Background As a numerical analyst, I've frequently taught the 'Introductory Numerical Analysis' class. Such courses are found in many major universities; the audience typically consists of reluctant engineering majors and some majors of mathematics.
The structure of the course is very similar in many of the institutions whose syllabi I've looked at: one begins with finite-precision arithmetic, then fixed-point methods for root-finding (usually 1-D problems),interpolation by polynomials, quadrature, numerical differentiation, some standard ODE methods, and perhaps some finite difference methods for PDE. Any rationale for this particular sequence of topics is obscured in the course.
The truly deep and interesting aspects – approximation theory, error analysis, computational complexity – are either not discussed, or not dwelt on. Instead, the typical introductory course is a collection of algorithms for problems which seem contrived.
This is a pity. The stronger mathematics student comes away believing numerical analysis is boring and shallow, and the engineer comes away thinking mathematics has nothing to offer a real problem.
The question: Are there examples (links to course outlines or course webpages preferred) of introductory numerical analysis courses which avoid the above-described tedium, and which have a history of attracting strong mathematics students?
The constraints: The courses should be aimed at students with a background in multivariate calculus, linear algebra, undergraduate dynamical systems and PDE. One example per answer, please.
The motivation: The eventual goal is to compile such a list, and based on these courses suggest a better curriculum at my institution.
Best Answer
John Hubbard tends to take sort of the opposite track, in that he likes to bring a more serious numerical analysis perspective into the 1st and 2nd courses on calculus and differential equations, rather than assuming the students come out of a standard service-stream calculus, differential equations, linear algebra sequence of courses. Usually this includes a discussion of various ways of representing numbers on computers, like floating-point numbers, round-off errors, perhaps even topics like interval arithmatic.
For example, once the idea of ODEs are set up he likes to talk about "fences". I don't know if this is standard terminology anywhere or just his, but it's basically like a Lyapanov function but for time-dependent ODEs. So it gives you regions that trap orbits, but the region may move with time. He gets students used to thinking in this way gradually, by cooking up fences in the 1-dimensional time-dependent ODE case first. Then he moves on to things like the Gronwall inequality, applying it for things like the Euler approximations to ODE solutions to observe error growth rates. He also proves Kantorovich's theorem, which he uses for the implicit and inverse function theorems. He has quite a lot of success getting 1st and 2nd year physics and engineering students thinking about these things. But it's known as the "challenging" calculus stream at Cornell, and less ambitious students have other options. I don't know what their numbers are now, but when I was a TA for the course I think he was getting around 80 students per year in the course.