By using the expansions of piecewise-linear polynomial functions, the Sturm-Liouville eigenvalue problem can be dealt with as a set of linear algebraic equations. Owing to the available recursive algorithm, these linear algebraic equations can be solved by straightforward substitution. An iterative improvement is used for finding eigenvalues in order to speed up the convergence. The usefulness of this method is demonstrated by an example and the results are satisfactory.