Abstract
Brezinski, C. and H. Sadok, Lanczos-type algorithms for solving systems of linear equations, Applied Numerical Mathematics 11 (1993) 443–473. In this paper, a synthesis of the various Lanczos-type algorithms for solving systems of linear equations is given. It is based on formal orthogonal polynomials and the various algorithms consist in using various recurrence relations for computing these orthogonal polynomials. Moreover new algorithms are easily obtained from the theory. New formulae and a new interpretation of the conjugate gradient squared (CGS) algorithm are also derived and a new formula for the second topological ε-algorithm. The theory of orthogonal polynomials also enables us to avoid breakdown in Lanczos-type methods and in the CGS. The case of near-breakdown can be treated similarly.
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