I need to use some heavy-duty numerical methods - quadrature, interpolation, ODE solution with expensive derivative terms, etc. What's the best book on the subject?
I will be using packages where I can, but I can already tell there will be times when I need to write my own routines...
If you consider Numerical Recipes, you should probably be aware of some criticism: Why not use Numerical Recipes?
Some alternatives are listed here.
Numerical Recipes is a good reference, but if I were to recommend a book to read cover to cover, it would be Numerical Methods that Work. It covers the kinds of problems you listed, but more importantly it teaches you how to think like a numerical analyst. Some of the examples are a bit dated, but the principles are very applicable.
The venerable and unmatched Stroud.
Engineering Mathematics and Advanced Engineering Mathematics.
Even if there are critisism, I write heavy-duty numerical methods all the day and Numerical Recipes is a great book, which teaches you a lot if you are willing to read the chapter and not copy and paste the code directly.
For production code, please use canned routines for linear algebra, but otherwise, I never had any problems with the code in NR regarding integration and ODE, and the interpolation classes in the third version of the book is my daily bread and butter.
But please be aware that the code in NR has to be adapted to your needs, since it is generally poor production code (but the design is generally good in the C++ 3rd version).