An Analysis of Lanczos Algorithms for Symmetric Matrices,

The Lanczos algorithm is presented as a way of generating bases for a sequence of Krylov subspaces. Explicit expressions are given for the departure of the bases from orthogonality. These relations enable one to comprehend the behavior of the algorithm in practice with a minimum of conventional error analysis. In particular this approach sheds light on the central, and difficult problem of ascertaining the right moment to stop the algorithm. Reorthogonalization and block versions are also examined.