# MATHS 770 : Advanced Numerical Analysis

## Science

### Course Prescription

Covers the use, implementation and analysis of efficient and reliable numerical algorithms for solving several classes of mathematical problems. The course assumes students have done an undergraduate course in numerical methods and can use Matlab or other high-level computational language.

### Course Overview

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.​

### Course Requirements

Prerequisite: B- in MATHS 270, 340 and 361

### Capabilities Developed in this Course

 Capability 3: Knowledge and Practice Capability 4: Critical Thinking Capability 5: Solution Seeking Capability 6: Communication Capability 7: Collaboration Capability 8: Ethics and Professionalism

### Learning Outcomes

By the end of this course, students will be able to:
1. Solve examples of the types of problems covered in the course using advanced numerical methods. (Capability 3, 4 and 5)
2. Demonstrate how to check the accuracy of numerical solutions calculated using the methods. (Capability 3, 4, 6 and 8)
3. Critique alternative methods for solving a given problem. (Capability 3, 4, 6, 7 and 8)
4. Demonstrate the limitations of the methods. (Capability 3, 4, 5, 6 and 7)
5. Demonstrate the use of the methods in a high-level software package. (Capability 3 and 7)

### Assessments

Assessment Type Percentage Classification
Assignments 65% Group & Individual Coursework
Quizzes 35% Individual Coursework
1 2 3 4 5
Assignments
Quizzes

### Tuākana

Tuākana Science is a multi-faceted programme for Māori and Pacific students providing topic specific tutorials, one-on-one sessions, test and exam preparation and more. Explore your options at
https://www.auckland.ac.nz/en/science/study-with-us/pacific-in-our-faculty.html
https://www.auckland.ac.nz/en/science/study-with-us/maori-in-our-faculty.html

Whanaungatanga and manaakitanga are fundamental principles of our Tuākana Mathematics programme which provides support for Māori and Pasifika students who are taking mathematics courses. The Tuākana Maths programme consists of workshops and drop-in times, and provides a space where Māori and Pasifika students are able to work alongside our Tuākana tutors and other Māori and Pasifika students who are studying mathematics.

### Key Topics

1. Linear algebra: eigenvalue problem; factoring — QR, SVD; solving banded systems.
2. Conjugate gradient method for large sparse systems of linear algebraic equations.
3. Approximation theory: minimax problem; continuous least squares problem.
4. 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.
5. Nonlinear algebraic systems: parameter continuation methods; pseudo-arclength continuation; quasi-Newton methods; approximation to the Jacobian.
6. Interpolation: Hermite polynomial interpolation, Legendre polynomials, cubic splines.
7. Solutions of ordinary differential equations: initial value problems; variable stepsize; stiff problems; two-point boundary value problems; shooting, collocation.
8. Fourier transforms in one dimension: DFT versus FFT; calculating FFTs.

### Special Requirements

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 use it remotely via https://www.auckland.ac.nz/flexit/. You can also download MATLAB and install it on your own computer or laptop  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.

### Delivery Mode

#### Campus Experience

• 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.

### Learning Resources

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)

### Student Feedback

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 for potential plagiarism or other forms of academic misconduct, using computerised detection mechanisms.

### Class Representatives

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.

### Inclusive Learning

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

### Special Circumstances

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 .

This should be done as soon as possible and no later than seven days after the affected test or exam date.

### Learning Continuity

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 .

### Disclaimer

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.

Published on 31/10/2023 10:53 a.m.