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On the existence of extremal positive definite solutions of the nonlinear matrix equation. (English) Zbl 1203.15011
The authors consider the matrix equation X r + j=1 m A j * X δ j A j =I, -1<δ j <0, where A j are nonsingular n×n matrices, I is the identity matrix, r and m are positive integers, A j * is the transpose conjugate of A j . The authors derive a necessary condition for the existence of a positive definite solution and, based on the Banach fixed point theorem, a sufficient condition for the existence of a unique such solution. Iterative methods for obtaining the extremal (maximal-minimal) positive definite solutions of this equation are proposed. The rate of convergence of some proposed algorithms is proved and numerical examples are given to illustrate their performance and effectiveness.
15A24Matrix equations and identities
65F30Other matrix algorithms
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