15 - Solution of Nonlinear Equations

Summary

This chpater is dedicated to the solution of nonlinear systems of equations, that is finding roots of functions. We begin with a classification of roots into simple and non-simple, and a few words about their stability. Then we begin with some of the simplest methods for root finding: bisection, false position, fixed point iterations, and its variants. For all these schemes, we provide sufficient conditions for them to be well defined and convergent. A detailed analysis of Newton’s method, and its variants (collectively known as quasi-Newton methods), in one dimension is then presented. We show sufficient conditions for its local and global quadratic convergence,as well as how to proceed in the case of non simple roots. Then we present Newton’s method in several dimensions, and show its local quadratic convergence, including the celebrated Kantorovich’s theorem.