ACS234 Mathematics and Data Modelling

Module Description

This module provides an introduction to the use of analytical mathematical techniques and numerical methods and algorithms for subsequent higher level module studies and for solving a wide range of engineering problems as well. Students will develop their skills in the theory and application of core mathematics tools required for systems engineering and the application of these in system simulation and data based modelling. A brief summary of topics covered includes: complex variables and Fourier transforms, analysis of matrices and systems represented by matrices, optimisation of functions of many variables, probability, numerical integration techniques and data modelling and analysis. The module is embedded throughout with engineering examples using the mathematical techniques.

Credits: 20 (Academic Year)

Module Leader

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Dr Hua-Liang Wei
Amy Johnson Building

If you have any questions about the module please talk to us during the lectures or the labs in the first instance. It is likely that other students will learn from any questions you ask as well, so don’t be afraid to ask questions.

Outside of lectures please contact one of us via email, or drop in to see one of us.

Other teaching staff

Professor Lucy Wyatt
Email: l.wyatt@sheffield.ac.uk

Dr Eleanor Stillman
Email: e.c.stillman@sheffield.ac.uk

Learning Outcomes

Learning Outcomes

By the end of the module students will be able to:

  1. Solve systems and control engineering-related problems using fundamental and advanced mathematical methods. [SM2m, SM3p]
  2. Explain and apply fundamental statistical methods that underpin the solution of systems and control engineering problems. [SM2p, D3p]
  3. Explain and choose between numerical methods for simulating problems in systems and control engineering. [SM2p, EA3p]

This module satisfies the AHEP3 (Accreditation of Higher Education Programmes, Third Edition) Learning Outcomes that are listed in brackets after each learning outcome above. For further details on AHEP3 Learning Outcomes, see the downloads section of our accreditation webpage.

Syllabus


The module will cover the following topics/themes:

Probability
Sample space, event, algebra of events, probability definition (axiomatic and linked to relative frequency), conditionality, Bayes’ Theorem, the notion of a random variable, discrete and continuous distributions, density functions, joint distributions (bi-variate, general multivariate), bivariate and multivariate Normal density, marginal density, expectation, moments and their key properties (esp. correlation), Bayes and conditionality for continuous RVs

Matrix Algebra – review
Linear equations and matrices. Matrix manipulation, determinants, inverse, rank.

Linear Algebra and Vector Spaces
Definition of a vector space. Examples. Linear independence/independence and basis. Dimension of a space.

Linear Transformations
Matrix representation of linear transformations. Examples. Null space and range space.

Solution of Linear Equations
Conditions for existence and uniqueness of solutions, in terms of the null and range space, in terms of rank. Least squares solution.

Eigenvalues and Eigenvectors
Characteristic polynomial. Evaluation of eigenvalues and eigenvectors. Jordan canonical form. Scalar product, norm distance. Orthogonality. Symmetric matrices and orthogonality.

Matrix Polynomials and Exponential
Evaluation in terms of eigenvalues. Cayley-Hamilton theorem. Solution of systems of linear equations. Applications in state space modelling.

Optimization
Quadratic forms and definiteness criteria. Partial differentiation. Maxima and minima of functions of two variables. Conditions for stationary points/maxima/minima/saddle points. Extension to functions of many variables.

Least squares modelling
Application of matrix algebra and optimisation to solve real engineering problems.

Polynomial interpolation methods
Lagrange interpolating polynomials; Newton’s interpolating polynomials; Splines; Problem solving with Matlab (in Lab).

Linear regression
Statistical review; Simple linear regression; Linear least squares method; Multiple linear regression; Problem solving with Matlab (in Lab).

Generalised linear regression
Polynomial regression; General linear least squares; Problem solving with Matlab (in Lab)

Nonlinear regression
Nonlinear regression; Nonlinear least squares and nonlinear; Simple nonlinear optimisation; Problem solving with Matlab (in Lab).

Numerical methods for ordinary differential equations
Euler’s method; Improved Euler’s method; Runge-Kutta methods; Problem solving with Matlab.

Numerical integration formulas
Newton-Cotes formulas; the Trapezoidal rule; Simpson’s rules; Higher-order Newton-Cotes formulas; Open methods and multiple integrals; Problem solving with Matlab.

Numerical differentiation
High-accuracy differentiation formulas; Richardson extrapolation; Derivatives of unequally spaced data; Partial derivatives; Problem solving with Matlab.

Teaching Methods

Learning and Teaching Methods

Lectures: 44 hours
Tutorials: 16 hours
Independent Study: 140 hours

Teaching Materials

Learning and Teaching Materials

All teaching materials will be available via MOLE.

Assessment

Assessment

Coursework (20%)
One 2 hour written examination (80%)

The resit for this module is usually by examination only.

Feedback

Feedback

  • Tutorials
  • Question-and-answer sessions
  • Other communications (email, MOLE, etc)
Student Evaluation

Student Evaluation

Students are encouraged to provide feedback during the module direct to the lecturer. Students will also have the opportunity to provide formal feedback via the Faculty of Engineering Student Evaluation Survey at the end of the module.

Recommended Reading

Recommended Reading

  • K.A. Stroud, Advanced engineering mathematics (5th edition), Palgrave Macmillan, 2011 [available in Information Commons, 510.2462 (S)]
  • E. Kreyszig, Erwin Advanced Engineering Mathematics, Wiley, 2011 [available in Information Commons, 510.2462 (K)]
  • Steve C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists. The McGraw-Hill Companies, 2008 (2nd ed.) [available in Information Commons, 518.0285 (C)]
  • Yang, W. Y., Cao, W., Chung, T. S., Morris, J., Applied Numerical Methods Using MATLAB. John Wiley & Sons, Inc., Hoboken, New Jersey, 2005 [available in Information Commons, 518.0285 (Y)]