5 - Zeros and Minima

Summary

Chapter 5 starts out with a physics motivation, as well as a mathematical statement of the problems that will be tackled in later sections. Several methods are introduced to solve a single nonlinear equation in one variable: fixed-point iteration, the bisection method, Newton’s method, the secant method, and Ridders’ method. After providing some advice about advantages and disadvantages of each approach, the text then studies how to find zeros of polynomials, employing two different techniques. The sophistication is then increased, by tackling systems of nonlinear equations and examining the corresponding challenges; in addition to Newton’s method, the text derives the equations behind Broyden’s method. A related subject is then broached, minimization in one or several dimensions; this includes the gradient-descent method, as well as detailed analysis of critical points; the second edition includes extensive new material on derivative-free optimization (golden-section search and Powell’s method).The chapter is rounded out by a physics project, the extremization of the action in classical mechanics, and a problem set. The physics project shows Hamilton’s principle in... action, translated into a multidimensional minimization problem.