Hermitian Positive Definite Solutions Of Two Classes Of Nonlinear Matrix Equations

In this paper, Hermitian positive definite solutions of the nonlinear matrix equa-tions X±A^*X^-qA = Q are studied, where q≥1, A is an n×n nonsingular complexmatrix and Q is an n×n positive definite matrix. The nonlinear matrix equationshave been widely used in many fields such as control theory, dynamic programming,statistics, stochastic filtering, queueing theory and ladder networks. Solving the matrixequations X±A^*X^-qA = Q is one of the important study fields of the numerical alge-bra. Here, we study three basic problems: (1) the problem of the existence of solutions;(2) the methods for solving the numerical solution; (3) the perturbation analysis of thesolution.The main results are as follows:1. The properties of solutions are studied. Some new necessary and su?cientconditions for the existence of the positive definite solutions of the nonlinear matrixequation X + A^*X^-qA = Q are obtained. The new upper boundary, lower boundaryand existence interval of the positive definite solutions are given. From this, the rangesof eigenvalues of Q?12XQ?21 are obtained. Furthermore, it's proved that the nonlinearmatrix equation X ? A^*X^-qA = Q always has a positive definite solution. The con-ditions that matrix equation X ? A^*X^-qA = Q has the unique the positive definitesolution are given.2. The methods for solving the numerical solution are constructed. By the fixedpoint theory, the iterative methods are presented to compute the smallest and the quasilargest positive definite solutions of the matrix equation X + A^*X^-qA = Q and thepositive definite solution of the matrix equation X ? A^*X^-qA = Q . The convergencetheorem is also given. When A is normal and A commutes with Q, the direct method forsolving these two class matrix equations are presented. The corresponding expressionof the positive definite solution are obtained.3. Some new perturbation bounds are derived. When A and Q are both perturbed,the perturbation bounds of the positive definite solutions of the matrix equation X±A^*X^-qA = Q are given. Based on theory of Rice condition number, the sensitivityanalysis of the positive definite solutions are studyed. A backward error bound of thepositive definite solution is obtained when Q is perturbed.4. The numerical examples are given. The numerical examples are given toillustrate the correctness of theoretical results and the e?ectiveness of iterative methods. At the same time, they also show that the number of the positive definite solutions ofthe matrix equation X + A^*X^-qA = Q is uncertain, and the positive definite solutionof the matrix equation X - A^*X^-qA = Q (q > 1) is not unique.