bluuebunny/crossref_metadata_2025 huggingface dataset:
Reduction of the eigenproblem for Hermitian persymmetric matrices
Authors: Goldstein Marvin J.
Published: June 2010

Both computer time and storage can be saved in the calculation of the eigensystem of Hermitian persymmetric matrices if advantage is taken of their special structure.
Applications of a continued fraction algorithm to some class number problems
Authors: Hendy M. D.
Published: June 2010

We make extensive use of Lagrange’s algorithm for the evaluation of the quotients in the continued fraction expansion of the quadratic surd

ω
\omega

, where

ω
=
√
d

\omega = \surd d

for

d
≡
2
,
3

(
mod

4
)

d \equiv 2,3 \pmod 4

and

(
√
d
−
1
)

/

2

(\surd d - 1)/2

for

d
≡
1

(
mod

4
)

d \equiv 1 \pmod 4

. The recursively generated terms

Q
n

{Q_n}

in his algorithm lead to all norms of primitive algebraic integers of

Q
(
√
d
)

Q(\surd d)

less than

√
(
D

/

4
)

\surd (D/4)

, D being the discriminant. By ensuring that the values

Q
n

{Q_n}

contain at most one small prime, we are able to generate sequences of determinants d of real quadratic fields whose genera usually contain more than one ideal class. Formulae for their fundamental units are given.
On Chebyshev-type quadratures
Authors: Gautschi Walter , Yanagiwara Hiroki
Published: June 2010

According to a result of S. N. Bernstein, n-point Chebyshev quadrature formulas, with all nodes real, do not exist when

n
=
8

n = 8

or

n
≧
10

n \geqq 10

. Modifications of such quadrature formulas have recently been suggested by R. E. Barnhill, J. E. Dennis, Jr. and G. M. Nielson, and by D. Kahaner. We establish here certain empirical observations made by these authors, notably the presence of multiple nodes. We also show how some of the quadrature rules proposed can be constructed by solving single algebraic equations, and we compute the respective nodes to 25 decimal digits. The same formulas also arise in recent work of P. Rabinowitz and N. Richter as limiting cases of optimal Chebyshev-type quadrature rules in a Hilbert space setting.
Global solution of the generalized Abel integral equation by implicit interpolation
Authors: Brunner H.
Published: June 2010

The construction of a (global) approximate solution for a given generalized Abel integral equation may be viewed as a problem of (implicit) interpolation in a prescribed linear space. In this paper, piecewise polynomials (extended spline functions) of a given degree and of class C are used to generate such an approximating function. Results on convergence and error bounds are given, and the practical application of this method is illustrated by a numerical example.
Computation of modified Bessel functions and their ratios
Authors: Amos D. E.
Published: June 2010

An efficient algorithm for calculating ratios
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r Subscript v Baseline left-parenthesis x right-parenthesis equals upper I Subscript v plus 1 Baseline left-parenthesis x right-parenthesis slash upper I Subscript v Baseline left-parenthesis x right-parenthesis comma v greater-than-over-equals 0 comma x greater-than-over-equals 0">

r
v

(
x
)
=

I

v
+
1

(
x
)

/

I
v

(
x
)
,
v
≧
0
,
x
≧
0

{r_v}(x) = {I_{v + 1}}(x)/{I_v}(x),v \geqq 0,x \geqq 0

, is presented. This algorithm in conjunction with the recursion relation for

r
v

(
x
)

{r_v}(x)

gives an alternative to other recursive methods for

I
v

(
x
)

{I_v}(x)

when approximations for low-order Bessel functions are available. Sharp bounds on

r
v

(
x
)

{r_v}(x)

and

I
v

(
x
)

{I_v}(x)

are also established in addition to some monotonicity properties of

r
v

(
x
)

{r_v}(x)

and

r

v

′

(
x
)

r’_{v}(x)

.
Factoring large integers
Authors: Lehman R. Sherman
Published: June 2010

A modification of Fermat’s difference of squares method is used for factoring large integers. This modification permits factoring n in

O
(

n

1

/

3

)

O({n^{1/3}})

elementary operations, where addition, subtraction, multiplication, division, or the extraction of a square root is considered as an elementary operation. A principal part is played by the use of a dissection of the continuum similar to the Farey dissection. This has been programmed for

n
≦
1.05
×

10

20

n \leqq 1.05 \times {10^{20}}

on the CDC 6400.
An implicit fourth order difference method for viscous flows
Authors: Watanabe Daniel S. , Flood J. Richard
Published: June 2010

A new implicit finite-difference scheme for viscous flows is presented. The scheme is based on Simpson’s rule and two-point Hermite interpolation, is uniformly accurate to fourth order in time and space, and is unconditionally stable according to a Fourier stability analysis. Numerical solutions of Burger’s equation are presented to illustrate the order and accuracy of the scheme.
The minimum root separation of a polynomial
Authors: Collins George E. , Horowitz Ellis
Published: June 2010

The minimum root separation of a complex polynomial A is defined as the minimum of the distances between distinct roots of A. For polynomials with Gaussian integer coefficients and no multiple roots, three lower bounds are derived for the root separation. In each case, the bound is a function of the degree n of A and the sum d of the absolute values of the coefficients of A. The notion of a seminorm for a commutative ring is defined, and it is shown how any seminorm can be extended to polynomial rings and matrix rings, obtaining a very general analogue of Hadamard’s determinant theorem.
Inversion of Toeplitz band matrices
Authors: Trench William F.
Published: June 2010

An algorithm for inverting Toeplitz matrices is simplified for Toeplitz band matrices. In some cases, the simplification yields formulas for the elements in the first row and column of the inverse, from which the remaining elements can be easily calculated. Two examples are given. In any case, the simplification yields a recursive method for computing the first row and column of the inverse of an nth order Toeplitz band matrix with

O
(
n
)

O(n)

operations, where

O
(

n
2

)

O({n^2})

are required with the more general algorithm.
Recursive collocation for the numerical solution of stiff ordinary differential equations
Authors: Brunner H.
Published: June 2010

The exact solution of a given stiff system of nonlinear (homogeneous) ordinary differential equations on a given interval I is approximated, on each subinterval

σ
k

{\sigma _k}

corresponding to a partition

π
N

{\pi _N}

of I, by a linear combination

U
k

(
x
)

{U_k}(x)

of exponential functions. The function

U
k

(
x
)

{U_k}(x)

will involve only the "significant" eigenvalues (in a sense to be made precise) of the approximate Jacobian for

σ
k

{\sigma _k}

. The unknown vectors in

U
k

(
x
)

{U_k}(x)

are computed recursively by requiring that

U
k

(
x
)

{U_k}(x)

satisfy the given system at certain suitable points in

σ
k

{\sigma _k}

(collocation), with the additional condition that the collection of these functions

{

U
k

}

\{ {U_k}\}

represent a continuous function on I satisfying the given initial conditions.
Brun’s constant
Authors: Shanks Daniel , Wrench John W.
Published: June 2010

This note reviews previous work and presents new numerical data and analytical development concerning a constant that arises in Brun’s famous theorem about twin primes.
Some primitive polynomials of the third kind
Authors: Beard Jacob T. B. , West Karen I.
Published: June 2010

This paper gives the first primitive polynomial of the third kind of degree n over

GF

(

p
d

)

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

for each p, d, n satisfying

p
>

10
2

,

p
d

>

10
3

,

p

d
n

>

10
6

p > {10^2},{p^d} > {10^3},{p^{dn}} > {10^6}

.
A continued fraction expansion, with a truncation error estimate, for Dawson’s integral
Authors: McCabe J. H.
Published: June 2010

A continued fraction expansion for Dawson’s integral is considered and an estimate of the truncation errors of the convergents of this continued fraction is provided. The continued fraction is shown to provide a sequence of rational approximations to the integral which have good convergence for both small and large values of the argument.
A search for elliptic curves with large rank
Authors: Penney David E. , Pomerance Carl
Published: June 2010

A search procedure is described for finding elliptic curves with rational coefficients for which the group of rational points has large rank. Specific examples are given of elliptic curves with rank

≧
6

\geqq 6

.
Error estimates for the approximation of a class of variational inequalities
Authors: Falk Richard S.
Published: June 2010

In this paper, we prove a general approximation theorem useful in obtaining order of convergence estimates for the approximation of the solutions of a class of variational inequalities. The theorem is then applied to obtain an "optimal" rate of convergence for the approximation of a second-order elliptic problem with convex set

K
=
{
υ
∈

H
0
1

(
Ω
)
:
υ
⩾
χ

K = \{ \upsilon \in H_0^1(\Omega ):\upsilon \geqslant \chi

a.e. in

Ω
\Omega

}.
The Rayleigh quotient iteration and some generalizations for nonnormal matrices
Authors: Parlett B. N.
Published: June 2010

The Rayleigh Quotient Iteration (RQI) was developed for real symmetric matrices. Its rapid local convergence is due to the stationarity of the Rayleigh Quotient at an eigenvector. Its excellent global properties are due to the monotonic decrease in the norms of the residuals. These facts are established for normal matrices. Both properties fail for nonnormal matrices and no generalization of the iteration has recaptured both of them. We examine methods which employ either one or the other of them.
On the stability of uniformly asymptotically diagonal systems
Authors: Anderssen R. S. , Omodei B. J.
Published: June 2010

In a number of papers ([1], [2]), Delves and Mead have derived some useful (though limited) rate of convergence results which can be applied to variational approximations for the solution of linear positive definite operator equations when the coordinate system is uniformly asymptotically diagonal. Independently, Mikhlin [5] has examined the stability of such variational approximations in the case of positive definite operators and concluded that the use of strongly minimal coordinate systems is a necessary and sufficient condition for their stability. Since, in general, the Delves and Mead results will only be applicable to actual variational approximations when their uniformly asymptotically diagonal system is at least strongly minimal, we examine the properties of uniformly asymptotically diagonal systems in terms of the minimal classification of Mikhlin. We show that (a) a normalized uniformly asymptotically diagonal system is either nonstrongly minimal or almost orthonormal; (b) the largest eigenvalue of a normalized uniformly asymptotically diagonal system is bounded above, independently of the size of the system; (c) the special property of normalized uniformly asymptotically diagonal systems mentioned in (b) is often insufficient to prevent their yielding unstable results when these systems are not strongly minimal.
A class of Hessenberg matrices with known pseudoinverse and Drazin inverse
Authors: Singh Inderjit , Poole George , Boullion Thomas
Published: June 2010

In this paper, a class of Hessenberg matrices is presented for adoption as test matrices. The Moore-Penrose inverse and the Drazin inverse for each member of this class are determined explicitly.
Complex zeros of the Jonquière or polylogarithm function
Authors: Fornberg B. , Kölbig K. S.
Published: June 2010

Complex zero trajectories of the function\[F(x,s)=∑k=1∞xkksF(x,s) = \sum \limits _{k = 1}^\infty {\frac {{{x^k}}}{{{k^s}}}}\]are investigated for realxwith|x|>1|x| > 1in the complexs-plane. It becomes apparent that there exist several classes of such trajectories, depending on their behaviour for|x|→1|x| \to 1. In particular, trajectories are found which tend towards the zeros of the Riemann zeta functionζ(s)\zeta (s)asx→−1x \to - 1, and approach these zeros closely asx→1−ρx \to 1 - \rhofor small but finiteρ>0\rho > 0. However, the latter trajectories appear to descend to the points=1s = 1asρ→0\rho \to 0. Both, forx→−1x \to - 1andx→1x \to 1, there are trajectories which do not tend towards zeros ofζ(s)\zeta (s). The asymptotic behaviour of the trajectories for|x|→0|x| \to 0is discussed. A conjecture of Pickard concerning the zeros ofF(x,s)F(x,s)is shown to be false.
On the prime factors of (²ⁿ_{𝑛})
Authors: Erdős P. , Graham R. L. , Ruzsa I. Z. , Straus E. G.
Published: June 2010

Several quantitative results are given expressing the fact that

(
n

2
n

)

(_n^{2n})

is usually divisible by a high power of the small primes. On the other hand, it is shown that for any two primes p and q, there exist infinitely many n for which

(

(
n

2
n

)
,
p
q
)
=
1

((_n^{2n}),pq) = 1

.
On the osculatory rational interpolation problem
Authors: Wuytack Luc
Published: June 2010

The problem of the existence and construction of a table of osculating rational functions

r

1
,
m

{r_{1,m}}

for

1
,
m
⩾
0

1,m \geqslant 0

is considered. First, a survey is given of some results from the theory of osculatory rational interpolation of order

s
i

−
1

{s_i} - 1

at points

x
i

{x_i}

for

i
⩾
0

i \geqslant 0

. Using these results, we prove the existence of continued fractions of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c 0 plus c 1 dot left-parenthesis x minus y 0 right-parenthesis plus ellipsis plus c Subscript k Baseline dot left-parenthesis x minus y 0 right-parenthesis ellipsis left-parenthesis x minus y Subscript k minus 1 Baseline right-parenthesis plus StartFraction c Subscript k plus 1 Baseline dot left-parenthesis x minus y 0 right-parenthesis ellipsis left-parenthesis x minus y Subscript k Baseline right-parenthesis Over 1 EndFraction plus StartFraction c Subscript k plus 2 Baseline dot left-parenthesis x minus y Subscript k plus 1 Baseline right-parenthesis Over 1 EndFraction plus StartFraction c Subscript k plus 3 Baseline dot left-parenthesis x minus y Subscript k plus 2 Baseline right-parenthesis Over 1 EndFraction plus ellipsis comma">

c
0

+

c
1

⋅
(
x
−

y
0

)
+
…
+

c
k

⋅
(
x
−

y
0

)
…
(
x
−

y

k
−
1

)
+

c

k
+
1

⋅
(
x
−

y
0

)
…
(
x
−

y
k

)

1

+

c

k
+
2

⋅
(
x
−

y

k
+
1

)

1

+

c

k
+
3

⋅
(
x
−

y

k
+
2

)

1

+
…
,

{c_0} + {c_1} \cdot (x - {y_0}) + \ldots + {c_k} \cdot (x - {y_0}) \ldots (x - {y_{k - 1}}) + \frac {{{c_{k + 1}} \cdot (x - {y_0}) \ldots (x - {y_k})}}{1} + \frac {{{c_{k + 2}} \cdot (x - {y_{k + 1}})}}{1} + \frac {{{c_{k + 3}} \cdot (x - {y_{k + 2}})}}{1} + \ldots ,

\]
with the

y
k

{y_k}

suitably selected from among the

x
i

{x_i}

, whose convergents form the elements

r

k
,
0

,

r

k
+
1
,
0

,

r

k
+
1
,
1

,

r

k
+
2
,
1

,
…

{r_{k,0}},{r_{k + 1,0}},{r_{k + 1,1}},{r_{k + 2,1}}, \ldots

of the table. The properties of these continued fractions make it possible to derive an algorithm for constructing their coefficients

c
i

{c_i}

for

i
⩾
0

i \geqslant 0

. This algorithm is a generalization of the qd-algorithm.
A difficulty in Williams’ algorithm for interpolating rationals
Authors: Dunham Charles B.
Published: June 2010

J. Williams has developed a theory and an algorithm for best Chebyshev approximation of decay-type functions by an oscillation factor times a negative power of a linear form (in particular a polynomial). It is shown that the levelling equations of the algorithm may not have an admissible solution.
A Fourier method for the numerical solution of Poisson’s equation
Authors: Sköllermo Gunilla
Published: June 2010

A method for the solution of Poisson’s equation in a rectangle, based on the relation between the Fourier coefficients for the solution and those for the right-hand side, is developed. The Fast Fourier Transform is used for the computation and its influence on the accuracy is studied. Error estimates are given and the method is shown to be second order accurate under certain general conditions on the smoothness of the solution. The accuracy is found to be limited by the lack of smoothness of the periodic extension of the inhomogeneous term. Higher order methods are then derived with the aid of special solutions. This reduces the problem to a case with sufficiently smooth data. A comparison of accuracy and efficiency is made between our Fourier method and the Buneman algorithm for the solution of the standard finite difference formulae.
On the largest prime divisor of an odd perfect number. II
Authors: Hagis Peter , McDaniel Wayne L.
Published: June 2010

It is proved here that every odd perfect number has a prime factor greater than 100110.
An algorithm for winding numbers for closed polygonal paths
Authors: Leland Kenneth O.
Published: June 2010

A winding number algorithm for closed polygonal paths (not necessarily simple) based on the notion of counting the number of oriented "signed cuts" of the negative x axis by the path is given. The algorithm is justified by a theory of integer-valued analogues of the complex log function. The algorithm is much simpler than those of J. V. Petty [this journal, v. 27, 1973, pp. 333-337] and H. R. P. Ferguson [Notices Amer. Math. Soc., v. 20, 1973, p. A-211] and leads to faster computation.
An equation of Mordell
Authors: Bremner Andrew
Published: June 2010

All integer solutions of the Diophantine equation

6

y
2

=
(
x
+
1
)
(

x
2

−
x
+
6
)

6{y^2} = (x + 1)({x^2} - x + 6)

are found.
Some properties of orthogonal polynomials
Authors: Hunter D. B.
Published: June 2010

Some results are obtained concerning the signs of the coefficients in the expansions in powers of

x

−
1

,

(
1
+
x

)

−
1

{x^{ - 1}},{(1 + x)^{ - 1}}

or

(
1
−
x

)

−
1

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

of

1

/

p
n

(
x
)

1/{p_n}(x)

and

q
n

(
x
)

{q_n}(x)

, where

p
n

(
x
)

{p_n}(x)

is the polynomial of degree n in the orthogonal sequence associated with a given weight-function

w
(
x
)

w(x)

over

(
−
1
,
1
)

( - 1,1)

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q Subscript n Baseline left-parenthesis x right-parenthesis equals integral Subscript negative 1 Superscript 1 Baseline w left-parenthesis t right-parenthesis p Subscript n Baseline left-parenthesis t right-parenthesis left-parenthesis x minus t right-parenthesis Superscript negative 1 Baseline d t">

q
n

(
x
)
=

∫

−
1

1

w
(
t
)

p
n

(
t
)

(
x
−
t

)

−
1

d
t

{q_n}(x) = \smallint _{ - 1}^1w(t){p_n}(t){(x - t)^{ - 1}}dt

.
Gershgorin theory for the generalized eigenvalue problem 𝐴𝑥=𝜆𝐵𝑥
Authors: Stewart G. W.
Published: June 2010

A generalization of Gershgorin’s theorem is developed for the eigenvalue problem

A
x
=
λ
B
x

Ax = \lambda Bx

and is applied to obtain perturbation bounds for multiple eigenvalues. The results are interpreted in terms of the chordal metric on the Riemann sphere, which is especially convenient for treating infinite eigenvalues.
On Bairstow’s method for the solution of polynomial equations
Authors: Brodlie Kenneth W.
Published: June 2010

We show that Bairstow’s method is just one member of a family of similar algorithms for determining a quadratic factor of a polynomial. We suggest a way of choosing an appropriate member of this family for a particular problem. Numerical results indicate that our strategy compares favorably with the classical Bairstow algorithm.
A necessary and sufficient condition for transcendency
Authors: Mahler K.
Published: June 2010

As has been known for many years (see, e.g., K. Mahler, J. Reine Angew. Math., v. 166, 1932, pp. 118-150), a real or complex number

ζ
\zeta

is transcendental if and only if the following condition is satisfied. To every positive number

ω
\omega

there exists a positive integer n and an infinite sequence of distinct polynomials
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace p Subscript r Baseline left-parenthesis z right-parenthesis right-brace equals left-brace p Subscript r 0 Baseline plus p Subscript r 1 Baseline z plus midline-horizontal-ellipsis plus p Subscript r Sub Subscript n Subscript Baseline z Superscript n Baseline right-brace">

{

p
r

(
z
)
}
=
{

p

r
0

+

p

r
1

z
+
⋯
+

p

r
n

z
n

}

\{ {p_r}(z)\} = \{ {p_{{r_0}}} + {p_{{r_1}}}z + \cdots + {p_{{r_n}}}{z^n}\}

at most of degree n with integral coefficients, such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than StartAbsoluteValue p Subscript r Baseline left-parenthesis zeta right-parenthesis EndAbsoluteValue less-than-or-slanted-equals left-brace p Subscript r 0 Superscript 2 Baseline plus p Subscript r 1 Superscript 2 Baseline plus midline-horizontal-ellipsis plus p Subscript r Sub Subscript n Subscript Superscript 2 Baseline right-brace Superscript negative omega Baseline f o r a l l r period">

0
>

|

p
r

(
ζ
)

|

⩽

{

p

r
0

2

+

p

r
1

2

+
⋯
+

p

r
n

2

}

−
ω

f
o
r

a
l
l

r
.

0 > |{p_r}(\zeta )| \leqslant {\{ p_{{r_0}}^2 + p_{{r_1}}^2 + \cdots + p_{{r_n}}^2\} ^{ - \omega }}\quad for\;all\quad r.

\]
In the present note I prove a simpler test which makes the transcendency of

ζ
\zeta

depend on the approximation behaviour of a single sequence of distinct polynomials of arbitrary degrees with integral coefficients.
New primality criteria and factorizations of 2^{𝑚}±1
Authors: Brillhart John , Lehmer D. H. , Selfridge J. L.
Published: June 2010

A collection of theorems is developed for testing a given integer N for primality. The first type of theorem considered is based on the converse of Fermat’s theorem and uses factors of

N
−
1

N - 1

. The second type is based on divisibility properties of Lucas sequences and uses factors of

N
+
1

N + 1

. The third type uses factors of both

N
−
1

N - 1

and

N
+
1

N + 1

and provides a more effective, yet more complicated, primality test. The search bound for factors of

N
±
1

N \pm 1

and properties of the hyperbola

N
=

x
2

−

y
2

N = {x^2} - {y^2}

are utilized in the theory for the first time. A collection of 133 new complete factorizations of

2
m

±
1

{2^m} \pm 1

and associated numbers is included, along with two status lists: one for the complete factorizations of

2
m

±
1

{2^m} \pm 1

; the other for the original Mersenne numbers.
A curiosity of low-order explicit Runge-Kutta methods
Authors: Oliver J.
Published: June 2010

By introducing an additional parameter into the first stage of the explicit Runge-Kutta process, new formulae of second and third order are derived, offering improved error bounds in the second-order case.
On lower and upper bounds of the difference between the arithmetic and the geometric mean
Authors: Tung S. H.
Published: June 2010

Lower and upper bounds of the difference between the arithmetic and the geometric mean of n quantities are given here in terms of n, the smallest value a and the largest value A of given n quantities. Also, an upper bound for the difference, independent of n, is given in terms of a and A. All the bounds obtained are sharp.
Sums of distinct elements from a fixed set
Authors: Kløve Torleiv
Published: June 2010

A sequence of natural numbers is complete if every large integer is a sum of distinct elements of the sequence. The greatest integer which is not such a sum is called the threshold of completeness. Richert developed a method to compute the threshold of completeness. We prove that Richert’s method applies to a large class of complete sequences. Further, we consider in some detail these concepts for the sequences of powers (with fixed exponents) and give numerical results.
Elliptic curves over finite fields. II
Authors: Borosh I. , Moreno C. J. , Porta H.
Published: June 2010

The class groups of certain elliptic function fields without complex multiplications are computed. Questions about the structure of these groups and the arithmetical nature of their orders are considered.
A Galerkin method for a nonlinear Dirichlet problem
Authors: Douglas Jim , Dupont Todd
Published: June 2010

A Galerkin method due to Nitsche for treating the Dirichlet problem for a linear second-order elliptic equation is extended to cover the nonlinear equation

∇
⋅
(
a
(
x
,
u
)
∇
u
)
=
f

\nabla \cdot (a(x,u)\nabla u) = f

. The asymptotic error estimates are of the same form as in the linear case. Newton’s method can be used to solve the nonlinear algebraic equations.
Characteristic 𝑚-sequences
Authors: Willett Michael
Published: June 2010

The initial k-tuple of the characteristic m-sequence associated with a primitive polynomial of degree k over

G
F
(
2
)

GF(2)

is given for

2
⩽
k
⩽
168

2 \leqslant k \leqslant 168

.
Inequalities for hypergeometric functions
Authors: Buschman R. G.
Published: June 2010

The upper and lower bounds for the determinant of a dominant diagonal matrix have been used recently to obtain bounds on the classical orthogonal polynomials. Similar methods are used here on the hypergeometric functions of Gauss and of Kummer.
A uniform distribution question related to numerical analysis
Authors: Niederreiter Harald , Osgood Charles F.
Published: June 2010

Using the theory of uniform distribution modulo one, it is shown that under certain conditions on the real-valued functions

α
(
x
)

\alpha (x)

and

g
(
x
)

g(x)

on [0,1 ],
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h sigma-summation Underscript gamma equals 1 Overscript left-bracket h Superscript negative 1 Baseline right-bracket Endscripts left-brace h Superscript negative 1 Baseline alpha left-parenthesis gamma h right-parenthesis right-brace Superscript m Baseline g left-parenthesis gamma h right-parenthesis equals left-parenthesis m plus 1 right-parenthesis Superscript negative 1 Baseline integral Subscript 0 Superscript 1 Baseline g left-parenthesis x right-parenthesis d x plus o left-parenthesis h Superscript 1 slash 3 Baseline log StartFraction 1 Over h EndFraction right-parenthesis as h right-arrow 0 plus comma">

h

∑

γ
=
1

[

h

−
1

]

{

h

−
1

α
(
γ
h
)
}

m

g
(
γ
h
)
=

(
m
+
1
)

−
1

∫
0
1

g
(
x
)

d
x
+
o

(

h

1

/

3

log
⁡

1
h

)

as

h
→
0
+
,

h\;\sum \limits _{\gamma = 1}^{[{h^{ - 1}}]} {{{\{ {h^{ - 1}}\alpha (\gamma h)\} }^m}g(\gamma h) = {{(m + 1)}^{ - 1}}\int _0^1 {g(x)\;dx + o\left ( {{h^{1/3}}\log \frac {1}{h}} \right )} \quad {\text {as}}\;h \to 0 + ,}

\]
where

m
>
0

m > 0

and x denotes the fractional part of x. The conditions are as follows:

α
(
x
)

\alpha (x)

exists for all but finitely many points in [0, 1], changes sign at most finitely often, and is bounded away in absolute value from both 0 and

∞
\infty

, whereas

g
(
x
)

g(x)

is of bounded variation on [0,1]. Also, under these conditions on

α
(
x
)

\alpha (x)

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h sigma-summation Underscript gamma equals 1 Overscript left-bracket h Superscript negative 1 Baseline right-bracket Endscripts left-brace h Superscript negative 1 Baseline alpha left-parenthesis gamma h right-parenthesis right-brace Superscript m Baseline equals left-parenthesis m plus 1 right-parenthesis Superscript negative 1 Baseline plus o left-parenthesis h Superscript 1 slash 3 Baseline right-parenthesis as h right-arrow 0 plus period">

h

∑

γ
=
1

[

h

−
1

]

{

h

−
1

α
(
γ
h
)
}

m

=

(
m
+
1
)

−
1

+
o
(

h

1

/

3

)

as

h
→
0
+
.

h\;\sum \limits _{\gamma = 1}^{[{h^{ - 1}}]} {{{\{ {h^{ - 1}}\alpha (\gamma h)\} }^m} = {{(m + 1)}^{ - 1}} + o({h^{1/3}})\quad {\text {as}}\;h \to 0 + .}

\]
These results, which are, in fact, deduced from somewhat more general propositions, answer questions of Feldstein connected with discretization methods for differential equations.
Pseudospline interpolation for space curves
Authors: Thomas D. H.
Published: June 2010

A method for interpolating a curve through points in space is described. It is the direct analogue of Fowler-Wilson or pseudospline interpolation for plane curves in that local coordinate systems, cubic polynomials of suitable parameters, and mildly nonlinear equations are used to obtain a continuous interpolating curve with continuous tangent and curvature vectors.
On measures of ill-conditioning for nonlinear equations
Authors: Rheinboldt Werner C.
Published: June 2010

Let

x
∗

{x^\ast }

be a solution of the nonlinear equation

F
x
=
b

Fx = b

on a normed linear space and

y
∗

{y^\ast }

that of a perturbed equation

G
x
=
c

Gx = c

. Estimates for the relativized error between

x
∗

{x^\ast }

and

y
∗

{y^\ast }

are derived which extend a known estimate for the corresponding matrix case. The condition number of F depends now also on the domain, and special considerations are needed to determine the existence of the solution of the perturbed equation. For differentiable F, when the domain shrinks to a point, the condition number of F is shown to reduce to that of the derivative at that point.
Rational approximation with a vanishing weight function and with a fixed value at zero
Authors: Dunham Charles B.
Published: June 2010

Chebyshev approximation by ordinary rational functions with respect to a vanishing weight function is considered. A best approximation is characterized by alternation and is unique but may not exist. The problem arises in Kuki’s technique for rational approximation with interpolation at zero and with Williams’ interpolating rationals.
On an algorithm of Billevich for finding units in algebraic fields
Authors: Steiner Ray , Rudman Ronald
Published: June 2010

The well-known algorithm of Billevich for finding units in algebraic number fields is derived by algebraic methods. Some tables of units in cubic and quartic fields are given.
The QR algorithm and Hyman’s method on vector computers
Authors: Ward Robert C.
Published: June 2010

The implementation on vector computers of the QR algorithm and of iterative schemes based on obtaining the determinant and its derivatives by Hyman’s method are presented. It is shown that iterative schemes based on Hyman’s method will probably be more efficient than the QR algorithm on vector computers for large matrices. A theoretical comparison of the Laguerre iterative scheme with the QR algorithm is presented using the latest available CDC STAR-100 instruction execution times. In addition, the results of several test cases run on the Laguerre-Hyman algorithm on a serial computer are reported.
An error functional expansion for 𝑁-dimensional quadrature with an integrand function singular at a point
Authors: Lyness J. N.
Published: June 2010

Let If be the integral of

f
(

x
→

)

f(\vec x)

over an N-dimensional hypercube and

Q

(
m
)

f

{Q^{(m)}}f

be the approximation to If obtained by subdividing the hypercube into

m
N

{m^N}

equal subhypercubes and applying the same quadrature rule Q to each. In order to extrapolate efficiently for If on the basis of several different approximations

Q

(

m
i

)

f

{Q^{({m_i})}}f

, it is necessary to know the form of the error functional

Q

(
m
)

f
−
I
f

{Q^{(m)}}f - If

as an expansion in m. When

f
(

x
→

)

f(\vec x)

has a singularity, the conventional form (with inverse even powers of m) is not usually valid. In this paper we derive the expansion in the case in which

f
(

x
→

)

f(\vec x)

has the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis ModifyingAbove x With right-arrow right-parenthesis equals r Superscript alpha Baseline phi left-parenthesis ModifyingAbove theta With right-arrow right-parenthesis h left-parenthesis r right-parenthesis g left-parenthesis ModifyingAbove x With right-arrow right-parenthesis comma alpha greater-than negative upper N comma">

f
(

x
→

)
=

r
α

φ
(

θ
→

)
h
(
r
)
g
(

x
→

)
,

α
>
−
N
,

f(\vec x) = {r^\alpha }\varphi (\vec \theta )h(r)g(\vec x),\quad \alpha > - N,

\]
the only singularity being at the origin, a vertex of the unit hypercube of integration. Here

(
r
,

θ
→

)

(r,\vec \theta )

represents the hyperspherical coordinates of

(

x
→

)

(\vec x)

. It is shown that for this integrand the error function expansion includes only terms
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript alpha plus upper N plus t Baseline slash m Superscript alpha plus upper N plus t Baseline comma upper B Subscript t Baseline slash m Superscript t Baseline comma upper C Subscript alpha plus upper N plus t Baseline ln m slash m Superscript alpha plus upper N plus t Baseline comma t equals 1 comma 2 comma ellipsis">

A

α
+
N
+
t

/

m

α
+
N
+
t

,

B
t

/

m
t

,

C

α
+
N
+
t

ln
⁡
m

/

m

α
+
N
+
t

,
t
=
1
,
2
,
…

{A_{\alpha + N + t}}/{m^{\alpha + N + t}},{B_t}/{m^t},{C_{\alpha + N + t}}\ln m/{m^{\alpha + N + t}},t = 1,2, \ldots

. The coefficients depend only on the integrand function

f
(

x
→

)

f(\vec x)

and the quadrature rule Q. For several easily recognizable classes of integrand function and for most familiar quadrature rules some of these coefficients are zero. An analogous expansion for the error functional with integrand function

F
(

x
→

)
=
ln
⁡
r
f
(

x
→

)

F(\vec x) = \ln rf(\vec x)

is also derived.
Tridiagonal fourth order approximations to general two-point nonlinear boundary value problems with mixed boundary conditions
Authors: Stepleman Robert S.
Published: June 2010

This paper develops fourth order discretizations to the two-point boundary value problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row y Superscript left-parenthesis 2 right-parenthesis Baseline left-parenthesis t right-parenthesis equals f left-parenthesis t comma y left-parenthesis t right-parenthesis comma y Superscript left-parenthesis 1 right-parenthesis Baseline left-parenthesis t right-parenthesis right-parenthesis comma 2nd Row alpha 0 y left-parenthesis 0 right-parenthesis minus beta 0 y Superscript left-parenthesis 1 right-parenthesis Baseline left-parenthesis 0 right-parenthesis equals delta 0 comma alpha 1 y left-parenthesis 1 right-parenthesis plus beta 1 y Superscript left-parenthesis 1 right-parenthesis Baseline left-parenthesis 1 right-parenthesis equals delta 1 period EndLayout">

y

(
2
)

(
t
)
=
f
(
t
,
y
(
t
)
,

y

(
1
)

(
t
)
)
,

α
0

y
(
0
)
−

β
0

y

(
1
)

(
0
)
=

δ
0

,

α
1

y
(
1
)
+

β
1

y

(
1
)

(
1
)
=

δ
1

.

\begin {array}{*{20}{c}} {{y^{(2)}}(t) = f(t,y(t),{y^{(1)}}(t)),} \\ {{\alpha _0}y(0) - {\beta _0}{y^{(1)}}(0) = {\delta _0},\quad {\alpha _1}y(1) + {\beta _1}{y^{(1)}}(1) = {\delta _1}.} \\ \end {array}

\]
These discretizations have the desirable properties that they are tridiagonal and of "positive type".
The convergence of the Ben-Israel iteration for nonlinear least squares problems
Authors: Boggs Paul T.
Published: June 2010

Ben-Israel [1] proposed a method for the solution of the nonlinear least squares problem

min

x
∈
D

‖

F
(
x
)

‖

2

{\min _{x \in D}}{\left \| {F(x)} \right \|_2}

where

F
:
D
⊂

R
n

→

R
m

F:D \subset {R^n} \to {R^m}

. This procedure takes the form

x

k
+
1

=

x
k

−

F
′

(

x
k

)
+

F
(

x
k

)

{x_{k + 1}} = {x_k} - F’{({x_k})^ + }F({x_k})

where

F
′

(

x
k

)
+

F’{({x_k})^ + }

denotes the Moore-Penrose generalized inverse of the Fréchet derivative of F. We give a general convergence theorem for the method based on Lyapunov stability theory for ordinary difference equations. In the case where there is a connected set of solution points, it is often of interest to determine the minimum norm least squares solution. We show that the Ben-Israel iteration has no predisposition toward the minimum norm solution, but that any limit point of the sequence generated by the Ben-Israel iteration is a least squares solution.
A class of accelerated conjugate direction methods for linearly constrained minimization problems
Authors: Best Michael J. , Ritter Klaus
Published: June 2010

A class of algorithms are described for the minimization of a function of n variables subject to linear inequality constraints. Under weak conditions convergence to a stationary point is demonstrated. The method uses a mixture of conjugate direction constructing and accelerating steps. Any mixture, for example alternation, may be used provided that the subsequence of conjugate direction constructing steps is infinite. The mixture of steps may be specified so that under appropriate assumptions the rate of convergence of the method is two-step superlinear or

(
n
−
p
+
1
)

(n - p + 1)

-step cubic where p is the number of constraints active at a stationary point. The accelerating step is always superlinearly convergent. A condition is given under which the alternating policy is every step superlinear. Computational results are given for several test problems.
A necessary condition for 𝐴-stability of multistep multiderivative methods
Authors: Jeltsch Rolf
Published: June 2010

The region of absolute stability of multistep multiderivative methods is studied in a neighborhood of the origin. This leads to a necessary condition for A-stability. For methods where

ρ
(
ζ
)

/

(
ζ
−
1
)

\rho (\zeta )/(\zeta - 1)

has no roots of modulus 1 this condition can be checked very easily. For Hermite interpolatory and Adams type methods a necessary condition for A-stability is found which depends only on the error order and the number of derivatives used at

(

x

n
+
k

,

y

n
+
k

)

({x_{n + k}},{y_{n + k}})

.
Optimal 𝐿^{∞} estimates for the finite element method on irregular meshes
Authors: Scott Ridgway
Published: June 2010

Uniform estimates for the error in the finite element method are derived for a model problem on a general triangular mesh in two dimensions. These are optimal if the degree of the piecewise polynomials is greater than one. Similar estimates of the error are also derived in

L
p

{L^p}

. As an intermediate step, an

L
1

{L^1}

estimate of the gradient of the error in the finite element approximation of the Green’s function is proved that is optimal for all degrees.
Asymptotic formulas related to free products of cyclic groups
Authors: Newman Morris
Published: June 2010

Asymptotic formulas for the number of subgroups of a given index of the free product of finitely many cyclic groups are given. The classical modular group

Γ
\Gamma

is discussed in detail, and a table of the number of subgroups of

Γ
\Gamma

of index n is given for

1
⩽
n
⩽
100

1 \leqslant n \leqslant 100

.
Subtracting out complex singularities in numerical integration
Authors: Lether F. G.
Published: June 2010

This paper is concerned with the numerical approximation of definite integrals over

[
−
1
,
1
]

[ - 1,1]

, in which the function f to be integrated has isolated singularities near

[
−
1
,
1
]

[ - 1,1]

. Complex variable techniques are used to study the effectiveness of the method of subtracting out complex singularities.
A generalization of the Jenkins-Traub method
Authors: Ford J. A.
Published: June 2010

A class of methods for finding zeros of polynomials is derived which depends upon an arbitrary parameter

ρ
\rho

. The Jenkins-Traub algorithm is a special case, corresponding to the choice

ρ
=
∞

\rho = \infty

. Global convergence is proved for large and small values of

ρ
\rho

and a duality between pairs of members is exhibited. Finally, we show that many members of the class (including the Jenkins-Traub method) converge with R-order at least 2.618..., which improves upon the result obtained by Jenkins and Traub [3].
One-step piecewise polynomial multiple collocation methods for initial value problems
Authors: Hennart J. P.
Published: June 2010

New methods are proposed for the numerical solution of systems of first-order differential equations. On each subinterval of a given mesh of size h, a polynomial of degree l is constructed, its parameters being determined by a multiple collocation technique. The resulting piecewise polynomial approximation is of order

O
(

h

l
+
1

)

O({h^{l + 1}})

at the mesh points and between them. In addition, the jth derivatives of the approximation on each subinterval provide approximations of order

O
(

h

l
+
1
−
j

)

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

,

j
=
1
,
…
,
l

j = 1, \ldots ,l

. Some of the methods proposed are shown to be A-stable or even strongly A-stable.
Multi-level adaptive solutions to boundary-value problems
Authors: Brandt Achi
Published: June 2010

The boundary-value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes. Interactions between these levels enable us (i) to solve the possibly nonlinear system of n discrete equations in

O
(
n
)

O(n)

operations (40n additions and shifts for Poisson problems); (ii) to conveniently adapt the discretization (the local mesh size, local order of approximation, etc.) to the evolving solution in a nearly optimal way, obtaining "

∞
\infty

-order" approximations and low n, even when singularities are present. General theoretical analysis of the numerical process. Numerical experiments with linear and nonlinear, elliptic and mixed-type (transonic flow) problems-confirm theoretical predictions. Similar techniques for initial-value problems are briefly discussed.
Higher order local accuracy by averaging in the finite element method
Authors: Bramble J. H. , Schatz A. H.
Published: June 2010

Let

u
h

{u_h}

be a Ritz-Galerkin approximation, corresponding to the solution u of an elliptic boundary value problem, which is based on a uniform subdivision in the interior of the domain. In this paper we show that by "averaging" the values of

u
h

{u_h}

in the neighborhood of a point x we may (for a wide class of problems) construct an approximation to

u
(
x
)

u(x)

which is often a better approximation than

u
h

(
x
)

{u_h}(x)

itself. The "averaging" operator does not depend on the specific elliptic operator involved and is easily constructed.
Certain pure cubic fields with class-number one
Authors: Williams H. C.
Published: June 2010

A description is given of the results of some calculations performed to determine the class number of each of the pure cubic fieldsQ(q3)Q(\sqrt [3]{q}), whereq(≡−1(mod3))q\;( \equiv - 1\;\pmod 3)is a prime andq>35,100q > 35,100. The stability of the percentage of these fields having class-number one is examined.
Computation of the solution of 𝑥³+𝐷𝑦³=1
Authors: Williams H. C. , Holte R.
Published: June 2010

A computer technique for finding integer solutions of
\[

x
3

+
D

y
3

=
1

{x^3} + D{y^3} = 1

\]
is described, and a table of all integer solutions of this equation for all positive

D
⩽
50000

D \leqslant 50000

is presented. Some theoretic results which describe certain values of D for which the equation has no nontrivial solution are also given.
Computation of the regular continued fraction for Euler’s constant
Authors: Brent Richard P.
Published: June 2010

We describe a computation of the first 20,000 partial quotients in the regular continued fractions for Euler’s constant

γ
=
0.577
…

\gamma = 0.577 \ldots

and

exp
⁡
(
γ
)
=
1.781
…
.

\exp (\gamma ) = 1.781 \ldots .

A preliminary step was the calculation of

γ
\gamma

and

exp
⁡
(
γ
)

\exp (\gamma )

to 20,700D. It follows from the continued fractions that, if

γ
\gamma

or

exp
⁡
(
γ
)

\exp (\gamma )

is of the form

P

/

Q

P/Q

for integers P and Q, then

|

Q

|

>

10

10000

|Q| > {10^{10000}}

.
Absorbing boundary conditions for the numerical simulation of waves
Authors: Engquist Bjorn , Majda Andrew
Published: June 2010

In practical calculations, it is often essential to introduce artificial boundaries to limit the area of computation. Here we develop a systematic method for obtaining a hierarchy of local boundary conditions at these artificial boundaries. These boundary conditions not only guarantee stable difference approximations but also minimize the (unphysical) artificial reflections which occur at the boundaries.
Roots of two transcendental equations involving spherical Bessel functions
Authors: Pexton Robert L. , Steiger Arno D.
Published: June 2010

Roots of the transcendental equations
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j Subscript l Baseline left-parenthesis alpha lamda right-parenthesis y Subscript l Baseline left-parenthesis lamda right-parenthesis equals j Subscript l Baseline left-parenthesis lamda right-parenthesis y Subscript l Baseline left-parenthesis alpha lamda right-parenthesis">

j
l

(
α
λ
)

y
l

(
λ
)
=

j
l

(
λ
)

y
l

(
α
λ
)

{j_l}(\alpha \lambda ){y_l}(\lambda ) = {j_l}(\lambda ){y_l}(\alpha \lambda )

and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket x j Subscript l Baseline left-parenthesis x right-parenthesis right-bracket prime Subscript x equals alpha eta Baseline left-bracket x y Subscript l Baseline left-parenthesis x right-parenthesis right-bracket Subscript x equals eta Superscript prime Baseline equals left-bracket x j Subscript l Baseline left-parenthesis x right-parenthesis right-bracket prime Subscript x equals eta Baseline left-bracket x y Subscript l Baseline left-parenthesis x right-parenthesis right-bracket prime Subscript x equals alpha eta">

[
x

j
l

(
x
)

]

x
=
α
η

′

[
x

y
l

(
x
)

]

x
=
η

′

=

[
x

j
l

(
x
)

]

x
=
η

′

[
x

y
l

(
x
)

]

x
=
α
η

′

{[x{j_l}(x)]’_{x = \alpha \eta }}{[x{y_l}(x)]’_{x = \eta }} = {[x{j_l}(x)]’_{x = \eta }}{[x{y_l}(x)]’_{x = \alpha \eta }}

\]
for the spherical Bessel functions of the first and second kind,

j
l

(
z
)

{j_l}(z)

and

y
l

(
z
)

{y_l}(z)

, have been computed. The ranges for the parameter

α
\alpha

, the order l and the root index n are:

α
=
0.1
(
0.1
)
0.7

\alpha = 0.1(0.1)0.7

,

l
=
1
(
1
)
15

l = 1(1)15

,

n
=
1
(
1
)
30

n = 1(1)30

.
High order local approximations to derivatives in the finite element method
Authors: Thomée Vidar
Published: June 2010

Consider the approximation of the solution u of an elliptic boundary value problem by means of a finite element Galerkin method of order r, so that the approximate solution

u
h

{u_h}

satisfies

u
h

−
u
=
O
(

h
r

)

{u_h} - u = O({h^r})

. Bramble and Schatz (Math. Comp., v. 31, 1977, pp. 94-111) have constructed, for elements satisfying certain uniformity conditions, a simple function

K
h

{K_h}

such that

K
h

∗

u
h

−
u
=
O
(

h

2
r
−
2

)

{K_h}\; \ast \;{u_h} - u = O({h^{2r - 2}})

in the interior. Their result is generalized here to obtain similar superconvergent order interior approximations also for derivatives of u.
Some new primes of the form 𝑘⋅2ⁿ+1
Authors: Matthew G. , Williams H. C.
Published: June 2010

All primes of the form

k
⋅

2
n

+
1
,
k

k \cdot {2^n} + 1,k

odd, for

9
⩽
k
⩽
99
,
512
⩽
n
⩽
1000

9 \leqslant k \leqslant 99, 512 \leqslant n \leqslant 1000

and for

101
⩽
k
⩽
129
,
1
⩽
n
⩽
1000

101 \leqslant k \leqslant 129, 1 \leqslant n \leqslant 1000

are determined and factors are found for the Fermat numbers

F

744

{F_{744}}

and

F

556

{F_{556}}

.
Existence of a solution to the discrete Theodorsen equation for conformal mappings
Authors: Gutknecht Martin H.
Published: June 2010

The discrete Theodorsen equation is the basis of efficient numerical methods for conformal mappings of the unit disk onto Jordan regions that are starlike with respect to the origin. Applying Brouwer’s fixed point theorem we show here that there always exists a solution to this equation.