MAT-2201 Numerical Methods - 10 stp

Generell studiekompetanse + REALFA + MAT-1003 Calkulus 3 and MAT-1004 Linear algebra or equal .

Søknadskode 9336.

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.

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.

Ordinary examination in the teaching free semester (early exam):
A new ordinary examination will be arranged provided that it already will be given a postponed or a re-sit exam for the course.

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

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

Curriculum for MAT-2201 Numerical Methods, autumn 2012

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