Nonlinear Equations: Methods and Applications

This lecture focuses on nonlinear equations, specifically methods for finding their zeros. The instructor begins by defining nonlinear equations and the objective of finding values where the function equals zero. Various methods are discussed, including the bisection method, which is explained in detail with graphical representations. The algorithm for bisection is presented, emphasizing the importance of continuity and sign changes in the function. The lecture also covers error criteria and stopping conditions for iterative methods. Several examples are provided, including approximating zeros for specific functions and analyzing convergence rates. The Newton-Raphson method is introduced as another approach for finding zeros, along with its convergence properties. The instructor discusses fixed-point methods and their applications, highlighting the significance of convergence criteria. The lecture concludes with a discussion on the order of convergence and the implications of different methods for solving nonlinear equations, providing a comprehensive overview of the topic.