BSc Mathematics with Placement Year

This module introduces numerical methods for approximating functions and data, evaluating integrals and solving ordinary differential equations. It continues the introduction to numerical analysis begun in MATH20602. It provides theoretical analysis of the problems along with algorithms for their solution. Insight into the algorithms will be given through MATLAB illustrations, but the course does not require any programming.

To introduce students to theoretical and practical aspects of best approximation, quadrature, and the numerical solution of ordinary differential equations.

On completion of the module, students should be able to:

• characterise the best approximation of a function using different norms;
• derive orthogonal polynomials with respect to a weight function and use them to derive best L2 approximations and Gauss quadrature rules;
• compute and evaluate Padé approximations of a prescribed degree of accuracy;
• derive quadrature rules and their error bounds;
• apply the Trapezium rule, Gauss quadrature and adaptive quadrature to compute integrals;
• describe the Romberg scheme in the context of extrapolation;
• analyse and apply one-step, multi-step, and the Euler method for solving ordinary differential equations (ODE);
• solve ODE numerically using Runge-Kutta, Trapezium and higher-order methods;
• quantify the error and convergence of numerical solvers for ODE.

1.Approximation and Curve Fitting: Best approximation in the infinity-norm. Weierstrass' theorem, Chebyshev’s equioscillation theorem, Chebyshev polynomials. Best approximation in the 2-norm. Orthogonal polynomials. Rational approximation, Padé approximants. [8]

2.Numerical Integration: Interpolatory rules. Gaussian quadrature. Adaptive quadrature. The Romberg scheme: extrapolation using the Euler-Maclaurin summation formula. [6]

3.Initial Value Problems for ODEs: Introduction and existence theorem. Numerical methods: one step methods and multistep methods. Euler's method. Local truncation error, convergence, global error. Taylor series method. Runge-Kutta methods. Trapezium rule. Functional iteration and predictor-corrector PE(CE)m implementations. Absolute stability. Linear multistep methods. Higher order systems. [8]

• Mid-semester test: weighting 20%
• End of semester examination: weighting 80%

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

1.Endre Süli and David F. Mayers. An Introduction to Numerical Analysis. Cambridge University Press, Cambridge, UK, 2003. ISBN 0-521-00794-1. x+433 pp.

2.Richard L. Burden and J. Douglas Faires. Numerical Analysis. Brooks/Cole, Pacific Grove, CA, USA, seventh edition, 2001. ISBN 0-534-38216-9. xiii+841 pp.

3.James L. Buchanan and Peter R. Turner. Numerical Methods and Analysis. McGraw-Hill, New York, 1992. ISBN 0-07-008717-2, 0-07-112922-7 (international paperback edition). xv+751 pp.

4.David Kincaid and Ward Cheney. Numerical Analysis: Mathematics of Scientific Computing. Brooks/Cole, Pacific Grove, CA, USA, third edition, 2002. ISBN 0-534-38905-8. xiv+788 pp.

5.David Nelson, editor. The Penguin Dictionary of Mathematics. Penguin, London, fourth edition, 2008. ISBN 978-0-141-03023-4. 480 pp.

The independent study hours will normally comprise the following. During each week of the taught part of the semester:
 
•         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently
•         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week
•         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises
•         In some weeks you may be preparing coursework or revising for mid-semester tests
 
Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.
The remaining independent study time comprises revision for and taking the end-of-semester assessment.
 
The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.