# Mathematics

## ECM2704 - Numerics and Optimisation (2015)

MODULE TITLE CREDIT VALUE Numerics and Optimisation 15 ECM2704 Dr Bob Beare (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 weeks 0 0
 Number of Students Taking Module (anticipated) 145
DESCRIPTION - summary of the module content

When designing any kind of system, the relationship between the system data and the actual physical performance of the system can be very complicated. For this reason, engineers use numerical optimisation to deliver a a system structure. The process of identifiying an objective, variables and constraints on system design is known as modeling. Next, engineers apply an optimisation algorithm. Since there are different kinds of algorithms, this course will acquaint you with some of the standard approaches to finding a solution to your question.

An example of this in everyday business would be a chemical company whose objective is to deliver a specific product from its three factories to a dozen retail outlets. The variables would include plant capacity, weekly demand, and shipping costs. The constraints might be distance, product stability, market timing, or seasonal demand fluctuations.

Building on the Matlab skills you acquired in ECM1704, you will study nonlinear equations, systems of linear equations, time-stepping of ordinary differential equations, and the finding of minima or functions of many variables.

Prerequisite module: ECM1701, ECM1705 or NSC1002 (Natural Science Students) or equivalent

AIMS - intentions of the module

This module explores the use of computers to solve mathematical problems by means of numerical approximation. The techniques discussed form the basis of the numerical simulation and computer modelling of problems in science and business. Topics to be covered include solving nonlinear equations, solving systems of linear equations, time-stepping of ordinary differential equations, and finding minima of functions of many variables. IMPORTANT: the module builds on the Matlab programming from the first year Scientific Computing module (ECM1704): for students who have taken this module, there will be revision of Matlab in the first week. However, the Scientific Computing module is not a prerequisite, so, for students who have not taken it, there will be an intensive introduction to Matlab in the first week computer classes.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:
1 demonstrate a working knowledge of the theory and practical implementation of basic numerical methods;
2 explore applications and ideas underpinning more advanced methods that are developed in third/fourth stage modules and project work;
Discipline Specific Skills and Knowledge:
3 show knowledge of the subject material of the module, which will provide the basis for future study of numerical analysis and its application to all areas of science and business: for example, meteorology, control theory, operations research;
4 understand computation as a natural method for tackling such problems;
Personal and Key Skills
5 demonstrate theoretical and practical mathematical skills, including programming.
6. formulate and solve problems.
7. communicate computer results and mathematical derivations effectively.
8. work in teams and use a variety of sources to write reports.

SYLLABUS PLAN - summary of the structure and academic content of the module

- revision/introduction to Matlab as a tool for numerical programming;
- what numerical analysis is: examples and applications;
- nonlinear scalar equations;
- Newton-Raphson;
- bisection method;
- iterative techniques;
- systems of equations;
- applications in Matlab;
- systems of linear equations;
- Gaussian elimination and the LU decomposition of matrices;
- Gauss-Seidel and Jacobi iteration;
- matrix eigenvalue problems and iterative schemes;
- power method, shifted and inverse power method;
- introduction to matrix norms and condition numbers;
- illustrations in Matlab;
- numerics for ODEs;
- forwards and backwards Euler;
- implicit schemes;
- Runge-Kutta schemes;
- multi-step methods;
- Adams-Bashforth techniques;
- local/global truncation errors;
- stability analysis;
- Matlab-based simulations;
- what optimisation is: examples and applications;
- revision of Hessians;
quadratic forms, eigenvalues and convexity;
- optimisation in one dimension;
- bracketing, golden section search;
- optimisation in two dimensions;
- one at a time method;
- steepest descents method, Newton-Raphson method;
- conjugate gradients; Matlab simulations;
- search-based methods;
- informal discussion of methods such as genetic algorithms, simulated annealing, and swarm-based optimisation;
- computer simulations.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 44 106 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 24 Lectures Scheduled learning and teaching activities 2 Tests Scheduled learning and teaching activities 3 Feedback Scheduled learning and teaching activities 4 Discussion sesssions Scheduled learning and teaching activities 11 Tutorials Guided independent study 106 Lecture and assessment preparation; wider reading

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Not applicable

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 20 80
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 80 2 hours 1, 3, 5, 6, 7 Available on request
Coursework – based on questions submitted for formative assessment 20 2 problem sheets 1, 2, 3, 4, 5, 6, 7, 8 Written comments on returned coursework, customized marksheet

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All above Written exam (100%) All August Ref/Def period

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk

Reading list for this module:

Type Author Title Edition Publisher Year ISBN Search
Set Gerald C.F. & Wheatley P.O. Applied Numerical Analysis 7th Anderson-Wesley 2004 978-8131717400 [Library]
Set Kharab A. & Guenther R.B. An Introduction To Numerical Methods: a MATLAB Approach Chapman & Hall 2012 978-1439868997 [Library]
Set Adby, P.R. & Dempster, M.A.H Introduction to Optimization Methods Chapman & Hall 1974 0-412-11040-7 [Library]
Set Press, W.H., Flannery, B.P., Teukolsky, S.A. & Vetterling, W.T Numerical Recipes: the Art of Scientific Computing 3rd edition Cambridge University Press 2007 13: 9780521880688 [Library]
Extended Iserles A. A first course in numerical analysis of differential equations Cambridge University Press 1996 000-0-521-55376-8 [Library]
Extended Yang, X-S Introduction to Computational Mathematics World Scientific 2008 13-978-981-281-81 [Library]
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES ECM1701, ECM1705
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 5 No Friday 09 January 2015 Friday 06 November 2015
KEY WORDS SEARCH Numerical analysis; differential equations; optimisation; minimisation; matrices; Gaussian elimination; MATLAB.