A “Look-around Lanczos” algorithm for solving a system of linear equations

Two algorithms for the solution of a large sparse linear system of equations are proposed. The first is a modification of Lanczos' method and the second is based on one of Brezinski's methods. Both the latter methods are iterative and they can break down. In practical situations, serious numerical error is far more likely to occur because an ill-conditioned pair of polynomials is (implicitly) used in the calculation rather than complete breakdown arising because a large square block of exactly defective polynomials is encountered. The algorithms proposed use a method based on selecting well-conditioned pairs of neighbouring polynomials (in the associated Padé table), and the method is equivalent to going round the blocks instead of going across them, as is done in the well-known look-ahead methods.