## Numerical Analysis II

 Unit code: MATH36022 Credit Rating: 10 Unit level: Level 3 Teaching period(s): Semester 2 Offered by School of Mathematics Available as a free choice unit?: N

#### Requisites

Prerequisite

MATH36022 pre-requisites

#### Aims

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

#### Overview

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.

#### Learning outcomes

On completion of the module, students will be familiar with:

• characterise the best approximation of a function using different norms,
• compute Padé approximations and evaluate their quality,
• 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,
• recognize some of the difficulties that can occur in the numerical solution of problems arising in science and engineering.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

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

#### Syllabus

1.Approximation and Curve Fitting. Best approximation in the 1-norm. Weierstrass theorem, equioscillation theorem, Chebyshev polynomials. Best approximation in the 2-norm. Orthogonal polynomials. Rational approximation; Pad approximants. [6]

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

3.Initial Value Problems for ODEs Introduction and existence theorem. Numerical methods: one step methods and multistep methods. Eulers method. Local truncation error, convergence, local 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. [10]

1.Endre Sli 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.

#### Feedback methods

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.

#### Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

#### Teaching staff

Catherine Powell - Unit coordinator