Background: first year graduate student in applied math, currently taking an introductory numerical analysis course which will be covered on the qualifying exams come fall. The instructor is quite often unclear and teaches out of his own notes, so I’m looking for a reference that can present the motivation and analysis a bit more clearly. We were assigned Trefethen’s Numerical Linear Algebra, but only covered about a quarter of it (the best part of the class). Most of the other material is pretty standard I think: numerical integration and differentiation, interpolation, root finding, initial and boundary value problems, etc.
Burden & Faires is the standard undergraduate level introduction. It might be beneath the level you’re looking for, but you should at least take a quick look at it. The intro numerical methods class at my university uses Beers’ “Numerical Methods for Chemical Engineering”, which apparently doesn’t assume you know anything about chemical engineering. I haven’t taken that one yet, so I can’t recommend the book, but it might also be worth looking at.