Stability analysis of algorithms for solving confluent Vandermonde-like systems

Higham, Nicholas J. (1990) Stability analysis of algorithms for solving confluent Vandermonde-like systems. SIAM Journal On Matrix Analysis And Applications, 11 (1). pp. 23-41. ISSN 1095-7162

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Abstract

A confluent Vandermonde-like matrix $P(\alpha _0 ,\alpha _1 , \cdots ,\alpha _n )$ is a generalisation of the confluent Vandermonde matrix in which the monomials are replaced by arbitrary polynomials. For the case where the polynomials satisfy a three-term recurrence relation algorithms for solving the systems $Px = b$ and $P^T a = f$ in $O(n^2 )$ operations are derived. Forward and backward error analyses that provide bounds for the relative error and the residual of the computed solution are given. The bounds reveal a rich variety of problem-dependent phenomena, including both good and bad stability properties and the possibility of Xextremely accurate solutions. To combat potential instability, a method is derived for computing a “stable” ordering of the points $\alpha _i $; it mimics the interchanges performed by Gaussian elimination with partial pivoting, using only $O(n^2)$ operations. The results of extensive numerical tests are summarised, and recommendations are given for how to use the fast algorithms to solve Vandermonde-like systems in a stable manner.

Item Type:	Article
Uncontrolled Keywords:	Vandermonde matrix, orthogonal polynomials, Hermite interpolation, Clenshaw recurrence, forward error analysis, backward error analysis, stability, iterative refinement
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Ms Lucy van Russelt
Date Deposited:	04 Jul 2006
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/355

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