Elementwise Minimal Nonnegative Solutions for a Class of Nonlinear Matrix Equations

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Abstract

The existence of elementwise minimal nonnegative solutions of the nonlinear matrix equations 
                                               $A$$T$$X$2$A$ − $X$ + $I$ = 0,

                                               $A$$T$$X$$n$$A$ − $X$ + $I$ = 0, $n$ > 2

are studied. Using Newton's method with the zero initial guess, we show that under suitable conditions the corresponding iterations monotonically converge to the elementwise minimal nonnegative solutions of the above equations. Numerical experiments confirm theoretical results and the efficiency of the method.

Keywords:

Elementwise minimal nonnegative solution Newton’s method monotonic convergence nonlinear matrix equation.
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DOI

10.4208/eajam.300518.120119