Superexponentially Convergent Algorithm for an Abstract Eigenvalue Problem with Applications to Ordinary Differential Equations

Abstract

A new algorithm for the solution of eigenvalue problems for linear operators of the form A = A + B (with a special application to high-order ordinary differential equations) is proposed and justified. The algorithm is based on the approximation of A by an operator \( \overline{A}=A+\overline{B} \) such that the eigenvalue problem for Ā is supposed to be simpler than for A: The algorithm for this eigenvalue problem is based on the homotopy idea and, for a given eigenpair number, recursively computes a sequence of approximate eigenpairs that converges to the exact eigenpair with a superexponential convergence rate. The eigenpairs can be computed in parallel for all prescribed indexes. The case of multiple eigenvalues of the operator Ā is emphasized. Examples of eigenvalue problems for the high-order ordinary differential operators are presented to support the theory.