Broyden's Iterative Method for Solving Nonlinear Equation Systems

Broyden's iterative method is an effective approach for solving nonlinear equation systems, belonging to the quasi-Newton method family. It improves upon Newton's method by eliminating the need for recalculation of the Jacobian matrix at each iteration, significantly enhancing computational efficiency.

In MATLAB development environment, implementing Broyden's method typically requires creating M-files. The core concept involves approximating and updating the Jacobian matrix or its inverse rather than recalculating it entirely. Key implementation steps include:

Initialization: Provide initial solution guess and initial Jacobian matrix (or its approximation). Iterative Computation: - Calculate iteration direction using the current approximate Jacobian matrix - Update solution vector and compute new function values - Update Jacobian matrix or inverse matrix using current solution and function value differences Convergence Check: Terminate iteration when solution changes or function values become sufficiently small.

Compared to Newton's method, Broyden's method only requires function evaluations at each step without Jacobian recalculation, making it particularly advantageous for large-scale nonlinear systems. However, its convergence rate may be slightly slower than Newton's method, especially when the approximated matrix lacks accuracy.

In MATLAB, custom M-file implementation can optimize performance through vectorized computations and loop structures. The built-in matrix operation capabilities efficiently handle matrix update steps, while functions like 'norm()' can monitor convergence criteria. A typical implementation would structure the algorithm using while-loops for iteration control and employ rank-1 updates for Jacobian approximation using Sherman-Morrison formula equivalents.