Breakdowns in the implementation of the Lánczos method for solving linear systems

AbstractThe Lánczos method for solving systems of linear equations is based on formal orthogonal polynomials. Its implementation is realized via some recurrence relationships between polynomials of a family of orthogonal polynomials or between those of two adjacent families of orthogonal polynomials. A division by zero can occur in such recurrence relations, thus causing a breakdown in the algorithm which has to be stopped. In this paper, two types of breakdowns are discussed. The true breakdowns which are due to the nonexistence of some polynomials and the ghost breakdowns which are due to the recurrence relationship used. Among all the recurrence relationships which can be used and all the algorithms for implementing the Lánczos method which came out from them, the only reliable algorithm is Lánczos/Orthodir which can only suffer from true breakdowns. It is shown how to avoid true breakdowns in this algorithm. Other algorithms are also discussed and the case of near-breakdown is treated. The same treatment applies to other methods related to Lánczos'

smt: a Matlab structured matrices toolbox

We introduce the smt toolbox for Matlab. It implements optimized storage and fast arithmetics for circulant and Toeplitz matrices, and is intended to be transparent to the user and easily extensible. It also provides a set of test matrices, computation of circulant preconditioners, and two fast algorithms for Toeplitz linear systems.Comment: 19 pages, 1 figure, 1 typo corrected in the abstrac

Dagstuhl Seminar Proceedings 07071

Abstract. An important problem in Web search is to determine the importance of each page. This problem consists in computing, by the power method, the left principal eigenvector (the PageRank vector) of a matrix depending on a parameter c which has to be chosen close to 1. However, when c is close to 1, the problem is ill-conditioned, and the power method converges slowly. So, the idea developed in this paper consists in computing the PageRank vector for several values of c, and then to extrapolate them, by a conveniently chosen rational function, at a point near 1. The choice of this extrapolating function is based on the mathematical considerations about the PageRank vector

Transpose-free Lanczos-type algorithms for nonsymmetric linear systems

The method of Lanczos for solving systems of linear equations is implemented by using recurrence relationships between formal orthogonal polynomials. A drawback is that the computation of the coefficients of these recurrence relationships usually requires the use of the transpose of the matrix of the system. Due to the indirect addressing, this is a costly operation. In this paper, a new procedure for computing these coefficients is proposed. It is based on the recursive computation of the products of polynomials appearing in their expressions and it does involve the transpose of the matrix. Moreover, our approach allows to implement simultaneously and at a low price a Lanczos--type product method such as the CGS or the BiCGSTAB

On The Zeros Of Various Kinds Of Orthogonal Polynomials

Recently, several generalizations of the notion of orthogonal polynomials appeared in the literature. The aim of this paper is to study their zeros

Block Projection Methods for Linear Systems

The aim of this paper is to provide a theory of block projection methods for the solution of a system of linear equations with multiple right{hand sides. Our approach allows to obtain recursive algorithms for the implementation of these methods