K. Atkinson, An INTRODUCTION to NUMERICAL ANALYSIS, Wiley, 1987

S. Conte & C. deBoor, ELEMENTARY NUMERICAL ANALYSIS, McGraw-Hill User friendly; Shows how 'it' works; Proofs, exercises and notes

G. Dahlquist & A. Bjorck, NUMERICAL METHODS, Prentice-Hall, User friendly; Shows how 'it' works; Exercises

A. Ralston & P. Rabinowitz, FIRST COURSE in
NUMERICAL ANALYSIS, 2nd ed., McGraw-hill, Detailed; Scholarly written; Comprehensive; Proofs exercises and notes

J. Stoer & R. Bulrisch, INTRODUCTION TO NUMERICAL ANALYSIS, 2nd ed., Springer detailed account on approximation, linear solvers & eigen-solvers,
ODE solvers,..

APPROXIMATION THEORY

E. W. Cheney, INTRODUCTION TO APPROXIMATION THEORY Classical

P. Davis, INTERPOLATION & APPROXIMATION, Dover Very readable

T. Rivlin, AN INTRODUCTION to the APPROXIMATION of FUNCTIONS Classical

R. DeVore & G. Lorentz, CONSTRUCTIVE APPROXIMATION, Springer A detailed account from classical theory to the modern theory; everything; Proofs exercises and notes

NUMERICAL INTEGRATION

F. Davis & P. Rabinowitz, NUMERICAL INTEGRATION, Everything...

NUMERICAL SOLUTION Of INITIAL-VALUE PROBLEMS

E. Hairer, S.P. Norsett and G. Wanner, SOLVING ODEs I: NONSTIFF PROBLEMS,
Springer-Verlag, Berlin. 1991, (2nd ed) Everything - the modern version

A. Iserles, A FIRST COURSE in the NUMERICAL ANALYSIS of DEs,
Cambridge Texts

W. Gear, NUMERICAAL INITIAL VALUE PROBLEMS in ODEs, 1971 The classical reference on theory and applications

Lambert, COMPUTATIONAL METHODS for ODEs, 1991 Detailed discussion of ideas and practical implementation

Shampine and Gordon, COMPUTER SOLUTION of ODES, 1975 Adams methods and practial implementation of ODE "black box" solvers

Butcher, NUMERICAL ANALYSIS of ODEs, 1987 Comprehensive discussion on Runge-Kutta methods

(mainly) ITERATIVE SOLUTION OF LINEAR SYSTEMS

A. Householder, THE THEORY OF MATRICES IN NUMERICAL ANALYSIS The theoretical part by one of
the grand masters; Outdated in some aspects

G. H. Golub & Van Loan, MATRIX COMPUTATIONS, The basic modern reference