This course is organized by the NGSSC Graduate School.

It consists of five modules, the first three of which will be given at the Department of Information Technology, Uppsala University,
and the last two will be given at the Department of Computing Science, Umeå University.

As a part of the work to be done within the course, a self-study part is included, to take place during Week 2 (Jan 10-14) 2011.

Review material for Week 2's self study, related to Modules I-III

Module I (Systems of linear equations):
In order to follow the material, basic Linear Algebra concepts and definitions have to be reviewed: vectors and matrices, vector spaces, linear independence, rank of a matrix,
vector and matrix norms, eigenvalues and eigenvectors, Gauss elimination. Recall also the notion of numerical error (absolute and relative) from your first course in Numerical Methods.

Module II (The Finite Difference method):
Refresh your knowledge on Partial and Ordinary Differential Equations. One possibility is to consult the following two web-sources: PDEs and ODEs .

Module III (Monte Carlo methods):
Refresh your basic knowledge in mathematical statistics e.g. from the book R. J. Larsen, M. L. Marx, An Introduction to Mathematical Statistics and its Applications, Prentice Hall, 1981,
or some other introductory text book. Of importance are concepts such as random variable, probability density function, cumulative distribution function, uniform and normal distributions,
central limit theorem. An introduction to Monte Carlo methods is found in G. Dahlquist, Å. Björck, Numerical Methods, Prentice-Hall, 1974, or Dover, 2003, ch 11.

Detailed time schedule

Week	Date	Topic(s)	Time	Location
		Module I: Systems of linear equations
3	Jan 17	Direct solution methods for linear systems. Numerical stability, pivoting. Dense matrices, structured matrices, general sparse matrices and the impact on the matrix structure on the performance of the direct methods. Computational complexity	9:15-12:00	1412
		Computer lab	13:15-17:00	1412
	Jan 18	Iterative solution methods. Projection methods. The Conjugate Gradient method, rate of convergence, condition number. The GMRES algorithm. Accelerating the convergence, the notion of preconditioning. Computational complexity	9:15-12:00	1412
		Computer lab	13:15-17:00	1412
		Module II: The Finite Difference Method
	Jan 19	Initial and boundary value problems for ordinary differential equations. Runge-Kutta methods and linear multistep methods for initial values problems. Accuracy and stability. Methods for systems of ordinary differential equations	9:15-12:00	1412
		Computer lab	13:15-17:00	1412
	Jan 20	Finite difference methods for partial differential equations. Boundary conditions. Methods for elliptic problems. Methods for parabolic and hyperbolic problems. Accuracy and stability. Von Neumann analysis	9:15-12:00	1412
		Computer lab	13:15-17:00	1412
		Module III: Monte Carlo methods
	Jan 21	Random numbers. Pseudo-random numbers. Law of large numbers and the central limit theorem. Stochastic algorithms. Monte Carlo method for high-dimensional integrals	9:15-12:00	1412
		Computer lab	13:15-17:00	1412
5	Jan 24-28	Work on Assignment (Modules I--III) (download here)
6	Jan 31--Feb 4	Umeå week

Recommended books

S.C. Charpa	Applied Numerical Methods with MATLAB for Engineers and Scientists	McGraW-Hill International Edition, 2005. (Bacis level)
J.H. Mathew, K.D. Fink	Numerical Methods Using MATLAB	Pearson Education International, 2004 (Basic level)
W.Y. Yang, W. Cao, T.-S. Chung and J. Moris	Applied Numerical Methods using MATLAB	J. Wiley & Sons Ltd, 2005. (Advanced level)

Material to downloads

Day 1: Some slides	here
Day 2: Some slides	here
Computer lab no. 1	here	Matlab codes and data
Computer lab no. 2	here	Matlab codes and data
Computer lab no. 3	here	Matlab codes
Computer lab no. 4	here	Matlab codes
Computer lab no. 5	here
Project no. 1	here	Matlab code
Project no. 2	here
Course evaluation form	here

Organization issues:
Here are some instructions how to find us.