Construction and implementation of asymptotic expansions for Jacobi--type orthogonal polynomials · arXivDesk
1502.07191Feb 25, 201539 pages, 5 figures, 35 references. The article mentioned is arXiv:math/0111252 and the implementation is available on http://nines.cs.kuleuven.be/software/JACOBI/. The final publication is available at Springer via http://dx.doi.org/10.1007/s10444-015-9442-z
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)
Nearby in the stack
,
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. 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