MATLAB Implementation of Newton's Iteration Method

The Newton-Raphson method is a classical algorithm in numerical analysis used to find approximate solutions to equations or systems of equations. It utilizes first derivative information to rapidly converge toward equation roots, offering the advantage of quadratic convergence rate. Implementing Newton's method in MATLAB involves the following key programming steps: 1. Define the objective function: First, express the target equation as a function, e.g., f(x)=0. For equation systems, define a vector-valued function using anonymous functions (@) or separate function files. 2. Compute the Jacobian matrix: For single-variable equations, calculate the derivative f'(x) analytically or numerically. For multivariable systems, construct the Jacobian matrix containing partial derivatives of each equation, which can be automated using MATLAB's symbolic toolbox with jacobian() function. 3. Iterative solving: Implement the Newton iteration formula x_{n+1} = x_n - f(x_n)/f'(x_n) using a while or for loop. Include convergence criteria checking within each iteration cycle. 4. Convergence verification: After each iteration, check if the solution update is sufficiently small (using tolerance comparison like abs(x_new - x_old) < tol) or if the function value approaches zero (abs(f(x_new)) < tol). For equation systems, MATLAB efficiently handles matrix operations using backslash operator (\) for solving linear systems instead of explicit matrix inversion, avoiding numerical instability. The implementation should include proper initial value selection since poor initialization may cause divergence. In practical implementation, developers can: - Use MATLAB's symbolic math toolbox for automatic differentiation with diff() function - Implement numerical differentiation using finite difference methods when analytical derivatives are unavailable - Add damping factors (modified Newton method) for ill-conditioned systems - Incorporate maximum iteration counters to prevent infinite loops - Consider switching to quasi-Newton methods (like Broyden's method) for better stability in challenging cases The algorithm typically requires 5-10 lines of core MATLAB code, combining function evaluation, derivative calculation, and iterative updates with convergence checks.