Rational Krylov methods

Most of the iterative methods for computing eigenvalues of structured/sparse matrices are based on the so-called Krylov subspaces. Given a matrix $A$ whose eigenvalues one would like to compute and an initial vector $\mathbf{v}$ we have the following Krylov subspace

$$\mathcal{K}^j(A,\mathbf{v}) = \{ \mathbf{v}, A\mathbf{v}, \ldots A^{j-1}\mathbf{v}\}.$$

The idea of using rational functions in $A$ instead of the standard powers of $A$ can be found in [10]. The idea is to work with the following Krylov sequence, in which all $\varphi_i(\lambda)$ are rational functions in $\lambda$:

$$\mathcal{F}^j(A,\mathbf{v}) = \{ \varphi_1(A)\mathbf{v}, \varphi_2(A)\mathbf{v}, \ldots \varphi_3(A)\mathbf{v}\}.$$

The idea in Rational Krylov methods is to choose the functions in an intelligent way, to speed up the convergence of the iterative methods.

Similarly one chooses a good shift to speed up convergence in case of the $QR$-method. In this manuscript we will develop a technique to perform rational $QR$-steps onto matrices, in order to speed up the convergence towards the eigenvalues.

How rational Krylov methods work can be found in, e.g., [10,11,12]

To conclude, we would like to mention the following. When considering a tridiagonal matrix $T$ and the standard Krylov subspace, it is well-known that there is a relation between the $QR$-factorization of the matrix $T$ and this Krylov sequence (see [4]). Similarly there is a relation between rational Krylov sequences and semiseparable plus diagonal matrices (see [13]).

The interested reader can find more information on rational Krylov methods in the references above.