bluuebunny/crossref_metadata_2025 huggingface dataset:
Numerical stability in evaluating continued fractions
Authors: Jones William B. , Thron W. J.
Published: June 2010

A careful analysis of the backward recurrence algorithm for evaluating approximants of continued fractions provides rigorous bounds for the accumulated relative error due to rounding. Such errors are produced by machine operations which carry only a fixed number v of significant digits in the computations. The resulting error bounds are expressed in terms of the machine parameter v. The derivation uses a basic assumption about continued fractions, which has played a fundamental role in developing convergence criteria. Hence, its appearance in the present context is quite natural. For illustration, the new error bounds are applied to two large classes of continued fractions, which subsume many expansions of special functions of physics and engineering, including those represented by Stieltjes fractions. In many cases, the results insure numerical stability of the backward recurrence algorithm.
A combinatorial theorem in group theory
Authors: Straus E. G.
Published: June 2010

There is an anti-Ramsey theorem for inhomogeneous linear equations over a field, which is essentially due to R. Rado [2]. This theorem is generalized to groups to get sharper quantitative and qualitative results. For example, it is shown that for any Abelian group A (written additively) and any mappings

f
1

,
⋯
,

f
n

{f_1}, \cdots ,{f_n}

of A into itself there exists a k-coloring

χ
\chi

of A so that the inhomogeneous equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript i equals 1 Overscript n Endscripts left-parenthesis f Subscript i Baseline left-parenthesis x Subscript i Baseline right-parenthesis minus f Subscript i Baseline left-parenthesis y Subscript i Baseline right-parenthesis right-parenthesis equals b comma b not-equals 0">

∑

i
=
1

n

(

f
i

(

x
i

)
−

f
i

(

y
i

)
)
=
b
,

b
≠
0

\sum \limits _{i = 1}^n {({f_i}({x_i}) - {f_i}({y_i})) = b,\quad b \ne 0}

\]
has no solutions

x
i

,

y
i

{x_i},{y_i}

with

χ
(

x
i

)
=
χ
(

y
i

)

\chi ({x_i}) = \chi ({y_i})

for all

i
=
1
,
⋯
,
n

i = 1, \cdots ,n

. Here the number of colors k can be chosen bounded by

(
3
n

)

n
−
1

{(3n)^{n - 1}}

which depends on n alone and not on the

f
i

{f_i}

or b. For non-Abelian groups an analogous qualitative result is proven when b is "residually compact". Applications to anti-Ramsey results in Euclidean geometry are given.
Polynomial interpolation to boundary data on triangles
Authors: Barnhill R. E. , Gregory J. A.
Published: June 2010

Boolean sum interpolation theory is used to derive a polynomial interpolant which interpolates a function

F
∈

C
N

(

T
¯

)

F \in {C^N}(\bar T)

, and its derivatives of order N and less, on the boundary

∂
T

\partial T

of a triangle T. A triangle with one curved side is also considered.
Two-parameter, arbitrary order, exponential approximations for stiff equations
Authors: Ehle Byron L. , Picel Zdenek
Published: June 2010

A two-parameter family of approximations to the exponential function is considered. Constraints on the parameters are determined which guarantee the approximations are A-acceptable. The suitability of these approximations for 2-point A-stable exponential fitting is established. Several numerical methods, which produce these approximations when solving

y
′

=
λ
y

y’ = \lambda y

, are presented.
A nodal basis for 𝐶¹ piecewise polynomials of degree 𝑛≥5
Authors: Morgan John , Scott Ridgway
Published: June 2010

A basis for the space of

C
1

{C^1}

piecewise polynomials in two variables of degree

n
⩾
5

n \geqslant 5

is constructed. The basis is parametrized by "nodal variables," namely, the values and derivatives of the basis functions at a discrete set of points.
Linear filtering
Authors: Shapiro Ralph
Published: June 2010

A simple linear smoothing operator of general order p is constructed for discretized functions. It is particularly suited for problems in numerical integration where it is necessary or desirable to suppress two-grid-interval waves. The operator is ideal in the sense that for the class of damping operators of order p, which remove two-grid-interval waves without changing the phase of any wave component, this operator produces the least damping of the amplitudes of all other waves.
Higher order compact implicit schemes for the wave equation
Authors: Ciment Melvyn , Leventhal Stephen H.
Published: June 2010

Higher order finite-difference techniques have associated large star systems which engender complications near the boundary. In the numerical solution of hyperbolic equations, such boundary conditions require careful treatment since errors or instabilities generated there will in general pollute the entire calculation. To circumvent this difficulty, we use a class of implicit schemes suggested by H.-O. Kreiss, which achieves the highest order of accuracy possible on the smallest (most compact) mesh system. Here we develop a scheme which approximates the wave equation,
\[

U

t
t

=
a
(
x
,
y
,
t
)

U

x
x

+
b
(
x
,
y
,
t
)

U

y
y

,

{U_{tt}} = a(x,y,t){U_{xx}} + b(x,y,t){U_{yy}},

\]
with fourth order accuracy in space and time. After an appropriate factorization, the resulting set of equations are tridiagonal and hence easily solved. The tridiagonal nature also indicates that the boundary conditions do not create special difficulties. Numerical experiments demonstrate the expected order of convergence and fulfill our expectations on the treatment of boundary conditions. An experimental computation also demonstrates that our results hold on L-shaped domains.
Sharper bounds for the Chebyshev functions 𝜃(𝑥) and 𝜓(𝑥)
Authors: Rosser J. Barkley , Schoenfeld Lowell
Published: June 2010

The authors demonstrate a wider zero-free region for the Riemann zeta function than has been given before. They give improved methods for using this and a recent determination that the first 3,502,500 zeros lie on the critical line to develop better bounds for functions of primes.
Some fourth degree integration formulas for simplexes
Authors: Stroud A. H.
Published: June 2010

A fourth degree integration formula is given for the n-dimensional simplex for

n
=
3
,
4
,
5
,
6
,
7
,
8
,
10
,
11
,
12

n = 3,4,5,6,7,8,10,11,12

. The formula contains

(

n
2

+
3
n
+
4
)

/

2

({n^2} + 3n + 4)/2

points.
Converting interpolation series into Chebyshev series by recurrence formulas
Authors: Salzer Herbert E.
Published: June 2010

Interpolation series (divided difference, Gregory-Newton, Gauss, Stirling, Bessel) are converted into Chebyshev (or Jacobi) series by applying a previously derived general five-term recurrence formula [3]. It employs the coefficients in three-term linear recurrence formulas (same kind as for orthogonal polynomials) which have been found for the mth degree nonorthogonal polynomial coefficients of the differences used in the interpolation series. In the Gauss, Stirling and Bessel series, the coefficients in the recurrence formulas vary with the parity of m. The basic five-term recurrence formula is applicable also to: (1) inter- and intraconversion of power series in

a
x
+
b

ax + b

, divided difference and equal-interval interpolation series (including subtabulation), and Chebyshev series, (2) obtaining Chebyshev series for solutions of difference equations, (3) the derivation of formulas for Chebyshev coefficients in terms of differences, and (4) the conversion of interpolation series into Chebyshev series, for more than one variable.
Triples of sixth powers with equal sums
Authors: Brudno Simcha
Published: June 2010

The diophantine equation

x
6

+

y
6

+

z
6

=

u
6

+

v
6

+

w
6

{x^6} + {y^6} + {z^6} = {u^6} + {v^6} + {w^6}

is shown to have a two-parameter solution which is homogeneous of degree four. The solution also satisfies

x
2

+

y
2

+

z
2

=

u
2

+

v
2

+

w
2

{x^2} + {y^2} + {z^2} = {u^2} + {v^2} + {w^2}

; and in addition,

3
x
+
y
+
z
=
3
u
+
v
+
w

3x + y + z = 3u + v + w

.
Rational Chebyshev approximations for the inverse of the error function
Authors: Blair J. M. , Edwards C. A. , Johnson J. H.
Published: June 2010

This report presents near-minimax rational approximations for the inverse of the error function inverf x, for

0
⩽
x
⩽
1
−

10

−
10000

0 \leqslant x \leqslant 1 - {10^{ - 10000}}

, with relative errors ranging down to

10

−
23

{10^{ - 23}}

. An asymptotic formula for the region

x
→
1

x \to 1

is also given.
Primal hybrid finite element methods for 2nd order elliptic equations
Authors: Raviart P.-A. , Thomas J. M.
Published: June 2010

The paper is devoted to the construction of finite element methods for 2nd order elliptic equations based on a primal hybrid variational principle. Optimal error bounds are proved. As a corollary, we obtain a general analysis of nonconforming finite element methods.
Interior maximum norm estimates for finite element methods
Authors: Schatz A. H. , Wahlbin L. B.
Published: June 2010

Interior a priori error estimates in the maximum norm are derived from interior Ritz-Galerkin equations which are common to a class of methods used in approximating solutions of second order elliptic boundary value problems. The estimates are valid for a large class of piecewise polynomial subspaces used in practice, which are defined on quasi-uniform meshes. It is shown that the error in an interior domain

Ω
1

{\Omega _1}

can be estimated with the best order of accuracy that is possible locally for the subspaces used plus the error in a weaker norm over a slightly larger domain which measures the effects from outside of the domain

Ω
1

{\Omega _1}

.
The infinity norm of a certain type of symmetric circulant matrix
Authors: Hoskins W. D. , Meek D. S.
Published: June 2010

An attainable bound for the infinity norm of the inverse of a whole class of symmetric circulant Toeplitz matrices is found. The class of matrices includes those arising from interpolation with both odd and even degree periodic polynomial splines on a uniform mesh.
Optimizing the arrangement of points on the unit sphere
Authors: Berman Joel , Hanes Kit
Published: June 2010

This paper is concerned with the problem of placing N points on the unit sphere in

E
3

{E^3}

so as to maximize the sum of their mutual distances. A necessary condition is proved which led to a computer algorithm. This in turn led to the apparent best arrangements for values of N from 5 to 10 inclusive.
Finite differences of the partition function
Authors: Gupta Hansraj
Published: June 2010

From the Hardy-Ramanujan-Rademacher formula for

p
(
n
)

p(n)

—the number of unrestricted partitions of n, it is not difficult to deduce that there exists a least positive integer

n
0

(
r
)

{n_0}(r)

such that

V
r

p
(
n
)
⩾
0

{V^r}p(n) \geqslant 0

for each

n
⩾

n
0

(
r
)

n \geqslant {n_0}(r)

, where

V
p
(
n
)
=
p
(
n
)
−
p
(
n
−
1
)

Vp(n) = p(n) - p(n - 1)

and

V
r

p
(
n
)
=
V
{

V

r
−
1

p
(
n
)
}

{V^r}p(n) = V\{ {V^{r - 1}}p(n)\}

. In this note, we give values of

n
0

(
r
)

{n_0}(r)

for each

r
⩽
10

r \leqslant 10

and conjecture that

n
0

(
r
)

/

r
3

∼
1

{n_0}(r)/{r^3}\sim 1

.
Accelerated overrelaxation method
Authors: Hadjidimos Apostolos
Published: June 2010

This paper describes a method for the numerical solution of linear systems of equations. The method is a two-parameter generalization of the Successive Overrelaxation (SOR) method such that when the two parameters involved are equal it coincides with the SOR method. Finally, a numerical example is given to show the superiority of the new method.
Analysis of some difference approximations for a singular perturbation problem without turning points
Authors: Kellogg R. Bruce , Tsan Alice
Published: June 2010

Some three point difference schemes are considered for a singular perturbation problem without turning points. Bounds for the discretization error are obtained which are uniformly valid for all h and

ε
>
0

\varepsilon > 0

. The degeneration of the order of the schemes at

ε
=
0

\varepsilon = 0

is considered.
The artificial compression method for computation of shocks and contact discontinuities. III. Self-adjusting hybrid schemes
Authors: Harten Amiram
Published: June 2010

This paper presents a new computational method for the calculation of discontinuous solutions of hyperbolic systems of conservation laws, which deal effectively with both shock and contact discontinuities. The method consists of two stages: in the first stage a standard finite-difference scheme is hybridized with a nonoscillatory first order accurate method to provide for the monotonic variation of the solution near discontinuities, and in the second stage artificial compression is applied to sharpen transitions at discontinuities. This modification of a standard finite-difference method results in a scheme which preserves the order of truncation error of the original method and yet yields a sharp and oscillation free transition for both shocks and contact discontinuities. The modification can be easily implemented in existing computer codes.
Monte Carlo methods for index computation (𝑚𝑜𝑑𝑝)
Authors: Pollard J. M.
Published: June 2010

We describe some novel methods to compute the index of any integer relative to a given primitive root of a prime p. Our first method avoids the use of stored tables and apparently requires

O
(

p

1

/

2

)

O(p^{1/2})

operations. Our second algorithm, which may be regarded as a method of catching kangaroos, is applicable when the index is known to lie in a certain interval; it requires

O
(

w

1

/

2

)

O(w^{1/2})

operations for an interval of width w, but does not have complete certainty of success. It has several possible areas of application, including the factorization of integers.
Block implicit one-step methods
Authors: Watanabe Daniel S.
Published: June 2010

A new class of block implicit one-step methods for ordinary differential equations is presented. The methods are based on quadrature and generate function values at nonmesh points through Hermite interpolation. A general convergence theorem for block implicit methods is given, and the stability of the new class of methods is analyzed. The class contains A-stable, stiffly stable, strongly A-stable, and strongly stiffly stable methods. Numerical results demonstrating the efficiency and effectiveness of a particular block method are presented.
Numerical treatment of eigenvalue problems for differential equations with discontinuous coefficients
Authors: Babuška I. , Osborn J. E.
Published: June 2010

The eigenvalues of a second order differential equation are approximated by "factoring" the second order equations into a first order system and then applying the Ritz-Galerkin method to this system. Convergence results and error estimates are derived. These error estimates are based on the application of Sobolev spaces with variable order.
Gauss type quadrature rules for Cauchy principal value integrals
Authors: Elliott David , Paget D. F.
Published: June 2010

Two quadrature rules for the approximate evaluation of Cauchy principal value integrals, with nodes at the zeros of appropriate orthogonal polynomials, are discussed. An expression for the truncation error, in terms of higher order derivatives, is given for each rule. In addition, two theorems, containing sufficient conditions for the convergence of the sequence of quadrature rules to the integral, are proved.
Convergence of a finite element method for the approximation of normal modes of the oceans
Authors: Luskin Mitchell
Published: June 2010

This paper gives optimal order error estimates for the approximation of the spectral properties of a variant of the shallow water equations by a finite element procedure recently proposed by Platzman. General results on the spectral approximation of unbounded, selfadjoint operators are also given in this paper.
A collocation solver for mixed order systems of boundary value problems
Authors: Ascher U. , Christiansen J. , Russell R. D.
Published: June 2010

Implementation of a spline collocation method for solving boundary value problems for mixed order systems of ordinary differential equations is discussed. The aspects of this method considered include error estimation, adaptive mesh selection, B-spline basis function evaluation, linear system solution and nonlinear problem solution. The resulting general purpose code, COLSYS, is tested on a number of examples to demonstrate its stability, efficiency and flexibility.
Stability of two-dimensional initial-boundary value problems using leap-frog type schemes
Authors: Abarbanel Saul , Gottlieb David
Published: June 2010

The paper discusses stable boundary conditions for the two-dimensional Leap-Frog scheme and the Modified Leap-Frog scheme introduced by the authors in a former work.
Alternatives to the exponential spline in tension
Authors: Pruess Steven
Published: June 2010

A general setting is given for smooth interpolating splines depending on a parameter such that as this parameter approaches infinity the spline converges to the piecewise linear interpolant. The theory includes the standard exponential spline in tension, a rational spline, and several cubic splines. An algorithm is given for one of the cubics; the parameter for this example controls the spacing of new knots which are introduced.
Estimating the largest eigenvalue of a positive definite matrix
Authors: O’Leary Dianne P. , Stewart G. W. , Vandergraft James S.
Published: June 2010

The power method for computing the dominant eigenvector of a positive definite matrix will converge slowly when the dominant eigenvalue is poorly separated from the next largest eigenvalue. In this note it is shown that in spite of this slow convergence, the Rayleigh quotient will often give a good approximation to the dominant eigenvalue after a very few iterations-even when the order of the matrix is large.
A combinatorial interpretation for the Schett recurrence on the Jacobian elliptic functions
Authors: Dumont Dominique
Published: June 2010

The coefficients introduced by Alois Schett containing the Taylor series expansions of the Jacobian elliptic functions are proved to count certain classes of permutations.
On a relationship between the convergents of the nearest integer and regular continued fractions
Authors: Adams William W.
Published: June 2010

In this paper we derive a relation concerning the speed of convergence of the nearest integer and regular continued fractions. If

A
n

/

B
n

{A_n}/{B_n}

,

p
k

/

q
k

{p_k}/{q_k}

denote the convergents of the nearest integer and regular continued fractions of an irrational number

α
\alpha

, then for all n there is a

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)}}

. It is shown that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow normal infinity Endscripts StartFraction n Over k left-parenthesis n right-parenthesis EndFraction equals StartStartFraction log left-parenthesis StartFraction 1 plus StartRoot 5 EndRoot Over 2 EndFraction right-parenthesis OverOver log 2 EndEndFraction">

lim

n
→
∞

n

k

(
n
)

=

log
⁡

(

1
+

5

2

)

log

2

\lim \limits _{n \to \infty } \;\frac {n}{{k\left ( n \right )}} = \frac {{\log \left ( {\frac {{1 + \sqrt 5 }}{2}} \right )}}{{\log \;2}}

\]
for almost all

α
\alpha

. This problem is reduced to a special case of a general result concerning the frequency of partial quotients in the regular continued fraction (Theorem 2).
On least squares exponential sum approximation with positive coefficients
Authors: Evans John W. , Gragg William B. , LeVeque Randall J.
Published: June 2010

An algorithm is given for finding optimal least squares exponential sum approximations to sampled data subject to the constraint that the coefficients appearing in the exponential sum are positive. The algorithm employs the divided differences of exponentials to overcome certain problems of ill-conditioning and is suitable for data sampled at noninteger times.
On the coupling of boundary integral and finite element methods
Authors: Johnson Claes , Nédélec J.-Claude
Published: June 2010

Let

Ω
c

{\Omega ^c}

be the complementary of a bounded regular domain in

R

2

{{\mathbf {R}}^2}

of boundary

Γ
\Gamma

. We consider the problem (1)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column normal upper Delta u equals f semicolon 2nd Column a m p semicolon in normal upper Omega Superscript c Baseline comma 2nd Row 1st Column u vertical-bar Subscript normal upper Gamma Baseline equals u Subscript 0 comma Baseline 2nd Column a m p semicolon EndLayout">

{

Δ
u
=
f
;

a
m
p
;

in

Ω
c

,

u

|

Γ

=

u

0
,

a
m
p
;

\left \{ {\begin {array}{*{20}{c}} {\Delta u = f;} & {{\text {in}}\;{\Omega ^c},} \\ {u{|_\Gamma } = {u_{0,}}} & {} \\ \end {array} } \right .

\]
where f has its support in a bounded subdomain

Ω
1

{\Omega _1}

of

Ω
c

{\Omega ^c}

. Let

Γ
2

{\Gamma _2}

be the common boundary of

Ω
1

{\Omega _1}

and

Ω
2

=

Ω
c

−

Ω
1

{\Omega _2} = {\Omega ^c} - {\Omega _1}

. We solve the problem (1) by using an equivalent system of equations involving an integral equation on

Γ
2

{\Gamma ^2}

coupled with the equation: (2)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column normal upper Delta u equals f 2nd Column a m p semicolon in normal upper Omega 1 comma 2nd Row 1st Column u vertical-bar Subscript normal upper Gamma Baseline equals u 0 comma 2nd Column a m p semicolon 3rd Row 1st Column u vertical-bar Subscript normal upper Gamma 2 Baseline equals lamda period 2nd Column a m p semicolon EndLayout">

{

Δ
u
=
f

a
m
p
;

in

Ω
1

,

u

|

Γ

=

u
0

,

a
m
p
;

u

|

Γ
2

=
λ
.

a
m
p
;

\left \{ {\begin {array}{*{20}{c}} {\Delta u = f} \hfill & {{\text {in}}\;{\Omega _1},} \hfill \\ {u{|_\Gamma } = {u_0},} \hfill & {} \hfill \\ {u{|_{{\Gamma _2}}} = \lambda .} \hfill & {} \hfill \\ \end {array} } \right .

\]
We introduce a finite element approximation of Eq. (2) and of the integral equation and we prove optimal error estimates.
The exact degree of precision of generalized Gauss-Kronrod integration rules
Authors: Rabinowitz Philip
Published: June 2010

It is shown that the Kronrod extension to the

n
n

-point Gauss integration rule, with respect to the weight function

(
1
−

x
2

)

μ
−
1

/

2

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

,

0
>
μ
⩽
2

0 > \mu \leqslant 2

,

μ
≠
1

\mu \ne 1

, is of exact precision

3
n
+
1

3n + 1

for n even and

3
n
+
2

3n + 2

for n odd. Similarly, for the

(
n
+
1
)

(n + 1)

-point Lobatto rule, with

−
1

/

2
>
μ
⩽
1

-1/2> \mu \leqslant 1

,

μ
≠
0

\mu \ne 0

, the exact precision is

3
n

3n

for

n
n

odd and

3
n
+
1

3n + 1

for n even.
Gaussian quadrature of integrands involving the error function
Authors: Vigneron J. P. , Lambin Ph.
Published: June 2010

Orthogonal polynomials corresponding to the weight function

1
−

erf

(
x
)

1 - {\operatorname {erf}}(x)

and defined on the positive real axis are constructed. Abscissas and weight factors for the associated Gaussian quadrature are then deduced (up to 12-point formulas). The stability of the algorithm used for this particular computation is discussed. An example is provided to test the efficiency of the new Gaussian rule.
Modulus and phase of the reduced logarithmic derivative of the cylindrical Bessel function
Authors: Cruz Andrés , Sesma Javier
Published: June 2010

The modulus and phase of the reduced logarithmic derivative of the cylindrical Bessel function
\[

z

J

′
v

(
z
)

/

J
v

(
z
)
,

z{J\prime _v}(z)/{J_v}(z),

\]
for complex variable z and real order v, are investigated. Special attention is paid to the location of saddle points and their trajectories as v varies.
The first occurrence of certain large prime gaps
Authors: Brent Richard P.
Published: June 2010

The first occurrence of a string of

2
r
−
1

2r - 1

consecutive composite numbers between two primes (denoted by

f
(
r
)

f(r)

and

f
(
r
)
+
2
r

f(r) + 2r

) is tabulated for

f
(
r
)

f(r)

in the range

2.6
×

10

12

>
f
(
r
)
⩽
4.444
×

10

12

2.6 \times {10^{12}} > f(r) \leqslant 4.444 \times {10^{12}}

. This extends earlier computations in the range

f
(
r
)
⩽
2.6
×

10

12

f(r) \leqslant 2.6 \times {10^{12}}

.
An optimal order process for solving finite element equations
Authors: Bank Randolph E. , Dupont Todd
Published: June 2010

A k-level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed. The work estimate for this procedure is proportional to the number of unknowns, an optimal order result. General geometry is permitted for the underlying domain, but the shape plays a role in the rate of convergence through elliptic regularity. Finally, a short discussion of the use of this method for parabolic problems is presented.
On a dimensional reduction method. I. The optimal selection of basis functions
Authors: Vogelius M. , Babuška I.
Published: June 2010

This paper is the first in a series of three, which analyze an adaptive approximate approach for solving

(
n
+
1
)

(n + 1)

-dimensional boundary value problems by replacing them with systems of equations in n-dimensional space. In this approach the unknown functions of

(
n
+
1
)

(n + 1)

variables are projected onto finite linear combinations of functions of just n variables. This paper shows how the coefficients of these linear combinations can be chosen optimally.
On a dimensional reduction method. III. A posteriori error estimation and an adaptive approach
Authors: Vogelius M. , Babuška I.
Published: June 2010

This paper is the last in a series of three which analyze an adaptive approximate approach for solving

(
n
+
1
)

(n + 1)

-dimensional boundary value problems by replacing them with systems of equations in n-dimensional space. In this paper we show how to find reliable a posteriori estimates for the error and how these can also be used in the design of an adaptive strategy. Various numerical examples are contained in the paper.
Cubic splines and approximate solution of singular integral equations
Authors: Jen Erica , Srivastav R. P.
Published: June 2010

Of concern here is the numerical solution of singular integral equations of Cauchy type; i.e., equations involving principal value integrals. The unknown function is expressed as the product of an appropriate weight function and a cubic spline. The problem is reduced to a system of linear algebraic equations which is solved for the approximate values of the function at the knots. An estimate is provided for the maximum error of the approximate solution. Numerical results from the spline method are compared with those obtained using other methods.
The number of polyhedral (3-connected planar) graphs
Authors: Duijvestijn A. J. W. , Federico P. J.
Published: June 2010

Data is presented on the number of 3-connected planar graphs, isomorphic to the graphs of convex polyhedra, with up to 22 edges. The numbers of such graphs having the same number of edges, and the same number of vertices and faces, are tabulated. Conjectured asymptotic formulas by W. T. Tutte and by R. C. Mullin and P. J. Schellenberg are discussed. Additional data beyond 22 edges are given enabling the number of 10-hedra to be presented for the first time, as well as estimates of the number of 11-hedra and dodecahedra.
Numerical quadrature rules for some infinite range integrals
Authors: Sidi Avram
Published: June 2010

Recently the present author has given a new approach to numerical quadrature and derived new numerical quadrature formulas for finite range integrals with algebraic and/or logarithmic endpoint singularities. In the present work this approach is used to derive new numerical quadrature formulas for integrals of the form

∫
0
∞

x
α

e

−
x

f
(
x
)

d
x

\smallint _0^\infty {x^\alpha }{e^{ - x}}f(x)\,dx

and

∫
0
∞

x
α

E
p

(
x
)
f
(
x
)

d
x

\smallint _0^\infty {x^\alpha }{E_p}(x)f(x)\,dx

, where

E
p

(
x
)

{E_p}(x)

is the exponential integral. It turns out the new rules are of interpolatory type, their abscissas are distinct and lie in the interval of integration and their weights, at least numerically, are positive. For fixed

α
\alpha

the new integration rules have the same set of abscissas for all p. Finally, the new rules seem to be at least as efficient as the corresponding Gaussian quadrature formulas. As an extension of the above, numerical quadrature formulas for integrals of the form

∫

−
∞

+
∞

|

x

|

β

e

−

x
2

f
(
x
)

d
x

\smallint _{ - \infty }^{ + \infty }|x{|^\beta }{e^{ - {x^2}}}f(x)\,dx

too are considered.
Numerical comparisons of nonlinear convergence accelerators
Authors: Smith David A. , Ford William F.
Published: June 2010

As part of a continuing program of numerical tests of convergence accelerators, we have compared the iterated Aitken’s

Δ
2

{\Delta ^2}

method, Wynn’s

ε
\varepsilon

algorithm, Brezinski’s

θ
\theta

algorithm, and Levin’s u transform on a broad range of test problems: linearly convergence alternating, monotone, and irregular-sign series, logarithmically convergent series, power method and Bernoulli method sequences, alternating and monotone asymptotic series, and some perturbation series arising in applications. In each category either the

ε
\varepsilon

algorithm or the u transform gives the best results of the four methods tested. In some cases differences among methods are slight, and in others they are quite striking.
Recurrence relations for the indefinite integrals of the associated Legendre functions
Authors: DiDonato A. R.
Published: June 2010

Two recurrence relations are derived for the computation of the integral of the associated Legendre functions of real argument and integer order and degree.
Vortex methods. II. Higher order accuracy in two and three dimensions
Authors: Beale J. Thomas , Majda Andrew
Published: June 2010

In an earlier paper the authors introduced a new version of the vortex method for three-dimensional, incompressible flows and proved that it converges to arbitrarily high order accuracy, provided we assume the consistency of a discrete approximation to the Biot-Savart Law. We prove this consistency statement here, and also derive substantially sharper results for two-dimensional flows. A complete, simplified proof of convergence in two dimensions is included.
The sequence of radii of the Apollonian packing
Authors: Boyd David W.
Published: June 2010

We consider the distribution function

N
(
x
)

N(x)

of the curvatures of the disks in the Apollonian packing of a curvilinear triangle. That is,

N
(
x
)

N(x)

counts the number of disks in the packing whose curvatures do not exceed x. We show that

log
⁡
N
(
x
)

/

log
⁡
x

\log N(x)/\log x

approaches the limit S as x tends to infinity, where S is the exponent of the packing. A numerical fit of a curve of the form

y
=
A

n
s

y = A{n^s}

to the values of

N
−

(
1000
n
)

{N^ - }(1000n)

for

n
=
1
,
2
,
…
,
6400

n = 1,2, \ldots ,6400

produces the estimate

S
≈
1.305636

S \approx 1.305636

which is consistent with the known bounds

1.300197
>
S
>
1.314534

1.300197 > S > 1.314534

.
Analysis of a multilevel iterative method for nonlinear finite element equations
Authors: Bank Randolph E. , Rose Donald J.
Published: June 2010

The multilevel iterative technique is a powerful technique for solving the systems of equations associated with discretized partial differential equations. We describe how this technique can be combined with a globally convergent approximate Newton method to solve nonlinear partial differential equations. We show that asymptotically only one Newton iteration per level is required; thus the complexity for linear and nonlinear problems is essentially equal.
Additive Runge-Kutta methods for stiff ordinary differential equations
Authors: Cooper G. J. , Sayfy A.
Published: June 2010

Certain pairs of Runge-Kutta methods may be used additively to solve a system of n differential equations

x
′

=
J
(
t
)
x
+
g
(
t
,
x
)

x’ = J(t)x + g(t,x)

. Pairs of methods, of order

p
⩽
4

p \leqslant 4

, where one method is semiexplicit and A-stable and the other method is explicit, are obtained. These methods require the LU factorization of one

n
×
n

n \times n

matrix, and p evaluations of g, in each step. It is shown that such methods have a stability property which is similar to a stability property of perturbed linear differential equations.
Odd perfect numbers not divisible by 3. II
Authors: Kishore Masao
Published: June 2010

We prove that odd perfect numbers not divisible by 3 have at least eleven distinct prime factors.
On the bisection method for triangles
Authors: Adler Andrew
Published: June 2010

Let UVW be a triangle with vertices U, V, and W. It is "bisected" as follows: choose a longest edge (say VW) of UVW, and let A be the midpoint of VW. The UVW gives birth to two daughter triangles UVA and UWA. Continue this bisection process forever. We prove that the infinite family of triangles so obtained falls into finitely many similarity classes, and we obtain sharp estimates for the longest jth generation edge.
Implicit Runge-Kutta methods of optimal order for Volterra integro-differential equations
Authors: Brunner Hermann
Published: June 2010

Implicit Runge-Kutta methods with m stages and optimal order

p
=
2
m

p = 2m

for the approximate solution of Volterra integro-differential equations can be viewed as fully discretized collocation methods in certain polynomial spline spaces. The choice of the quadrature formulas needed for the full discretization of the collocations is investigated, and it is shown that, in contrast to ordinary differential equations, there exist (for fixed m) several optimal methods.
The arithmetic-harmonic mean
Authors: Foster D. M. E. , Phillips G. M.
Published: June 2010

Consider two sequences generated by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a Subscript n plus 1 Baseline equals upper M left-parenthesis a Subscript n Baseline comma b Subscript n Baseline right-parenthesis comma b Subscript n plus 1 Baseline equals upper M prime left-parenthesis a Subscript n plus 1 Baseline comma b Subscript n Baseline right-parenthesis comma">

a

n
+
1

=
M
(

a
n

,

b
n

)
,

b

n
+
1

=

M
′

(

a

n
+
1

,

b
n

)
,

{a_{n + 1}} = M({a_n},{b_n}),\quad {b_{n + 1}} = M’ ({a_{n + 1}},{b_n}),

\]
where the

a
n

{a_n}

and

b
n

{b_n}

are positive and M and M’ are means. The paper discusses the nine processes which arise by restricting the choice of M and M’ to the arithmetic, geometric and harmonic means, one case being that used by Archimedes to estimate

π
\pi

. Most of the paper is devoted to the arithmetic-harmonic mean, whose limit is expressed as an infinite product and as an infinite series in two ways.
Explicit estimates for the error term in the prime number theorem for arithmetic progressions
Authors: McCurley Kevin S.
Published: June 2010

We give explicit numerical estimates for the Chebyshev functions

ψ
(
x
;
k
,
l
)

\psi (x;k,l)

and

θ
(
x
;
k
,
l
)

\theta (x;k,l)

for certain nonexceptional moduli k. For values of

ε
\varepsilon

and b, a constant c is tabulated such that

|

ψ
(
x
;
k
,
l
)
−
x

/

φ
(
k
)

|

>
ε
x

/

φ
(
k
)

|\psi (x;k,l) - x/\varphi (k)| > \varepsilon x/\varphi (k)

, provided

(
k
,
l
)
=
1

(k,l) = 1

,

x
⩾
exp
⁡
(
c

log
2

k
)

x \geqslant \exp (c{\log ^2}k)

, and

k
⩾

10
b

k \geqslant {10^b}

. The methods are similar to those used by Rosser and Schoenfeld in the case

k
=
1

k = 1

, but are based on explicit estimates of

N
(
T
,
χ
)

N(T,\chi )

and an explicit zero-free region for Dirichlet L-functions.
Galerkin methods for singular boundary value problems in one space dimension
Authors: Eriksson Kenneth , Thomée Vidar
Published: June 2010

Two Galerkin type piecewise polynomial approximation procedures based on bilinear forms with different weight functions are analyzed and compared. Optimal order error estimates are proved and numerical results are presented.
A priori estimates and analysis of a numerical method for a turning point problem
Authors: Berger Alan E. , Han Hou De , Kellogg R. Bruce
Published: June 2010

Bounds are obtained for the derivatives of the solution of a turning point problem. These results suggest a modification of the El-Mistikawy Werle finite difference scheme at the turning point. A uniform error estimate is obtained for the resulting method, and illustrative numerical results are given.
Rates of convergence of Gaussian quadrature for singular integrands
Authors: Lubinsky D. S. , Rabinowitz P.
Published: June 2010

The authors obtain the rates of convergence (or divergence) of Gaussian quadrature on functions with an algebraic or logarithmic singularity inside, or at an endpoint of, the interval of integration. A typical result is the following: For a bounded smooth weight function on

[
−
1
,
1
]

[ - 1,1]

, the error in n-point Gaussian quadrature of

f
(
x
)
=

|

x
−
y

|

−
δ

f(x) = |x - y{|^{ - \delta }}

is

O
(

n

−
2
+
2
δ

)

O({n^{ - 2 + 2\delta }})

if

y
=
±
1

y = \pm 1

and

O
(

n

−
1
+
δ

)

O({n^{ - 1 + \delta }})

if

y
∈
(
−
1
,
1
)

y \in ( - 1,1)

, provided we avoid the singularity. If we ignore the singularity y, the error is
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis n Superscript negative 1 plus 2 delta Baseline left-parenthesis log n right-parenthesis Superscript delta Baseline left-parenthesis log log n right-parenthesis Superscript delta left-parenthesis 1 plus epsilon right-parenthesis Baseline right-parenthesis">

O
(

n

−
1
+
2
δ

(
log
⁡
n

)
δ

(
log
⁡
log
⁡
n

)

δ
(
1
+
ε
)

)

O({n^{ - 1 + 2\delta }}{(\log n)^\delta }{(\log \log n)^{\delta (1 + \varepsilon )}})

for almost all choices of y. These assertions are sharp with respect to order.
Decay rates for inverses of band matrices
Authors: Demko Stephen , Moss William F. , Smith Philip W.
Published: June 2010

Spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices. The rate of decay of

A

−
1

{A^{ - 1}}

can be bounded in terms of the (essential) spectrum of

A

A
∗

A{A^\ast }

for general A and in terms of the (essential) spectrum of A for positive definite A. In the positive definite case the bound can be attained. These results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices. We also establish decay rates for certain generalized inverses.
The application of Runge-Kutta schemes to singular initial value problems
Authors: de Hoog Frank , Weiss Richard
Published: June 2010

A theory for explicit Runge-Kutta schemes applied to the initial value problem for a first-order system of differential equations with a singularity of the first kind is developed. It is shown that, in general, the order of convergence is at most two but that the usual order up to a logarithmic term can be obtained for level three and four schemes applied to specific problems.
A simplified version of the fast algorithms of Brent and Salamin
Authors: Newman D. J.
Published: June 2010

We produce more elementary algorithms than those of Brent and Salamin for, respectively, evaluating

e
x

{e^x}

and

π
\pi

. Although the Gauss arithmetic-geometric process still plays a central role, the elliptic function theory is now unnecessary.
Some a posteriori error estimators for elliptic partial differential equations
Authors: Bank R. E. , Weiser A.
Published: June 2010

We present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations. The estimators are based on solving local Neumann problems in each element. The estimators differ in how they enforce consistency of the Neumann problems. We prove that as the mesh size decreases, under suitable assumptions, two of the error estimators approach upper bounds on the norm of the true error, and all three error estimators are within multiplicative constants of the norm of the true error. We present numerical results in which one of the error estimators appears to converge to the norm of the true error.
Elliptic curves over finite fields and the computation of square roots mod 𝑝
Authors: Schoof René
Published: June 2010

In this paper we present a deterministic algorithm to compute the number of

F

q

{{\mathbf {F}}_q}

-points of an elliptic curve that is defined over a finite field

F

q

{{\mathbf {F}}_q}

and which is given by a Weierstrass equation. The algorithm takes

O
(

log
9

q
)

O({\log ^9}q)

elementary operations. As an application we give an algorithm to compute square roots

mod
p

\bmod p

. For fixed

x
∈

Z

x \in {\mathbf {Z}}

, it takes

O
(

log
9

p
)

O({\log ^9}p)

elementary operations to compute

x

mod
p

\sqrt x \bmod p

.
An algorithm for nonsmooth convex minimization with errors
Authors: Kiwiel Krzysztof C.
Published: June 2010

A readily implementable algorithm is given for minimizing any convex, not necessarily differentiable, function f of several variables. At each iteration the method requires only one approximate evaluation of f and its

ε
\varepsilon

-subgradient, and finds a search direction by solving a small quadratic programming problem. The algorithm generates a minimizing sequence of points, which converges to a solution whenever f has any minimizers.
Constructing integral lattices with prescribed minimum. I
Authors: Plesken W. , Pohst M.
Published: June 2010

Methods for computing integral laminated lattices with prescribed minimum are developed. Laminating is a process of stacking layers of an

(
n
−
1
)

(n - 1)

-dimensional lattice as densely as possible to obtain an n-dimensional lattice. Our side conditions are: All scalar products of lattice vectors are rational integers, and all lattices are generated by vectors of prescribed minimum (square) length m. For

m
=
3

m = 3

all such lattices are determined.
Recurrence relations for hypergeometric functions of unit argument
Authors: Lewanowicz Stanisław
Published: June 2010

We show that the generalized hypergeometric function
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript n Baseline colon equals Subscript p plus 3 Baseline upper F Subscript p plus 2 Baseline left-parenthesis StartLayout 1st Row negative n comma n plus lamda comma a Subscript p Baseline comma 1 2nd Row b Subscript p plus 2 Baseline EndLayout vertical-bar 1 right-parenthesis left-parenthesis n greater-than-or-slanted-equals 0 right-parenthesis">

P
n

:

=

p
+
3

F

p
+
2

(

−
n
,
n
+
λ
,

a
p

,
1

b

p
+
2

|

1

)

(
n
⩾
0
)

{P_n}:{ = _{p + 3}}{F_{p + 2}}\left ( {\left . {\begin {array}{*{20}{c}} { - n,n + \lambda ,{a_p},1} \\ {{b_{p + 2}}} \\ \end {array} } \right |1} \right )\quad (n \geqslant 0)

\]
satisfies a nonhomogeneous recurrence relation of order

p
+
σ

p + \sigma

, where

σ
=
0

\sigma = 0

when

p
+
3

F

p
+
2

(
1
)

_{p + 3}{F_{p + 2}}(1)

is balanced, and

σ
=
1

\sigma = 1

otherwise. Also, for
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript n Baseline colon equals StartFraction left-parenthesis c Subscript q plus 1 Baseline right-parenthesis Subscript n Baseline Over left-parenthesis d Subscript q Baseline right-parenthesis Subscript n Baseline left-parenthesis n plus lamda right-parenthesis Subscript n Baseline EndFraction Subscript q plus 2 Baseline upper F Subscript q plus 1 Baseline left-parenthesis StartLayout 1st Row n plus c Subscript q plus 2 Baseline 2nd Row n plus d Subscript q Baseline comma 2 n plus lamda plus 1 EndLayout vertical-bar 1 right-parenthesis left-parenthesis n greater-than-or-slanted-equals 0 right-parenthesis">

U
n

:=

(

c

q
+
1

)

n

(

d
q

)

n

(
n
+
λ
)

n

q
+
2

F

q
+
1

(

n
+

c

q
+
2

n
+

d
q

,
2
n
+
λ
+
1

|

1

)

(
n
⩾
0
)

{U_n}: = \frac {{{{({c_{q + 1}})}_n}}}{{{{({d_q})}_n}{{(n + \lambda )}_n}}}{\,_{q + 2}}{F_{q + 1}}\left ( {\left . {\begin {array}{*{20}{c}} {n + {c_{q + 2}}} \\ {n + {d_q},2n + \lambda + 1} \\ \end {array} } \right |1} \right )\quad (n \geqslant 0)

\]
a homogeneous recurrence relation of order

q
+
1

q + 1

is given.
Some practical Runge-Kutta formulas
Authors: Shampine Lawrence F.
Published: June 2010

A new selection is made of the most practical of the many explicit Runge-Kutta formulas of order 4 which have been proposed. A new formula is considered, formulas are modified to improve their quality and efficiency in agreement with improved understanding of the issues, and formulas are derived which permit interpolation. It is possible to do a lot better than the pair of Fehlberg currently regarded as "best".
Julia sets and Mandelbrot-like sets associated with higher order Schröder rational iteration functions: a computer assisted study
Authors: Vrscay Edward R.
Published: June 2010

Schröder iteration functionsSm(z){S_m}(z), a generalization of Newton’s method (for whichm=2m = 2), are constructed so that the sequencezn+1=Sm(zn){z_{n + 1}} = {S_m}({z_n})converges locally to a rootz∗{z^\ast }ofg(z)=0g(z) = 0asO(|zn−z∗|m)O(|{z_n} - {z^\ast }{|^m}). Forg(z)g(z)a polynomial, this involves the iteration of rational functions over the complex Riemann sphere, which is described by the classical theory of Julia and Fatou and subsequent developments. The Julia sets for theSm(z){S_m}(z), as applied to the simple casesgn(z)=zn−1{g_n}(z) = {z^n} - 1, are examined for increasingmwith the help of microcomputer plots. The possible types of behavior ofzn{z_n}iteration sequences are catalogued by examining the orbits of free critical points of theSm(z){S_m}(z), as applied to a one-parameter family of cubic polynomials.
Some evaluations for the generalized hypergeometric series
Authors: Lavoie J.-L.
Published: June 2010

Whipple’s theorem on the sum of a

3

F
2

(
1
)

_3{F_2}(1)

plays a key role in obtaining a family of summation formulas for the generalized hypergeometric series of unit argument.
Explicit determination of nontrivial torsion structures of elliptic curves over quadratic number fields
Authors: Reichert Markus A.
Published: June 2010

We determine equations of the modular curves

X
1

(
N
)

{X_1}(N)

for

N
=
11
,
13
,
14
,
15
,
16
,
17

N = 11,13,14,15,16,17

and 18. Except for

N
=
17

N = 17

, these are the only existing elliptic or hyperelliptic

X
1

(
N
)

{X_1}(N)

. Applying these

X
1

(
N
)

{X_1}(N)

, we calculate tables of elliptic curves E over quadratic fields K with torsion groups of one of the following isomorphism types:
\[

E

tor

(
K
)
≅

Z

/

m

Z

,

m
=
11
,
13
,
14
,
15
,
16

and

18.

{E_{{\operatorname {tor}}}}(K) \cong {\mathbf {Z}}/m{\mathbf {Z}},\quad m = 11,13,14,15,16\;{\text {and}}\;18.

\]
A computer-assisted investigation of Ramanujan pairs
Authors: Blecksmith Richard , Brillhart John , Gerst Irving
Published: June 2010

Four new Ramanujan pairs

{

a
i

}

\{ {a_i}\}

,

{

b
j

}

\{ {b_j}\}

are given along with the theorem that no such pairs exist with

a
1

=
1

{a_1} = 1

and

a
2

=
s

{a_2} = s

for any

s
⩾
5

s \geqslant 5

. All finite Ramanujan pairs are determined and their significance in bounding the local branching degree in the search tree for such pairs is discussed. The search techniques and programs that were used are also described. The parity of the coefficients in the power series is determined in two of the new identities. Partition interpretations of the six recent identities are also given.
Numerical approximation of Mindlin-Reissner plates
Authors: Brezzi F. , Fortin M.
Published: June 2010

We consider a finite element approximation of the so-called Mindlin-Reissner formulation for moderately thick elastic plates. We show that stability and optimal error bounds hold independently of the value of the thickness.