A METHOD FOR DETERMINING EIGENSOLUTIONS OF LARGE, SPARSE, SYMMETRIC MATRICES BY THE PRECONDITIONED CONJUGATE GRADIENT METHOD IN THE GENERALIZED EIGENVALUE PROBLEM

Abstract

This paper presents a conjugate gradient method for the eigensolutions of large, sparse and symmetric matrices using Sylvester's law of inertia and the quadratic form. Since the proposed method is a PCG-based method, it is particularly effective when a sequence of either the smallest or largest eigensolutions of a large and sparse matrix are required. Moreover, the proposed method does not find unnecessary solutions, and it thus minimizes the required computational memory capacity. The proposed method is a method requiring only a relatively small computational time. The accuracy and stability of this method are confirmed by considering several numerical examples. The numerical results are of high accuracy even for the systems with multiple eigenvalues.