TITLE: There must be a good introductory numerical analysis course out there!
QUESTION [57 upvotes]: 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.

REPLY [3 votes]: I am teaching an experimental offering at UVic that goes part-way to addressing your concerns.  
The goal of the course is to get 2nd year students comfortable with writing mathematical software in a high-level computer language.  This semester we are using Python but the specific language is the choice of the instructor. 
The main part of the course is about building students' confidence up, writing small scripts to test mathematical ideas. 
But along the way we teach them about various elements from numerical analysis and their limitations.  We largely do not teach any theory in this course.  The course is about learning by example.  So students see first-hand the issues that come from round-off error. They see first-hand arbitrary precision floats and integers, and how they can help (and hinder) an investigation.  
We also touch on a variety of topics not specific to numerical analysis.