CTMS-MAT-13: Numerical Methods

This site will be the primary source of information and resources for the course.

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:

1. Describe the basic principles of discretization which are used in the numerical treatment of continuous problems;

2. 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;

3. 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;

4. 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, $n$-dimensional Euclidean vector space, rotation, translation, 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
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---
- 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 Equations

Assessment

Examination Type

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.

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

By submitting homework assignments, via Teams as a single pdf, you can improve this grade by up to 0.66 points, as bonus achievements. The bonus marks are only added if the exam is passed, and no more than 100% can be achieved.

Homework submission is voluntary, although highly recommended. It is possible to get a 100% final grade without submitting homework.

Homework 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. However, the submitted assignments should be written individually. No copying is allowed!

The two lowest homework scores will be discarded before the final homework score is calculated. This rule covers short illness, excursions, late adding of the course, and similar situations.

Note that all homework assignments carries equal marks.

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.

1. 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.

2. 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.