A Lanczos-subspace method for generalized eigenproblems

Abstract

The Lanczos algorithm produces partial eigensolutions at the two extreme ends of the eigenspectrum. The inverse Lanczos algorithm with shift s̀ produces partial eigensolutions nearest to the shift. It generates iteratively one Lanczos vector at a time and the elements of a tridiagonal matrix whose eigenvalues approximate the required eigenvalues. The Lanczos vectors are orthogonal and span the same subspace of the required eigenvectors theoretically. Orthogonality of the Lanczos vectors is lost during iteration owing to the attraction of the round-off errors by some initially found Lanczos vectors, and reorthogonalization is required. Full reorthogonalization by the Gram-Schmidt process is expensive and unstable. An economical and stable method of reorthogonalization by subspace iteration is introduced. The Lanczos vectors without reorthogonalization are taken as initial trial vectors in the subspace method to produce the required eigensolutions. The operation counts of the combined method compare favorably with the Lanczos method with full reorthogonalization for the same accuracy of solutions. Application of the Sturm sequence check is recommended to make sure that no solutions are missed in the required spectrum.