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

Abstract

We conduct a numerical study of the Legendre-Galerkin method for the evaluation of the prolate spheroidal wave functions (PSWFs) viewed as the eigenfunctions of the prolate differential operator with boundary conditions of continuity. Our experiments indicate that the minimal dimension N of the Legendre-Galerkin matrix for the evaluation of the nth prolate with precision ? is O n + nc $\mathcal {O}( n + \sqrt {nc} ) $ , as n , c ? ? $n, c \rightarrow \infty $ , where c>0 is the bandwidth parameter. The behavior of N, when either c or n is held constant, is also examined. As a consequence, we obtain an upper bound on the complexity of the evaluation of the prolates. We also study the condition number of the approximate Legendre coefficients, computed as an eigenvector of the Legendre-Galerkin matrix. We observe experimentally that for fixed precision ?, an error estimate based on this condition number is O n + c $\mathcal {O}( n + c ) $ as n , c ? ? $n, c \rightarrow \infty $ . We conclude that the Legendre-Galerkin method is accurate for fairly large values of n and c.