Newton's Method for Solving Nonlinear Equations

Newton's method is a fundamental numerical computation technique for finding roots of nonlinear equations. Based on calculus principles, this approach rapidly converges to the solution through iterative approximation.

The core concept utilizes linear approximation via Taylor expansion. For a given nonlinear equation f(x)=0, the algorithm starts with an initial guess x₀. At each iteration step, it computes the next approximation xₙ₊₁ using both the function value f(xₙ) and its derivative f'(xₙ).

A typical MATLAB implementation follows these steps: Define the objective function f(x) and its derivative f'(x) as separate function handles Set initial guess x₀ and convergence criteria (tolerance levels) Implement iterative loop using the update formula: xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ) Check convergence conditions (e.g., difference between iterations |xₙ₊₁ - xₙ| or function value |f(xₙ)| below threshold)

Key implementation considerations in MATLAB include: requiring precise derivative expressions to maintain quadratic convergence; initial value selection significantly impacts convergence behavior; and implementing iteration limits prevents infinite loops. The algorithm typically uses while loops with condition checks combining absolute and relative tolerances.

While Newton's method achieves fast quadratic convergence, it has limitations: requires derivative information; sensitivity to initial guesses; potential failure in regions where derivatives approach zero. Practical applications often combine it with other methods, such as using bisection method first to narrow the search interval before applying Newton's method.

For more complex scenarios, modified approaches like the modified Newton method or quasi-Newton methods can address challenges with derivative calculation difficulties or expensive matrix inversions. These alternatives approximate derivatives using finite differences or maintain approximation matrices to reduce computational costs.