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

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.