# NUMERICAL AND COMPUTATIONAL METHODS - 2023/4

Module code: MAT2001

## Module Overview

When an analytical approach is not known or practical for solving a mathematical problem, which is the case for most real-world problems, a numerical approach can be useful in finding approximate solutions that are as close as possible to the exact one. This module introduces a selection of numerical methods for the solution of systems of linear and nonlinear equations, for finding a function that interpolates or approximates a set of data points, for finding numerical values of derivatives and integrals, and for solving initial value problems.

For each numerical method, we will consider the error that results from using approximations and introduce some theories of quantifying the error, which then indicates the accuracy of a numerical solution. We also analyse the complexity of numerical methods in order to estimate the computational resources required for achieving a given accuracy. Students will learn and practice implementing some of the methods in Python.

### Module provider

Mathematics

KLUS Stefan (Maths)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 95

Lecture Hours: 22

Seminar Hours: 6

Laboratory Hours: 5

Captured Content: 22

Semester 2

## Prerequisites / Co-requisites

MAT1041, MAT1030, MAT1034

## Module content

The module will consider the following:

• Algorithms and complexity

• Singular value decomposition

• Systems of linear equations (direct methods)

• Systems of nonlinear equations

• Interpolation and approximation

• Numerical integration

• Numerical solution of ordinary differential equations

• Systems of linear equations (iterative methods)

## Assessment pattern

Assessment type Unit of assessment Weighting
Coursework ASSESSED COURSEWORK 20
Examination EXAMINATION (2 hours) 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of and ability to derive, devise and analyse numerical methods.

• Subject knowledge through the recall of key definitions, theorems and their proofs.

• Analytical ability through the solution of unseen problems in the test and exam.

• Practical skills of implementing numerical methods in Python, and ability to understand and interpret given code.

Thus, the summative assessment for this module consists of:

• One piece of Assessed Coursework, worth 20% module mark

• One final Examination (2 hours), worth 80% of the module mark.

Formative assessment and feedback

Students receive written feedback via a number of marked coursework assignments which include computation projects over an 11 week period.  In addition, verbal feedback is provided by lecturer/teaching assistant at lab sessions.

## Module aims

• The aim of this module is to introduce students to a selection of numerical methods in terms of their derivation, their accuracy and efficiency and their implementation.

## Learning outcomes

 Attributes Developed 001 Demonstrate knowledge and literacy of the taught numerical methods; K 002 For basic numerical methods, prove their convergence and error bounds, and demonstrate understanding of their efficiency; KC 003 Derive and devise numerical schemes with specific details for a range of mathematical problems; KCT 004 Apply the above knowledge to determine the most suitable numerical method(s) for a practical problem; CPT 005 Implement some numerical Linear Algebra methods in Python, and gain proficiency in basic programming structures and methodology that can be used for general coding purposes. KPT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

• A detailed introduction to the derivation of numerical methods and the concepts of accuracy and efficiency

• Experience (through demonstration) of the methods used to find approximate solutions of basic mathematical problems

The learning and teaching methods include:

• 2 x 1 hour lectures per week x 11 weeks with examples to supplement the module lecture notes and Q&A opportunities for students.

• 1 x 1 hour seminar session every 2 weeks (weeks 1, 3, 5, 7, 9, 11) to go through additional examples.

• 1 x 1 hour lab session every 2 weeks (weeks 2, 4, 6, 8, 10) to provide hands-on learning experience of implementing numerical methods in Python.  Students are given lab sheets and template codes in order to solve practical problems. Instructor and teaching assistant(s) provide real time guidance, and students can also discuss with peers.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MAT2001

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics and Physics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MPhys 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics MMath 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Music BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2023/4 academic year.