Low-Rank Modification of the Unsymmetric Lanczos Algorithm

The unsymmetric Lanczos algorithm is an important method for eigenvalue estimation and for solving linear equations. Unfortunately, the algorithm may break down without providing useful information; this is referred to as a serious breakdown in the literature. Here, we introduce a low-rank modification of the original matrix A in the case of a serious breakdown. This modification can be used to cure a serious breakdown as long as we have orthogonality of the already computed Lanczos vectors. We can switch to a new rank-1 modified matrix $\tilde A = A + a{b^T}$ such that - the Lanczos algorithm has no serious breakdown in this step when applied on $\tilde A$, - the already computed variables in the Lanczos algorithm for A and $\tilde A$ coincide, - using a Lanczos-based iterative solver, e.g. BCG or QMR, with start vectors ${x_0} = 0$ and ${v_1} = f$, we have ${A^{ - 1}}f = {\tilde A^{ - 1}}f$, and thus by continuing the Lanczos algorithm with $\tilde A$ we automatically get the desired solution ${A^{ - 1}}f$. Also, if the Lanczos vectors have lost their orthogonality, we show theoretically and by numerical examples that the modified Lanczos method has the same convergence behavior as the Lanczos method without breakdown. Thus, in the case of a serious breakdown we only have to compute the new rank-1 modified matrix $\tilde A$ and step further in the original algorithm now using $\tilde A$.