MA2K4 Numerical Methods and Computing
Lecturer: Tobias Grafke
Term(s): Term 2
Status for Mathematics students: List A (note that 3rd years who have taken MA261 Differential Equations: Modelling and Numerics previously cannot register for this module)
Commitment: 30 one-hour lectures
Assessment: 100% by coursework
Formal registration prerequisites: None
Assumed knowledge:
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Basic knowledge on solving differential equations and the structure of solutions for systems of ODEs and DEs as provided e.g. by MA146 Methods of Mathematical Modelling 1 or MA147 Mathematical Methods and Modelling 1 or MA133 Differential Equations.
- Programming in Python as provided, for instance, by MA124 Maths by Computer
- Concepts like Taylor expansion and continuity of multivariable functions.
Useful background: Good working knowledge in linear algebra and analysis
Synergies:
- MA117 Programming for Scientists
- MA269 Asymptotics and Integral Transforms
- MA256 Introduction to Systems Biology
Leads to: The following modules will have this module listed as assumed knowledge or useful background:
- MA3J4 Mathematical Modelling with PDE
- MA398 Matrix Analysis and Algorithms
- MA3K1 Mathematics of Machine Learning
- MA3H0 Numerical Analysis and PDEs
- MA4M1 Epidemiology by Example
Aims: This module focuses on fundamental concepts of the analysis of numerical methods and mathematical modelling involving for example ordinary differential equations showcased using typical examples from physics, biology, and other areas of science and engineering. Basic numerical approximation methods will be presented including for example methods for solving systems of differential equations, the solution to algebraic equations, polynomial interpolation and .extrapolation, and quadrature. Concepts like stability, consistency, and convergence will be covered in this module, with the aim of introducing the approximation techniques used in tackling mathematical problems which do not yield to closed form analytic formulae. Other application will include for example graphics and visualization or medical imaging.
Content: Numerical approximations, Derivation of explicit and implicit Runge Kutta and multistep methods, Butcher tableau, Newton’s method, polynomial interpolation/extrapolation, linear regression, optimization, and quadrature, stability, consistency, and convergence analysis.
Derivation and analysis of numerical methods for complex real word application is a focus of this module. Concepts like stability, consistency, and convergence will be covered in this module, with the aim of introducing the approximation techniques used in tackling mathematical problems which do not yield to closed form analytic formulae.
Basic numerical approximation methods will be presented for solving complex systems of differential equations like Runge-Kutta and multistep methods. The derivation and implementation of these methods will require studying a wide range of other approximation problems like polynomial interpolation, approximations to derivatives and integrals (finite differences and quadratures) and the solution to algebraic problems. Examples for the application of these methods will be taken from physics, biology but also for example from computer graphics/visualization, medical imagi
Objectives: By the end of the module, students should be able to:
- derive and analyse fundamental numerical methods
- implement and test numerical methods using a scripting language
- design and evaluate problem solving strategies for real-world application
Books:
- D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Scientific Computing
- L. N. Trefethen, Approximation Theory and Approximation Practice
- E. Suli and D. F. Mayers, An Introduction to Numerical Analysis
- D. F. F. Griffiths and D. J. Higham, Numerical Methods for Ordinary Differential Equations
- T. Witelski, M. Bowen, Methods of Mathematical Modelling: Continuous System and Differential Equations
- R. L. Burden, J. D. Faires, A.M. Burden, Numerical Analysis