A new Lanczos type algorithm for system of linear equations

TL;DR

This paper introduces a novel Lanczos-type algorithm based on a higher degree polynomial recurrence relation, demonstrating improved stability over existing algorithms through theoretical development and numerical testing.

Contribution

A new Lanczos-type algorithm utilizing a higher degree polynomial recurrence relation, offering enhanced stability compared to existing methods.

Findings

Shows superior stability in numerical tests

Performs better than A5/B10, A8/B10, and Arnoldi algorithms

Effective on standard test problems

Abstract

Lanczos-type algorithms are efficient and easy to implement. Unfortunately they breakdown frequently and well before convergence has been achieved. These algorithms are typically based on recurrence relations which involve formal orthogonal polynomials of low degree. In this paper, we consider a recurrence relation that has not been studied before and which involves a relatively higher degree polynomial. Interestingly, it leads to an algorithm that shows superior stability when compared to existing Lanczos-type algorithms. This new algorithm is derived and described. It is then compared to the best known algorithms of this type, namely A5/B10, A8/B10, as well as Arnoldi's algorithm, on a set of standard test problems. Numerical results are included.