Fast inversion of vandermonde-like matrices involving orthogonal polynomials

Let {q}j=0n−1 be a family of polynomials that satisfy a three-term recurrence relation and let {tk}k=1n be a set of distinct nodes. Define the Vandermonde-like matrixWn=[wjk]k,j=1n,wjk=qj−1(tk). We describe a fast algorithm for computing the elements of the inverse ofWn inO(n2) 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(n3) arithmetic operations.

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