An Efficient Implementation of Non Symmetric Lanczos Algorithm,
CALIFORNIA UNIV BERKELEY CENTER FOR PURE AND APPLIED MATHEMATICS
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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. 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 justifies 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 benefit 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 significantly more efficient than Arnoldis method.
- Numerical Mathematics
- Operations Research