MATHS 770 : Advanced Numerical Analysis
2023 Semester Two (1235) (15 POINTS)
MATHS 770 is a core applied mathematics course for the BSc(Hons), BAdvSci and PGDipSci programmes and will also be of interest to students majoring in Computer Science, Engineering Science, Physics or Statistics. It is intended for all students who need to use or learn about advanced numerical methods for computational problems in applied mathematics. After successfully completing MATHS 770 students will be well prepared to decide which advanced method to use for a specific problem and to assess if they are using the method correctly. The course has been designed so that students do not need specialist knowledge to take it, but students are expected to have a working knowledge of a high-level computer language such as MATLAB or Python.
Capabilities Developed in this Course
|Capability 1:||Disciplinary Knowledge and Practice|
|Capability 2:||Critical Thinking|
|Capability 3:||Solution Seeking|
|Capability 4:||Communication and Engagement|
|Capability 5:||Independence and Integrity|
- Solve examples of the types of problems covered in the course using advanced numerical methods. (Capability 1, 2, 3 and 5)
- Demonstrate how to check the accuracy of numerical solutions calculated using the methods. (Capability 1, 2, 3 and 4)
- Critique alternative methods for solving a given problem. (Capability 1, 2, 4 and 5)
- Demonstrate the limitations of the methods. (Capability 1, 2, 3 and 4)
- Demonstrate the use of the methods in a high-level software package. (Capability 1, 4 and 5)
|Assignments||65%||Group & Individual Coursework|
|Assessment Type||Learning Outcome Addressed|
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- Linear algebra: eigenvalue problem; factoring — QR, SVD; solving banded systems.
- Conjugate gradient method for large sparse systems of linear algebraic equations.
- Approximation theory: minimax problem; continuous least squares problem.
- Optimisation: linear least squares — full-rank and rank deficient cases; nonlinear least squares — convergence results, Gauss-Newton, Levenberg-Marquardt method; quadratic objective function with linear constraints.
- Nonlinear algebraic systems: parameter continuation methods; pseudo-arclength continuation; quasi-Newton methods; approximation to the Jacobian.
- Interpolation: Hermite polynomial interpolation, Legendre polynomials, cubic splines.
- Solutions of ordinary differential equations: initial value problems; variable stepsize; stiff problems; two-point boundary value problems; shooting, collocation.
- Fourier transforms in one dimension: DFT versus FFT; calculating FFTs.
Students are expected to have a working knowledge of a high-level computer language such as MATLAB or Python. MATLAB is available in the University of Auckland's computer labs, or you can download MATLAB for home use by following the instructions here: https://www.software.auckland.ac.nz/en/matlab.html
This course is a standard 15-point course and students are expected to spend 10 hours per week involved in each 15-point course that they are enrolled in. For each week of this course, you can expect 3 hours of lectures, 3 hours of reading and thinking about the content and 4 hours of work on assignments.
- Attendance is expected at scheduled lectures, which will not normally be available as recordings.
- The activities for the course are scheduled as a standard weekly timetable.
Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).
Please remember that the recording of any class on a personal device requires the permission of the instructor.
A comprehensive reading list and additional lecture notes will be supplied. Part of the course will follow Chapter 1 of the textbook,
- Numerical continuation methods for dynamical systems: path following and boundary value problems edited by Bernd Krauskopf, Hinke M. Osinga & Jorge Galán-Vioque (Springer, 2007)
and several chapters from the textbook,
- Numerical computing with MATLAB by Cleve B. Moler (SIAM, 2004; or 2008)
During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.
At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.
Your feedback helps to improve the course and its delivery for all students.
The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.
Class representatives are students tasked with representing student issues to departments, faculties, and the wider university. If you have a complaint about this course, please contact your class rep who will know how to raise it in the right channels. See your departmental noticeboard for contact details for your class reps.
The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.
You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.
All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.
Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website http://disability.auckland.ac.nz
If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.
If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.
This should be done as soon as possible and no later than seven days after the affected test or exam date.
In the event of an unexpected disruption, we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, and if disruption occurs you should refer to the university website for information about how to proceed.
The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.
Student Charter and Responsibilities
The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html.
Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.
In this course students may be asked to submit coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.