Taylor Series (convergence proofs, truncated series)

Error Characterization (relative, absolute, big ``O'')

Computer Arithmetic
Floating point representations
Machine epsilon, roundoff error, loss of significance

Root Finding f:R --> R
Bisection method
Newton's method
Secant method
Fixed point method
Convergence theorems, modes of failure
Linear, quadratic, superlinear convergence
A priori convergence theorems

Root Finding f:Rⁿ --> Rⁿ
Newton's method

Solution of Linear Systems
System Theory
Solvability
Matrix norms and condition number of a matrix
Ill-conditioned systems, real symmetric matrices
Special systems
Diagonal
Upper/lower triangular (diagonal dominance)
Tridiagonal and banded systems
Symmetric positive definite
Solution methods
Gaussian elimination: scaling, scaled partial pivoting
Backward/Forward substitution
LU factorization
Computational complexity: O(n^p)
Power method and convergence analysis

General Least Squares
Least squares in Rⁿ
Normal equations
Singular value decomposition: pseudoinverse and related theory

Interpolation
Lagrangian interpolants
Existence and uniqueness theorems
Error analysis for even grids

Splines
Linear, quadratic, cubic, degree
Natural spline algorithm
Error analysis for natural cubic splines

Integration
Trapezoid, Simpson's rules
Composite rules
Error analysis
Romberg extrapolation
n-point Gaussian quadrature: basic rules, error analysis
Degree of accuracy
Orthogonal polynomial derivation of Gauss n-point rules

Initial Value Problems: Existence, uniqueness, blowup

Initial Value Problem Methods: y'(t) = f(y(t),t)
Taylor series methods
Adams-Bashforth methods (explicit)
Adams-Moulton methods (implicit)
Runge-Kutta (RK) methods (explicit)
Backward difference methods
Euler, midpoint, implicit trapezoid, backward Euler

Initial Value Problem Methods: Theory
Error analysis
Absolute stability, A-stable
Local and global error

Second-Order Boundary Value Problems
Conversion to systems
Uniqueness and nonuniqueness of solutions
Shooting methods: Newton/secant
Finite difference methods
Finite element methods

References:

Numerical Analysis, Richard L. Burden and J. Douglas Faires, PWS-Kent Publishing Company.
Numerical Analysis, David Kincaid and Ward Cheney, Brooks/Cole.