Module manager: Dr Toni Lassila
Email: T.Lassila@leeds.ac.uk
Taught: Semester 1 (Sep to Jan) View Timetable
Year running 2024/25
This module is not approved as a discovery module
This module covers a range of numerical algorithms for computational problems that can be formulated as continuous nonlinear equations and large linear equation systems. Starting from standard algorithms, such as the Newton-Raphson method, we cover the various properties and difficulties faced by such numerical algorithms (robustness, complexity, generality) and ways to tackle these problems reliably. By introducing the solution of nonlinear partial differential equations through numerical discretisation techniques, we demonstrate how state-of-the-art numerical methods can be designed to achieve maximum efficiency and allow the solution of large, sparse systems of linear and nonlinear systems of equations.
On completion of this module, students should be able to:
- understand the role of computational methods in scientific computing and the importance of robustness, computational efficiency, stability and accuracy of numerical methods
- demonstrate awareness of the state-of-the-art in scientific computing algorithms for the solution of nonlinear and linear problems and identify the best methods for a given problem
- use sparsity and sparse data structures to develop and analyse efficient implementation of numerical methods
- perform discretisation to systems of differential equations to approximate their solutions numerically
- understand the practical issues associated with code implementation of numerical methods and program basic implementations of numerical methods themselves
- develop awareness of typical applications for numerical analysis software in engineering, computer science and computational mathematics
- analyse the computational complexity of a numerical method for a given system of equations
1. Formulate and solve systems of nonlinear equations to solve challenging real-world problems arising from engineering and computational science.
2. Analyse a given nonlinear equation and choose and implement the best numerical method for its solution.
3. Implement algorithmic solutions to solve computational differential equation problems based on mathematical theory.
4. Analyse computational linear algebra problems to identify and implement the most efficient and scalable solution algorithm to apply for large problems.
5. Develop a broad understanding of the theory of numerical analysis and its applications to the solution of nonlinear and differential equations.
Solving one nonlinear equation: Newton's method, bisection method, secant method, Dekker’s method; Issues: divergence of Newton's method, finding an initial guess, local convergence; Solving systems of equations: Jacobian computation, Gradient descent method, Newton with line-search, continuation methods; Partial differential equations: discretisation in space and time; time-stepping algorithms, sparse Jacobian computation; Linear solvers for large systems: Sparsity: computational complexity, pivoting, reordering; Direct methods: LU factorisation; Iterative linear solvers: Gauss-Seidel, Jacobi, conjugate gradient; Inexact Newton-Krylov method.
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Laboratory | 10 | 2 | 20 |
| Lectures | 20 | 1 | 20 |
| Independent online learning hours | 20 | ||
| Private study hours | 90 | ||
| Total Contact hours | 40 | ||
| Total hours (100hr per 10 credits) | 150 | ||
Private study consists of 3 hours of review of the lecture/tutorial materials per week (total 30 hours), 20 hours of review for the final exam, and 60 hours for each of the two coursework assessments.
Attendance and formative coursework.
| Assessment type | Notes | % of formal assessment |
|---|---|---|
| In-course Assessment | Coursework 1 | 20 |
| In-course Assessment | Coursework 2 | 20 |
| Total percentage (Assessment Coursework) | 40 | |
Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated
| Exam type | Exam duration | % of formal assessment |
|---|---|---|
| Standard exam (closed essays, MCQs etc) | 2.0 Hrs 0 Mins | 60 |
| Total percentage (Assessment Exams) | 60 | |
The module will be reassessed by exam only.
The reading list is available from the Library website
Last updated: 9/25/2024
Errors, omissions, failed links etc should be notified to the Catalogue Team