I agree with many of your sentiments.  Here is a view from "inside" (I do PDE solvers from a mathematics and computational physics perspective, have written open-source models, and contribute to a popular open-source solver library).<p>1. Very little of numerical analysis is about propagation of rounding errors.  I think you underestimate the reliability of floating point numbers in this context.  The challenge of numerical analysis is not in representing the continuous one-dimensional object (real numbers), but in representing an infinite dimensional space in finite dimensions.  You might appreciate Trefethen's "The definition of numerical analysis" (<a href="http://www.comlab.ox.ac.uk/nick.trefethen/publication/PDF/1992_55.pdf" rel="nofollow">http://www.comlab.ox.ac.uk/nick.trefethen/publication/PDF/19...</a>).<p>2. We can prove convergence of iterative methods for some problems of practical interest.  More frequently, we can prove that if the iterative method converges, then it converges to a meaningful solution.  Such results are unavailable for some systems, and this is where we enter the domain of Verification (the code converges to exact continuum solutions at the appropriate rate) and Validation (the continuum equations approximate reality).  V&#38;V is a rapidly growing field which offers a number of mature tools, but is still under-appreciated in many disciplines.  The fact that V&#38;V is rarely taught at university, even to applied mathematics graduate students, let alone physicists, doesn't help.<p>3. It takes effort to turn a one-off code that only runs in your special environment into portable, distributable software. Funding agencies and institutions place little emphasis on this process, so it is regularly neglected.  It is exacerbated by the fact that few physicists have any background or interest in software engineering, thus making it more challenging to release non-broken software.