
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 industrystandard 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 builtin ones. Be fluent in exploring Matlab’s capabilities, such as using matrices as the fundamental datastorage unit, array manipulation, control flow, script and function mfiles, function handles, graphical output. Write programs for solving linear and nonlinear systems, including those arising from boundary value problems and integral equations, and for rootfinding 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 builtin and userdefined 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 GaussSeidel. 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 piecewisepolynomial interpolants, Local and low order interpolants (basis of FEM), Leastsquares, 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, predictorCorrector and RungeKutta 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.