Sylvester's equation: Accuracy and computational stability
Two problems related to the equation \(AX + XB = C\), known as the Sylvester matrix equation, are investigated. The Sylvester equation in which \(A\), \(B\) and \(C\) are \(m \times m\), \(n \times n\) and \(m \times n\) matrices, respectively, appears frequently in the theory of linear dynamic systems in relation to problems of control and stability. The first problem deals with the accuracy of the computed solution \(\widehat{X}\). The second handles the stability aspects of the solution and discusses ways for computing the backward error. A scheme is proposed using the residual \(R\) formed as \(A\widehat{X} + \widehat{X} B - C\) to ameliorate the solution of Sylvester's equation based on an iterative improvement technique similar to the famous one used in linear equations' solvers. The convergence of the proposed scheme is proved. The results apply automatically to the special case when \(A = B^T\) (Lyapunov equation).
scientific article; zbMATH DE number 825278
scientific article