General Texts in Numerical Analysis

K. Atkinson: An Introduction to Numerical Analysis, Wiley, (2nd ed.), 1989.
P.G. Ciarlet and J. L. Lions (eds), Handbook of Numerical Analysis,
     North Holland, 1990.
E. W. Cheney and D. R. Kincaid: Numerical Mathematics and Computing,
     Brooks Cole, 6 edition, 2007.
S. Conte and C. de Boor: Elementary Numerical Analysis, McGraw-Hill, 1980. 
G. Dahlquist and A. Bjorck: Numerical Methods, Prentice Hall, 1974.
P. Deuflhard and A. Hohmann, Numerical Analysis in Modern Scientific
      Computing, 2nd ed., Springer, 2003.
W. Gautschi: Numerical Analysis: an introduction, Birkhauser, 1997.
M. T. Heath, Scientific Computing: An Introductory Survey, 2nd ed., 
      McGraw-Hill, 2002.
E. Isaacson and H. Keller: Analysis of Numerical Methods, Wiley, 1966
     (or Dover 1994).
D. Kahaner, C. Moler, and S. Nash: Numerical Methods and Software,
     Prentice-Hall, 1989.
D. Kincaid and W. Cheney: Numerical Analysis: Mathematics of Scientific
     Computing, American Mathematical Society, 3rd edition, 2001.
A. Quarteroni, R. Sacco, and F. Saleri: Numerical Mathematics, 2nd Edition,
     Springer, 2004.
A. Ralston and P. Rabinowitz: A First Course in Numerical Analysis, 
     McGraw-Hill, 1978.
L. Shampine, Allen, and Pruess: Fundamentals of Numerical Computing,
G. W. Stewart: Afternotes on Numerical Analysis, SIAM, 1996.
G. W. Stewart:  Afternotes Goes to Graduate School:
     Lectures on Advanced Numerical Analysis, SIAM, 1998
J. Stoer and R. Bulirsch: Introduction to Numerical Analysis, Springer,
     3rd edition, 2002.
E. Suli and D. Mayers: An Introduction to Numerical Analysis, Cambridge, 2003.

Numerical Solution of Ordinary Differential Equations

U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential
     Equations and Differential-Algebraic Equations, SIAM, 1998
J. C. Butcher, Numerical Methods for Ordinary Differential Equations, 2nd
     ed., Wiley, 2003.
J. C. Butcher: The Numerical Analysis of Ordinary Differential
     Equations: Runge-Kutta and general linear methods, Wiley, 1987.
K. Dekker and J. G. Verwer, Stability of Runge-Kutta methods for stiff
     nonlinear differential equations, North Holland, 1984.
P. Deuflhard and F. Bornemann, Scientific Computing with Ordinary
     Differential Equations, Springer, 2002
S. O. Fatunla: Numerical Methods for Initial Value Problems in Ordinary
     Differential Equations, Academic Press, 1988.
C. W. Gear: Numerical Initial Problems in Ordinary Differential  Equations,
     Prentice Hall, 1971.
E. Hairer, S. P. Norsett, and G. Wanner, Solving Ordinary Differential
     Equations, Springer-Verlag,  I: nonstiff problems (1993), II (1991).
P. Henrici: Discrete Variable Methods in Ordinary Differential Equations,
     Wiley, 1962.
I. Iserles, A First Course in the Numerical Analysis of Differential
     Equations, Cambridge University Press, 1996.
J. D. Lambert, Numerical Methods for Ordinary Differential Systems:
     The Initial Value Problem, Wiley, 1991.
J. Lambert, Computational Methods in Ordinary Differential Equations, 1973.
L. Shampine: Numerical solution of ordinary differential equations,
     Chapman & Hall, 1994.
L. Shampine and M. Gordon: Computer Solution of Ordinary Differential
     Equations, Freeman, 1975.
L. Shampine, I. Gladwell, and S. Thompson, Solving ODEs with MATLAB,
     Cambridge, 2003.

Solution of Nonlinear Systems of Equations and Optimization

J. E. Dennis and J. More: Quasi-Newton methods, motivation, and theory,
     SIAM Review, vol 19 #1, Jan. 1977.
J. E. Dennis and R. B. Schnabel: Numerical Methods for Unconstrained
     Optimization and Nonlinear Equations, Prentice-Hall, 1983, SIAM, 1996.
P. Deuflhard: Newton Methods for Nonlinear Problems: Affine Invariance and
     Adaptive Algorithms, Springer, 2004.
C. T. Kelley: Iterative Methods for Linear and Nonlinear Equations, SIAM, 1995.
C. T. Kelley, Iterative Methods for Optimization, SIAM, 1999.
C. T. Kelley: Solving Nonlinear Equations with Newton's Method, SIAM, 2003.
J. Nocedal and S.J. Wright: Numerical Optimization, Springer, 2000.
J. Ortega and W. Rheinboldt: Iterative Solution of Nonlinear Equations
     in Several Variables, Academic Press, 1970.

Numerical Linear Algebra

O. Axelsson: Iterative Solution Methods, Cambridge University Press, 1994.
J. W. Demmel: Applied Numerical Linear Algebra, SIAM, 1997.
G. Forsythe and C. Moler:  Computer Solution of Linear Algebraic Systems,
     Prentice Hall, 1967.
G. Golub and C. Van Loan: Matrix Computations, (3rd ed.), Johns Hopkins
     University Press, 1996
A. Gourley and G. Watson:  Computational Methods for Matrix Eigenproblems,
     Wiley, 1973.
W. W. Hager: Applied Numerical Linear Algebra, Prentice-Hall, 1988.
G. W. Stewart:  Introduction to Matrix Computations, Academic Press, 1973.
G. W. Stewart: Matrix Algorithms: Basic Decompositions, SIAM, 1998
G. W. Stewart: Matrix Algorithms, Volume II: Eigensystems, SIAM, 2001
G. W. Stewart, J-q. Sun: Matrix Perturbation Theory, Academic Press, 1990 
L. N. Trefethen and D. Bau, Numerical Linear Algebra, SIAM, 1997.
J. H. Wilkinson: The Algebraic Eigenvalue Problem, Oxford, 1988.
J. H. Wilkinson and C. Reinsch:  Linear Algebra

Finite Element, Finite Volume, and Spectral Methods

D. Braess: Finite Elements: Theory, fast solvers, and applications in
     solid mechanics, Cambridge University Press, 3 edition, 2007.
S. Brenner and L. R. Scott: The Mathematical Theory of Finite Element
     Methods, Springer-Verlag, 3rd edition, 2007.
D. Boffi, F. Brezzi, and M. Fortin, Mixed Finite Element Methods and
     Applications, Springer Series in Computational Mathematics, Vol 44, 2013.
P. G. Ciarlet:  The Finite Element Method for Elliptic Problems,
     SIAM, 2002 (Originally published by North Holland, 1980).
K. Eriksson, D. Estep, P. Hansbo, C. Johnson: Computational Differential
     Equations, Cambridge University Press, 1996.
A. Ern and J-L. Guermond: Theory and Practice of Finite Elements,
     Springer, 2004.
V. A. Girault and P-A. Raviart: Finite Element Methods for Navier-Stokes
     Equations, Springer-Verlag, 1986.
M. S. Gockenbach: Understanding and Implementing the Finite Element
     Method, SIAM, 2006.
C. Grossmann, H-G. Roos, and M. Stynes: Numerical Treatment of Partial
     Differential Equations, Springer, 2007.
M. Gunzburger, Finite Element Methods for Viscous Incompressible Flows,
     Academic Press, 1989.
T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic
     Finite Element Analysis, Prentice Hall, 1987.
C. Johnson: Numerical Solutions of Partial Differential Equations by
     the Finite Element Method, Cambridge University Press, 1987.
S. Larsson and V. Thomee, Partial Differential Equations with
     Numerical Methods, Springer, 2008.
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge, 2002.
G. Strang and G. Fix: An Analysis of the Finite Element Method,
     Prentice-Hall, 1973.
B. Szabo and I. Babuska: Finite Element Analysis, Wiley, 1991.
L. N. Trefethen: Spectral Methods in MATLAB, SIAM 2000.
O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method
     (4th edition), McGraw Hill, 1989.

Finite Difference Methods for Two-Point Boundary Value Problems and Partial Differential Equations

S. K. Godunov and V. S. Ryabenki, Difference Schemes: an introduction
     to the underlying theory, North Holland, 1987.
C. Grossmann, H-G. Roos, and M. Stynes: Numerical Treatment of Partial
     Differential Equations, Springer, 2007.
C. Hall and T. Porsching: Numerical Analysis of Partial Differential
     Equations, Prentice Hall, 1990.
R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential
     Equations, Steady State and Time Dependent Problems, SIAM, 2007.
R. J. LeVeque, Numerical methods for conservation laws, Birkhauser Verlag,
     1992.
I. Iserles, A First Course in the Numerical Analysis of Differential
     Equations, Cambridge University Press, 1996.
H. Keller: Numerical Methods for Two-Point Boundary Value Problems,
     SIAM, 1976.
R. D. Richtmyer and K. W. Morton:  Difference Methods for Initial Value
     Problems, Wiley-Interscience, 1967.
G. Sod: Numerical Methods in Fluid Mechanics: Initial and
     Initial Boundary Value Problems, Cambridge University Press, 1985.
J. Strikwerda: Finite Difference Schemes and Partial Differential
     Equations, Second Edition, SIAM, 2004.

MATLAB

T. Davis and K. Sigmon: MATLAB Primer, Seventh Edition, Chapman and Hall, 2004.
D. J. Higham and N. J. Higham: MATLAB Guide, Second Edition, SIAM, 2005.
C. Moler: Numerical Computing with MATLAB, SIAM, 2004.
A. Quarteroni and F. Saleri, Scientific Computing with Matlab, Springer, 2003.