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View all- Van Buggenhout N(2023)On generating Sobolev orthogonal polynomialsNumerische Mathematik10.1007/s00211-023-01379-3155:3-4(415-443)Online publication date: 1-Dec-2023
Generation of orthogonal rational functions by procedures for structured matrices
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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Publication History
Published: 01 February 2022
Accepted: 27 April 2021
Received: 08 March 2021
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View all- Van Buggenhout N(2023)On generating Sobolev orthogonal polynomialsNumerische Mathematik10.1007/s00211-023-01379-3155:3-4(415-443)Online publication date: 1-Dec-2023
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