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2017•Advances in Linear Algebra &amp Matrix TheoryOpen access

Iterative Methods for Solving the Nonlinear Matrix Equation X-A*XpA-B*X-qB=I (0<p,q<1)

Dongjie Gao

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Abstract

Consider the nonlinear matrix equation X-A*XpA-B*X-qB=I (0p,q1). By using the fixed point theorem for mixed monotone operator in a normal cone, we prove that the equation with 0p,q1 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.

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Consider the nonlinear matrix equation X-A*XpA-B*X-qB=I (0p,q1). By using the fixed point theorem for mixed monotone operator in a normal cone, we prove that the equation with 0p,q1 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.

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Available abstract

Consider the nonlinear matrix equation X-A*XpA-B*X-qB=I (0p,q1). By using the fixed point theorem for mixed monotone operator in a normal cone, we prove that the equation with 0p,q1 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.

Key concepts: Iterative method, Mathematics, Monotone polygon, Fixed-point theorem, Matrix (chemical analysis), Nonlinear system, Mathematical analysis, Fixed point

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