Important Techniques and Concepts
 Taylor's theorem and the mean value theorem
 Vector and matrix norms, spectral radius
 Convergence and convergence rate
 Algorithm complexity and operation count

Error Analysis
 Sources of errors
 Absolute and relative error
 Effects of finite precision arithmetic
 Floating point representation
 Rounding
 Machine epsilon
 Cancellation
 IEEE floating point, including subnormals
 Conditioning of problems and condition number
 Backward error (and forward error)
 Stability of algorithms
 Perturbation analysis for linear systems

Nonlinear Equations
 Bracketing methods
  Bisection
  Regula falsi 
  Modified regula falsi
 Newton's method
  Graphical, Taylor series, fixed-point method derivations
  Newton's method with double roots
  Newton's method for systems of nonlinear equations
 Interpolatory methods
  Secant method
   Derivation of convergence rate
  Muller's method and inverse quadratic interpolation
 Safeguarded methods
  Matlab's fzero and how to find an initial bracket
 Fixed-point iteration
  Convergence rate derivation
  Existence and uniqueness of a fixed point
  Fixed-point iteration for systems of nonlinear equations
 
Linear Equations
 Special linear systems
  Triangular matrices
  Tridiagonal and banded matrices
  Sparse matrices and the structure of 1-D, 2-D, and 3-D problems
 Direct methods
  LU factorization in its various forms
  Cholesky factorization and SPD matrices
  Elementary elimination matrices
  Partial pivoting and complete pivoting
  Reorderings for reducing fill-in for direct methods
 Iterative methods
  Jacobi
  Gauss-Seidel
  SOR
  Conjugate gradient method and preconditioning
  Convergence of iterative methods