Root-Finding for Nonlinear Equations with MATLAB Implementation

Root-finding for nonlinear equations represents a fundamental numerical computation approach that can be implemented through various algorithms. Among these, the bisection method and Newton-Raphson iteration method stand as widely adopted techniques. The corresponding MATLAB programs can effectively solve diverse nonlinear equations, including polynomial equations, trigonometric equations, and other complex functional forms. The bisection method operates by repeatedly bisecting an interval and selecting the subinterval where the root must exist, ensuring convergence through continuous interval halving. In MATLAB implementation, this involves setting initial boundaries [a,b] satisfying f(a)*f(b)<0, then iteratively updating the interval based on function sign changes. The Newton-Raphson method utilizes linear approximation through derivative information to achieve quadratic convergence near roots. The MATLAB implementation requires defining both the function f(x) and its derivative df(x), followed by iterative updates using x_{n+1} = x_n - f(x_n)/df(x_n). Program performance can be optimized by adjusting key parameters such as maximum iteration counts and convergence tolerance thresholds. For the bisection method, convergence is guaranteed when |b-a| falls below a specified tolerance, while Newton's method typically uses |f(x)| < tolerance or |x_{n+1}-x_n| < tolerance as stopping criteria. These parameter adjustments significantly enhance solution efficiency and computational accuracy for various equation types.