bluuebunny/crossref_metadata_2025 huggingface dataset:
Approximations for elliptic integrals
Authors: Luke Yudell L.
Published: June 2010

Closed-form approximations are derived for the three kinds of incomplete elliptic integrals by using the Padé approximations for the square root. An effective analytical representation of the error is presented. Approximations for the complete integrals based on trapezoidal-type integration formulae are also developed.
Differentiation formulas for analytic functions
Authors: Lyness J. N.
Published: June 2010

In a previous paper (Lyness and Moler

[
1
]

[1]

), several closely related formulas of use for obtaining a derivative of an analytic function numerically are derived. Each of these formulas consists of a convergent series, each term being a sum of function evaluations in the complex plane. In this paper we introduce a simple generalization of the previous methods; we investigate the "truncation error" associated with truncating the infinite series. Finally we recommend a particular differentiation rule, not given in the previous paper.
Numerical solution of symmetric positive differential equations
Authors: Katsanis Theodore
Published: June 2010

A finite-difference method for the solution of symmetric positive linear differential equations is developed. The method is applicable to any region with piecewise smooth boundaries. Methods for solution of the finite-difference equations are discussed. The finite-difference solutions are shown to converge at essentially the rateO(h1/2)O({h^{1/2}})ash→0,hh \to 0,h, being the maximum distance between adjacent mesh-points. An alternate finite-difference method is given with the advantage that the finite-difference equations can be solved iteratively. However, there are strong limitations on the mesh arrangements which can be used with this method.
Generalized finite-difference schemes
Authors: Swartz Blair , Wendroff Burton
Published: June 2010

Finite-difference schemes for initial boundary-value problems for partial differential equations lead to systems of equations which must be solved at each time step. Other methods also lead to systems of equations. We call a method a generalized finite-difference scheme if the matrix of coefficients of the system is sparse. Galerkin’s method, using a local basis, provides unconditionally stable, implicit generalized finite-difference schemes for a large class of linear and nonlinear problems. The equations can be generated by computer program. The schemes will, in general, be not more efficient than standard finite-difference schemes when such standard stable schemes exist. We exhibit a generalized finite-difference scheme for Burgers’ equation and solve it with a step function for initial data.
On a method to subtract off a singularity at a corner for the Dirichlet or Neumann problem
Authors: Wigley Neil M.
Published: June 2010

Let

D
D

be a plane domain partly bounded by two line segments which meet at the origin and form there an interior angle

π
α
>
0

\pi \alpha > 0

. Let

U
(
x
,
y
)

U(x,y)

be a solution in

D
D

of Poisson’s equation such that either

U
U

or

∂
U

/

∂
n

\partial U/\partial n

(the normal derivative) takes prescribed values on the boundary segments. Let

U
(
x
,
y
)

U(x,y)

be sufficiently smooth away from the corner and bounded at the corner. Then for each positive integer

N
N

there exists a function

V
N

(
x
,
y
)

{V_N}(x,y)

which satisfies a related Poisson equation and which satisfies related boundary conditions such that

U
−

V
N

U - {V_N}

is

N
N

-times continuously differentiable at the corner. If

1

/

α

1/\alpha

is an integer

V
N

{V_N}

may be found explicitly in terms of the data of the problem for

U
U

.
On relatively prime odd amicable numbers
Authors: Hagis Peter
Published: June 2010

Whether or not a relatively prime pair of amicable numbers exists is still an open question. In this paper some necessary conditions for

m
m

and

n
n

to be a pair of odd relatively prime amicable numbers are proved. In particular, lower bounds for

m
m

,

n
n

,

m
n

mn

and the number of prime divisors of

m
n

mn

are established. The arguments are based on an extensive case study carried out on the CDC 6400 at the Temple University Computing Center.
On computation of the bivariate normal distribution
Authors: Amos D. E.
Published: June 2010

A quadrature and two series representations are given as limiting cases of a bivariate

t
t

-distribution. The quadrature is taken over the complementary error function and the series are sums of Bessel functions and incomplete beta functions, respectively. Comparisons with some known results are made in terms of accuracy and computer time.
Chebyshev polynomial expansion of Bose-Einstein functions of orders 1 to 10
Authors: Ng Edward W. , Devine C. J. , Tooper R. F.
Published: June 2010

Chebyshev series approximations are given for the complete Bose-Einstein functions of orders 1 to 10. This paper also gives an exhaustive presentation of the relation of this function to other functions, with the emphasis that some Fermi-Dirac functions and polylogarithms are readily computable from the given approximations. The coefficients are given in 21 significant figures and the maximal relative error for function representation ranges from

2
×

10

−
20

2 \times {10^{ - 20}}

to

3
×

10

−
19

3 \times {10^{ - 19}}

. These expansions are fast convergent; for example, typically six terms gives an accuracy of

10

−
8

{10^{ - 8}}

.
Gaussian quadratures for the integrals ₀^{∞}𝑒𝑥𝑝(-𝑥²)𝑓(𝑥)𝑑𝑥 and ₀^{𝑏}𝑒𝑥𝑝(-𝑥²)𝑓(𝑥)𝑑𝑥
Authors: Steen N. M. , Byrne G. D. , Gelbard E. M.
Published: June 2010

Gaussian quadratures are developed for the evaluation of the integrals given in the title. The weights and abscissae for the semi-infinite integral are given for two through fifteen points with fifteen places. For

b
=
1

b = 1

, the weights and abscissae are given for two through ten points with fifteen places.
On divisibility by nine of the sums of even amicable pairs
Authors: Lee Elvin
Published: June 2010

Most known even amicable pairs have sums divisible by nine [9]. The general form of the exceptions to the rule of divisibility by nine (Gardner’s rule) is deduced and the results expressed in the form of a theorem. A computer search based on a corollary to the theorem is described and six new exceptions to Gardner’s rule are found.
Perfectly symmetric two-dimensional integration formulas with minimal numbers of points
Authors: Rabinowitz Philip , Richter Nira
Published: June 2010

Perfectly symmetric integration formula of degrees 9–15 with a minimal number of points are computed for the square, the circle and the entire plane with weight functions exp

(
−
(

x
2

+

y
2

)
)

( - ({x^2} + {y^2}))

and exp

(
−

(

x
2

+

y
2

)

1

/

2

)

( - {({x^2} + {y^2})^{1/2}})

. These rules were computed by solving a large system of nonlinear algebraic equations having a special structure. In most cases where the minimal formula has a point exterior to the region or where some of the weights are negative, ’good’ formulas, which consist only of interior points and have only positive weights, are given which contain more than the minimal number of points.
Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function
Authors: Kölbig K. S.
Published: June 2010

Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function are calculated as functions of the upper limit

λ
\lambda

of the definition integrals. It becomes apparent that not all, but only some, of the zero trajectories of the incomplete Riemann zeta function join a zero of the Riemann zeta function

ζ
(
s
)

\zeta (s)

for

λ
→
∞

\lambda \to \infty

. The remaining trajectories at least in the region considered, approach the zero trajectories of the incomplete gamma function.
Marginal stability and stabilization in the numerical integration of ordinary differential equations
Authors: Brunner H.
Published: June 2010

Strongly stable and consistent multistep methods with maximum order are subject to marginal (or weak) stability. In this paper we introduce modified multistep methods whose coefficients depend linearly on the stepsize

h
h

and a parameter

L
L

in such a way that the order of the original method is not decreased. By choosing

L
L

in a suitable manner (depending essentially on

f
y

(
x
,
y
)

{f_y}(x,y)

of the differential equation

y
′

=
f
(
x
,
y
)

y’ = f(x,y)

and on the growth parameters of the multistep method), marginal stability can be eliminated.
Optimal approximation in Hilbert spaces with reproducing kernel functions
Authors: Larkin F. M.
Published: June 2010

Characterisations of optimal linear estimation rules are given in terms of the reproducing kernel function of a suitable Hilbert space. The results are illustrated by means of three different, useful function spaces, showing, among other things, how Gaussian quadrature rules, and the Whittaker Cardinal Function, relate to optimal linear estimation rules in particular spaces.
Formulas for bivariate hyperosculatory interpolation
Authors: Salzer Herbert E.
Published: June 2010

For a given function

f
(
x
,
y
)

f(x,y)

, bivariate hyperosculatory interpolation formulas are obtained by employing a suitably constructed binary nic

p
n

(
x
,
y
)

{p_n}(x,y)

that is fitted to the values of

f
(
x
,
y
)

f(x,y)

and its first and second partial derivatives at the m points

(

x
i

,

y
i

)

({x_i},{y_i})

of a rectangular

h
×
k

h \times k

Cartesian grid, where (

(

x
i

,

y
i

)
=
(

x
0

+

p
i

h
,

y
0

+

q
i

k
)
,

p
i

({x_i},{y_i}) = ({x_0} + {p_i}h,{y_0} + {q_i}k),{p_i}

and

q
i

{q_i}

are small integers

≧
0
,
i
=
0
(
1
)
m
−
1
,
m
≧
2

\geqq 0,i = 0(1)m - 1,m \geqq 2

. In terms of the variables (p, q), where

x
=

x
0

+
p
h
,
y
=

y
0

+
q
k

x = {x_0} + ph,y = {y_0} + qk

(and

f
(
x
,
y
)
=
F
(
p
,
q
)

f(x,y) = F(p,q)

), we have

p
n

(
x
,
y
)
=

P
n

(
p
,
q
)

{p_n}(x,y) = {P_n}(p,q)

. Often, for

P
n

(
p
,
q
)

{P_n}(p,q)

having a specified desirable form, this problem turns out to be insoluble for every configuration of the points

(

x
i

,

y
i

)

({x_i},{y_i})

. When this is not the case, it generally requires considerable investigation to find a practical configuration of points

(

x
i

,

y
i

)

({x_i},{y_i})

for which there is a solution of the form

P
n

(
p
,
q
)

{P_n}(p,q)

. Formulas are found for choices of

P
n

(
p
,
q
)

{P_n}(p,q)

, and soluble configurations of points

(

x
i

,

y
i

)

({x_i},{y_i})

, that have dominant remainder terms in
\[

h
r

k
s

f

x
…
x
(
r

times

)
y
…
y
(
s

times

)

(

x
0

,

y
0

)

{h^r}{k^s}{f_{x \ldots x(r\;{\text {times}})y \ldots y(s\;{\text {times}})}}({x_0},{y_0})

\]
whose orders

r
+
s

r + s

are as high as possible. Three two-point formulas, two three-point formulas and one four-point formula, including all remainder terms through the order
\[

r
+
s
=

(

n
,

a
m
p
;

for

m
=
2

n
+
1
,

a
m
p
;

for

m
=
3
,
4

)

,

r + s = \left ( {\begin {array}{*{20}{c}} {n,} \hfill & {{\text {for}}\;m = 2} \hfill \\ {n + 1,} \hfill & {{\text {for}}\;m = 3,4} \hfill \\ \end {array} } \right ),

\]
are given here in convenient matrix form.
The convergence of an algorithm for solving sparse nonlinear systems
Authors: Broyden C. G.
Published: June 2010

A new algorithm for solving systems of nonlinear equations where the Jacobian is known to be sparse is shown to converge locally if a sufficiently good initial estimate of the solution is available and if the Jacobian satisfies a Lipschitz condition. The results of numerical experiments are quoted in which systems of up to 600 equations have been solved by the method.
A fourth-order finite-difference approximation for the fixed membrane eigenproblem
Authors: Kuttler J. R.
Published: June 2010

The fixed membrane problem

Δ
u
+
λ
u
=
0

\Delta u + \lambda u = 0

in

Ω
,
u
=
0

\Omega ,u = 0

on

∂
Ω

\partial \Omega

, for a bounded region

Ω
\Omega

of the plane, is approximated by a finite-difference scheme whose matrix is monotone. By an extension of previous methods for schemes with matrices of positive type,

O
(

h
4

)

O({h^4})

convergence is shown for the approximating eigenvalues and eigenfunctions, where h is the mesh width. An application to an approximation of the forced vibration problem

Δ
u
+
q
u
=
f

\Delta u + qu = f

in

Ω
,
u
=
0

\Omega ,u = 0

in

∂
Ω

\partial \Omega

, is also given.
The finite element method for infinite domains. I
Authors: Babuška Ivo
Published: June 2010

Numerical methods (finite element methods) for the approximate solution of elliptic partial differential equations on unbounded domains are considered, and error bounds, with respect to the number of unknowns which have to be determined, are proven.
A quasi-Newton method with no derivatives
Authors: Greenstadt John
Published: June 2010

The Davidon formula and others of the “quasi-Newton” class, which are used in the unconstrained minimization of a function

f
f

, provide a (generally) convergent sequence of approximations to the Hessian of

f
f

. These formulas, however, require the independent calculation of the gradient of

f
f

. In this paper, a set of new formulas is derived—using a previously described variational approach—which successively approximates the gradient as well as the Hessian, and uses only function values. These formulas are incorporated into an algorithm which, although still crude, works quite well for various standard test functions. Extensive numerical results are presented.
The asymptotic expansions of Hankel transforms and related integrals
Authors: MacKinnon Robert F.
Published: June 2010

In this paper, the asymptotic expansion of integrals of the form

∫
0
∞

F
(
k
r
)
f
(
k
)
d
k

\int _0^\infty {F(kr)f(k)dk}

s considered, as

r
r

tends to infinity, and where

F
(
k
r
)

F(kr)

are Bessel functions of the first and second kind, or functions closely related to these. Asymptotic expansions for several functions of this type are presented under suitable restrictions on

f
(
k
)

f(k)

. The expansion given by Willis for Hankel transforms is seen to be valid under conditions of

f
(
k
)

f(k)

less restrictive than those imposed by that author.
Norms of the successive overrelaxation method
Authors: Kincaid David R.
Published: June 2010

Norms, primarily of theoretical interest, of the successive overrelaxation method, with a relaxation parameter other than the optimum one, are studied. New results on this subject are presented and compared with those of Wachspress [8].
Discrete Galerkin and related one-step methods for ordinary differential equations
Authors: Hulme Bernie L.
Published: June 2010

New techniques for numerically solving systems of first-order ordinary differential equations are obtained by finding local Galerkin approximations on each subinterval of a given mesh. Different classes of methods correspond to different quadrature rules used to evaluate the innerproducts involved. At each step, a polynomial of degree

n
n

is constructed and the arcs are joined together continuously, but not smoothly, to form a piecewise polynomial of degree

n
n

and class

C
0

{C^0}

. If the

n
n

-point quadrature rule used for the innerproducts is of order

ν
+
1
,
ν
≧
n

\nu + 1,\nu \geqq n

, then the Galerkin method is of order

ν
\nu

at the mesh points. In between the mesh points, the

j
j

th derivatives have accuracy of order

O
(

h

min
(
ν
,
n
+
1
)

)

O({h^{\min (\nu ,n + 1)}})

, for

j
=
0

j = 0

and

O
(

h

n
−
j
+
1

)

O({h^{n - j + 1}})

for

1
≦
j
≦
n

1 \leqq j \leqq n

.
Stability theory of difference approximations for mixed initial boundary value problems. II
Authors: Gustafsson Bertil , Kreiss Heinz-Otto , Sundström Arne
Published: June 2010

A stability theory is developed for general difference approximations to mixed initial boundary value problems. The results are applied to certain commonly used difference approximations which are stable for the Cauchy problem, and different ways of defining boundary conditions are analyzed.
A note on the computation of integrals involving products of trigonometric and Bessel functions
Authors: Linz Peter , Kropp T. E.
Published: June 2010

A method useful in the numerical computation of integrals involving Bessel functions is extended to the case where the integrand contains products of trigonometric and Bessel functions.
A lower bound for the set of odd perfect numbers
Authors: Hagis Peter
Published: June 2010

It is proved here that if n is odd and perfect, then

n
>

10

50

n > {10^{50}}

.
The determination of Galois groups
Authors: Stauduhar Richard P.
Published: June 2010

A technique is described for the nontentative computer determination of the Galois groups of irreducible polynomials with integer coefficients. The technique for a given polynomial involves finding high-precision approximations to the roots of the polynomial, and fixing an ordering for these roots. The roots are then used to create resolvent polynomials of relatively small degree, the linear factors of which determine new orderings for the roots. Sequences of these resolvents isolate the Galois group of the polynomial. Machine implementation of the technique requires the use of multiple-precision integer and multiple-precision real and complex floating-point arithmetic. Using this technique, the writer has developed programs for the determination of the Galois groups of polynomials of degree

N
≦
7

N \leqq 7

. Two exemplary calculations are given.
Some probability density functions and their characteristic functions
Authors: McNolty Frank
Published: June 2010

This paper presents, without derivation, several generalized density functions together with their characteristic functions. The densities are expressed variously in terms of special functions such as:

I
v

(
x
)

{I_v}(x)

, the modified Bessel function of the first kind of order v;

K
v

(
x
)

{K_v}(x)

, the modified Bessel function of the second kind of order v;

1

F
1

(
a
;
b
;
x
)

_1{F_1}(a;b;x)

, the confluent hypergeometric function;

2

F
1

(
a
,
b
;
c
;
x
)

_2{F_1}(a,b;c;x)

, the hypergeometric function;

W

a
,
b

(
x
)

{W_{a,b}}(x)

, Whittaker’s function;

Φ
3

(
β
;
γ
;
b
x
,
c
x
)

{\Phi _3}(\beta ;\gamma ;bx,cx)

, a generalized hypergeometric function (type I);
\[

Φ
2

(
b
,
c
,
d
;
γ
;
λ
x
,
τ
x
,
β
x
)
,

{\Phi _2}(b,c,d;\gamma ;\lambda x,\tau x,\beta x),

\]
a generalized hypergeometric function (type II); and

ϕ
λ
μ

(
b

v
μ

)

\phi _\lambda ^\mu (b{v^\mu })

, a generalized Bessel type function. The first five cases are summarized from the work of Laha [7], Pearson [25] and Raj [26] while Cases 13 through 19 have not previously appeared in the literature of statistics or Fourier transforms. In what follows, the usual notation

f
(
x
)

f(x)

, for a density function, and

φ
(
t
)

\varphi (t)

, for a characteristic function, will be used with all parameters considered as real quantities:
\[

φ
(
t
)
=

∫

−
∞

∞

exp
⁡
(
i
t
x
)
f
(
x
)

d
x
.

\varphi (t) = \int _{ - \infty }^\infty {\exp (itx)f(x)\;dx.}

\]
On integral groups. III. Normalizers
Authors: Brown H. , Neubüser J. , Zassenhaus H.
Published: June 2010

Methods for determining a generating set for the normalizer of a finite group ofn×nn \times nintegral matrices, i.e., ann-dimensional crystallographic point group, are discussed. Necessary and sufficient conditions for the finiteness of such a normalizer are derived, and several examples of the application of the methods to cases when the normalizer is infinite are presented.
Spline approximation to the solution of the Volterra integral equation of the second kind
Authors: Netravali Arun N.
Published: June 2010

A cubic spline approximation in

C
2

{C^2}

to the solution of a general Volterra integral equation of the second kind is constructed. Under certain conditions, convergence of the approximation and its first two derivatives is proved and error bounds are obtained. The question of stability is not examined.
Solving linear boundary value problems by approximating the coefficients
Authors: Pruess Steven A.
Published: June 2010

A method for solving linear boundary value problems is described which consists of approximating the coefficients of the differential operator. Error estimates for the approximate solutions are established and improved results are given for the case of approximation by piecewise polynomial functions. For the latter approximations, the resulting problem can be solved by Taylor series techniques and several examples of this are given.
Efficient computer manipulation of tensor products with applications to multidimensional approximation
Authors: Pereyra V. , Scherer G.
Published: June 2010

The objective of this paper is twofold: (a) To make it possible to perform matrix-vector operations in tensor product spaces, using only the factors (

n
⋅

p
2

n \cdot {p^2}

words of information for

⊗

i
=
1

n

A
i

,

A
i

∈

L

(

E
p

,

E
p

)

\otimes _{i = 1}^n{A_i},{A_i} \in \mathcal {L}({E^p},{E^p})

) instead of the tensor-product operators themselves (

(

p
2

)
n

{({p^2})^n}

words of information). (b) To produce efficient algorithms for solving systems of linear equations with coefficient matrices being tensor products of nonsingular matrices, with special application to the approximation of multidimensional linear functionals.
On the instability of leap-frog and Crank-Nicolson approximations of a nonlinear partial differential equation
Authors: Fornberg B.
Published: June 2010

It is well known that nonlinear instabilities may occur when the partial differential equations, describing, for example, hydrodynamic flows, are approximated by finite-difference schemes, even if the corresponding linearized equations are stable. A scalar model equation is studied, and it is proved that methods of leap-frog and Crank-Nicolson type are unstable, unless the differential equation is rewritten to make the approximations quasi-conservative. The local structure of the instabilities is discussed.
Some definite integrals of the product of two Bessel functions of the second kind: (order zero)
Authors: Glasser M. L.
Published: June 2010

A new integral representation is derived for the expression

J
0

(
z
)

J
0

(
Z
)
+

Y
0

(
z
)
⋅

Y
0

(
Z
)

{J_0}(z){J_0}(Z) + {Y_0}(z) \cdot {Y_0}(Z)

and used to evaluate a number of integrals containing

Y
0

(
a
x
)

Y
0

(
b
x
)

{Y_0}(ax){Y_0}(bx)

. A supplementary table of integrals involving the function

K
0

(
x
)

{K_0}(x)

in the integrand appears in the microfiche section of this issue.
Fourth order difference methods for the initial boundary-value problem for hyperbolic equations
Authors: Oliger Joseph
Published: June 2010

Centered difference approximations of fourth order in space and second order in time are applied to the mixed initial boundary-value problem for the hyperbolic equation

u
t

=
−
c

u
x

{u_t} = - c{u_x}

. A method utilizing third order uncentered differences at the boundaries is shown to be stable and to retain an overall fourth order convergence estimate. Several computational examples illustrate the success of these methods for problems with one and two spacial dimensions. Further examples illustrate the effects of approximations of various orders of accuracy used at the boundaries.
Finite element methods for parabolic equations
Authors: Zlámal Miloš
Published: June 2010

The initial-boundary value problem for a linear parabolic equation with the Dirichlet boundary condition is solved approximately by applying the finite element discretization in the space dimension and three types of finite-difference discretizations in time: the backward, the Crank-Nicolson and the Calahan discretization. New error bounds are derived.
The distribution of crossings of chords joining pairs of 2𝑛 points on a circle
Authors: Riordan John
Published: June 2010

This paper, in the first place, calls attention to an extraordinarily compact solution of the problem in the title, given (a trifle hidden) in the work of the late Jacques Touchard. Its main weight, however, is on properties of the several kinds of number sequences appearing.
Spectral approximation for compact operators
Authors: Osborn John E.
Published: June 2010

In this paper a general spectral approximation theory is developed for compact operators on a Banach space. Results are obtained on the approximation of eigenvalues and generalized eigenvectors. These results are applied in a variety of situations.
On the global convergence of Broyden’s method
Authors: Moré J. J. , Trangenstein J. A.
Published: June 2010

We consider Broyden’s 1965 method for solving nonlinear equations. If the mapping is linear, then a simple modification of this method guarantees global and Q-superlinear convergence. For nonlinear mappings it is shown that the hybrid strategy for nonlinear equations due to Powell leads to R-superlinear convergence provided the search directions form a uniformly linearly independent sequence. We then explore this last concept and its connection with Broyden’s method. Finally, we point out how the above results extend to Powell’s symmetric version of Broyden’s method.
Dissipative two-four methods for time-dependent problems
Authors: Gottlieb David , Turkel Eli
Published: June 2010

A generalization of the Lax-Wendroff method is presented. This generalization bears the same relationship to the two-step Richtmyer method as the Kreiss-Oliger scheme does to the leapfrog method. Variants based on the MacCormack method are considered as well as extensions to parabolic problems. Extensions to two dimensions are analyzed, and a proof is presented for the stability of a Thommen-type algorithm. Numerical results show that the phase error is considerably reduced from that of second-order methods and is similar to that of the Kreiss-Oliger method. Furthermore, the (2, 4) dissipative scheme can handle shocks without the necessity for an artificial viscosity.
Approximations for hand calculators using small integer coefficients
Authors: Derenzo Stephen E.
Published: June 2010

Methods are presented for deriving approximations containing small integer coefficients. This approach is useful for electronic hand calculators and programmable calculators, where it is important to minimize the number of keystrokes necessary to evaluate the function. For example, the probability

P
(
x
)

P(x)

of exceeding x standard deviations of either sign (Gaussian probability integral) is approximated by
\[

P
(
x
)
≈

EXP

[

−

(
83
x
+
351
)
x
+
562

703

/

x
+
165

]

P(x) \approx {\text {EXP}}\left [ { - \frac {{(83x + 351)x + 562}}{{703/x + 165}}} \right ]

\]
with a relative error less than 0.042
On truncatable primes
Authors: Angell I. O. , Godwin H. J.
Published: June 2010

Truncatable primes are those that yield a sequence of primes when digits are removed always from the left or always from the right. The sizes of the largest truncatable primes in a given number base are estimated by a probabilistic argument and compared with computed values.
Quasi-amicable numbers
Authors: Hagis Peter , Lord Graham
Published: June 2010

If

m
=
σ
(
n
)
−
n
−
1

m = \sigma (n) - n - 1

and

n
=
σ
(
m
)
−
m
−
1

n = \sigma (m) - m - 1

, the integers m and n are said to be quasi-amicable numbers. This paper is devoted to a study of such numbers.
Computational complexity of Fourier transforms over finite fields
Authors: Preparata F. P. , Sarwate D. V.
Published: June 2010

In this paper we describe a method for computing the Discrete Fourier Transform (DFT) of a sequence of n elements over a finite field

GF

(

p
m

)

{\text {GF}}({p^m})

with a number of bit operations

O
(
n
m

log
⁡
(
n
m
)
⋅
P
(
q
)
)

O(nm\,\log (nm) \cdot P(q))

where

P
(
q
)

P(q)

is the number of bit operations required to multiply two q-bit integers and

q
≅
2

log
2

n
+
4

log
2

m
+
4

log
2

p

q \cong 2\,{\log _2}n + 4\,{\log _2}m + 4\,{\log _2}p

. This method is uniformly applicable to all instances and its order of complexity is not inferior to that of methods whose success depends upon the existence of certain primes. Our algorithm is a combination of known and novel techniques. In particular, the finite-field DFT is at first converted into a finite field convolution; the latter is then implemented as a two-dimensional Fourier transform over the complex field. The key feature of the method is the fusion of these two basic operations into a single integrated procedure centered on the Fast Fourier Transform algorithm.
Computing the strict Chebyshev solution of overdetermined linear equations
Authors: Abdelmalek Nabih N.
Published: June 2010

A method for calculating the strict Chebyshev solution of overdetermined systems of linear equations using linear programming techniques is described. This method provides: (1) a way to determine, for the majority of cases, all the equations belonging to the characteristic set, (2) an efficient method to obtain the inverse of the matrix needed to calculate the strict Chebyshev solution, and (3) a way of recognizing when an element of the Chebyshev solution equals a corresponding element of the strict Chebyshev solution. As a result, in general, the computational effort is considerably reduced. Also the present method deals with full rank as well as rank deficient cases. Numerical results are given.
Discretization of Volterra integral equations of the first kind
Authors: Brunner Hermann
Published: June 2010

We show that various (discrete) methods for the approximate solution of Volterra (and Abel) integral equations of the first kind correspond to some discrete version of the method of (recursive) collocation in the space of (continuous) piecewise polynomials. In a collocation method no distinction has to be made between equations with regular or weakly singular kernels; the regularity or nonregularity of the given integral operator becomes only relevant when selecting a discretization procedure for the moment integrals resulting from collocation. Similar results hold for equations of the second kind.
A numerical conception of entropy for quasi-linear equations
Authors: le Roux A. Y.
Published: June 2010

A family of difference schemes solving the Cauchy problem for quasi-linear equations is studied. This family contains well-known schemes such as the decentered, Lax, Godounov or Lax-Wendroff schemes. Two conditions are given, the first assures the convergence to a weak solution and the second, more restrictive, implies the convergence to the solution in Kružkov’s sense, which satisfies an entropy condition that guarantees uniqueness. Some counterexamples are proposed to show the necessity of such conditions.
Shepard’s method of “metric interpolation” to bivariate and multivariate interpolation
Authors: Gordon William J. , Wixom James A.
Published: June 2010

Shepard developed a scheme for interpolation to arbitrarily spaced discrete bivariate data. This scheme provides an explicit global representation for an interpolant which satisfies a maximum principle and which reproduces constant functions. The interpolation method is basically an inverse distance formula which is generalized to any Euclidean metric. These techniques extend to include interpolation to partial derivative data at the interpolation points.
On the convergence of a quasi-Newton method for sparse nonlinear systems
Authors: Lam Binh
Published: June 2010

It is shown that an algorithm for solving a system of nonlinear equations where the Jacobian is known to be sparse, converges locally and Q-superlinearly.
The irregular primes to 125000
Authors: Wagstaff Samuel S.
Published: June 2010

We have determined the irregular primes below 125000 and tabulated their distribution. Two primes of index five of irregularity were found, namely 78233 and 94693. Fermat’s Last Theorem has been verified for all exponents up to 125000. We computed the cyclotomic invariants

μ
p

{\mu _p}

,

λ
p

{\lambda _p}

,

ν
p

{\nu _p}

, and found that

μ
p

=
0

{\mu _p} = 0

for all

p
>
125000

p > 125000

. The complete factorizations of the numerators of the Bernoulli numbers

B

2
k

{B_{2k}}

for

2
k
⩽
60

2k \leqslant 60

and of the Euler numbers

E

2
k

{E_{2k}}

for

2
k
⩽
42

2k \leqslant 42

are given.
Symbolic generation of finite difference formulas
Authors: Keller H. B. , Pereyra V.
Published: June 2010

Tables of coefficients for high order accurate, compact approximations to the first ten derivatives on and at the midpoints of uniform nets are presented. The exact rational weights are generated and tested by means of symbolic manipulation implemented through MACSYMA. These weights are required in the application of deferred corrections to new methods for solving higher order two point boundary value problems.
Error analysis for spline collocation methods with application to knot selection
Authors: Christiansen J. , Russell R. D.
Published: June 2010

Some collocation schemes used to solve mth order ordinary differential equations are known to display superconvergence at the mesh points. Here we show that some such schemes have additional superconvergence points for the approximate solution and all of its derivatives. Using such points, we argue that a mesh selection scheme introduced by Dodson can be expected to perform well under general circumstances. A numerical example is given to demonstrate the new superconvergence results.
A quasi-projection analysis of Galerkin methods for parabolic and hyperbolic equations
Authors: Douglas Jim , Dupont Todd , Wheeler Mary F.
Published: June 2010

Superconvergence phenomena are demonstrated for Galerkin approximations of solutions of second order parabolic and hyperbolic problems in a single space variable. An asymptotic expansion of the Galerkin solution is used to derive these results and, in addition, to show optimal order error estimates in Sobolev spaces of negative index in multiple dimensions.
Superconvergence and reduced integration in the finite element method
Authors: Zlámal Miloš
Published: June 2010

The finite elements considered in this paper are those of the Serendipity family of curved isoparametric elements. There is given a detailed analysis of a superconvergence phenomenon for the gradient of approximate solutions to second order elliptic boundary value problems. An approach is proposed how to use the superconvergence in practical computations.
The Fourier method for nonsmooth initial data
Authors: Majda Andrew , McDonough James , Osher Stanley
Published: June 2010

Application of the Fourier method to very general linear hyperbolic Cauchy problems having nonsmooth initial data is considered, both theoretically and computationally. In the absence of smoothing, the Fourier method will, in general, be globally inaccurate, and perhaps unstable. Two main results are proven: the first shows that appropriate smoothing techniques applied to the equation gives stability; and the second states that this smoothing combined with a certain smoothing of the initial data leads to infinite order accuracy away from the set of discontinuities of the exact solution modulo a very small easily characterized exceptional set. A particular implementation of the smoothing method is discussed; and the results of its application to several test problems are presented, and compared with solutions obtained without smoothing.
Inverse linear multistep methods for the numerical solution of initial value problems of ordinary differential equations
Authors: Alfeld Peter
Published: June 2010

The well-known explicit linear multistep methods for the numerical solution of ordinary differential equations advance the numerical solution fromxn+k−1{x_{n + k - 1}}toxn+k{x_{n + k}}by computing some numerical approximation from back values and then evaluating the problem defining function to obtain an approximation of the derivative. In this paper similar methods are proposed that first compute an approximation to the derivative and then compute an approximation to the exact solution, either by evaluating a suitable function, or by solving a nonlinear system of equations. The methods can be applied to initial value problems where the exact solution is explicitly given in terms of the derivative. They can also be applied in the context of the CDS technique for certain stiff initial value problems of ordinary differential equations, introduced in [1 ] and [2]. Local accuracy and stability of the methods are defined and investigated, and specific methods, containing free parameters, are given. The methods are not convergent, but they yield very good numerical results if applied to the type of problem they are designed for. Their major advantage is that they significantly reduce the amount of implicitness necessary in the numerical solution of certain problems.
The Lanczos algorithm with selective orthogonalization
Authors: Parlett B. N. , Scott D. S.
Published: June 2010

The simple Lanczos process is very effective for finding a few extreme eigenvalues of a large symmetric matrix along with the associated eigenvectors. Unfortunately, the process computes redundant copies of the outermost eigenvectors and has to be used with some skill. In this paper it is shown how a modification called selective orthogonalization stifles the formation of duplicate eigenvectors without increasing the cost of a Lanczos step significantly. The degree of linear independence among the Lanczos vectors is controlled without the costly process of reorthogonalization.
Analysis of optimal finite-element meshes in 𝑅¹
Authors: Babuška I. , Rheinboldt W. C.
Published: June 2010

A theory of a posteriori estimates for the finite-element method was developed earlier by the authors. Based on this theory, for a two-point boundary value problem the existence of a unique optimal mesh distribution is proved and its properties analyzed. This mesh is characterized in terms of certain, easily computable local error indicators which in turn allow for a simple adaptive construction of the mesh and also permit the computation of a very effective a posteriori error bound. While the error estimates are asymptotic in nature, numerical experiments show the results to be excellent already for 10
On the efficient implementation of implicit Runge-Kutta methods
Authors: Varah J. M.
Published: June 2010

Extending some recent ideas of Butcher, we show how one can efficiently implement general implicit Runge-Kutta methods, including those based on Gaussian quadrature formulas which are particularly useful for stiff equations. With this implementation, it appears that these methods are more efficient than the recently proposed semiexplicit methods and their variants.
A three-dimensional analogue to the method of bisections for solving nonlinear equations
Authors: Sikorski Krzysztof
Published: June 2010

This paper deals with a three-dimensional analogue to the method of bisections for solving a nonlinear system of equations

F
(
X
)
=
θ
=

(
0
,
0
,
0

)
T

F(X) = \theta = {(0,0,0)^T}

, which does not require the evaluation of derivatives of F. We divide the original parallelepiped (Figure 2.1) into 8 tetrahedra (Figure 2.2), and then bisect the tetrahedra to form an infinite sequence of tetrahedra, whose vertices converge to

Z
∈

R
3

Z \in {R^3}

such that

F
(
Z
)
=
θ

F(Z) = \theta

. The process of bisecting a tetrahedron

>

|

>

E
1

E
2

E
3

E
4

> | > {E_1}{E_2}{E_3}{E_4}

with vertices

E
i

{E_i}

is defined as follows. We first locate the longest edge

E
i

E
j

,
i
≠
j

{E_i}{E_j},i \ne j

, set

D
=
(

E
i

+

E
j

)

/

2

D = ({E_i} + {E_j})/2

, and then define two new tetrahedra

>

|

>

E
i

D

E
k

E
l

> | > {E_i}D{E_k}{E_l}

and

>

|

>
D

E
j

E
k

E
l

> | > D{E_j}{E_k}{E_l}

, where

j
≠
l
,
l
≠
i
,
i
≠
k
,
k
≠
j

j \ne l,l \ne i,i \ne k,k \ne j

and

k
≠
l

k \ne l

. We give sufficient conditions for convergence of the algorithm. The results of our numerical experiments show that the required storage may be large in some cases.
Transitive graphs with fewer than twenty vertices
Authors: McKay Brendan D.
Published: June 2010

A graph is called transitive if its automorphism group acts transitively on the vertex set. We list the 1031 transitive graphs with fewer than 20 vertices, together with many of their properties.
On the zeros of the Riemann zeta function in the critical strip
Authors: Brent Richard P.
Published: June 2010

We describe a computation which shows that the Riemann zeta function

ζ
(
s
)

\zeta (s)

has exactly 75,000,000 zeros of the form

σ
+
i
t

\sigma + it

it in the region

0
>
t
>
32
,
585
,
736.4

0 > t > 32,585,736.4

; all these zeros are simple and lie on the line

σ
=
1

/

2

\sigma = 1/2

. (A similar result for the first 3,500,000 zeros was established by Rosser, Yohe and Schoenfeld.) Counts of the number of Gram blocks of various types and the number of failures of "Rosser’s rule" are given.
Numerical integrators for stiff and highly oscillatory differential equations
Authors: Fatunla Simeon Ola
Published: June 2010

Some L-stable fourth-order explicit one-step numerical integration formulas which require no matrix inversion are proposed to cope effectively with systems of ordinary differential equations with large Lipschitz constants (including those having highly oscillatory solutions). The implicit integration procedure proposed in Fatunla [11] is further developed to handle a larger class of stiff systems as well as those with highly oscillatory solutions. The same pair of nonlinear equations as in [11] is solved for the stiffness/oscillatory parameters. However, the nonlinear systems are transformed into linear forms and an efficient computational procedure is developed to obtain these parameters. The new schemes compare favorably with the backward differentiation formula (DIFSUB) of Gear [13], [14] and the blended linear multistep methods of Skeel and Kong [24], and the symmetric multistep methods of Lambert and Watson [17].
Calculation of the regulator of a pure cubic field
Authors: Williams H. C. , Cormack G. , Seah E.
Published: June 2010

A description is given of a modified version of Voronoi’s algorithm for obtaining the regulator of a pure cubic field

Q
(

D
3

)

Q(\sqrt [3]{D})

. This new algorithm has the advantage of executing relatively rapidly for large values of D. It also eliminates a computational problem which occurs in almost all algorithms for finding units in algebraic number fields. This is the problem of performing calculations involving algebraic irrationals by using only approximations of these numbers. The algorithm was implemented on a computer and run on all values of

D

(
⩽

10
5

)

D\;( \leqslant {10^5})

such that the class number of

Q

(

D
3

)

Q\;(\sqrt [3]{D})

is not divisible by 3. Several tables summarizing the results of this computation are also presented.
Collocation approximation to eigenvalues of an ordinary differential equation: the principle of the thing
Authors: de Boor Carl , Swartz Blair
Published: June 2010

It is shown that simple eigenvalues of an mth order ordinary differential equation are approximated within

O

(

|

Δ

|

2
k

)

\mathcal {O}(|\Delta {|^{2k}})

by collocation at Gauss points with piecewise polynomial functions of degree

>
m
+
k

> m + k

on a mesh

Δ
\Delta

. The same rate is achieved by certain averages in case the eigenvalue is not simple. The argument relies on an extension and simplification of Osborn’s recent results concerning the approximation of eigenvalues of compact linear maps.
Analysis of convergence of the 𝑇-transformation for power series
Authors: Sidi Avram
Published: June 2010

Recently the present author has given some convergence theorems of general nature for Levin’s nonlinear sequence transformations. In this work these theorems are extended and sharpened to cover the case of power series, both inside and on their circle of convergence. It is shown that one of the two limiting processes considered in the previous work can be used for analytic continuation and a realistic estimate of its rate of convergence is given. Three illustrative examples are also appended.
Outline of a proof that every odd perfect number has at least eight prime factors
Authors: Hagis Peter
Published: June 2010

An argument is outlined which demonstrates that every odd perfect number is divisible by at least eight distinct primes.
On faster convergence of the bisection method for all triangles
Authors: Stynes Martin
Published: June 2010

Let

Δ
A
B
C

\Delta ABC

be a triangle with vertices A, B, and C. It is "bisected" as follows: choose a/the longest side (say AB) of

Δ
A
B
C

\Delta ABC

, let D be the midpoint of AB, then replace

Δ
A
B
C

\Delta ABC

by two triangles

Δ
A
D
C

\Delta ADC

and

Δ
D
B
C

\Delta DBC

. Let

Δ

01

{\Delta _{01}}

be a given triangle. Bisect

Δ

01

{\Delta _{01}}

into two triangles

Δ

11

{\Delta _{11}}

and

Δ

12

{\Delta _{12}}

. Next bisect each

Δ

1
i

,

i
=
1
,
2

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

, forming four new triangles

Δ

2
i

,

i
=
1
,
2
,
3
,
4

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

. Continue thus, forming an infinite sequence

T
j

,

j
=
0
,
1
,
2
,
…

{T_j},\;j = 0,1,2, \ldots

, of sets of triangles, where

T
j

=

{

Δ

j
i

:
1
⩽
i
⩽

2
j

}

{T_j} = \left \{ {{\Delta _{ji}}:1 \leqslant i \leqslant {2^j}} \right \}

. Let

m
j

{m_j}

denote the mesh of

T
j

{T_j}

. It is shown that there exists

N
=
N
(

Δ

01

)

N = N({\Delta _{01}})

such that, for

j
⩾
N

j \geqslant N

,

m

2
j

⩽

(

3

/

2

)
N

(
1

/

2

)

j
−
N

m
0

{m_{2j}} \leqslant {(\sqrt 3 /2)^N}{(1/2)^{j - N}}{m_0}

, thus greatly improving the previous best known bound of

m

2
j

⩽

(

3

/

2

)
j

m
0

{m_{2j}} \leqslant {(\sqrt 3 /2)^j}{m_0}

. It is also shown that only a finite number of distinct shapes occur among the triangles produced, and that, as the method proceeds,

Δ

01

{\Delta _{01}}

tends to become covered by triangles which are approximately equilateral in a certain sense.
Reciprocal polynomials having small measure
Authors: Boyd David W.
Published: June 2010

The measure of a monic polynomial is the product of the absolute value of the roots which lie outside and on the unit circle. We describe an algorithm, based on the root-squaring method of Graeffe, for finding all polynomials with integer coefficients whose measures and degrees are smaller than some previously given bounds. Using the algorithm, we find all such polynomials of degree at most 16 whose measures are at most 1.3. We also find all polynomials of height 1 and degree at most 26 whose measures satisfy this bound. Our results lend some support to Lehmer’s conjecture. In particular, we find no noncyclotomic polynomial whose measure is less than the degree 10 example given by Lehmer in 1933.
The determination of all imaginary, quartic, abelian number fields with class number 1
Authors: Setzer Bennett
Published: June 2010

In this paper, it is proved that there are just seven imaginary number fields, quartic cyclic over the rational field, and having class number 1. These are the quartic, cyclic imaginary subfields of the cyclotomic fields generated by thefth roots of unity, wherefis 16 or is a prime less than 100. This completes the list of imaginary, quartic, abelian number fields with class number 1. There are 54 such fields, with maximal conductor 67.163.
Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable
Authors: Arnold Douglas N. , Douglas Jim , Thomée Vidar
Published: June 2010

A standard Galerkin method for a quasilinear equation of Sobolev type using continuous, piecewise-polynomial spaces is presented and analyzed. Optimal order error estimates are established in various norms, and nodal superconvergence is demonstrated. Discretization in time by explicit single-step methods is discussed.
Product integration for weakly singular integral equations
Authors: Schneider Claus
Published: June 2010

The product integration method is used for the numerical solution of weakly singular integral equations of the second kind. These equations often have solutions which have derivative singularities at the endpoints of the range of integration. Therefore, the order of convergence results of de Hoog and Weiss for smooth solutions do not hold in general. In this paper it is shown that their results may be regained for the general case by using an appropriate nonuniform mesh. The spacing of the knot points is defined by the behavior of the solution at the endpoints. If the solution is smooth enough the mesh becomes uniform. Numerical examples are given.