A Stepsize Variation Strategy for the Solution of Regular Sturm-Liouville Problems

In this note we show how a simple stepsize variation strategy improves the solution algorithm of regular Sturm-Liouville problems. We suppose the eigenvalue problem is approximated by variable stepsize finite difference schemes and the obtained algebraic eigenvalue problem is solved by a matrix method estimating the first eigenvalues and eigenvectors of sparse matrices. The variable stepsize strategy is based on an equidistribution of the error (approximated by two methods with different orders). The results show a marked reduction of the number of points and, consequently, a much lower computational cost, with respect to the algorithm obtained using constant stepsize.