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

Lanczos methods for solving $\textit{A}\textbf{x}=\textbf{b}$ consist in constructing a sequence of vectors $(\textbf{x}_k), k=1,...$ such that $\textbf{r}_{k}=\textbf{b}-\textit{A}\textbf{x}_{k}=\textit{P}_{k}(\textit{A})\textbf{r}_{0}$,, where $\textit{P}_{k}$ is the orthogonal polynomial of degree at most $k$ with respect to the linear functional $c$ defined as $c(ξ^i)=(\textbf{y},\textit{A}^i\textbf{r}_{0})$. Let $\textit{P}^{(1)}_{k}$ be the regular monic polynomial of degree $k$ belonging to the family of formal orthogonal polynomials (FOP) with respect to $c^{(1)}$ defined as $c^{(1)}(ξ^{i})=c(ξ^{i+1})$. All Lanczos-type algorithms are characterized by the choice of one or two recurrence relationships, one for $\textit{P}_{k}$ and one for $\textit{P}^{(1)}_{k}$. We shall study some new recurrence relations involving $\textit{P}_{k}$ and $\textit{P}^{(1)}_{k}$ 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.