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.