autumn 2013
MAT-2201 Numerical Methods - 10 ECTS

Type of course

The course is included in the Master's degree program in applied physics and and mathematics, and in the Bachelor's degree program in mathematics and statistics. It may also be taken independent of study program.

Course overlap

If you pass the examination in this course, you will get an reduction in credits (as stated below), if you previously have passed the following courses:

MA-224 Numerical calculations 10 stp
FYS-2011 Numerical simulations 10 stp

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.

Recommended prerequisites

MAT-1003 Calculus 3, MAT-1004 Linear algebra

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 Taylor’s 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 Simpson’s 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 and examination

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. Letter grades (A-F).

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

Recommended reading/syllabus

Curriculum for MAT-2201 Numerical Methods, autumn 2013
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

 

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  • About the course
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  • ECTS: 10
  • Course code: MAT-2201
  • Tidligere år og semester for dette emnet