Stability analysis of parallel evaluation of finite series of orthogonal polynomials
by
Roberto Barrio
Dpto. Matematica Aplicada, Univ. Zaragoza, E-50009 Zaragoza, Spain
Coauthors: Plamen Yalamov (Center of Applied Mathematics and Informatics, University of Rousse, Bulgaria)

In this communication we analyse the stability of parallel algorithms for the evaluation of polynomials written as a finite series of a general family of orthogonal polynomials. For the parallel evaluation of such series recently two kinds of methods have been proposed: two methods based on the sequential Forsythe's algorithm and two based on the Clenshaw's one. For every method the first algorithm uses parallel techniques in matrix methods by means of the matrix formulation of the sequential algorithm, whereas the second algorithm is based on a recurrence matrix product formulation of the sequential recurrence. Some efficiency tests of these algorithms on a CRAY T3D are presented.

In the paper we present rounding error bounds of the first Clenshaw's and Forsythe's parallel algorithms. The basic part of the computation is the solution of a triangular tridiagonal linear system. This fact allows a more detailed analysis which is obtained here. The analysis shows that the parallel algorithms are almost as stable as their sequential counterparts for practical applications. Extensive numerical experiments confirm the theoretical conclusions.

Date received: February 2, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeb-96.