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

Farooq, M and Salhi, A (2012) 'New recurrence relationships between orthogonal polynomials which lead to new Lanczos-type algorithms.' Journal of Prime Research in Mathematics, 8. 61 - 75. ISSN 1817-3462

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Abstract

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.

Item Type: Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science and Health > Mathematical Sciences, Department of
Depositing User: Jim Jamieson
Date Deposited: 06 Aug 2013 10:58
Last Modified: 30 Jan 2019 16:17
URI: http://repository.essex.ac.uk/id/eprint/7257

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