A Breakdown-free Variation of the Nonsymmetric Lanczos Algorithms

The nonsymmetric Lanczos tridiagonalization algorithm is essentially the Gram-Schmidt biorthogonalization method for generating biorthogonal bases of a pair of Krylov subspaces. It suffers from breakdown and instability when a pivot at some step is zero or nearly zero, which is often the result of mismatch of the two Krylov subspaces. In this paper, we propose to modify one of the two Krylov subspaces by introducing a "new-start" vector when a pivot is small. This new-start vector generates another Krylov subspace, which we add to the old one in an appropriate way so that the Gram-Schmidt method for the modified subspaces yields a recurrence similar to the Lanczos algorithm. Our method enforces the pivots to be above a certain threshold and can handle both exact breakdown and near-breakdown. In particular, we recover lookahead Lanczos algorithms and Arnoldi's algorithm as two special cases. We also discuss theoretical and practical issues concerning the new-start procedure and present a convergence analysis as well as some numerical examples.