The convergence of diagonal elements of an irreducible symmetric tridiagonal matrix under QL algorithm with some kinds of shift is discussed. It is proved that if α₁—σ→0 and βj→0, j = 1, 2, ···, m, ...

The two-sided Lanczos algorithm sometimes suffers from serious breakdowns. These occur when the associated moment matrix does not permit triangular factorization. We modify the algorithm slightly so ...