The $LZ$-Algorithm to Solve the Generalized Eigenvalue Problem

In this paper, we will present and analyze an algorithm for finding x and λ such that \[ A{ x} = λ B{ x},\] where A and B are n × n matrices. The algorithm does not require matrix inversion, and may be used when either or both matrices are singular. Our method is a generalization of Rutishauser’s $LR$-method [20] for the standard eigenvalue problem A x = λ x and closely resembles the $QZ$-algorithm given by Moler and Stewart [13] for the generalized problem given above. Unlike the $QZ$-algorithm, which uses orthogonal transformations, our method, the $LZ$-algorithm, uses elementary transformations. When either A or B is complex, our method should be more efficient.