CTMS-MAT-13: Numerical Methods
This site will be the primary source of information and resources for the course.
Lectures will take place in Conrad Naber Lecture Hall, Reimer Lüst Hall, at Wednesdays from 14:15
Lectures
All information regarding scheduling, as well as a link to this site, will also be found on CampusNet.
Course notes can be found here.
Intended Learning Outcomes
By the end of the module, students will be able to:
Describe the basic principles of discretization which are used in the numerical treatment of continuous problems;
Command the methods described in the content section of this module description to the extent that they can solve standard text-book problems reliably and with confidence;
Recognize mathematical terminology used in textbooks and research papers on numerical methods in the quantitative sciences, engineering, and mathematics to the extent that they fall into the content categories covered in this module;
- Understand the documentation of standard numerical library code and understand potential limitations and caveats of such algorithms.
Knowledge, Abilities, or Skills
Knowledge of calculus: functions, inverse functions, sets, real numbers, sequences and limits, polynomials, rational functions, trigonometric functions, logarithm and exponential function, parametric equations, tangent lines, graphs, derivatives, anti-derivatives, elementary techniques for solving equations
Knowledge of linear algebra: vectors, matrices, lines, planes, rotations, translations, $n$-dimensional Euclidean vector spaces, dot product (scalar product), cross product, normal vectors, eigenvalues, eigenvectors, elementary techniques for solving systems of linear equations
Some examples will be presented as python notebooks, but no knowledge of python is required, nor will there be any assessment of ability to code in the course.
Recommendations for Preparation
Taking “Calculus and Elements of Linear Algebra II” (CTMS-MAT-10) before taking this module is recommended, but is not a pre-requisite.
A thorough review of “Calculus and Elements of Linear Algebra” (CTMS-MAT-09), with emphasis on the topics listed as “Content and Educational Aims” is recommended.
Usability and Relationship to other Modules
The module is a mandatory / mandatory elective module of the Methods and Skills area that is part of the Constructor Track (Methods and Skills modules; Community Impact Project module; Language modules; Big Questions modules).
This module is a co-recommendation for the module “Applied Dynamical Systems Lab” (CA-S-MAT-810) in which the actual implementation in a high level programming language of the methods will be covered.
Mandatory for ECE. Mandatory elective for CS and RIS.
Text Books
There is no primary or required book, but there are many suitable books, such as:
“An Introduction to Numerical Methods and Analysis” - J. F. Epperson (2013) Wiley 2nd Edition. ISBN: 978-1118367599.
“Numerical Analysis” - R. L. Burden and J. D. Faires (2011) Brooks/Cole. 9th Edition. ISBN: 978-0538733519.
Course Outline
The following topics will be covered:
- Taylor series
- number representations
- Gaussian elimination
- LU decomposition
- Cholesky decomposition
- iterative methods
- bisection method
- Newton’s method
- secant method
- polynomial interpolation
- Aitken’s algorithm
- Lagrange interpolation
- Newton interpolation
- piecewise interpolation
- spline interpolation
- B-splines
- least squares approximation
- difference schemes
- Richardson extrapolation
- quadrature rules: the trapezium and Simpson’s methods
- Romberg method
- time stepping schemes for ordinary differential equations
- Runge-Kutta schemes
- finite difference method for partial differential equations
Itinerary
Provisionally, the course will run as follows
Structure
The course content has the following structure:
---
markmap:
zoom: false
pan: false
---
- Numerical Methods
- Errors
- [Taylor Series](/docs/numericalmethods/part1/taylor/)
- [Number Representations](docs/numericalmethods24/part1/number-representations/)
- Algebraic Equations
- Linear Equations
- [Direct Methods](/docs/numericalmethods/part1/linear-equations/#direct-methods)
- Gaussian Elimination
- Partial Pivoting
- LU Decomposition
- Cholesky Decomposition
- [Iterative Methods](/docs/numericalmethods/part1/linear-equations/#indirect-methods)
- Jacobi
- Gauss-Seidel
- SOR
- [Nonlinear Equations](/docs/numericalmethods/part1/nonlinear-equations/)
- Bisection Method
- Newton Method
- Secant Method
- Approximations
- [Interpolation](/docs/numericalmethods/part2/interpolation/)
- Polynomial
- Lagrange
- Newton
- B-splines
- [Integration](/docs/numericalmethods/part2/integration/#numerical-integration)
- Simpson
- Trapezium
- Romberg Algorithm
- Gaussian Quadrature
- [Solutions](/docs/numericalmethods/part3/odes/)
- Difference Schemes
- Ordinary Differential Equations
- Partial Differential EquationsAssessment
There will be two short quizzes held, in person, covering the course material. This is as a bonus achievement, so that you can improve this grade by up to 0.66 points.
The duration of the tests will be 1 hour. Only the highest score will count as a bonus mark. This rule covers short illnesses, excursions, late adding of the course, and similar situations.
Taking part in the quizzes is voluntary, although highly recommended.
The bonus marks are only added if the exam is passed, and no more than 100% can be achieved.
Details of when the quizzes will be held will be announced in lectures and online.
Examination Type
| Examination Type: | Module examination |
| Assessment Type: | Written examination |
| Scope: | All intended learning outcomes of this module |
| Duration: | 120 min |
| Weight: | 100% |
You have three attempts to pass the module. Once you pass the module, no further retakes of the exam are possible. (See Academic Policies for more details). The exams in this course are offered twice per year: firstly in May and then a resit in August.
Completion of the quizzes is not required to take the exam.
No supplementary material can be brought to the exam. A calculator is necessary however. Note that graphical and scientific calculators are permitted.
The pass mark is 45%.
Previous exams, as well as sample questions and solutions, can be found here.
Assignments
Problem sheets will be assigned approximately every two weeks. Homework assignments are posted on here and on Teams approximately ten days before the due date.
You are encouraged to discuss homework between each other and in the tutorials online.
The problems on the exam will be similar to the ones from homework. So, by doing homework you prepare for the exam - maths is not a spectator sport.
Academic Integrity
All involved parties (lecturers, instructors and students) are expected to abide by the word and spirit of the Code of Academic Integrity. Violations of the Code should be brought to the attention of the Academic Integrity Committee.
Artificial Intelligence Use Policy
This policy covers any generative AI tool, such as ChatGPT, Elicit, etc. This includes text, slides, artwork/graphics/video/audio and other products.
You are discouraged from using AI tools unless under direct instruction from your instructor to do so. Please contact your instructor if you are unsure or have questions before using AI for any assignment.
Note that the material generated by these programs may be inaccurate, incomplete, or otherwise problematic. Their use may also stifle your own independent thinking and creativity. Accordingly, reduction in the grade is likely when using AI.
If any part of this AI policy is confusing or uncertain, please reach out to your instructor for a conversation before submitting your work.
In the words of Terence Tao, when asked as to why there is value in a human approach:
These problems are like distant locations that you would hike to. And in the past, you would have to go on a journey. You can lay down trail markers that other people could follow, and you could make maps.
AI tools are like taking a helicopter to drop you off at the site. You miss all the benefits of the journey itself. You just get right to the destination, which actually was only just a part of the value of solving these problems.