MA 539: Methods of Scientific Computing.
MWF 1-2pm, Fall 2005
Course Description:
This course offers a survey of methods for
computing numerical solutions to a variety of standard mathematical
problems, and is likely to be of interest to students across a broad range
of quantitative disciplines, including mathematics, computer science,
statistics, engineering, finance, geography, physics, chemistry,
and computational biology. Emphasis will
be on a balance of mathematical, algorithmic, and computing issues.
Material in the course is organized around three modules: computational
linear algebra, numerical methods for curves and surfaces, and methods of
stochastic simulation. Substantial usage of computers will be made in this
course, both inside and outside of the classroom.
For more details, see the
course syllabus.
A Taste of the Course:
Here are three examples of the types of issues and problems we consider in
MA539.
- Matrix Multiplication: Just how difficult can it be to multiply
two matrices? In fact, we will see that there are at least three
methods that, although mathematically equivalent,
are algorithmically different. This distinction has important
implications when the computer enters the picture, and is one of the
first of many such issues we will study in the module on computational
linear algebra.
- Searching for a Minimum: The figure above shows a rendering of
Rosenbrock's so-called "banana" function, a standard test case in
the numerical optimization literature. This function has a unique
minimum, as can be demonstrated analytically, but it in fact proves
to be somewhat challenging to find for most methods of numerical
optimization. We will consider this example and such methods in detail
towards the end of our module on numerical methods for curves and
surfaces.
- A Random Path: Brownian motion is a stochastic phenomena that
lies at the heart of a wide range of models, ranging from physics
to finance. Being random by nature, it would seem to be
impossible to generate examples for analysis and use using such a
deterministic machine as the computer. Yet there are methods
whereby so-called "pseudo-random" sequences of numbers may be
produced by the computer to do just that, resulting in objects such as
that shown in the figure to the left . The generation, evaluation,
and usage of such sequences is the topic of our third and final module
on methods of stochastic simulation.
Course Related Materials: