The Modified Trust-region Algorithm For The System Of Nonlinear Equations

We present a new modified trust-region algorithm for nonlinear equations with the trust region converging to zero. The new algorithm preserves the global con-vergence of the traditional trust region algorithm in which the trust region ra-dius is larger than a positive constant. We obtain the convergence order of the new algorithm under the local error bound condition which is weaker than the nonsingularity. Numerical results show that the new algorithm is very efficient. Furthermore,we give a modified regularized Newton algorithm for monotone non-linear equations with its application to the unconstrained convex optimization with singular solutions. Some numerical results are also given.