Høst 2015

MAT-2201 Numerical Methods - 10 stp

The course is administrated by

Institutt for matematikk og statistikk

Type of course

The course is mandatory i the study program Mathematics and Finance - master, and it's included in the study programs Mathematics and Statistics - bachelor and Applied Physics and Mathematics - master (5-years). It may also be taken independent of study program.

Course overlap

FYS-2011 Numerical simulations 10 stp

Course contents

This course gives an introduction to basic concepts and issues of numerical computation. The topics treated include: Binary representation and floating point numbers, round-off errors, conditioning, rates of convergence, truncation and discretization errors, best approximation, numerical stability, and complexity analysis. Selected methods will be covered for some of these classes of problems: Linear systems of equations, nonlinear equations, overdetermined linear systems, numerical differentiation and integration, and numerical solution of differential equations.

Application deadline

Applicants from Nordic countries: 1 June for the autumn semester and 1 December for the spring semester. Applicants from outside the Nordic countries: 1 October for the spring semester and 15 April for the autumn semester.

Admission requirements

Applicants from Nordic countries: Generell studiekompetanse + Matematikk R1 eller (S1+S2) og enten Matematikk (R1+R2) eller Fysikk (1+2) eller Kjemi (1+2) eller Biologi (1+2) eller Informasjonsteknologi( 1+2) eller Geologi (1+2) eller Teknologi og forskningslære (1+2).

International applicants: Higher Education Entrance Qualification and certified language requirements in English. It is a requirement that students have some prior knowledge of biology and ecology, chemistry and mathematics (Participants must have taken introductory level university courses, and achieved pass grades, in these subjects).

A list of the requirements for the Higher Education Entrance Qualification in Norway can be found on the Norwegian Agency for Quality Assurance in Education website - nokut.no

Recommended prerequsites is MAT-1003 Calkulus 3 and MAT-1004 Linear algebra or equal. Application code is 9336.

Application code: 9336 (Nordic applicants).

Objective of the course

After the course the student should:

Be able to analyze methods for numerical calculations with respect to errors and complexity
Have mathematical understanding for the methods they apply
Know the main features in IEEE-standards for binary number representation
Be able to use iterative methods, like the Jacobi-method for systems of linear equations, and Newtons method for non-linear equations, and be able to describe convergence properties.
Be able to describe Gaussian elimination and LU factorization, and know QR factorization, and how this is used to find least squares solutions.
Know the problem of polynomial interpolation, how to solve it, and how to prove unqueness. They should be able to use Chebychev polynomials as tools.
Use Taylors theorem to find errors of discretization when calculating dericatives and finite difference.
Know simple methods for numerical calculation of integrals, such as the Trapezoid method and Simpsons formula, and general results about global errors, when local errors are known.
Know the simplest algorithms for stepwise numerical solution of initial value problems for systems of first order differential equations, and know how to reformulate a higher order differential equation to such a system.

Language of instruction

The language of instruction and the syllabus is English. Examination questions will be given in English, but may be answered either in English or a Scandinavian language.

Teaching methods

Lectures: 40 h
Coursework: 30 h

Assessment

Written final exam of 4 hours duration, counting 100 %.

Assessment scale: Letter grades A-F.

Re-sit examination:
Students having failed the last ordinary examination are offered a re-sit examination early in the following semester, if the course is compulsory in their study programme.

Postponed examination:
Students with valid grounds for absence will be offered a postponed examination early in the following semester.

For further details see:
- Utfyllende bestemmelser for eksamener ved Fakultet for naturvitenskap og teknologi (only in Norwegian)
- Regulations for examinations at the University of Tromsø

Coursework requirements
A passing grade is required on the mandatory homework sets for permission to take the exam.

Date for examination

One written 09.12.2015

The date for the exam can be changed. The final date will be announced at your faculty early in May and early in November.

Recommended reading/syllabus

Curriculum for MAT-2201 Numerical Methods, autumn 2015
UiT The Arctic University of Norway, Department of Mathematics and Statistics

Textbook: T.Sauer: Numerical Analysis. Pearson 2006

Ch. 0. Fundamentals. The whole chapter

Ch. 1. Solving equations
1.1 The bisection method
1.2 Fixed point iteration
In addition: An extended treatment of iteration, see text "A note on the method of successive approximations" laid out on Fronter.
1.3 Limit of accuracy
1.4 Newton's method

Ch. 2. System of equations
2.1 Gaussian elimination
2.2 The LU factorization
2.3 Sources of error
2.4 The PA=LU factorization
2.5 Interative methods. Confer also the text "A note on the method of successive approximations" referred to above.
2.7.1 Nonelinear systems of equations/Multivariate Newton's method

Ch. 3. Interpolation
3.1 Data and interpolating Functions (exept 3.1.2 Newton's divided differences)
3.2 Interpolation error

Ch. 4. Least squares
4.1 Least squares and the normal equations
4.2 A survey of models
4.3 QR factorization

Ch. 5. Numerical differentiation and Intergration
5.1 Numerical Differentiation
5.2 Newton-Cotes formulas (except 5.2.4 open Newton-Cotes methods)
5.5 Gaussian quadrature

Ch. 6. Ordinary differential equations
6.1 Initial value problem
6.2 Analysis of IVP solvers
6.3 Systems of Ordynary Differential Equations (except 6.3.2 and 6.3.3)
6.4 Runge-Kutta Methods (except 6.4.2 and 6.4.3)
Multistep methods: a simplified discussion of the second order Adam-Bashforth method lectured.

Ch. 7. Bonudary value problems
7.2.1 Finite difference methods/Linear boundary value problems