A Lanczos-type method for multiple starting vectors

Given a square matrix and single right and left, starting vectors, the clas sical 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 multi ple starting vectors. Given a square matrix and two blocks of right and lef t starting vectors, the algorithm generates two sequences of biorthogonal b asis vectors for the right and left block Krylov subspaces induced by the g iven data. The algorithm can handle the most general case of right and left starting blocks of arbitrary sizes, while all previously proposed extensio ns of the Lanczos process are restricted to right and left starting blocks of identical sizes. Other features of our algorithm include a built-in defl ation procedure to detect and delete linearly dependent vectors in the bloc k Krylov sequences, and the option to employ look-ahead to remedy the poten tial breakdowns that may occur in nonsymmetric Lanczos-type methods.