Among the iterative methods for solving large linear systems with a sparse (or, possibly, structured) nonsymmetric matrix, those that are based on the Lanczos process feature short recurrences for the generation of the Krylov space. This means low cost and low memory requirement. This review article introduces the reader not only to the basic forms of the Lanczos process and some of the related theory, but also describes in detail a number of solvers that are based on it, including those that are considered to be the most efficient ones. Possible breakdowns of the algorithms and ways to cure them by look-ahead are also discussed.
Orthogonal s-step methods for nonsymmetric linear systems of equations