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Cited by 55Mathematical functions and polynomialsMatrix Theory and AlgorithmsPolynomial and algebraic computation

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Chebyshev-Vandermonde systems

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Reichel, Lothar, Opfer, Gerhard·1991·Mathematics of Computation
DOI 10.1090/s0025-5718-1991-1094957-9
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Abstract

A Chebyshev-Vandermonde matrix \[ V = [ p j ( z k ) ] j , k = 0 n ∈ C ( n + 1 ) × ( n + 1 ) V = [{p_j}({z_k})]_{j,k = 0}^n \in {\mathbb {C}^{(n + 1) \times (n + 1)}} \] is obtained by replacing the monomial entries of a Vandermonde matrix by Chebyshev polynomials p j {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 {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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Metadata

Authors
Reichel, L., & Opfer, G.
Venue
Mathematics of Computation
Publisher
American Mathematical Society (AMS)
Published
1991
Volume
57
Issue
196
Pages
703-721
ISSN
  • 0025-5718 (ISSN-L)
  • 1088-6842
DOI
10.1090/s0025-5718-1991-1094957-9
OpenAlex
W1989782579
Cited by
55
References
21
Citations by year
2025: 1 · 2024: 2 · 2023: 4 · 2022: 4 · 2021: 1 · 2020: 1 · 2019: 2 · 2018: 2 · 2016: 2 · 2013: 2 · 2012: 1
FWCI
4.53
Type
journal article
Language
en
Open access
Locations
Mathematics of Computation (publishedVersion)
Funders
National Science Foundation (8704196)
Topics
  • Physical Sciences › Mathematics › Applied Mathematics › Mathematical functions and polynomials
  • Physical Sciences › Computer Science › Computational Theory and Mathematics › Matrix Theory and Algorithms
  • Physical Sciences › Computer Science › Computational Theory and Mathematics › Polynomial and algebraic computation
Keywords
Vandermonde systems, node allocation, matrix conditioning, progressive algorithms, Newton interpolation, condition number bounds, right-hand side perturbation, polynomial interpolation

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CoChat. (2026). Citation check: Chebyshev-Vandermonde systems. https://app.cochat.ai/papers/10.1090/s0025-5718-1991-1094957-9
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Reichel, L., & Opfer, G. (1991). Chebyshev-Vandermonde systems. Mathematics of Computation. https://doi.org/10.1090/s0025-5718-1991-1094957-9