The lz-algorithm to solve the generalized eigenvalue problem.

TLDR: The method is a generalization of Rutishauser’s $LR$-method for the standard eigenvalue problem and closely resembles the $QZ$-algorithm given by Moler and Stewart for the generalized problem given above.

Abstract: In this paper, we will present and analyze an algorithm for finding ${\bf x}$ and $\lambda$ such that \[ A{\bf x} = \lambda B{\bf x},\] where A and B are $n \times n$ 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 $A{\bf x} = \lambda {\bf x}$ 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.