An efficient implementation of the nonsymmetric Lanczos algorithm

Lanczos vectors computed in finite 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. Far the symmetric case, there is a compromise approach. This compromise, known as maintaining semiorthogonality, minimizes the cost of reorthogonalization. This paper extends the compromise to the true-sided Lanczos algorithm and justifies the new algorithm. The compromise is called maintaining semiduality. An advantage of maintaining semiduality 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 benefit is that the simple two-sided Gram-Schmidt procedure is a viable way to correct for loss of duality. A numerical experiment is included in which our Lanczos code is significantly more efficient than Arnoldi's method.