15 - Iterations for Nonlinear Equations

Summary

One can find a lot of mathematical examples involving iterations and the solution of nonlinear equations. Such equations arise from solving nonlinear problems – either differential or statistical. Real world problems are often nonlinear and may involve more than one independent variable, although techniques may resemble or reduce to the one dimensional case. The main reason for iterations is that direct (i.e. analytical) solutions of nonlinear equations are in general difficult to find, and numerical solutions need more than one step to converge. This chapter mainly considers equations with real coefficients; see Chapter 13 for the complex case.

Aims of the project

The purpose of this investigation is to study if, when and how numerical methods work in the context of solving nonlinear equations. In particular, the important issues of accuracy and convergence speed of iterative methods are considered.

Mathematical ideas used

This project involves vectors and matrices. The method is based on iterations and linearisations of nonlinear equations. We first consider one equation in one unknown (prefixed by ID for simplicity), and then consider systems of equations in multiple unknowns (prefixed by 2D).