Solving Nonlinear Equation Systems Using Broyden's Iteration Method

In the MATLAB development environment, Broyden's iteration method can be employed to solve systems of nonlinear equations. Broyden's method is a numerical computation technique that approximates solutions to nonlinear equation systems through iterative refinement. The implementation involves creating an M-file - a MATLAB program file containing code and algorithms written in MATLAB scripting language. The algorithm works by maintaining an approximation of the Jacobian matrix and updating it using rank-one modifications in each iteration, avoiding the need for costly Jacobian recalculations. Key implementation aspects include: initial guess specification, convergence criteria setup, and iterative matrix updates using Broyden's formula. The M-file typically contains functions for evaluating the system equations, implementing the Broyden update step, and checking convergence conditions. Through proper M-file implementation of Broyden's iteration method, users can efficiently obtain solutions to nonlinear equation systems while balancing computational efficiency and numerical stability. The method is particularly valuable for problems where analytical Jacobian calculation is complex or computationally expensive.