week 1 Lecture 2 Bisection Method

Master the foundational algorithm of numerical root-finding. This lecture breaks down the Bisection Method, the most reliable bracketing technique in numerical analysis, which leverages the Intermediate Value Theorem to guarantee convergence to a root. Whether you are solving non-linear equations by hand or preparing to code your first root-finding loop in MATLAB, this comprehensive guide covers the theory, the step-by-step algorithm, and the error analysis required to execute it flawlessly. What You Will Learn The Mathematical Foundation: How continuous functions and the Intermediate Value Theorem guarantee the existence of a root between two initial guesses. The Bisection Algorithm: A mechanical, 5-step process for halving the bracket interval and zeroing in on the exact root. Error Tracking: How to calculate the absolute relative approximate error to determine exactly when to stop iterating. Common Pitfalls: Understanding the limitations of bracketing, including handling multiple roots and false-positive sign intervals. Step-by-Step Examples: Two fully worked manual examples, including a cubic equation and an exponential equation, demonstrating linear convergence in action.