Welcome to the CSE/Math 441, Approximation Theory, home page. You're probably here for information, so let's start with the vital statistics of the course.

### Essential Information

• Announcements Announcements about current class activities.
• Class Policy Statement
• Course Syllabus
• Course Materials Some useful downloads for this course, including printable copies of the Class Policy Statement and Syllabus.
• Assignments Assignments and due dates. Check this periodically for the most current information.
• Blackboard Home Link to the Blackboard course site, if you are not already on the site -- Blackboard contains this home page and more, e.g., you may access your current grades via Blackboard.

### Approximation Theory Resources

• Course Description So what is this "approximation theory" all about?
• Netlib A huge repository of mathematical software, papers, and databases.
• Maple Math Site Has lots of information about the Maple software which is available on math department computers.
• SIAM Home Page Every serious student of applied mathematics should consider joining the Society for Industrial and Applied Mathematics!
As you know, a staggering amount of information can be found on the web. Of course, some of it helps, some confuses, and some is downright wrong -- caveat emptor! Go to your favorite search engine and try searching on "numerical linear algebra". See how many web pages you hit and visit a few interesting looking sites.

## Notes and FAQ

8/20/09:(Just to get the FAQ started) About significant digits...
I've been asked to explain what "significant digits of an approximation" to a (nonzero) number means. There are several interpretations that one commonly sees. The "absolute" definition, which is perhaps more intuitive, goes as follows: to get the number of significant digits, first *subtract* (rather than just looking at the numbers) the two (may as well be larger - smaller), then find the position of the leading digit of the error relative to the position of leading digit of the exact answer. If the difference in that position is less than 5, then number of significant digits is one less than the difference, else two less.

For example if 3.14 and 3.15 are used to approximate 3.14159, calculate 3.14159 - 3.14 = 0.00159 and 3.15 - 3.14159 = 0.00841. Notice I put a zero in front of the decimal to start counting from the right position. There is a nonzero digit at the 4th position with each approximation, counting from the (base 10) position of the leading digit of 3.14159. The size of this digit is at most 5 in the first case, so this approximation has 3 significant digits. In the second case, the digit is larger than 5, so the approximation only has 2 significant digits.

The "relative" definition is usually preferred in numerical analysis and it goes as follows: if xapprox is used to approximate xtrue, the number of significant digits is largest integer n such that the relative error, |(xapprox-xtrue)/xtrue)| is no larger than 5x10^(-n-1). In our previous example the relevant quotients are 0.000506 and 0.002677, respectively. The best bound in both cases is 5x10^(-3). Thus, this definition will give only 2 significant digits in both cases. In general, the "relative" definition is stingier than the "absolute" one. Hope this helps.

### Class Policy Statement

Course: CSE/Math 441, Approximation Theory

Place/Time: AvH 12, 3:30-4:45 TR, Fall 2009

Preq: CSE/Math 340, Math 221 and 314 or equivalent, or permission.

Objectives: To help students achieve competence in the following areas:

• • Basic principles of approximation of functions, including existence and uniqueness.
• • Important types of approximations, including polynomial, trigonometric and splines.
• • Analysis and computer implementation of algorithms for approximations.
Instructor: Dr. Thomas Shores

Telephone: Office 472-7233   Home 489-0560

Email: tshores1@math.unl.edu

Office Hours: Monday 10:00-12:00, Tuesday 1:30-3:00, Wednesday 2:00-4:00, Thursday 9:00-10:30, Friday 9:30-10:30, and by appointment. Office: 229 AvH
Note: Circumstances may necessitate occassional changes in office hours. Consult the course home page or Blackboard for the most current times.

Class Attendance: Is required. If absent, it is incumbent upon the student to determine what has been missed as soon as possible. It is advisable to consult with the instructor.

Homework/Projects: Homework will be assigned in class and collected in accordance with the syllabus, and will be usually returned within one week. Although collaboration in solving most problems is allowed, it is strictly forbidden to copy someone else's homework. It is expected that co-collaborators and other sources for the homework will be duly acknowledged. Assignments will be due approximately every two weeks, for a total of six assignments. For some specified problems no collaboration will be allowed. Matlab (Octave) is the official programming language for this course. Prior programming experience with it is not required. Current information about the course will be available through Blackboard and the 441 homepage. Using the web is strongly recommended for keeping track of current activities in the course.

Reading Assignment: Read the sections of the texts as, or before, they are covered in class lectures. This is a standing assignment throughout the semester.

Grade: One midterm will be given and will account for 130 points. The final exam will count 140 points. Each exam may have a take home component. In-class exams are closed book with calculators. Homework will count 230 points. The final grade will be based on these 500 points.

Final Exam: To be given on Tuesday, December 15, 8:30 - 10:30 pm in AvH 12.

Grades of "I", "W" or "P": These grades will be given in strict accordance with University policy. (See any Schedule of Classes for the relevant information and dates.)

Keep This Information!!!