The students become familiar with basic concepts of Numerical Mathematics. They will learn to analyze and develop basic schemes in Numerical Analysis of Linear and Nonlinear systems.
- Error analysis: Floating-point arithmetic (round-off errors, error propagation)
- Vectors and matrices (norms, elementary linear algebra, BLAS package)
- Initial value problems (Runge-Kutta integration)
- Fast Fourier transformation
- Finite element methods
- Diffusion equation
- Random number generators and distributions
- Monte Carlo integration
- Probability, important distributions and their properties, expectation values, RMS, correlation, parameter estimation, maximum likelihood, confidence intervals, detector unfolding, special methods (Bootstrap, Jackknife), parameterization.
- Linear systems (Gaussian elimination & W decomposition, Cholesky, condition number round-off errors, least-squares, elementary iteration schemes)
- Nonlinear equations (Univariate case, Newton method in Rn, fixed point theorems, Gauss-Newton, modified Newton)
- Direct and iterative methods for solving linear systems and eigenvalue problems. The methods are analyzed with respect to stability, convergence, and complexity. Their application in different context is discussed.
- Familiarity with one of: Matlab, Octave, Python (numerical/scientific python), R or a similar high-level scripting language to complete some of the coursework
Bibliography and teaching material includes:
- Course notes
- Numerical Recipes: The Art of Scientific Computing
- Numerical Methods with Applications, A. Kaw, E. Kalu, D. Nguyen
- Iterative Methods for Linear and Nonlinear Equations, C. T. Kelley
- The following assessment methods will be combined for the final grade:
- Homework exercises
- In-class exercises
- A final examination