Construction and implementation of asymptotic expansions for Jacobi-type orthogonal polynomials

We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree n goes to ∞. These are defined on the interval [−1, 1] with weight function: w(x)=(1−x)α(1+x)βh(x),α,β>−1 and h(x) a real, analytic and strictly positive function on [−1, 1]. This information is available in the work of Kuijlaars et al. (Adv. Math. 188, 337–398 2004), where the authors use the Riemann–Hilbert formulation and the Deift–Zhou non-linear steepest descent method. We show that computing higher-order terms can be simplified, leading to their efficient construction. The resulting asymptotic expansions in every region of the complex plane are implemented both symbolically and numerically, and the code is made publicly available. The main advantage of these expansions is that they lead to increasing accuracy for increasing degree of the polynomials, at a computational cost that is actually independent of the degree. In contrast, the typical use of the recurrence relation for orthogonal polynomials in computations leads to a cost that is at least linear in the degree. Furthermore, the expansions may be used to compute Gaussian quadrature rules in O(n) operations, rather than O(n2) based on the recurrence relation

Asymptotic computation of classical orthogonal polynomials

The classical orthogonal polynomials (Hermite, Laguerre and Jacobi) are involved in a vast number of applications in physics and engineering. When large degrees n are needed, the use of recursion to compute the polynomials is not a good strategy for computation and a more efficient approach, such as the use of asymptotic expansions,is recommended. In this paper, we give an overview of the asymptotic expansions considered in [8] for computing Laguerre polynomials L(α)n(x) for bounded values of the parameter α. Additionally, we show examples of the computational performance of an asymptotic expansion for L(α)n(x) valid for large values of α and n. This expansion was used in [6] as starting point for obtaining asymptotic approximations to the zeros. Finally, we analyze the expansions considered in [9], [10] and [11] to compute the Jacobi polynomials for large degrees n

Construction and implementation of asymptotic expansions for Jacobi--type orthogonal polynomials

We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree n goes to ∞. These are defined on the interval [-1, 1] with weight function w(x) = (1-x) α (1+x) β h(x), α,β > -1 and h(x) a real, analytic and strictly positive function on [-1, 1]. This information is available in the work of Kuijlaars, McLaughlin, Van Assche and Vanlessen, where the authors use the Riemann-Hilbert formulation and the Deift-Zhou non-linear steepest descent method. We show that computing higher order terms can be simplified, leading to their efficient construction. The resulting asymptotic expansions in every region of the complex plane are implemented both symbolically and numerically, and the code is made publicly available.nrpages: 36status: publishe