, Volume 33, Issue 3, pp 473-484

Fast inversion of vandermonde-like matrices involving orthogonal polynomials

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Abstract

Let {q} j =0n−1 be a family of polynomials that satisfy a three-term recurrence relation and let {t k } k =1n be a set of distinct nodes. Define the Vandermonde-like matrixW n =[w jk ] k,j =1n ,w jk =q j−1(t k ). We describe a fast algorithm for computing the elements of the inverse ofW n inO(n 2) arithmetic operations. Our algorithm generalizes a scheme presented by Traub [22] for fast inversion of Vandermonde matrices. Numerical examples show that our scheme often yields higher accuracy than the LINPACK subroutine SGEDI for inverting a general matrix. SGEDI uses Gaussian elimination with partial pivoting and requiresO(n 3) arithmetic operations.

Dedicated to Gene H. Golub on his 60th birthday
Research supported by NSF grant DMS-9002884.