LANCZOS-TYPE ALGORITHMS FOR SOLVING SYSTEMS OF LINEAR-EQUATIONS

In this paper, a synthesis of the various Lanczos-type algorithms for solving systems of linear equations is given. It is based on formal or thogonal polynomials and the various algorithms consist in using vario us recurrence relations for computing these orthogonal polynomials. Mo reover new algorithms are easily obtained from the theory. New formula e and a new interpretation of the conjugate gradient squared (CGS) alg orithm are also derived and a new formula for the second topological e psilon-algorithm. The theory of orthogonal polynomials also enables us to avoid breakdown in Lanczos-type methods and in the CGS. The case o f near-breakdown can be treated similarly.