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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.
MSC:
15A24Matrix equations and identities
65F30Other matrix algorithms
References:
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