求解非线性方程组的一种新的全局收敛的Levenberg-Marquardt算法

A NEW GLOBALLY CONVERGENT LEVENBERG-MARQUARDT METHOD FOR SOLVING NONLINEAR SYSTEM OF EQUATIONS

本文提出了求解非线性方程组的一种新的全局收敛的Levenberg-Marquardt算法,即μk=ακ(θ||F_k||+(1-θ)||J_k^TF_k||),θ∈[0,1],其中ακ利用信赖域技巧来修正.在不必假设雅可比矩阵非奇异的局部误差界条件下,证明了该算法是全局收敛和局部二次收敛的.数值试验表明该算法能有效地求解奇异非线性方程组问题.

In this paper, we propose a new globally convergent Leveberg-Marquardt method for solving nonlinear systems of equations, i.e. μk=αk（θ‖Fk‖＋（1-θ）‖Jk^TFk‖）.θ∈[0,1], where αk is updated by trust region techniques. Global and local convergence of this new method are proved without the nonsingularity assumption of the Jacobian matrix. Numerical results show that this new method performs very well for the singular nonlinear systems of equations.