35006 Numerical Methods
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Credit points: 6 cp
Result type: Grade and marks
Requisite(s): (33230 Mathematics 2 OR 33290 Statistics and Mathematics for Science OR 37132 Introduction to Mathematical Analysis and Modelling) AND (37171 Introduction to Programming OR 41039 Programming 1)
These requisites may not apply to students in certain courses. See access conditions.
Description
This subject covers the theory and practice of numerical methods in modern mathematical computing. The subject addresses the main numerical techniques in calculus, minimisation and optimisation, numerical integration and Monte Carlo methods, the numerical solution of large linear systems, and the numerical solution to ordinary and partial differential equations. Students learn the theoretical underpinnings of these algorithms, with a focus given throughout the subject on practical implementation and coding. The Python programming language is used throughout the subject.
Subject learning objectives (SLOs)
Upon successful completion of this subject students should be able to:
| 1. | Describe the main algorithms for solving a range of important mathematical problems. |
|---|---|
| 2. | Design and implement code to solve a range of problems numerically. |
| 3. | Explain why various numerical algorithms work, and what their limitations are. |
Course intended learning outcomes (CILOs)
This subject also contributes specifically to the development of following course intended learning outcomes:
- Demonstrate theoretical and technical knowledge of mathematical sciences including calculus, discrete mathematics, linear algebra, probability, statistics and quantitative management. (1.1)
- Evaluate mathematical and statistical approaches to problem solving, analysis, application, and critical thinking to make mathematical arguments, and conduct experiments based on analytical, numerical, statistical, algorithms to solve new problems. (2.1)
- Work autonomously or in teams to demonstrate professional and responsible analysis of real-life problems that require application of mathematics and statistics. (3.1)
- Design creative solutions to contemporary mathematical sciences-related issues by incorporating innovative methods, reflective practices and self-directed learning. (4.1)
- Use succinct and accurate presentation of reasoning and conclusions to communicate mathematical solutions, and their implications, to a variety of audiences, using a variety of approaches. (5.1)
Contribution to the development of graduate attributes
GA1: Disciplinary knowledge
This will be achieved through the analysis of the effectiveness of various numerical methods, and through the writing of the appropriate code for the solution of problems to a specified accuracy.
GA2: Research, Inquiry and Critical Thinking
The check-in Assignments will require the students to synthesize solutions to unseen problems. They will also be required to analyse the accuracy of their own solutions.
GA3: Professional, Ethical and Social Responsibilty
Students will be expected to apply high-level professional conduct and responsibility in the performance of their own work throughout the semester and especially in the computer labs.
GA4: Reflection, Innovation and Critical Thinking
The final project will require the students to apply solutions learned throughout the semester to an original and authentic problem. They will also be required to analyse the performance of their project code, as well as critically assess code for structure and accuracy.
GA5: Communication
Students will present their final project in the form of an oral report, which must clearly outline their problem, how this was approached and how their code works.
Teaching and learning strategies
Subject content will be presented in a combination of online materials and in the workshops.
Workshops will take the form of an interactive session (2 hours per week) where the material is covered in depth.
Students are expected to revise the online material before each workshop.
Computer labs (2 hours per week) will focus on the practical implementation of numerical methods.
Direct feedback will be provided during the computer labs. Further feedback on progress will be provided using the check-in Assignments which are spaced throughout the semester.
Students will be encouraged to develop code-sharing practices in the computer labs, and to tackle problems collaboratively, as well as being able to work on solving problems individually. A central aim of this is to prepare students for real-world coding environments, which consist of a mix of collaboration with intense periods of individual work.
The final project will enable the students to tackle an authentic and challenging problem in science or mathematics that can be approached using the methods given in this subject.
Content (topics)
- Root finding and Numerical solution to nonlinear equations. Newton’s method. Bisection and related methods.
- Numerical differentiation. Minimisation and maximisation in 1D
- Numerical optimisation in higher dimensions – gradient descent, simplex methods
- Interpolation and extrapolation
- Integration in 1D
- Integration in multiple dimensions; Monte Carlo methods
- Numerical solution of linear systems
- Methods for sparse linear systems
- Solution to ODEs – Euler, Runge Kutta, Predictor-Corrector methods
- Numerical solution of ODEs and PDEs by finite differences
- The Fast Fourier Transform (time permitting)
Assessment
Assessment task 1: Check-in Assignments (4 throughout the semester)
| Intent: | This Task will address Graduate attributes 1: Disciplinary Knowledge 2: Research, Inquiry and Critical Thinking 3: Professional, Ethical and Social Responsibilty |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2 and 3 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1 and 3.1 |
| Type: | Exercises |
| Groupwork: | Individual |
| Weight: | 50% |
| Criteria: | Correct implementation and explanation of numerical algorithms |
Assessment task 2: Final Project
| Intent: | This task will address Graduate Attributes: 1: Disciplinary Knowledge 2: Research, Inquiry and Critical Thinking 4: Reflection, Innovation and Critical Thinking 5: Communication |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2 and 3 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 4.1 and 5.1 |
| Type: | Project |
| Groupwork: | Individual |
| Weight: | 40% |
| Criteria: | Appropriate approach chosen for the problem Effectiveness of practical implementation Accuracy of the solution Comprehensibility of explanation |
Assessment task 3: Computer Labs
| Intent: | The Labs will address graduate attributes: 2. Research, inquiry and critical thinking 3. Professional, ethical and social responsibility |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1 and 2 This assessment task contributes to the development of course intended learning outcome(s): 2.1 and 3.1 |
| Type: | Laboratory/practical |
| Groupwork: | Individual |
| Weight: | 10% |
| Criteria: | Engagement with computational methods Reflection of effectiveness of practical implementation |
Minimum requirements
Students must obtain a minimum of 50% to pass the subject.