 Mathematics
Te Tari Pāngarau me te Tatauranga
Department of Mathematics & Statistics

## COMO303 Numerical Methods

 First Semester
18 points

### Paper details

This paper introduces methods and theory for computational applied mathematics and modelling, with an emphasis on practical applications and modelling. You will learn a useful collection of numerical techniques for solving a wide variety of mathematical problems. In particular, we discuss solving systems of equations using matrix decompositions, curve fitting, numerical integration and differential equations. For some methods, detailed derivations are given, so you will also obtain an understanding of why the methods work, when they will not work, and of difficulties that can arise. For other methods, the focus will be on applying them in practical situations. For the computational side, we will use the numerical computing environment MATLAB. Previous experience with MATLAB is useful, but not required. An introduction will be provided in the first labs. At the end of this paper, you will have a good understanding of how to solve various problems numerically, how to choose the best method for a given problem, and how to interpret the solutions found in the context of error bounds and stability.

### Potential students

This paper should appeal to a wide group of students, including those majoring in Mathematics, Statistics, Physics, Engineering, Computer Science and Economics, or any other field where one often needs to use numerical approximations to solve real world problems.

### Main topics

#### Introduction to numerical algorithms:

Theory: Algorithms; numerical and measurement error; stability

Computation: Introduction to MATLAB programming

Applications: Examples of catastrophic numerical error

#### Matrix decompositions and their uses:

Theory: Standard matrix decompositions, their advantages and uses

Computation: Implementing matrix algorithms; making use of built-in methods; exploring conditioning and stability in practice

Applications: Image deblurring; image compressions (SVD)

#### Iterative methods for solving linear systems:

Theory:Stationary iterative methods; relaxation; the conjugate gradient method

Computation: Computing with sparse matrices; implementation of iterative methods

Applications: Solving large systems linear of equations

#### Least-squares fitting and applications:

Theory: Necessary and sufficient conditions for an unconstrained optimum; exact solutions for least squares; Newton's method, properties and extensions

Computation: Implementation of steepest descent and Newton's method; use of built-in MATLAB optimisers; fitting polynomials, splines and other functions to data

Applications: Clustering; numerical integration

#### Modelling with ordinary differential equations:

Theory: Runge-Kutta and predictor-corrector methods, multistep methods; boundary value problems; finite difference methods

Computation: Symbolic calculation of simple analytic solutions; implementation of basic iterative solvers; use of built-in solvers; exploration of stability

Applications: Population growth models; predator-prey models; epidemiology

### Prerequisites

MATH 202.

COMO 204 is recommended.

### Lecturer

Dr Jörg Hennig, room 215.

Office hours: by arrangement (or just pop in if I am in my office).

### Lectures

Mon, Wed and alternating Fri, 1-2pm, MA241.

### Computer labs

Mo, 3-5pm, MA242

Weekly labs starting in the first week of lectures.

### Assessment

Your final mark F contains the following:

• 15%: 5 fortnightly assignments
• 15%: midterm test
• 10%: computer labs
• 60%: final exam

### Final mark

Your final mark F in the paper will be calculated according to this formula:

F = 0.15A + 0.15M + 0.10L + 0.6E

where:

• E is the Exam mark
• A is the Assignments mark
• M is the Midterm test mark
• L is the Labs mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

Academic integrity means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

Academic misconduct is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

#### Plagiarism

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

• Although not intended, unintentional plagiarism is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided.
• Intentional plagiarism is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

#### Unauthorised Collaboration

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

#### Impersonation

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

#### Falsiﬁcation

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

#### Use of Unauthorised Materials

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

#### Assisting Others to Commit Academic Misconduct

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.

Further information

While we strive to keep details as accurate and up-to-date as possible, information given here should be regarded as provisional. Individual lecturers will confirm teaching and assessment methods. Computer Aided Tomography (CAT) scans, such as this one of a pelvic bone with clover-shaped tumor, are made possible by careful measurements, sophisticated algorithms and some powerful mathematics. The image is constructed by solving a massive linear equation. The circumference $C$ of an ellipse with axes $a$ and $b$ is $$C=\int_0^{2\pi} \sqrt{a^2\sin^2 t+b^2\cos^2 t}\;dt.$$ For a circle (where $a=b=r$) this becomes easy, but what for $a eq b$? Then there is no answer in terms of elementary functions! However, for given values of $a$ and $b$, we can use numerical integration to find arbitrarily good approximations for the circumference. The motion of a swinging pendulum of length $l$ is described by the differential equation $$\ddot\phi(t)+\frac{g}{l}\sin\phi(t)=0,$$ where $g\approx 9.81m/s^2$ is the gravitational constant. For small amplitudes, we can use the approximation $\sin\phi\approx \phi$ and solve the problem exactly. For arbitrary amplitudes, we can use various numerical methods to approximate the solution.