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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Chebyshev-Vandermonde systems

Authors: Lothar Reichel and Gerhard Opfer
Journal: Math. Comp. 57 (1991), 703-721
MSC: Primary 65F30; Secondary 65D10
MathSciNet review: 1094957
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Abstract: A Chebyshev-Vandermonde matrix

$\displaystyle V = [{p_j}({z_k})]_{j,k = 0}^n \in {\mathbb{C}^{(n + 1) \times (n + 1)}}$

is obtained by replacing the monomial entries of a Vandermonde matrix by Chebyshev polynomials $ {p_j}$ for an ellipse. The ellipse is also allowed to be a disk or an interval. We present a progressive scheme for allocating distinct nodes $ {z_k}$ on the boundary of the ellipse such that the Chebyshev-Vandermonde matrices obtained are reasonably well-conditioned. Fast progressive algorithms for the solution of the Chebyshev-Vandermonde systems are described. These algorithms are closely related to methods recently presented by Higham. We show that the node allocation is such that the solution computed by the progressive algorithms is fairly insensitive to perturbations in the right-hand side vector. Computed examples illustrate the numerical behavior of the schemes. Our analysis can also be used to bound the condition number of the polynomial interpolation operator defined by Newton's interpolation formula. This extends earlier results of Fischer and the first author.

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Keywords: Vandermonde-like matrix, ordering of nodes, numerical conditioning, progressive algorithm, Newton interpolation formula
Article copyright: © Copyright 1991 American Mathematical Society

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