Solving Nonlinear Equations Using Newton's Iteration Method with MATLAB Implementation

Newton's iteration method is a fundamental numerical technique for solving nonlinear equations through iterative approximation of roots. The MATLAB implementation below demonstrates this approach with proper convergence safeguards and parameter controls: function [x, k] = newton(f, df, x0, tol, maxiter) k = 0; while k < maxiter fx = f(x0); if abs(fx) < tol break; end dfx = df(x0); if abs(dfx) < tol break; end x = x0 - fx / dfx; if abs(x - x0) < tol break; end k = k + 1; x0 = x; end end The function parameters include: f (the target equation function), df (the derivative function of f), x0 (initial guess), tol (error tolerance), and maxiter (maximum iteration count). The algorithm implements three convergence criteria: function value tolerance, derivative value tolerance, and successive approximation difference. The core Newton-Raphson formula x = x0 - f(x0)/f'(x0) updates the approximation at each iteration. It's crucial to note that Newton's method doesn't guarantee convergence for all cases, requiring appropriate parameter selection to balance computational accuracy and efficiency. The function returns both the root approximation x and the actual iteration count k for convergence monitoring.