TUTORIAL no. 1Random numbers

G. LattanziUniversità degli studi di Bari “Aldo Moro”

Trieste, July 9th, 2012

CECAM school on Numerical simulationsF. Becca & G. Lattanzi

SISSA

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/

Exercises with random numbers (1)1. Casuality test: consider the function, being r a random variable uniformlydistributed between 0 and 1:

In the large N limit:

€

ψ r1,r2,K ,rN( ) =1

Nri

i=1

N

∑

€

ψ =0.5

€

σ 2 = ψ 2 − ψ2=1

12N

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.

Modules

To see the list of available modules: module availTo load a specific module: module load name_module

LOCATED AT id001719CALLED random.c

Real(8) drand1

Iseed=142527Call rand_init(iseed)

R=drand1()

Exercises with random numbers (2)2. Random integration. Consider the following integral:

€

I = f (x)dx = 1− x 2dx0

1

∫0

1

∫

We will evaluate the integral exploiting random numbers: we generate Nnumbers xi in the interval [0,1] and calculate:

€

F =1

Nf (x i) =

1

N1− x i

2

i=1

N

∑i=1

N

∑

Verify that the average value is, as expected, π/4. What about the variance? What is its expected value?

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 ρΜΑΧ.

Method:I. We extract two series of random numbers uniformly distributed between

0 and 1: we call them ηi and θi.II. We use the ηi’s to obtain a sequence of random numbers xi uniformly

distributed between a and b:

III. We use the θi’s to generate random numbers yi uniformly distributed between 0 and ρΜΑΧ:

IV. We accept the (xi,yi) if and only if: €

x i = (b− a)η i + a

€

y i = ρMAXθ i

€

y i ≤ ρ(x i)

Exercises with random numbersAlgorithm:1. Extract ηi

2. Obtain xi=(b-a)ηi+a3. Extract θi 4. Obtain yi=θi*ρΜΑΧ

5. Check: yi ≤ ρ(xi)? If yes, accept (xi, yi) . If not, reject it.

Exercise 3. Use the above specified algorithm to generate random numbersdistributed between -1 and 1 with probability density:

€

ρ(x) =2

π1− x 2

Having divided the interval [-1,1] in bins, plot the histogram of the valuesgenerated with this procedure. Plot also in the plane (x,y) the points obtained.

Exercises with random numbersJustification of the algorithm

The probability to obtain the sequence of numbers yi is the product of the probabilityto extract xi with the given uniform probability distribution:

And the acceptance probability

€

u(x)dx =1

b− adx

€

α(x) =f (x)

fMAXThis function is exactly ρ(x), apart from a normalization factor:

€

α(x)u(x)dx =1

b− a

1

ρMAXρ(x)