Restarting the Nonsymmetric Lanczos Algorithm for Eigenvalues and Linear Equations Including Multiple Right-Hand Sides

A restarted nonsymmetric Lanczos algorithm is given for computing eigenvalues and both right and left eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Restarting also makes it possible to deal with roundoff error in new ways. We give a scheme for avoiding near-breakdown and discuss maintaining biorthogonality. A system of linear equations can be solved simultaneously with the eigenvalue computations. Deflation from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting. The right and left eigenvectors generated while solving the linear equations can be used to help solve systems with multiple right-hand sides.