bluuebunny/crossref_metadata_2025 huggingface dataset:
Improved formulas for complete and partial summation of certain series.
Authors: Salzer Herbert E. , Kimbro Genevieve M.
Published: June 2010

In two previous articles one of the authors gave formulas, with numerous examples, for summing a series either to infinity (complete) or up to a certain number n of terms (partial) by considering the sum of the first j terms

S
j

{S_j}

, or some suitable modification

S
j
′

{S’_j}

, closely related to

S
j

{S_j}

, as a polynomial in 1/j. Either

S
∞

{S_\infty }

or

S
n

{S_n}

was found by m-point Lagrangian extrapolation from

S

j

0

{{S}_{{{j}_{0}}}}

,

S

j

0

−
1

{{S}_{{{j}_{0}}-1}}

,

⋅
⋅
⋅

\cdot \cdot \cdot

,

S

j

0

−
m
+
1

{{S}_{{{j}_{0}}-m+1}}

to 1/j = 0 or 1/j = 1/n respectively. This present paper furnishes more accurate m-point formulas for sums (or sequences)

S
j

{S_j}

which behave as even functions of 1/j. Tables of Lagrangian extrapolation coefficients in the variable

1

/

j
2

1/{j^2}

are given for: complete summation, m = 2(1)7,

j
0

{j_0}

= 10, exactly and 20D, m = 11,

j
0

{j_0}

= 20, 30D; partial summation, m = 7,

j
0

{j_0}

= 10, n = 11(1)25(5)100, 200, 500, 1000, exactly. Applications are made to calculating

π
\pi

or the semi-perimeters of many-sided regular polygons, Euler’s constant,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 plus sigma-summation Underscript r equals 1 Overscript j Endscripts left-brace StartFraction negative 1 Over left-parenthesis 4 r minus 1 right-parenthesis squared EndFraction plus StartFraction 1 Over left-parenthesis 4 r plus 1 right-parenthesis squared EndFraction right-brace equals 1 minus StartFraction 1 Over 3 squared EndFraction plus StartFraction 1 Over 5 squared EndFraction minus midline-horizontal-ellipsis for j equals normal infinity">

1
+

∑

r
=
1

j

{

−
1

(
4
r
−
1
)

2

+

1

(
4
r
+
1
)

2

}

=
1
−

1

3
2

+

1

5
2

−
⋯

for

j
=
∞

1 + \sum \limits _{r = 1}^j {\left \{ {\tfrac {{ - 1}}{{{{(4r - 1)}^2}}} + \tfrac {1}{{{{(4r + 1)}^2}}}} \right \} = 1 - \tfrac {1}{{{3^2}}} + \tfrac {1}{{{5^2}}} - \cdots } \,{\text {for}}\,j = \infty

\]
(Catalan’s constant), calculation of a definite integral as the limit of a suitably chosen sequence, determining later zeros of

J
v

(
x
)

{J_v}(x)

from earlier zeros for suitable v, etc. A useful device in many cases involving sums of odd functions, is to replace

S
j

{S_j}

by a trapezoidal-type

S
j
′

{S’_j}

which is seen, from the Euler-Maclaurin formula, to be formally a series in

1

/

j
2

1/{j^2}

. In almost every example, comparison with the earlier (1/j)-extrapolation showed these present formulas, for 7 points, to improve results by anywhere from around 4 to 9 places.
Expansions of hypergeometric functions in hypergeometric functions
Authors: Fields Jerry L. , Wimp Jet
Published: June 2010

In [1] Luke gave an expansion of the confluent hypergeometric function in terms of the modified Bessel functions

I
v

(
z
)

{I_v}(z)

. The existence of other, similar expansions implied that more general expansions might exist. Such was the case. Here multiplication type expansions of low-order hypergeometric functions in terms of other hypergeometric functions are generalized by Laplace transform techniques.
Note on the round-off errors in iterative processes
Authors: Descloux J.
Published: June 2010

This paper discusses round-off errors in iterative processes for solving equations. Let

x

n
+
1

=

x
n

+
F
(

x
n

)

{x_{n + 1}} = {x_n} + F({x_n})

be a scalar iterative converging process; the different values

x
n

{x_n}

are represented in a computer with a certain precision; when

x
n

{x_n}

is close to the limit,

F
(

x
n

)

F({x_n})

is small and can perhaps be obtained easily with a higher absolute precision than

x
n

{x_n}

; consequently, the addition

x
n

+
F
(

x
n

)

{x_n} + F({x_n})

will practically involve a rounding operation. Besides some general remarks, it will be shown that for a fixed-point computer an appropriate rounding method can provide a more accurate solution to the problem; analogous results are given in Appendix I for a floating-point computer; Appendix II deals with Aitken’s

δ
2

{\delta ^2}

process. The author is indebted to A. H. Taub for many suggestions and stimulating discussions.
On the evaluation of certain determinants
Authors: Lavoie Jean-L.
Published: June 2010

Using the properties of the generalized Hilbert matrix and familiar results from the theory of hypergeometric series, we evaluate the determinants of certain matrices whose general terms are known explicitly. In certain cases it is even possible to find the analytic expression for the general terms of the inverses.
A method of calculation of integrals of products of radial atomic wave functions
Authors: Ormonde Stephan , Dick Jerald S.
Published: June 2010

A simple computational procedure for the calculation of certain useful integrals in atomic physics is described. The procedure has been compared with other techniques and was found to be superior.
Bounds for the two-dimensional discrete harmonic Green’s function
Authors: Mangad Moshe
Published: June 2010

Estimation of bounds for the two-dimensional discrete harmonic Green’s function is obtained. These estimations can then be used to obtain approximate solutions to two-dimensional boundary problems associated with the harmonic difference operator.
Computer investigation of Landau’s theorem
Authors: Chiang P. S.
Published: June 2010

Let

f
(
z
)
=

a
0

+

a
1

z
+
⋯

f(z) = {a_0} + {a_1}z + \cdots

be regular for

|
z
|

>
1

\left | z \right | > 1

and never take the values

0
0

and

1
1

; then

|

a
1

|

\left | {{a_1}} \right |

has a bound depending only on

a
0

{a_0}

. J. A. Jenkins gave an explicit bound (Canad. J. Math. 8 (1956), 423–425)

|

a
1

|

≦
2

|

a
0

|

{

|

log

|

a
0

‖

+
5.94

}

\left | {{a_1}} \right | \leqq 2\left | {{a_0}} \right |\left \{ {\left | {\log } \right |\left . {{a_0}} \right \| + 5.94} \right \}

. The author investigates the shapes for the curves

|

a
1

|

≦
L

(

a
0

)

\left | {{a_1}} \right | \leqq L{\text {(}}{a_0}{\text {)}}

for given

a
0

{a_0}

by the aid of a computer and shows that although Jenkins’ result is about right when

a
0

{a_0}

is negative, 4.38 will be the best possible constant in his form and that a much smaller estimate should be available when

a
0

{a_0}

is positive or complex.
On the computation of a bivariate 𝑡-distribution
Authors: Amos D. E. , Bulgren W. G.
Published: June 2010

The cumulative bivariate

t
t

-distribution associated with random variables

T
1

=

X
1

/

(
S

/

k

)

1

/

2

{T_1} = {X_1}/{(S/k)^{1/2}}

,

T
2

=

X
2

/

(
S

/

k

)

1

/

2

{T_2} = {X_2}/{(S/k)^{1/2}}

is considered where

X
1

{X_1}

,

X
2

{X_2}

are bivariate normal with correlation coefficient

ρ
\rho

and

S
S

is an independent

χ
2

{\chi ^2}

random variable with

k
k

degrees of freedom. Representations in terms of series and simple, one-dimensional quadratures are presented together with efficient computational procedures for the special functions used in numerical evaluation.
Integration formulae involving derivatives
Authors: Patterson T. N. L.
Published: June 2010

A method, developed by Hammer and Wicke, for deriving high precision integration formulae involving derivatives is modified. It is shown how such formulae may be simply derived in terms of well-known polynomials.
Compressible fluid flow and the approximation of iterated integrals of a singular function
Authors: Richman P. L.
Published: June 2010

A computer implementation of Bergman’s solution to the initial value problem for the partial differential equation of compressible fluid flow is described. This work necessitated the discovery of an efficient approximation to the iterated indefinite integrals of an implicitly defined real function of a real variable with a singularity which is not included in the possible domains of integration. The method of approximation used here and the subsequently derived error bounds appear to have rather general applications for the approximation of the iterated integrals of a singular function of one real variable.
Monotone type discrete analogue for the mixed boundary value problem.
Authors: Thuraisamy V.
Published: June 2010

This paper is concerned with the formulation of finite-difference analogues of mixed boundary value problems for Poisson’s equation. A discrete approximation to the normal derivative is devised such that the matrix of the resulting system is of monotone type. This enables us to prove that the rate of convergence is

O
(

h
2

)

O({h^2})

, where h is the mesh constant.
A note on a generalisation of a method of Douglas
Authors: Fairweather Graeme
Published: June 2010

In this note, the high-order correct method of Douglas [1] for the diffusion equation in one space variable is extended to

q
≦
3

q \leqq 3

space variables. The resulting difference equations are then solved using the A. D. I. technique of Douglas and Gunn [3]. When

q
=
2

q = 2

, this method is equivalent to that of Mitchell and Fairweather [5] while

q
=
3

q = 3

provides a method which is similar to Samarskiï’s method [6] and of higher accuracy than that of Douglas [2]. When the proposed methods are used to solve the diffusion equation with timeindependent boundary conditions, they have the advantage that no boundary modification (see [4]) is required to maintain accuracy.
Calculation of Dirichlet 𝐿-functions
Authors: Spira Robert
Published: June 2010

A method for calculating Dirichlet

L
L

-series is presented along with the theory of residue class characters and their automatic generation. Tables are given of zeros of

L
L

series for moduli
A method for solving nonlinear Volterra integral equations of the second kind
Authors: Linz Peter
Published: June 2010

The approach given in this paper leads to numerical methods for Volterra integral equations which avoid the need for special starting procedures. Formulae for a typical fourth-order method are derived and some numerical examples presented. A convergence theorem is given for the method described.
Simultaneous approximation of a set of bounded real functions
Authors: Diaz J. B. , McLaughlin H. W.
Published: June 2010

The problem of simultaneous Chebyshev approximation of a set

F
F

of uniformly bounded, real-valued functions on a compact interval

I
I

. by a set

P
P

of continuous functions is equivalent to the problem of simultaneous approximation of two real-valued functions

F
+

(
x
)
,

F
−

(
x
)

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

, with

F
−

(
x
)
≦

F
+

(
x
)

{F^ - }(x) \leqq {F^ + }(x)

, for all

x
x

in

I
I

, where

F
−

{F^ - }

is lower semicontinuous and

F
−

{F^ - }

is upper semicontinuous.
Factorization of polynomials over finite fields
Authors: McEliece Robert J.
Published: June 2010

If

f
(
x
)

f(x)

is a polynomial over

G
F
(
q
)

GF(q)

, we observe (as has Berlekamp) that if

h

(
x

)
q

≡
h
(
x
)
(
mod
f
(
x
)
)

h{(x)^q} \equiv h(x)(\bmod f(x))

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis equals product Subscript a Baseline Subscript element-of upper G upper F left-parenthesis q right-parenthesis Baseline gcd left-parenthesis f left-parenthesis x right-parenthesis comma h left-parenthesis x right-parenthesis minus a right-parenthesis">

f
(
x
)
=
∏

a

∈
G
F
(
q
)

gcd
(
f
(
x
)
,
h
(
x
)
−
a
)

f(x) = \prod {{{_a}_{ \in GF(q)}}\gcd (f(x),h(x) - a)}

. The object of this paper is to give an explicit construction of enough such

h
h

’s so that the repeated application of this result will succeed in separating all irreducible factors of

f
f

. The

h
h

’s chosen are loosely defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript i Baseline left-parenthesis x right-parenthesis identical-to x Superscript i Baseline plus x Superscript i q Baseline plus x Superscript i q squared Baseline plus midline-horizontal-ellipsis left-parenthesis mod f left-parenthesis x right-parenthesis right-parenthesis">

h
i

(
x
)
≡

x
i

+

x

i
q

+

x

i

q
2

+
⋯
(
mod
f
(
x
)
)

{h_i}(x) \equiv {x^i} + {x^{iq}} + {x^{i{q^2}}} + \cdots (\bmod f(x))

. A detailed example over

G
F
(
2
)

GF(2)

is given, and a table of the factors of the cyclotomic polynomials
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi Subscript n Baseline left-parenthesis x right-parenthesis left-parenthesis mod p right-parenthesis for p equals 2 comma n less-than-over-equals 250 semicolon p equals 3 comma n less-than-over-equals 100 semicolon p equals 5 comma 7 comma n less-than-over-equals 50">

Φ
n

(
x
)
(
mod
p
)

 for 

p
=
2
,
n
≦
250
;
p
=
3
,
n
≦
100
;
p
=
5
,
7
,
n
≦
50

{\Phi _n}(x)(\bmod p){\text { for }}p = 2,n \leqq 250;p = 3,n \leqq 100;p = 5,7,n \leqq 50

, is included.
Nonnegative matrix equations having positive solutions
Authors: Walters Jerry A.
Published: June 2010

Suppose Ã is a nonnegative invertible matrix with a positive diagonal Diag

(

A
¯

)
>
0

(\bar A) > 0

and

y
¯

>
0

\bar y > 0

is a positive vector. Let

A
=

D

−
1

A
~

A = {D^{ - 1}}\tilde A

and

y
=

D

−
1

y
~

y = {D^{ - 1}}\tilde y

. If

0
>
2
y
−
A
y

0 > 2y - Ay

, then

2
y
−
A
y
≦
x
≦
y

2y - Ay \leqq x \leqq y

, where

A

−
1

y

{A^{ - 1}}y

.
Integer sequences having prescribed quadratic character
Authors: Lehmer D. H. , Lehmer Emma , Shanks Daniel
Published: June 2010

For the odd primes

p
1

=
3
,

{p_1} = 3,

,

p
2

=
5
,
⋯
,

{p_2} = 5, \cdots ,

we determine integer sequences

N
p

{N_p}

such that the Legendre symbol

(

N

/

N

p
i

p
i

)
=
±
1

({N \left / {\vphantom {N {{p_i}}}} \right . {{p_i}}}) = \pm 1

for all

p
i

≦
p

{p_i} \leqq p

for a prescribed array of signs

±
1

\pm 1

; (i.e., for a prescribed quadratic character). We examine six quadratic characters having special interest and applications. We present tables of these

N
p

{N_p}

and examine some applications, particularly to questions concerning extreme values for the smallest primitive root (of a prime

N
N

), the class number of the quadratic field

R
(
√
−
N
)

R(\surd - N)

, the real Dirichlet

L
L

functions, and quadratic character sums.
Note on the distribution of Ramanujan’s tau function
Authors: Lehmer D. H.
Published: June 2010

According to a conjecture of Sato and Tate, the angle

θ
\theta

whose cosine is

1
2

τ
(
p
)

p

−
11

/

2

\tfrac {1} {2}\tau (p){p^{ - 11/2}}

, where

τ
\tau

is Ramanujan’s function and

p
p

a prime, is distributed over

[
0
,
π
]

[0,\pi ]

according to a

sin
2

θ

{\sin ^2}\theta

law. The paper reports on a test of this conjecture for the 1229 primes under 10000. Extreme values of

θ
\theta

are also given.
Movable singularities and quadrature
Authors: Goodrich R. F. , Stenger F.
Published: June 2010

A general procedure is described for treating a movable singularity in an integral. This enabĺes us to change the original integral

I
0

{I_0}

into

G

I
1

G{I_1}

, where

G
G

depends only on the parameters of the singularity and

I
1

{I_1}

, is a new integral which exists for all values of the parameters. The results are then applied to the particular problem of evaluating
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript negative 1 Superscript 1 Baseline StartFraction f left-parenthesis x right-parenthesis d x Over left-brace left-parenthesis 1 minus x squared right-parenthesis left-parenthesis 1 minus k squared x squared right-parenthesis right-brace Superscript 1 slash 2 Baseline EndFraction comma">

∫

−
1

1

f
(
x
)
d
x

{
(
1
−

x
2

)
(
1
−

k
2

x
2

)
}

1

/

2

,

\int _{ - 1}^1 {\frac {{f(x)dx}} {{{{\{ (1 - {x^2})(1 - {k^2}{x^2})\} }^{1/2}}}}} ,

\]
where

f
f

is entire and

k
k

varies between

0
0

and

1
1

. Some new quadrature schemes and new effective methods of evaluating incomplete elliptic integrals are derived.
New error coefficients for estimating quadrature errors for analytic functions
Authors: Rabinowitz Philip , Richter Nira
Published: June 2010

Since properly normalized Chebyshev polynomials of the first kind

T
~

n

(
Z
)

{\tilde T_n}(Z)

satisfy
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper T overTilde Subscript m Baseline comma upper T overTilde Subscript n Baseline right-parenthesis equals integral Underscript element-of rho Endscripts ModifyingAbove upper T With tilde Subscript m Baseline left-parenthesis z right-parenthesis ModifyingAbove upper T Subscript n Baseline left-parenthesis z right-parenthesis With bar StartAbsoluteValue 1 minus z squared EndAbsoluteValue Superscript negative 1 slash 2 Baseline StartAbsoluteValue d z EndAbsoluteValue equals delta Subscript m n">

(

T
~

m

,

T
~

n

)
=

∫

∈
ρ

T
~

m

(

z

)

T
n

(

z

)

¯

|

1
−

z

2

|

−
1

/

2

|

d

z

|

=

δ

m
n

({\tilde T_m},{\tilde T_n}) = \int _{ \in \rho } {{{\tilde T}_m}({\text {z}})} \overline {{T_n}({\text {z}})} |1 - {{\text {z}}^2}{|^{ - 1/2}}|d{\text {z}}| = {\delta _{mn}}

\]
for ellipses

∈
ρ

\in \rho

with foci at

±
1

\pm 1

, any function analytic in

∈
ρ

\in \rho

has an expansion,

f
(

z

)
=
∑

a
n

T
~

n

(

z

)

f({\text {z}}) = \sum {{a_n}{{\tilde T}_n}({\text {z}})}

with

a
n

=
(
f
,

T
~

n

)

{a_n} = (f,{\tilde T_n})

. Applying the integration error operator

E
E

yields

E
(
f
)
=
∑

a
n

E
(

T
~

n

)

E(f) = \sum {{a_n}E({{\tilde T}_n})}

. Applying the Cauchy-Schwarz inequality to

E
(
f
)

E(f)

leads to the inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper E left-parenthesis f right-parenthesis StartAbsoluteValue less-than-over-equals sigma-summation StartAbsoluteValue a Subscript n Baseline EndAbsoluteValue squared sigma-summation vertical-bar upper E left-parenthesis upper T overTilde Subscript n Baseline right-parenthesis EndAbsoluteValue squared squared equals EndAbsoluteValue StartAbsoluteValue f EndAbsoluteValue StartAbsoluteValue StartAbsoluteValue EndAbsoluteValue upper E EndAbsoluteValue squared vertical-bar Superscript 2 Baseline period">

|

E
(
f
)

|

2

≦
∑

|

a
n

|

2

∑

|

E
(

T
~

n

)

|

2

=

|

|

f

|

|

2

|

|

E

|

|

2

.

|E(f){|^2} \leqq \sum {|{a_n}{|^2}} \sum {|E({{\tilde T}_n})} {|^2} = ||f|{|^2}||E|{|^2}.

\]
.

|

|

E

|

|

||E||

can be computed for any integration rule and approximated quite accurately for Gaussian integration rules. The bound for

|

E
(
f
)

|

|E(f)|

using this norm is compared to that using a previously studied norm based on Chebyshev polynomials of the second kind and is shown to be superior in practical situations. Other results involving the latter norm are carried over to the new norm.
On the numerical computation of eigenvalues and eigenvectors of symmetric integral equations
Authors: Linz Peter
Published: June 2010

The well-known error estimates for the numerical computation of eigenvalues of symmetric integral equations are extended to the computation of the eigenvectors. The results are used to justify the application of an improvement method to obtain an efficient algorithm for solving the eigenvalue problem.
On the existence of regions with minimal third degree integration formulas
Authors: Fritsch F. N.
Published: June 2010

A. H. Stroud has shown that

n
+
1

n + 1

is the minimum possible number of nodes in an integration formula of degree three for any region in

E
n

{E_n}

. In this paper, in answer to the question of the attainability of this minimal number, we exhibit for each

n
n

a region that possesses a third degree formula with

n
+
1

n + 1

nodes. This is accomplished by first deriving an

(
n
+
2
)

(n + 2)

-point formula of degree three for an arbitrary region that is invariant under the group of affine transformations that leave an

n
n

-simplex fixed. The formula is then applied to a one-parameter family of such regions, and a value of the parameter is determined for which the weight at the centroid vanishes.
Some calculations related to Riemann’s prime number formula
Authors: Riesel Hans , Göhl Gunnar
Published: June 2010

The objective of this paper is to study the relation of the complex zeros of the Riemann zeta function to the distribution of prime numbers. This relation arises from a formula of Riemann, which is studied here by extensive machine calculations. To establish the validity of the computations, reasonable upper bounds for the various errors involved are deduced. The analysis makes use of a formula, (32), which seems to be quite new.
Remark on a conjecture of Erdős on binomial coefficients
Authors: Mąkowski Andrzej
Published: June 2010

A conjecture attributed to Erdös concerning the Diophantine equation
\[

2

(

x
+
n
−
1

n

)

=

(

y
+
n
−
1

n

)

2\left ( {\begin {array}{*{20}{c}} {x + n - 1} \\ n \\ \end {array} } \right ) = \left ( {\begin {array}{*{20}{c}} {y + n - 1} \\ n \\ \end {array} } \right )

\]
is shown to be false.
Computing irreducible representations of groups
Authors: Dixon John D.
Published: June 2010

How can you find a complete set of inequivalent irreducible (ordinary) representations of a finite group? The theory is classical but, except when the group was very small or had a rather special structure, the actual computations were prohibitive before the advent of high-speed computers; and there remain practical difficulties even for groups of relatively small orders

(
≦
100
)

( \leqq 100)

. The present paper describes three techniques to help solve this problem. These are: the reduction of a reducible unitary representation into its irreducible components; the construction of a complete set of irreducible unitary representations from a single faithful representation; and the calculation of the precise values of a group character from values which have only been computed approximately.
Chebyshev-type integration rules of minimum norm
Authors: Rabinowitz Philip , Richter-Dyn Nira
Published: June 2010

Equal-weight integration rules are studied in the context of certain families of Hilbert spaces of analytic functions defined in a family of confocal ellipses containing the interval of integration. Rules which minimize the norm of the error functional in these spaces are shown to exist and several such rules are tabulated. Asymptotic properties of these rules are studied for ellipses shrinking to the integration interval and for ellipses expanding to cover the entire plane. In the latter case, an algebraic formulation for these asymptotic rules is given and it is shown that they agree with the classical Chebyshev integration rules whenever such rules exist.
Exponential Chebyshev approximation on finite subsets of [0,1]
Authors: Rosman Bernard H.
Published: June 2010

In this note the convergence of best exponential Chebyshev approximation on finite subsets of [0,1] to a best approximation on the interval is proved when the function to be approximated is continuous and when the union of the finite subsets is dense in [0, 1].
On the convergence of Broyden’s method for nonlinear systems of equations
Authors: Dennis J. E.
Published: June 2010

This paper uses majorant techniques to study the convergence of Broyden’s single-rank update method for nonlinear systems of equations. It also contains a very elementary proof of the local convergence of the method. The heart of the method is a procedure for generating an approximation to the Jacobian of the system using only information on hand and not requiring partial derivatives.
Least squares methods for 2𝑚th order elliptic boundary-value problems
Authors: Bramble J. H. , Schatz A. H.
Published: June 2010

In this paper we consider a general class of boundary-value problems for 2mth order elliptic equations including nonhomogeneous essential boundary conditions and nonselfadjoint problems. Approximation methods involving least squares approximation of the data are presented and corresponding error estimates are proved. These methods can be considered in the category of Rayleigh-Ritz-Galerkin methods and have the special feature that the trial functions need not satisfy the boundary conditions. A special case of the trial functions which is studied are spline functions defined on a uniform mesh of width h (or more generally piecewise polynomial functions). For a given "well set" boundary-value problem for a 2mth order operator the theory presented will provide a method with any prescribed order of accuracy r which is optimal in the sense that the best approximation in the underlying subspace is of order of accuracy r.
A note on Chowla’s function
Authors: Lal M. , Forbes A.
Published: June 2010

Iterates of a number-theoretic function, defined by

L
(
n
)
=
σ
(
n
)
−
(
1
+
n
)

L(n) = \sigma (n) - (1 + n)

, are investigated empirically, for

n
≦

10
5

n \leqq {10^5}

. This search has yielded 9 reduced amicable pairs.
Table for third-degree spline interpolation using equi-spaced knots
Authors: Hoskins W. D.
Published: June 2010

A table is given for the calculation of the parameters of a third-degree natural spline with n data points

(
n
>
2
)

(n > 2)

using a minimum number of multiplications and divisions. In addition, an example is given that demonstrates the method of use and enables comparisons to be made with a method previously described.
Miniaturized tables of Bessel functions. II
Authors: Luke Yudell L.
Published: June 2010

In a previous study, we discussed the expansion of two-parameter functions in a double series of Chebyshev polynomials, and, in particular, we presented coefficients for the evaluation of the modified Bessel function

(
2
z

/

π

)

1

/

2

e
z

K
v

(
z
)

{(2z/\pi )^{1/2}}{e^z}{K_v}(z)

to 20 decimals for all

z
≧
5

z \geqq 5

and all

v
,
0
≦
v
≦
1

v,0 \leqq v \leqq 1

. In the present study, we give similar coefficients for the evaluation of

g

e

−
z

z

−
μ

I
v

(
z
)

g{e^{ - z}}{z^{ - \mu }}{I_v}(z)

to at least 20 decimals where

I
v

(
z
)

{I_v}(z)

is the modified Bessel function of the first kind and g and

μ
\mu

are certain constants which depend on the range of the parameter and variable for four different situations. The ranges are
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 right-parenthesis 0 greater-than z less-than-over-equals 8 comma 0 less-than-over-equals v less-than-over-equals 4 semicolon left-parenthesis 2 right-parenthesis 0 greater-than z less-than-over-equals 8 comma 4 less-than-over-equals v less-than-over-equals 8 semicolon left-parenthesis 3 right-parenthesis z greater-than-over-equals 8 comma negative 1 less-than-over-equals v less-than-over-equals 0 semicolon left-parenthesis 4 right-parenthesis z greater-than-over-equals 8 comma 0 less-than-over-equals v less-than-over-equals 1">

(
1
)

0
>
z
≦
8
,
0
≦
v
≦
4
;
(
2
)

0
>
z
≦
8
,
4
≦
v
≦
8
;
(
3
)

z
≧
8
,
−
1
≦
v
≦
0
;
(
4
)

z
≧
8
,
0
≦
v
≦
1

(1)\;0 > z \leqq 8,0 \leqq v \leqq 4;(2)\;0 > z \leqq 8,4 \leqq v \leqq 8;(3)\;z \geqq 8, - 1 \leqq v \leqq 0;(4)\;z \geqq 8,0 \leqq v \leqq 1

.
Minimal quadratures for functions of low-order continuity
Authors: Johnson L. W. , Riess R. D.
Published: June 2010

An analog of Wilf’s quadrature is developed for functions of low-order continuity. This analog is used to demonstrate that the order of convergence of Wilf’s quadrature is at least

1

/

n

1/n

.
Rational approximations to 𝜋
Authors: Choong K. Y. , Daykin D. E. , Rathbone C. R.
Published: June 2010

Using an IBM 1130 computer, we have generated the first 20,000 partial quotients in the ordinary continued-fraction representation ofπ\pi.
Algorithms for Hermite and Smith normal matrices and linear Diophantine equations
Authors: Bradley Gordon H.
Published: June 2010

New algorithms for constructing the Hermite normal form (triangular) and Smith normal form (diagonal) of an integer matrix are presented. A new algorithm for determining the set of solutions to a system of linear diophantine equations is presented. A modification of the Hermite algorithm gives an integer-preserving algorithm for solving linear equations with real-valued variables. Rough bounds for the number of operations are cubic polynomials involving the order of the matrix and the determinant of the matrix. The algorithms are valid if the elements of the matrix are in a principal ideal domain.
Difference analogues of quasi-linear elliptic Dirichlet problems with mixed derivatives
Authors: Stepleman Robert S.
Published: June 2010

In this paper we consider a class of difference approximations to the Dirichlet problem for second-order quasi-linear elliptic operators with mixed derivative terms. The main result is that for this class of discretizations and bounded g (the right-hand side) a solution to the difference equations exists. We also explicitly exhibit a discretization of this type for a class of operators.
Infinite sums of roots for a class of transcendental equations and Bessel functions of order one-half
Authors: Liron N.
Published: June 2010

The roots of Bessel functions of order one-half are special cases of roots of transcendental equations of the form

tan
⁡
z
=
A
(
z
)

/

B
(
z
)

\tan z = A(z)/B(z)

, where

A
(
z
)
,
B
(
z
)

A(z),B(z)

are polynomials and

A
(
z
)

/

B
(
z
)

A(z)/B(z)

is odd. We prove that the function

f
(
z
)
=
B
(
z
)
sin
⁡
z
−
A
(
z
)
cos
⁡
z
,
f
(
z
)

f(z) = B(z)\sin z - A(z)\cos z,f(z)

even or odd, satisfies the conditions of Hadamard’s factorization theorem, and derive recurrences for sums of the form

S
l

=

∑

k
=
1

∞

α
k

−
2
l

,
l
=
1
,
2
,
⋯

{S_l} = \sum \nolimits _{k = 1}^\infty {\alpha _k^{ - 2l},l = 1,2, \cdots }

, where the

α
k

{\alpha _k}

’s are the nonzero roots of

f
(
z
)

f(z)

. We also prove under what conditions on

A
(
z
)

A(z)

and

B
(
z
)

B(z)

is

S
l

=

π

−
2
l
−
2

ζ
(
2
l
+
2
)

{S_l} = {\pi ^{ - 2l - 2}}\zeta (2l + 2)

or

S
l

=

π

−
2
l
−
2

ζ
(
2
l
+
2
)
(

2

2
l
+
2

−
1
)

{S_l} = {\pi ^{ - 2l - 2}}\zeta (2l + 2)({2^{2l + 2}} - 1)

, where

ζ
\zeta

is the Riemann zeta function. We prove that, although Bessel functions of positive half-order,

J

l
+
1

/

2

{J_{l + 1/2}}

, have only real roots, perturbation of any one of its coefficients introduces nonreal roots for

l
>
2

l > 2

.
On the convergence rates of variational methods. I. Asymptotically diagonal systems
Authors: Delves L. M. , Mead K. O.
Published: June 2010

We consider the problem of estimating the convergence rate of a variational solution to an inhomogeneous equation. This problem is not soluble in general without imposing conditions on both the class of expansion functions and the class of problems considered; we introduce the concept of “asymptotically diagonal systems,” which is particularly appropriate for classical variational expansions as applied to elliptic partial differential equations. For such systems, we obtain a number of a priori estimates of the asymptotic convergence rate which are easy to compute, and which are likely to be realistic in practice. In the simplest cases these estimates reduce the problem of variational convergence to the simpler problem of Fourier series convergence, which is considered in a companion paper. We also produce estimates for the convergence rate of the individual expansion coefficients

a
i

(
n
)

a_i^{(n)}

, thus categorising the convergence completely.
Stability of bounded solutions of linear functional equations
Authors: Franklin Joel N.
Published: June 2010

The weak sequential compactness of reflexive Banach spaces is used to explain the fact that certain ill-posed, linear problems become well-posed if the solutions are required to satisfy a prescribed bound. Applications are made to the computability of solutions of ill-posed problems associated with elliptic and parabolic partial differential equations.
A useful approximation to 𝑒⁻𝑡2
Authors: Bellman Richard , Kashef B. G. , Vasudevan R.
Published: June 2010

Using differential approximation, we obtain a remarkably accurate representation of

e

−

t
2

{e^{ - {t^2}}}

as a sum of three exponentials.
A note concerning the two-step Lax-Wendroff method in three dimensions
Authors: Eilon B.
Published: June 2010

The two-step Lax-Wendroff method in three spatial dimensions is discussed and, dealing with its linear stability in the hydrodynamic case, the sufficiency of the von Neumann condition is proved.
A rank two algorithm for unconstrained minimization
Authors: Bass Ronald
Published: June 2010

A stable second-order unconstrained minimization algorithm with quadratic termination is given. The algorithm does not require any one-dimensional minimizations. Computational results presented indicate that the performance of this algorithm compares favorably with other well-known unconstrained minimization algorithms.
Bateman’s constants reconsidered and the distribution of cubic residues
Authors: Shanks Daniel , Lal Mohan
Published: June 2010

We analyze the computation of certain slowly convergent infinite products involving cubic characters. A first-order analysis gives a

2

D

2{\text {D}}

or

3

D

3{\text {D}}

answer immediately, but extensive computation of cubic residues only improves this to

5D

{\text {5D}}

or

6D

{\text {6D}}

. To do better, one must examine the distribution of cubic residues or evaluate certain Dedekind Zeta functions. Both are done. The constants thus obtained are used to examine a variant of the Hardy-Littlewood Conjecture

K

{\text {K}}

concerning primes of the form

n
3

+
a

{n^3} + a

. Some related mathematics needed and developed includes an answer to this: Which

p
p

, satisfying

x
3

≡
a

(
mod

p
)

{x^3} \equiv a\pmod p

, have two solutions

x
x

that differ by

k

(
mod

p
)

k\pmod p

?
A method for computing Bessel function integrals
Authors: Linz Peter
Published: June 2010

Infinite integrals involving Bessel functions are recast, by means of an Abel transform, in terms of Fourier integrals. As there are many efficient numerical methods for computing Fourier integrals, this leads to a convenient way of approximating Bessel function integrals.
Some properties of a class of band matrices
Authors: Hoskins W. D. , Ponzo P. J.
Published: June 2010

Let

A
(
2
r
+
1
,
n
)

A(2r + 1,n)

denote the

n
×
n

n \times n

band matrix, of bandwidth

2
r
+
1

2r + 1

, with the binomial coefficients in the expansion of

(
x
−
1

)

2
r

{(x - 1)^{2r}}

as the elements in each row and column. Using the fact that the rows of

A
(
2
r
+
1
,
n
)

A(2r + 1,n)

provide the coefficients for the

2
r

2r

th central difference, a number of properties of

A
(
2
r
+
1
,
n
)

A(2r + 1,n)

are obtained for all positive integers

r
r

and

n
n

. These include obtaining explicit formulas for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="det upper A left-parenthesis 2 r plus 1 comma n right-parenthesis comma upper A Superscript negative 1 Baseline left-parenthesis 2 r plus 1 comma n right-parenthesis comma StartAbsoluteValue EndAbsoluteValue upper A Superscript negative 1 Baseline left-parenthesis 2 r plus 1 comma n right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript normal infinity Baseline">

det
A
(
2
r
+
1
,
n
)
,

A

−
1

(
2
r
+
1
,
n
)
,

|

|

A

−
1

(
2
r
+
1
,
n
)

|

|

∞

\det A(2r + 1,n),{A^{ - 1}}(2r + 1,n),||{A^{ - 1}}(2r + 1,n)|{|_\infty }

and an upper triangular matrix

U
U

such that

A
(
2
r
+
1
,
n
)
U

A(2r + 1,n)U

is lower triangular.
A convergence criterion for a class of integration methods
Authors: Butcher J. C.
Published: June 2010

A criterion is established for the convergence of sequences of a general type of Runge-Kutta method. The criterion is expressed in terms of “weak convergence", a property defined in the paper.
The numerical solution of hyperbolic systems using bicharacteristics
Authors: Johnston R. L. , Pal S. K.
Published: June 2010

This paper is concerned with deriving methods for solving numerically a firstorder quasilinear hyperbolic system. The basic principle of the method is to integrate along one of the bicharacteristics of the system to obtain an equivalent integral system. The numerical methods are then obtained by making suitable approximations to this integral system. Stability and convergence properties are analyzed in some detail. The methods are relatively easy to implement and have been successfully applied to problems in one, two and three space dimensions in such areas as magnetohydrodynamics and dynamic elasticity.
Minimal error constant numerical differentiation (N.D.) formulas
Authors: Pelios A. , Klopfenstein R. W.
Published: June 2010

In this paper, we consider a class of

k
k

-step linear multistep methods in the form (1.1) of numerical differentiation (N.D.) formulas. For each

k
k

, we have required the property of

A
A

-stability which implies at most second order for the associated operator. Among such second-order operators, the parameters of the formulas have been selected to minimize the error constant consistent with the

A
A

-stability property. It is shown that the error constant approaches that of the trapezoidal rule as

k
→
∞

k \to \infty

and that significant reductions occur for quite modest

k
k

. Thus, these results have significance in practical applications.
Coupled harmonic equations, SOR, and Chebyshev acceleration
Authors: Ehrlich L. W.
Published: June 2010

A coupled pair of harmonic equations is solved by the application of Chebyshev acceleration to the Jacobi, Gauss-Seidel, and related iterative methods, where the Jacobi iteration matrix has purely imaginary (or zero) eigenvalues. Comparison is made with a block SOR method used to solve the same problem.
A unified presentation of certain classical polynomials
Authors: Srivastava H. M. , Singhal J. P.
Published: June 2010

This paper attempts to present a unified treatment of the classical orthogonal polynomials, viz. Jacobi, Laguerre and Hermite polynomials, and their generalizations introduced from time to time. The results obtained here include a number of linear, bilinear and bilateral generating functions and operational formulas for the polynomials
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper T Subscript n Superscript left-parenthesis alpha comma beta right-parenthesis Baseline left-parenthesis x comma a comma b comma c comma d comma p comma r right-parenthesis vertical-bar n equals 0 comma 1 comma 2 comma midline-horizontal-ellipsis right-brace">

{

T
n

(
α
,
β
)

(
x
,
a
,
b
,
c
,
d
,
p
,
r
)

|

n
=
0
,
1
,
2
,
⋯
}

\{ T_n^{(\alpha ,\beta )}(x,a,b,c,d,p,r)|n = 0,1,2, \cdots \}

, defined by Eq. (3) below.*
Endpoint formulas for interpolatory cubic splines
Authors: Seidman Thomas I. , Korsan Robert J.
Published: June 2010

In the absence of known endpoint derivatives, the usual procedure is to use a “natural” spline interpolant which Kershaw has shown to have

O

(

h
4

)

\mathcal {O}({h^4})

error except near the endpoints. This note observes that either the use of appropriate finite-difference approximations for the endpoint derivatives or a proposed modification of the interpolation algorithm leads to

O

(

h
4

)

\mathcal {O}({h^4})

error uniformly in the interval of approximation.
Some prime numbers of the forms 2𝐴3ⁿ+1 and 2𝐴3ⁿ-1
Authors: Williams H. C. , Zarnke C. R.
Published: June 2010

All primes of the form

2
A

3
n

+
1

2A{3^n} + 1

and of the form

2
A

3
n

−
1

2A{3^n} - 1

, where

1
≦
A
≦
50

1 \leqq A \leqq 50

and

1
≦
n
≦
325

1 \leqq n \leqq 325

, are found. Some large twin primes are also determined.
On the equation 𝜙(𝑛)=𝜙(𝑛+𝑘)
Authors: Lal M. , Gillard P.
Published: June 2010

The number of solutions of the equation

(
n
)
=
(
n
+
k
)

(n) = (n + k)

, for

k
≦
30

k \leqq 30

, at intervals of 10

4

^{4}

to 10

5

^{5}

are given. The values of

n
n

for which

(
n
)
=
(
n
+
k
)
=
(
n
+
2
k
)

(n) = (n + k) = (n + 2k)

, and for which

ϕ
(
n
)
=
ϕ
(
n
+
k
)
=
ϕ
(
n
+
2
k
)
=
ϕ
(
n
+
3
k
)

\phi (n) = \phi (n + k) = \phi (n + 2k) = \phi (n + 3k)

, are also tabulated.
On the solution of block-tridiagonal systems arising from certain finite-difference equations
Authors: Varah J. M.
Published: June 2010

We consider the solution of the linear systems arising from certain implicit finite-difference approximations to systems of linear differential equations. In particular, we consider those schemes which lead to matrices of block-tridiagonal form. There are two common methods for solving such equations: using a block-tridiagonal factorization (blocksolve), or treating the matrix as a band matrix (bandsolve). First, we discuss conditions for ensuring the numerical stability of the block-tridiagonal factorization for general matrices of this form. Then, we compare the two methods for general block-tridiagonal matrices (including matrices arising from the Crank-Nicolson scheme for systems of parabolic equations) and for a more specialized block-tridiagonal matrix which arises from schemes of H. B. Keller for systems of two-point boundary value problems and parabolic equations.
Inversion of isoclinal matrices
Authors: Derr Leroy J.
Published: June 2010

Triangular matrices (over the reals), whose elements satisfy

a

i
,
j

=

a

i
+
1
,
j
+
1

{a_{i,j}} = {a_{i + 1,j + 1}}

, can be mapped algebraically into the rational functions to facilitate determination of inverses and factors of the matrices. It is further shown in this note that these methods can be extended to the nontriangular forms to obtain inverses or demonstrate singular cases. Applications are made to certain matrices of Hessenberg type.
Computation of best monotone approximations
Authors: Lewis James T.
Published: June 2010

A numerical procedure to compute the best uniform approximation to a given continuous function by algebraic polynomials with nonnegative

r
r

th derivative is presented and analyzed. The method is based on discretization and linear programming. Several numerical experiments are discussed.
Exit criteria for Simpson’s compound rule
Authors: Rowland J. H. , Varol Y. L.
Published: June 2010

In many automated numerical algorithms, the calculations are stopped when the difference between two successive approximations is less than a preassigned tolerance. The dependability of this procedure for Simpson’s compound rule has been investigated. Classes of functions have been determined for which the above criterion is (a) always valid, and (b) asymptotically valid. A new exit rule is proposed which appears to be less conservative than the standard technique.
On the largest prime divisor of an odd perfect number
Authors: Hagis Peter , McDaniel Wayne L.
Published: June 2010

It is shown here that if n is odd and perfect, then n has a prime divisor which exceeds 11200.
An asymptotic expansion of 𝑊_{𝑘,𝑚}(𝑧) with large variable and parameters
Authors: Wong R.
Published: June 2010

In this paper, we obtain an asymptotic expansion of the Whittaker function

W

k
,
m

(
z
)

{W_{k,m}}(z)

when the parameters and variable are all large but subject to the growth restrictions that

k
=
o
(
z
)

k = o(z)

and

m
=
o
(

z

1

/

2

)

m = o({z^{1/2}})

as

z
→
∞

z \to \infty

. Here, it is assumed that k and m are real and

|

arg
⁡
z

|

≦

π
−
δ

|\arg z|\; \leqq \;\pi - \delta

.
On a method of Carasso and Laurent for constructing interpolating splines
Authors: Munteanu M. J. , Schumaker L. L.
Published: June 2010

Carasso and Laurent studied a method for computing natural polynomial splines interpolating simple data. We discuss several similar methods which can be applied to numerical construction of more general interpolating splines, including Lg-splines interpolating Extended-Hermite-Birkhoff data.
Polynomial approximation of a function and its first derivative in near minimax norms
Authors: Cohen Edgar A.
Published: June 2010

Two near minimax norms for polynomial approximation are presented. They are designed for approximation of both a function and its first derivative uniformly by polynomials over a given finite interval. The first one is a convex combination of two integrals, one involving the function and the other the derivative, and the second is the sum of the square of the value of the function at one point and an integral involving the derivative. For any smooth function defined on a finite closed interval, one forms a generalized Chebyshev polynomial expansion to approximate both the function and derivative uniformly.
Cubature error bounds for analytic functions
Authors: Lether F. G.
Published: June 2010

In this paper, two-dimensional cubature error bounds are developed. It is assumed that the function to be integrated is analytic, and that the domain of integration is contained in

[
−
1
,
1
]
×
[
−
1
,
1
]

[ - 1,1] \times [ - 1,1]

. Tables of error constants for several cubature rules and domains of integration are included.
A modified Bairstow method for multiple zeros of a polynomial
Authors: Carrano F. M.
Published: June 2010

A modification of Bairstow’s method to find multiple quadratic factors of a polynomial is presented. The nonlinear system of equations of the Bairstow method is replaced by high order partial derivatives of that system. The partials are computed by a repetition of the Bairstow recursion formulas. Numerical results demonstrate that the modified method converges in many cases where the Bairstow method fails due to the multiplicity of the quadratic factor. Rall [4] has described a generalization of Newton’s method for simultaneous nonlinear equations with multiple roots. This may be applied to solve the nonlinear Bairstow equations; however, it fails in some cases due to near-zero divisors. Examples are presented which illustrate the behavior of the author’s algorithm as well as the methods of Rall and Bairstow.
Computational problems in 𝑆-function theory
Authors: Morris P. A.
Published: June 2010

In this paper we discuss some computational problems associated with Schur-functions. A well-known algorithm for the ordinary product is described and adapted for a computer. A theorem of Todd is discussed in the same way, and these methods are combined to produce a general program for the plethysm (wreath product) of two S-functions.
Convergence for a vortex method for solving Euler’s equation
Authors: Dushane Theodore E.
Published: June 2010

We consider a new vortex approximation for solving the initial-value problem for the Euler equations in two dimensions. We assume there exists a smooth solution to these equations and that the vorticity has compact support. Then we show that our approximation to the velocity field converges uniformly in space and time for a short time interval.
A note on the stability of an iterative finite-difference method for hyperbolic systems
Authors: Goldberg Moshe
Published: June 2010

In this note, we find analytically the linear stability criteria for two finite-difference methods for hyperbolic systems in conservation-law form, presented recently by S. Abarbanel and G. Zwas and by S. Abarbanel and M. Goldberg.
On computing the minimal number of defining relations for finite groups
Authors: Sag T. W. , Wamsley J. W.
Published: June 2010

This paper describes a method for computing the Schur multiplicator of a finite supersolvable group G, given by some fixed generating system chosen from a cyclic series for G, and hence a lower bound for the minimal number of relations needed to define G.
The Euler-Maclaurin expansion for the simplex
Authors: Lyness J. N. , Puri K. K.
Published: June 2010

A natural extension of the one-dimensional trapezoidal rule to the simplex

0
≦

x
i

≦
1
,
∑

x
i

≦
1

0 \leqq {x_i} \leqq 1, \sum {x_i} \leqq 1

, is a rule Rf which uses as abscissas all those points on a hyper-rectangular lattice of spacing

h
=
1

/

m

h = 1/m

which lie within the simplex, assigning an equal weight to each interior point. In this paper, rules of this type are defined and some of their properties are derived. In particular, it is shown that the error functional satisfies an Euler-Maclaurin expansion of the type
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R f minus upper I f tilde upper A 1 h plus upper A 2 h squared plus midline-horizontal-ellipsis plus upper A Subscript p Baseline h Superscript p Baseline plus upper O left-parenthesis h Superscript p plus 1 Baseline right-parenthesis">

R
f
−
I
f
∼

A
1

h
+

A
2

h
2

+
⋯
+

A
p

h
p

+
O
(

h

p
+
1

)

Rf - If \sim {A_1}h + {A_2}{h^2} + \cdots + {A_p}{h^p} + O({h^{p + 1}})

\]
so long as

f
(

x

)

f({\text {x}})

and its partial derivatives of order up to p are continuous. Conditions under which this asymptotic series terminates are given, together with the condition for odd terms to drop out leaving an even expansion. The application to Romberg integration is discussed.