A Lanczos-type method for multiple starting vectors

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Abstract
Given a square matrix and single right and left starting vectors, the classical nonsymmetric Lanczos process generates two sequences of biorthogonal basis vectors for the right and left Krylov subspaces induced by the given matrix and vectors. In this paper, we propose a Lanczos-type algorithm that extends the classical Lanczos process for single starting vectors to multiple starting vectors. Given a square matrix and two blocks of right and left starting vectors, the algorithm generates two sequences of biorthogonal basis vectors for the right and left block Krylov subspaces induced by the given data. The algorithm can handle the most general case of right and left starting blocks of arbitrary sizes, while all previously proposed extensions of the Lanczos process are restricted to right and left starting blocks of identical sizes. Other features of our algorithm include a built-in deflation procedure to detect and delete linearly dependent vectors in the block Krylov sequences, and the option to employ look-ahead to remedy the potential breakdowns that may occur in nonsymmetric Lanczos-type methods.