Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 
 
 

 

Explicit barycentric formulae for osculatory interpolation at roots of classical orthogonal polynomials


Authors: Przemysław Rutka and Ryszard Smarzewski
Journal: Math. Comp. 86 (2017), 2409-2427
MSC (2010): Primary 41A05, 33C45, 65D05
DOI: https://doi.org/10.1090/mcom/3184
Published electronically: February 15, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we extend the recent results of H. Wang et al. [Math. Comp. 81 (2012) and 83 (2014), pp. 861-877 and 2893-2914, respectively], on barycentric Lagrange interpolation at the roots of Hermite, Laguerre and Jacobi orthogonal polynomials, not only to all classical distributions, but also to osculatory Fejér and Hermite interpolation at the roots $ \left (x_{\nu }\right )_{1}^{n}$ of orthogonal polynomials $ p_{n}\left (x\right )$, generated by these distributions. More precisely, we present comparatively simple unified proofs of representations for barycentric weights of Fejér, Hermite and Lagrange type in terms of values $ p_{n-1}\left (x_{\nu }\right )$, $ p_{n}'\left ( x_{\nu }\right )$ and Christoffel numbers $ \lambda _{\nu }$ without any additional assumptions on the classical distributions. The first two representations enable us to design a general $ O\left (n^{2}\right )$-algorithm to simultaneous computations of barycentric weights and Christoffel numbers, which is based on the stable and efficient divide-and-conquer $ O\left (n^{2}\right )$-algorithm for the symmetric tridiagonal eigenproblem due to M. Gu and S. C. Eisenstat [SIAM J. Matrix Anal. Appl. 16 (1995), pp. 172-191]. On the other hand, the third representations can be used to compute all classical barycentric weights in the faster $ O\left ( n\right ) $ way proposed for the Lagrange interpolation at the roots of Hermite, Laguerre and Jacobi orthogonal polynomials by H. Wang et al. in the second cited paper. Such an essential accelaration requires one to use the $ O\left ( n\right ) $-algorithm of A. Glaser et al. [SIAM J. Sci. Comput. 29 (2007), pp. 1420-1438] to compute the roots $ x_{\nu }$ and Christoffel numbers $ \lambda _{\nu }$ by applying the Runge-Kutta and Newton methods to solve the Sturm-Liouville differential problem, which is generic for classical orthogonal polynomials. Finally, in the four special important cases of Jacobi weights $ w\left ( x\right ) =\left ( 1-x\right )^{\alpha }\left ( 1+x\right ) ^{\beta }$ with $ \alpha =\pm \frac {1}{2}$ and $ \beta =\pm \frac {1}{2}$, that is, of the Chebyshev and Szegő weights of the first and second kind, we present explicit representations of the Fejér and Hermite barycentric weights, which yield an $ O\left ( 1\right )$-algorithm.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 41A05, 33C45, 65D05

Retrieve articles in all journals with MSC (2010): 41A05, 33C45, 65D05


Additional Information

Przemysław Rutka
Affiliation: Institute of Mathematics and Computer Science, The John Paul II Catholic University of Lublin, ul. Konstantynów 1H, 20-708 Lublin, Poland
Email: rootus@kul.pl

Ryszard Smarzewski
Affiliation: Institute of Mathematics and Computer Science, The John Paul II Catholic University of Lublin, ul. Konstantynów 1H, 20-708 Lublin, Poland
Email: rsmax@kul.pl

DOI: https://doi.org/10.1090/mcom/3184
Keywords: Pearson and Sturm-Liouville differential equations, classical orthogonal polynomials, interpolation of Fej\'er, Hermite and Lagrange type, barycentric weights, Christoffel numbers, tridiagonal symmetric eigenvalue problems, fast barycentric algorithms.
Received by editor(s): September 25, 2015
Received by editor(s) in revised form: April 1, 2016
Published electronically: February 15, 2017
Article copyright: © Copyright 2017 American Mathematical Society