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1.
Summary. In this paper, the adaptive filtering method is introduced and analysed. This method leads to robust algorithms for the solution
of systems of linear equations which arise from the discretisation of partial differential equations with strongly varying
coefficients. These iterative algorithms are based on the tangential frequency filtering decompositions (TFFD). During the
iteration with a preliminary preconditioner, the adaptive test vector method calculates new test vectors for the TFFD. The
adaptive test vector iterative method allows the combination of the tangential frequency decomposition and other iterative
methods such as multi-grid. The connection with the TFFD improves the robustness of these iterative methods with respect to
varying coefficients. Interface problems as well as problems with stochastically distributed properties are considered. Realistic
numerical experiments confirm the efficiency of the presented algorithms.
Received June 26, 1996 / Revised version received October 7, 1996 相似文献
2.
Christian Wagner 《Numerische Mathematik》1997,78(1):119-142
Summary. The tangential frequency filtering decomposition (TFFD) is introduced. The convergence theory of an iterative scheme based
on the TFFD for symmetric matrices is the focus of this paper. The existence of the TFFD and the convergence of the induced
iterative algorithm is shown for symmetric and positive definite matrices. Convergence rates independent of the number of
unknowns are proven for a smaller class of matrices. Using this framework, the convergence independent of the number of unknowns
is shown for Wittum's frequency filtering decomposition. Some characteristic properties of the TFFD are illustrated and results
of several numerical experiments are presented.
Received April 1, 1996 / Revised version July 4, 1996 相似文献
3.
Fierro Ricardo D. Hansen Per Christian Hansen Peter Søren Kirk 《Numerical Algorithms》1999,20(2-3):165-194
We describe a Matlab 5.2 package for computing and modifying certain rank-revealing decompositions that have found widespread
use in signal processing and other applications. The package focuses on algorithms for URV and ULV decompositions, collectively
known as UTV decompositions. We include algorithms for the ULLV decomposition, which generalizes the ULV decomposition to
a pair of matrices. For completeness a few algorithms for computation of the RRQR decomposition are also included. The software
in this package can be used as is, or can be considered as templates for specialized implementations on signal processors
and similar dedicated hardware platforms.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
4.
Miloud Sadkane 《Numerische Mathematik》1993,64(1):195-211
Summary We present two methods for computing the leading eigenpairs of large sparse unsymmetric matrices. Namely the block-Arnoldi method and an adaptation of the Davidson method to unsymmetric matrices. We give some theoretical results concerning the convergence and discuss implementation aspects of the two methods. Finally some results of numerical tests on a variety of matrices, in which we compare these two methods are reported. 相似文献
5.
Thomas Huckle 《Numerische Mathematik》1998,79(2):213-229
The problem of solving linear equations with a Toeplitz matrix appears in many applications. Often is positive definite but ill-conditioned with many small eigenvalues. In this case fast and superfast algorithms may show
a very poor behavior or even break down. In recent papers the transformation of a Toeplitz matrix into a Cauchy-type matrix
is proposed. The resulting new linear equations can be solved in operations using standard pivoting strategies which leads to very stable fast methods also for ill-conditioned systems. The
basic tool is the formulation of Gaussian elimination for matrices with low displacement rank. In this paper, we will transform
a Hermitian Toeplitz matrix into a Cauchy-type matrix by applying the Fourier transform. We will prove some useful properties of and formulate a symmetric Gaussian elimination algorithm for positive definite . Using the symmetry and persymmetry of we can reduce the total costs of this algorithm compared with unsymmetric Gaussian elimination. For complex Hermitian , the complexity of the new algorithm is then nearly the same as for the Schur algorithm. Furthermore, it is possible to include
some strategies for ill-conditioned positive definite matrices that are well-known in optimization. Numerical examples show
that this new algorithm is fast and reliable.
Received March 24, 1995 / Revised version received December 13, 1995 相似文献
6.
The paper considers the problem of constructing a basic iterative scheme for solving systems of linear algebraic equations with unsymmetric and indefinite coefficient matrices. A new GMRES-type algorithm with explicit restarts is suggested. When restarting, this algorithm takes into account the spectral/singular data transferred using orthogonal matrix relations in the so-called QR form, which arise when performing inner iterations of Arnoldi type. The main idea of the algorithm developed is to organize inner iterations and the filtering of directions before restarting in such a way that, from one restart to another, matrix relations effectively accumulate information concerning the current approximate solution and, simultaneously, spectral/singular data, which allow the algorithm to converge with a rate comparable with that of the GMRES algorithm without restarts. Convergence theory is provided for the case of nonsingular, unsymmetric, and indefinite matrices. A bound for the rate of decrease of the residual in the course of inner Arnoldi-type iterations is obtained. This bound depends on the spectral/singular characterization of the subspace spanned by the directions retained upon filtering and is used in developing efficient filtering procedures. Numerical results are provided. Bibliography: 9 titles. 相似文献
7.
8.
Marko Huhtanen 《Linear algebra and its applications》2006,418(1):347-361
The Kronecker product in the real linear matrix analytic setting is studied. More versatile operations are proposed. Such generalizations are of interest for the same reasons the standard Kronecker product is. To give an example, new preconditioning ideas are suggested. In connection with this, several formulae for the inverse are devised. Orthogonal decompositions of real-entried matrices are derived through introducing new Kronecker product SVDs. Matrix equations are given to illustrate how the Kronecker product structures introduced can arise. 相似文献
9.
In this paper, we introduce a new class of frequency‐filtering IBLU decompositions that use continued‐fraction approximation for the diagonal blocks. This technique allows us to construct efficient frequency‐filtering preconditioners for discretizations of elliptic partial differential equations on domains with non‐trivial geometries. We prove theoretically for a class of model problems that the application of the proposed preconditioners leads to a convergence rate of up to 1?O(h1/4) of the CG iteration. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
10.
Summary. In this work we address the issue of integrating
symmetric Riccati and Lyapunov matrix differential equations. In
many cases -- typical in applications -- the solutions are positive
definite matrices. Our goal is to study when and how this property
is maintained for a numerically computed solution.
There are two classes of solution methods: direct and
indirect algorithms. The first class consists of the schemes
resulting from direct discretization of the equations. The second
class consists of algorithms which recover the solution by
exploiting some special formulae that these solutions are known to
satisfy.
We show first that using a direct algorithm -- a one-step scheme or
a strictly stable multistep scheme (explicit or implicit) -- limits
the order of the numerical method to one if we want to guarantee
that the computed solution stays positive definite. Then we show two
ways to obtain positive definite higher order approximations by
using indirect algorithms. The first is to apply a symplectic
integrator to an associated Hamiltonian system. The other uses
stepwise linearization.
Received April 21, 1993 相似文献
11.
Richard E. Ewing 《BIT Numerical Mathematics》1989,29(4):850-866
The simulation of large-scale fluid flow applications often requires the efficient solution of extremely large nonsymmetric linear and nonlinear sparse systems of equations arising from the discretization of systems of partial differential equations. While preconditioned conjugate gradient methods work well for symmetric, positive-definite matrices, other methods are necessary to treat large, nonsymmetric matrices. The applications may also involve highly localized phenomena which can be addressed via local and adaptive grid refinement techniques. These local refinement methods usually cause non-standard grid connections which destroy the bandedness of the matrices and the associated ease of solution and vectorization of the algorithms. The use of preconditioned conjugate gradient or conjugate-gradient-like iterative methods in large-scale reservoir simulation applications is briefly surveyed. Then, some block preconditioning methods for adaptive grid refinement via domain decomposition techniques are presented and compared. These techniques are being used efficiently in existing large-scale simulation codes. 相似文献
12.
Summary. A new algorithm for triangularizing an Toeplitz matrix is presented. The algorithm is based on the previously developed recursive algorithms that exploit the Toeplitz
structure and compute each row of the triangular factor via updating and downdating steps. A forward error analysis for this
existing recursive algorithm is presented, which allows us to monitor the conditioning of the problem, and use the method
of corrected semi-normal equations to obtain higher accuracy for certain ill-conditioned matrices. Numerical experiments show
that the new algorithm improves the accuracy significantly while the computational complexity stays in .
Received April 30, 1995 / Revised version received February 12, 1996 相似文献
13.
Here are considered matrices represented as a sum of diagonal and semiseparable ones. These matrices belong to the class of structured matrices which arises in numerous applications. FastO(N) algorithms for their inversion were developed before under additional restrictions which are a source of instability. Our aim is to eliminate these restrictions and to develop reliable and stable numerical algorithms. In this paper we obtain such algorithms with the only requirement that the considered matrix is invertible and its determinant is not close to zero. The case of semiseparable matrices of order one was considered in detail in an earlier paper of the authors. 相似文献
14.
A generalization of the QR algorithm proposed by Francis [2] for square matrices is introduced for the singular values decomposition of arbitrary rectangular matrices. Geometrically the algorithm means the subsequent orthogonalization of the image of orthonormal bases produced in the course of the iteration. Our purpose is to show how to get a series of lower triangular matrices by alternate orthogonal-upper triangular decompositions in different dimensions and to prove the convergence of this series. 相似文献
15.
Conjugate gradient type methods are discussed for unsymmetric and inconsistent system of equations. For unsymmetric problems, besides conjugate gradient methods based on the normal equations, we also present a (modified) minimal residual (least square) method, which converges for systems with matrices that have a positive definite symmetric part. For inconsistent problems, for completeness we discuss briefly various (well-known) versions of the conjugate gradient method. Preconditioning and rate of convergence are also discussed. 相似文献
16.
Daniel Kressner 《Numerische Mathematik》2006,103(3):461-483
Stewart's recently introduced Krylov-Schur algorithm is a modification of the implicitly restarted Arnoldi algorithm which
employs reordered Schur decompositions to perform restarts and deflations in a numerically reliable manner. This paper describes
a variant of the Krylov-Schur algorithm suitable for addressing eigenvalue problems associated with products of large and
sparse matrices. It performs restarts and deflations via reordered periodic Schur decompositions and, by taking the product
structure into account, it is capable of achieving qualitatively better approximations to eigenvalues of small magnitude.
Supported by DFG Research Center Matheon, Mathematics for key technologies, in Berlin. 相似文献
17.
Georg Heinig 《Linear algebra and its applications》2011,435(1):1-59
The paper gives a self-contained survey of fast algorithms for solving linear systems of equations with Toeplitz or Hankel coefficient matrices. It is written in the style of a textbook. Algorithms of Levinson-type and Schur-type are discussed. Their connections with triangular factorizations, Padè recursions and Lanczos methods are demonstrated. In the case in which the matrices possess additional symmetry properties, split algorithms are designed and their relations to butterfly factorizations are developed. 相似文献
18.
In this paper, the singular value decompositions of anti-symmetric matrices and anti-bisymmetric matrices are presented, and a class of parameterized inverse singular value problem for anti-bisymmetric matrices is solved. Numerical examples which confirm our theory are presented. 相似文献
19.
Dario Andrea Bini 《Numerical Functional Analysis & Optimization》2013,34(1-2):47-66
Computations with univariate polynomials, like the evaluation of product, quotient, remainder, greatest common divisor, etc, are closely related to linear algebra computations performed with structured matrices having the Toeplitz-like or the Hankel-like structures. The discrete Fourier transform, and the FFT algorithms for its computation, constitute a powerful tool for the design and analysis of fast algorithms for solving problems involving polynomials and structured matrices. We recall the main correlations between polynomial and matrix computations and present two recent results in this field: in particular, we show how Fourier methods can speed up the solution of a wide class of problems arising in queueing theory where certain Markov chains, defined by infinite block Toeplitz matrices in generalized Hessenberg form, must be solved. Moreover, we present a new method for the numerical factorization of polynomials based on a matrix generalization of Koenig's theorem. This method, that relies on the evaluation/interpolation technique at the Fourier points, reduces the problem of polynomial factorization to the computation of the LU decomposition of a banded Toeplitz matrix with its rows and columns suitably permuted. Numerical experiments that show the effectiveness of our algorithms are presented 相似文献
20.
Fast inversion of Chebyshev--Vandermonde
matrices 总被引:2,自引:0,他引:2
Summary.
This paper contains two fast algorithms for
inversion of
Chebyshev--Vandermonde
matrices of the first and second kind. They are
based on special representations of the Bezoutians of Chebyshev polynomials of
both kinds. The paper also contains the results of numerical
experiments which show that the algorithms proposed here are not only much
faster, but also more stable than other algorithms available. It is also
efficient to use the above two algorithms for solving Chebyshev--Vandermode
systems of equations with preprocessing.
Received January 23, 1993/Revised version received May 21, 1993 相似文献