Block-Arnoldi and Davidson methods for unsymmetric large eigenvalue problems

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.     

A block Arnoldi-Chebyshev method for computing the leading eigenpairs of large sparse unsymmetric matrices

Summary In this paper we describe a block version of Arnoldi's method for computing a few eigenvalues with largest or smallest real parts. The method is accelerated via Chebyshev iteration and a procedure is developed to identify the optimal ellipse which encloses the spectrum. A parallel implementation of this method is investigated on the eight processor Alliant FX/80. Numerical results and comparisons with simultaneous iteration on some Harwell-Boeing matrices are reported.     

On generalized successive overrelaxation methods for augmented linear systems

For the augmented system of linear equations, Golub, Wu and Yuan recently studied an SOR-like method (BIT 41(2001)71–85). By further accelerating it with another parameter, in this paper we present a generalized SOR (GSOR) method for the augmented linear system. We prove its convergence under suitable restrictions on the iteration parameters, and determine its optimal iteration parameters and the corresponding optimal convergence factor. Theoretical analyses show that the GSOR method has faster asymptotic convergence rate than the SOR-like method. Also numerical results show that the GSOR method is more effective than the SOR-like method when they are applied to solve the augmented linear system. This GSOR method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation, matrix relaxation and inexact iteration. Besides, we also demonstrate a complete version about the convergence theory of the SOR-like method. Subsidized by The Special Funds For Major State Basic Research Projects (No. G1999032803) and The National Natural Science Foundation (No. 10471146), P.R. China     

New look-ahead Lanczos-type algorithms for linear systems

Summary. A breakdown (due to a division by zero) can arise in the algorithms for implementing Lanczos' method because of the non-existence of some formal orthogonal polynomials or because the recurrence relationship used is not appropriate. Such a breakdown can be avoided by jumping over the polynomials involved. This strategy was already used in some algorithms such as the MRZ and its variants. In this paper, we propose new implementations of the recurrence relations of these algorithms which only need the storage of a fixed number of vectors, independent of the length of the jump. These new algorithms are based on Horner's rule and on a different way for computing the coefficients of the recurrence relationships. Moreover, these new algorithms seem to be more stable than the old ones and they provide better numerical results. Numerical examples and comparisons with other algorithms will be given. Received September 2, 1997 / Revised version received July 24, 1998     

A Jacobi–Davidson type method for the product eigenvalue problem

We propose a Jacobi–Davidson type method to compute selected eigenpairs of the product eigenvalue problem _Am?_A1x=λx,$A_m?A_{1}x=\lambda x,$ where the matrices may be large and sparse. To avoid difficulties caused by a high condition number of the product matrix, we split up the action of the product matrix and work with several search spaces. We generalize the Jacobi–Davidson correction equation and the harmonic and refined extraction for the product eigenvalue problem. Numerical experiments indicate that the method can be used to compute eigenvalues of product matrices with extremely high condition numbers.     

UTV Tools: Matlab templates for rank-revealing UTV decompositions

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.     

A preconditioner for the h-p version of the finite element method in two dimensions

Summary. A preconditioner, based on a two-level mesh and a two-level orthogonalization, is proposed for the - version of the finite element method for two dimensional elliptic problems in polygonal domains. Its implementation is in parallel on the subdomain level for the linear or bilinear (nodal) modes, and in parallel on the element level for the high order (side and internal) modes. The condition number of the preconditioned linear system is of order , where is the diameter of the -th subdomain, and are the diameter of elements and the maximum polynomial degree used in the subdomain. This result reduces to well-known results for the -version (i.e. ) and the -version (i.e. ) as the special cases of the - version. Received August 15, 1995 / Revised version received November 13, 1995     

On the regularizing properties of the GMRES method

Summary. The GMRES method is a popular iterative method for the solution of large linear systems of equations with a nonsymmetric nonsingular matrix. However, little is known about the behavior of this method when it is applied to the solution of nonsymmetric linear ill-posed problems with a right-hand side that is contaminated by errors. We show that when the associated error-free right-hand side lies in a finite-dimensional Krylov subspace, the GMRES method is a regularization method. The iterations are terminated by a stopping rule based on the discrepancy principle. Received November 10, 2000 / Revised version received April 11, 2001 / Published online October 17, 2001     

How bad are Hankel matrices?

Summary. Considered are Hankel, Vandermonde, and Krylov basis matrices. It is proved that for any real positive definite Hankel matrix of order , its spectral condition number is bounded from below by . Also proved is that the spectral condition number of a Krylov basis matrix is bounded from below by . For , a Vandermonde matrix with arbitrary but pairwise distinct nodes , we show that ; if either or for all , then . Received January 24, 1993/Revised version received July 19, 1993     

Incomplete block factorization preconditioning for indefinite elliptic problems

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