logic for random

1

C PROGRAMMING: RANDOM NUMBER GENERATION

We’re about to start the core of our cluster: random walks. We first

need to learn how to

generate

random numbers.

Some people object to calling numbers random when they are

generated with a computer code which has definite steps. These pe

ople prefer the name

pseudo-random numbers.

I’m not sure it’s such a big deal, really. After all, you could mak

e

a similar objection to calling a coin toss or die roll a random pr

ocess, since, really, the motion

of a spinning coin or die is completely determined by Newton’s l

aws of motion and so the

outcome is not really random either.

We will look at two random number generators. The first is not as g

ood a generator. (There

are various tests of how ‘random’ the generator really is, and t

he first one scores less well

on these tests.) It has the virtue however that you can see exactly

what it is doing. The

second is better, and it’s the one we will therefore use. However

, it is a ‘black box’: we just

call some code someone else wrote and we cannot peak inside. So it’

s good to start with the

more open one.

The first code, on page two, generates (pseudo) random numbers t

his way: you enter a

seed

integer

i

to start the process. Then you multiply

i

by 7

5

and calculate the remainder when

it is divided by 2

31

−

1. Finally, you divide the result by 2

31

−

1. (In the last step, make

sure you do this as real number arithmetic and not as integer ar

ithmetic. This is why the

code has separate variables

M

and

rM

.) That is

i

= 7

5

∗

i

i

=

i

%(2

31

−

1)

i

=

i/

(2

31

−

1)

To get another random number you repeat the process using your n

ew

i

as the seed for the

next step. You can continue this process as long as you like. Wel

l, not really: Eventually

the (pseudo) random numbers start to repeat. This is one reason t

his generator is not such

a good one. In fact, can you see the maximum number of different n

umber of randoms that

can be generated with this algorithm? Hint:

Think about how many possible remainders

you can get when you divide by something.

2

/* linear congruential random number generator */

#include <stdio.h>

#include <math.h>

int main(void)

{

double r,rM;

long unsigned int a,b,i,M,j,N;

printf("\nEnter seed");

printf("\n");

scanf("%i",&i);

printf("\nEnter number of iterations");

printf("\n");

scanf("%i",&N);

a=7*7*7*7*7;

M= 2*2*2*2*2*2*2*2;

M=M*2*2*2*2*2*2*2*2;

M=M*2*2*2*2*2*2*2*2;

M=M*2*2*2*2*2*2*2 -1;

rM=M;

for (j=0;j<N;j=j+1){

i=a*i;

i=i%M;

r=i;

r=r/rM;

printf("\n%12.8lf",r);

}

printf("\n");

return 0;

}

Comments:

[1]

Type in the code and generate 20 random numbers to see what they

look like.

3

/* This program generates N random numbers between 0 and 1 */

#include <stdio.h>

#include <time.h>

#include <stdlib.h>

int main(){

srand(time(NULL));

int i,N;

double R;

printf("Enter the number of random numbers you want ");

scanf("%d",&N);

for(i=0;i<N;i++)

{

R=(double)rand()/RAND_MAX;

printf(" %lf\n",R);

}

return 0;

}

[2]

This random numer generator uses the computer’s clock to prov

ide a seed. Hence the

need for the extra header file ‘time.h’.

[3]

Type in the code and generate 20 random numbers to see what they

look like.

[4]

Our random numbers are uniform on [0

,

1]. That is they obey 0

< r <

1 and the

r

are

equally likely to be anywhere in the interval. There are othe

r random number generators

one can construct, but this is the most common type.