$$ \def\eb{\boldsymbol{e}} \def\fb{\boldsymbol{f}} \def\hb{\boldsymbol{h}} \def\xb{\boldsymbol{x}} \def\Rb{\boldsymbol{R}} \def\Real{\mathbb{R}} \def\bfzero{\boldsymbol{0}} \newcommand{\ddy}[2]{\frac{\partial{#1}}{\partial{#2}}} $$
Introduction
Welcome to Computational Mathematics II!
This course aims to help you build skills and knowledge in using modern computational methods to do and apply mathematics. It will involve a blend of hands-on computing work and mathematical theory—this theory will include aspects of numerical analysis, computational algebra, and other topics within scientific computing. These areas consist of studying the mathematical properties of the computational representations of mathematical objects (numerical values as well as symbolic manipulations). The computing skills developed in this module will be valuable in all subsequent courses in your degree at Durham and well beyond. We will also introduce you to the use (and abuse) of various computational tools invaluable for doing mathematics, such as AI and searchable websites. While we will encourage you throughout to use all the tools at your disposal, it is imperative that you understand the details and scope of what you are doing! You will also develop your communication, presentation, and group-work skills through the various assessments involved in the course – more on that below!
This module has no final exam. In fact, there are no exams of any kind. Instead, the summative assessment and associated final grade are entirely based on coursework undertaken during the term. This means that you should expect to spend more time on this course during the term relative to your other modules. We believe this workload distribution is a better way to train the skills we are trying to develop, and as a bonus, you will not need to worry about this course any further once the term ends!
Content
The module’s content is divided into six chapters of roughly equal length; some will focus slightly more on theory, while others have a more practical and hands-on nature.
- Chapter 1: Introduction to Computational Mathematics
- Programming basics (including GitHub, and numerical versus symbolic computation)
- LaTeX, Overleaf, and presenting lab reports
- Finite-precision arithmetic, rounding error, symbolic representations
- Chapter 2: Continuous Functions
- Interpolation using polynomials – fitting curves to data (Lagrange polynomials, error estimates, convergence, and Chebyshev nodes)
- Solving nonlinear equations (bisection, fixed-point iteration, Newton’s method)
- Chapter 3: Linear Algebra
- Solving linear systems numerically (LU decomposition, Gaussian elimination, conditioning) and symbolically
- Applications: PageRank, computer graphics
- Chapter 4: Calculus
- Numerical differentiation (finite differences)
- Numerical integration (quadrature rules, Newton-Cotes formulae)
- Chapter 5: Ordinary Differential Equations (ODEs)
- Numerically approximating solutions of ODEs
- Timestepping: explicit and implicit methods
- Stability and convergence order
- Chapter 6: Selected Further Topics
- Intro. to random numbers and stochastic processes
- Intro. to partial differential equations
Weekly workflow and summative assessment
The final grade for this module is determined as follows:
- Weekly lab reports (weeks 1-6) – 20%
- Weekly e-assessments (weeks 1-6) – 30%
- Project (weeks 7-10) – 50%
Lab reports
Each week for the first six weeks of the course, we will release a short set of exercises based on the lectures from the previous week. Students will be expected to submit a brief report (1-2 pages A4, including figures) with their solutions to the set of exercises – the report will consist of written answers and figures/plots. The reports will be evaluated for correctness and quality of the presentation and communication (quality of figures, clarity of argumentation, etc.).
The lab report for a given week will be due at noon on Monday of the following week (e.g., week one’s lab report is due on Monday of week two and so on). Solutions and generalised feedback will be provided to the class on common mistakes and issues arising in each report. Students can also seek detailed feedback on their submission from the lecturers during drop-in sessions and office hours. There will be six lab reports in total, and your mark is based on your four highest-scoring submissions.
E-assessments
Each week for the first six weeks of the course, we will release a set of e-assessments based on the lectures from the previous week. These exercises are designed to complement the lab reports by focusing exclusively on coding skills. The e-assessments will involve submitting code auto-marked by an online grading tool, and hence give immediate feedback. As with the lab reports, the set of e-assessments for a given week will be due at noon on Monday of the following week. There will be six sets of e-assessments in total, and your mark is based on your four highest-scoring submissions.
Project
The single largest component of the assessment for this module is the project. Weeks 7-10 of this course focus exclusively on project work with lectures ending in Week 6. We will be releasing more detailed instructions on the project submission format and assessment criteria separately, but briefly, the main aspects of the project are as follows:
- There will be approximately eight different project options to choose from across different areas of mathematics (e.g., pure, applied, probability, mathematical physics, etc.); each project has a distinct member of the Maths Department as supervisor.
- Students will submit their preferred project options (ranked choice preferences) in Week 4 of the term and be allocated to projects by the end of Week 6 (there are maximum subscription numbers for each option to ensure equity of supervision).
- Each project consists of two parts: a guided component that is completed as part of a small group and an extension component that is open-ended and completed as an individual. Group allocations will be done by the lecturers.
- Each group will jointly submit a five-page report for the guided component of the project, and this is worth 60% of the project grade.
- Each student will also submit a three-page report and a six-minute video presentation on their extension component. This submission is worth 40% of the project grade.
In Weeks 7-10 of the term, lectures will be replaced by project workshop sessions during which students can discuss their project with the designated supervisor. This will be an opportunity to discuss progress, ask questions, and seek clarification. Each student only needs to attend the one project drop-in weekly session relevant to their project. Computing drop-in sessions will continue as scheduled in the first six weeks to provide additional support for coding pertinent tasks for the projects – there will be two timetabled computing drop-ins per week and students are encouraged to attend at least one of them.
Lectures, computing drop-ins & project workshops
Lectures will primarily present, explain, and discuss new material (especially theory), but will also feature computer demonstrations of the algorithms and numerical methods. As such, students are encouraged to bring their laptops to lectures to run the examples themselves. Students must bring a laptop or device capable of running code to the computer drop-ins to work on the e-assessments and lab reports.
| Week 1 |
Introductory lecture, 2 lectures |
Chapter 1 |
| Week 2 |
3 lectures, 1 computing drop-in |
Chapter 2 |
| Week 3 |
3 lectures, 1 computing drop-in |
Chapter 3 |
| Week 4 |
3 lectures, 1 computing drop-in |
Chapter 4 |
| Week 5 |
3 lectures, 1 computing drop-in |
Chapter 5 |
| Week 6 |
3 lectures, 1 computing drop-in |
Chapter 5/6 |
| Week 7 |
0 lectures, 1 project workshop |
Project |
| Week 8 |
0 lectures, 1 project workshop |
Project |
| Week 9 |
0 lectures, 1 project workshop |
Project |
| Week 10 |
0 lectures, 1 project workshop |
Project |
Acknowledgements
We are indebted to Prof. Anthony Yeates (Durham) whose numerical analysis notes formed the basis of several chapters of the coures notes.