Abstract
In this paper, an approach to orthogonal polynomials based on matrix calculus is proposed. Known and new basic results are given, such as recurrence relations and determinant forms. New algorithms, similar, but not identical, to the Chebyshev one, for practical calculation of the polynomials are presented. The cases of monic and symmetric orthogonal polynomial sequences and the case of orthonormal polynomial sequences have been considered. Some classical and non-classical examples are given. The work is framed in a broader perspective, already started by the authors. It provides for the determination of properties of a general sequence of polynomials and, therefore, their applicability to special classes of the most important polynomials.
Similar content being viewed by others
References
Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)
Askey, R.: Orthogonal Polynomials and Special Functions. Society for Industrial and Applied Mathematics, Philadelphia (1975)
Brezinski, C.: Padé-type Approximation and General Orthogonal Polynomials. Springer, Berlin (1980)
Brezinski, C.: Computational Aspects of Linear Control, vol. 1. Springer Science and Business Media, New York (2013)
Caira, R., Costabile, F.: Two steps methods of Runge-Kutta type for initial value problem \(y^{\prime \prime }=f(x, y)\). Rend. Mat. Appl. 7(6), 441–465 (1986)
Chebychev, M.: Sur les fractions continues. Journal de Mathématiques Pures et Appliquées 2e série 3, 289–323 (1858)
Chihara, T.S.: An Introduction to Orthogonal Polynomials. Courier Corporation (2011)
Costabile, F., Luceri, R.: On the order of a Runge-Kutta type method for the initial value problem. Rendiconti di Matematica e delle sue Applicazioni 6(4), 548–553 (1986)
Costabile, F.A.: Modern umbral calculus. An elementary introduction with applications to linear interpolation and operator approximation theory, De Gruyter Studies in Mathematics, vol. 72. De Gruyter (2019)
Costabile, F.A., Caira, R., Gualtieri, M.I.: A block hybrid method for non-linear second order boundary value problems. Mediterr. J. Math.16(1), Art. 17 (2019)
Costabile, F.A., Gualtieri, M.I., Napoli, A.: Recurrence relations and determinant forms for general polynomial sequences. Application to Genocchi polynomials. Integr. Transforms Spec. Funct. 30(2), 112–127 (2019)
Costabile, F.A., Gualtieri, M.I., Napoli, A.: Polynomial sequences: elementary basic methods and application hints. A survey. RACSAM (2019). https://doi.org/10.1007/s13398-019-00682-9
Dominici, D.: Matrix factorizations and orthogonal polynomials. Random Matrices: Theory and Applications. Published on line 11 June (2019). https://doi.org/10.1142/S2010326320400031
Draux, A.: Polynômes orthogonaux formels-applications, vol. 974. Springer, Berlin (2006)
Freud, G.: Orthogonal Polynomials. Elsevier, Amsterdam (2014)
Garza, L.G., Garza, L.E., Marcellán, F., Pinzón-Cortés, N.C.: A matrix approach for the semiclassical and coherent orthogonal polynomials. Appl. Math. Comput. 256, 459–471 (2015)
Gautschi, W.: Computational aspects of three-term recurrence relations. SIAM Rev. 9(1), 24–82 (1967)
Gautschi, W.: On generating orthogonal polynomials. SIAM J. Sci. Stat. Comput. 3(3), 289–317 (1982)
Gautschi, W.: Orthogonal polynomials constructive theory and applications. J. Comput. Appl. Math. 12, 61–76 (1985)
Gautschi, W.: On the sensitivity of orthogonal polynomials to perturbations in the moments. Numer. Math. 48(4), 369–382 (1986)
Gautschi, W.: Computational aspects of orthogonal polynomials. In: Orthogonal Polynomials, pp. 181–216. Springer (1990)
Gautschi, W.: Algorithm 726: ORTHPOL-a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Softw. (TOMS) 20(1), 21–62 (1994)
Gautschi, W.: Orthogonal polynomials: applications and computation. Acta Numer. 5, 45–119 (1996)
Gautschi, W.: Orthogonal polynomials: computation and approximation. Numerical Mathematics and Scientific Computation. Oxford Science Publications. Oxford University Press, New York (2004)
Gautschi, W.: Orthogonal polynomials (in Matlab). J. Comput. Appl. Math. 178(1–2), 215–234 (2005)
Gautschi, W.: Orthogonal polynomials, quadrature, and approximation: computational methods and software (in Matlab). In: Orthogonal Polynomials and Special Functions, pp. 1–77. Lecture Notes in Mathematics, vol. 1883. Springer, Berlin (2006)
Gautschi, W., Golub, G., Opfer, G.: Applications and computation of orthogonal polynomials. In: Conference at the Mathematical Research Institute Oberwolfach, Germany March 22–28, 1998. International Series in Numerical Analysis, vol. 131 (1999)
Geronimus, Ya L.: Orthogonal Polynomials: Estimates, Asymptotic Formulas, and Series of Polynominals Orthogonal on the Unit Circle and on an Interval. Consultants Bureau, New York (1961)
Gragg, W.B.: Matrix interpretations and applications of the continued fraction algorithm. Rocky Mt. J. Math. 4(2), 213–225 (1974)
Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002)
Ismail, M.E.H.: Classical and quantum orthogonal polynomials in one variable. Encyclopedia of Mathematics and its Applications, vol. 98. Cambridge University Press, Cambridge (2005)
Koekoek, R., Lesky, P.A., Swarttouw, R.: Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer Science and Business Media, New York (2010)
Krall, A.M.: Hilbert space, boundary value problems and orthogonal polynomials. In: Operator Theory: Advances and Applications, vol. 133. Birkhauser Verlag, Basel (2002)
Legendre, A.M.: Sur l’attraction des spheroides. Mémoires de Mathématiques et de Physique, Presentés à l”Académie Royale des Sciences par divers savants, vol. 10 (1785)
Levin, E., Lubinsky, D.S.: Bounds and Asymptotics for Orthogonal Polynomials for Varying Weights. Springer, Berlin (2018)
Macdonald, I.G.: Symmetric functions and orthogonal polynomials, University Lecture Series, vol. 12. American Mathematical Society, Providence (1998)
Maroni, P.: Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques. Orthogonal Polynomials and their Applications (Erice, 1990), IMACS Ann. Comput. Appl. Math, vol. 9, pp. 95–130 (1991)
Milovanović, G.V.: Orthogonal polynomials on the real line. In: Walter Gautschi, vol. 2, pp. 3–16. Springer (2014)
Nikiforov, A.F., Uvarov, V.B., Suslov, S.K.: Classical Orthogonal Polynomials of a Discrete Variable. Translated from the Russian. Springer Series in Computational Physics. Springer, Berlin (1991)
Rivlin, T.J.: The Chebyshev Polynomials. Wiley, New York (1974)
Sack, R.A., Donovan, A.F.: An algorithm for Gaussian quadrature given modified moments. Numer. Math. 18(5), 465–478 (1971)
Schoutens, W.: Stochastic processes and orthogonal polynomials, Lecture Notes in Statistics, vol. 146. Springer Science and Business Media, New York (2012)
Stahl, H., Steel, J., Totik, V.: General orthogonal polynomials, Encyclopedia of Mathematics and its Applications, vol. 43. Cambridge University Press, Cambridge (1992)
Szegô, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Colloquium Publications, vol. 23, Providence (1975)
Van Assche, W.: Orthogonal polynomials and Painlevé equations. Australian Mathematical Society Lecture Series. Cambridge University Press, Cambridge (2017)
Verde-Star, L.: Characterization and construction of classical orthogonal polynomials using a matrix approach. Linear Algebra Appl. 438(9), 3635–3648 (2013)
Wheeler, J.C.: Modified moments and Gaussian quadratures. Rocky Mt. J. Math. 4(2), 287–296 (1974)
Acknowledgements
The authors would like to thank anonymous referees for their valuable comments. One of the authors wishes to thank the support of INdAM-GNCS project 2019.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Costabile, F.A., Gualtieri, M.I. & Napoli, A. Matrix Calculus-Based Approach to Orthogonal Polynomial Sequences. Mediterr. J. Math. 17, 118 (2020). https://doi.org/10.1007/s00009-020-01555-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-020-01555-x