共查询到20条相似文献,搜索用时 15 毫秒
1.
Kai Diethelm 《Numerische Mathematik》1996,73(1):53-63
Summary.
We show that, if
(),
the error term of
every modified positive interpolatory quadrature rule for
Cauchy principal value integrals of the type
,
, fulfills
uniformly for all
, and hence it is
of optimal
order of magnitude in the classes
().
Here, is a weight function with the property
.
We give explicit upper bounds for the Peano-type error
constants of such rules.
This improves and completes earlier results by
Criscuolo and Mastroianni
(Calcolo 22 (1985), 391–441 and Numer. Math.
54 (1989), 445–461)
and Ioakimidis (Math. Comp. 44 (1985), 191–198).
For the special case of the Gaussian rule, we
show that the restriction
can be dropped.
The results are based on a new representation of the
Peano kernels of these formulae via the Peano kernels of the underlying
classical quadrature formulae. This representation may also be
useful in connection with some different problems.
Received November 21, 1994 相似文献
2.
Summary. In this paper, interpolatory quadrature formulas based upon the roots of unity are studied for certain weight functions.
Positivity of the coefficients in these formulas is deduced along with computable error estimations for analytic integrands.
A comparison is made with Szeg? quadrature formulas. Finally, an application to the interval [-1,1] is also carried out.
Received February 29, 2000 / Published online August 17, 2001 相似文献
3.
Quadrature formulae with free nodes for periodic
functions 总被引:3,自引:0,他引:3
Dimiter P. Dryanov 《Numerische Mathematik》1994,67(4):441-464
Summary. The problem of existence and uniqueness of a quadrature formula with
maximal trignonometric degree of precision for
2-periodic functions with
fixed number of free nodes of fixed different multiplicities at each
node is considered. Our approach is based on some properties of the
topological degree of a mapping with respect to an open bounded set and
a given point. The explicit expression for the quadrature formulae with maximal
trignometric degree of precision in the 2-periodic case of
multiplicities is obtained. An error analysis for the quadrature with maximal
trigonometric degree of precision is given.
Received April 16, 1992/Revised version received June 21, 1993 相似文献
4.
Summary. For analytic functions the remainder term of quadrature rules can be represented as a contour integral with a complex kernel function. From this representation different remainder term estimates involving the kernel are obtained. It is studied in detail how polynomial biorthogonal systems can be applied to derive sharp bounds for the kernel function. It is shown that these bounds are practical to use and can easily be computed. Finally, various numerical examples are presented. Received March 11, 1998 / Revised version January 22, 1999/ Published online November 17, 1999 相似文献
5.
P. Köhler 《Numerische Mathematik》1995,72(1):93-116
Summary.
We show that, for integrals with arbitrary integrable weight functions,
asymptotically best quadrature formulas with equidistant nodes can be
obtained by applying a certain scheme of piecewise polynomial interpolation
to the function
to be integrated, and then integrating this interpolant.
Received August 7, 1991 相似文献
6.
Masaaki Sugihara 《Numerische Mathematik》1997,75(3):379-395
Summary. In the light of the functional analysis theory we establish the optimality of the double exponential formula. The argument
consists of the following three ingredients: (1) introduction of a number of spaces of functions analytic in a strip region
about the real axis, each space being characterized by the decay rate of their elements (functions) in the neighborhood of
the infinity; (2) proof of the (near-) optimality of the trapezoidal formula in each space introduced in (1) by showing the
(near-) equality between an upper estimate for the error norm of the trapezoidal formula and a lower estimate for the minimum
error norm of quadratures; (3) nonexistence theorem for the spaces, the characterizing decay rate of which is more rapid than
the double exponential.
Received September 15, 1995 / Accepted December 14, 1995 相似文献
7.
Hong Jiang 《Numerische Mathematik》1994,67(3):345-364
Summary. This paper studies polynomials used in polynomial
preconditioning
for solving linear systems of equations. Optimum preconditioning
polynomials are obtained by solving some constrained minimax
approximation
problems. The resulting residual polynomials are referred to as
the de Boor-Rice and
Grcar polynomials. It will be shown in this paper that the
de Boor-Rice and Grcar polynomials are orthogonal polynomials
over several intervals. More specifically, each de Boor-Rice or
Grcar polynomial belongs to an orthogonal family, but the
orthogonal
family varies with the polynomial.
This orthogonality property is important,
because it enables one to generate the
minimax preconditioning polynomials by three-term recursive
relations.
Some results on the convergence properties of certain
preconditioning
polynomials are also presented.
Received February 1, 1992/Revised version received July 7, 1993 相似文献
8.
Luca F. Pavarino 《Numerische Mathematik》1994,69(2):185-211
Summary.
In some applications, the accuracy of the numerical solution of an
elliptic problem needs to be increased only in certain parts of the
domain. In this paper, local refinement is introduced for an overlapping
additive Schwarz algorithm for the $-version finite element method.
Both uniform and variable degree refinements are considered.
The resulting algorithm is highly parallel and scalable.
In two and three dimensions,
we prove an optimal bound for the condition number of the iteration
operator under certain hypotheses on the refinement region.
This bound is independent of the degree $, the number of
subdomains $ and the mesh size $.
In the general two dimensional case, we prove an almost optimal bound
with polylogarithmic growth in $.
Received February 20, 1993 / Revised version received January
20, 1994 相似文献
9.
In this paper we compare G(p), the Mellin transform (together with its analytic continuation), and , the related Hadamard finite-part integral of a function g(x), which decays exponentially at infinity and has specified singular behavior at the origin. Except when p is a nonpositive integer, these coincide. When p is a nonpositive integer, is well defined, but G(p) has a pole. We show that the terms in the Laurent expansion about this pole can be simply expressed in terms of the Hadamard
finite-part integral of a related function. This circumstance is exploited to provide a conceptually uniform proof of the
various generalizations of the Euler-Maclaurin expansion for the quadrature error functional.
Received June 11, 1997 / Revised version received December 15, 1997 相似文献
10.
Summary. This paper is concerned with the convergence of product quadrature formulas of interpolatory type based on the zeros of Jacobi
polynomials for the approximation of integrals of the type
is supposed to be of the form not an integer, . The kernel can be a smooth one or it can contain an algebraic or a logarithmic singularity.
Received January 20, 1995 相似文献
11.
Giovanni Monegato 《Numerische Mathematik》1984,43(2):161-173
Summary We consider product rules of interpolatory type for the numerical approximation of certain two-dimensional Cauchy principal value integrals. We present convergence results which generalize those known in the one-dimensional case.Work sponsored by the Ministero della Pubblica Istruzione of Italy 相似文献
12.
Summary. We prove the existence of a Gaussian quadrature formula for Tchebycheff systems, based on integrals over non-overlapping
subintervals of arbitrary fixed lengths and the uniqueness of this formula in the case the subintervals have equal lengths.
Received July 6, 1999 / Published online August 24, 2000 相似文献
13.
Summary. We construct a quadrature formula for integration on the unit disc which is based on line integrals over distinct chords in the disc and integrates exactly all polynomials in two variables of total degree .
Received August 8, 1996 / Revised version received July 2, 1997 相似文献
14.
Michael Rosier 《Numerische Mathematik》1995,72(2):263-283
Summary.
The concept of singular value decompositions is a valuable tool
in the examination of ill-posed inverse problems
such as the inversion of the Radon transform. A singular value
decomposition depends on the determination of suitable orthogonal systems
of eigenfunctions of the operators
, .
In this paper we consider a new approach which generalizes this concept.
By application of biorthogonal instead of orthogonal functions we
are able to apply a larger class of function sets in order to
account for the structure of the eigenfunction spaces. Although it is
preferable to use eigenfunctions it is still possible to consider
biorthogonal function systems which are not eigenfunctions of the operator.
With respect to the Radon transform for functions with support in the
unit ball we apply the system of Appell polynomials which is a natural
generalization of the univariate system of Gegenbauer (ultraspherical)
polynomials to the multivariate case. The corresponding biorthogonal
decompositions show some advantages in comparison with the known
singular value decompositions. Vice versa by application of our
decompositions we are able to prove new properties of the Appell
polynomials.
Received October 19, 1993 相似文献
15.
Summary. The cascade algorithm with mask a and dilation M generates a sequence by the iterative process
from a starting function where M is a dilation matrix. A complete characterization is given for the strong convergence of cascade algorithms in Sobolev spaces
for the case in which M is isotropic. The results on the convergence of cascade algorithms are used to deduce simple conditions for the computation
of integrals of products of derivatives of refinable functions and wavelets.
Received May 5, 1999 / Revised version received June 24, 1999 / Published online June 20, 2001 相似文献
16.
Summary. Starting with some results of Lyness concerning the Euler-Maclaurin expansion of Cauchy principal value integrals over it is shown how, by the use of sigmoidal transformations, good approximations can be found for the Hadamard finite-part integral
where The analysis is illustrated by some numerical examples.
Received March 13, 1996 相似文献
17.
Summary. We study the additive and multiplicative Schwarz domain decomposition methods for elliptic boundary value problem of order
2 r based on an appropriate spline space of smoothness . The finite element method reduces an elliptic boundary value problem to a linear system of equations. It is well known that
as the number of triangles in the underlying triangulation is increased, which is indispensable for increasing the accuracy
of the approximate solution, the size and condition number of the linear system increases. The Schwarz domain decomposition
methods will enable us to break the linear system into several linear subsystems of smaller size. We shall show in this paper
that the approximate solutions from the multiplicative Schwarz domain decomposition method converge to the exact solution
of the linear system geometrically. We also show that the additive Schwarz domain decomposition method yields a preconditioner
for the preconditioned conjugate gradient method. We tested these methods for the biharmonic equation with Dirichlet boundary
condition over an arbitrary polygonal domain using cubic spline functions over a quadrangulation of the given domain. The computer experiments agree with our theoretical results.
Received December 28, 1995 / Revised version received November 17, 1998 / Published online September 24, 1999 相似文献
18.
Carlos F. Borges 《Numerische Mathematik》1994,67(3):271-288
Summary. We consider a problem that arises in the evaluation of computer graphics
illumination models. In particular, there is a need to find a finite
set of wavelengths at which the illumination model should be evaluated.
The result of evaluating the illumination model at these points is a
sampled representation of the spectral power density of light emanating
from a point in the scene. These values are then used to determine the
RGB coordinates of the light by evaluating three definite integrals,
each with a common integrand (the SPD) and interval of integration but
with distinct weight functions. We develop a method for selecting the
sample wavelengths in an optimal manner.
More abstractly, we examine the problem of numerically evaluating a set
of definite integrals taken with respect to
distinct weight
functions but related by a common integrand and interval of integration.
It is shown that when it is not efficient
to use a set of
Gauss rules because valuable information is wasted. We go on to extend
the notions used in Gaussian quadrature to find an optimal set of
shared abcissas that maximize precision in a well-defined sense.
The classical Gauss rules come out as the special case
and some
analysis is given concerning the existence of these rules when
. In particular, we give conditions on the
weight functions that are
sufficient to guarantee that the shared abcissas are real, distinct, and
lie in the interval of integration. Finally, we examine some
computational strategies for constructing these rules.
Received July 15, 1991 相似文献
19.
Summary. A Galerkin approximation of both strongly and hypersingular boundary integral equation (BIE) is considered for the solution
of a mixed boundary value problem in 3D elasticity leading to a symmetric system of linear equations. The evaluation of Cauchy
principal values (v. p.) and finite parts (p. f.) of double integrals is one of the most difficult parts within the implementation
of such boundary element methods (BEMs). A new integration method, which is strictly derived for the cases of coincident elements
as well as edge-adjacent and vertex-adjacent elements, leads to explicitly given regular integrand functions which can be
integrated by the standard Gauss-Legendre and Gauss-Jacobi quadrature rules. Problems of a wide range of integral kernels
on curved surfaces can be treated by this integration method. We give estimates of the quadrature errors of the singular four-dimensional
integrals.
Received June 25, 1995 / Revised version received January 29, 1996 相似文献
20.
Yuan Xu 《Numerische Mathematik》1994,69(2):233-241
Summary.
The existence of Gaussian cubature for a given measure
depends on whether the corresponding multivariate orthogonal polynomials have
enough common zeros. We examine a class of orthogonal
polynomials of two variables generated from that of one variable.
Received February 9, 1993 / Revised version received
January 18, 1994 相似文献