Boundary Value Methods as an extension of Numerovʹs method for Sturm–Liouville eigenvalue estimates

In this paper a class of Boundary Value Methods obtained as an extension of the Numerovʹs method is proposed for the numerical approximation of the eigenvalues of regular Sturm–Liouville problems subject to Dirichlet boundary conditions. It is proved that the error in the so obtained estimate of the kth eigenvalue behaves as O ( k p + 1 h p − 1 2 ) + O ( k p + 2 h p ) , where p is the order of accuracy of the method and h is the discretization stepsize. Numerical results comparing the performances of the new matrix methods with that of the corrected Numerovʹs method are also reported.