bluuebunny/crossref_metadata_2025 huggingface dataset:
An iterative solution method for linear systems of which the coefficient matrix is a symmetric 𝑀-matrix
Authors: Meijerink J. A. , van der Vorst H. A.
Published: June 2010

A particular class of regular splittings of not necessarily symmetric M-matrices is proposed. If the matrix is symmetric, this splitting is combined with the conjugate-gradient method to provide a fast iterative solution algorithm. Comparisons have been made with other well-known methods. In all test problems the new combination was faster than the other methods.
Common zeros of two Bessel functions
Authors: Benton T. C. , Knoble H. D.
Published: June 2010

There is a theorem that two Bessel functions

J
μ

(
x
)

{J_\mu }(x)

and

J
ν

(
x
)

{J_\nu }(x)

can have no common positive zeros if

μ
\mu

is an integer and

ν
=
μ
+
m

\nu = \mu + m

where m is an integer, but this does not preclude the possibility that for unrestricted real positive

μ
\mu

and

ν
\nu

not differing by an integer, the two functions

J
μ

(
x
)

{J_\mu }(x)

and

J
ν

(
x
)

{J_\nu }(x)

can have common zeros. An example is found where two such functions have two positive zeros in common.
Fast Poisson solvers for problems with sparsity
Authors: Banegas Alexandra
Published: June 2010

Fast Poisson solvers, which provide the numerical solution of Poisson’s equation on regions that permit the separation of variables, have proven very useful in many applications. In certain of these applications the data is sparse and the solution is only required at relatively few mesh points. For such problems this paper develops algorithms that allow considerable savings in computer storage as well as execution speed. Results of numerical experiments are given.
Roots of two transcendental equations as functions of a continuous real parameter
Authors: Pexton Robert L. , Steiger Arno D.
Published: June 2010

The roots,

λ
\lambda

and

η
\eta

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

j
l

(
α
λ
)

y
l

(
λ
)
=

j
l

(
λ
)

y
l

(
α
λ
)

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

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

[
x

j
l

(
x
)

]

x
=
α
η

′

[
x

y
l

(
x
)

]

x
=
η

′

=
[
x

j
l

(
x
)

]

x
=
η

′

[
x

y
l

(
x
)

]

x
=
α
η

′

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

\]
where

l
=
1
,
2
,
…

l = 1,2, \ldots

are considered as functions of the continuous real parameter

α
\alpha

. The symbols

j
l

{j_l}

and

y
l

{y_l}

denote the spherical Bessel functions of the first and second kind. The two transcendental equations are invariant under the transformations

λ
→
−
λ

\lambda \to - \lambda

and

η
→
−
η

\eta \to - \eta

, respectively. Therefore, only positive roots are discussed. All the

λ
\lambda

-roots increase monotonically as

α
\alpha

increases in the open interval (0, 1). For each order l, the smallest

η
\eta

-root decreases monotonically as

α
\alpha

increases in (0, 1), tending towards

l
(
l
+
1
)

\sqrt {l(l + 1)}

as

α
\alpha

approaches unity. For

α
∈
(
0
,
1
)

\alpha \in (0,1)

all the other

η
\eta

-roots have a minimum value equal to

l
(
l
+
1
)

/

α

\sqrt {l(l + 1)} /\alpha \,

.
An improved multivariate polynomial factoring algorithm
Authors: Wang Paul S.
Published: June 2010

A new algorithm for factoring multivariate polynomials over the integers based on an algorithm by Wang and Rothschild is described. The new algorithm has improved strategies for dealing with the known problems of the original algorithm, namely, the leading coefficient problem, the bad-zero problem and the occurrence of extraneous factors. It has an algorithm for correctly predetermining leading coefficients of the factors. A new and efficient p-adic algorithm named EEZ is described. Basically it is a linearly convergent variable-by-variable parallel construction. The improved algorithm is generally faster and requires less store then the original algorithm. Machine examples with comparative timing are included.
The prime number graph
Authors: Pomerance Carl
Published: June 2010

Let

p
n

{p_n}

denote the nth prime. The prime number graph is the set of lattice points

(
n
,

p
n

)

(n,{p_n})

,

n
=
1
,
2
,
…

n = 1,2, \ldots

. We show that for every k there are k such points that are collinear. By considering the convex hull of the prime number graph, we show that there are infinitely many n such that

2

p
n

>

p

n
−
i

+

p

n
+
i

2{p_n} > {p_{n - i}} + {p_{n + i}}

for all positive

i
>
n

i > n

. By a similar argument, we show that there are infinitely many n for which

p
n
2

>

p

n
−
i

p

n
+
i

p_n^2 > {p_{n - i}}{p_{n + i}}

for all positive

i
>
n

i > n

, thus verifying a conjecture of Selfridge. We make some new conjectures.
Hadamard matrices, finite sequences, and polynomials defined on the unit circle
Authors: Yang C. H.
Published: June 2010

If a

(
∗
)

(\ast )

-type Hadamard matrix of order 2n (i.e. a pair (A, B) of

n
×
n

n \times n

circulant (1,-1) matrices satisfying

A
A
′
+
B
B
′
=
2
n
I

AA\prime + BB\prime = 2nI

) exists and a pair of Golay complementary sequences (or equivalently, two-symbol

δ
\delta

-code) of length m exists, then a

(
∗
)

(\ast )

-type Hadamard matrix of order 2mn also exists. If a Williamson matrix of order 4n (i.e. a quadruple (W, X, Y, Z) of

n
×
n

n \times n

symmetric circulant (1,-1) matrices satisfying

W
2

+

X
2

+

Y
2

+

Z
2

=
4
n
I

{W^2} + {X^2} + {Y^2} + {Z^2} = 4nI

) exists and a four-symbol

δ
\delta

-code of length m exists, then a Goethals-Seidel matrix of order 4mn (i.e. a quadruple (A, B, C, D) of

m
n
×
m
n

mn \times mn

circulant (1, -1) matrices satisfying

A
A
′
+
B
B
′
+
C
C
′
+
D
D
′
=
4
m
n
I

AA\prime + BB\prime + CC\prime + DD\prime = 4mnI

) also exists. Other related topics are also discussed.
A polynomial representation of hybrid methods for solving ordinary differential equations
Authors: Gupta G. K.
Published: June 2010

A polynomial representation of the hybrid methods for solving ordinary differential equations is presented. The advantages of the representation are briefly discussed. Also it is shown that one step taken using a hybrid method is equivalent to two steps of the usual multistep methods; one step taken using an explicit method and the other taken using an implicit method. Therefore, the hybrid methods are really a special case of cyclic methods.
Monotone difference approximations for scalar conservation laws
Authors: Crandall Michael G. , Majda Andrew
Published: June 2010

A complete self-contained treatment of the stability and convergence properties of conservation-form, monotone difference approximations to scalar conservation laws in several space variables is developed. In particular, the authors prove that general monotone difference schemes always converge and that they converge to the physical weak solution satisfying the entropy condition. Rigorous convergence results follow for dimensional splitting algorithms when each step is approximated by a monotone difference scheme. The results are general enough to include, for instance, Godunov’s scheme, the upwind scheme (differenced through stagnation points), and the Lax-Friedrichs scheme together with appropriate multi-dimensional generalizations.
On the 𝐿^{∞}-convergence of Galerkin approximations for second-order hyperbolic equations
Authors: Baker Garth A. , Dougalis Vassilios A.
Published: June 2010

It is shown that certain classes of high order accurate Galerkin approximations for homogeneous second-order hyperbolic equations, known to possess optimal order rate of convergence in

L
2

{L^2}

, also possess optimal order rate of convergence in

L
∞

{L^\infty }

. This is attainable with particular smoothness assumptions on the initial data. We establish sufficient conditions for optimal

L
∞

{L^\infty }

-convergence of the approximations to the solution and also the approximation to its time derivative. This is done for both semidiscrete approximations and for single-step fully discrete approximations generated by rational functions.
The pseudoprimes to 25⋅10⁹
Authors: Pomerance Carl , Selfridge J. L. , Wagstaff Samuel S.
Published: June 2010

The odd compositen⩽25⋅109n \leqslant 25 \cdot {10^9}such that2n−1≡1(modn){2^{n - 1}} \equiv 1\;\pmod nhave been determined and their distribution tabulated. We investigate the properties of three special types of pseudoprimes: Euler pseudoprimes, strong pseudoprimes, and Carmichael numbers. The theoretical upper bound and the heuristic lower bound due to Erdös for the counting function of the Carmichael numbers are both sharpened. Several new quick tests for primality are proposed, including some which combine pseudoprimes with Lucas sequences.
The rate of convergence of Hermite function series
Authors: Boyd John P.
Published: June 2010

Let

α
>
0

\alpha > 0

be the least upper bound of

γ
\gamma

for which
\[

f
(
z
)
∼
O
(

e

−
q

|

z

|

γ

)

f(z) \sim O({e^{ - q|z|\gamma }})

\]
for some positive constant q as

|

z

|

→
∞

|z| \to \infty

on the real axis. It is then proved that at least an infinite subsequence of the coefficients

{

a
n

}

\{ {a_n}\}

in
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis equals e Superscript minus z squared slash 2 Baseline sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts a Subscript n Baseline upper H Subscript n Baseline left-parenthesis z right-parenthesis comma">

f
(
z
)
=

e

−

z
2

/

2

∑

n
=
0

∞

a
n

H
n

(
z
)
,

f(z) = {e^{ - {z^2}/2}}\sum \limits _{n = 0}^\infty {{a_n}{H_n}(z),}

\]
where the

H
n

{H_n}

are the normalized Hermite polynomials, must satisfy certain lower bounds. The theorems show two striking facts. First, the convergence rate of a Hermite series depends not only upon the order

ρ
\rho

for an entire function or the location of the nearest singularity for a singular function as for a power series but also upon

α
\alpha

, thus making the convergence theory of Hermitian series more complicated (and interesting) than that for any ordinary Taylor expansion. Second, the poorer the match between the asymptotic behavior of

f
(
z
)

f(z)

and

exp
⁡
(
−
1

/

2

z
2

)

\exp (-1/2 z^2)

the poorer the convergence of the Hermite series will be.
Lucas pseudoprimes
Authors: Baillie Robert , Wagstaff Samuel S.
Published: June 2010

We define several types of pseudoprimes with respect to Lucas sequences and prove the analogs of various theorems about ordinary pseudoprimes. For example, we show that Lucas pseudoprimes are rare and we count the Lucas sequences modulonwith respect to whichnis a Lucas pseudoprime. We suggest some powerful new primality tests which combine Lucas pseudoprimes with ordinary pseudoprimes. Since these tests require the evaluation of the least numberf(n)f(n)for which the Jacobi symbol(f(n)/n)(f(n)/n)is less than 1, we evaluate the average order of the functionf.
Asymptotically fast factorization of integers
Authors: Dixon John D.
Published: June 2010

The paper describes a "probabilistic algorithm" for finding a factor of any large composite integer n (the required input is the integer n together with an auxiliary sequence of random numbers). It is proved that the expected number of operations which will be required is

O
(
exp
⁡
{
β

(
ln
⁡
n
ln
⁡
ln
⁡
n

)

1

/

2

}
)

O(\exp \{ \beta {(\ln n\ln \ln n)^{1/2}}\} )

for some constant

β
>
0

\beta > 0

. Asymptotically, this algorithm is much faster than any previously analyzed algorithm for factoring integers; earlier algorithms have all required

O
(

n
α

)

O({n^\alpha })

operations where

α
>
1

/

5

\alpha > 1/5

.
Eigenvalue approximation by mixed and hybrid methods
Authors: Mercier B. , Osborn J. , Rappaz J. , Raviart P.-A.
Published: June 2010

Rate of convergence estimates are derived fro the approximation of eigenvalues and eigenvectors by mixed and hybrid methods. Several closely related abstract results on spectral approximation are proved. These results are then applied to a variety of finite element methods of mixed and hybrid type: a mixed method for 2nd order problems, mixed methods for 4th order problems, a hybrid method for 2nd order problems, and two mixed methods for the Stokes eigenvalue problem.
Minimal solutions of three-term recurrence relations and orthogonal polynomials
Authors: Gautschi Walter
Published: June 2010

We observe that the well-known recurrence relation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript n plus 1 Baseline left-parenthesis z right-parenthesis equals left-parenthesis z minus a Subscript n Baseline right-parenthesis p Subscript n Baseline left-parenthesis z right-parenthesis minus b Subscript n Baseline p Subscript n minus 1 Baseline left-parenthesis z right-parenthesis">

p

n
+
1

(
z
)
=
(
z
−

a
n

)

p
n

(
z
)
−

b
n

p

n
−
1

(
z
)

{p_{n + 1}}(z) = (z - {a_n}){p_n}(z) - {b_n}{p_{n - 1}}(z)

for orthogonal polynomials admits a "minimal solution" if z is outside the spectrum of the mass distribution

d
s
(
t
)

ds(t)

with respect to which the polynomials are orthogonal and if the moment problem for this distribution is determined. The minimal solution, indeed is

f
n

(
z
)
=
∫

p
n

(
t
)

d
s
(
t
)

/

(
z
−
t
)

{f_n}(z) = \smallint {p_n}(t)\,ds(t)/(z - t)

, and can be computed accurately by means of the author’s continued fraction algorithm. An application is made to special Gauss-type quadrature formulas.
A new algorithm for factoring polynomials over finite fields
Authors: Cantor David G. , Zassenhaus Hans
Published: June 2010

We present a new probabilistic algorithm for factoring polynomials over finite fields.
A numerical investigation into the length of the period of the continued fraction expansion of √𝐷
Authors: Williams H. C.
Published: June 2010

Let

p
(
D
)

p(D)

be the length of the period of the continued fraction expansion of

D

\sqrt D

, where D is a positive square-free integer. In this paper it is suggested that

p
(
D
)
=
O
(

D

log
⁡
log
⁡
D
)

p(D) = O(\sqrt D \log \log D)

and several tables of numerical results, which support this suggestion, are provided.
Factorization of the eighth Fermat number
Authors: Brent Richard P. , Pollard John M.
Published: June 2010

We describe a Monte Carlo factorization algorithm which was used to factorize the Fermat number

F
8

=

2

256

+
1

{F_8} = {2^{256}} + 1

. Previously

F
8

{F_8}

was known to be composite, but its factors were unknown.
Krylov subspace methods for solving large unsymmetric linear systems
Authors: Saad Y.
Published: June 2010

Some algorithms based upon a projection process onto the Krylov subspace

K
m

=
Span
⁡
(

r
0

,
A

r
0

,
…
,

A

m
−
1

r
0

)

{K_m} = \operatorname {Span}({r_0},A{r_0}, \ldots ,{A^{m - 1}}{r_0})

are developed, generalizing the method of Conjugate gradients to unsymmetric systems. These methods are extensions of Arnoldi’s algorithm for solving eigenvalue problems. The convergence is analyzed in terms of the distance of the solution to the subspace

K
m

{K_m}

and some error bounds are established showing, in particular, a similarity with the conjugate gradient method (for symmetric matrices) when the eigenvalues are real. Several numerical experiments are described and discussed.
Scattered data interpolation: tests of some methods
Authors: Franke Richard
Published: June 2010

This paper is concerned with the evaluation of methods for scattered data interpolation and some of the results of the tests when applied to a number of methods. The process involves evaluation of the methods in terms of timing, storage, accuracy, visual pleasantness of the surface, and ease of implementation. To indicate the flavor of the type of results obtained, we give a summary table and representative perspective plots of several surfaces.
Some optimal error estimates for piecewise linear finite element approximations
Authors: Rannacher Rolf , Scott Ridgway
Published: June 2010

It is shown that the Ritz projection onto spaces of piecewise linear finite elements is bounded in the Sobolev space,

W
^

p
1

\hat {W}_p^1

, for

2
⩽
p
⩽
∞

2 \leqslant p \leqslant \infty

. This implies that for functions in

W
^

p
1

∩

W
p
2

\hat {W}_p^1 \cap W_p^2

the error in approximation behaves like

O
(
h
)

O(h)

in

W
p
1

W_p^1

, for

2
⩽
p
⩽
∞

2 \leqslant p \leqslant \infty

, and like

O
(

h
2

)

O({h^2})

in

L
p

{L_p}

, for

2
⩽
p
>
∞

2 \leqslant p > \infty

. In all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur.
An upper bound for the first zero of Bessel functions
Authors: Chambers Ll. G.
Published: June 2010

It is shown, using the Rayleigh-Ritz method of the calculus of variations, that an upper bound for the first zero

j
v

{j_v}

, of

z

−
v

J
v

(
z
)

{z^{ - v}}{J_v}(z)

,

v
>
−
1

v > - 1

, is given by
\[

(
v
+
1

)

1

/

2

{

(
v
+
2

)

1

/

2

+
1
}
,

{(v + 1)^{1/2}}\{ {(v + 2)^{1/2}} + 1\} ,

\]
and that for large v,

v
,

j
v

=
v
+
O
(

v

1

/

2

)

v,{j_v} = v + O({v^{1/2}})

.
Runge-Kutta theory for Volterra integral equations of the second kind
Authors: Brunner H. , Hairer E. , Nørsett S. P.
Published: June 2010

The present paper develops the theory of general Runge-Kutta methods for Volterra integral equations of the second kind. The order conditions are derived by using the theory of P-series, which for our problem reduces to the theory of V-series. These results are then applied to two special classes of Runge-Kutta methods introduced by Pouzet and by Beĺtyukov.
Asymptotic expansion of the Lebesgue constants associated with polynomial interpolation
Authors: Shivakumar P. N. , Wong R.
Published: June 2010

An infinite asymptotic expansion is obtained for the Lebesgue constants associated with the polynomial interpolation at the zeros of the Chebyshev polynomials. The error due to truncation is shown to be bounded in absolute value by, and of the same sign as, the first neglected term.
A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains
Authors: Goldstein Charles I.
Published: June 2010

A finite element method is described for solving Helmholtz type boundary value problems in unbounded regions, including those with infinite boundaries. Typical examples include the propagation of acoustic or electromagnetic waves in waveguides. The radiation condition at infinity is based on separation of variables and differs from the classical Sommerfeld radiation condition. It is shown that the problem may be replaced by a boundary value problem on a fixed bounded domain. The behavior of the solution near infinity is incorporated in a nonlocal boundary condition. This problem is given a weak or variational formulation, and the finite element method is then applied. It is proved that optimal error estimates hold.
Galerkin methods for second kind integral equations with singularities
Authors: Graham Ivan G.
Published: June 2010

This paper discusses the numerical solution of Fredholm integral equations of the second kind which have weakly singular kernels and inhomogeneous terms. Global convergence estimates are derived for the Galerkin and iterated Galerkin methods using splines on arbitrary quasiuniform meshes as approximating subspaces. It is observed that, due to the singularities present in the solution being approximated, the resulting convergence may be slow. It is then shown that convergence will be improved greatly by the use of splines based on a mesh which has been suitably graded to accommodate these singularities. In fact, it is shown that, under suitable conditions, the Galerkin method converges optimally and the iterated Galerkin method is superconvergent. Numerical llustrations are given.
Complex Chebyshev polynomials on circular sectors with degree six or less
Authors: Grothkopf U. , Opfer G.
Published: June 2010

Let

T
n
α

T_n^\alpha

denote the nth Chebyshev polynomial on the circular sector

S
α

=
{
z
:

|

z

|

⩽
1
,

|

arg
⁡
z

|

⩽
α
}

{S^\alpha } = \{ z:|z| \leqslant 1,|\arg z| \leqslant \alpha \}

. This paper contains numerical values of

‖

T
n
α

‖

∞

{\left \| {T_n^\alpha } \right \|_\infty }

and the corresponding coefficients of

T
n
α

T_n^\alpha

for

n
=
1
(
1
)
6

n = 1(1)6

and

α
=

0
∘

(

5
∘

)

180
∘

\alpha = {0^ \circ }({5^ \circ }){180^ \circ }

. Also all critical angles for

T
n
α

,
n
=
1
(
1
)
6

T_n^\alpha ,n = 1(1)6

are listed, where an angle is called critical when the number of absolute maxima of

|

T
n
α

|

|T_n^\alpha |

changes at that angle. All figures are given to six places. The positions (and hence the number) of extremal points of

T
n
α

,
n
=
1
(
1
)
6

T_n^\alpha ,n = 1(1)6

are presented graphically. The method consists of a combination of semi-infinite linear programming, finite linear programming, and Newton’s method.
On the number of Markoff numbers below a given bound
Authors: Zagier Don
Published: June 2010

According to a famous theorem of Markoff, the indefinite quadratic forms with exceptionally large minima (greater than13\frac {1}{3}of the square root of the discriminant) are in 1 : 1 correspondence with the solutions of the Diophantine equationp2+q2+r2=3pqr{p^2} + {q^2} + {r^2} = 3pqr. By relating Markoffs algorithm for finding solutions of this equation to a problem of counting lattice points in triangles, it is shown that the number of solutions less thanxequalsClog23x+O(log⁡xlog⁡log2x)C{\log ^2}3x + O(\log x\log {\log ^2}x)with an explicitly computable constantC=0.18071704711507…C = 0.18071704711507 \ldotsNumerical data up to101300{10^{1300}}is presented which suggests that the true error term is considerably smaller.
On convergence of monotone finite difference schemes with variable spatial differencing
Authors: Sanders Richard
Published: June 2010

Monotone finite difference schemes used to approximate solutions of scalar conservation laws have the advantage that these approximations can be proved to converge to the proper solution as the mesh size tends to zero. The greatest disadvantage in using such approximating schemes is the computational expense encountered since monotone schemes can have at best first order accuracy. Computation savings and effective accuracy could be gained if the spatial mesh were refined in regions of expected rapid solution variation. In this paper we prove that standard monotone difference schemes, (satisfying a fairly unrestrictive CFL condition), converge to the "correct" physical solution even in the case when a nonuniform spatial mesh is employed.
Preconditioning and two-level multigrid methods of arbitrary degree of approximation
Authors: Axelsson O. , Gustafsson I.
Published: June 2010

Let h be a mesh parameter corresponding to a finite element mesh for an elliptic problem. We describe preconditioning methods for two-level meshes which, for most problems solved in practice, behave as methods of optimal order in both storage and computational complexity. Namely, per mesh point, these numbers are bounded above by relatively small constants for all

h
⩾

h
0

h \geqslant {h_0}

, where

h
0

{h_0}

is small enough to cover all but excessively fine meshes. We note that, in practice, multigrid methods are actually solved on a finite, often even a fixed number of grid levels, in which case also these methods are not asymptotically optimal as

h
→
0

h \to 0

. Numerical tests indicate that the new methods are about as fast as the best implementations of multigrid methods applied on general problems (variable coefficients, general domains and boundary conditions) for all but excessively fine meshes. Furthermore, most of the latter methods have been implemented only for difference schemes of second order of accuracy, whereas our methods are applicable to higher order approximations. We claim that our scheme could be added fairly easily to many existing finite element codes.
Numerical methods based on additive splittings for hyperbolic partial differential equations
Authors: LeVeque Randall J. , Oliger Joseph
Published: June 2010

We derive and analyze several methods for systems of hyperbolic equations with wide ranges of signal speeds. These techniques are also useful for problems whose coefficients have large mean values about which they oscillate with small amplitude. Our methods are based on additive splittings of the operators into components that can be approximated independently on the different time scales, some of which are sometimes treated exactly. The efficiency of the splitting methods is seen to depend on the error incurred in splitting the exact solution operator. This is analyzed and a technique is discussed for reducing this error through a simple change of variables. A procedure for generating the appropriate boundary data for the intermediate solutions is also presented.
Computation of Faber series with application to numerical polynomial approximation in the complex plane
Authors: Ellacott S. W.
Published: June 2010

Kövari and Pommerenke [19], and Elliott [8], have shown that the truncated Faber series gives a polynomial approximation which (for practical values of the degree of the polynomial) is very close to the best approximation. In this paper we discuss efficient Fast Fourier Transform (FFT) and recursive methods for the computation of Faber polynomials, and point out that the FFT method described by Geddes [13], for computing Chebyshev coefficients can be generalized to compute Faber coefficients. We also give a corrected bound for the norm of the Faber projection (that given in Elliott [8], being unfortunately slightly in error) and very briefly discuss a possible extension of the method to the case when the mapping function, which is required to compute the Faber series, is not known explicitly.
A note on the semi-infinite programming approach to complex approximation
Authors: Streit Roy L. , Nuttall Albert H.
Published: June 2010

Several observations are made about a recently proposed semi-infinite programming (SIP) method for computation of linear Chebyshev approximations to complex-valued functions. A particular discretization of the SIP problem is shown to be equivalent to replacing the usual absolute value of a complex number with related estimates, resulting in a class of quasi-norms on the complex number field

C

\mathbf {C}

, and consequently a class of quasi-norms on the space

C
(
Q
)

C(Q)

consisting of all continuous functions defined on

Q
⊂

C

Q \subset {\mathbf {C}}

, Q compact. These quasi-norms on

C
(
Q
)

C(Q)

are estimates of the

L
∞

{L_\infty }

norm on

C
(
Q
)

C(Q)

and are useful because the best approximation problem in each quasi-norm can be solved by solving (i) an ordinary linear program if Q is finite or (ii) a simplified SIP if Q is not finite.
Runge-Kutta theory for Volterra and Abel integral equations of the second kind
Authors: Lubich Ch.
Published: June 2010

The present paper develops the local theory of general Runge-Kutta methods for a broad class of weakly singular and regular Volterra integral equations of the second kind. Further, the smoothness properties of the exact solutions of such equations are investigated.
On smooth multivariate spline functions
Authors: Chui Charles K. , Wang Ren Hong
Published: June 2010

In this paper the dimensions of bivariate spline spaces with simple cross-cut grid partitions are determined and expressions of their basis functions are given. Consequently, the closures of these spaces over all partitions of the same type can be determined. A somewhat more detailed study on bivariate splines with rectangular grid partitions is included. The results in this paper can be applied to problems on interpolation and approximation by bivariate spline functions.
Analysis of some mixed finite element methods for plane elasticity equations
Authors: Pitkäranta J. , Stenberg R.
Published: June 2010

We analyze some mixed finite element methods, based on rectangular elements, for solving the two-dimensional elasticity equations. We prove error estimates for a method proposed by Taylor and Zienkiewicz and for some new variants of the known equilibrium methods. A numerical example is given demonstrating the performance of the various algorithms considered.
Determination of the solutions of the Navier-Stokes equations by a set of nodal values
Authors: Foias Ciprian , Temam Roger
Published: June 2010

We consider the Navier-Stokes equations of a viscous incompressible fluid, and we want to see to what extent these solutions can be determined by a discrete set of nodal values of these solutions. The results presented here are exact results and not approximate ones: we show, in several cases, that the solutions are entirely determined by their values on a discrete set, provided this set contains enough points and these points are sufficiently densely distributed (in a sense described in the article). Two typical results are the following ones: two stationary solutions coincide if they coincide on a set sufficiently dense but finite; similarly if the large time behavior of the solutions to the Navier-Stokes equations is known on an appropriate discrete set, then the large time behavior of the solution itself is totally determined.
Numerical viscosity and the entropy condition for conservative difference schemes
Authors: Tadmor Eitan
Published: June 2010

Consider a scalar, nonlinear conservative difference scheme satisfying the entropy condition. It is shown that difference schemes containing more numerical viscosity will necessarily converge to the unique, physically relevant weak solution of the approximated conservative equation. In particular, entropy satisfying convergence follows for E schemes—those containing more numerical viscosity than Godunov’s scheme.
Finding all solutions to a system of polynomial equations
Authors: Wright Alden H.
Published: June 2010

Given a polynomial equation of degree d over the complex domain, the Fundamental Theorem of Algebra tells us that there are d solutions, assuming that the solutions are counted by multiplicity. These solutions can be approximated by deforming a standard nth degree equation into the given equation, and following the solutions through the deformation. This is called the homotopy method. The Fundamental Theorem of Algebra can be proved by the same technique. In this paper we extend these results and methods to a system of n polynomial equations in n complex variables. We show that the number of solutions to such a system is the product of the degrees of the equations (assuming that infinite solutions are included and solutions are counted by multiplicity)

∗

^\ast

. The proof is based on a homotopy, or deformation, from a standard system of equations with the same degrees and known solutions. This homotopy provides a computational method of approximating all solutions. Computational results demonstrating the feasibility of this method are also presented.
On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals and their derivatives
Authors: Ioakimidis N. I.
Published: June 2010

The convergence of the aforementioned quadrature rules for integrands possessing Holder-continuous derivatives of an appropriate order is proved to be uniform and not only pointwise. The rate of convergence is also established and an application to the numerical solution of singular integral equations is made.
Formulas for higher derivatives of the Riemann zeta function
Authors: Apostol Tom M.
Published: June 2010

The functional equation for

ζ
(
s
)

\zeta (s)

is used to obtain formulas for all derivatives

ζ

(
k
)

(
s
)

{\zeta ^{(k)}}(s)

. A closed form evaluation of

ζ

(
k
)

(
0
)

{\zeta ^{(k)}}(0)

is given, and numerical values are computed to 15D for

k
=
0
(
1
)
18

k = 0(1)18

.
Improved accuracy by adapted mesh-refinements in the finite element method
Authors: Eriksson Kenneth
Published: June 2010

For appropriately adapted mesh-refinements, optimal order error estimates are proved for the finite element approximate solution of the Neumann problem for the second-order elliptic equation

L
u
=
δ

Lu = \delta

, where

δ
\delta

is the Dirac distribution.
The generalized integro-exponential function
Authors: Milgram M. S.
Published: June 2010

The generalized integro-exponential function is defined in terms of the exponential integral (incomplete gamma function) and its derivatives with respect to order. A compendium of analytic results is given in one section. Rational minimax approximations sufficient to permit the computation of the first six first-order functions are reported in another section.
On the accuracy of stable schemes for 2D scalar conservation laws
Authors: Goodman Jonathan B. , LeVeque Randall J.
Published: June 2010

We show that any conservative scheme for solving scalar conservation laws in two space dimensions, which is total variation diminishing, is at most first-order accurate.
A general equivalence theorem in the theory of discretization methods
Authors: Sanz-Serna J. M. , Palencia C.
Published: June 2010

The Lax-Richtmyer theorem is extended to work in the framework of Stetter’s theory of discretizations. The new result applies to both initial and boundary value problems discretized by finite elements, finite differences, etc. Several examples are given, together with a comparison with other available equivalence theorems. The proof relies on a generalized Banach-Steinhaus theorem.
Boundary value techniques for initial value problems in ordinary differential equations
Authors: Axelsson A. O. H. , Verwer J. G.
Published: June 2010

The numerical solution of initial value problems in ordinary differential equations by means of boundary value techniques is considered. We discuss a finite-difference method which was already investigated by Fox in 1954 and Fox and Mitchell in 1957. Hereby we concentrate on explaining the fundamentals of the method because for initial value problems the boundary value method seems to be fairly unknown. We further propose and discuss new Galerkin methods for initial value problems along the lines of the boundary value approach.
Factorization of multivariate polynomials over finite fields
Authors: von zur Gathen J. , Kaltofen E.
Published: June 2010

We present a probabilistic algorithm that finds the irreducible factors of a bivariate polynomial with coefficients from a finite field in time polynomial in the input size, i.e., in the degree of the polynomial and log (cardinality of field). The algorithm generalizes to multivariate polynomials and has polynomial running time for densely encoded inputs. A deterministic version of the algorithm is also discussed, whose running time is polynomial in the degree of the input polynomial and the size of the field.
On some high-order accurate fully discrete Galerkin methods for the Korteweg-de Vries equation
Authors: Dougalis Vassilios A. , Karakashian Ohannes A.
Published: June 2010

We construct and analyze fully discrete Galerkin (finite-element) methods of high order of accuracy for the numerical solution of the periodic initial-value problem for the Korteweg-de Vries equation. The methods are based on a standard space discretization using smooth periodic splines on a uniform mesh. For the time stepping, we use two schemes of third (resp. fourth) order of accuracy which are modifications of well-known, diagonally implicit Runge-Kutta methods and require the solution of two (resp. three) nonlinear systems of equations at each time step. These systems are solved approximately by Newton’s method. Provided the initial iterates are chosen in a specific, accurate way, we show that only one Newton iteration per system is needed to preserve the stability and order of accuracy of the scheme. Under certain mild restrictions on the space mesh length and the time step we prove

L
2

{L^2}

-error estimates of optimal rate of convergence for both schemes.
Deformations of the bifurcation diagram due to discretization
Authors: Bigge J. , Bohl E.
Published: June 2010

With a singular perturbation problem occurring in chemical reaction processes, substantial changes of the bifurcation diagram due to discretization are demonstrated. It is shown that a discrete system can possess any number of solutions, whereas the underlying continuous problem has exactly one solution. In addition to that, there is no way to favor one of the various discrete solutions as the one approximating the continuous solution.
Accuracy in random number generation
Authors: Monahan John F.
Published: June 2010

The generation of continuous random variables on a digital computer encounters a problem of accuracy caused by approximations and discretization error. These in turn impose a bias on the simulation results. An ideal discrete approximation of a continuous distribution and a measure of error are proposed. Heuristic analysis of common methods for transforming uniform deviates to other continuous random variables is discussed. Comments and recommendations are made for the design of algorithms to reduce the bias and avoid overflow problems.
On the numerical solution of singular boundary value problems of second order by a difference method
Authors: Weinmüller Ewa
Published: June 2010

The standard three-point discretization applied to the numerical solution of nonlinear boundary value problems for second-order systems with a singularity of the first kind is investigated. The results are extended to the boundary value problems arising in practical problems from mechanics and chemistry. A number of numerical examples illustrating the theoretical results is presented.
A convergent 3-D vortex method with grid-free stretching
Authors: Beale J. Thomas
Published: June 2010

We prove the convergence of a vortex method for three-dimensional, incompressible, inviscid flow without boundaries. This version differs from an earlier one whose convergence was shown in [4] in that the calculation does not depend explicitly on the arrangement of the vorticity elements in a Lagrangian frame. Thus, it could be used naturally in a more general context in which boundaries and viscosity are present. It is also shown that previous estimates for the velocity approximation can be improved by taking into account the fact that the integral kernel has average value zero. Implications for the design of the method are discussed.
Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations
Authors: Johnson Claes , Saranen Jukka
Published: June 2010

We present and analyze extensions of the streamline diffusion finite element method to the time-dependent two-dimensional Navier-Stokes equations for an incompressible fluid in the case of high Reynolds numbers. The limit case with zero viscosity, the Euler equations, is also considered.
Galerkin approximations of abstract parabolic boundary value problems with rough boundary data—𝐿_{𝑝} theory
Authors: Lasiecka I.
Published: June 2010

Galerkin approximations of an abstract parabolic boundary value problem with "rough" boundary data are considered. The optimal rates of convergence in

L
p

[
0
T
;

L
2

(
Ω
)
]

{L_p}[0T;{L_2}(\Omega )]

norms for

L
p

[
0
T
;

L
2

(
Γ
)
]

{L_p}[0T;{L_2}(\Gamma )]

boundary terms are derived.
Root neighborhoods of a polynomial
Authors: Mosier Ronald G.
Published: June 2010

The root neighborhoods of

p
(
z
)

p(z)

, a polynomial over the complex field, are the sets of complex numbers that are the roots of polynomials which are near to

p
(
z
)

p(z)

. The term ’near’ means that the coefficients of the polynomials are within some fixed magnitude of the coefficients of

p
(
z
)

p(z)

. A necessary and sufficient condition for a complex number to be in the root neighborhoods is given and it is proved that each root neighborhood contains at least one root of

p
(
z
)

p(z)

and the same number of roots of each near polynomial. Finally, a necessary and sufficient condition is given for a root neighborhood to contain more than one root of

p
(
z
)

p(z)

, and consequently more than one root of any of the near polynomials.
An asymptotic expansion for the first derivative of the generalized Riemann zeta function
Authors: Elizalde E.
Published: June 2010

An asymptotic expansion for the partial derivative

∂
ζ
(
z
,
q
)

/

∂
z

\partial \zeta (z,q)/\partial z

of the generalized Riemann zeta function

ζ
(
z
,
q
)

\zeta (z,q)

, all negative integer values of z, is obtained.
Rates of convergence of Gauss, Lobatto, and Radau integration rules for singular integrands
Authors: Rabinowitz Philip
Published: June 2010

Rates of convergence (or divergence) are obtained in the application of Gauss, Lobatto, and Radau integration rules to functions with an algebraic or logarithmic singularity inside, or at an endpoint of, the interval of integration. A typical result is the following: For a generalized Jacobi weight function on

[
−
1
,
1
]

[ - 1,1]

, the error in applying an n-point rule to

f
(
x
)
=

|

x
−
y

|

−
δ

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

is

O
(

n

−
2
+
2
δ

)

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

, if

y
=
±
1

y = \pm 1

and

O
(

n

−
1
+
δ

)

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

if

y
∈
(
−
1
,
1
)

y \in ( - 1,1)

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

O
(

n

−
1
+
2
δ

(
log
⁡
n

)
δ

(
log
⁡
log
⁡
n

)

δ
(
1
+
ε
)

)

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

for almost all choices of y. These assertions are sharp with respect to order.
On the distribution of spacings between zeros of the zeta function
Authors: Odlyzko A. M.
Published: June 2010

A numerical study of the distribution of spacings between zeros of the Riemann zeta function is presented. It is based on values for the first105{10^5}zeros and for zeros number1012+1{10^{12}} + 1to1012+105{10^{12}} + {10^5}that are accurate to within±10−8\pm {10^{ - 8}}, and which were calculated on the Cray-1 and Cray X-MP computers. This study tests the Montgomery pair correlation conjecture as well as some further conjectures that predict that the zeros of the zeta function behave like eigenvalues of random Hermitian matrices. Matrices of this type are used in modeling energy levels in physics, and many statistical properties of their eigenvalues are known. The agreement between actual statistics for zeros of the zeta function and conjectured results is generally good, and improves at larger heights. Several initially unexpected phenomena were found in the data and some were explained by relating them to the primes.
Multigrid solution of monotone second-order discretizations of hyperbolic conservation laws
Authors: Spekreijse Stefan
Published: June 2010

This paper is concerned with two subjects: the construction of second-order accurate monotone upwind schemes for hyperbolic conservation laws and the multigrid solution of the resulting discrete steady-state equations. By the use of an appropriate definition of monotonicity, it is shown that there is no conflict between second-order accuracy and monotonicity (neither in one nor in more dimensions). It is shown that a symmetric block Gauss-Seidel underrelaxation (each block is associated with 4 cells) has satisfactory smoothing rates. The success of this relaxation is due to the fact that, by coupling the unknowns in such blocks, the nine-point stencil of a second-order 2D upwind discretization changes into a five-point block stencil.
Discrete solution of the Plateau problem and its convergence
Authors: Tsuchiya Takuya
Published: June 2010

In this paper we define a discrete solution of the Plateau problem and we prove theorems for the convergence of the discrete solution when the contour satisfies a certain condition. A numerical example is given.
The Faber polynomials for circular sectors
Authors: Coleman John P. , Smith Russell A.
Published: June 2010

The Faber polynomials for a region of the complex plane, which are of interest as a basis for polynomial approximation of analytic functions, are determined by a conformal mapping of the complement of that region to the complement of the unit disc. We derive this conformal mapping for a circular sector

{

z
:

|

z

|

⩽
1
,

|

arg
⁡
z

|

⩽
π

/

α

}

\{ {z:|z|\; \leqslant 1,\;|\arg z|\; \leqslant \pi /\alpha } \}

, where

α
>
1

\alpha > 1

, and obtain a recurrence relation for the coefficients of its Laurent expansion about the point at infinity. We discuss the computation of the coefficients of the Faber polynomials of degree 1 to 15, which are tabulated here for sectors of half-angle

5
∘

{5^ \circ }

,

10
∘

{10^\circ }

,

15
∘

{15^ \circ }

,

30
∘

{30^ \circ }

,

45
∘

{45^\circ }

, and

90
∘

{90^ \circ }

, and we give explicit expressions, in terms of

α
\alpha

, for the polynomials of degree

⩽
3

\leqslant 3

. The norms of Faber polynomials are tabulated and are compared with those of the Chebyshev polynomials for the same regions.
Tables of Fibonacci and Lucas factorizations
Authors: Brillhart John , Montgomery Peter L. , Silverman Robert D.
Published: June 2010

We list the known prime factors of the Fibonacci numbersFn{F_n}forn≤999n \leq 999and Lucas numbersLn{L_n}forn≤500n \leq 500. We discuss the various methods used to obtain these factorizations, and primality tests, and give some history of the subject.
The probability that a numerical analysis problem is difficult
Authors: Demmel James W.
Published: June 2010

Numerous problems in numerical analysis, including matrix inversion, eigenvalue calculations and polynomial zerofinding, share the following property: The difficulty of solving a given problem is large when the distance from that problem to the nearest "ill-posed" one is small. For example, the closer a matrix is to the set of non-invertible matrices, the larger its condition number with respect to inversion. We show that the sets of ill-posed problems for matrix inversion, eigenproblems, and polynomial zerofinding all have a common algebraic and geometric structure which lets us compute the probability distribution of the distance from a "random" problem to the set. From this probability distribution we derive, for example, the distribution of the condition number of a random matrix. We examine the relevance of this theory to the analysis and construction of numerical algorithms destined to be run in finite precision arithmetic.
On 𝐴⁴+𝐵⁴+𝐶⁴=𝐷⁴
Authors: Elkies Noam D.
Published: June 2010

We use elliptic curves to find infinitely many solutions to

A
4

+

B
4

+

C
4

=

D
4

{A^4} + {B^4} + {C^4} = {D^4}

in coprime natural numbers A, B, C, and D, starting with
\[

2682440
4

+

15365639
4

+

18796760
4

=

20615673
4

.

{2682440^4} + {15365639^4} + {18796760^4} = {20615673^4}.

\]
We thus disprove the

n
=
4

n = 4

case of Euler’s conjectured generalization of Fermat’s Last Theorem. We further show that the corresponding rational points

(
±
A

/

D
,
±
B

/

D
,
±
C

/

D
)

( \pm A/D, \pm B/D, \pm C/D)

on the surface

r
4

+

s
4

+

t
4

=
1

{r^4} + {s^4} + {t^4} = 1

are dense in the real locus. We also discuss the smallest solution, found subsequently by Roger Frye.