suitably modified. Numerical examples are given for (a) singularities on the real axis, (b) singularities in derivatives higher than those involved in the differential equation, and (c) singularities near the real axis. These examples show that the pole vaulting method merits further study for some special problems for which it is competitive with standard methods.
Some very large primes of the form 𝑘⋅2^{𝑚}+1
Authors: Cormack G. V. , Williams H. C.
Published: June 2010

Several large primes of the form

k
⋅

2
m

+
1

k \cdot {2^m} + 1

with

3
⩽
k
⩽
29

3 \leqslant k \leqslant 29

and

m
>
1500

m > 1500

are tabulated and four new factors of Fermat numbers are presented.
A stable finite element method for initial-boundary value problems for first-order hyperbolic systems
Authors: Winther Ragnar
Published: June 2010

A nonstandard finite element method for initial-boundary value problems for first-order hyperbolic systems in one space dimension with general boundary conditions is analyzed. The method can be considered as a generalization of the box scheme. We first establish a stability result for the method and then derive several error estimates.
A new method for Chebyshev approximation of complex-valued functions
Authors: Glashoff K. , Roleff K.
Published: June 2010

In this paper we are concerned with a formulation of the Chebyshev approximation problem in the complex plane as a problem of linear optimization in the presence of infinitely many constraints. It is shown that there exist stable and fast algorithms for the solution of optimization problems of this type. Some numerical examples are presented.
An 𝑂(𝑛^{1/10.89}) primality testing algorithm
Authors: Adleman Leonard , Leighton Frank Thomson
Published: June 2010

In this paper, we describe an

O
(

n

1

/

10.89

)

O({n^{1/10.89}})

deterministic algorithm to decide primality. The algorithm incorporates several recent results in complexity theory.
A method of calculation of lifting flows around 2-dimensional corner-shaped bodies
Authors: Djaoua M.
Published: June 2010

We study the flow problem of a perfect fluid around a two-dimensional corner-shaped body. By adding a singular function to the basis, we obtain better accuracy for the velocities, and we determine the lift with precision.
Hydrocode subcycling stability
Authors: Hicks D. L.
Published: June 2010

The method of artificial viscosity was originally designed by von Neumann and Richtmyer for calculating the propagation of waves in materials that were hydrodynamic and rate-independent (e.g., ideal gas law). However, hydrocodes (such as WONDY) based on this method continue to expand their repertoire of material laws even unto material laws that are rate-dependent (e.g., Maxwell’s material law). Restrictions on the timestep required for stability with material laws that are rate-dependent can be considerably more severe than restrictions of the Courant-Friedrichs-Lewy (CFL) type that are imposed in these hydrocodes. These very small timesteps can make computations very expensive. An alternative is to go ahead and integrate the conservation laws with the usual CFL timestep while subcycling (integrating with a smaller timestep) the integration of the stress-rate equation. If the subcycling is done with a large enough number of subcycles (i.e., with a small enough subcycle timestep), then the calculation is stable. Specifically, the number of subcycles must be one greater than the ratio of the CFL timestep to the relaxation time of the material.
A block-by-block method for Volterra integro-differential equations with weakly-singular kernel
Authors: Makroglou Athena
Published: June 2010

The theory of a block-by-block method for solving Volterra integro-differential equations with continuous kernels (see Makroglou [4], [5]) is adapted to Volterra integro-differential equations with weakly-singular kernels, and a rate of convergence is given.
Surfaces generated by moving least squares methods
Authors: Lancaster P. , Salkauskas K.
Published: June 2010

An analysis of moving least squares (m.l.s.) methods for smoothing and interpolating scattered data is presented. In particular, theorems are proved concerning the smoothness of interpolants and the description of m.l.s. processes as projection methods. Some properties of compositions of the m.l.s. projector, with projectors associated with finiteelement schemes, are also considered. The analysis is accompanied by examples of univariate and bivariate problems.
The rate of convergence of Chebyshev polynomials for functions which have asymptotic power series about one endpoint
Authors: Boyd John P.
Published: June 2010

The theorem proved here extends Chebyshev theory into what has previously been no man’s land: functions which have an infinite number of bounded derivatives on the expansion interval [a, b] but which are singular at one endpoint. The Chebyshev series in

1

/

x

1/x

for all the familiar special functions fall into this category, so this class of functions is very important indeed. In words, the theorem shows that the more slowly the asymptotic power series about the singular point converges, the slower the convergence of the corresponding Chebyshev series must be. More formally, if

f
(
x
)

f(x)

, analytic on [a, b), is singular at

x
=
b

x = b

in such a way that it has an asymptotic power series

f
(
x
)
∼
Σ

a
n

(
x
−
b

)
n

f(x) \sim \Sigma \,{a_n}{(x - b)^n}

about that endpoint, then, if
\[

lim

n
→
∞

¯

log
⁡

|

a
n

|

n
log
⁡
n

=
r
,

\overline {\lim \limits _{n \to \infty } } \frac {{\log |{a_n}|}}{{n\log n}} = r,

\]
it is proved that the coefficients of the convergent Chebyshev polynomial series on [a, b],

f
(
x
)
∼
Σ

b
n

T
n

(
y
)

f(x) \sim \Sigma \,{b_n}{T_n}(y)

where

y
=
2
[
x
−
0.5
(
b
+
a
)
]

/

(
b
−
a
)

y = 2[x - 0.5(b + a)]/(b - a)

, satisfy the inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow normal infinity Endscripts Overscript bar Endscripts StartFraction log StartAbsoluteValue left-parenthesis log StartAbsoluteValue b Subscript n Baseline EndAbsoluteValue right-parenthesis EndAbsoluteValue Over log n EndFraction less-than-or-slanted-equals StartFraction 2 Over r plus 2 EndFraction">

lim

n
→
∞

¯

log
⁡

|

(
log
⁡

|

b
n

|

)

|

log
⁡
n

⩽

2

r
+
2

\overline {\lim \limits _{n \to \infty } } \frac {{\log |(\log |{b_n}|)|}}{{\log n}} \leqslant \frac {2}{{r + 2}}

\]
.
Automatic selection of sequence transformations
Authors: Delahaye J.-P.
Published: June 2010

Some methods of automatic selection of sequence transformations for accelerating the convergence of sequences are presented. Two theorems show the efficiency of these methods. This is confirmed by numerical experiments.
The problem of Sierpiński concerning 𝑘⋅2ⁿ+1
Authors: Baillie Robert , Cormack G. , Williams H. C.
Published: June 2010

Let

k
0

{k_0}

be the least odd value of k such that

k
⋅

2
n

+
1

k \cdot {2^n} + 1

is composite for all

n
⩾
1

n \geqslant 1

. In this note, we present the results of some extensive computations which restrict the value of

k
0

{k_0}

to one of 119 numbers between 3061 and 78557 inclusive. Some new large primes are also given.
Perturbation theory for evaluation algorithms of arithmetic expressions
Authors: Stummel F.
Published: June 2010

The paper presents the theoretical foundation of a forward error analysis of numerical algorithms under data perturbations, rounding error in arithmetic floating-point operations, and approximations in ’built-in’ functions. The error analysis is based on the linearization method that has been proposed by many authors in various forms. Fundamental tools of the forward error analysis are systems of linear absolute and relative a priori and a posteriori error equations and associated condition numbers constituting optimal bounds of possible accumulated or total errors. Derivations, representations, and properties of these condition numbers are studied in detail. The condition numbers enable simple general, quantitative definitions of numerical stability, backward analysis, well- and ill-conditioning of a problem and an algorithm. The well-known illustration of algorithms and their linear error equations by graphs is extended to a method of deriving condition numbers and associated bounds. For many algorithms the associated condition numbers can be determined analytically a priori and be computed numerically a posteriori. The theoretical results of the paper have been applied to a series of concrete algorithms, including Gaussian elimination, and have proved to be very effective means of both a priori and a posteriori error analysis.
A class of Bessel summations
Authors: Glasser M. L.
Published: June 2010

A number of infinite series for which the general term contains the product of two Bessel functions are summed in closed form.
The finite element method in the family of improperly posed problems
Authors: Han Hou De
Published: June 2010

The numerical solution of the Cauchy problem for elliptic equations is considered. We reformulate the original problem as a variational inequality problem, which we solve using the finite element method. Moreover, we prove the convergence of the approximate solution.
Spline interpolation at knot averages on a two-sided geometric mesh
Authors: Marsden M. J.
Published: June 2010

For splines of degree

k
⩾
1

k \geqslant 1

with knots

−

t
i

=

t

2
m
+
1
−
i

=
1
+
q
+

q
2

+
⋯
+

q

m
−
i

- {t_i} = {t_{2m + 1 - i}} = 1 + q + {q^2} + \cdots + {q^{m - i}}

,

i
=
1
,
…
,
m

i = 1, \ldots ,m

, where

0
>
q
>
1

0 > q > 1

, it is shown that spline interpolation to continuous functions at nodes

τ
i

=

Σ
1
k

w
j

t

i
+
j

{\tau _i} = \Sigma _1^k{w_j}{t_{i + j}}

,

i
=
1
,
…
,
n
=
2
m
−
k
−
1

i = 1, \ldots ,n = 2m - k - 1

, has operator norm

‖
P
‖

\left \| P \right \|

which is bounded independently of q and m as q tends to zero if and only if

(
1
−

w
1

)
k

>

1
2

{(1 - {w_1})^k} > \frac {1}{2}

,

(
1
−

w
k

)
k

>

1
2

{(1 - {w_k})^k} > \frac {1}{2}

, and

w
j

>
0

{w_j} > 0

,

j
=
1
,
…
,
k

j = 1, \ldots ,k

. The choice of nodes

τ
i

=

Σ
0

k
+
1

w
j

t

i
+
j

{\tau _i} = \Sigma _0^{k + 1}{w_j}{t_{i + j}}

and the growth rate of

‖
P
‖

\left \| P \right \|

with k are also discussed.
Integrals of Jacobi functions
Authors: Kalla Shyam L. , Conde Salvador , Luke Yudell L.
Published: June 2010

In this paper, we study
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript negative 1 Superscript 1 Baseline left-parenthesis 1 minus x right-parenthesis Superscript a Baseline left-parenthesis 1 plus x right-parenthesis Superscript b Baseline upper P Subscript nu Superscript left-parenthesis alpha comma beta right-parenthesis Baseline left-parenthesis x right-parenthesis d x">

∫

−
1

1

(
1
−
x

)
a

(
1
+
x

)
b

P
ν

(
α
,
β
)

(
x
)

d
x

\smallint _{ - 1}^1{(1 - x)^a}{(1 + x)^b}P_\nu ^{(\alpha ,\beta )}(x)\,dx

and its partial derivatives with respect to a and b, where

P
ν

(
α
,
β
)

(
x
)

P_\nu ^{(\alpha ,\beta )}(x)

is the Jacobi function. Our expressions generalize the work of Blue, Gautschi and Gatteschi. The results are useful to derive integration formulas for integrands with algebraic and logarithmic singularities.
Evaluation of integrals of Howland type involving a Bessel function
Authors: Ling Chih Bing , Wu Ming Jing
Published: June 2010

This paper presents a method of evaluation of four integrals of Howland type, which involve a Bessel function in the integrands. With the aid of tabulated values, they are evaluated to 10D. Two of the four Howland integrals needed in the evaluation are evaluated anew to 20D in order to provide adequate accuracy.
A lower bound for odd triperfects
Authors: Beck Walter E. , Najar Rudolph M.
Published: June 2010

If n is odd and

σ
(
n
)
=
3
n

\sigma (n) = 3n

, then n exceeds

10

50

{10^{50}}

.
On effective computation of fundamental units. I
Authors: Pohst Michael , Zassenhaus Hans
Published: June 2010

The new method for efficient computation of the fundamental units of an algebraic number field developed by the authors in an earlier paper is considerably improved with respect to (Section 1) utilization to best advantage of the element of choice inherent in the method and the mastery of the linear programming techniques involved, (Section 2) ideal factorization, and (Section 3) the determination of sharper upper bounds for the index of

U
ε

{U_\varepsilon }

in

U
F

{U_F}

.
A noniterative method for the generation of orthogonal coordinates in doubly-connected regions
Authors: Warsi Z. U. A. , Thompson J. F.
Published: June 2010

In this paper a noniterative method for the numerical generation of orthogonal curvilinear coordinates for plane annular regions between two arbitrary smooth closed curves has been developed. The basic generating equation is the Gaussian equation for a Euclidean space under the constraint of orthogonality. The resulting equation has been solved analytically for the case of isothermic coordinates. A coordinate transformation then yields nonisothermic coordinates in which the outgoing coordinate can be distributed in any desired manner. The method has been applied in many cases and these test results demonstrate that the proposed method can be readily applied in a variety of problems.
Time discretization in the backward solution of parabolic equations. I
Authors: Eldén Lars
Published: June 2010

The problem of solving a parabolic partial differential equation backwards in time by a method related to the Tikhonov-Phillips regularization method is considered. Time discretizations based on Padé approximations of the exponential function are studied, and a priori estimates of the step length are given, which guarantee an almost optimal error bound. The computational efficiency of different discretizations is discussed. Some numerical examples are given. In Part II of this paper we study the backward beam method, and the same error estimates are obtained. A new scheme for time descretization based on Padé approximation is discussed.
Mesh modification for evolution equations
Authors: Dupont Todd
Published: June 2010

Finite element methods for which the underlying function spaces change with time are studied. The error estimates produced are all in norms that are very naturally associated with the problems. In some cases the Galerkin solution error can be seen to be quasi-optimal. K. Miller’s moving finite element method is studied in one space dimension; convergence is proved for the case of smooth solutions of parabolic problems. Most, but not all, of the analysis is done on linear problems. Although second order parabolic equations are emphasized, there is also some work on first order hyperbolic and Sobolev equations.
Computation of bounds for the positive eigenvector of a nonnegative irreducible matrix by monotone iteration
Authors: Bunse W. , Bunse-Gerstner A.
Published: June 2010

A method for the computation of iterative bounds for the positive eigenvector of a nonnegative irreducible matrix is presented. It is based on the P-boundedness of the corresponding fixed point operator. Admissible initial bounds can be obtained by conditional preiteration.
Strong primality tests that are not sufficient
Authors: Adams William , Shanks Daniel
Published: June 2010

A detailed investigation is given of the possible use of cubic recurrences in primality tests. No attempt is made in this abstract to cover all of the many topics examined in the paper. Define a doubly infinite set of sequencesA(n)A(n)by\[A(n+3)=rA(n+2)−sA(n+1)+A(n)A(n + 3) = rA(n + 2) - sA(n + 1) + A(n)\]withA(−1)=sA( - 1) = s,A(0)=3A(0) = 3, andA(1)=rA(1) = r. Ifnis prime,A(n)≡A(1)(modn)A(n) \equiv A(1)\;\pmod n. Perrin asked if any composite satisfies this congruence ifr=0r = 0,s=−1s = - 1. The answer is yes, and our first example leads us to strengthen the condition by introducing the "signature" ofn:\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A left-parenthesis negative n minus 1 right-parenthesis comma upper A left-parenthesis negative n right-parenthesis comma upper A left-parenthesis negative n plus 1 right-parenthesis comma upper A left-parenthesis n minus 1 right-parenthesis comma upper A left-parenthesis n right-parenthesis comma upper A left-parenthesis n plus 1 right-parenthesis">A(−n−1),A(−n),A(−n+1),A(n−1),A(n),A(n+1)A( - n - 1),A( - n),A( - n + 1),A(n - 1),A(n),A(n + 1)\]modn\bmod n. Primes have three types of signatures depending on how they split in the cubic field generated byx3−rx2+sx−1=0{x^3} - r{x^2} + sx - 1 = 0. Composites with "acceptable" signatures do exist but are very rare. TheS-type signature, which corresponds to the completely split primes, has a very special role, and it may even be thatIandQtype composites do not occur in Perrin’s sequence even though theIandQprimes comprise5/65/6ths of all primes.A(n)(modn)A(n)\;\pmod nis easily computable inO(log⁡n)O(\log n)operations. The paper closes with ap-adic analysis. This powerful tool sets the stage for our [12] which will be Part II of the paper.
On optimal integration methods for Volterra integral equations of the first kind
Authors: Gladwin C. J.
Published: June 2010

Families of methods depending on free parameters are constructed for the solution of nonsingular Volterra integral equations of the first kind in [5]. These parameters are restricted to certain regions in order that a certain polynomial satisfies both a stability and a consistency condition. In this note an optimal choice of the free parameters is outlined in order that the

l
2

{l_2}

-norm of the roots of the polynomial is minimized.
On the uniqueness of best 𝐿₂[0,1] approximation by piecewise polynomials with variable breakpoints
Authors: Chow Jeff
Published: June 2010

In this paper a sufficient condition for the uniqueness of best

L
2

[
0
,
1
]

{L_2}[0,1]

approximation by piecewise polynomials of order k with variable breakpoints is generalized from that of order 2. Other extensions included here are nonuniqueness and eventual uniqueness results.
Zeros of the Hankel function of real order and of its derivative
Authors: Cruz Andrés , Sesma Javier
Published: June 2010

The trajectories followed in the complex plane by all the zeros of the Hankel function and those of its derivative, when the order varies continuously along real values, are discussed.
Computing in permutation and matrix groups. I. Normal closure, commutator subgroups, series
Authors: Butler Gregory , Cannon John J.
Published: June 2010

This paper is the first in a series which discusses computation in permutation and matrix groups of very large order. The fundamental concepts are defined, and some algorithms which perform elementary operations are presented. Algorithms to compute normal closures, commutator subgroups, derived series, lower central series, and upper central series are presented.
On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions
Authors: Schatz A. H. , Wahlbin L. B.
Published: June 2010

Second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered. The behavior of the ordinary Galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed.
Sharp error estimates for a finite element-penalty approach to a class of regulator problems
Authors: Chen Goong , Mills Wendell H. , Sun Shun Hua , Yost David A.
Published: June 2010

Quadratic cost optimal controls can be solved by penalizing the governing linear differential equation [2], [9]. In this paper, we study the numerical analysis of this approach using finite elements. We formulate the geometric condition (H) which requires that pairs of certain related finite-dimensional approximation spaces form “angles” which are bounded away from the “

180
∘

180^\circ

angle”. Under condition (H), we prove that the penalty parameter

ε
\varepsilon

and the discretization parameter h are independent in the error bounds, thereby giving sharp asymptotic error estimates. This condition (H) is shown to be also a necessary condition for such independence. Examples and numerical evidence are also provided.
Quadratic pencils and least-squares piecewise-polynomial approximation
Authors: Mityagin Boris
Published: June 2010

For a partition

ξ
=
(
0
=

ξ
0

>

ξ
1

>
⋯
>

ξ
n

>

ξ

n
+
1

=
1
)

\xi = (0 = {\xi _0} > {\xi _1} > \cdots > {\xi _n} > {\xi _{n + 1}} = 1)

of the unit interval,

S
ξ

k
m

S_\xi ^{km}

,

k
>
m

k > m

, denotes the space of piecewise-polynomials of order k and of smoothness

m
−
1

m - 1

; this space can be represented as the graph of the appropriate linear operator between two finite-dimensional Hilbert spaces. It gives an approach to the C. de Boor problem, 1972, on uniform boundedness (with respect to

ξ
\xi

) in the

L
∞

{L_\infty }

-norm of the orthogonal projections onto

S
ξ

k
m

S_\xi ^{km}

, and we give the detailed analysis of a quadratic pencil (matrix-valued polynomial of the second degree) which appears in the case of a geometric mesh

ξ
\xi

if

2
m
⩽
k

2m \leqslant k

. The explicit calculations and estimates of zeros of the "characteristic" polynomial show that in the case

S

ξ
(
x
)

63

S_{\xi (x)}^{63}

,

ξ
(
x
)

\xi (x)

me geometric mesh with the parameter x,

0
>
x
>
∞

0 > x > \infty

, the orthogonal projectors are uniformly bounded.
On the convergence behavior of continued fractions with real elements
Authors: Gautschi Walter
Published: June 2010

We define the notion of transient (geometric) convergence rate for infinite series and continued fractions. For a class of continued fractions with real elements we prove a monotonicity property for such convergence rates which helps explain the effectiveness of certain continued fractions known to converge "only" sublinearly. This is illustrated in the case of Legendre’s continued fraction for the incomplete gamma function.
Divisors of Mersenne numbers
Authors: Wagstaff Samuel S.
Published: June 2010

We add to the heuristic and empirical evidence for a conjecture of Gillies about the distribution of the prime divisors of Mersenne numbers. We list some large prime divisors of Mersenne numbersMp{M_p}in the range17000>p>10517000 > p > {10^5}.
Sketch of a proof that an odd perfect number relatively prime to 3 has at least eleven prime factors
Authors: Hagis Peter
Published: June 2010

An argument is outlined which demonstates that every odd perfect number which is not divisible by 3 has at least eleven distinct prime factors.
Twenty-fourth power residue difference sets
Authors: Evans Ronald J.
Published: June 2010

It is proved that if p is a

prime

≡
1

(
mod

2
)
4

{\text {prime}} \equiv 1\;\pmod 24

such that either 2 is a cubic residue or 3 is a quartic residue

(
mod

p
)

\pmod p

, then the twenty-fourth powers

(
mod

p
)

\pmod p

do not form a difference set or a modified difference set.
A performance analysis of a simple prime-testing algorithm
Authors: Wunderlich M. C.
Published: June 2010

This paper gives an empirical performance analysis of a prime-proving program designed and implemented by the author and J. L. Selfridge in 1974. The algorithm has been commonly referred to as the "down algorithm" because of its recursive characteristics. It is shown, among other things, that of the 2270 primes tested, 94
An extension of Ortiz’ recursive formulation of the tau method to certain linear systems of ordinary differential equations
Authors: Crisci M. R. , Russo E.
Published: June 2010

Ortiz’ step-by-step recursive formulation of the Lanczos tau method is extended to the numerical solution of linear systems of differential equations with polynomial coefficients. Numerical comparisons are made with Gear’s and Enright’s methods.
Unigrid for multigrid simulation
Authors: McCormick S. F. , Ruge J. W.
Published: June 2010

This paper develops unigrid which is an algorithm that results from interpreting multigrid as it directly affects the fine grid approximation. Unigrid is theoretically equivalent to multigrid under certain assumptions, yet has different computational characteristics. Though it is generally less efficient, unigrid is very useful as a software tool for testing the feasibility of applying multigrid to a given problem. This is illustrated with several numerical examples.
The quotient-difference algorithm and the Padé table: an alternative form and a general continued fraction
Authors: McCabe J. H.
Published: June 2010

The quotient-difference algorithm is applied to a given power series in a modified way, and various continued fractions provided by the algorithm are described in terms of their relationships with the Padé table for the power series. In particular a general continued fraction whose convergents form any chosen combination of horizontal or vertical connected sequences of Padé approximants is introduced.
On the largest zeroes of orthogonal polynomials for certain weights
Authors: Lubinsky D. S. , Sharif A.
Published: June 2010

The asymptotic growth of the largest zero of the orthogonal polynomials for the weights
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W left-parenthesis x right-parenthesis equals StartAbsoluteValue x EndAbsoluteValue Superscript b Baseline exp left-parenthesis minus k StartAbsoluteValue log EndAbsoluteValue x StartAbsoluteValue EndAbsoluteValue Superscript c Baseline right-parenthesis">

W
(
x
)
=

|

x

|

b

exp
⁡
(
−
k

|

log
⁡

|

x

|

|

c

)

W(x) = |x{|^b}\exp ( - k|\log |x|{|^c})

is investigated.
An application of matrices over finite fields to algebraic number theory
Authors: Gerth Frank
Published: June 2010

This paper ultilizes properties of matrices over finite fields to obtain information about the rank of the p-class group of certain algebraic number fields.
The use of positive matrices for the analysis of the large time behavior of the numerical solution of reaction-diffusion systems
Authors: Galeone Luciano
Published: June 2010

In this paper we study the numerical solution of nonlinear reaction-diffusion systems with homogeneous Neumann boundary conditions, via the known

θ
\theta

-method. We show that if conditions for the positivity of solutions are imposed, then the study of the asymptotic behavior of the numerical solution can be done by means of the theory of stochastic matrices. In this way it is shown that the numerical solution reproduces the asymptotic behavior of the corresponding theoretical one. In particular, we obtain the decay of the solution to its mean value. An analysis of the asymptotic stability of the equilibrium points and the convergence of the numerical scheme is given based on the use of M-matrices. Finally we consider the case in which the nonlinear term satisfies a condition of quasimonotonicity.
Some extensions of W. Gautschi’s inequalities for the gamma function
Authors: Kershaw D.
Published: June 2010

It has been shown by W. Gautschi that if

0
>
s
>
1

0 > s > 1

, then for

x
⩾
1

x \geqslant 1

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript 1 minus s Baseline greater-than StartFraction normal upper Gamma left-parenthesis x plus 1 right-parenthesis Over normal upper Gamma left-parenthesis x plus s right-parenthesis EndFraction greater-than exp left-bracket left-parenthesis 1 minus s right-parenthesis psi left-parenthesis x plus 1 right-parenthesis right-bracket period">

x

1
−
s

>

Γ
(
x
+
1
)

Γ
(
x
+
s
)

>
exp
⁡
[
(
1
−
s
)
ψ
(
x
+
1
)
]
.

{x^{1 - s}} > \frac {{\Gamma (x + 1)}}{{\Gamma (x + s)}} > \exp [(1 - s)\psi (x + 1)].

\]
The following closer bounds are proved:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="exp left-bracket left-parenthesis 1 minus s right-parenthesis psi left-parenthesis x plus s Superscript 1 slash 2 Baseline right-parenthesis right-bracket greater-than StartFraction normal upper Gamma left-parenthesis x plus 1 right-parenthesis Over normal upper Gamma left-parenthesis x plus s right-parenthesis EndFraction greater-than exp left-bracket left-parenthesis 1 minus s right-parenthesis psi left-parenthesis x plus StartFraction s plus 1 Over 2 EndFraction right-parenthesis right-bracket">

exp
⁡
[
(
1
−
s
)
ψ
(
x
+

s

1

/

2

)
]
>

Γ
(
x
+
1
)

Γ
(
x
+
s
)

>
exp
⁡

[

(
1
−
s
)
ψ

(

x
+

s
+
1

2

)

]

\exp [(1 - s)\psi (x + {s^{1/2}})] > \frac {{\Gamma (x + 1)}}{{\Gamma (x + s)}} > \exp \left [ {(1 - s)\psi \left ( {x + \frac {{s + 1}}{2}} \right )} \right ]

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket x plus StartFraction s Over 2 EndFraction right-bracket Superscript 1 minus s Baseline greater-than StartFraction normal upper Gamma left-parenthesis x plus 1 right-parenthesis Over normal upper Gamma left-parenthesis x plus s right-parenthesis EndFraction greater-than left-bracket x minus one half plus left-parenthesis s plus one fourth right-parenthesis Superscript 1 slash 2 Baseline right-bracket Superscript 1 minus s Baseline period">

[

x
+

s
2

]

1
−
s

>

Γ
(
x
+
1
)

Γ
(
x
+
s
)

>

[

x
−

1
2

+

(

s
+

1
4

)

1

/

2

]

1
−
s

.

{\left [ {x + \frac {s}{2}} \right ]^{1 - s}} > \frac {{\Gamma (x + 1)}}{{\Gamma (x + s)}} > {\left [ {x - \frac {1}{2} + {{\left ( {s + \frac {1}{4}} \right )}^{1/2}}} \right ]^{1 - s}}.

\]
These are compared with each other and with inequalities given by T. Erber and J. D. Kečkić and P. M. Vasić.
Eighteen primes in arithmetic progression
Authors: Pritchard Paul A.
Published: June 2010

There is an arithmetic progression of 18 primes, viz.

107928278317
+
k
⋅
9922782870

107928278317 + k \cdot 9922782870

,

k
=
0
,
1
,
…
,
17

k = 0,1, \ldots ,17

.
Anderson-Björck for linear sequences
Authors: King Richard F.
Published: June 2010

The proposed one-point method for finding the limit of a slowly converging linear sequence features an Anderson-Björck extrapolation step that had previously been applied to the Regula Falsi problem. Convergence is of order 1.839 as compared to

2

\sqrt 2

for the well-known Aitken-Steffensen

δ
2

{\delta ^2}

-process, and to 1.618 for another one-point extrapolation procedure of King. There are examples for computing a polynomial’s mutiple root with Newton’s method and for finding a fixed point of a nonlinear function.
A modified method for reconstructing periodic Jacobi matrices
Authors: Boley Daniel , Golub Gene H.
Published: June 2010

In this note, we discuss the reconstruction of periodic Jacobi matrices from spectral data. The method combines ideas and techniques from the algorithms given by Boley and Golub [1], [2], and Ferguson [3], resulting in a numerically stable algorithm applicable to a larger class of problems. The number of initial data items needed for this method equals the number of items in the resulting matrix, namely 2n.
A note on the numerical solution of the wave equation with piecewise smooth coefficients
Authors: Brown David L.
Published: June 2010

The numerical solution of the initial value problem for the wave equation is considered for the case when the equation coefficients are piecewise smooth. This problem models linear wave propagation in a medium in which the properties of the medium change discontinuously at interfaces. Convergent difference approximations can be found that do not require the explicit specification of the boundary conditions at interfaces in the medium and hence are simple to program. Although such difference approximations typically can only be expected to be first-order accurate, the numerical phase velocity has the same accuracy as the difference approximation would if the coefficients in the differential equation were smoooth. This is proved for the one-dimensional case and demonstrated numerically for an example in two space dimensions in which the interface is not aligned with the computational mesh.
A collocation-𝐻⁻¹-Galerkin method for some elliptic equations
Authors: Nakao Mitsuhiro
Published: June 2010

A collocation-

H

−
1

{H^{ - 1}}

-Galerkin method is defined for some elliptic boundary value problems on a rectangle. The method uses tensor products of discontinuous piecewise polynomial spaces and collocation based on Jacobi points with weight function

x
2

(
1
−
x

)
2

{x^2}{(1 - x)^2}

. Optimal order of

L
2

{L^2}

rates of convergence is established for the approximation solution. A numerical example which confirms these results is presented.
On the convergence properties of the quadrilateral elements of Sander and Beckers
Authors: Shi Zhong Ci
Published: June 2010

A class of nonconforming quadrilateral membrane elements, introduced by Sander and Beckers, is discussed in all details. It is proved that two elements that do not pass the patch test may still yield convergence under a suitable condition on mesh subdivisions, whereas one such element is found to be divergent for all mesh subdivisions. All other finite elements in this class that pass the patch test are convergent. The mathematical analysis provides a clear explanation of the convergence behavior appearing in the numerical examples of Sander and Beckers.