Short recurrences for computing extended Krylov bases for Hermitian and unitary matrices

Abstract

It is well known that the projection of a matrix $A$ onto a Krylov subspace span $\left\{ \mathbf {h}, A\mathbf {h}, A^2\mathbf {h}, \ldots , A^{k-1}\mathbf {h}\right\}$, with $A \in \mathbb {C}^{n \times n}$ and $\mathbf {h} \in \mathbb {C}^n$, results in a Hessenberg matrix. We show that the projection of the matrix $A$ onto an extended Krylov subspace, which is of the form span $\left\{ A^{-k_r}\mathbf {h}, \ldots , A^{-2}\mathbf {h},A^{-1}\mathbf {h}, \mathbf {h}, A\mathbf {h}, A^2 \mathbf {h}, \ldots , A^{k_\ell } \mathbf {h} \right\}$, is a matrix of so-called extended Hessenberg form which can be characterized uniquely by its $QR$-factorization. This $QR$-factorization will be presented by means of a pattern of $2 \times 2$ unitary rotations. We will show how this rotation pattern leads to new insights and allows to elegantly predict the structure of the matrix. In case $A$ is Hermitian or unitary, this extended Hessenberg matrix is banded and structured, allowing the design of short recurrence relations. For the unitary case, coupled two term recurrence relations are derived of which the coefficients capture all information necessary for a sparse factorization of the corresponding extended Hessenberg matrix.