An alternative derivation of a new Lanczos-type algorithm for systems of linear equations

Various recurrence relations between formal orthogonal polynomials can be used to derive Lanczos-type algorithms. In this paper, we consider recurrence relation A12 for the choice Ui(x) = Pi(x), where Ui is an auxiliary family of polynomials of exact degree i. It leads to a Lanczos-type algorithm that shows superior stability when compared to existing Lanczos-type algorithms. The new algorithm is derived and described. It is then computationally compared to the most robust algorithms of this type, namely A12, A5/B10 and A8/B10, on the same test problems. Numerical results are included