Skip to main content
Log in

On generating Sobolev orthogonal polynomials

Numerische Mathematik Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Algorithm 1
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

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 \(i=1,\dots ,k_j\).

References

  1. Martínez-Finkelshtein, A.: Analytic aspects of Sobolev orthogonal polynomials revisited. In: Numerical Analysis 2000. Vol. V: Quadrature and Orthogonal Polynomials. J. Comput. Appl. Math., 127(1-2), pp. 255–266 (2001). https://doi.org/10.1016/S0377-0427(00)00499-4

  2. Marcellán, F., Xu, Y.: On Sobolev orthogonal polynomials. Expo. Math. 33(3), 308–352 (2015). https://doi.org/10.1016/j.exmath.2014.10.002

    Article  MathSciNet  MATH  Google Scholar 

  3. López Lagomasino, G., Pijeira Cabrera, H., Pérez Izquierdo, I.: Sobolev orthogonal polynomials in the complex plane. In: Numerical Analysis 2000. Vol. V: Quadrature and Orthogonal Polynomials. J. Comput. Appl. Math. 127(1–2), pp. 219–230 (2001). https://doi.org/10.1016/S0377-0427(00)00498-2

  4. Forsythe, G.E.: Generation and use of orthogonal polynomials for data-fitting with a digital computer. J. Soc. Ind. Appl. Math. 5(2), 74–88 (1957). https://doi.org/10.1137/0105007

    Article  MathSciNet  MATH  Google Scholar 

  5. Szegő, G.: Orthogonal Polynomials, 4th edn. Americal Mathematical Society Colloquium Publications, vol. 23. American Mathematical Society, Providence, RI (1975)

  6. Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation. Oxford Science Publications. Oxford University Press, New York (2004)

  7. Golub, G.H., Meurant, G.: Matrices, Moments and Quadrature with Applications. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2009)

  8. Gragg, W.B., Harrod, W.J.: The numerically stable reconstruction of Jacobi matrices from spectral data. Numer. Math. 44(3), 317–335 (1984). https://doi.org/10.1007/BF01405565

    Article  MathSciNet  MATH  Google Scholar 

  9. Golub, G.H., Welsch, J.H.: Calculation of Gauss quadrature rules. Math. Comput. 23(106), 221–230 (1969). https://doi.org/10.1090/S0025-5718-69-99647-1

    Article  MathSciNet  MATH  Google Scholar 

  10. Brubeck, P.D., Nakatsukasa, Y., Trefethen, L.N.: Vandermonde with Arnoldi. SIAM Rev. 63(2), 405–415 (2021). https://doi.org/10.1137/19M130100X

    Article  MathSciNet  MATH  Google Scholar 

  11. Reichel, L.: Fast \(QR\) decomposition of Vandermonde-like matrices and polynomial least squares approximation. SIAM J. Matrix Anal. Appl. 12(3), 552–564 (1991). https://doi.org/10.1137/0612041

    Article  MathSciNet  MATH  Google Scholar 

  12. de Boor, C., Golub, G.H.: The numerically stable reconstruction of a Jacobi matrix from spectral data. Linear Algebra Appl. 21(3), 245–260 (1978). https://doi.org/10.1016/0024-3795(78)90086-1

    Article  MathSciNet  MATH  Google Scholar 

  13. Reichel, L., Ammar, G., Gragg, W.: Discrete least squares approximation by trigonometric polynomials. Math. Comput. 57(195), 273–289 (1991). https://doi.org/10.1090/S0025-5718-1991-1079030-8

    Article  MathSciNet  MATH  Google Scholar 

  14. Laurie, D.P.: Accurate recovery of recursion coefficients from Gaussian quadrature formulas. J. Comput. Appl. Math. 112(1–2), 165–180 (1999). https://doi.org/10.1016/S0377-0427(99)00228-9

    Article  MathSciNet  MATH  Google Scholar 

  15. Liesen, J., Strakoš, Z.: Krylov Subspace Methods. Principles and Analysis. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013)

  16. Meurant, G.: The Lanczos and Conjugate Gradient Algorithms. From Theory to Finite Precision Computations. Software, Environments, and Tools, vol. 19. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2006)

  17. Kuijlaars, A.B.J.: Convergence analysis of Krylov subspace iterations with methods from potential theory. SIAM Rev. 48(1), 3–40 (2006). https://doi.org/10.1137/S0036144504445376

    Article  MathSciNet  MATH  Google Scholar 

  18. Stahl, H., Totik, V.: General Orthogonal Polynomials. Encyclopedia of Mathematics and its Applications, vol. 43. Cambridge University Press, Cambridge (1992)

  19. Althammer, P.: Eine Erweiterung des Orthogonalitätsbegriffes bei Polynomen und deren Anwendung auf die beste Approximation. J. Reine Angew. Math. 211, 192–204 (1962). In German. https://doi.org/10.1515/crll.1962.211.192

  20. Gröbner, W.: Orthogonale Polynomsysteme, die gleichzeitig mit \(f(x)\) auch deren Ableitung \(f^{\prime }(x)\) approximieren. In: Funktionalanalysis, Approximationstheorie, Numerische Mathematik (Oberwolfach, 1965). Internationale Schriftenreihe zur Numerischen Mathematik, vol. 7, pp. 24–32. Birkhäuser Verlag, Basel-Stuttgart (1967). In German. https://doi.org/10.1007/978-3-0348-5821-2_3

  21. Iserles, A., Koch, P.E., Nørsett, S.P., Sanz-Serna, J.M.: Orthogonality and approximation in a Sobolev space. In: Algorithms for Approximation. II (Shrivenham 1988), pp. 117–124. Chapman and Hall, London (1990)

  22. Niu, Q., Zhang, H., Zhou, Y.: Confluent Vandermonde with Arnoldi. Appl. Math. Lett. 135, 108420 (2023). https://doi.org/10.1016/j.aml.2022.108420

    Article  MathSciNet  MATH  Google Scholar 

  23. Bernardi, C., Maday, Y.: Polynomial interpolation results in Sobolev spaces. J. Comput. Appl. Math. 43(1–2), 53–80 (1992). https://doi.org/10.1016/0377-0427(92)90259-Z

    Article  MathSciNet  MATH  Google Scholar 

  24. Yu, X., Wang, Z., Li, H.: Jacobi–Sobolev orthogonal polynomials and spectral methods for elliptic boundary value problems. Commun. Appl. Math. Comput. 1(2), 283–308 (2019). https://doi.org/10.1007/s42967-019-00016-x

    Article  MathSciNet  MATH  Google Scholar 

  25. Fernández, L., Marcellán, F., Pérez, T.E., Piñar, M.A.: Sobolev orthogonal polynomials and spectral methods in boundary value problems. Appl. Numer. Math. (2023). https://doi.org/10.1016/j.apnum.2023.07.027

    Article  Google Scholar 

  26. Rutishauser, H.: On Jacobi rotation patterns. In: Experimental Arithmetic, High Speed Computing and Mathematics. Proceedings of Symposia in Applied Mathematics, vol. XV, pp. 241–258. American Mathematical Society, Providence, RI (1963)

  27. Gautschi, W., Zhang, M.: Computing orthogonal polynomials in Sobolev spaces. Numer. Math. 71(2), 159–183 (1995). https://doi.org/10.1007/s002110050140

    Article  MathSciNet  MATH  Google Scholar 

  28. López Lagomasino, G., Pijeira Cabrera, H.: Zero location and \(n\)th root asymptotics of Sobolev orthogonal polynomials. J. Approx. Theory 99(1), 30–43 (1999). https://doi.org/10.1006/jath.1998.3318

    Article  MathSciNet  MATH  Google Scholar 

  29. Marcellán, F., Osilenker, B.P., Rocha, I.A.: On Fourier series of a discrete Jacobi–Sobolev inner product. J. Approx. Theory 117(1), 1–22 (2002). https://doi.org/10.1006/jath.2002.3681

    Article  MathSciNet  MATH  Google Scholar 

  30. Cachafeiro, A., Marcellán, F.: The characterization of the quasi-typical extension of an inner product. J. Approx. Theory 62(2), 235–242 (1990). https://doi.org/10.1016/0021-9045(90)90036-P

    Article  MathSciNet  MATH  Google Scholar 

  31. Saylor, P.E., Smolarski, D.C.: Computing the roots of complex orthogonal and kernel polynomials. SIAM J. Sci. Stat. Comput. 9(1), 1–13 (1988). https://doi.org/10.1137/0909001

    Article  MathSciNet  MATH  Google Scholar 

  32. Marcellán, F., Pérez, T.E., Piñar, M.A.: Laguerre–Sobolev orthogonal polynomials. J. Comput. Appl. Math. 71(2), 245–265 (1996). https://doi.org/10.1016/0377-0427(95)00234-0

    Article  MathSciNet  MATH  Google Scholar 

  33. Hermoso, C., Huertas, E.J., Lastra, A., Marcellán, F.: Higher-order recurrence relations, Sobolev-type inner products and matrix factorizations. Numer. Algorithms 92(1), 665–692 (2023). https://doi.org/10.1007/s11075-022-01402-y

    Article  MathSciNet  MATH  Google Scholar 

  34. Higham, N.J.: Functions of Matrices: Theory and Computation. SIAM, Philiadelphia (2008)

    Book  MATH  Google Scholar 

  35. Chu, M.T., Golub, G.H.: Structured inverse eigenvalue problems. Acta Numer. 11, 1–71 (2002). https://doi.org/10.1017/S0962492902000016

    Article  MathSciNet  MATH  Google Scholar 

  36. Ammar, G., Gragg, W., Reichel, L.: Constructing a unitary Hessenberg matrix from spectral data. In: Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms (Leuven 1988). NATO Adv. Sci. Inst. Ser. F: Comput. Systems Sci., vol. 70, pp. 385–395. Springer, Berlin (1991)

  37. Van Buggenhout, N.: Structured matrix techniques for orthogonal rational functions and rational Krylov methods. Ph.D. thesis, KU Leuven (2021)

  38. Rutishauser, H.: Der Quotienten-Differenzen-Algorithmus. Mitt. Inst. Angew. Math. Zürich 1957(7), 74 pp. (1957). In German

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

    Article  MathSciNet  MATH  Google Scholar 

  40. Arnoldi, W.E.: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quart. Appl. Math. 9, 17–29 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  41. Van Buggenhout, N.: SOP. GitHub (2022). https://github.com/nielvb/SOP

  42. Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Nat. Bur. Standards 45(4), 255–282 (1950)

    Article  MathSciNet  Google Scholar 

  43. Pozza, S., Pranić, M.S., Strakoš, Z.: The Lanczos algorithm and complex Gauss quadrature. Electron. Trans. Numer. Anal. 50, 1–19 (2018). https://doi.org/10.1553/etna_vol50s1

    Article  MathSciNet  MATH  Google Scholar 

  44. Kailath, T., Sayed, A.H.: Fast Reliable Algorithms for Matrices with Structure. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999)

    Book  MATH  Google Scholar 

  45. Parlett, B.N., Reinsch, C.: Balancing a matrix for calculation of eigenvalues and eigenvectors. Numer. Math. 13(4), 293–304 (1969). https://doi.org/10.1007/BF02165404

    Article  MathSciNet  MATH  Google Scholar 

  46. Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Monographs on Numerical Analysis. The Clarendon Press, Oxford University Press, New York (1988). 1st ed. published 1963

  47. Toh, K.-C., Trefethen, L.N.: Pseudozeros of polynomials and pseudospectra of companion matrices. Numer. Math. 68(3), 403–425 (1994). https://doi.org/10.1007/s002110050069

    Article  MathSciNet  MATH  Google Scholar 

  48. Day, D., Romero, L.: Roots of polynomials expressed in terms of orthogonal polynomials. SIAM J. Numer. Anal. 43(5), 1969–1987 (2005). https://doi.org/10.1137/040609847

    Article  MathSciNet  MATH  Google Scholar 

  49. Nakatsukasa, Y., Noferini, V.: On the stability of computing polynomial roots via confederate linearizations. Math. Comput. 85(301), 2391–2425 (2016). https://doi.org/10.1090/mcom3049

    Article  MathSciNet  MATH  Google Scholar 

  50. Faber, V., Manteuffel, T.: Necessary and sufficient conditions for the existence of a conjugate gradient method. SIAM J. Numer. Anal. 21(2), 352–362 (1984). https://doi.org/10.1137/0721026

    Article  MathSciNet  MATH  Google Scholar 

  51. Barth, T., Manteuffel, T.: Multiple recursion conjugate gradient algorithms. I. Sufficient conditions. SIAM J. Matrix Anal. Appl. 21(3), 768–796 (2000). https://doi.org/10.1137/S089547989833245X

  52. Evans, W.D., Littlejohn, L.L., Marcellán, F., Markett, C., Ronveaux, A.: On recurrence relations for Sobolev orthogonal polynomials. SIAM J. Math. Anal. 26(2), 446–467 (1995). https://doi.org/10.1137/S0036141092226922

    Article  MathSciNet  MATH  Google Scholar 

  53. Kressner, D.: Numerical Methods for General and Structured Eigenvalue Problems. Lecture Notes in Computational Science and Engineering, vol. 46. Springer, Berlin (2005)

  54. Olver, S., Slevinsky, R.M., Townsend, A.: Fast algorithms using orthogonal polynomials. Acta Numer. 29, 573–699 (2020). https://doi.org/10.1017/S0962492920000045

    Article  MathSciNet  MATH  Google Scholar 

  55. Laudadio, T., Mastronardi, N., Van Dooren, P.: On computing modified moments for half-range Hermite weights. Numer. Algorithms (2023). https://doi.org/10.1007/s11075-023-01615-9

    Article  MATH  Google Scholar 

  56. Zhang, M.: Orthogonal polynomials in Sobolev spaces: Computational method. Ph.D. thesis, Purdue University (1993)

  57. Zhang, M.: Sensitivity analysis for computing orthogonal polynomials of Sobolev type. In: Approximation and Computation (West Lafayette, IN, 1993). International Series of Numerical Mathematics, vol. 119, pp. 563–576. Birkhäuser Boston, Boston, MA (1994). https://doi.org/10.1007/978-1-4684-7415-2_38

  58. Meijer, H.G., de Bruin, M.G.: Zeros of Sobolev orthogonal polynomials following from coherent pairs. J. Comput. Appl. Math. 139(2), 253–274 (2002). https://doi.org/10.1016/S0377-0427(01)00421-6

    Article  MathSciNet  MATH  Google Scholar 

  59. Knizhnerman, L.: Sensitivity of the Lanczos recurrence to Gaussian quadrature data: How malignant can small weights be? J. Comput. Appl. Math. 233(5), 1238–1244 (2010). https://doi.org/10.1016/j.cam.2007.12.028

    Article  MathSciNet  MATH  Google Scholar 

  60. Durán, A.J., Van Assche, W.: Orthogonal matrix polynomials and higher-order recurrence relations. Linear Algebra Appl. 219, 261–280 (1995). https://doi.org/10.1016/0024-3795(93)00218-O

    Article  MathSciNet  MATH  Google Scholar 

Download references

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.

Funding

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

Author information

Authors and Affiliations

Authors

Contributions

Not applicable (single-authored paper).

Corresponding author

Correspondence to Niel Van Buggenhout.

Ethics declarations

Conflict of interest

The author declares that there is no conflict of interest nor competing interests.

Code and data availability

All code and numerical experiments are made publicly available https://github.com/nielvb/SOP.

Ethics approval

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-023-01379-3

Mathematics Subject Classification

Navigation