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Generation of orthogonal rational functions by procedures for structured matrices

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Abstract

The problem of computing recurrence coefficients of sequences of rational functions orthogonal with respect to a discrete inner product is formulated as an inverse eigenvalue problem for a pencil of Hessenberg matrices. Two procedures are proposed to solve this inverse eigenvalue problem, via the rational Arnoldi iteration and via an updating procedure using unitary similarity transformations. The latter is shown to be numerically stable. This problem and both procedures are generalized by considering biorthogonal rational functions with respect to a bilinear form. This leads to an inverse eigenvalue problem for a pencil of tridiagonal matrices. A tridiagonal pencil implies short recurrence relations for the biorthogonal rational functions, which is more efficient than the orthogonal case. However, the procedures solving this problem must rely on nonunitary operations and might not be numerically stable.

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Funding

The research of the first author was funded by the Research Council KU Leuven, C1-project C14/17/073 (Numerical Linear Algebra and Polynomial Computations), project C14/16/056 (Inverse-free Rational Krylov Methods: Theory and Applications); the research of the second author by the Research Council KU Leuven, C1-project C14/17/073 (Numerical Linear Algebra and Polynomial Computations), by the Fund for Scientific Research–Flanders (Belgium), EOS Project no 30468160; and the research of the third author by the Research Council KU Leuven, project C14/16/056 (Inverse-free Rational Krylov Methods: Theory and Applications).

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  1. Department of Computer Science, KU Leuven, University of Leuven, 3001, Leuven, Belgium

    Niel Van Buggenhout, Marc Van Barel & Raf Vandebril

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  1. Niel Van Buggenhout
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  2. Marc Van Barel
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  3. Raf Vandebril
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Correspondence to Niel Van Buggenhout.

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Van Buggenhout, N., Van Barel, M. & Vandebril, R. Generation of orthogonal rational functions by procedures for structured matrices. Numer Algor 89, 551–582 (2022). https://doi.org/10.1007/s11075-021-01125-6

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  • DOI: https://doi.org/10.1007/s11075-021-01125-6

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