The aim of the course is to present several numerical methods that can be used in different scientific areas and implement them using Python. The course starts from the basic idea of an algorithm and evolves discussing, for instance, numerical methods to compute derivatives and integrals of functions, to solve linear systems, and to integrate ordinary and partial differential equations. Each lecture will have an initial part of theory and a final part of Python demo.
Calculus, linear algebra, ordinary and partial differential equations. Some basic concepts of probability, mechanics and fluid mechanics might be used during the examples.
- how to install Python and basic commands, definition of an algorithm, evaluation of the square root
- root finding algorithms: bisection and more advanced methods
- solutions of linear systems, direct and indirect methods
- derivatives of a function using finite differences, methods of finding the function extremes
- Lagrange polynomials and splines
- integration of single variable functions with rectangles and other methods
- Monte Carlo method to compute multivariable integrals
- solutions of ODEs using Euler and Runge-Kutta methods
- integration of PDEs using finite difference algorithms
- fast Fourier transforms (FFTs) and their use in solving PDEs with periodic boundary conditions