autumn 2017

## MAT-2201 Numerical Methods - 10 ECTS

## 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).

## Course content

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.## Objectives 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.

## Assessment

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

Assessment scale: Letter grades A-F.

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) - http://en.uit.no/students/article?p_document_id=172032

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

Written test 06.12.2017The date for the exam can be changed. The final date will be announced in the StudentWeb early in May and early in November.

## Recommended reading/syllabus

**Curriculum for MAT-2201 Numerical Methods, autumn 2017**

UiT The Arctic University of Norway, Department of Mathematics and Statistics

**Textbook**: T. Sauer: Numerical Analysis. 2nd edition. Pearson:

US edition (alternative **1**):

ISBN-13: 978-0321783677

ISBN-10: 0321783670

**or **International edition (alternative **2**):**ISBN13:** 9781292023588

Ch. 0. Fundamentals

Ch. 1. Solving equations

1.1 The bisection method

1.2 Fixed-point iteration

1.3 Limits of accuracy

1.4 Newton's method

Ch. 2. Systems of equations

2.1 Gaussian elimination

2.2 The LU factorization

2.3 Sources of error

2.4 The PA=LU factorization

2.5 Iterative methods

Ch. 3. Interpolation

3.1 Data and interpolating functions

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 integration

5.1 Numerical Differentiation

5.2 Newton-Cotes formulas (except 5.2.4 open Newton-Cotes methods)

Ch. 6. Ordinary differential equations

6.1 Initial value problem

6.2 Analysis of IVP solvers

6.3 Systems of Ordinary Differential Equations (except 6.3.2 and 6.3.3)

6.4 Runge-Kutta Methods (except 6.4.2 and 6.4.3)

Ch. 7. Boundary value problems

7.1 Shooting Method

7.2 Finite difference methods

Appendix A. Matrix Algebra

Compendium "*p***-Norm of vectors and matrices**" is also a part of the syllabus

Appendix B (**for those who use MATLAB**): Introduction to MATLAB