Tridiagonal matrix algorithm

Abstract: 1. Introduction. In recent years, a number of methods have been proposed for finding the eigenvalues of real, symmetric matrices. The methods of Lanczos [8], Givens [3], and Householder [5, 14] reduce the original matrix to a tridiagonal matrix whose eigenvalues are the same as those of the original matrix. The problem reduces then to finding the eigenvalues of a tridiagonal form. Givens has suggested the use of Sturm sequences, and others have used Muller's method [9]. In this paper, Rutishauser's LR algorithm and its variants [10, 11] will be considered for tridiagonal symmetric matrices. Henrici [4] has shown that for this case the LR algorithm is equivalent to the QD algorithm. It will be shown that during the iteration procedure, it is possible to determine bounds on the eigen-values. Wilkinson [12] has recently considered the problem of determining rigorous error bounds after the eigensystem has been computed.