*)N.B. if there are students who do not speak Portuguese the language is English.
1. The student should become aware of the limitations of pseudo-random numbers and of the different architectures of random number generators.
2. He must understand how Monte Carlo simulation works and how and when to apply it.
3. He should be able to simulate from a sample and to anticipate, through simulation, the response of a system.
4. He should also be able to model a physical process to predict and reproduce the outcome of a system
Monte Carlo method: Definition and implementation.
Random numbers ? requirements. Types of random number generators. Tests.
Probabilities: discrete, continuous and cumulative. Uniform distribuition. Non-uniform distributions: exponential and Gaussian.
Change of probability density: inversion and Box-Muller methods. Change of variables.
Monte Carlo integration.
Finding the root of equations: successive substitutions, Newton-Cotes, half intervals and regula-falsi methods.
Interpolation methods: polynomial interpolation, Lagrange formula and piecewise interpolation.
Finding the roots of functions: Newton-Raphson, secant, bisection and regula-falsi
Integration methods: Newton-Cotes quadrature (open and closed), middle point, trapezoid and Simpson rules. Gaussian quadrature.
Integration of differential equations: Euler, Neuer and Runge-Kutta methods.
Random walks, Markov chains, Metropolis algorithm. Examples and applications.
Programming capabilities at an intermediate level.
Generic skills to reach
. Computer Skills for the scope of the study; . Competence to solve problems; . Critical thinking; . Creativity; . Research skills; . Competence in analysis and synthesis; . Competence in oral and written communication; . Adaptability to new situations; . Quality concerns; . Self-criticism and self-evaluation; (by decreasing order of importance)
Teaching hours per semester
total of teaching hours
apresentação teórica de tema relacionado com a matéria %
à escolha do aluno %
10/presentation by the student of a topic included in the syllabus %
Bibliography of reference
- Knuth, The Art of Computer Programming, 3rd vol, Addison-Wesley, 1999.
- Press et al., Numerical Recipies in c, Camb. Univ. Press, 1992.
- Wong, Computational Methods in Physics and Engineering, 2nd ed, Prentice-Hall, 1997.
- R. Gaylord, P. Wellin, Computer Simulations with Mathematica, Springer, 1995.
Lectures use the blackboard and occasionally slide projection. They intend to be a discussion of the subjects and they include examples; students are encouraged to participate in these discussions. Examples discussed in lectures can and will, whenever possible, include case studies and typical applications, either in Physics or in other subjects.
We also aim to develop students creativity and curiosity by encouraging them to suggest ideas, themes, problems to be solved, etc.
Typical Monte Carlo dealt situations are also described and studied.