New implementations of Lanczos method

Lanczos method for solving Ax = b consists in constructing the sequence of vectors (xk) such that rk = b − Axk = Pk(A)r0, where Pk is the orthogonal polynomial of degree at most k with respect to the linear functional c whose moments are c(ξi) = (y, Air0). Let Pk(1) de the regular monic polynomial of degree k belonging to the family of formal orthogonal polynomials with the respect to the functional c(1) defined by c(1)(ξi) = c(ξi + 1). The new algorithms are characterized by the choice of one or two recurrence relationships: one for Pk and one for Pk(1). We shall study all these formulae and all the possible combinations to obtain Lanczos algorithms. The implementation of these new algorithms is discussed. Numerical examples are given.