A Switching Approach to Avoid Breakdown in Lanczos-type Algorithms

TL;DR

This paper introduces a novel switching strategy between different Lanczos-type algorithms to prevent breakdowns caused by non-existent orthogonal polynomials, improving stability during computations.

Contribution

It proposes a switching approach among various Lanczos-type algorithms, specifically analyzing the ST2 strategy, to enhance numerical stability and avoid breakdowns.

Findings

Switching algorithms prevents breakdowns in Lanczos-type methods.

Numerical results demonstrate improved stability with the proposed approach.

The ST2 strategy effectively maintains algorithmic progress without breakdowns.

Abstract

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 $A_4$, $A_{12}$, $A_5/B_{10}$ and $A_5/B_8$ will be presented and then deployed in the switching framework. In this paper, only strategy ST2 will be investigated. Numerical results will be presented.