MATLAB Code Implementation of Newton's Method Iteration

Newton's method is a classical numerical iterative technique primarily used for finding roots of nonlinear equations. In numerical analysis courses, it is often employed to demonstrate rapid convergence to exact solutions through local linear approximations. MATLAB, with its powerful matrix computation capabilities, is particularly well-suited for implementing such algorithms.

The core concept of Newton's method relies on first-order Taylor series approximation. During each iteration, the algorithm constructs a tangent line using the current point's function value and derivative information, then takes the intersection of this tangent with the x-axis as the new approximate solution. Implementation typically requires three key components: the objective function, its derivative expression, and an initial guess. In MATLAB, these mathematical expressions can be defined using function handles - for example, f = @(x) x^2 - 2 for the function and df = @(x) 2*x for its derivative.

The termination criteria for the iterative process are generally set as either the difference between consecutive iterations falling below a preset tolerance threshold, or reaching a maximum iteration count to prevent infinite loops. For robust implementation, mature code should include validation checks for near-zero derivatives to avoid numerical instability issues. A practical implementation would typically involve a while loop structure with conditional breaks based on these criteria.

For extended exploration in numerical analysis assignments, one could compare the convergence rates of Newton's method with bisection and secant methods, or investigate how initial value selection affects convergence behavior. In more complex scenarios such as multi-variable root finding, Newton's method can be extended using Jacobian matrix formulations, which lays foundation for subsequent studies in optimization algorithms. MATLAB's vectorization capabilities make such extensions particularly efficient through matrix operations.