New Recurrence Relationships between Orthogonal Polynomials which Lead to New Lanczos-type Algorithms

Abstract

Lanczos methods for solving $\text{A}\text{x}=\text{b}$ consist in constructing a sequence of vectors $({\text{x}}_k),k=1,...$ such that ${\text{r}}_k=\text{b}-\text{A}{\text{x}}_k={\text{P}}_k(\text{A}){\text{r}}_0$,, where ${\text{P}}_k$ is the orthogonal polynomial of degree at most $k$ with respect to the linear functional $c$ defined as $c({\xi }^i)=(\text{y},{\text{A}}^i{\text{r}}_0)$. Let ${\text{P}}_k^{(1)}$ be the regular monic polynomial of degree $k$ belonging to the family of formal orthogonal polynomials (FOP) with respect to $c^{(1)}$ defined as $c^{(1)}({\xi }^i)=c({\xi }^{i+1})$. All Lanczos-type algorithms are characterized by the choice of one or two recurrence relationships, one for ${\text{P}}_k$ and one for ${\text{P}}_k^{(1)}$. We shall study some new recurrence relations involving ${\text{P}}_k$ and ${\text{P}}_k^{(1)}$ and their possible combination 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.

Cited by in corpus (1)

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