Approximation theory is concerned with approximating functions of a given class, or data of a given type, using functions from another, usually more elementary, class. A simple example is the problem of approximating a function such as ex by means of polynomial functions. The efficient solution of such problems is of great importance for computing such approximations, and this module will introduce the mathematical theory behind many approximation methods in common use. This intermediate-level module is based on the set book Approximation Theory and Methods by M. J. D. Powell.
What you will study
The subject of approximation theory lies at the frontier between applied mathematics and pure mathematics. Practical problems, such as the computer calculation of special functions like ex, lead naturally to theoretical problems, such as ‘how well can we approximate by a given method?’ or ‘how fast does a given algorithm converge?’.
The module is based on Approximation Theory and Methods by M. J. D. Powell (Cambridge University Press, 1981). This book provides an excellent introduction to these theoretical problems, covering the basic theory of a wide range of approximation methods
You will learn
Successful study of this module should enhance your skills in understanding complex mathematical texts, constructing solutions to problems logically and communicating mathematical ideas clearly.
Note you must study this module if you wish to take the ‘Advances in approximation theory’ topic for your Dissertation in mathematics (M840).
Entry
To study this module you must declare the MSc in Mathematics (or another qualification towards which the module can count) as your qualification intention.
Normally, you should have also completed at least one of the entry modules for the MSc in Mathematics (F04), Calculus of variations and advanced calculus (M820) or Analytic number theory I (M823).
The subject of approximation theory lies at the frontier between applied mathematics and pure mathematics, since practical problems such as how to calculate special functions on a computer lead to theoretical problems such as ‘which approximation method is best?’. Therefore you will need some familiarity with real analysis and linear algebra, such as that developed in typical undergraduate courses, and knowing the basic properties of metric spaces would also be useful.
All teaching is in English and your proficiency in the English language should be adequate for the level of study you wish to take. We strongly recommend that students have achieved an IELTS (International English Language Testing System) score of at least 7. To assess your English language skills in relation to your proposed studies you can visit the IELTS website.
If you have any doubt about the suitability of the module, please speak to an adviser.
Qualifications
M832 is an optional module in our:
If you have a disability
The material contains small print and diagrams, which may cause problems if you find reading text difficult. You will also need to be able to use a scientific calculator.
Study materials
What's included
You’ll have access to a module website, which includes:
- a week-by-week study planner
- course-specific module materials
- audio and video content
- assessment details and submission section
- online tutorial access
- access to student and tutor group forums.
You’ll be provided with printed materials covering the content of the module, including explanations, examples and activities to aid your understanding of the concepts and associated skills and techniques that are contained in the set book. You will need to obtain your own copy of the set book.
You will need
A scientific calculator.
Computing requirements
You’ll need broadband internet access and a desktop or laptop computer with an up-to-date version of Windows (10 or 11) or macOS Ventura or higher.
Any additional software will be provided or is generally freely available.
To join in spoken conversations in tutorials, we recommend a wired headset (headphones/earphones with a built-in microphone).
Our module websites comply with web standards, and any modern browser is suitable for most activities.
Our OU Study mobile app will operate on all current, supported versions of Android and iOS. It’s not available on Kindle.
It’s also possible to access some module materials on a mobile phone, tablet device or Chromebook. However, as you may be asked to install additional software or use certain applications, you’ll also require a desktop or laptop, as described above.
Teaching and assessment
Support from your tutor
Throughout your module studies, you’ll get help and support from your assigned module tutor. They’ll help you by:
- Marking your assignments (TMAs) and providing detailed feedback for you to improve.
- Guiding you to additional learning resources.
- Providing individual guidance, whether that’s for general study skills or specific module content.
The module has a dedicated and moderated forum where you can join in online discussions with your fellow students. There are also online module-wide tutorials. While these tutorials won’t be compulsory for you to complete the module, you’re strongly encouraged to take part. If you want to participate, you’ll likely need a headset with a microphone.
Assessment
The assessment details can be found in the facts box.
Future availability
Approximation theory (M832) starts every other year – in October.
This page describes the module that will start in October 2022.
We expect it to start for the last time in October 2022.
Regulations
As a student of The Open University, you should be aware of the content of the academic regulations which are available on our
Student Policies and Regulations website.