Gauss–Arnoldi quadrature for $\bigl\langle(zI-A)^{-1}\varphi,\varphi\bigr\rangle$ and rational Padé-type approximation for Markov-type functions

The efficiency of Gauss–Arnoldi quadrature for the calculation of the quantity $\bigl\langle(zI-A)^{-1}\varphi,\varphi\bigr\rangle$ is studied, where $A$ is a bounded operator in a Hilbert space and $\varphi$ is a non-trivial vector in this space. A necessary and a sufficient conditions are found for the efficiency of the quadrature in the case of a normal operator. An example of a non-normal operator for which this quadrature is inefficient is presented.
It is shown that Gauss–Arnoldi quadrature is related in certain cases to rational Padé-type approximation (with the poles at Ritz numbers) for functions of Markov type and, in particular, can be used for the localization of the poles of a rational perturbation. Error estimates are found, which can also be used when classical Padé approximation does not work or it may not be efficient.
Theoretical results and conjectures are illustrated by numerical experiments.
Bibliography: 44 titles.