Holy Cross Mathematics and Computer Science

Mathematics 371 -- Numerical Analysis

Syllabus, Fall 2001

Professor: John Little
Office: Swords 335
Office Phone: 793-2274
Email: little@mathcs.holycross.edu (preferred) or jlittle@holycross.edu
Course Homepage: http://mathcs.holycross.edu/~little/Num01/Num01.html
Office Hours: MWF 11-12, TR 1-3, and by appointment

Course Description

The subject matter of numerical analysis is the mathematics of computation of:

approximate solutions of algebraic (linear and polynomial) and transcendental equations,
numerical approximations to values of functions, their derivatives and integrals,
approximate solutions of differential equations

The numerical computation of solutions of systems of linear equations, eigenvalues and eigenvectors of matrices, and other quantities from linear algebra is another part of the subject. That material is covered in the Numerical Linear Algebra course (MATH 372). Although they are related, these courses do not form a linked sequence for the Mathematics major requirements.

Numerical analysis is as old as calculus itself--many of its techniques were first introduced as practical methods for hand calculations by the same mathematicians who developed the theoretical tools of derivatives and integrals, matrices and linear algebra.

The advent of computers has changed the field in several ways. First, the size and difficulty of the problems that can be attacked by numerical methods have increased dramatically. Indeed, almost all of the ``real world'' applications of mathematics in science and engineering, from the calculation of space shuttle orbits to the design of automobile bodies, make heavy use of ideas from numerical analysis. But second, the way arithmetic is implemented in computer hardware has introduced new levels of subtlety, since roundoff and computational errors cannot be avoided.

In this course, we will study a selection of the most important basic ideas in this field, paying attention both to the ``why'' and the ``how'' of numerical methods. That is, we will place roughly equal emphasis on

the underlying theoretical reasons why they work (in the cases that they do work), and the potential pitfalls of numerical approaches (for the cases where they do not work)
the mechanics of the calculations themselves.

In addition, we will study some important applications of the methods we develop, and we will consider how computer programs are constructed to implement these techniques.

The topics we will study this semester are:

Unit I: Floating-point arithmetic, algorithms, Maple programming (about 5 class days)
Unit II: Solving non-linear equations in one variable (about 9 class days)
Unit III: Polynomial and spline interpolation (about 7 class days)
Unit IV: Numerical methods for differentiation and integration (about 12 class days)
Unit V: Rudiments of numerical methods for ODE initial value problems (about 4 class days)

There is a more detailed day-by-day schedule on the course homepage. As always, some additions, deletions, or rearrangement of topics may become necessary as we progress through the course. Any changes will be announced in class and on the class homepage.

Text

The text for the course is Numerical Analysis, 7th ed. by R. Burden and J. D. Faires. We will cover most of the material in Chapters 1-4 and the first part of Chapter 5 this term.

Course Format

Since I expect this will be a rather small class, most of the class meetings will be structured as lecture/discussions. I will expect you to come to class well-prepared and to participate by asking and answering questions during class time. Some of the assignments may be designed as group work. I will probably also assign an in-class presentation for each student later in the semester based on the assigned topic for the day.

Computer Work

We will be using Maple on the departmental Sun workstation network quite extensively throughout the course to implement the techniques we discuss and to generate numerical solutions to problems. Several class meetings will take place in the SW 219 computer lab and some of the individual problem sets will include problems for which you will need to use Maple. You can use either the SW 219 workstations or the HA 408 lab PC's for this work, since both labs give you access to Maple.

Grading

The assignments for the course will consist of:

Two midterm exams, together worth 35% of the course grade. Both of these will be given as take-home problem sets. Tentative dates:
- Take-home out Friday, September 28, due Friday, October 5
- Take-home out Friday, November 2, due Friday November 9.
Final exam worth 30% of the course grade. This will also be given as a take-home exam, due the date of the scheduled exam period for this class.
Weekly problem sets and labs, worth 25% of the course grade.
Class participation, presentation, worth 10% of the course grade.

If you ever have a question about the grading policy, or about your standing in the course, please feel free to consult with me.