A New Approach to Sparse Matrix Eigenvalues

Abstract

A new approach to the eigenvalue problem of large sparse symmetric matrices is developed in this paper. It is based on scaling and translating the matrix such that it can be regarded as the cosine of another symmetric positive definite matrix. FFT is used to obtain the different eigenvalues. To compute all the n eigenvalues and eigenvectors, where n is the size of the matrix, this method requires O(n) matrix vector multiplications. When only the eigenvalues are required, the memory space needed is on the order of n. This method provides a practical alternative to other iterative methods.