Sven C. Koehler has started to read "Numerical Methods for Scientists and Engineers". He was so kind to send me a few quotes he likes from the introduction.

 Most books on computing stress the estimation of roundoff,
 especially the bounding of roundoff, but we shall concentrate
 on the avoidance of roundoff.  It seems better to avoid roundoff
 than to estimate what did not have to occur if common sense and
 few simple rules had been followed before the problem was put on
 the machine.

 Another standard algorithmic problem both in mathematics and in the use
 of computation to solve problems is the solution of simultaneous linear
 equations.  Unfortunately much of what is commonly taught is usually not
 relevant to the problem as it occurs in practice; nor is any completely
 statisfactory method of solution known at present.  Because the solution
 of simultaneous linear equations is so often a standard library package
 supplied by the computing center and because the corresponding
 description is so often misleading, it is necessary to discuss the
 limitations (and often the plain foolishness) of the method used by the
 package.  Thus it is necessary to examine carefully the obvious flaws and
 limitations, rather than pretending they do not exist.

update: (sorry, German only;) A big thanks to Sven C. Koehler for this summary

 Ich habe es nun in den groessten Teilen ueberflogen.  Die Ideen sind nicht
 wirklich neu: Umformen von Gleichungen, Vermeiden ungefaehr gleichgrosse
 Zahlen von einander abzuziehen, Approximation.  Beim Loesen von
 Gleichungssystemen schlägt er z.B. vor, ein Verfahren einzusetzen, das
 kein wiederholtes Dividieren benoetigt, dann wird's auch nicht ungenau.
 Trotzdem mag ich das Buch, weil es in mir den Eindruck weckt, dass es
 sehr fundiert ist.  Es ist voll von mathematischen Formeln, die ich alle
 nicht wirklich verstanden habe, aber ich werde in jedem Fall wieder darin
 nach Erklaerungen suchen, wenn ich mal wieder ein Numerik-Problem habe.

 Ich glaube für dich ist as Buch eher langweilig, das meiste kennst du
 bestimmt aus dem Studium. :-)