bluuebunny/crossref_metadata_2025 huggingface dataset:
Existence of Gauss interpolation formulas for the one-dimensional heat equation
Authors: Barrow David L.
Published: June 2010

Let

C
=
{
(
x
(
s
)
,
t
(
s
)
)
:
a
⩽
s
⩽
b
}

C = \{ (x(s),t(s)):a \leqslant s \leqslant b\}

be a Jordan arc in the x-t plane satisfying
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x left-parenthesis a right-parenthesis comma t left-parenthesis a right-parenthesis right-parenthesis equals left-parenthesis a comma t Subscript asterisk Baseline right-parenthesis comma left-parenthesis x left-parenthesis b right-parenthesis comma t left-parenthesis b right-parenthesis right-parenthesis equals left-parenthesis b comma t Subscript asterisk Baseline right-parenthesis">

(
x
(
a
)
,
t
(
a
)
)
=
(
a
,

t
∗

)
,
(
x
(
b
)
,
t
(
b
)
)
=
(
b
,

t
∗

)

(x(a),t(a)) = (a,{t_ \ast }),(x(b),t(b)) = (b,{t_\ast })

, and

t
(
s
)
>

t
∗

t(s) > {t_\ast }

when

a
>
s
>
b

a > s > b

. Let

a
>

x
∗

>
b

a > {x_\ast } > b

. We prove the existence of Gauss interpolation formulas for C and the point

(

x
∗

,

t
∗

)

({x_\ast },{t_\ast })

, for solutions u of the one-dimensional heat equation,

u
t

=

u

x
x

{u_t} = {u_{xx}}

. Such formulas approximate

u
(

x
∗

,

t
∗

)

u({x_\ast },{t_\ast })

in terms of a linear combination of its values on C. The formulas are characterized by the requirement that they are exact for as many basis functions (the heat polynomials) as possible.
The 𝑇_{+𝑚} translformation
Authors: Streit Roland F.
Published: June 2010

This paper discusses a nonlinear sequence-to-sequence transformation, known as the

T

+
m

{T_{ + m}}

transform, which is used to accelerate the convergence of an infinite series. A brief history of the transform is given; a number of theorems are established which enable one to make effective use of the transform, and several examples are presented to illustrate this effectiveness.
Factorized variable metric methods for unconstrained optimization
Authors: Goldfarb Donald
Published: June 2010

Several efficient methods are given for updating the Cholesky factors of a symmetric positive definite matrix when it is modified by a rank-two correction which maintains symmetry and positive definiteness. These ideas are applied to variable metric (quasi-Newton) methods to produce numerically stable algorithms.
Hybrid difference methods for the initial boundary-value problem for hyperbolic equations
Authors: Oliger Joseph
Published: June 2010

The use of lower order approximations in the neighborhood of boundaries coupled with higher order interior approximations is examined for the mixed initial boundary-value problem for hyperbolic partial differential equations. Uniform error can be maintained using smaller grid intervals with the lower order approximations near the boundaries. Stability results are presented for approximations to the initial boundary-value problem for the model equation

u
t

+
c

u
x

=
0

{u_t} + c{u_x} = 0

which are fourth order in space and second order in time in the interior and second order in both space and time near the boundaries. These results are generalized to a class of methods of this type for hyperbolic systems. Computational results are presented and comparisons are made with other methods.
A note on extended Gaussian quadrature rules
Authors: Monegato Giovanni
Published: June 2010

Extended Gaussian quadrature rules of the type first considered by Kronrod are examined. For a general nonnegative weight function, simple formulas for the computation of the weights are given, together with a condition for the positivity of the weights associated with the new nodes. Examples of nonexistence of these rules are exhibited for the weight functions

(
1
−

x
2

)

λ
−
1

/

2

(1-x^2)^{\lambda - 1/2}

,

e

−

x
2

e^{-x^2}

and

e

−
x

e^{-x}

. Finally, two examples are given of quadrature rules which can be extended repeatedly.
On the condition number of local bases for piecewise cubic polynomials
Authors: Varah J. M.
Published: June 2010

The condition number of the Gram matrix associated with piecewise polynomial finite element bases is discussed in general, and computed explicitly for cubic splines and cubic Hermite polynomials. In the latter case, we discuss the inherent ambiguity in the basis, and find the minimum condition number.
Prime factors of cyclotomic class numbers
Authors: Lehmer D. H.
Published: June 2010

Let p be an odd prime. The "first factor"

h
∗

(
p
)

{h^\ast }(p)

of the class number of the field of pth roots of unity has been the subject of many investigations beginning with Kummer (1861). In the present paper it is shown how the theory of a function introduced by T. A. Pierce (1917) can be used to find the prime factors of

h
∗

(
p
)

{h^\ast }(p)

.
An extension of the applicability of iterated deffered corrections
Authors: Frank Reinhard , Hertling Joerg , Ueberhuber Christoph W.
Published: June 2010

A new way of estimating local discretization errors (based on an idea due to P. E. Zadunaisky) is introduced. If error estimates obtained by this method are used in connection with the general class of iterated deferred correction algorithms, they lead to an extension of the domain of applicability, when compared with the variants used by Fox and Pereyra.
On the 𝑙² convergence of an algorithm for solving finite element equations
Authors: Nicolaides R. A.
Published: June 2010

An iterative method of multiple grid type is proposed for solving general finite element systems. It is proved that the method can produce a solution to the equations in

O
(
N
)

O(N)

arithmetical operations where N is the number of unknowns.
Evaluation of the integral ∫^{∞}₀𝑡^{2^{𝛼}-1}𝐽𝜈(𝜒√(1+𝑡²))/(1+𝑡²)^{𝛼+𝛽-1}𝑑𝑡
Authors: Schmidt Paul W.
Published: June 2010

Methods are developed for evaluating the integral
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript nu Superscript alpha beta Baseline left-parenthesis x right-parenthesis equals integral Subscript 0 Superscript normal infinity Baseline StartFraction t Superscript 2 alpha minus 1 Baseline upper J Subscript nu Baseline left-parenthesis x StartRoot 1 plus t squared EndRoot right-parenthesis Over left-parenthesis 1 plus t squared right-parenthesis Superscript alpha plus beta minus 1 Baseline EndFraction d t comma">

I
ν

α
β

(
x
)
=

∫
0
∞

t

2
α
−
1

J
ν

(
x

1
+

t
2

)

(
1
+

t
2

)

α
+
β
−
1

d
t
,

I_\nu ^{\alpha \beta }(x) = \int _0^\infty {\frac {{{t^{2\alpha - 1}}{J_\nu }(x\sqrt {1 + {t^2}} )}}{{{{(1 + {t^2})}^{\alpha + \beta - 1}}}}dt,}

\]
where

J
ν

(
t
)

{J_\nu }(t)

is the Bessel function of the first kind and order

ν
\nu

,

α
>
0

\alpha > 0

,

β
>
1

/

4

\beta > 1/4

, and

ν
\nu

is real. Only

I
ν

1

/

2
,
1

(
x
)

I_\nu ^{1/2,1}(x)

and

I
ν

α
,
ν

/

2
+
1
−
α

(
x
)

I_\nu ^{\alpha ,\nu /2 + 1 - \alpha }(x)

are included in previously published tables of integrals of Bessel functions. The integrals

I
1

1

/

2
,
1

/

2

(
x
)

I_1^{1/2,1/2}(x)

and

I
2

1

/

2
,
1

(
x
)

I_2^{1/2,1}(x)

are used in a technique developed by I. S. Fedorova for calculating the diameter distribution of long circular cylinders from small-angle x-ray, light, or neutron scattering data. The

I
ν

α
β

(
x
)

I_\nu ^{\alpha \beta }(x)

are shown to be proportional to a G function. From this result, power series expansions and recurrence relations are developed for use in evaluating the

I
ν

α
β

(
x
)

I_\nu ^{\alpha \beta }(x)

. A convenient expression is obtained for the quantity required in Fedorova’s method for computing diameter distributions.
Stability of rounded off inverses under iteration
Authors: Diamond Harold G.
Published: June 2010

Let f be a monotone and strictly convex (or concave) function on a real interval and let g be the inverse function. Let

I
(
x
)
=
x

I(x) = x

. For

ϕ
\phi

a real valued function and N a positive integer let

ϕ
N

(
x
)

{\phi _N}(x)

denote the rounding of

ϕ
(
x
)

\phi (x)

to N significant figures. Let

h
=

g
N

∘

f
N

h = {g_N} \circ {f_N}

, the composition of

f
N

{f_N}

and

g
N

{g_N}

. It is shown that
\[

h
∘
h
∘

I
N

=
h
∘
h
∘
h
∘

I
N

,

h \circ h \circ {I_N} = h \circ h \circ h \circ {I_N},

\]
and that equality can fail for fewer iterations.
Global approximations to solutions of initial value problems
Authors: Kramarz Luis
Published: June 2010

A wide class of implicit one-step methods for the construction of global approximations to solutions of initial value problems is studied. Approximations more general than piecewise polynomials can be constructed to exploit certain characteristics of the differential equation. Error bounds are given for the general class of methods but emphasis is placed on methods based on Hermite interpolation, for which higher rates of convergence are obtained for special choices of interpolation points. Computational examples are presented.
Rational Chebyshev approximations for the Bickley functions 𝐾𝑖_{𝑛}(𝑥)
Authors: Blair J. M. , Edwards C. A. , Johnson J. H.
Published: June 2010

This report presents near-minimax rational approximations for the Bickley functions

K

i
1

(
x
)

K{i_1}(x)

for

x
⩾
0
,
K

i
2

(
x
)

x \geqslant 0, K{i_2}(x)

and

K

i
3

(
x
)

K{i_3}(x)

for

0
⩽
x
⩽
6

0 \leqslant x \leqslant 6

, and

K

i
8

(
x
)

K{i_8}(x)

,

K

i
9

(
x
)

K{i_9}(x)

and

K

i

10

(
x
)

K{i_{10}}(x)

for

x
⩾
6

x \geqslant 6

, with relative errors ranging down to

10

−
23

{10^{ - 23}}

. The approximations, combined with the recurrence relation, yield a stable method of computing

K

i
n

(
x
)

K{i_n}(x)

,

n
=
1
,
2
,
…
,
10

n = 1,2, \ldots ,10

, for the complete range of the argument.
On some families of imaginary quadratic fields
Authors: Diaz y Diaz F.
Published: June 2010

This paper gives a method of obtaining imaginary quadratic fields whose class groups have at least three invariants divisible by 3. Complementary calculations have yielded a large number of imaginary quadratic fields having class groups with four invariants divisible by 3. Some numerical examples, previously unknown, are included.
Finding zeroes of maps: homotopy methods that are constructive with probability one
Authors: Chow Shui Nee , Mallet-Paret John , Yorke James A.
Published: June 2010

We illustrate that most existence theorems using degree theory are in principle relatively constructive. The first one presented here is the Brouwer Fixed Point Theorem. Our method is "constructive with probability one" and can be implemented by computer. Other existence theorems are also proved by the same method. The approach is based on a transversality theorem.
Numerical solution of an exterior Neumann problem using a double layer potential
Authors: Giroire J. , Nédélec J.-C.
Published: June 2010

We give here a variational formulation in

H

1

/

2

(
Γ
)

/

R

{H^{1/2}}(\Gamma )/R

of the exterior Neumann problem for the Laplace operator using a double layer potential. This formulation is then applied to the construction of a finite element method. Optimal error estimates are given.
Simultaneous approximation in scales of Banach spaces
Authors: Bramble James H. , Scott Ridgway
Published: June 2010

The problem of verifying optimal approximation simultaneously in different norms in a Banach scale is reduced to verification of optimal approximation in the highest order norm. The basic tool used is the Banach space interpolation method developed by Lions and Peetre. Applications are given to several problems arising in the theory of finite element methods.
Free subgroups of the free product of cyclic groups
Authors: Stothers W. W.
Published: June 2010

Two kinds of recurrence relation for the number of subgroups of finite index in a free product of finitely many cyclic groups are given. An asymptotic formula is obtained from the first of these relations.
Maximum norm estimates in the finite element method on plane polygonal domains. I
Authors: Schatz A. H. , Wahlbin L. B.
Published: June 2010

The finite element method is considered when applied to a model Dirichlet problem on a plane polygonal domain. Rate of convergence estimates in the maximum norm, up to the boundary, are given locally. The rate of convergence may vary from point to point and is shown to depend on the local smoothness of the solution and on a possible pollution effect. In one of the applications given, a method is proposed for calculating the first few coefficients (stress intensity factors) in an expansion of the solution in singular functions at a corner from the finite element solution. In a second application the location of the maximum error is determined. A rather general class of non-quasi-uniform meshes is allowed in our present investigations. In a subsequent paper, Part 2 of this work, we shall consider meshes that are refined in a systematic fashion near a corner and derive sharper results for that case.
Error analysis of finite difference schemes applied to hyperbolic initial-boundary value problems
Authors: Sköllermo Gunilla
Published: June 2010

A technique for the complete error analysis of finite difference schemes for hyperbolic initial boundary value problems is developed. The error analysis is split into subproblems so that different boundary approximations or different initial approximations easily can be compared for a given interior scheme. The steps of the theoretical analysis are demonstrated on the leapfrog scheme for a simple model equation. The technique is applied to several choices of initial approximations and boundary conditions for leapfrog with second and fourth order accuracy in space. A comparison with two implicit schemes is also made. The theoretical error estimates are shown to agree very well with computational results.
A collocation-Galerkin method for Poisson’s equation on rectangular regions
Authors: Díaz Julio César
Published: June 2010

A collocation-Galerkin method is defined for Poisson’s equation on the unit square, using tensor products of continuous piecewise polynomials. Optimal

L
2

{L^2}

and

H
0
1

H_0^1

orders of convergence are established. This procedure requires fewer quadratures than the corresponding Galerkin procedure.
Computer solution and perturbation analysis of generalized linear least squares problems
Authors: Paige C. C.
Published: June 2010

A new formulation of the generalized linear least squares problem is given. This is based on some ideas in estimation and allows complete generality in that there are no restrictions on the matrices involved. The formulation leads directly to a numerical algorithm involving orthogonal decompositions for solving the problem. A perturbation analysis of the problem is obtained by using the new formulation and some of the decompositions used in the solution. A rounding error analysis is given to show that the algorithm is numerically stable.
A bench mark experiment for minimization algorithms
Authors: Lyness J. N.
Published: June 2010

In this paper we suggest a single bench mark problem family for use in evaluating unconstrained minimization algorithms or routines. In essence, this problem consists of measuring, for each algorithm, the rate at which it descends an unlimited helical valley. The periodic nature of the problem allows us to exploit affine scale invariance properties of the algorithm. As a result, the capacity of the algorithm to minimize a wide range of helical valleys of various scales may be summarized by calculating a single valued function

g
Q

(

X
1

)

{g_Q}({X_1})

. The measurement of this function is not difficult, and the result provides information of a simple, general character for use in decisions about choice of algorithm.
A fast Cauchy-Riemann solver
Authors: Ghil Michael , Balgovind Ramesh
Published: June 2010

We present a solution algorithm for a second-order accurate discrete form of the inhomogeneous Cauchy-Riemann equations. The algorithm is comparable in speed and storage requirements with fast Poisson solvers. Error estimates for the discrete approximation of sufficiently smooth solutions of the problem are established; numerical results indicate that second-order accuracy obtains even for solutions which do not have the required smoothness. Different combinations of boundary conditions are considered and suitable modifications of the solution algorithm are described and implemented.
On stable calculation of linear functionals
Authors: Gustafson Sven-Ȧke
Published: June 2010

In this paper we discuss the recurrent task of evaluating a linear functional defined by (generally infinitely many) linear constraints. We develop a theory for the stability of this problem and suggest a regularization procedure, based on orthogonal expansions. Simple and efficient computational schemes for evaluating the functional numerically are given.
Quadrature rule methods for Volterra integral equations of the first kind
Authors: Gladwin Charles J.
Published: June 2010

A new class of quadrature rule methods for solving nonsingular Volterra integral equations of the first kind are introduced; these methods are based on an appropriate modification of the higher-order Newton-Gregory methods which are known to be divergent. Methods up to order six are constructed explicitly and illustrated with numerical examples.
Monte Carlo functional expansion calculation of free molecular flows
Authors: Petschek A. G. , Williamson R. E. , Krauser W. J. , White P. C.
Published: June 2010

A Monte Carlo technique for Knudsen flow, light transmission in tubes, the flow of cold neutrons, phonon transmission, heat transfer calculations and other problems involving free streaming in cylindrical enclosures is described. The coefficients of a functional expansion of the view factors are estimated by Monte Carlo, are then perturbed in a least squares minimum sense to satisfy some consistency conditions, and finally are used to calculate the collision density on the boundary and the transmission of the enclosure. Without importance sampling or other variance reduction techniques, the calculational work required for a given accuracy is reduced by a factor frequently near 10 when compared to a simple calculation in which particles are followed from entrance to exit.
Tables for the Gaussian computation of ∫₀^{∞}𝑥^{𝛼}𝑒^{-𝑥}𝑓(𝑥)𝑑𝑥 for values of 𝛼 varying continuously between -1 and +1
Authors: Lambin P. , Vigneron J. P.
Published: June 2010

Tables of coefficients of Chebyshev expansions for computing to 11 places the abscissas and weight factors of the 12-point generalized Gauss-Laguerre quadrature formula are presented.
Capacitance matrix methods for the Helmholtz equation on general three-dimensional regions
Authors: O’Leary Dianne P. , Widlund Olof
Published: June 2010

Capacitance matrix methods provide techniques for extending the use of fast Poisson solvers to arbitrary bounded regions. These techniques are further studied and developed with a focus on the three-dimensional case. A discrete analogue of classical potential theory is used as a guide in the design of rapidly convergent iterative methods. Algorithmic and programming aspects of the methods are also explored in detail. Several conjugate gradient methods are discussed for the solution of the capacitance matrix equation. A fast Poisson solver is developed which is numerically very stable even for indefinite Helmholtz equations. Variants thereof allow substantial savings in primary storage for problems on very fine meshes. Numerical results show that accurate solutions can be obtained at a cost which is proportional to that of the fast Helmholtz solver in use.
On the numerical evaluation of a particular singular two-dimensional integral
Authors: Monegato G. , Lyness J. N.
Published: June 2010

We investigate the possibility of using two-dimensional Romberg integration to approximate integrals, over the square

0
⩽
x

0 \leqslant x

,

y
⩽
1

y \leqslant 1

, of integrand functions of the form

g
(
x
,
y
)

/

(
x
−
y
)

g(x,y)/(x - y)

where

g
(
x
,
y
)

g(x,y)

is, for example, analytic in x and y. We show that Romberg integration may be properly justified so long as it is based on a diagonally symmetric rule and function values on the singular diagonal, if required, are defined in a particular way. We also investigate the consequences of ignoring fhese function values (i.e. setting them to zero) in the context of such a calculation. We also derive the asymptotic expansion on which extrapolation methods can be based when

g
(
x
,
y
)

g(x,y)

has a point singularity of a specified nature at the origin.
A weak discrete maximum principle and stability of the finite element method in 𝐿_{∞} on plane polygonal domains. I
Authors: Schatz Alfred H.
Published: June 2010

Let

Ω
\Omega

be a polygonal domain in the plane and

S
r
h

(
Ω
)

S_r^h(\Omega )

denote the finite element space of continuous piecewise polynomials of degree

⩽
r
−
1

(
r
⩾
2
)

\leqslant r - 1\;(r \geqslant 2)

defined on a quasi-uniform triangulation of

Ω
\Omega

(with triangles roughly of size h). It is shown that if

u
h

∈

S
r
h

(
Ω
)

{u_h} \in S_r^h(\Omega )

is a "discrete harmonic function" then an a priori estimate (a weak maximum principle) of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar u Subscript h Baseline double-vertical-bar Subscript upper L Sub Subscript normal infinity Subscript left-parenthesis normal upper Omega right-parenthesis Baseline less-than-or-slanted-equals upper C double-vertical-bar u Subscript h Baseline double-vertical-bar Subscript upper L Sub Subscript normal infinity Subscript left-parenthesis partial-differential normal upper Omega right-parenthesis">

‖

u
h

‖

L
∞

(
Ω
)

⩽
C

‖

u
h

‖

L
∞

(
∂
Ω
)

{\left \| {{u_h}} \right \|_{{L_\infty }(\Omega )}} \leqslant C{\left \| {{u_h}} \right \|_{{L_\infty }(\partial \Omega )}}

\]
holds. Now let u be a continuous function on

Ω
¯

\bar \Omega

and

u
h

{u_h}

be the usual finite element projection of u into

S
r
h

(
Ω
)

S_r^h(\Omega )

(with

u
h

{u_h}

interpolating u at the boundary nodes). It is shown that for any

χ
∈

S
r
h

(
Ω
)

\chi \in S_r^h(\Omega )

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar u minus u Subscript h Baseline double-vertical-bar Subscript upper L Sub Subscript normal infinity Subscript left-parenthesis normal upper Omega right-parenthesis Baseline less-than-or-slanted-equals c left-parenthesis ln StartFraction 1 Over h EndFraction right-parenthesis Superscript r overbar Baseline double-vertical-bar u minus chi double-vertical-bar Subscript upper L Sub Subscript normal infinity Subscript left-parenthesis normal upper Omega right-parenthesis Baseline comma where r overbar equals StartLayout Enlarged left-brace 1st Row 1st Column 1 2nd Column a m p semicolon if r equals 2 comma 2nd Row 1st Column 0 2nd Column a m p semicolon if r greater-than-or-slanted-equals 3 period EndLayout">

‖

u
−

u
h

‖

L
∞

(
Ω
)

⩽
c

(

ln
⁡

1
h

)

r
¯

‖

u
−
χ

‖

L
∞

(
Ω
)

,

where

r
¯

=

{

1

a
m
p
;

if

r
=
2
,

0

a
m
p
;

if

r
⩾
3.

{\left \| {u - {u_h}} \right \|_{{L_\infty }(\Omega )}} \leqslant c{\left ( {\ln \frac {1}{h}} \right )^{\bar r}}{\left \| {u - \chi } \right \|_{{L_\infty }(\Omega )}},\quad {\text {where}}\;\bar r = \left \{ {\begin {array}{*{20}{c}} 1 & {{\text {if}}\;r = 2,} \\ 0 & {{\text {if}}\;r \geqslant 3.} \\ \end {array} } \right .

\]
This says that (modulo a logarithm for

r
=
2

r = 2

) the finite element method is bounded in

L
∞

{L_\infty }

on plane polygonal domains.
Computational complexity of one-step methods for systems of differential equations
Authors: Werschulz Arthur G.
Published: June 2010

The problem is to calculate an approximate solution of an initial value problem for an autonomous system of N ordinary differential equations. Using fast power series techniques, we exhibit an algorithm for the pth-order Taylor series method requiring only

O
(

p
N

ln
⁡
p
)

O({p^N}\ln p)

arithmetic operations per step as

p
→
+
∞

p \to + \infty

. (Moreover, we show that any such algorithm requires at least

O
(

p
N

)

O({p^N})

operations per step.) We compute the order which minimizes the complexity bounds for Taylor series and linear Runge-Kutta methods and show that in all cases this optimal order increases as the error criterion

ε
\varepsilon

decreases, tending to infinity as

ε
\varepsilon

tends to zero. Finally, we show that if certain derivatives are easy to evaluate, then Taylor series methods are asymptotically better than linear Runge-Kutta methods for problems of small dimension N.
Quadrature error functional expansions for the simplex when the integrand function has singularities at vertices
Authors: Lyness J. N. , Monegato G.
Published: June 2010

In a previous paper by Lyness, an asymptotic expansion for the quadrature error functional for the n-dimensional simplex, valid when the integrand is an analytic function, was given. In this paper we develop the analogue of this result to cover the cases in which the integrand function has singularities of a specified nature at one or more vertices of the simplex.
On the primality of 𝑘!+1 and 2∗3 ∗5∗⋯∗𝑝+1
Authors: Templer Mark
Published: June 2010

In this paper the results of an investigation of

k
!
+
1

k! + 1

and

2
∗
3
∗
5
∗
⋯
∗
p
+
1

2 \ast 3 \ast 5 \ast \cdots \ast p + 1

are reported. Values of

k
=
1
(
1
)
230

k = 1(1)230

and

2
⩽
p
⩽
1031

2 \leqslant p \leqslant 1031

were investigated. Five new primes were discovered.
Residue classes of the partition function
Authors: MacLean D. W.
Published: June 2010

The results of computations of

p
(
n
)

(
mod

p
k

)

p(n)\;\pmod {p^k}

for primes

p
>
100

p > 100

are summarized in frequency tables of residue classes.
A note on Olver’s algorithm for the solution of second-order linear difference equations
Authors: Cash J. R.
Published: June 2010

An extension of Olver’s algorithm is described. This allows the solution of certain linear difference equations which are in a sense ill-conditioned for solution by Olver’s method. The new algorithm is illustrated by considering a particular problem.
Recurrence relations for the coefficients in Chebyshev series solutions of ordinary differential equations
Authors: Horner T. S.
Published: June 2010

Systematic methods are presented for obtaining recurrence relations for the coefficients in Chebyshev series solutions of linear differential equations, of first to fourth order, with polynomial coefficients. Polynomial approximations to certain rational functions are also discussed.
Optimal partitioning of Newton’s method for calculating roots
Authors: Meinardus Günter , Taylor G. D.
Published: June 2010

In this paper, an algorithm is given for calculating roots via Newton’s method initialized with a piecewise best starting approximation. The piecewise best starting approximation corresponds to an optimal partitioning of the interval of the domain of Newton’s method. Explicit formulas are given when piecewise linear polynomials are used for the best starting approximations. Specific tables are given for square roots, cube roots and reciprocal square roots.
On an accelerated overrelaxation iterative method for linear systems with strictly diagonally dominant matrix
Authors: Martins M. Madalena
Published: June 2010

We consider a linear system

A
x
=
b

Ax = b

of n simultaneous equations, where A is a strictly diagonally dominant matrix. We get bounds for the spectral radius of the matrix

L

r
,
ω

{L_{r,\omega }}

, which is accociated with the Accelerated Overrelaxation iterative method (AOR). Sufficient conditions for the convergence of that method will be given, which improve the results of Theorem 3, Section 4 of [2], applied to this type of matrices.
An efficient one-point extrapolation method for linear convergence
Authors: King Richard F.
Published: June 2010

For iteration sequences otherwise converging linearly, the proposed one-point extrapolation method attains a convergence rate and efficiency of 1.618. This is accomplished by retaining an estimate of the linear coefficient from the previous step and using the estimate to extrapolate. For linear convergence problems, the classical Aitken-Steffensen

δ
2

{\delta ^2}

-process has an efficiency of just

2

\sqrt 2

, while a recently proposed fourth-order method reaches an efficiency of 1.587. Not only is the method presented here more efficient, but it is also quite straightforward. Examples given are for Newton’s method in finding multiple polynomial roots and for locating a fixed point of a nonlinear function.
An efficient algorithmic solution of the Diophantine equation 𝑢²+5𝑣²=𝑚
Authors: Wilker Peter
Published: June 2010

The determination of irreducible elements of the domain

Z
[

−
5

]

Z[\sqrt { - 5} ]

requires the solution of the Diophantine equation

u
2

+
5

v
2

=
m

{u^2} + 5{v^2} = m

, where m represents certain primes or products of two primes. An algorithm of order

log
⁡
m

\log m

is given for the solution of the equation.
Approximation of the spectrum of closed operators: the determination of normal modes of a rotating basin
Authors: Descloux Jean , Luskin Mitchell , Rappaz Jacques
Published: June 2010

This paper gives a theory of spectral approximation for closed operators in Banach spaces. The perturbation theory developed in this paper is applied to the study of a finite element procedure for approximating the spectral properties of a differential system modeling a fluid in a rotating basin.
A rapidly convergent series for computing 𝜓(𝑧) and its derivatives
Authors: McCullagh Peter
Published: June 2010

We derive a series expansion for

ψ
(
z
)

\psi (z)

in which the terms of the expansion are simple rational functions of z. From a computational viewpoint, the new series is of interest in that it converges for all z not necessarily real valued, and is particularly rapid for values of z near the origin. From a mathematical viewpoint the series is of interest in that, although

ψ
(
z
)

\psi (z)

has poles at the negative integers and zero, the series is uniformly convergent in any finite interval

a
>
Re
⁡
(
z
)
>
b

a > \operatorname {Re} (z) > b

.
Some large primes and amicable numbers
Authors: Borho W.
Published: June 2010

Some new large primes of the form

3
⋅

2
n

−
1

3 \cdot {2^n} - 1

and

9
⋅

2
n

−
1

9 \cdot {2^n} - 1

, related to amicable numbers, are given. Two new large amicable number pairs are found by the method of so-called "Thabit rules".
On a fourth order accurate implicit finite difference scheme for hyperbolic conservation laws. I. Nonstiff strongly dynamic problems
Authors: Harten Amiram , Tal-Ezer Hillel
Published: June 2010

An implicit finite difference method of fourth order accuracy (in space and time) is introduced for the numerical solution of one-dimensional systems of hyperbolic conservation laws. The basic form of this method is a straightforward generalization of the Crank-Nicholson scheme: it is a two-level scheme which is unconditionally stable and nondissipative. The scheme is compact, i.e., it uses only 3 mesh points at level t and 3 mesh points at level

t
+
Δ
t

t + \Delta t

. In this paper, the first in a series, we present a dissipative version of the basic method which is conditionally stable under the CFL (Courant-Friedrichs-Lewy) condition. This version is particularly useful for numerical solution of problems with strong but nonstiff dynamic features, where the CFL restriction is reasonable on accuracy grounds. Numerical results are presented to illustrate properties of the proposed scheme.
Calculation of the Taylor series expansion coefficients of the Jacobian elliptic function 𝑠𝑛(𝑥,𝑘)
Authors: Wrigge Staffan
Published: June 2010

The Taylor series expansion coefficients of the Jacobian elliptic function

sn

(
x
,
k
)

{\text {sn}}(x,k)

and its power

sn

2

(
x
,
k
)

{\text {sn}}^2(x,k)

are studied. Recurrence formulae are given, and tables of the coefficients constructed. Using Lagrange’s inversion formula, these coefficients can be expressed in terms of Legendre polynomials.
The finite element method with Lagrange multipliers for domains with corners
Authors: Pitkäranta Juhani
Published: June 2010

We study the convergence of the finite element method with Lagrange multipliers for approximately solving the Dirichlet problem for a second-order elliptic equation in a plane domain with piecewise smooth boundary. Assuming mesh refinements around the corners, we construct families of boundary subspaces that are compatible with triangular Lagrange elements in the interior, and we carry out the error analysis of the resulting approximations in weighted Sobolev spaces.
Rate of convergence of discretization in Chebyshev approximation
Authors: Dunham C. B. , Williams Jack
Published: June 2010

The paper treats, in a particularly simple fashion, the practical problem of the rate of convergence of discretization in real and complex Chebyshev approximation. Both linear and nonlinear approximations are discussed and, subject to certain conditions, quadratic convergence of the discretizations is obtained along with an explicit rate constant which can be estimated numerically.
On the measure of totally real algebraic integers. II
Authors: Smyth C. J.
Published: June 2010

For a certain measure

Ω
(
α
)

\Omega (\alpha )

, defined for totally real algebraic integers

α
≠
0

\alpha \ne 0

,

±
1

\pm 1

, we find the four smallest values of

Ω
(
α
)

\Omega (\alpha )

. The methods used involve linear programming, and the results are verified using Interval Arithmetic.
Finite element analysis of a scattering problem
Authors: Aziz A. K. , Kellogg R. Bruce
Published: June 2010

A finite element method for the solution of a scattering problem for the reduced wave equation is formulated and analyzed. The method involves a reformulation of the problem on a bounded domain with a nonlocal boundary condition. The space of trial functions includes piecewise polynomial functions and functions arising from spherical harmonics.
Some convergence estimates for semidiscrete type schemes for time-dependent nonselfadjoint parabolic equations
Authors: Huang Ming You , Thomée Vidar
Published: June 2010

L
2

{L_2}

-norm error estimates are shown for semidiscrete (continuous in time) Galerkin finite element type approximations to solutions of general time-dependent nonselfadjoint second order parabolic equations under Dirichlet boundary conditions. The semidiscrete solutions are defined in terms of given methods for the corresponding elliptic problem such as the standard Galerkin method in which the boundary conditions are satisfied exactly but also methods for which this is not necessary. The results are proved by energy arguments and include estimates for the homogeneous equation with both smooth and nonsmooth initial data.
A note on the Taylor series expansion coefficients of the Jacobian elliptic function 𝑠𝑛(𝑥,𝑘)
Authors: Wrigge Staffan
Published: June 2010

By considering

sn

(
x
,
k
)

{\text {sn}}(x,k)

and

sn

2

(
x
,
k
)

{\text {sn}}^2(x,k)

as power series in the modulus k, closed expressions are obtained for some of the integers occurring in the polynomials considered.
A method for finding permanents of 0,1 matrices
Authors: Kallman Ralph
Published: June 2010

Certain row operations are used in a method for computing permanents of 0, 1 matrices. Machine execution times for this method are compared with those for the Ryser and Nijenhuis-Wilf algorithms.
On Stieltjes integral transforms involving Γ-functions
Authors: Belevitch V. , Boersma J.
Published: June 2010

After some methodological remarks on the theory of Stieltjes transforms, a systematic classification of transforms involving

Γ
\Gamma

-functions is presented As a consequence, many new transforms are established and much simpler proofs for a few known transforms are obtained.
Upwind difference schemes for hyperbolic systems of conservation laws
Authors: Osher Stanley , Solomon Fred
Published: June 2010

We derive a new upwind finite difference approximation to systems of nonlinear hyperbolic conservation laws. The scheme has desirable properties for shock calculations. Under fairly general hypotheses we prove that limit solutions satisfy the entropy condition and that discrete steady shocks exist which are unique and sharp. Numerical examples involving the Euler and Lagrange equations of compressible gas dynamics in one and two space dimensions are given.
𝐴-stability of a class of methods for the numerical integration of certain linear systems of ordinary differential equations
Authors: Crisci M. R. , Russo E.
Published: June 2010

This paper is concerned with the analysis of the stability of a class of one-step integration methods, originated by the Lanczos tau method and applicable to particular linear differential systems. It is proved that these methods are A-stable for every order.
Evaluation of generalized Howland integrals
Authors: Ling Chih Bing
Published: June 2010

This paper presents a method of evaluation of the generalized Howland integrals. The values are tabulated to 10D.
Density problems involving 𝑝ᵣ(𝑛)
Authors: Costello Patrick J.
Published: June 2010

Lower bounds on the density of zeros of

p
r

(
n
)

{p_r}(n)

are provided for certain values of r.
Primes of the form 𝑛!±1 and 2⋅3⋅5⋯𝑝±1
Authors: Buhler J. P. , Crandall R. E. , Penk M. A.
Published: June 2010

All primes less than

10

1000

{10^{1000}}

of the form

n
!
±
1

n! \pm 1

or

2
⋅
3
⋅
5
⋯
p
±
1

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

are determined. Results of Brillhart, Lehmer, and Selfridge are used together with a fast algorithm that applies to primality tests of integers N for which many factors of

N
±
1

N \pm 1

are known.
Vortex methods. I. Convergence in three dimensions
Authors: Beale J. Thomas , Majda Andrew
Published: June 2010

Recently several different approaches have been developed for the simulation of three-dimensional incompressible fluid flows using vortex methods. Some versions use detailed tracking of vortex filament structures and often local curvatures of these filaments, while other methods require only crude information, such as the vortex blobs of the two-dimensional case. Can such "crude" algorithms accurately account for vortex stretching and converge? We answer this question affirmatively by constructing a new class of "crude" three-dimensional vortex methods and then proving that these methods are stable and convergent, and can even have arbitrarily high order accuracy without being more expensive than other "crude" versions of the vortex algorithm.
Type-insensitive ODE codes based on implicit 𝐴(𝛼)-stable formulas
Authors: Shampine L. F.
Published: June 2010

Previous work on A-stable formulas is extended to

A
(
α
)

A(\alpha )

-stable formulas, which are far more important in practice. Some important improvements in technique based on another interation method and an idea of Enright for the efficient handling of Jacobians are proposed. Implementation details and numerical examples are provided for a research-grade code.
Some new convergence acceleration methods
Authors: Brezinski Claude
Published: June 2010

The E-algorithm is a general extrapolation method which includes most of the sequence transformations actually known. Some new convergence acceleration methods are derived from the E-algorithm by applying the so-called

θ
\theta

-procedure. The algorithms thus obtained are studied. Some theoretical results are proved and numerical examples are given.
On certain extrapolation methods for the numerical solution of integro-differential equations
Authors: Chang S. H.
Published: June 2010

Asymptotic error expansions have been obtained for certain numerical methods for linear Volterra integro-differential equations. These results permit the application of extrapolation procedures. Computational examples are presented.
The number of partitions of the integer 𝑁 into 𝑀 nonzero positive integers
Authors: Colman W. J. A.
Published: June 2010

The function

p
m

(
n
)

{p_m}(n)

is defined as the number of partitions of the integer n into exactly m nonzero positive integers where the order is irrelevant. A series in which the leading terms alternate in sign is given for

p
m

(
n
)

{p_m}(n)

which yields good numerical estimates.
On Euler Lehmer pseudoprimes and strong Lehmer pseudoprimes with parameters 𝐿, 𝑄 in arithmetic progressions
Authors: Rotkiewicz A.
Published: June 2010

Let

U
n

=
(

α
n

−

β
n

)

/

(
α
−
β
)

{U_n} = ({\alpha ^n} - {\beta ^n})/(\alpha - \beta )

for n odd and

U
n

=
(

α
n

−

β
n

)

/

(

α
2

−

β
2

)

{U_n} = ({\alpha ^n} - {\beta ^n})/({\alpha ^2} - {\beta ^2})

for even n, where

α
\alpha

and

β
\beta

are distinct roots of the trinomial

f
(
z
)
=

z
2

−

L

z
+
Q

f(z) = {z^2} - \sqrt L z + Q

and

L
>
0

L > 0

and Q are rational integers.

U
n

{U_n}

is the nth Lehmer number connected with

f
(
z
)

f(z)

. Let

V
n

=
(

α
n

+

β
n

)

/

(
α
+
β
)

{V_n} = ({\alpha ^n} + {\beta ^n})/(\alpha + \beta )

for n odd, and

V
n

=

α
n

+

β
n

{V_n} = {\alpha ^n} + {\beta ^n}

for n even denote the nth term of the associated recurring sequence. An odd composite number n is a strong Lehmer pseudoprime with parameters L, Q (or

slepsp

(
L
,
Q
)

{\text {slepsp}}(L,Q)

) if

(
n
,
D
Q
)
=
1

(n,DQ) = 1

, where

D
=
L
−
4
Q
≠
0

D = L - 4Q \ne 0

, and with

δ
(
n
)
=
n
−
(
D
L

/

n
)
=
d
⋅

2
s

\delta (n) = n - (DL/n) = d \cdot {2^s}

, d odd, where

(
D
L

/

n
)

(DL/n)

is the Jacobi symbol, we have either

U
d

≡
0

(
mod

n
)

{U_d} \equiv 0\,\pmod n

or

V

d
⋅

2
r

≡
0

(
mod

n
)

{V_{d \cdot {2^r}}} \equiv 0\,\pmod n

, for some r with

0
⩽
r
>
s

0 \leqslant r > s

. Let

D
=
L
−
4
Q
>
0

D = L - 4Q > 0

. Then every arithmetic progression

a
x
+
b

ax + b

, where a, b are relatively prime integers, contains an infinite number of odd (composite) strong Lehmer pseudoprimes with parameters L, Q. Some new tests for primality are also given.
Two classes of internally 𝑆-stable generalized Runge-Kutta processes which remain consistent with an inaccurate Jacobian
Authors: Day J. D. , Murthy D. N. P.
Published: June 2010

Generalized Runge-Kutta Processes for stiff systems of ordinary differential equations usually require an accurate evaluation of a Jacobian at every step. However, it is possible to derive processes which are Internally S-stable when an accurate Jacobian is used but still remain consistent and highly stable if an approximate Jacobian is used. It is shown that these processes require at least as many function evaluations as an explicit Runge-Kutta process of the same order, and second and third order processes are developed. A second class of Generalized Runge-Kutta is introduced which requires that the Jacobian be evaluated accurately less than once every step. A third order process of this class is developed, and all three methods contain an error estimator similar to those of Fehlberg or England.
A note on Haselgrove’s method for numerical integration
Authors: Sugihara Masaaki , Murota Kazuo
Published: June 2010

An alternative set of weights is proposed for Niederreiter’s generalization of Haselgrove’s method for numerical integration.
On the speed of convergence of the nearest integer continued fraction
Authors: Jager H.
Published: June 2010

Let

p
n

/

q
n

{p_n}/{q_n}

and

A
n

/

B
n

{A_n}/{B_n}

denote the convergents of, respectively, the regular and the nearest integer continued fraction expansion of the irrational number x. There exists a function

k
(
n
)

k(n)

such that

A
n

/

B
n

=

p

k
(
n
)

/

q

k
(
n
)

{A_n}/{B_n} = {p_{k(n)}}/{q_{k(n)}}

. Adams proved that for almost all x one has

lim
k
(
n
)

/

n
=
log
⁡
2

/

log
⁡
G

\lim k(n)/n = \log 2/\log G

,

G
=

1
2

(
1
+

5

)

G = \frac {1}{2}(1 + \sqrt 5 )

. Here we present a shorter proof of this result, based on a simple expression for

k
(
n
)

k(n)

and the ergodicity of the shift operator, connected with the nearest integer continued fraction.
Formulas for factorial 𝑁
Authors: Murray Francis J.
Published: June 2010

Burnside’s and Stirling’s formulas for factorial N are special cases of a family of formulas with corresponding asymptotic series given by E. W. Barnes in 1899. An operational procedure for obtaining these formulas and series is presented which yields both convergent and divergent series and error estimates in the latter case. Two formulas of this family have superior accuracy and the geometric mean is better than either.
Chebyshev expansions for the Bessel function 𝐽_{𝑛}(𝑧) in the complex plane
Authors: Coleman J. P. , Monaghan A. J.
Published: June 2010

Polynomial-based approximations for

J
0

(
z
)

{J_0}(z)

and

J
1

(
z
)

{J_1}(z)

are presented. The first quadrant of the complex plane is divided into six sectors, and separate approximations are given for

|

z

|

⩽
8

|z| \leqslant 8

and for

|

z

|

⩾
8

|z| \geqslant 8

on each sector. Each approximation is based on a Chebyshev expansion in which the argument of the Chebyshev polynomials is real on the central ray of the sector. The errors involved in extrapolation off the central ray are discussed. The approximation obtained for

|

z

|

⩾
8

|z| \geqslant 8

can also be used to evaluate the Bessel functions

Y
0

(
z
)

{Y_0}(z)

and

Y
1

(
z
)

{Y_1}(z)

and the Hankel functions of the first and second kinds.
On the numerical construction of ellipsoidal wave functions
Authors: Arscott F. M. , Taylor P. J. , Zahar R. V. M.
Published: June 2010

The ellipsoidal wave equation, which is the most general equation derived by separation of the Helmholtz equation in confocal coordinates, presents unusual computational difficulties, and its solutions, despite their importance for physical applications, have not hitherto been effectively computed. This paper describes a successful technique, which involves the solution of a four-term recursion and the simultaneous handling of two eigenparameters.
A computational study of finite element methods for second order linear two-point boundary value problems
Authors: Keast P. , Fairweather G. , Diaz J. C.
Published: June 2010

A computational study of five finite element methods for the solution of a single second order linear ordinary differential equation subject to general linear, separated boundary conditions is described. In each method, the approximate solution is a piecewise polynomial expressed in terms of a B-spline basis, and is determined by solving a system of linear algebraic equations with an almost block diagonal structure. The aim of the investigation is twofold: to determine if the theoretical orders of convergence of the methods are realized in practice, and to compare the methods on the basis of cost for a given accuracy. In this study three parametrized families of test problems, containing problems of varying degrees of difficulty, are used. The conclusions drawn are rather straightforward. Collocation is the cheapest method for a given accuracy, and the easiest to implement. Also, for solving the linear algebraic equations, the use of a special purpose solver which takes advantage of the structure of the equations is advisable.
On the integral ∫^{∞}₀𝑒^{-𝜇𝑡}𝑡^{𝜈-1}𝑙𝑜𝑔^{𝑚}𝑡𝑑𝑡
Authors: Kölbig K. S.
Published: June 2010

A recurrence relation is given for the integral in the title. Formulae which allow easy evaluation by formula manipulation are given for integer and half-integer values of

ν
\nu

. Explicit expressions for these integrals for small values of m and

ν
=
n

\nu = n

,

ν
=
n
+

1
2

\nu = n + \frac {1}{2}

are also presented.
Factoring large numbers with a quadratic sieve
Authors: Gerver Joseph L.
Published: June 2010

The quadratic sieve algorithm was used to factor a 47-digit number into primes. A comparison with Wagstaff’s results using the continued fraction early abort algorithm suggests that QS should be faster than CFEA when the number being factored exceeds 60 digits (plus or minus ten or more digits, depending on details of the hardware and software).
The calculation of a large cubic class number with an application to real cyclotomic fields
Authors: Seah Eric , Washington Lawrence C. , Williams Hugh C.
Published: June 2010

The class number of the cubic subfield of the pth cyclotomic field is calculated for the prime

p
=
11290018777

p = 11290018777

. This is used to construct an example where the class number of the pth real cyclotomic field is larger than p.