autumn 2013
MAT-2201 Numerical Methods - 10 ECTS
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.
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
Error rendering component
- About the course
- Campus: |
- ECTS: 10
- Course code: MAT-2201
- Responsible unit
- Institutt for matematikk og statistikk
- Kontaktpersoner
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