2024/25 Undergraduate Module Catalogue

MATH1013 Computational Mathematics and Modelling

20 creditsClass Size: 450

Module manager: Daniel Read; Rob Sturman; Grant Lythe
Email: d.j.read@leeds.ac.uk; R.Sturman@leeds.ac.uk

Taught: Semesters 1 & 2 (Sep to Jun) View Timetable

Year running 2024/25

Pre-requisite qualifications

Grade B in A-level Mathematics or equivalent.

This module is not approved as a discovery module

Module summary

This module introduces students to computational techniques, algorithms and numerical solutions, as well as to the mathematics of discrete systems. Students will learn basic programming using the language Python and apply computational techniques to the solution of mathematical problems.

Objectives

This module will give students a grounding in computational techniques used to address mathematical problems, as well as a field of mathematics that is very well-suited to computational modelling and solution.

Learning outcomes
On successful completion of the module students will have demonstrated the following learning outcomes relevant to the subject:
1. Demonstrate a broad understanding of the concepts, information, practical competencies and techniques of computational mathematics.
2. Demonstrate a reasonable level of skill in calculation and manipulation within this basic body of knowledge.
3. Apply core concepts and principles in well-defined contexts.
4. Appreciate the coherence, logical structure and broad applicability of mathematics.
5. Demonstrate an awareness of skills in comprehending problems, formulating them mathematically and obtaining solutions by appropriate methods.
6. Use a range of techniques to initiate and undertake problem solving.
7. Formulate, analyse and solve difference equations in practical settings.
8. Describe how complex behaviour can emerge from apparently simple discrete systems.
9. Show how discrete systems can arise as a means of solving continuum systems.
10. Use a computer programming language to solve and visualise solutions of discrete systems.
11. Conduct a computational investigation and communicate the results.

Skills Learning Outcomes
SOL1. Be able to formulate problems mathematically and use technical skills to solve them.
SOL2. Write and use computer programs to solve problems.
SOL3. Understand how complex behaviour can manifest in systems.
SOL4. Find and utilise existing solutions in your own work.

Syllabus

The following topics will be covered:
1. Introduction to the principles of computational mathematics.
Fundamentals of syntax, structure, and file management in Python.
2. Loops, functions, control flow statements.
3. Data types (how a computer understands intergers, rationals, irrationals).
4. Coding of simple algorithms, such as Euclid's algorithm, the Sieve of Eratosthenes, continued fraction algorithm.
5. Computational solution to more advanced problems, such as sorting algorithms and random number generation.
6. Plotting and visualisation.
7. Limitations of computational methods.
8. First-order linear difference equations, including general solutions for various inhomogeneous cases. Applications (e.g., loans, mortgages).
9. First-order nonlinear difference equations. Cobweb diagrams, fixed points, stability, cycles. The logistic equation, and introduction to chaos.
10. Computational implementation in python.
11. Link to numerical algorithms. Newton-Raphson method, including rate of convergence. Simple discretisation of ODEs.
12. Systems in the complex plane. Visualisation of fractals using python.
13. Second-order difference equations (linear and nonlinear).
14. Fibonacci sequence. Applications in biology, physics, and finance.
15. Matrix difference equations, and solution using eigenvectors. Applications.

Additional topics that build on these may be covered as time allows. Further details of possible topics will be delivered closer to the time that the module runs.

Methods of Assessment

We are currently refreshing our modules to make sure students have the best possible experience. Full assessment details for this module are not available before the start of the academic year, at which time details of the assessment(s) will be provided.

Assessment for this module will consist of;

4 x Coursework
2 x Exam

Teaching methods

Delivery type	Number	Length hours	Student hours
Lecture	22	1.00	33.00
Practical	12	2.00	24.00
Seminar	5	1.00	5.00
Independent online learning hours			46.00
Private study hours			92.00
Total Contact hours			62.00
Total hours (100hr per 10 credits)			200.00

Opportunities for Formative Feedback

Regular practicals and example sheets.

Reading list

The reading list is available from the Library website

Last updated: 02/09/2024

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