Numerical Analysis: Math 488/688
Numerical Analysis is a branch of mathematics that studies various aspects of numerical approximations of solutions of mathematical problems that are hard (or impossible) to solve otherwise. These aspects include
understanding the mathematics of the original problem;
replacing the original problem with some approximation;
analyzing the theoretical properties of the approximation (error analysis);
devising efficient algorithms to solve the approximate problem;
implementing the algorithms using a programming language.
In this course my plan is, by using relatively few simple mathematical problems, touch on every aspect mentioned above.
Nowadays we have access to a lot of powerful software (such as MatLab or Mathematica, to mention just two), in which most basic numerical methods are already implemented, and the users are frequently applying these methods as black boxes. It is not my intent for this course to discuss various built-in functions of the existing software, but rather my intent is, in a sense, to look inside these black boxes, understand how they function, and learn how to create these black boxes ourselves from scratch. This is why this course will have a significant programming component (I will use Python), but no programming experience is expected. On the mathematical side I plan to include: solving scalar nonlinear equations; a bit of numerical linear algebra, interpolation and numerical integration, and solving the initial value problems for ordinary differential equations.