bluuebunny/crossref_metadata_2025 huggingface dataset:
A note on Chambers’ method for finding a zero of a function
Authors: Cox M. G.
Published: June 2010

A class of methods due to Chambers is compared with the secant method and the method of inverse quadratic interpolation.
A continued fraction algorithm for real algebraic numbers
Authors: Cantor David G. , Galyean Paul H. , Zimmer Horst G.
Published: June 2010

Let a denote a real algebraic number that is a root of a polynomial

f
(
x
)
∈

Z

[
x
]

f(x) \in {\text {Z}}[x]

. The purpose of this paper is to state an algorithm for finding the simple continued fraction expansion of

α
\alpha

. Furthermore, an application of the algorithm to sign determination in real algebraic number fields is given.
Anharmonic frequency analysis
Authors: Paul A. K.
Published: June 2010

A new numerical method of frequency analysis is described, designed mainly to search for discrete frequencies in a time series. An integral transform is applied twice to the data for different reference times. A complex amplitude within a selected narrow frequency band is obtained for each transform. The frequency is then determined from the phase change of the complex amplitude over the difference of the two reference times. Very high precision is obtained, which is demonstrated in two examples.
An elliptic integral identity
Authors: Wrigge H. S.
Published: June 2010

The identity
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-parenthesis tau right-parenthesis equals StartFraction 1 Over left-parenthesis 2 pi right-parenthesis Superscript 1 slash 2 Baseline EndFraction integral Subscript negative normal infinity Superscript normal infinity Baseline integral Subscript negative normal infinity Superscript normal infinity Baseline exp left-bracket minus one half left-parenthesis x Superscript 4 Baseline minus 2 left-parenthesis 2 tau squared minus 1 right-parenthesis x squared y squared plus y Superscript 4 Baseline right-parenthesis right-bracket d x d y comma">

K
(
τ
)
=

1

(
2
π
)

1

/

2

∫

−
∞

∞

∫

−
∞

∞

exp
⁡

[

−

1
2

(

x
4

−
2
(
2

τ
2

−
1
)

x
2

y
2

+

y
4

)

]

d
x

d
y
,

K(\tau ) = \frac {1}{{{{(2\pi )}^{1/2}}}}\int _{ - \infty }^\infty {\int _{ - \infty }^\infty {\exp \left [ { - \frac {1}{2}({x^4} - 2(2{\tau ^2} - 1){x^2}{y^2} + {y^4})} \right ]dx\;dy,} }

\]
where

K
(
τ
)

K(\tau )

is the complete elliptic integral of the first kind, is used to prove that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-parenthesis square-root 2 minus 1 right-parenthesis equals pi Superscript 3 slash 2 Baseline left-parenthesis 2 plus square-root 2 right-parenthesis Superscript 1 slash 2 Baseline slash 4 normal upper Gamma left-parenthesis five eighths right-parenthesis normal upper Gamma left-parenthesis seven eighths right-parenthesis">

K
(
√
2
−
1
)
=

π

3

/

2

(
2
+
√
2

)

1

/

2

/

4
Γ

(

5
8

)

Γ

(

7
8

)

K(\surd 2 - 1) = {\pi ^{3/2}}{(2 + \surd 2)^{1/2}}/4\Gamma \left ( {\frac {5}{8}} \right )\Gamma \left ( {\frac {7}{8}} \right )

.
Improved computation of cubic natural splines with equi-spaced knots
Authors: MacLeod Malcolm A.
Published: June 2010

An improved algorithm is given for the computation of the coefficients of the interpolating polynomials for cubic natural splines with equi-spaced knots. By solving the continuity equation recursively, a gain in computation efficiency is obtained and the requirement of previous techniques for exact computation is eliminated.
Algorithms for triangular decomposition of block Hankel and Toeplitz matrices with application to factoring positive matrix polynomials
Authors: Rissanen J.
Published: June 2010

Algorithms are given for calculating the block triangular factors

A
,

A
^

,
B
=

A

−
1

A,\hat A,B = {A^{ - 1}}

and

B
^

=

A
^

−
1

\hat B = {\hat A^{ - 1}}

and the block diagonal factor D in the factorizations

R
=
A
D

A
^

R = AD\hat A

and

B
R

B
^

=
D

BR\hat B = D

of block Hankel and Toeplitz matrices R. The algorithms require

O
(

p
3

n
2

)

O({p^3}{n^2})

operations when R is an

n
×
n

n \times n

-matrix of

p
×
p

p \times p

-blocks. As an application, an iterative method is described for factoring

p
×
p

p \times p

-matrix valued positive polynomials

R
=

∑

i
=
−
m

m

R
i

x
i

,

R

−
i

=

R
i
′

R = \sum \nolimits _{i = - m}^m {R_i}{x^i},{R_{ - i}} = {R’_i}

, as

A
¯

(
x
)

A
¯

′

(

x

−
1

)

\bar A(x)\bar A’({x^{ - 1}})

, where

A
¯

(
x
)

\bar A(x)

is outer.
Self-scaling variable metric algorithms without line search for unconstrained minimization
Authors: Oren Shmuel S.
Published: June 2010

This paper introduces a new class of quasi-Newton algorithms for unconstrained minimization in which no line search is necessary and the inverse Hessian approximations are positive definite. These algorithms are based on a two-parameter family of rank two, updating formulae used earlier with line search in self-scaling variable metric algorithms. It is proved that, in a quadratic case, the new algorithms converge at least weak superlinearly. A special case of the above algorithms was implemented and tested numerically on several test functions. In this implementation, however, cubic interpolation was performed whenever the objective function was not satisfactorily decreased on the first "shot" (with unit step size), but this did not occur too often, except for very difficult functions. The numerical results indicate that the new algorithm is competitive and often superior to previous methods.
Symmetrization of the fluid dynamic matrices with applications
Authors: Turkel Eli
Published: June 2010

The matrices occurring in the equations of inviscid fluid dynamics are simultaneously symmetrized by a similarity transformation. The resulting matrices decompose into several lower-dimensional blocks. In addition these blocks are more sparse than previously obtained. These properties are then used to find a sufficiency proof for an improved version of the two-step Richtmyer method.
A mollifier useful for approximations in Sobolev spaces and some applications to approximating solutions of differential equations
Authors: Hilbert Stephen
Published: June 2010

For a given uniform grid of

E
N

{E^N}

(N-dimensional Euclidean space) with mesh h, a class of smoothing functions (mollifiers) is constructed. If a function is an element of the Sobolev space

H
2
m

H_2^m

, then the error made by replacing the given function by a smoother

(

C
∞

)

({C^\infty })

function (which is the given function convolved with one of the mollifiers) is bounded by a constant times

h
m

{h^m}

. This result is used to construct approximations for functions using Hermite or spline interpolation, even though the function to be approximated need not satisfy the continuity conditions necessary for the existence of a Hermite or spline interpolate. These techniques are used to find approximations to the generalized solution of a second order elliptic Neumann problem.
A first order method for differential equations of neutral type
Authors: Castleton R. N. , Grimm L. J.
Published: June 2010

A first order method is presented for solution of the initial-value problem for a differential equation of neutral type with implicit delay in the critical case where the time-lag is zero and the method of stepwise integration does not apply. A convergence theorem is proved, and numerical examples are given.
On computing some extremal periodic positive-definite functions
Authors: Page R. L.
Published: June 2010

This paper is concerned with maximizing the square integral over certain classes of periodic positive-definite functions. The question arose at Jet Propulsion Laboratory in connection with maximizing the average power of the received signal in radar exploration of the planets. We present computational evidence that the maximizing function exists and, in most but not all cases, is unique. Upper and lower bounds for the maximum square integral are computed, and formulas are conjectured for the maximizing functions in several cases.
12,758
Authors: Dressler Robert E. , Parker Thomas
Published: June 2010

It is known that 128 is the largest integer which is not expressible as a sum of distinct squares. Here, a computer is used to prove that 12,758 is the largest integer which is not expressible as a sum of distinct cubes.
Triangular factorization and inversion by fast matrix multiplication
Authors: Bunch James R. , Hopcroft John E.
Published: June 2010

The fast matrix multiplication algorithm by Strassen is used to obtain the triangular factorization of a permutation of any nonsingular matrix of order n in

>

C
1

n

log

2

7

> {C_1}{n^{{{\log }_2}7}}

operations, and, hence, the inverse of any nonsingular matrix in

>

C
2

n

log

2

7

> {C_2}{n^{{{\log }_2}7}}

operations.
Computational techniques based on the Lanczos representation
Authors: Lyness J. N.
Published: June 2010

In his bookDiscourse on Fourier Series, Lanczos deals in some detail with representations off(x)f(x)of the typef(x)=hp−1(x)+gp(x)f(x) = {h_{p - 1}}(x) + {g_p}(x)wherehp−1(x){h_{p - 1}}(x)is a polynomial of degreep−1p - 1andgp(x){g_p}(x)has the property that its full range Fourier coefficients converge at the rater−p{r^{ - p}}. In Part I, some properties ofhp(x){h_p}(x)and of the series{hp(x)}1∞\{ {h_p}(x)\} _1^\inftyare described. These properties are used here to provide criteria for the convergence or divergence of the Euler-Maclaurin series, in the case whenf(x)f(x)is an analytic function. The similarities and differences between this series and the Lidstone and other two-point series are briefly mentioned. In Part II, the Lanczos representation is employed to derive an approximate representationF(x)F(x)for an analytic functionf(x)f(x)on the interval [0, 1] is derived. This has the form\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis x right-parenthesis equals sigma-summation Underscript q equals 1 Overscript p minus 1 Endscripts lamda Subscript q minus 1 Baseline upper B Subscript q Baseline left-parenthesis x right-parenthesis slash q factorial plus 2 sigma-summation Underscript r equals 0 Overscript m slash 2 Endscripts left-parenthesis mu Subscript r Baseline cosine 2 pi r x plus nu Subscript r Baseline sine 2 pi r x right-parenthesis">F(x)=∑q=1p−1λq−1Bq(x)/q!+2∑r=0m/2(μrcos⁡2πrx+νrsin⁡2πrx)F(x) = \sum \limits _{q = 1}^{p - 1} {{\lambda _{q - 1}}{B_q}(x)/q! + 2\sum \limits _{r = 0}^{m/2} {({\mu _r}\cos 2\pi rx + {\nu _r}\sin 2\pi rx)} }\]and requires for its determination the values of the derivatives<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Superscript left-parenthesis q minus 1 right-parenthesis Baseline left-parenthesis 1 right-parenthesis minus f Superscript left-parenthesis q minus 1 right-parenthesis Baseline left-parenthesis 0 right-parenthesis left-parenthesis q equals 1 comma 2 comma midline-horizontal-ellipsis p minus 1 right-parenthesis">f(q−1)(1)−f(q−1)(0)(q=1,2,⋯p−1){f^{(q - 1)}}(1) - {f^{(q - 1)}}(0)\;(q = 1,2, \cdots p - 1)and the regularly spaced function valuesf(j/m)(j=0,1,⋯,m)f(j/m)\;(j = 0,1, \cdots ,m). It involves replacinggp(x)g_p(x)by a discrete Fourier expansion based on trapezoidal rule approximations to its Fourier coefficients. This representation is a powerful one. The drawback is that it requires derivatives. Most of Part II is devoted to the effect of using only approximate derivatives. It is shown that when these are successively less accurate with increasing order (the sort of behaviour encountered using finite difference formula), then the representation is still powerful and reliable. In a computational context the only penalty for using inaccurate derivatives is that a larger value ofmmay—or may not—be required to attain a specific accuracy.
Error analysis of a computation of Euler’s constant
Authors: Beyer W. A. , Waterman M. S.
Published: June 2010

A complete error analysis of a computation of

γ
\gamma

, Euler’s constant, is given. The results have been used to compute

γ
\gamma

to 7114 places and this value has been deposited in the UMT file.
A note on Chambers’ method
Authors: Blackburn J. A. , Beaudoin Y.
Published: June 2010

A correction is given for one of Chambers’ second-order iteration formulae. It is shown that composition of the secant method with itself exhibits a convergence exponent of 2.414, whereas composition of the iteration function with itself yields an exponent of 2.831.
A note on the optimal addition of abscissas to quadrature formulas of Gauss and Lobatto type
Authors: Piessens Robert , Branders Maria
Published: June 2010

An improved method for the optimal addition of abscissas to quadrature formulas of Gauss and Lobatto type is given.
On the 3-rank of quadratic fields and the Euler product
Authors: Neild Carol , Shanks Daniel
Published: June 2010

This paper covers many (closely related) topics: the distribution of the 3-Sylow subgroups of imaginary quadratic fields; the possibility of finding 3-ranks greater than 4; some questions concerning

a
3

=

b
2

+

c
2

D

{a^3} = {b^2} + {c^2}D

; and the convergence of Euler products and its relation to the extended Riemann hypothesis. Two programs that were used in this investigation are described.
Computing in 𝐺𝐹(𝑞)
Authors: Beard Jacob T. B.
Published: June 2010

This paper gives an elementary deterministic algorithm for completely factoring any polynomial over

GF

(
q
)
,
q
=

p
d

{\text {GF}}(q),q = {p^d}

, criteria for the identification of three types of primitive polynomials, an exponential representation for

GF

(
q
)

{\text {GF}}(q)

which permits direct rational calculations in

GF

(
q
)

{\text {GF}}(q)

as opposed to modular arithmetic over

GF

[
p
,
x
]

{\text {GF}}[p,x]

, and a matrix representation for

GF

¯

(
p
)

\overline {{\text {GF}}} (p)

which admits computer computations. The third type of primitive polynomial examined permits the given representation of

GF

(
q
)

{\text {GF}}(q)

to display a primitive normal basis over

GF

(
p
)

{\text {GF}}(p)

. The techniques developed require only the usual addition and multiplication of square matrices over

GF

(
p
)

{\text {GF}}(p)

. Partial tables from computer programs based on certain of these results will appear in later papers.
A class of 𝐴-stable advanced multistep methods
Authors: Williams Jack , de Hoog Frank
Published: June 2010

A class of A-stable advanced multistep methods is derived for the numerical solution of initial value problems in ordinary differential equations. The methods, of all orders of accuracy up to ten, only require values of y’ and are self starting. Results for the asymptotic behaviour of the discretization error and for estimating local truncation error are also obtained. The practical implementation of the fourth order method is described and the method applied to some stiff equations. Numerical comparisons are made with Gear’s method.
Some problems in optimally stable Lagrangian differentiation
Authors: Salzer Herbert E.
Published: June 2010

In many practical problems in numerical differentiation of a function

f
(
x
)

f(x)

that is known, observed, measured, or found experimentally to limited accuracy, the computing error is often much more significant than the truncating error. In numerical differentiation of the n-point Lagrangian interpolation polynomial, i.e.,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Superscript left-parenthesis k right-parenthesis Baseline left-parenthesis x right-parenthesis tilde normal upper Sigma Subscript i equals 1 Superscript n Baseline upper L Subscript i Superscript n left-parenthesis k right-parenthesis Baseline left-parenthesis x right-parenthesis f left-parenthesis x Subscript i Baseline right-parenthesis">

f

(
k
)

(
x
)
∼

Σ

i
=
1

n

L
i

n
(
k
)

(
x
)
f
(

x
i

)

{f^{(k)}}(x) \sim \Sigma _{i = 1}^nL_i^{n(k)}(x)f({x_i})

, a criterion for optimal stability is minimization of

Σ

i
=
1

n

|

L
i

n
(
k
)

(
x
)

|

\Sigma _{i = 1}^n|L_i^{n(k)}(x)|

. Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L identical-to upper L left-parenthesis n comma k comma x 1 comma ellipsis comma x Subscript n Baseline semicolon x or x 0 right-parenthesis equals normal upper Sigma Subscript i equals 1 Superscript n Baseline StartAbsoluteValue upper L Subscript i Superscript n left-parenthesis k right-parenthesis Baseline left-parenthesis x or x 0 right-parenthesis EndAbsoluteValue">

L
≡
L
(
n
,
k
,

x
1

,
…
,

x
n

;
x

or

x
0

)
=

Σ

i
=
1

n

|

L
i

n
(
k
)

(
x

or

x
0

)

|

L \equiv L(n,k,{x_1}, \ldots ,{x_n};x\;{\text {or}}\;{x_0}) = \Sigma _{i = 1}^n|L_i^{n(k)}(x\;{\text {or}}\;{x_0})|

. For

x
i

{x_i}

and fixed

x
=

x
0

x = {x_0}

in

[
−
1
,
1
]

[ - 1,1]

, one problem is to find the n

x
i

{x_i}

’s to give

L
0

≡

L
0

(
n
,
k
,

x
0

)
=
min
L

{L_0} \equiv {L_0}(n,k,{x_0}) = \min L

. When the truncation error is negligible for any

x
0

{x_0}

within

[
−
1
,
1
]

[ - 1,1]

, a second problem is to find

x
0

=

x
∗

{x_0} = {x^\ast }

to obtain

L
∗

≡

L
∗

(
n
,
k
)
=
min

L
0

=
min
min
L

{L^\ast } \equiv {L^\ast }(n,k) = \min {L_0} = \min \min L

. A third much simpler problem, for

x
i

{x_i}

equally spaced,

x
1

=
−
1
,

x
n

=
1

{x_1} = - 1,{x_n} = 1

, is to find

x
¯

\bar x

to give

L
¯

≡

L
¯

(
n
,
k
)
=
min
L

\bar L \equiv \bar L(n,k) = \min L

. For lower values of n, some results were obtained on

L
0

{L_0}

and

L
∗

{L^\ast }

when

k
=
1

k = 1

, and on

L
¯

\bar L

when

k
=
1

k = 1

and 2 by direct calculation from available tables of

L
i

n
(
k
)

(
x
)

L_i^{n(k)}(x)

. The relation of

L
0

,

L
∗

{L_0},{L^\ast }

and

L
¯

\bar L

to equally spaced points, Chebyshev points, Chebyshev polynomials

T
m

(
x
)

{T_m}(x)

for

m
⩽
n
−
1

m \leqslant n - 1

, minimax solutions, and central difference formulas, considering also larger values of n, is indicated sketchily.
The evaluation of determinants by expansion by minors and the general problem of substitution
Authors: Gentleman W. M. , Johnson S. C.
Published: June 2010

In this paper, we consider a class of substitution strategies which correspond to various minor expansion algorithms, and show that the usual expansion by minors, a row at a time, is optimal in this class.
Splines with nonnegative 𝐵-spline coefficients
Authors: de Boor C. , Daniel James W.
Published: June 2010

We consider the question of the approximation of nonnegative functions by nonnegative splines of order k (degree

>
k

> k

) compared with approximation by that subclass of nonnegative splines of order k consisting of all those whose B-spline coefficients are nonnegative; while approximation by the former gives errors of order

h
k

{h^k}

, the latter may yield only

h
2

{h^2}

. These results are related to certain facts about quasi-interpolants.
Optimal indexing of the vertices of graphs
Authors: FitzGerald Carl H.
Published: June 2010

The incidence matrices of various graphs are considered. By reordering the points, the bandwidth can be changed. In the cases of rectangular grids in the plane or cubic grids in three dimensions, the exact, minimum values of the bandwidth are determined. In certain numerical analysis problems, it is of interest to index the vertices of a graph in such a way that the matrix used to represent an associated system of linear equations is as close to diagonal as possible or, equivalently, to index the vertices of a graph in a way that minimizes the width of the band of nonzero terms in the incidence matrix for that graph. The purpose of this note is to present a few methods of determining the minimum possible width in some cases that arise in the numerical solution of Laplace’s equation. Of particular interest are the first proof of Theorem 2 which solves the problem for the common square grid and Theorem 3 which answers the problem for the cubic grid.
Stable approximations for hyperbolic systems with moving internal boundary conditions
Authors: Goldberg M. , Abarbanel S.
Published: June 2010

The work of Kreiss on the stability theory of difference schemes for the mixed initial boundary value problem for linear hyperbolic systems is extended to deal with the case of the pure initial value problem with an internal boundary. The case of an internal boundary

X
B

{X_B}

that moves with constant speed c is treated, i.e.,

X
B

=

X
0

+
c
t

{X_B} = {X_0} + ct

. In particular, the stability of "hybrid" schemes is studied by using the Lax-Wendroff scheme at points that are not on the internal boundary, while using a first order accurate scheme at the internal boundary points. Numerical evidence is given that the results of the linear stability analysis describes the qualitative behavior of such schemes for nonlinear cases, when the internal boundary is a shock.
𝐿^{𝑝} approximation of Fourier transforms and certain interpolating splines
Authors: Shreve David C.
Published: June 2010

We extend to

L
p

,
1
≦
p
≦
∞

{L^p},1 \leqq p \leqq \infty

, the

L
2

{L^2}

results of Bramble and Hilbert on convergence of discrete Fourier transforms and on approximation using smooth splines. The main tools are the estimates of [1] for linear functionals on Sobolev spaces and elementary results on Fourier multipliers.
Low-order approximations for the normal probability integral and the error function
Authors: Carta David G.
Published: June 2010

Rational fractions of the form

0.5

/

(
a
+
b
x
+
…

)

2
q

0.5/{(a + bx + \ldots )^{2q}}

are used to evaluate the function of interest. Polynomials of from third to sixth order are derived which achieve absolute errors ranging from 0.01 to 0.000001 for all (real) positive x, and relative errors of from 0.1 to 0.00001 for (real) positive x less than 3.1, 4.0, and 5.2. Denominator coefficients are calculated by linearizing the rational fraction about progressively improved nominal solutions and using linear programming to solve the resulting linear minimax problems.
Power sum method and the approximative solution of algebraic equations
Authors: Turán Paul
Published: June 2010

A universal algorithm is given for computing a root of an arbitrary polynomial with complex coefficients in a number of operations dependent only on the order n and relative error

ε
\varepsilon

. This paper is a brief English exposition of the author’s earlier "Algebrai egyenletek közelitö megoldásáról," Vol. XVIII of the Communications of the III Class of the Hungarian Academy of Sciences (1968), pp. 223-235.
Generalized 𝑀-matrices and applications
Authors: Poole George D.
Published: June 2010

Recently, two distinct directions have been taken in an attempt to generalize the definition of an M-matrix. Even for nonsingular matrices, these two generalizations are not equivalent. The role of these and other classes of recently defined matrices is indicated showing their usefulness in various applications.
Chebyshev approximation by interpolating rationals on [𝛼,∞]
Authors: Dunham Charles B.
Published: June 2010

Decay-type functions with a finite number of zeros are approximated on

[
α
,
∞
)

[\alpha ,\infty )

by an oscillation factor times a negative power of a polynomial. Best approximations are characterized and an algorithm indicated.
The design of slowly shrinking labelled squares
Authors: Berlekamp Elwyn R.
Published: June 2010

This paper considers the nonlinear operator T, which acts on 4-dimensional vectors with nonnegative real components. T transforms such a vector into a new vector whose components are the magnitudes of the differences of cyclically adjacent pairs of components of the previous vector. We show that T is not nilpotent.
An identity for simplifying certain generalized hypergeometric functions
Authors: Osler Thomas J.
Published: June 2010

In this paper, an identity is proved in which certain generalized hypergeometric functions

N
P

F

N
Q
+
N
−
1

_{NP}{F_{NQ + N - 1}}

are expressed as a linear combination of N functions of the

P

F
Q

_P{F_Q}

type. Several special cases of this identity are studied. The inverse of this identity is also given.
On comparing Adams and natural spline multistep formulas
Authors: Hill David R.
Published: June 2010

This paper presents two techniques for the comparison of Adams formulas and methods based on natural splines. A rigorous foundation to the claim that a pth order natural spline formula produces better results than a pth order Adams method, but not quite as good as a

(
p
+
1
)

(p + 1)

st order Adams formula is given for

p
=
2
,
3
,
4

p = 2,3,4

, which suggests the general case.
An efficient method for the discrete linear 𝐿₁ approximation problem
Authors: Abdelmalek Nabih N.
Published: June 2010

An improved dual simplex algorithm for the solution of the discrete linear

L
1

{L_1}

approximation problem is described. In this algorithm certain intermediate iterations are skipped. This method is comparable with an improved simplex method due to Barrodale and Roberts, in both speed and number of iterations. It also has the advantage that in case of ill-conditioned problems, the basis matrix can lend itself to triangular factorization and can thus ensure a stable solution. Numerical results are given.
On the distribution of the zeros of generalized Airy functions
Authors: Headley V. B. , Barwell V. K.
Published: June 2010

We give tables of zeros and values of the generalized Airy functions introduced by Swanson and Headley [SIAM J. Appl. Math., v. 15, 1967, pp. 1400-1412]. The tables enable us to sharpen substantially results on the distribution of the zeros. We show that the nonreal zeros are asymptotically close to the boundary rays of the sectors obtained in the paper cited. We conjecture from the numerical evidence that the zeros monotonically approach the rays.
A generalization of the method of averages for overdetermined linear systems
Authors: Anderson Ned
Published: June 2010

Necessary and sufficient conditions are established for an existence and uniqueness result for a generalization of the method of averages for overdetermined linear systems. Some practical schemes are given, and a connection with Galerkin’s method is indicated.
Some analytic or asymptotic confluent expansions for functions of several variables
Authors: Srivastava H. M. , Panda Rekha
Published: June 2010

This paper aims at presenting multivariable extensions of the recent results of J. L. Fields [5] and others (cf. [1]-[4]) on certain analytic or asymptotic confluent expansions for functions of one and two variables. It is also demonstrated how these extensions would apply, for instance, to derive an asymptotic confluent expansion for a certain class of the generalized Lauricella function of several variables.
Cubatures of precision 2𝑘 and 2𝑘+1 for hyperrectangles
Authors: Hunkins Dalton R.
Published: June 2010

It is well known that integration formulas of precision

2
k

(
2
k
+
1
)

2k\;(2k + 1)

for a region in n-space which is a Cartesian product of intervals can be obtained from one-dimensional Radau (Gauss) rules. The number of function evaluations in these product cubatures is

(
k
+
1

)
n

{(k + 1)^n}

. In this paper, an algorithm is given for obtaining cubatures for hyperrectangles in n-space of precision 2k, in many instances

2
k
+
1

2k + 1

, which uses

(
k
+
1
)

(
k

)

n
−
1

(k + 1){(k)^{n - 1}}

nodes. The weights and nodes of these new formulas are derived from one-dimensional generalized Radau rules.
Factoring multivariate polynomials over the integers
Authors: Wang Paul S. , Rothschild Linda Preiss
Published: June 2010

An algorithm for the irreducible factorization of any multivariate polynomial over the integers is given. It is much faster than the classical method ascribed to Kronecker. The algorithm begins by making substitutions for all but one of the variables with selected integers, giving a polynomial in just one variable. This univariate polynomial is then factored by a known method, which uses an algorithm of Berlekamp for factoring univariate polynomials over finite fields. The multivariate factors are constructed from the univariate ones by a kind of Hensel algorithm. The procedure has been implemented in the algebraic manipulation systems MACSYMA and SCRATCHPAD. A number of machine examples with timing are included.
Some numerical results on Fekete polynomials
Authors: Bateman Paul T. , Purdy George B. , Wagstaff Samuel S.
Published: June 2010

It is known that if

χ
\chi

is a real residue character modulo k with

χ
(
p
)
=
−
1

\chi (p) = - 1

for the first five primes p, then the corresponding Fekete polynomial

Σ

n
=
1

k

χ
(
n
)

x
n

\Sigma _{n = 1}^k\;\chi (n){x^n}

changes sign on (0, 1). In this paper it is shown that the condition that

χ
(
p
)

\chi (p)

be -1 for the first four primes p is not sufficient to guarantee such a sign change. More specifically, if

χ
\chi

is the real nonprincipal character modulo either 1277 or 1973, it is shown that the corresponding Fekete polynomial is positive throughout (0, 1) even though

χ
(
2
)
=
χ
(
3
)
=
χ
(
5
)
=
χ
(
7
)
=
−
1

\chi (2) = \chi (3) = \chi (5) = \chi (7) = - 1

.
The effect of interpolating the coefficients in nonlinear parabolic Galerkin procedures
Authors: Douglas Jim , Dupont Todd
Published: June 2010

Error estimates are derived for a class of Galerkin methods for a quasilinear parabolic equation. In these Galerkin methods, both continuous and discrete in time, the nonlinear coefficient in the differential equation is interpolated into a finite-dimensional function space in order to compute the integrals involved. Asymptotic error estimates of optimal order are produced.
Determination of the primality of 𝑁 by using factors of 𝑁²±1
Authors: Williams H. C. , Judd J. S.
Published: June 2010

Algorithms are developed which can be used to determine the primality of a large integer N when a sufficient number of prime factors of

N
2

+
1

{N^2} + 1

are known. A test for the primality of N which makes use of known factors of

N
−
1
,
N
+
1

N - 1,N + 1

and

N
2

+
1

{N^2} + 1

and the factor bounds on these numbers is also presented. In order to develop the necessary theory, the properties of some functions which are a generalization of Lehmer functions are used. Several examples of numbers proved prime by employing these tests are given.
A table of totally real cubic fields
Authors: Angell I. O.
Published: June 2010

In this paper the author describes the construction of a table of totally real cubic number fields. Each field is distinguished by the coefficients of a generating polynomial, the index of this polynomial over the field and the discriminant of the field. The class number and a fundamental pair of units is also given.
Maximal binary matrices and sum of two squares
Authors: Yang C. H.
Published: June 2010

A maximal

(
+
1
,
−
1
)

( + 1, - 1)

-matrix of order 66 is constructed by a method of matching two finite sequences. This method also produced many new designs for maximal

(
+
1
,
−
1
)

( + 1, - 1)

-matrices of order 42 and new designs for a family of H-matrices of order

26.2
n

{26.2^n}

. A nonexistence proof for a

(
∗
)

(\ast )

-type H-matrix of order 36, consequently for Golay complementary sequences of length 18, is also given.
Some algorithms for prime testing using generalized Lehmer functions
Authors: Williams H. C. , Judd J. S.
Published: June 2010

Let N be an odd integer thought to be prime. The properties of special functions which are generalizations of the functions of Lehmer (Ann. of Math., v. 31, 1930, pp. 419-448) are used to develop algorithms that produce information concerning the possible prime divisors of N. It is shown how the factors of

N
±
1
,

N
2

+
1
,

N
2

±
N
+
1

N \pm 1,{N^2} + 1,{N^2} \pm N + 1

, together with the factor bounds on these numbers, may all be used to calculate lower bounds for the possible prime divisors of N. Frequently, these bounds are large enough that N may be shown to be prime. These tests were implemented on an IBM/370-158 computer and run on the pseudoprime divisors of the first 385 Fibonacci and Lucas numbers.
On the numerical solution of Helmholtz’s equation by the capacitance matrix method
Authors: Proskurowski Wlodzimierz , Widlund Olof
Published: June 2010

In recent years the usefulness of fast Laplace solvers has been extended to problems on arbitrary regions in the plane by the development of capacitance matrix methods. The solution of the Dirichlet and Neumann problems for Helmholtz’s equation is considered. It is shown, that by an appropriate choice of the fast solver, the capacitance matrix can be generated quite inexpensively. An analogy between capacitance matrix methods and classical potential theory for the solution of Laplace’s equation is explored. This analogy suggests a modification of the method in the Dirichlet case. This new formulation leads to well-conditioned capacitance matrix equations which can be solved quite efficiently by the conjugate gradient method. A highly accurate solution can, therefore, be obtained at an expense which grows no faster than that for a fast Laplace solver on a rectangle when the mesh size is decreased.
Factoring multivariate polynomials over algebraic number fields
Authors: Wang Paul S.
Published: June 2010

The algorithm for factoring polynomials over the integers by Wang and Rothschild is generalized to an algorithm for the irreducible factorization of multivariate polynomials over any given algebraic number field. The extended method makes use of recent ideas in factoring univariate polynomials over large finite fields due to Berlekamp and Zassenhaus. The procedure described has been implemented in the algebraic manipulation system MACSYMA.

∗
∗

^{\ast \ast }

Some machine examples with timing are included.
Functional fitting–new family of schemes for integration of stiff O.D.E
Authors: Iserles Arieh
Published: June 2010

Following the ideas of Liniger and Willoughby (although acting along different lines), a new family of schemes, derived from the trapezoidal rule, is developed, which enables the fitting of the scheme to arbitrary sets of equations.
A least squares decomposition method for solving elliptic equations
Authors: Jespersen Dennis C.
Published: June 2010

This paper analyzes a numerical method for solving second-order elliptic partial differential equations. The idea is to write the equation as a lower-order system and solve the system using least squares techniques. Error estimates are derived for a model problem.
A collocation method for Fredholm integral equations of the second kind
Authors: Houstis E. N. , Papatheodorou T. S.
Published: June 2010

An interpolation scheme based on piecewise cubic polynomials with Gaussian points as interpolation points is analyzed and applied to the solution of Fredholm equations of the second kind.
A combined Remes-differential correction algorithm for rational approximation
Authors: Kaufman Edwin H. , Leeming David J. , Taylor G. D.
Published: June 2010

In this paper a hybrid Remes-differential correction algorithm for computing best uniform rational approximants on a compact subset of the real line is developed. This algorithm differs from the classical multiple exchange Remes algorithm in two crucial aspects. First of all, the solving of a nonlinear system to find a best approximation on a given reference set in each iteration of the Remes algorithm is replaced with the differential correction algorithm to compute the desired best approximation on the reference set. Secondly, the exchange procedure itself has been modified to eliminate the possibility of cycling that can occur in the usual exchange procedure. This second modification is necessary to guarantee the convergence of this algorithm on a finite set without the usual normal and sufficiently dense assumptions that exist in other studies.
Computer-generated van der Waerden partitions
Authors: Stevens R. S. , Shantaram R.
Published: June 2010

An exhaustive search made on a PDP-8 mini-computer shows the value of the van der Waerden number

n
(
2
,
5
)

n(2,5)

to be 178.
On de Vogelaere’s method for 𝑦′′=𝑓(𝑥,𝑦)
Authors: Coleman John P. , Mohamed Julie
Published: June 2010

Easily calculated truncation-error estimates are given which permit efficient automatic error control in computations based on de Vogelaere’s method. An upper bound for the local truncation error is established, the interval of absolute stability is found to be

[
−
2
,
0
]

[ - 2,0]

, and it is shown that the global truncation error is of order

h
4

{h^4}

where h is the steplength.
The resultant iteration for determining the stability of a polynomial
Authors: Howland James Lucien
Published: June 2010

The square root iteration

x

i
+
1

=
(

x
i

+

x
i

−
1

)

/

2

{x_{i + 1}} = ({x_i} + x_i^{ - 1})/2

is known to be convergent to

+
1

+ 1

from starting values

x
0

{x_0}

in the open right half-plane, and to

−
1

- 1

from starting values in the open left half-plane. The resultant of the quadratic

z
2

−
2
w
z
+
1

{z^2} - 2wz + 1

and a given polynomial

P
(
z
)

P(z)

, of proper degree n, is a polynomial

R
(
w
)

R(w)

whose zeros, expressed in terms of the zeros

z
i

{z_i}

of

P
(
z
)

P(z)

, are just

w
i

=
(

z
i

+

z
i

−
1

)

/

2

{w_i} = ({z_i} + z_i^{ - 1})/2

. The construction of

R
(
w
)

R(w)

is thus equivalent to n independent applications of the square root iteration, with unknown starting values

z
i

{z_i}

. Repetition of this construction generates a sequence of resultants whose zeros are independently convergent to either

+
1

+ 1

or

−
1

- 1

according as the initial zero of

P
(
z
)

P(z)

lies in the open right or left half-plane. When the given polynomial, and each of the resultants, is written as a linear combination of the

n
+
1

n + 1

polynomials
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis z plus 1 right-parenthesis Superscript n Baseline comma left-parenthesis z plus 1 right-parenthesis Superscript n minus 1 Baseline left-parenthesis z minus 1 right-parenthesis comma left-parenthesis z plus 1 right-parenthesis Superscript n minus 2 Baseline left-parenthesis z minus 1 right-parenthesis squared comma ellipsis comma left-parenthesis z plus 1 right-parenthesis left-parenthesis z minus 1 right-parenthesis Superscript n minus 1 Baseline comma left-parenthesis z minus 1 right-parenthesis Superscript n">

(
z
+
1

)
n

,

(
z
+
1

)

n
−
1

(
z
−
1
)
,

(
z
+
1

)

n
−
2

(
z
−
1

)
2

,
…
,
(
z
+
1
)

(
z
−
1

)

n
−
1

,

(
z
−
1

)
n

{(z + 1)^n},{(z + 1)^{n - 1}}(z - 1),{(z + 1)^{n - 2}}{(z - 1)^2}, \ldots ,(z + 1){(z - 1)^{n - 1}},{(z - 1)^n}

. the sequence of resultants is ultimately convergent to one of the elements

(
z
+
1

)

n
−
p

(
z
−
1

)
p

{(z + 1)^{n - p}}{(z - 1)^p}

of this basis, whence it follows that

P
(
z
)

P(z)

has p zeros with positive real part and

n
−
p

n - p

zeros with negative real part, and the Hurwitz stability problem for

P
(
z
)

P(z)

is solved. By an application of the principle of argument it is, in general, possible to determine p at a finite stage of the resultant iteration. When formulated in terms of this basis, the resultant iteration becomes formally identical with the root-squaring process. This fact may be exploited to establish the properties of the resultant iteration, and to show that root-squaring may be applied to solve the Schur stability problem for a given polynomial. Although precise results are not available, numerical results and the general properties of this iteration suggest that, for the purpose at hand, it is unaffected by the progressive deterioration of condition that sometimes occurs in other applications of root-squaring.
Some interior estimates for semidiscrete Galerkin approximations for parabolic equations
Authors: Thomée Vidar
Published: June 2010

Consider a solution u of the parabolic equation
\[

u
t

+
A
u
=
f

in

Ω
×
[
0
,
T
]
,

{u_t} + Au = f\quad {\text {in}}\quad \Omega \times [0,T],

\]
where A is a second order elliptic differential operator. Let

S
h

{S_h}

; h small denote a family of finite element subspaces of

H
1

(
Ω
)

{H^1}(\Omega )

which permits approximation of a smooth function to order

O
(

h
r

)

O({h^r})

. Let

Ω
0

⊂
Ω

{\Omega _0} \subset \Omega

and assume that

u
h

:
[
0
,
T
]
→

S
h

{u_h}:[0,T] \to {S_h}

is an approximate solution which satisfies the semidiscrete interior equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis u Subscript h comma t Baseline comma chi right-parenthesis plus upper A left-parenthesis u Subscript h Baseline comma chi right-parenthesis equals left-parenthesis f comma chi right-parenthesis for-all chi element-of upper S Subscript h Superscript 0 Baseline left-parenthesis normal upper Omega 0 right-parenthesis equals StartSet chi element-of upper S Subscript h Baseline comma supp chi subset-of normal upper Omega 0 EndSet comma">

(

u

h
,
t

,
χ
)
+
A
(

u
h

,
χ
)
=
(
f
,
χ
)

∀
χ
∈

S
h
0

(

Ω
0

)
=
{
χ
∈

S
h

,

supp

χ
⊂

Ω
0

}
,

({u_{h,t}},\chi ) + A({u_h},\chi ) = (f,\chi )\quad \forall \chi \in S_h^0({\Omega _0}) = \{ \chi \in {S_h},{\text {supp}}\chi \subset {\Omega _0}\} ,

\]
where

A
(
⋅
,
⋅
)

A( \cdot , \cdot )

denotes the bilinear form on

H
1

(
Ω
)

{H^1}(\Omega )

associated with A. It is shown that if the finite element spaces are based on uniform partitions in a specific sense in

Ω
0

{\Omega _0}

, then difference quotients of

u
h

{u_h}

may be used to approximate derivatives of u in the interior of

Ω
0

{\Omega _0}

to order

O
(

h
r

)

O({h^r})

provided certain weak global error estimates for

u
h

−
u

{u_h} - u

to this order are available. This generalizes results proved for elliptic problems by Nitsche and Schatz [9) and Bramble, Nitsche and Schatz [1].
How to make the Lanczos algorithm converge slowly
Authors: Scott D. S.
Published: June 2010

The Paige style Lanczos algorithm is an iterative method for finding a few eigenvalues of large sparse symmetric matrices. Some beautiful relationships among the elements of the eigenvectors of a symmetric tridiagonal matrix are used to derive a perverse starting vector which delays convergence as long as possible. Why such slow convergence is never seen in practice is also examined.
A simplified Galerkin method for hyperbolic equations
Authors: Chin R. C. Y. , Hedstrom G. W. , Karlsson K. E.
Published: June 2010

We modify a Galerkin method for nonlinear hyperbolic equations so that it becomes a simpler method of lines, which may be viewed as a collocation method. The high order of accuracy is preserved. We present a linear wave analysis of the scheme and discuss some aspects of nonlinear problems. Our numerical experiments indicate that the addition of a proper artificial viscosity makes the method competitive and the common difference schemes, even when the solution has discontinuities.
On faster convergence of the bisection method for certain 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

,

Δ

12

{\Delta _{11}},{\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} = \{ {\Delta _{ji}}:1 \leqslant i \leqslant {2^j}\}

. It is known that the mesh of

T
j

{T_j}

tends to zero as

j
→
∞

j \to \infty

. It is shown here that if

Δ

01

{\Delta _{01}}

satisfies any of four certain properties, the rate of convergence of the mesh to zero is much faster than that predicted by the general case.
New backward recurrences for Bessel functions
Authors: Thacher Henry C.
Published: June 2010

The recurrences for the coefficients of appropriate power series may be used with the Miller algorithm to evaluate

J
v

(
x
)

(

|

x

|

small

)

{J_v}(x)\;(|x|\;{\text {small}})

,

e
x

K
v

(
x
)

(
Re
⁡
x
>
0
,

|

x

|

large

)

{e^x}{K_v}(x)\;(\operatorname {Re} x > 0,\;|x|\;{\text {large}})

, and the modulus and phase of

H
v

(
1
)

(
x
)

(
Re
⁡
x
>
0
,

|

x

|

large

)

H_v^{(1)}(x)\;(\operatorname {Re} x > 0,\;|x|\;{\text {large}})

. The first converges slightly faster than the power series or the classical recurrence, but requires more arithmetic; the last three give both better ultimate precision and faster convergence than the corresponding asymptotic series. The analysis also leads to a formal continued fraction for

K

v
+
1

(
x
)

/

K
v

(
x
)

{K_{v + 1}}(x)/{K_v}(x)

the convergence of which increases with

|

x

|

|x|

. The procedures were tested numerically both for integer and fractional values of v, and for real and complex x.
Finite sum evaluation of the Gauss hypergeometric function in an important special case
Authors: Detrich John , Conn Robert W.
Published: June 2010

We present explicit results expressing the Gauss hypergeometric function

F
(
l
+
1
,
m
+
1

/

2
;
n
+
3

/

2
;
z
)

F(l + 1,m + 1/2;n + 3/2;z)

as a finite sum in the general case where l, m, and n are integers.
A marching technique for nonseparable equations
Authors: Ehrlich Louis W.
Published: June 2010

A multiple-shooting marching technique is described which is applicable to arbitrary block tridiagonal matrices derived from nonseparable difference equations which are solved many times. Comparison with other methods on a particular problem shows the method to be competitive with respect to time and storage.
A note on the Rosenbrock procedure
Authors: Bui T. D.
Published: June 2010

To be useful for extremely stiff systems of ordinary differential equations, A-stability and a maximally damped condition as

λ
h
→
−
∞

\lambda h \to - \infty

(i.e., L-stability) are desirable. This paper investigates the condition of L-stability for a class of Runge-Kutta methods known as the Rosenbrock procedure. This procedure requires only one computation of a Jacobian matrix per step of integration, L-stable Rosenbrock methods up to order four are derived.
Greatest of the least primes in arithmetic progressions having a given modulus
Authors: Wagstaff Samuel S.
Published: June 2010

We give a heuristic argument, supported by numerical evidence, which suggests that the maximum, taken over the reduced residue classes modulo k, of the least prime in the class, is usually about

ϕ
(
k
)
log
⁡
k
log
⁡
ϕ
(
k
)

\phi (k)\log k\log \phi (k)

, where

ϕ
\phi

is Euler’s phi-function.
Equivalent forms of multistep formulas
Authors: Skeel Robert D.
Published: June 2010

For uniform meshes it is shown that any linear k-step formula can be formulated so that only k values need to be saved between steps. By saving additional m values it is possible to construct local polynomial approximations of degree

k
+
m
−
1

k + m - 1

, which can be used as predictor formulas. Different polynomial bases lead to different equivalent forms of multistep formulas. In particular, local monomial bases yield Nordsieck formulas. An explicit one-to-one correspondence is established between Nordsieck formulas and k-step formulas of order at least k, and a strong equivalence result is proved for all but certain pathological cases. Equivalence is also shown for

P(EC)
∗

\text {P(EC)}^\ast

formulas but not for

P(EC)
∗

E

\text {P(EC)}^\ast \text {E}

formulas.
Initial value methods for parabolic control problems
Authors: Winther Ragnar
Published: June 2010

We study iterative methods for parabolic control problems with a Neumann boundary value control and where the observation is the final state. The methods are based on transforming the original control problem (which may have constraints on the control) into an equivalent problem of minimizing a strictly convex functional (no constraints). The methods are semidiscrete in the sense that we assume that parabolic initial value problems can be solved exactly.