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1.
Summary.
Large, sparse nonsymmetric systems of linear equations with a
matrix whose eigenvalues lie in the right half plane may be solved by an
iterative method based on Chebyshev polynomials for an interval in the
complex plane. Knowledge of the convex hull of the spectrum of the
matrix is required in order to choose parameters upon which the
iteration depends. Adaptive Chebyshev algorithms, in which these
parameters are determined by using eigenvalue estimates computed by the
power method or modifications thereof, have been described by Manteuffel
[18]. This paper presents an adaptive Chebyshev iterative method, in
which eigenvalue estimates are computed from modified moments determined
during the iterations. The computation of eigenvalue estimates from
modified moments requires less computer storage than when eigenvalue
estimates are computed by a power method and yields faster convergence
for many problems.
Received May 13, 1992/Revised version received May 13,
1993 相似文献
2.
Igor Moret 《Numerische Mathematik》1994,68(3):341-353
Summary. Certain types
of singular solutions of nonlinear parameter-dependent
operator equations were characterized by
Griewank and Reddien [5, 6] as regular solutions of
suitable augmented systems. For their numerical
approximation an approach based on the use of
Krylov subspaces is here presented. The application
to boundary value problems is illustrated by
numerical examples.
Received March 8, 1993 / Revised version received December 13,
1993 相似文献
3.
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
4.
In earlier papers Tyrtyshnikov [42] and the first author [14] considered the analysis of clustering properties of the spectra of specific Toeplitz preconditioned matrices obtained by means of the best known matrix algebras. Here we generalize this technique to a generic Banach algebra of matrices by devising general preconditioners related to “convergent” approximation processes [36]. Finally, as case study, we focus our attention on the Tau preconditioning by showing how and why the best matrix algebra preconditioners for symmetric Toeplitz systems can be constructed in this class. Received April 25, 1997 / Revised version received March 13, 1998 相似文献
5.
Summary. To solve 1D linear integral equations on bounded intervals with nonsmooth input functions and solutions, we have recently
proposed a quite general procedure, that is essentially based on the introduction of a nonlinear smoothing change of variable
into the integral equation and on the approximation of the transformed solution by global algebraic polynomials. In particular,
the new procedure has been applied to weakly singular equations of the second kind and to solve the generalized air foil equation
for an airfoil with a flap. In these cases we have obtained arbitrarily high orders of convergence through the solution of
very-well conditioned linear systems. In this paper, to enlarge the domain of applicability of our technique, we show how
the above procedure can be successfully used also to solve the classical Symm's equation on a piecewise smooth curve. The
collocation method we propose, applied to the transformed equation and based on Chebyshev polynomials of the first kind, has
shown to be stable and convergent. A comparison with some recent numerical methods using splines or trigonometric polynomials
shows that our method is highly competitive.
Received October 1, 1998 / Revised version received September 27, 1999 / Published online June 21, 2000 相似文献
6.
On the convergence of line iterative methods for cyclically
reduced non-symmetrizable linear systems
Summary. We derive analytic bounds on the convergence factors associated
with block relaxation methods for solving the discrete
two-dimensional convection-diffusion equation. The analysis
applies to the reduced systems derived when one step of block
Gaussian elimination is performed on red-black ordered
two-cyclic discretizations. We consider the case where centered
finite difference discretization is used and one cell Reynolds
number is less than one in absolute value and the other is
greater than one. It is shown that line ordered relaxation
exhibits very fast rates of convergence.
Received March 3, 1992/Revised version received July 2, 1993 相似文献
7.
Summary. We present new theoretical results on two classes of multisplitting methods for solving linear systems iteratively. These
classes are based on overlapping blocks of the underlying coefficient matrix which is assumed to be a band matrix. We show that under suitable conditions the spectral radius of the iteration matrix does not depend on the weights of the method even if these weights are allowed to be negative. For a certain class of splittings
we prove an optimality result for with respect to the weights provided that is an M–matrix. This result is based on the fact that the multisplitting method can be represented by a single splitting
which in our situation surprisingly turns out to be a regular splitting. Furthermore we show by numerical examples that weighting
factors may considerably improve the convergence.
Received July 18, 1994 / Revised version received November 20, 1995 相似文献
8.
Summary.
Hybrid methods for the solution of systems of linear equations
consist of a first phase where some information about the associated
coefficient matrix is acquired, and a second phase in which a
polynomial iteration designed with respect to this information is
used. Most of the hybrid algorithms proposed recently for the
solution of nonsymmetric systems rely on the direct use of
eigenvalue estimates constructed by the Arnoldi process in Phase I.
We will show the limitations of this approach and propose an
alternative, also based on the Arnoldi process, which approximates
the field of values of the coefficient matrix and of its inverse in
the Krylov subspace. We also report on numerical experiments
comparing the resulting new method with other hybrid algorithms.
Received May 27, 1993 / Revised version received
November 14, 1994 相似文献
9.
Summary.
Discretisation of the classical Stokes problem gives rise
to symmetric indefinite matrices with eigenvalues which,
in a precise way, are not symmetric about the origin, but which
do depend on a mesh size parameter. Convergence
estimates for the Conjugate Residual or Minimum Residual
iterative solution of such systems are given by best
minimax polynomial approximations on an inclusion set for the
eigenvalues.
In this paper, an analytic convergence estimate for such
problems is given in terms of an asymptotically small
mesh size parameter.
Received
November 16, 1993 / Revised version received August 2,
1994 相似文献
10.
Summary. We prove convergence results and error estimates for interpolatory product quadrature formulas for Cauchy principal value
integrals on the real line with Freud–type weight functions. The formulas are based on polynomial interpolation at the zeros
of orthogonal polynomials associated with the weight function under consideration. As a by–product, we obtain new bounds for
the derivative of the functions of the second kind for these weight functions.
Received July 15, 1997 / Revised version received August 25, 1998 相似文献
11.
Summary. The Schur complement of a model problem is considered as a preconditioner for the Uzawa type schemes for the generalized
Stokes problem (the Stokes problem with the additional term in the motion equation). The implementation of the preconditioned method requires for each iteration only one extra solution
of the Poisson equation with Neumann boundary conditions. For a wide class of 2D and 3D domains a theorem on its convergence
is proved. In particular, it is established that the method converges with a rate that is bounded by some constant independent
of . Some finite difference and finite element methods are discussed. Numerical results for finite difference MAC scheme are
provided.
Received May 2, 1997 / Revised version received May 10, 1999 / Published online May 8, 2000 相似文献
12.
Ch. Lubich 《Numerische Mathematik》1994,67(3):365-389
Summary. Convergence estimates in terms of the data are shown for
multistep methods applied to non-homogeneous linear initial-boundary
value problems. Similar error bounds are derived
for a
new class of time-discrete and
fully discrete approximation
schemes for boundary integral equations of such
problems, e.g., for the single-layer potential
equation of the wave equation. In both cases,
the results are obtained from convergence and
stability estimates for operational quadrature
approximations of convolutions.
These estimates, which are also proved here, depend on bounds of the
Laplace transform of the (distributional)
convolution kernel outside the stability region scaled
by the time stepsize, and on the smoothness of the
data.
Received
January 18, 1993 / Revised version received September 15,
1993 相似文献
13.
Summary.
It is well known that the zeros of a polynomial are equal to
the
eigenvalues of the associated companion matrix . In this paper
we take a
geometric view of the conditioning of these two problems and of the stability of
algorithms for polynomial zerofinding. The
is the set of zeros of all polynomials obtained by
coefficientwise perturbations of of size ;
this is a subset of the
complex plane considered earlier by Mosier, and is bounded by a
certain generalized lemniscate. The
is another subset of
defined as the set of eigenvalues of matrices
with ; it is bounded by a
level curve of the resolvent
of $A$. We find that if $A$ is first balanced in the usual EISPACK sense, then
and
are usually quite close to one another. It follows that the Matlab
ROOTS algorithm of balancing the companion matrix, then computing its eigenvalues, is a stable
algorithm for polynomial zerofinding. Experimental comparisons with the
Jenkins-Traub (IMSL) and
Madsen-Reid (Harwell) Fortran codes confirm that these three algorithms have roughly
similar stability properties.
Received June 15, 1993 相似文献
14.
On vector subdivision 总被引:8,自引:0,他引:8
In this paper we give a complete characterization of the convergence of stationary vector subdivision schemes and the regularity
of the associated limit function. These results extend and complete our earlier work on vector subdivision and its use in
the construction of multiwavelets.
Received March 19, 1997; in final form November 13, 1997 相似文献
15.
Chun-Hua Guo 《Numerische Mathematik》1999,83(4):621-639
Summary. The application of the finite difference method to approximate the solution of an indefinite elliptic problem produces a
linear system whose coefficient matrix is block tridiagonal and symmetric indefinite. Such a linear system can be solved efficiently
by a conjugate residual method, particularly when combined with a good preconditioner. We show that specific incomplete block
factorization exists for the indefinite matrix if the mesh size is reasonably small, and that this factorization can serve
as an efficient preconditioner. Some efforts are made to estimate the eigenvalues of the preconditioned matrix. Numerical
results are also given.
Received November 21, 1995 / Revised version received February 2, 1998 / Published online July 28, 1999 相似文献
16.
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
17.
Summary.
Methods for the numerical inversion of a Laplace transform
which
use a special bilinear transformation of are particularly
effective in
many cases and are widely used.
The main purpose of this paper is to analyze the convergence
and conditioning
properties of a special class of such methods, characterized
by the use of
Lagrange interpolation. The results derived apply both to
complex and real
inversion, and show that some known inversion methods are in
fact in this
class.
Received
June 21, 1993 / Revised version received March 10, 1994 相似文献
18.
Summary. We derive error bounds for bivariate spline interpolants which are calculated by minimizing certain natural energy norms.
Received March 28, 2000 / Revised version received June 23, 2000 / Published online March 8, 2002
RID="*"
ID="*" Supported by the National Science Foundation under grant DMS-9870187
RID="**"
ID="**" Supported by the National Science Foundation under grant DMS-9803340 and by the Army Research Office under grant DAAD-19-99-1-0160 相似文献
19.
20.
Michael-Ralf Skrzipek 《Numerische Mathematik》1998,79(4):601-613
We show a connection between the Clenshaw algorithm for evaluating a polynomial , expanded in terms of a system of orthogonal polynomials, and special linear combinations of associated polynomials. These
results enable us to get the derivatives of analogously to the Horner algorithm for evaluating polynomials in monomial representations. Furthermore we show how a polynomial
given in monomial (!) representation can be evaluated for using the Clenshaw algorithm without complex arithmetic. From this we get a connection between zeros of polynomials expanded
in terms of Chebyshev polynomials and the corresponding polynomials in monomial representation with the same coefficients.
Received January 2, 1995 / Revised version received April 9, 1997 相似文献