All projects and home works should be submitted
before 1/11/2008 (Friday).
None will be accepted after this date.
Homework 3. (Due 12/20)
Derive in
details from 2D Poisson’s (strong) problem with zero Dirichlet
boundary condition to its weak problem by using 2D divergence theorem.
Project 7. (Due 12/13)
Do Project
7.1 (CG).
Homework 2. (Due 11/29)
A. A complex square
matrix
A is a normal matrix if
A*A=AA*
where A* is the conjugate
transpose of A.
(If A is a real matrix, A*=AT and so it
is normal if ATA = AAT.) Show that if A
is normal then the condition number
where are maximal and minimal eigenvalues of A
respectively. Give an example for this theorem with a 3
by 3 matrix.
B. Prove the convergence result (7.23) in Lecture 7 for the method of
steepest descent.
Write your homework in English
and email it in pdf file to me.
Project 6. (Due 11/15)
Do Projects
4.1 (GS), 5.1 (SOR), and 6.1 (SSOR).
Homework 1. (Due 10/18)
A. Give a
detailed analysis on the convergence order (alpha) in (2.3) of Lecture 2. (Hint:
Taylor’s Theorem)
B. Explain why (under what conditions) JM will converge. (Hint: Spectral
analysis on the matrix B in (3.11) of Lecture 3.)
Write
your homework in English and email it in pdf file to me.
Project 5. (Due 10/11)
Do Project
3.1 (JM) in Lecture 3.
Project 4. (Due 10/1)
Do Project
2.1 (GE) in Lecture 2.
Submit
all files including I/O files. (From this project on.)
課程大綱
Lecture 1. 1D Poisson's
Equation and Finite Difference Method (FDM)
Lecture 2. Gaussian
Elimination (GE) for Ax=b
Lecture 3. Jacobi’s
Method (JM)
Templates for the
Solution of Linear Systems
(pdf)
Lecture 4. Gauss-Seidel
Method (GS)
Lecture 5. Successive
Overrelaxation Method (SOR)
Lecture 6.
Symmetric
Successive Overrelaxation Method (SSOR)
Lecture 7. Conjugate
Gradient Method (CG)
Lecture 8. Finite Element Method
(FEM) for 1D Poisson’s Problem
Lecture 9. 2D and 3D Poisson's Equation
Lecture 10. Convection-Diffusion-Reaction Model
Lecture 11. 1D Sturm-Liouville problem (pdf*)
Lecture 12. Power Method A (pdf*) Power Method
B (pdf**)
Lecture 13. QR METHOD
A (pdf*)
Lecture 14. A Brief Tour of Eigenproblems
教學目標
·
Learn
basic numerical methods in Partial Differential Equations (PDEs)
·
Learn
basic numerical methods in Numerical Linear Algebra
·
Learn
basic Physics behind PDEs
·
Learn
C++ programming in scientific computing
授課方式
·
Regular
Lecturing
評分標準
·
Programming
projects 60%
·
Homework
30%
·
In
class performance 10%
教科書
·
Jinn-Liang
Liu, Lecture Notes on Scientific Computing, 2007.
閱讀文獻
· [BB94] R. Barrett, M. Berry, T. F. Chan,
J. Demmel, J. Donato, J. Dongarra , V. Eijkhout, R. Pozo, C. Romine, and H. Van
der Vorst, Templates for the Solution of Linear
Systems: Building Blocks for Iterative Methods, 2nd Edition, SIAM,
1994, Philadelphia, PA.
· [BD00] Z. Bai, J. Demmel, J. Dongarra,
A. Ruhe, and H. van der Vorst, editors. Templates for the Solution of Algebraic Eigenvalue Problems: A Practical
Guide. SIAM, Philadelphia, 2000.
· [S00] Y. Saad, Iterative methods for sparse linear systems,
2000.
·
[S94] J. R. Shewchuk, An Introduction to the Conjugate Gradient
Method Without the Agonizing Pain, 1994.