Abstract
We introduce a gradient iterative scheme with an optimal convergent factor for solving a generalized Sylvester matrix equation ∑i=1pAiXBi=F, where Ai,Bi and F are conformable rectangular matrices. The iterative scheme is derived from the gradients of the squared norm-errors of the associated subsystems for the equation. The convergence analysis reveals that the sequence of approximated solutions converge to the exact solution for any initial value if and only if the convergent factor is chosen properly in terms of the spectral radius of the associated iteration matrix. We also discuss the convergent rate and error estimations. Moreover, we determine the fastest convergent factor so that the associated iteration matrix has the smallest spectral radius. Furthermore, we provide numerical examples to illustrate the capability and efficiency of this method. Finally, we apply the proposed scheme to discretized equations for boundary value problems involving convection and diffusion.
Highlights
It is well known that several problems in control and system theory are closely related to a generalized Sylvester matrix equation of the form p ∑ Ai XBi = F, (1)i =1 where Ai, Bi and F are given matrices of conforming dimensions
In Example 1, we show that our method is efficient matrices are non-square and we discuss the effect of changing the convergent factor τ
We propose a gradient-based iterative method with an optimal convergent factor for solving a generalized Sylvester matrix equation
Summary
It is well known that several problems in control and system theory are closely related to a generalized Sylvester matrix equation of the form p ∑ Ai XBi = F, (1). I =1 where Ai , Bi and F are given matrices of conforming dimensions. Equation (1) includes the following special cases: AX + XB = F, AX + XA T (2) (3) AXB + X = F, (4). Known respectively as the Sylvester equation, the Lyapunov equation, and the Kalman–Yakubovich equation. Equations (1)–(4) have important applications in stability analysis, optimal control, observe design, output regulation problem, and so on; see e.g., [1,2,3]. Equation (1) can be solved directly using the vector operator and the Kronecker product.
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