A new Lanczos-type algorithm for systems of linear equations

Muhammad Farooq
Department of Mathematics, University of Peshawar, Khyber Pakhtunkhwa, 25120, Pakistan.
Abdellah Salhi
Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, UK.

\(^{1}\)Corresponding Author: mfarooq@upesh.edu.pk

Copyright © 2014 Muhammad Farooq, Abdellah Salhi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Published: December, 2014.

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 \(A_5/B_{10}\), \(A_8/B_{10}\), as well as Arnoldi’s algorithm, on a set of standard test problems. Numerical results are included.

Keywords:

Lanczos algorithm; Arnoldi algorithm; Systems of Linear Equations; Formal Orthogonal Polynomials.