Due my writing an exam on numerical analysis I had the pleasure to look through lots and lots of books on numerical analysis, and here is a list of my favorite ones so far:

Afternotes on Numerical Analysis & Afternotes Goes to Graduate School by G. W. Stewart Both books are very readable and introduce many of the concepts in a light way that builds an intuitive understanding for them. It's possible to read the books cover to cover in a few days and you can learn a lot very quickly. They are also quite amusing:

"In the nineteenth century the Norwegian mathematician Niels Abel showed that no polynomial of degree five could be solved by a finite number of additions, multiplications, divisions, and root extractions. If we had a finite algorithm for finding eigenvalues of general matrices, we could apply it to companion matrices and make a fool out of Abel. Abel was no fool."

Numerical Linear Algebra by Trefethen Another very good book which I haven't used much personally, though

Numerical Methods by Kelley contains an okay introduction to CG and GMRES.

Numerische Mathematik I & Numerische Mathematik II are very good books, too. The first volume contains a good introduction to everything but ODEs and PDEs and the second volume consists of a very thorough overview of numerical algorithms for differential equations, including a nice introduction to the general theory of their solvability etc.

Numerical Analysis by Burden and Faires a very nice book that contains lots of visualizations and covers many topics

More to follow soon :-)

Cheers,

Andreas