Bi-Lanczos with partial orthogonalization

Abstract

In theory the bi-Lanczos algorithm is an attractive possibility for solving the unsymmetric eigenproblem. In practice, however, it turns out to be unstable due to loss of orthogonality of the iteration vectors. In this paper we discuss the possibilities of reorthogonalizing the bi-Lanczos iteration vectors. To save part of the advantage of the bi-Lanczos method over the other most commonly used Krylov subspace based method of Arnoldi (bi-Lanczos lacks the growth of the number of vector operations per iteration), we propose partial reorthogonalization. In that case reorthogonalization takes place if and only if too much orthogonality is lost, so that the results are accurate enough, whereas the algorithm is still less time consuming than Arnoldi. The theory presented here is an extension of the theory available for partial reorthogonalization of the symmetric Lanczos algorithm.