This thesis presents novel algorithms for the numerical evaluation of the discrete prolate spheroidal wave functions (DPSWFs) and their associated integral operator eigenvalues. The DPSWFs and associated eigenvalues arise in a variety of science and engineering applications including signal processing, communications technology, paleoclimatology, fluid dynamics, and wave phenomena.

Existing algorithms compute the integral operator eigenvalues to high relative accuracy when the eigenvalues are not close to zero. However, the integral operator eigenvalues computed by these algorithms lose all digits of relative accuracy when the eigenvalues are small. The new numerical algorithms compute the eigenvalues to high relative accuracy independent of their mangitude.

The proposed algorithms exploit the fact that the integral operator commutes with a second order linear differential operator. While this differential operator was identified in 1978, it has not been used in numerical algorithms to evaluate the DPSWFs nor the associated eigenvalues until this work. Numerical experiments demonstrate the accuracy of the proposed algorithms.

The design of the proposed algorithms exploits several properties of the DPSWFs to reduce computational cost. Furthermore, the use of high order numerical methods ensures that the algorithms are efficient. In addition to algorithms for computing the eigenvalues to high relative accuracy, several new properties of the DPSWFs are derived.