Principles of Numerical Mathematics

- well-posedness and condition number of a problem
- stability and convergence of numerical methods
- a-priori and a-posteriori analysis
- sources of error in computational models
- machine representation of numbers

Linear Operators on Normed Spaces

Rootfinding for Nonlinear Equations

- the bisection method, the methods of chord and secant, Newton's method
- fixed-point iterations (the Banach fixed-point theorem and convergence 
results)
- zeros of algebraic functions
- stopping criteria
- post-processing techniques for iterative methods
- solution of nonlinear systems of equations

Polynomial Interpolation

- Lagrange polynomial interpolation
- Newton form of the interpolating polynomial
- Hermite polynomial interpolation
- piecewise polynomial interpolation
- extension to the two-dimensional case
- approximation by splines
- B-splines

Numerical Integration

- midpoint, trapezoidal, and Simpson formulae
- Newton-Cote formulae
- Hermite quadrature formulae
- Richardson extrapolation
- singular integrals
- multidimensional numerical integration

Orthogonal Polynomials in Approximation Theory

- approximation of functions by generalized Fourier series
- the Chebyshev polynomials
- the Legendre polynomials
- Gaussian integration and interpolation
- Chebyshev integration and interpolation
- Legendre integration and interpolation
- Gaussian integration over unbounded intervals
- least-squares approximation
- best approximations
- Fourier trigonometric polynomials
- approximation of function derivatives (classical finite differences, 
compact finite differences, pseudo-spectral derivative)
- transforms and their applications
- wavelet approximations

Numerical Solution of ODEs

- one-step numerical methods
- analysis of one-step numerical methods
-multistep methods
- analysis of multistep methods
- predictor-corrector methods
- Runge-Kutta methods
- stiff problems