Root Finding Methods

Root Finding

Introduction

• Root finding is a numerical method used to find the roots of a given function f(x) = 0
• Also known as nonlinear equation solving or zero-finding
• Important in various fields such as physics, engineering, and economics

Methods

Bisection Method

• Simple and intuitive method
• Assumes function is continuous and changes sign over the interval [a, b]
• Repeatedly divides the interval in half and selects the subinterval where the function changes sign
• Converges slowly, but guaranteed to converge

Secant Method

• Improves upon the bisection method
• Uses the secant line to estimate the root
• Requires two initial guesses, x0 and x1
• Faster convergence than bisection method

Newton-Raphson Method

• Most popular and efficient method
• Uses the tangent line to estimate the root
• Requires an initial guess, x0
• Quadratic convergence, but may not converge if initial guess is far from the root

Fixed Point Iteration

• Finds the fixed point of a function g(x) = x
• Guaranteed to converge if |g'(x)| < 1 in the neighborhood of the root
• Can be used to find multiple roots

Brent's Method

• Hybrid method combining bisection, secant, and inverse interpolation
• Robust and efficient, but more complex to implement
• Often used in practice for its reliability and speed

Convergence and Error Analysis

• Convergence criteria: |x_n - x_{n-1}| < ε or |f(x_n)| < ε
• Error analysis: consider the number of iterations, computational effort, and numerical stability