A19/B6: A new Lanczos-type algorithm and its implementation

Lanczos-type algorithms are mostly derived using recurrence relationships between formal orthogonal polynomials. Various recurrence relations between these polynomials can be used for this purpose. In this paper, we discuss recurrence relations A 19 and B 6 for the choice Ui ( x ) = P(1)i(x), where Ui is an auxiliary family of polynomials of exact degree i. This leads to new Lanczos-type algorithm A19=B6 that shows superior stability when compared to existing algorithms of the same type. This new algorithm is derived and described here. Computational results obtained with it are compared to those of the most robust algorithms of this type namely A12, A new 12 A5=B10 and A8=B10 on the same test problems. These results are included.