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

The Levenberg-Marquardt trust region method is an optimization algorithm for solving systems of nonlinear equations. This algorithm combines the advantages of steepest descent and Gauss-Newton methods, featuring fast convergence and high precision. In MATLAB, this algorithm can be employed to solve nonlinear equation systems. Below is a detailed MATLAB implementation of the Levenberg-Marquardt trust region method: function [x, fval] = levenberg_marquardt(fun, x0, options) % fun: Objective function handle returning residual vector % x0: Initial guess for solution vector % options: Optimization parameters (maximum iterations, tolerances) % Returns: x - optimized solution, fval - final residual values % Algorithm initialization x = x0; fval = fun(x); nu = 2; % Trust region parameter rho = 0; % Actual-to-predicted reduction ratio max_iter = 100; % Maximum iteration limit tol_fun = 1e-6; % Function value tolerance tol_x = 1e-6; % Solution change tolerance % Main iteration loop implementing trust region strategy for k = 1:max_iter J = jacobian(fun, x); % Compute Jacobian matrix using finite differences H = J' * J; % Approximate Hessian matrix (Gauss-Newton) g = J' * fval; % Gradient vector fval_new = fun(x); % Evaluate current residuals d = fval_new - fval; % Residual change % Trust region adjustment based on performance ratio rho = d' * d / (g' * g); if rho < 0.25 nu = 2 * nu; % Reduce trust region for poor steps elseif rho > 0.75 nu = 0.5 * nu; % Expand trust region for good steps end % Update solution using damped Hessian matrix if rho > 0 x = x - (H + nu * eye(size(H))) g; % Levenberg-Marquardt update fval = fun(x); % Re-evaluate residuals end % Convergence checking using residual norm and solution change if norm(d) < tol_fun || norm(x - x0) < tol_x break; % Exit when convergence criteria met end end end function J = jacobian(fun, x) % Computes Jacobian matrix using central finite differences % fun: Objective function handle % x: Current point where Jacobian is evaluated % Returns: J - m x n Jacobian matrix (m equations, n variables) n = length(x); m = length(fun(x)); J = zeros(m, n); h = 1e-6; % Finite difference step size % Finite difference computation for each variable for j = 1:n xj = x(j); % Store original value x(j) = xj + h; % Forward step fx1 = fun(x); % Evaluate at x+h x(j) = xj - h; % Backward step fx2 = fun(x); % Evaluate at x-h J(:, j) = (fx1 - fx2) / (2 * h); % Central difference derivative x(j) = xj; % Restore original value end end This program solves nonlinear equation systems using the Levenberg-Marquardt trust region method by accepting objective functions, initial guesses, and optimization parameters. The implementation handles ill-conditioned problems through adaptive trust region adjustments and includes finite-difference Jacobian computation. For improved accuracy or faster convergence, users can modify tolerance parameters and step size settings in the options structure.