Newton's Method for Solving Nonlinear Equation Systems

Newton's Method is a classical iterative algorithm used to numerically solve systems of nonlinear equations. Its core principle involves using local linear approximations to progressively approach the true solution of the equation system. Implementation typically requires three key components: an equation system function, a Jacobian matrix calculation function, and a main iteration function.

For a given nonlinear equation system F(x)=0, Newton's method follows these iterative steps: First, compute the function value F(x_k) and Jacobian matrix JF(x_k) at the current point. Then solve the linear system JF(x_k)Δx=-F(x_k) to obtain the increment Δx. Finally update the solution using x_{k+1}=x_k+Δx. This process repeats until convergence criteria are satisfied.

In MATLAB implementations, code is typically organized into modular components: Equation system function F.m: Defines the left-hand side vector of the equation system, where each element corresponds to one equation Jacobian matrix function JF.m: Contains partial derivatives of each equation with respect to all variables, reflecting local rates of change Main function newdim.m: Implements the Newton iteration workflow including initial guess setup, iteration loop, and termination condition checking

Newton's method achieves quadratic convergence, but practical considerations include: initial point selection affects convergence, Jacobian matrix singularity may cause failure, and computational cost for large-scale problems can be high. To address these issues, modified Newton methods and quasi-Newton methods have been developed as improved algorithms.