## COMO303 Numerical Methods

First Semester |

### 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, 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, to choose the best method for a given problem, and 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, Computational Modelling, 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 condition 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; preconditioning

*Computation:* Computing with sparse matrices; implementation of iterative methods and preconditioning techniques

*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 optimizers; 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; 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; plotting solutions and direction fields

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

### Prerequisites

MATH 202.

COMO 204 (or MATH 262) 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.

### Useful reference

Cleve B. Moler, *Numerical Computing with MATLAB*, SIAM (2008).

A free web edition is available here.

### 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 + 0.1L + 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.