Introductory description

This module develops understanding of numerical methods that are used in many areas of the mathematical sciences, and it will improve Python programming skills.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Mathematical modelling in various applications in science and engineering can lead to problems that do not yield to closed form analytic solutions. Specifically, systems of nonlinear differential equations often cannot be explicitly integrated and thus require numerical methods to learn about solutions and their behaviour. This module is an introduction to such numerical approximation techniques and explains how, why, and when they can be expected to work. In particular, explicit and implicit Runge-Kutta and multistep methods will be discussed. The derivation of these methods and their implementation benefits from studying a wider range of related problems like polynomial interpolation of data points and its use to approximate functions, numerically differentiating functions with finite differences, integrating them using quadrature formulas, and solving algebraic problems with Newton's method. Fundamental concepts will be explained, namely conditioning of mathematical problems, but also consistency and stability of approximation techniques, which underpin the convergence analysis of approximation techniques. By implementing some methods theoretical convergence results will be validated.

Learning outcomes

By the end of the module, students should be able to:

• understand fundamental ideas that underpin numerical analysis
• derive and analyse fundamental numerical methods
• implement and test numerical methods using a scripting language

Indicative reading list

Reading lists can be found in Talis

Specific reading list for the module

Subject specific skills

Student will gain understanding of general principles that underpin the numerical analysis of approximation techniques.
Students will be comfortable in deriving, applying and evaluating approximation techniques involved in topics such as polynomial interpolation and extrapolation, numerical differentiation and integration.
Students will be able to numerically solve systems of ordinary differential equation using a variety of approximations methods and have a good understanding of the different properties of these methods.
This module will provide students with the foundations required for a wide range of topics in computational mathematics. The acquired skill set can then be applied to larger scale problems in various areas that require a mixture of robust and efficient numerical methods.

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.
They will also gain the ability to use analytical and numerical methods (including associated programming knowledge) in harmony, enhancing capabilities in tackling complex challenges.