A numerical study of the Legendre-Galerkin method for the evaluation of the prolate spheroidal wave functions

An $\mathcal{O}\left(1\right)$ algorithm for the numerical evaluation of the Sturm-Liouville eigenvalues of the spheroidal wave functions of order zero

“…The authors of [14], for instance, take it to be 1000 `n `t1.1γu in their implementation. The experiments of [15], however, suggest that the necessary dimension grows as O `n `?nγ ˘. It is difficult to find a simple formula which suffices in all cases of interest.…”

“…The Osipov-Xiao-Rokhlin method [20,14], which is the standard approach to the numerical calculation of Ps n pz; γq and the corresponding Sturm-Liouville eigenvalue χ n pγq, operates by representing a solution of (3) as a finite Legendre expansion. While the dependence of its running time on the parameters γ and n is not fully understood, the numerical experiments of [15] suggest that it grows as O `n `?nγ ˘, at least for large values of n and γ.…”

A Review of Prolate Spheroidal Wave Functions from the Perspective of Spectral Methods

“…Boyd [8] provided the algorithms and Matlab codes for computing the PSWFs, eigenvalues and their zeros etc.. The recent work [73] had deeper insights into the truncation of the infinite eigensystem where the Legendre-Galerkin method was applied to solve the Sturm-Liouville problem (2.21). One can refer to [66] for some delicate algorithms to enhance the codes in [8] for certain range of the parameters.…”

“…Schmutzhard et al [73] provided insightful observations on the choice of M for a prescribed accuracy. We next present some stable formulas for computing the eigenvalues {λ n (c)} of the integral operator (see [92]), where we note that the magnitude of λ n (c) with c > 0 is exponentially small for large n. Taking x = 0 in (2.13), and using (3.1) and the property…”

“…(i) Study of their analytic and asymptotic properties, numerical evaluations, prolate quadrature, interpolation and related issues (see, e.g., [4,6,8,25,37,40,43,60,66,70,73,91,[99][100][101]);…”