On Numerical Methods For Solving Nonlinear Matrix Equation X+A^*X^-nA=Q

In this work,we investigate positive definite solutions of the nonlinear matrix equation X+A^*X^-nA=Q,where A is a m×m matrix,Q is a m×m matrix and n is an interger.Solving nonlinear matrix equations is one of the important study fields of the numerical algebra.Here,we study the properties and methods for the minimal positive definite solution and extremal solution.In the first section,we state the research significance and development of nonlinear matrix equations.Moreover,we introduce some methods for solving the family matrix equations which has been investigated.The main results are in the second section.Firstly,two methods based on the fixed point theory are given.The minimal positive definite solution of matrix equation X+A^*X^-nA=Q with A is nonsingular can be obtained by the two methods.If A is singular,the condition for the two methods is derived.Secondly,an elegant property of the extremal solution to the matrix equation X+A^*X^-nA=Q is presented.The property shows that the extremal solution is well-conditioned.Two iteration methods for finding the extremal solution are proposed.An easy method similar to the Newton method for solving the matrix equation is given.The two methods can find the extremal solution.In the third section,we investigate the generalized Sylvester matrix equation results from the Newton method.Simple iterative method,Gmres and CG can be applied to this equation. We proved that they can find the unique Hermite solution.The numerical examples are given in the last section to illustrate the correctness of theoretical results and the effectiveness of iterative methods.Comparisons have been made between these methods.