New recurrence relationships between orthogonal polynomials which lead to new Lanczos-type algorithms

Lanczos methods for solving Ax = b consist in constructing a sequence of vectors (Xk),k = 1,... such that rk = b-AXk= Pk(A)r0, where Pk is the orthogonal polynomial of degree at most k with respect to the linear functional c defined as c(εi) = (y, Air0). Let P(1)k be the regular monic polynomial of degree k belonging to the family of formal orthogonal polynomials (FOP) with respect to c(1) defined as c(1)(εi) = c(εi+1). All Lanczos-type algorithms are characterized by the choice of one or two recurrence relationships, one for Pk and one for P(1)k. We shall study some new recurrence relations involving these two polynomials and their possible combinations to obtain new Lanczos-type algorithms. We will show that some recurrence relations exist, but cannot be used to derive Lanczos-type algorithms, while others do not exist at all

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'

A Switching Approach to Avoid Breakdown in Lanczos-Type Algorithms

Lanczos-type algorithms are well known for their inherent instability. They typically breakdown when relevant orthogonal polynomials do not exist. Current approaches to avoiding breakdown rely on jumping over the non-existent polynomials to resume computation. This jumping strategy may have to be used many times during the solution process. We suggest an alternative to jumping which consists in switching between different algorithms that have been generated using different recurrence relations between orthogonal polynomials. This approach can be implemented as three different strategies: ST1, ST2, and ST3.We shall briefly recall how Lanczos-type algorithms are derived. Four of the most prominent such algorithms namely A4, A12, A5/B10 and A5/B8 will be presented and then deployed in the switching framework. In this paper, only strategy ST2 will be investigated. Numerical results will be presented. © 2014 NSP Natural Sciences Publishing Cor

An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices

The nonsymmetric Lanczos method can be used to compute eigenvalues of large sparse non-Hermitian matrices or to solve large sparse non-Hermitian linear systems. However, the original Lanczos algorithm is susceptible to possible breakdowns and potential instabilities. An implementation is presented of a look-ahead version of the Lanczos algorithm that, except for the very special situation of an incurable breakdown, overcomes these problems by skipping over those steps in which a breakdown or near-breakdown would occur in the standard process. The proposed algorithm can handle look-ahead steps of any length and requires the same number of matrix-vector products and inner products as the standard Lanczos process without look-ahead

Progress on lattice QCD algorithms

I review recent progress on algorithms for calculating quark propagators and for simulating full QCD.Comment: 8 uuencoded PostScript pages, contribution to LAT95 (fig.1 simplified to conserve space; available upon request

Avoiding breakdown in variants of the bi-cgstab algorithm

AbstractThe bi-cg method and its variants such as cgs, bi-cgstab, and bi-cgstab2 for solving nonsymmetric linear systems of equations can suffer from breakdown. By using the theory of the formal orthogonal polynomials we present a breakdown-free bi-cgstab algorithm and a breakdown-free bi-cgstab2 algorithm. Only exact breakdowns are cured accurately

Restarting from Specific Points to Cure Breakdown in Lanczos-type Algorithms

Breakdown in Lanczos-type algorithms is a common phenomenon which is due to the non-existence of some orthogonal polynomials. It causes the solution process to halt. It is, therefore, important to deal with it to improve the resilience of the algorithms and increase their usability. In this paper, we consider restarting from a number of different approximate solutions that seem to be attractive starting points. They are: (a) the last iterate preceding breakdown, (b) the iterate with minimum residual norm found so far, and (c) the approximate solution whose entries are the median values of entries of all iterates generated by the Lanczos-type algorithm considered. Although it has been shown theoretically in the context of Arnoldi-type algorithms as well as Lanczos-type algorithms that restarting mitigates breakdown and allows the iterative process to continue and converge to good solutions, here we give an alternative theorem to that effect and a proof of it. However, emphasis is on the quality of the restarting points. Numerical results are included