Iterative Methods Applied to Matrix Equations Found in Calculating Spheroidal Functions

We look at iterative methods for solving matrix equations, particularly those matrices with small entries. Iterative methods aid computational stability by relying on the topological structure of Banach or Hilbert spaces rather than depending on a calculation's numerical precision. When applicable, they are also quicker than Gaussian elimination. As an example, we use these methods to tabulate the expansion of periodic spheroidal functions in associated Legendre functions, given arbitrary values of the parameters appearing in its defining differential equation. These functions appear in solutions to 3-D Helmholtz equations in oblate and prolate spheroidal coordinates as well as a 1-D Schrödinger equation.