Abstract
Consider the nonlinear matrix equation X-A*XpA-B*X-qB=I (0<p,q<1). By using the fixed point theorem for mixed monotone operator in a normal cone, we prove that the equation with 0<p,q<1 always has the unique positive definite solution. Two different iterative methods are given, including the basic fixed point iterative method and the multi-step stationary iterative method. Numerical examples show that the iterative methods are feasible and effective.
Highlights
We consider the matrix equation X − A* X p A − B* X −q B = I, 0 < p, q < 1 (1.1)where I is an n × n identity matrix, A, B are arbitrary mn × n matrices
By using the fixed point theorem for mixed monotone operator in a normal cone, we prove that the equation with 0 < p, q < 1 always has the unique positive definite solution
Numerical examples show that the iterative methods are feasible and effective
Summary
Where I is an n × n identity matrix, A, B are arbitrary mn × n matrices. A* , B* are used to denote the conjugate transpose of the matrix A and B, separately. Reurings [2] showed that the matrix equation X − A* X p A= I (0 < p < 1) and X − A* X −q A= I (0 < q < 1) always has a unique positive definite solution. Gao [3] provides a new proof for the uniqueness of the positive definite solution of the matrix equation. X − A*X q A= I (0 < q < 1) via the fixed point theorem. The method is shown much easier than the way of [1]. We prove that the Equation (1.1) always has the unique positive definite solution by using the fixed point theorem for mixed monotone operator in the normal cone P (n). Numerical examples show that the iterative methods are feasible and effective
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