Aims

This module aims to introduce new numerical and statistical solution techniques and
to apply previous mathematical methods in MATH199 and MATH299 to problems encountered
in engineering analysis. Computer programming and applications will be the key feature.
It will introduce students to the industry-standard programming environment MATLAB throughout the module.


Learning Outcomes

At the end of this unit students will be able to: 

Use Matlab as a convenient tool for solving a broad range of practical problems in engineering from simple models to real examples.
Write programs using first principles without automatic use of built-in ones.
Be fluent in exploring Matlab’s capabilities, such as  using matrices as the fundamental data-storage unit, array manipulation, control flow, script and function m-files, function handles, graphical output.
Write programs for solving linear and nonlinear systems, including those arising from boundary value problems and integral equations, and for root-finding and interpolation, including piecewise approximations.
Write Matlab functions for various solvers appropriately, including an understanding of choice of methods, control of accuracy, event detection
B e able to integrate functions of both one and many variables, paying particular attention to special cases such as interpolating or approximating functions derived from data, both using built-in and user-defined routines
Be confident about probabilistic and statistical analysis of engineering data.
Understand stability and conditioning issues in numerical methods
Make use of Maltab visual capabilities for all engineering applications.

Syllabus

36 	
Introduction to MATLAB and practical numerical analysis
Variables versus vectors: max, min, mean / median - matrices versus functional equations, vectorization, solving linear equations, looping & branching, plotting: 2D, 3D, histogram

Numerical solution of linear systems and Eigensystems.
Jacobi and Gauss-Seidel.

Power methods: eig and selected eigenvalues: eigs

Numerical solution of nonlinear algebraic equations
Interval bisection, Newton's method; programming for multivariable nonlinear systems. Jacobi matrix.

Interpolation, Curve fitting of  functions and Numerical integration
Global polynomial and piecewise-polynomial interpolants, Local and low order interpolants (basis of FEM), Least-squares, fas t Fourier transform. Trapezium and Simpson's rule.

Probability and regression analysis
Discrete events versus continuous variables. Probability distribution functions: Gaussian or normal, binomial, geometric and exponential. Applications and simulations.

Numerical differentiation and Numerical solution of DEs
PDEs: Finite difference schemes and simple boundary value problems. Poisson equations. Linear systems.

ODEs: Explicit and implicit Euler, midpoint rule, trapezoidal, predictor-Corrector and Runge-Kutta methods. Reduction to ODE systems. Implicit versus Nonlinear solvers.


Teaching and Learning Strategies

See http://www.liv.ac.uk/maths/Current_Student/DMS_Learning_&_Teaching_Strategy.html


Teaching Schedule

Study Hours	 24
The class is divided into 2 separate groups so that teaching is manageable, 
sharing the syllabus and CA/Exam questions. Tutorial classes will be manned 
by a mix of experimental officers and PG helpers (both Engineering Dept and Math Sci).
 Students cannot change groups to monitor attendance.
 There are 4 contact hours per week (2 lectures, 1 interactive lab lecture and 1H lab work). 
 	  	 24

 	  	48
Timetable (if known)	  	  	  	  	  	  	 

Private Study	102
TOTAL HOURS	150

Assessment

EXAM 52%
CA   48% from
6 class tests 	 
	First and second semester 	45 	2 hour class test in a PC centre 	Standard university policy 	
CA components of home assignments will be assessed using an Automated Matlab Marking system developed at Loughborough University. Home work will be simple engineering problems with contexts.  

Recommended Texts:
  To be advised later. No single book is suitable due to the spread of topics.

Reference Books:

[1] Griffiths D V and Smith I M, Numerical Methods for Engineers, Blackwell, 1991.

[2] Laurene Fausett, Applied Numerical Analysis Using MATLAB, Pearson 2008.

[3] Moin  P, Fundamentals of Engineering Numerical Analysis, Cambridge University Press, 2001.

[4] Wilson HB, Turcotte LH, Advanced mathematics and mechanics applications using MATLAB.  CRC Press, 1997

[5] Ke Chen, Peter Giblin and Alan Irving , Mathematical Exploration with MATLAB,  Cambridge University Press, 1999.