On an Iteration Method for Solving a Class of Nonlinear Matrix Equations

This paper treats a set of equations of the form X + AF(X)A = Q, where F maps positive definite matrices either into positive definite matrices or into negative definite 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 iteration 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. Current address: Department of Mathematics, Scientific Departments, Education College for Girls, Al-Montazah, Buradah, Al-Qassim, Kingdom of Saudi Arabia.