MATLAB Implementation of Levenberg-Marquardt Trust Region Method for Solving Nonlinear Equations

The Levenberg-Marquardt (LM) method is a classical algorithm for solving nonlinear least-squares problems. It combines the advantages of Gauss-Newton and gradient descent methods while enhancing numerical stability through a trust region mechanism. This method is particularly suitable for parameter estimation and optimization in nonlinear equation systems.

In MATLAB, the LM algorithm can be implemented using either the built-in `lsqnonlin` function or custom trust region iteration logic. The core implementation approach includes: Parameter Initialization: Set initial guess values, trust region radius, damping coefficient, and other parameters using structures like `options = optimoptions('lsqnonlin','Algorithm','levenberg-marquardt')`. Jacobian Matrix Calculation: Compute the Jacobian matrix of the objective function through numerical differentiation (using `finiteDifference`) or analytical methods for linear approximation. Trust Region Adjustment: Dynamically adjust the trust region radius based on the fit quality of the current approximation model, using ratio tests to ensure effective iteration directions. Step Size Selection: Adopt Gauss-Newton steps when model predictions match the actual function well; otherwise, increase damping terms to degenerate into gradient descent through lambda parameter adjustment. Convergence Criteria: Terminate iterations based on residual reduction, parameter changes, or gradient thresholds monitored via output functions.

Key considerations in practical implementation: Jacobian matrix accuracy directly affects convergence speed - consider using symbolic computation or complex-step differentiation for higher precision. Heuristic adjustment strategies for trust region radius (e.g., ratio-based control with typical thresholds of 0.25/0.75) can prevent无效 iterations. MATLAB's `optimoptions` function allows configuration of algorithm parameters such as MaxIterations, FunctionTolerance, and StepTolerance for optimal performance.

The method's strength lies in its strong adaptability - it can converge even with poor initial values, making it a practical choice for engineering optimization problems. The algorithm automatically balances between fast convergence and robustness through its adaptive damping mechanism.