SUPEREXPONENTIALLY CONVERGENT ALGORITHM FOR AN ABSTRACT EIGENVALUE PROBLEM WITH APPLICATIONS TO ORDINARY DIFFERENTIAL EQUATIONS

摘要

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 A = A + B such that the eigenvalue problem for A 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 .4 is emphasized. Examples of eigenvalue problems for the high-order ordinary differential operators are presented to support the theory.