On Newton's Method for Solving a System of Nonlinear Matrix Equations
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In this paper, we are concerned with the minimal positive solution to system of the nonlinear matrix equations A1X2+B1Y +C1 = 0 and A2Y 2 +B2X +C2 = 0, where Ai is a positive matrix or a nonnegative irreducible matrix, Ci is a nonnegative matrix and −Bi is a nonsingular M-matrix for i = 1,2. We apply Newton’s method to system and present a modified Newton’s iteration which is validated to be efficient in the numerical experiments. We prove that the sequences generated by the modified Newton’s iteration converge to the minimal positive solution to system of nonlinear matrix equations.
Abstract
In this paper, we are concerned with the minimal positive solution to system of the nonlinear matrix equations A1X2+B1Y +C1 = 0 and A2Y 2 +B2X +C2 = 0, where Ai is a positive matrix or a nonnegative irreducible matrix, Ci is a nonnegative matrix and −Bi is a nonsingular M-matrix for i = 1,2. We apply Newton’s method to system and present a modified Newton’s iteration which is validated to be efficient in the numerical experiments. We prove that the sequences generated by the modified Newton’s iteration converge to the minimal positive solution to system of nonlinear matrix equations.
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