Root Finding Methods: Secant, Newton, and Fixed Point Iteration

This lecture discusses various numerical methods for finding roots of functions, focusing on the secant method, Newton's method, and fixed point iteration. The instructor begins by explaining the limitations of the bisection method, emphasizing the need for more efficient techniques that utilize function values and derivatives. The Taylor series expansion is introduced as a tool for reformulating root-finding problems. The lecture then delves into the secant method, detailing its iterative process and convergence properties. Following this, the instructor covers Newton's method, highlighting its quadratic convergence and the importance of selecting appropriate initial guesses. The fixed point iteration method is also explored, demonstrating how to transform root-finding problems into fixed point problems. Throughout the lecture, the instructor provides practical examples and visualizations using GeoGebra to illustrate the iterative processes and convergence behaviors of these methods. The session concludes with a discussion on the theoretical underpinnings of these methods and their applications in solving real-world problems.