Least-Squares Approximate Solution of Overdetermined Sylvester Equations

We address the problem of computing a low-rank estimate Y of the solution X of the Lyapunov equation A X + X A' + Q = 0 without computing the matrix X itself. This problem has applications in both the reduced-order modeling and the control of large dimensional systems as well as in a hybrid algorithm for the rapid numerical solution of the Lyapunov equation via the alternating direction implicit method. While no known methods for low-rank approximate solution provide the two-norm optimal rank k estimate Xk of the exact solution X of the Lyapunov equation, our iterative algorithms provide an effective method for estimating the matrix Xk by minimizing the error ||A Y + Y A' + Q||F.

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