MA/CS/EGR 537: Numerical Analysis I Fall, 1997

Section of Professor C. C. Douglas

Class Meets on Tuesdays and Thursdays from 11:00 to 12:15 in CB 343

Graduate Handbook Description: MA 537, Numerical Analysis I

Computer arithmetic. Discussion of the various types of errors, solution of systems of linear algebraic equations - Gaussian elimination with partial pivoting and scaling. Iterative refinement. Polynomial and piecewise polynomial interpolation. Orthogonal polynomials. Method of least-squares. Numerical differentiation. Numerical integration: Newton Cotes formulas and Gaussian quadrature.

Prerequisite: MA/CS 321, or graduate standing, or consent of instructor. Knowledge of a procedural computer language is required. (Same as CS/EGR 537.)

This course will give the students a solid foundation in a number of topics in numerical analysis. Algorithms to solve problems will be emphasized. Theory and applications will be equally weighted. At the end of the course the students should know what type of algorithm to try to solve a problem, why it works, and how well it should have worked.

How to Locate the Instructor

I have 2 offices at UK, which complicates finding me. I spend most of my time on campus in McVey. If you need to leave me a phone message (versus an e-mail message which will get to me quicker), call the McVey number and leave a message with whoever picks up the line.

Building	Room	Phone
McVey	321A	257-2326
Patterson	761	257-6792

Web Usage

The course will use the web extensively. You must know how to use a version 3 (preferably a version 4) type browser like ones provided by Netscape or Microsoft. The syllabus currently makes up most, but not all, of the class home page. It is located at the URL

http://www.ccs.uky.edu/~douglas/ma537

The class web page has a number of hyperlinks that you will find either useful or essential. Included is a finger locator for trying to find which office I am in.

Homework, the Late Policy, Exams, Grading, and Dead Sources

I will hand out roughly 10 homework assignments during the semester. The pencil and paper parts should be turned in at the beginning of class on the due date. Codes should be e-mailed before class to

douglas@ccs.uky.edu

I will take late homework only if there is a compelling reason; please contact me in advance, if possible. I will give you an extension for serious health problems, job interviews, death of a relative, or a similar, serious situation. Do not come and tell me that so-and-so's course is more important than mine and you did their assignment or project instead of mine.

Grading will be very simple. Since this is a graduate level course, +'s and -'s will not be given. The homework will count 40%, the midterm (Tuesday, October 21) will count 20%, and the final (Thursday, December 18 from 8-10 AM) will count 40%.

All UK faculty are required to state in the syllabus the grading system. The system can change if I give you adequate warning. As a rule, having the following percent of the scaled points will earn a grade of

Grade	Minimum %
A	85
B	70
C	50

Textbook

John H. Mathews
Numerical Methods for Mathematics, Science and Engineering
Prentice-Hall, Englewood Cliffs, NJ (USA), 1992, ISBN 0-13-624990-6.

Outline of the Course

The material listed in the graduate handbook will be covered. In particular, the following topics will be covered.

1. Basics
(a) Round off errors
(b) Matrix algrebra review

2. Solution of linear systems of equations: solve Ax=b

(a) Gaussian elimination and factorization methods
(i) Tridiagonal matrices
(ii) General, dense matrices
(iii) Sparse matrices
(b) Iterative methods
(i) Relaxation methods
(ii) Conjugate gradients

3. Overdetermined systems

Least squares

4. Nonlinear equations: find roots of f(x)=0

(a) Bracketing methods
(b) Fixed point methods (chord, Newton, secant)
(c) Zeroes of polynomials

5. Interpolation and approximation: given {f(x₀), f(x₁), ..., f(x_N+1)}, where x₀ < x₁ < ... < x_N+1, what is f(x), x₀ <= x <= x_N+1?

(a) Polynomial, Lagrange, Hermite, piecewise polynomial, Chebyshev, tensor product methods
(b) Least squares fit

6. Numerical integration

(a) Classical quadrature rules
(b) Automatic quadrature rules

7. Richardson extrapolation

Companion Software

Numerical analysts need to know Fortran in order to read old codes and re-use them. Translators (e.g., f2c or c++2j) produce sufficiently bad code as to make the translation unusable from a wall clock point of view. This does not mean that all numerical analysts should program only in Fortran. Many applications are better suited to Ada, Matlab, Lisp, C, C++, or Java.

This course will use Matlab or a clone whenever a programming assignment is necessary unless stated otherwise. The Free Software Foundation supports a Matlab clone called Octave. Octave works on the following platforms: Linux, OSF/Alpha, HPUX, SUNOS, and AIX. If you have a Windows 95/NT based PC, you might consider purchasing the student edition of Matlab from Prentice-Hall. There are almost no Matlab software keys at UK except in the engineering division (where there are hundreds).

There is a lot of software associated with the textbook. A gzipped, tar file is available.

Cheers,
Craig C. Douglas