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 selfstudy part is included, to take place during Week 2 (Jan 1014) 2011.
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 websources:
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, PrenticeHall, 1974, or Dover, 2003, ch 11.
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  9:1512:00  1412 
Computer lab  13:1517:00  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  9:1512:00  1412 

Computer lab  13:1517:00  1412 

Module II: The Finite Difference Method  
Jan 19  Initial and boundary value problems for ordinary differential equations. RungeKutta methods and linear multistep methods for initial values problems. Accuracy and stability. Methods for systems of ordinary differential equations  9:1512:00  1412 

Computer lab  13:1517:00  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  9:1512:00  1412 

Computer lab  13:1517:00  1412 

Module III: Monte Carlo methods  
Jan 21  Random numbers. Pseudorandom numbers. Law of large numbers and the central limit theorem. Stochastic algorithms. Monte Carlo method for highdimensional integrals  9:1512:00  1412 

Computer lab  13:1517:00  1412 

5  Jan 2428  Work on Assignment (Modules IIII) (download here)  
6  Jan 31Feb 4  Umeå week 
Recommended books
S.C. Charpa  Applied Numerical Methods with MATLAB for Engineers and Scientists 
McGraWHill 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.