BI-LANCZOS WITH PARTIAL ORTHOGONALIZATION

In theory the bi-Lanczos algorithm is an attractive possibility for so lving 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 b i-Lanczos iteration vectors. To save part of the advantage of the bi-L anczos 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 ort hogonality is lost, so that the results are accurate enough, whereas t he algorithm is still less time consuming than Arnoldi. The theory pre sented here is an extension of the theory available for partial reorth ogonalization of the symmetric Lanczos algorithm.