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Fast factorization of rectangular Vandermonde matrices with Chebyshev nodes

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Abstract

The polynomial interpolation problem with distinct interpolation points and the polynomial represented in the power basis gives rise to a linear system of equations with a Vandermonde matrix. This system can be solved efficiently by exploiting the structure of the Vandermonde matrix with the aid of the Björck–Peyrera algorithm. We are concerned with polynomial least-squares approximation at the zeros of Chebyshev polynomials. This gives rise to a rectangular Vandermonde matrix. We describe fast algorithms for the factorization of these matrices. Both QR and QR-like factorizations are discussed. The situations when the nodes are extreme points of Chebyshev polynomials or zeros of some classical orthogonal polynomial also are considered.

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Funding

This research was supported in part by NSF grants DMS-1729509 and DMS-1720259.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, Kent, OH, 44242, USA

    Mykhailo Kuian & Lothar Reichel

  2. Liquid Crystal Institute, Kent State University, Kent, OH, 44242, USA

    Sergij V. Shiyanovskii

Authors
  1. Mykhailo Kuian
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  2. Lothar Reichel
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  3. Sergij V. Shiyanovskii
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Correspondence to Lothar Reichel.

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Kuian, M., Reichel, L. & Shiyanovskii, S.V. Fast factorization of rectangular Vandermonde matrices with Chebyshev nodes. Numer Algor 81, 547–559 (2019). https://doi.org/10.1007/s11075-018-0560-9

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  • DOI: https://doi.org/10.1007/s11075-018-0560-9

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