Description

This project will explore numerical methods for solving a system of two nonlinear equations in two variables. In the general problem, not only do we not know the locations of any roots, but we don't even know how many roots there are to find.

Visualization of a 2D function. Boundaries between colours indicate where either of the two function components change sign. Roots are located where all four colours intersect (there are 5 of them).

A general algorithm has two stages. First, one has to perform some kind of search for approximate locations of the root(s). Then with good enough initial guesses for each root, an iterative method can be used to find the root to the required precision. Either of these stages could form the basis of a project. For example, Newton's method for systems of equations was touched on in 2H Numerical Analysis, but you could investigate more sophisticated methods such as those of Broyden which aim to improve on its performance. Alternatively, you could look at "topological" techniques for improving the initial search, for example using the Poincare index to pin down the possible root locations.

This is a practical project in numerical analysis and you will be expected to write computer programs to implement the methods you study. Some guidance will be given but familiarity with simple computer programming is a prerequisite.

Prerequisites

Numerical Analysis II. For the practical aspect of the project you will need to be comfortable with either a programming language (e.g. Python/C/FORTRAN) or a mathematical package such as Maple/MATLAB.

Resources

Most of the recommended textbooks for 2H Numerical Analysis include some basic material on nonlinear systems of equations. For example, Chapter 9 of Burden & Faires, or Chapter 4 of Suli & Mayers. Other resources will be recommended depending on the direction of the project.