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, 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 endeavours at the University of Otago are built upon an essential commitment to academic integrity.

The two most common forms of academic misconduct are plagiarism and unauthorised collaboration.

Plagiarism is defined as:

• Copying or paraphrasing another person’s work and presenting it as your own.
• Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
• Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
• Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.

All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.

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 $a\neq b$? 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.