# NEWTON'S METHOD FOR SYMMETRIC AND BISYMMETRIC SOLVENTS OF THE NONLINEAR MATRIX EQUATIONS

• Han, Yin-Huan (School of Mathematics and Physics, Qingdao University of Science and Technology) ;
• Kim, Hyun-Min (Department of Mathematics, Pusan National University)
• Received : 2010.08.18
• Published : 2013.07.01
• 56 9

#### Abstract

One of the interesting nonlinear matrix equations is the quadratic matrix equation defined by $$Q(X)=AX^2+BX+C=0$$, where X is a $n{\times}n$ unknown real matrix, and A, B and C are $n{\times}n$ given matrices with real elements. Another one is the matrix polynomial $$P(X)=A_0X^m+A_1X^{m-1}+{\cdots}+A_m=0,\;X,\;A_i{\in}\mathbb{R}^{n{\times}n}$$. Newton's method is used to find the symmetric and bisymmetric solvents of the nonlinear matrix equations Q(X) and P(X). The method does not depend on the singularity of the Fr$\acute{e}$chet derivative. Finally, we give some numerical examples.

#### Keywords

quadratic matrix equation;matrix polynomial;solvent;Newton's method;iterative algorithm;symmetric;bisymmetric

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