Introduction to Numerical Analysis and its Applications

Cod y Modiwl: MA25200
Teitl y Modiwl: Introduction to Numerical Analysis and its Applications
Blwyddyn Academaidd: 2027/2028
Semester: Semester 1 Dysgwyd dros 2 semester
Cyd-gysylltydd y Modiwl: Dr Tudur Davies
Rhestr Ddarllen: Gweld ar Aspire
Rhagofynion: MA11110 neu MT11110
Anghymharus (Unrhyw Flwyddyn Acad): CC24520 neu CS24520
Anghymharus (Unrhyw Flwyddyn Acad): MT25220
Anghymharus (Unrhyw Flwyddyn Acad)
Staff Eraill sy'n Cyfrannu: Dr Tudur Davies

Dulliau Asesu

Math o Asesiad	Manylion Asesiad	Cyfran
Asesiad Semester	Coursework: Semester 1 Worksheets (30%)Semester 2 Assignments (20%)	50%
Arholiad Semester	Written Examination: 2 Awr	50%
Asesiad Ailsefyll	Supplementary Coursework: Supplementary worksheet	50%
Arholiad Ailsefyll	Written Examination: 2 Awr	50%

Canlyniadau Dysgu

Wedi cwblhau'r modiwl dylai'r myfyrwyr fedru:

Demonstrate an ability to write short programs in Python.
Demonstrate an awareness of good practice in developing computer programs for numerical computation.
Construct a polynomial or a spline that interpolates a set of data.
Derive and apply standard numerical methods for approximating an integral and analyse the errors of these methods.
Determine the root(s) of a non-linear equation using standard numerical methods, and demonstrate an awareness of the order of convergence of these methods.
Compute numerical approximations to the solution of initial value problems using one-step methods, and demonstrate an awareness of the convergence and stability of these methods.

Disgrifiad cryno

It is often impossible to find the exact solution of a mathematical problem using standard techniques. In these situations one has to resort to numerical techniques and the use of a computer. Numerical analysis is concerned with the development and analysis of methods for the numerical solution of practical problems. This course will provide an introduction to the Python programming language and, more broadly, the Scientific Python Stack. It will then introduce techniques for the numerical approximation of mathematical problems and the analysis of these techniques, using Python to provide practical application of the numerical methods.

Nod

To teach students programming skills and introduce them to numerical analysis and its applications

Cynnwys

Semester 1:

1. INTRODUCTION TO PYTHON: Types, variables, if-statements, for- and while-loops. The interpreter and the evaluation of simple expressions. Editing, saving and loading code.

2. DATA STRUCTURES IN PYTHON: Lists and numpy arrays.

3. FUNCTIONS IN PYTHON. Function definition and function calling.

4. ORGANISING PYTHON CODE. Generating documentation, catching and handling exceptions, organising code into modules.

5. PLOTTING IN PYTHON: Manipulate data and plot results.

Semester 2:

1. POLYNOMIAL INTERPOLATION: Lagrange's formula. Newton's formula and divided differences. The forward difference formula. Interpolation error. Cubic spline interpolation.

2. NUMERICAL INTEGRATION: Trapezoidal rule, Simpson's rule. Composite integration rules. Quadrature errors. Romberg integration. Gaussian quadrature rules.

3. SOLUTION OF NONLINEAR EQUATION IN A SINGLE VARIABLE: Bisection method. Fixed point methods and contraction mappings. Newton's method. Order of convergence.

4. INITIAL VALUE PROBLEMS: Existence and uniqueness of solutions. Euler's method. Local truncation error. Consistency. Convergence. Stability. General one-step methods. Trapezoidal method. Predictor-corrector methods.

Sgiliau Modiwl

Math o Sgiliau	Manylion Sgiliau
Cyfathrebu	Written answers to exercises and documenting of code must be clear and well-structured. Good listening skills are essential to successful progress in this course.
Datblygu personol a chynllunio gyrfa	Completion of tasks (worksheets and problem sheets) to set deadlines will aid personal development. Students will develop highly sought after programming skills.
Datrys Problemau	Exercises will be set and marked. These will involve the derivation and implementation of numerical methods and their errors.
Gwella dysgu a pherfformiad ei hun	Students will be expected to develop their own approach to time-management in their attitude to the completion of work on time, and in doing the necessary preparation between lectures and workshops. They should act on feedback, for example the in-practical feedback from demonstrators and from their marked work.
Rhifedd	Necessary throughout.
Sgiliau pwnc penodol	Programming skills, debugging skills, numeracy skills, numerical analysis skills.
Sgiliau ymchwil	Using a computer. Searching the language and library documentation.
Technoleg Gwybodaeth	Students will be set exercises in workshops that involve developing computer code. They will need to be able to use computers and library facilities throughout.

Nodau

Mae'r modiwl hwn yn cydymffurfio a FfCChC Lefel 5