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

Lanczos methods for solving Ax = b consist in constructing a sequence of vectors (X<inf>k</inf>),k = 1,... such that r<inf>k</inf> = b-AX<inf>k</inf>= P<inf>k</inf>(A)r<inf>0</inf>, where P<inf>k</inf> is the orthogonal polynomial of degree at most k with respect to the linear functional c defined as c(εⁱ) = (y, Aⁱr<inf>0</inf>). Let P⁽¹⁾<inf>k</inf> 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⁾(εⁱ) = c(εⁱ⁺¹). All Lanczos-type algorithms are characterized by the choice of one or two recurrence relationships, one for P<inf>k</inf> and one for P⁽¹⁾<inf>k</inf>. 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.