Matrix inverse-free algorithms for the general eigenvalue problem

Abstract

There are numerous applications in sciences, engineering and mathematics that give rise to problems involving the computation of orthogonal projections onto selective invariant subspaces of matrices. Conventional algorithms for subspace estimation based upon eigenvalue decomposition (EVD) or singular value decomposition (SVD) are, however, both expensive to compute, and difficult to make recursive or implement in parallel. In contrast, algorithms based on the QR factorization have established pipelinable architectures. In this paper, we introduce novel matrix-inverse free algorithms for block matrix decomposition. They involve a combination of Newton's method and the QR factorization. Some of these methods can be shown to have cubic or quadratic convergence rates, while others can be of any desirable convergent rate.