Journal of Prime Research in Mathematics

# $$A_19/B_6$$: A new lanczos-type algorithm and its implementation

Zakir Ullah
Department of Mathematics, University of Peshawar, Khyber Pakhtunkhwa, 25120, Pakistan.
$$^{1}$$Corresponding Author: zakirmath728@gmail.com
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 A19 and B6 for the choice $$U_i(x) = P_{i}^{(1)}$$, where $$U_i$$ is an auxiliary family of polynomials of exact degree $$i$$. This leads to new Lanczos-type algorithm $$A_19/B_6$$ 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 $$A_12$$,  (A^{new}_12\) $$A_5/B_{10}$$ and $$A_8/B_{10}$$ on the same test problems. These results are included.