**Journal of Prime Research in Mathematics**

Vol. 1 (2012), Issue 1, pp. 61 – 75

ISSN: 1817-3462 (Online) 1818-5495 (Print)

ISSN: 1817-3462 (Online) 1818-5495 (Print)

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

**Muhammad Farooq
**Department of Mathematics, University of Peshawar, Peshawar, Pakistan.

**Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester, United Kingdom.**

Abdellah Salhi

Abdellah Salhi

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

Copyright © 2012

**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, 2012.

### Abstract

Lanczos methods for solving \(Ax = b\) consist in constructing a sequence of vectors \((x_k)\), \(k = 1, …\) such that \(r_k = b − Ax_k = P_k(A)r_0\), where \(P_k\) is the orthogonal polynomial of degree at most k with respect to the linear functional c defined as c(ξ^i) = (y, A^ir_0)\). 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)(ξ ^{i}) = c^{(ξi+1)}\). All Lanczos-type algorithms are characterized by the choice of one or two recurrence relationships, one for \(P_k\) 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.

#### Keywords:

Lanczos algorithm, formal orthogonal polynomials, linear system, monic polynomials.