Coupled implementations of transpose-free Lanczos and Lanczos-type product methods

Algorithms for implementing Lanczos method usually require the transpose of the matrix of the system, sometimes a costly or impossible operation. Here we present a general way to avoid this drawback, based on the recursive computation of the products of polynomials appearing in the coefficients of the recurrences between formal orthogonal polynomials. Moreover, we can implement simultaneously a Lanczos--type product method. As an example, this technique will be applied to the Lanczos/Orthomin algorithm. A complete exposition of the algorithms could be found in [3].