# CoS 502: Frontiers in Numerical Methods

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.

Instructor:
Asst. Prof. Nikos Savva

Contents:

• 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.

Prerequisites:

• CoS501
• 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

Assessment methods:

• The following assessment methods will be combined for the final grade:
• Homework exercises
• In-class exercises
• A final examination