## Scientific Computing (I), Spring 2004.

A graduate course in Mathematics at the Department of Mathematics, National Tsing Hua University.

Instructor: Professor Wei-Cheng Wang,

Class meeting: ºî¤T 734. Meeting schedule: W567.

First meeting: Wednesday 2/18, 1:10 PM.

Grading: 3-4 sets of homework assignments, mostly programming.

Textbook and References: Typed lecture notes will be distributed in class. You can refer to any standard numerical analysis textbook for materials not covered in this course. For example: Cheney, E. W. and Kincaid, D: Numerical Mathematics and Computing. or Shampine, Allen and Pruess: Fundamentals of Numerical Computing.

Course description: This course is a continuation of the undergraduate course `Numerical analysis' with emphasis on practical applications including numerical methods for differential equations, impage processing and mathematical finance. Students are required to do independent homework assignments on programming using modern languages (any one of matlab, c, c++, fortran ...). Some knowledges on numerical programming. and a preliminary undergraduate course of numerical analysis (or equivalent) is preferred, but not required. Undergraduate students are welcome. If necessary, we will have a short tutorial on matlab so that everyone can proceed. We will also arrange computer lab hours to help you resolve programming problems.

Topics and Contents:

• Review of Basic Numerics and Programming Practice: sources of error, conditioning and stability, computer arithmetic, numerical resolution, order of accuracy, accuracy check, numerical differentiation, integration, interpolation, error expansions and Richardson extrapolation.
• Fast Fourier Transform, Wavelets and Applications: discrete Fourier transforms, fast algorithm, numerical solution of differential equations via FFT, wavelets, image processing, fast convolution, fast Poisson solver. Leture Note
• Monte Carlo Methods and Applications: basic probability and statistics: random variables, probability density function (pdf), cumulative distribution function (cdf), mean, variance, standard deviation, correlation, variance estimation, confidence intervals, central limit theorem. Markov process, etc; basic Monte-Carlo simulation: Pseudo random number generators, generation of non-uniformly distributed numbers, Box-Muller methods for normals, Brownian motion simulation and application to finance, Monte-Carlo integration, rejection methods, statistical analysis of simulation data and error bars; variance reduction: stratified sampling, importance sampling, control variates, and antithetic variates
• Optimization and/or Numerical integration of ODE (if time permits): unconstrained optimization: Newton's method, line searches (Golden section search, steepest descent) Global optimization; safeguards in Newton's method, ... explicit and implicit methods, basic facts about stability and convergence; applications.

Other sources of useful information:

Netlib is an excellent net resource. It contains tons of reliable numerical software that can be downloaded and links to most technical information on numerical analysis and scientific computing on the internet. The ftp mirror site in Taiwan is at the National Center for High Performance Computing.

Matlab Information

Matlab is an excellent environment for experimentation in scientific computing. It is not the most efficient environment for large-scale simulations, although with some effort, Matlab can be used in conjunction with C and Fortran routines. Matlab is a product of The Mathworks. For a tutorial, see A Free Matlab Online Tutorial or Another Tutorial or Matlab Primer
You can start from our class note on matlab before reading these materials.

### Announcement:

Here is an excellent note on IEEE arithmetic by Professor Michael Overton as a supplement reading material for Chap 1. It contains everthing you need and don't need to know about floating point arithmetics.

Here is a quick summary of the Fourier Transform and Applications .

### Homework assignments:

Postscript viewer: To view the Postscript files on your PC screen, you may need to install the GSview from here.

PDF viewer: To view the PDF files on your PC screen, you may need to install the Acrobat Reader from Adobe. (Other softwares, such as GSview, can be possible alternatives.) Acrobat Reader can be downloaded from the Adobe site

Assignment 1: Basic Numerics .