1. TUTORIAL no. 1 Random numbers G. Lattanzi Università degli studi di Bari “Aldo Moro” Trieste, July 9 th, 2012 CECAM school on Numerical simulations F. Becca & G. Lattanzi SISSA

2. Random numbers generators The programming language of your choice contains a built-in function that generates random numbers: Fortran: RAND C++: rand Perl: rand() Python: random There are also alternatives, based on different and more sophisticated algorithms, e.g. those proposed by Numerical Recipes http://www.nr.com/

3. Exercises with random numbers (1) 1.Casuality test: consider the function, being r a random variable uniformly distributed between 0 and 1: In the large N limit: Verify that your favourite built-in random number generator passes the confidence test at 5%, i.e. that 95% of the realizations fall in the interval (μ-2σ,μ+2σ). Plot the obtained values of ψ for N=10, 20, 50, 100.

4. Modules To see the list of available modules: module avail To load a specific module: module load name_module LOCATED AT id001719 CALLED random.c Real(8) drand1 Iseed=142527 Call rand_init(iseed) R=drand1()

5. Exercises with random numbers (2) 2. Random integration. Consider the following integral: We will evaluate the integral exploiting random numbers: we generate N numbers x i in the interval [0,1] and calculate: Verify that the average value is, as expected, π/4. What about the variance? What is its expected value?

6. Exercises with random numbers (3) 3. Rejection method. We want to obtain a sequence of random numbers distributed according to a probability distribution ρ defined in [a,b] with a maximum value ρ ΜΑΧ. Method: I.We extract two series of random numbers uniformly distributed between 0 and 1: we call them η i and θ i. II.We use the η i ’s to obtain a sequence of random numbers x i uniformly distributed between a and b: III.We use the θ i ’s to generate random numbers y i uniformly distributed between 0 and ρ ΜΑΧ : IV.We accept the (x i,y i ) if and only if:

7. Exercises with random numbers Algorithm: 1.Extract η i 2.Obtain x i =(b-a)ηi+a 3.Extract θ i 4.Obtain y i =θ i *ρ ΜΑΧ 5.Check: y i ≤ ρ(x i )? If yes, accept (x i, y i ). If not, reject it. Exercise 3. Use the above specified algorithm to generate random numbers distributed between -1 and 1 with probability density: Having divided the interval [-1,1] in bins, plot the histogram of the values generated with this procedure. Plot also in the plane (x,y) the points obtained.

8. Exercises with random numbers Justification of the algorithm The probability to obtain the sequence of numbers y i is the product of the probability to extract x i with the given uniform probability distribution: And the acceptance probability This function is exactly ρ(x), apart from a normalization factor: