Numerical Methods To Solve The Complex Symmetric Stabilizing Solution Of The Complex Matrix Equation X+A^TX^-1A=Q

When the matrices A and Q have special structure, structure-preserving algorithm was used to compute the stabilizing solution of the complex matrix equation X+ATX-1A= Q in [8]. In this paper, we study the more general matrix equation X+ATX-1A= Q when A is a complex square matrix and Q is complex symmetric. This equation plays an important role in nano research and the vibration analysis of fast trains. The existence of a unique complex symmetric stabilizing solution has been proved in [7] and it is required in practical application.We present the numerical methods which include the fixed-point method (FPI), the modified fixed-point method (MFPI), the structure-preserving algorithm (SPA) and Newton’s method (NW) to solve the complex symmetric stabilizing solution of the matrix equation X+ATX-1A= Q. In section 1, we review the development of this matrix equation. In section 2, we introduce some preliminary knowledge about the complex equation X+ATX-1A= Q, and the existence of the complex symmetric stabilizing solution is provided. In section 3, we perform the feasibility for FPI, MFPI, SPA and NW under an assumption, respectively. Furthermore, we give the convergence analysis of SPA. In section 4, numerical experiments show that the proposed algorithms are efficient. In section 5, some concluding remarks are given.