Abstract New approach to systems of polynomial recursions is developed based on the Carleman linearization procedure. The article is divided into two main sec tions: firstly, we focus on the case of uni-variable depth-one polynomial re currences. Subsequently, the systems of depth-one polynomial recurrence re lations are discussed. The corresponding transition matrix is constructed and upper triangularized. Furthermore, the powers of the transition matrix are calculated using the back substitution procedure. The explicit expression for a solution to a broad family of recurrence relations is obtained. We investigate to which recurrences the framework can be applied and construct sufficient conditions for the method to work. It is shown how introduction of auxiliary variables can be used to reduce arbitrary depth systems to the depth-one sys tem of recurrences dealt with earlier. Finally, the limitations of the method are discussed, outlining possible directions for future research .

Abstract In this paper, we present a compact version of the Heart iteration. One that requires less matrix-vector products per iteration and attains faster convergence. The Heart iteration is a new type of Restarted Krylov methods for calculating peripheral eigenvalues of symmetric matrices. The new framework avoids the Lanczos tridiagonalization process and the use of implicit restarts.This simplifies the restarting mechanism and allows the introduction of several modifications. Convergence is assured by a monotonicity property that pushes the computed Ritz values toward their limits. Numerical experiments illustrate the usefulness of the proposed approach.