The eigenvalues of a regular Sturm-Liouville system on with periodic and semi-periodic boundary conditions satisfy an equation (i) , where , are basic solutions of the differential equation satisfying initial conditions , , , .
The author applies the shooting method to generate a sequence , of numbers and functions , , , such that , converge to eigenvalues of the periodic and semi-periodic boundary value problems respectively. Starting values for , are based on the known behavior of .
The new algorithm is applied to three test problems, one of which is the Mathieu equation. Calculations made on a CYBER 860 computer for the first ten pairs of eigenvalues are accurate to .