On the numerical solution ofAX −XB =C

In this note we analyze the numerical solution ofAX −XB =C when a Galerkin method is applied, assuming thatB has much smaller size thanA. We show that the corresponding Galerkin equation can be obtained from the truncation of the original problem, if matrix polynomials are used for writing the analytical solutionX. Moreover, we provide some relations between the separation ofA andB in their natural space and that in the projected space. Experimental tests validate some of the theoretical results and show the rate of applicability of the method with respect to a standard linear system solver.

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