Fixed Point Theorem: Convergence of Newton's Method

This lecture discusses the fixed point theorem and the convergence of Newton's method. The instructor begins by reviewing the iteration function and its derivation from the starting function. They illustrate the method of Picard for finding fixed points and emphasize the importance of choosing an appropriate function for convergence. The lecture includes graphical representations of functions and their roots, demonstrating the convergence behavior of the method. The instructor explains the conditions under which the method converges, particularly focusing on the derivative's behavior. They introduce the concept of contraction mappings and how they relate to fixed points. The lecture culminates in a discussion of Newton's method as a specific case of fixed point iteration, highlighting its convergence properties and the significance of the derivative being less than one in absolute value. The instructor concludes by outlining future topics, including error analysis and additional computational tools for further exploration of these concepts.