The constant solution method for solving large-scale differential Sylvester matrix equations with time invariant coefficients.

This paper is mainly focused on the solution of Sylvester matrix differential equations with time-independent coefficients. We propose a new approach based on the construction of a particular constant solution which allows to construct an approximate solution of the differential equation from that of the corresponding algebraic equation. Moreover, when the matrix coefficients of the differential equation are large, we combine the constant solution approach with Krylov subspace methods for obtaining an approximate solution of the Sylvester algebraic equation, and thus form an approximate solution of the large-scale Sylvester matrix differential equation. We establish some theoretical results including error estimates and convergence as well as relations between the residuals of the differential and its corresponding algebraic Sylvester matrix equation. We also give explicit benchmark formulas for the solution of the differential equation. To illustrate the efficiency of the proposed approach, we perform numerous numerical tests and make various comparisons with other methods for solving Sylvester matrix differential equations. [ABSTRACT FROM AUTHOR]