The College of the Holy Cross

Mathematics 172 -- Numerical Analysis 2

Syllabus, Spring 1999

Professor: John Little
Office: Swords 335
Office Phone: 793-2274
email: little@math.holycross.edu or jlittle@holycross.edu
Office Hours: MWF 10-12, TR 1-3, and by appointment.

Course Description

As we said at the start of the fall semester, 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, and
eigenvalues and eigenvectors of matrices and other quantities from linear algebra

We covered (most of!) the topics we will discuss from points 1 and 2 in the fall; this semester we will concentrate on topics from points 3 and 4. In somewhat more detail, the topics we will study and the projected schedule this semester are:

Unit I: Additional numerical integration methods -- adaptive quadrature and Gaussian quadrature (about 3 class days; January 19, 21, 26; Sections 4.6 and 4.7 in Burden and Faires)
Unit II: Numerical techniques for ordinary differential equation initial value problems -- Euler, Taylor, and Runge-Kutta methods (about 5 class days; January 28, February 2,4,9,11; Sections 5.1-5.5 and 5.9 in Burden and Faires)
Unit III: Direct methods for systems of linear equations -- Gaussian Elimination interpreted as matrix factorization, pivoting strategies, techniques for special classes of matrices (about 4 class days; February 16,18,23,25; Sections 6.1-6.2,6.5-6.6 in Burden and Faires)
Unit IV: Iterative techniques for systems of linear equations -- Jacobi and Gauss-Seidel iteration, their analysis via matrix and vector norms, the spectral radius of a matrix (about 8 class days; March 2, 16, 18, 23, 25, 30, April 6, 8; Sections 7.1-7.4 in Burden and Faires)
Unit V: Numerical eigenvalue techniques -- the basic and shifted inverse power methods, Householder matrices and the QR algorithm (about 6 class days; April 13, 15, 20, 22, 27, 29; Chapter 9 in Burden and Faires)

The remaining day will be devoted to an in-class exam. See below. 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 the same as last semester -- Numerical Analysis, 6th ed. by R. Burden and J. D. Faires. We will cover most of the material in Chapters 5,6,7,9 this term.

Course Format

To get you more directly involved in the subject matter of this course, periodically during the semester, the class will break down into groups of 3 or 4 students for one or more days, and each group will work together for a portion of those class periods on a group discussion exercise. The exercises will be made up by me. I will be present and available for questions and other help during these periods. At the conclusion of some of these discussions, groups, the class as a whole will reconvene to talk about what has been done, to sum up the results, to hear short oral reports from each group, etc. Each group will be responsible for a write-up of solutions for the questions from each discussion day, and those will be graded and and returned with comments. Other meetings of the class will be structured as lectures or computer laboratory days when that seems appropriate.

Computer Work

We will again 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.

Grading

The assignments for the course will consist of:

One midterm in-class exam worth 20% of the course grade. Tentative date: Thursday, March 4.
Final exam worth 30% of the course grade. The final exam for this course will be held at 8:30 a.m. Thursday, May 13.
Group final project, worth 20% of the course grade.
Individual problem sets and lab reports, worth 15% of the course grade.
Group reports from discussion days, worth 15% of the course grade.

The final projects will give you the chance to explore additional topics extending what we do in class, implement some of the algorithms involved, and apply them to realistic problems. This should be an interesting way to see how numerical analysis is used ``in the real world''. More information about the projects, including some suggested topics and sources to consult as you work on them, will be distributed later in the semester in class and on the course homepage.

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