Semi-duality in the Two-sided Lanczos Algorithm

Lanczos vectors computed in nite precision arithmetic by the three-term recurrence tend to lose their mutual biorthogonality. One either accepts this loss and takes more steps or re-biorthogonalizes the Lanczos vectors at each step. For the symmetric case, there is a compromise approach. This compromise, known as maintaining semi-orthogonality, minimizes the cost of re-orthogonalization. This paper extends the compromise to the two-sided Lanczos algorithm, and justiies the new algorithm. The compromise is called maintaining semi-duality. An advantage of maintaining semi-duality is that the computed tridiagonal is a perturbation of a matrix that is exactly similar to the appropriate projection of the given matrix onto the computed subspaces. Another beneet is that the simple two-sided Gram-Schmidt procedure is a viable way to correct for loss of duality. Some numerical experiments show that our Lanczos code is signiicantly more eecient than Arnoldi's method.