A fast divide-and-conquer algorithm for computing the spectra of real symmetric tridiagonal matrices

Abstract

We propose a new fast algorithm for computing the spectrum of an N×N symmetric tridiagonal matrix in O(NlnN) operations. Such an algorithm may be combined with any of the existing methods for the determination of eigenvectors of a symmetric tridiagonal matrix with known eigenvalues. The underlying technique is a divide-and-conquer approach which determines eigenvalues of a larger tridiagonal matrix from those of constituent matrices by the use of relations of their characteristic polynomials. The evaluation of characteristic polynomials is accelerated by the use of a technique known as the fast multipole method. An implementation of the algorithm has been developed in Fortran, providing for a comparison with existing techniques in terms of running time and accuracy. We present numerical results which demonstrate the effectiveness of the method.