Corso di laurea magistrale in Fisica

The course is aimed at making the students familiar with the use of algorithms to enable them to face various numerical issues that they deal with during their studies. The implementation of various algorithms will be discussed using physical problems introduced in previous courses.

The student learns, by means of a programming language, to compile programmes for the resolution fo scientific problems.

The course provides fundamental techniques for solving scientific problems of various kinds.In particular, standard numerical techniques are illustrated throghout the following chapters:

1. Introduction: brief review of C ++ (functions, preprocessor), Linux shell (basic commands and compilation from terminal);
2. Arithmetic precision: floating point representation, single & double precision variables, machine precision. Uncertainty, accuracy and stability of algorithms.
3. Computation of definite integrals (numerical quadrature): Euler method, of the trapezoid. Simpson's rule. Gaussian quadrature. Hints of calculation in several dimensions.
4. Random numbers and Monte Carlo methods (brief outline);
5. Search for zeros of functions (nonlinear): bisection method, Newton-Raphson method, secant method and False Position. Convergence.
6. Numerical differentiation.
7. Numerical solution of ordinary differential equations (ODE, initial value problems): Euler method and second and fourth order Runge-Kutta methods. Symplectic methods. Convergence. Conservation of energy and reversibility of motion. Implicit methods (hints)
8. Numerical solution of ordinary differential equations ordering (boundary problems): "shooting" method, residual and matching point. Applications to the quantum harmonic oscillator.
9. Inversion of linear systems: review of multidimensional arrays. Gaussian Elimination. Solver for tridiagonal matrices. Problems.
10. Numerical solution of partial differential equations (the elliptical case): methods of  Jacobi, Gauss and Successive Over-Relaxation (SOR) method.

The course includes 60 hours of in-person teaching, typically divided into 20 lessons of 3 hours each.

The course does not include any oral exams.

During the course, several practical sessions are offered to facilitate the understanding of the algorithm and its implementation.

Additionally, during class hours, 'in-class assignments' or tests are conducted in which the student must solve a physics problem using the methods presented in the lectures. The results of these tests contribute to the determination of the base grade. After the course has ended, the student can optionally choose to submit an additional project (around ten pages) in which a physics problem is solved using numerical methods. The evaluation of the project can help improve the base grade

Different English textbooks are suggested

W.H. Press, S.A. Teukolsky, W. Vetterling and B. Flannery. Numerical Recipes in C (o Fortran). Cambridge University Press.

Donald B. KNUTH. "the Art of Computational Programming" Addison Wesley Ed.

Foley, van Dam, Feiner and Hughes.Computer Graphics (C o Pascal), Addison Wesely Ed.

Attendance is mandatory. Excused absences are considered only in case of illness / overlapping with other courses / public transport strikes / graduation sessions.

In any case, the number of absences has to be limited.

Examination methodology: Oral.
For further details please have a look on the personal webpage of the professor ( http://personalpages.to.infn.it/~mignone/Numerical_Algorithms/ )