Newton's Method For Hermitian Positive Definite Solutions Of Non-Linear Matrix Equation X+A^*X^-1A=P

The problem of solving the nonlinear matrix equation, is mainly to con-trolled the error order by iteration to get the approximate solution of the equa-tion. Because the Hermite definite solutions is widely applied in practice, so we only discuss these solutions. In this paper, we mainly discuss two kinds of New-ton's iterative methods on the nonlinear matrix equation X + A*X-1A = P.Algorithm 1 andAlgorithm 2 and then compare with several other iterative algorithms. The main conclu-sions are obtained as follows:Theorem1 Suppose (?), {Xn} is determined by the algo-rithm 1, then (?), and (?). Theorem2 For algorithm 1, if (?), then we haveTheorem3 Consider the algorithm 1, if (?), then the iter-ative sequence {Xn} converges to XL.Theorem4 If (?), the matrix sequence {Xn} in algorithm 1 satisfies whereTheorem5 If (?), the matrix sequence {Xn} and {Yn} in algorithm 2 satisfies (?) and Yn≤2P-1 for n = 1,2, ? ? ?.Theorem6 Suppose that (?), if we choose Y0∈(0, P-1], and the matrix sequence {Xn} and {Yn} are determined by algorithm 2, then Moreover, for any n =1,2,…, Xn≥XL and Yn≤XL-1. Then we can get (?) and (?).Theorem7 If (?), the matrix sequence {Xn} in algorithm 2 converge to XL in a quadratic rate, and it satisfies...