ABSTRACT
The shooting methods can be used in principle to find all the eigenvalues and eigenfunctions of the Sturm-Liouville problems. Two important features of the methods are: No roundoff errors accumulate when several eigenvalues and eigenfunctions are determined numerically because each eigenvalue and eigenfunction is found independently of the others. The methods handle both regular and singular problems with equal ease. A shooting method for solving eigenvalue problems is similar in spirit but leads to a nonlinear equation that must be solved by a root finding method, typically Newton's method or the bisection method. There is a natural conceptual and effective computational approach for finding the eigenvalues and eigenfunctions of a Sturm-Liouville problem via initial value problems and a shooting method. However, practical implementation of the method requires some educated guesswork to find a suitable pair of starting values for the bisection method or a suitable initial guess for Newton's method.
