Convergence of relaxed Newton method for order-convex matrix equations

Abstract

In this paper, we suggest singular monotone Newton theorem and a relaxed Newton method for a nonlinear matrix equation $$F(X) = 0$$ where F is an order-convex matrix function. At first, we introduce monotone Newton theorem from Ortega and Rheinboldt (SIAM, 2000), and we show singular monotone Newton theorem which shows that the Newton sequence converges to a solution of $$F(X) = 0$$ even if the Frechet derivative at the solution is singular. At last, we provide a relaxed Newton method for $$F(X) = 0$$ which is better than the pure Newton method and we give some numerical experiments for the relaxed Newton method.