An iterative algorithm for calculation of eigenvalues and eigenfunctions of the Sturm--Liouville problem

Abstract

An iterative algorithm for computing the $i$-th eigenvalue (e. v.) and the corresponding eigenfunction (e. f.) of the Sturm--Liouville problem on a finite interval is proposed. The algorithm uses the well-known asymptotic formulas for e. v and e. f. of the Sturm--Liouville problem. Each iteration of the algorithm requires the solution of the boundary value problem for a second-order differential equation. The left-hand side of this equation is the differential operator of the left-hand side of the Sturm--Liouville equation with some shift, and the right-hand side is an approximation to the desired e. f. An example is given in which the boundary value problem was solved by the finite elements method with trigonometric hat functions, defined on a uniform mesh. In this example, the proposed algorithm actually reduces to an iterative algorithm for determining the $i$-th e. v. of a finite-element approximation of the Sturm--Liouville problem, which is a generalized matrix problem on an eigenvalue, only the $ i $-th e. v. of which approximates e. v. of the original problem.