Journal of Prime Research in Mathematics

# New recurrence relationships between orthogonal polynomials which lead to new lanczos-type algorithms

$$^{1}$$Corresponding Author: mf arooq@upesh.edu.pk
Lanczos methods for solving $$Ax = b$$ consist in constructing a sequence of vectors $$(x_k)$$, $$k = 1, …$$ such that $$r_k = b − Ax_k = P_k(A)r_0$$, where $$P_k$$ is the orthogonal polynomial of degree at most k with respect to the linear functional c defined as c(ξ^i) = (y, A^ir_0)\). Let $$P^(1)_k$$ be the regular monic polynomial of degree k belonging to the family of formal orthogonal polynomials (FOP) with respect to $$c^(1)$$ defined as c^(1)(ξ ^{i}) = c^{(ξi+1)}\). All Lanczos-type algorithms are characterized by the choice of one or two recurrence relationships, one for $$P_k$$ and one for $$P^{(1)}_k$$. We shall study some new recurrence relations involving these two polynomials and their possible combinations to obtain new Lanczos-type algorithms. We will show that some recurrence relations exist, but cannot be used to derive Lanczos-type algorithms, while others do not exist at all.