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Iterative Methods for Solving the Nonlinear Matrix Equation <i>X</i>-<i>A</i>*<i>X</i><sup>p</sup><i>A</i>-<i>B</i>*<i>X</i><sup>-q</sup><i>B</i>=<i>I</i> (0<<i>p</i>,<i>q</i><1)
Department of Mathematics, Heze University, Heze, China
- 1 Department of Mathematics, Heze University, Heze, China
Advances in Linear Algebra & Matrix Theory·Volume 07 (2017)·Pages 72–78·Published 26 September 2017·DOI10.4236/alamt.2017.73007
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Abstract
Consider the nonlinear matrix equation X - A*X p A - B*X -q B = 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.
KeywordsNonlinear Matrix EquationPositive Definite SolutionIterative MethodNormal Cone
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