On a correction of Numerov-like eigenvalue approximations for Sturm-Liouville problems

For computing approximations for the eigenvalues of Sturm-Liouville problems, Numerov's method and other finite-difference methods are often used, giving rise to an algebraic eigenvalue problem of which the smallest eigenvalues approximate the fundamental eigenvalue and the first few harmonics of the original problem. Recently the authors have derived a modified Numerov method with step-dependent coefficients, which integrates exactly the functions 1, x, x2, x3, sin kx and cos kx. The parameter k is calculated by minimizing the error term associated to the above modified multistep method equation. We shall show that this modified Numerov method delivers much more accurate eigenvalues than the classical Numerov method which is based on a same grid pattern. Several numerical experiments will be presented.For computing approximations for the eigenvalues of Sturm-Liouville problems, Numerov's method and other finite-difference methods are often used, giving rise to an algebraic eigenvalue problem of which the smallest eigenvalues approximate the fundamental eigenvalue and the first few harmonics of the original problem. Recently the authors have derived a modified Numerov method with step-dependent coefficients, which integrates exactly the functions 1, x, x2, x3, sin kx and cos kx. The parameter k is calculated by minimizing the error term associated to the above modified multistep method equation. We shall show that this modified Numerov method delivers much more accurate eigenvalues than the classical Numerov method which is based on a same grid pattern. Several numerical experiments will be presented.A