Variations on Lanczos' tridiagonalization process

Lanczos' tridiagonalization processes transform a matrix into an equivalent tridiagonal one. In this paper, we propose several ways of implementing these procedures. They are based on different choices of the auxiliary polynomials which appear in the underlying theory of formal orthogonal polynomials and on a change in their normalization. We also give transpose-free variants of Lanczos processes in the spirit of the CGS and BiCGStab algorithms for solving a system of linear equations. Numerical experiments show that some of the variants proposed have a better numerical behavior than the original algorithms.