On Some Numerical Methods for Solving Large Differential Nonsymmetric Stein Matrix Equations

Abstract

In this paper, we propose a new numerical method based on the extended block Arnoldi algorithm for solving large-scale differential nonsymmetric Stein matrix equations with low-rank right-hand sides. This algorithm is based on projecting the initial problem on the extended block Krylov subspace to obtain a low-dimensional differential Stein matrix equation. The obtained reduced-order problem is solved by the backward differentiation formula (BDF) method or the Rosenbrock (Ros) method, the obtained solution is used to build the low-rank approximate solution of the original problem. We give some theoretical results and report some numerical experiments.

Keywords

• extended block Krylov
• low-rank approximate solutions
• differential Stein matrix equations
• BDF method
• Rosenbrock method