Pre-requisite: MAST4006: Mathematical Methods 1, MAST4010: Real Analysis 1
OverviewThis module is an introduction to the methods, tools and ideas of numerical computation. In mathematics, one often encounters standard problems for which there are no easily obtainable explicit solutions, given by a closed formula. Examples might be the task of determining the value of a particular integral, finding the roots of a certain non-linear equation or approximating the solution of a given differential equation. Different methods are presented for solving such problems on a modern computer, together with their applicability and error analysis. A significant part of the module is devoted to programming these methods and running them in MATLAB.
Introduction: Importance of numerical methods; short description of flops, round-off error, conditioning
Solution of linear and non-linear equations: bisection, Newton-Raphson, fixed point iteration
Interpolation and polynomial approximation: Taylor polynomials, Lagrange interpolation, divided differences, splines
Numerical integration: Newton-Cotes rules, Gaussian rules
Numerical differentiation: finite differences
Introduction to initial value problems for ODEs: Euler methods, trapezoidal method, Runge-Kutta methods.
This module appears in:
Method of assessment
80% Examination, 20% Coursework
R. L. Burden, J. D. Faires, A. M. Burden. Numerical Analysis. 2016.
J. H. Matthews, K. D. Fink. Numerical methods using MATLAB. Pearson, 2004.
W. Gautschi. Numerical Analysis. Birkhäuser Boston, 2012 (ebook available from the Library)
The intended subject specific learning outcomes. On successfully completing the module students will be able to:
1 demonstrate knowledge and critical understanding of the well-established principles within a wide range of basic numerical methods, including iterative methods, interpolation, quadrature, finite difference approximation of initial-value problems for ordinary differential equations (ODEs);
2 demonstrate the capability to use a range of established techniques and a reasonable level of skill in calculation and manipulation of the material to solve problems in the following areas: root finding, interpolation, numerical quadrature, finite differences, initial-value problems for ODEs;
3 apply the concepts and principles in basic numerical approximation in well-defined contexts beyond those in which they were first studied, showing the ability to evaluate critically the appropriateness of different tools and techniques;
4 make appropriate use of MATLAB commands to implement numerical methods.
The intended generic learning outcomes. On successfully completing the module students will be able to:
Demonstrate an increased ability to:
1 manage their own learning and make use of appropriate resources;
2 understand logical arguments, identifying the assumptions made and the conclusions drawn;
3 communicate straightforward arguments and conclusions reasonably accurately and clearly;
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working;
5 solve problems relating to qualitative and quantitative information;
6 make use of information technology skills such as online resources (Moodle), internet communication;
7 communicate technical material competently;
8 demonstrate an increased level of skill in numeracy and computation.