In a shooting algorithm the solution of a boundary value problem is obtained by solving a set of related initial value problems. By applying a shooting algorithm to eigenvalue problems the eigenvalues are computed via numerical integration of the associated initial value problems with varying approximations for the eigenvalues. Shooting algorithms are one of the popular methods for Sturm-Liouville eigenvalue problems with separated boundary conditions.
The authors consider Sturm-Liouville eigenvalue problems with periodic coefficient functions and periodic boundary conditions. By applying some results of the well-known Floquet theory the original problem can be recasted into initial value problems. A shooting algorithm used in conjunction with a Newton method is then applied to solve the resulting initial value problems and hence the original eigenvalue problem.
The paper contains a detailed convergence analysis and also general guidelines to provide the starting eigenvalues (this is the main problem of the method, especially when no information on the eigenvalue distribution are known). The performance of the shooting algorithm is illustrated by computational results.