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On generating Sobolev orthogonal polynomials

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Abstract

Sobolev orthogonal polynomials are polynomials orthogonal with respect to a Sobolev inner product, an inner product in which derivatives of the polynomials appear. They satisfy a long recurrence relation that can be represented by a Hessenberg matrix. The problem of generating a finite sequence of Sobolev orthogonal polynomials can be reformulated as a matrix problem, that is, a Hessenberg inverse eigenvalue problem, where the Hessenberg matrix of recurrences is generated from certain known spectral information. Via the connection to Krylov subspaces we show that the required spectral information is the Jordan matrix containing the eigenvalues of the Hessenberg matrix and the normalized first entries of its eigenvectors. Using a suitable quadrature rule the Sobolev inner product is discretized and the resulting quadrature nodes form the Jordan matrix and associated quadrature weights are the first entries of the eigenvectors. We propose two new numerical procedures to compute Sobolev orthonormal polynomials based on solving the equivalent Hessenberg inverse eigenvalue problem.

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Notes

  1. The term discrete inner product is avoided because it is already used for a specific type of Sobolev inner product, e.g., see [29] and Sect. 2.4.2.

  2. Note that we misuse the term Jordan block, since \(J_{j,k_j}\) is only a Jordan block if \(\alpha _{i}^{(j)}=1\) for \(i=1,\dots ,k_j\).

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Acknowledgements

The author would like to thank Francisco Marcellán and Stefano Pozza for comments on an earlier draft of this paper which improved its presentation greatly and Petr Tichý for suggesting valuable references. The author is also grateful to two anonymous referees for their valuable comments that contributed to improving the quality of this paper.

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The research of the author was supported by Charles University Research program No. PRIMUS/21/SCI/009.

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  1. Department of Numerical Mathematics, Charles University, Sokolovská 83, 186 75, Prague 8, Czech Republic

    Niel Van Buggenhout

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Van Buggenhout, N. On generating Sobolev orthogonal polynomials. Numer. Math. 155, 415–443 (2023). https://doi.org/10.1007/s00211-023-01379-3

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