On an iteration method for solving a class of nonlinear matrix equations. (English) Zbl 1002.65061

El-Sayed, Salah M.; Ran, André C. M.

SIAM J. Matrix Anal. Appl. 23, No. 3, 632-645 (2002).

Authors’ abstract: This paper treats a set of equations of the form \(X+ A^*F(X)A= Q\), where \(F\) maps positive definite matrices either into positive matrices or into negative matrices, and satisfies some monotonicity property. Here \(A\) is arbitrary and \(Q\) is a positive definite matrix. It is shown that under some conditions an iterative method converges to a positive definite solution. An estimate for the rate of convergence is given under additional conditions, and some numerical results are given. Special cases are considered, which cover also particular cases of the discrete algebraic Riccati equation.

Reviewer: A.Meister (Hamburg)

MSC:

65H10	Numerical computation of solutions to systems of equations
15A24	Matrix equations and identities
65F30	Other matrix algorithms (MSC2010)

Keywords:

nonlinear matrix equations; iteration methods; monotone operator functions; Hermitian positive definite matrices; convergence; positive definite solution; numerical results; discrete algebraic Riccati equation