Random Numbers

1. Random Numbers Dick Steflik

2. Pseudo Random Numbers • In most cases we do not want truly random numbers • most applications need the idea of repeatability to be able to debug • if we used truly random numbers how would we be able to debug a program, every time we would run the program it would be a different problem • What we really want is something that will appear to be random but will be able to reproduce a sequence

3. Generation Uniform Random Numbers • Consider a methos for generating a sequence of random fractions (Random real numbers ) Un • uniformly distributed between 0 and 1 • Since a computer can only represent a real number with finite accuracy we’ll actually generate integers Xn between 0 and some number m • The fraction Un = Xn / m • will always be between 0 and 1 • Usually m is the word size of the computer so that Xn can be regarded as the integer content of of a computer word with the radix point assumed at the far-right. • Un can be regarded as the contents of the same word with the radix point at the far-left.

4. The Linear Congruential Method By far the most popular random number generators in use today are special cases of the following scheme, introduced by D.H.Lehmer in 1949. Choose 4 “magic numbers: m, the modulus; m > 0 a, the multiplier; 0 <= a < m c, the increment; 0 <= c < m X0 the Starting value; 0 <= Xo < m The desired sequence is then: Xn+1 = (aXn + c) mod m, n >= 0 This is called a linear congruential sequence

5. Example For m = 10 ; X0 = a = c = 7 the sequence is 7, 6, 9, 0, 7, 6, 9, 0 notice: that the sequence has a period of 4; I.e. it repeats and will do so forever. This will happen for any linear congruential generator.

6. Sample #include <iostream.h>int seed; int random() { int num = (13*seed+11) % 11; seed = num; return num; } int main() { cout << "input a seed number = "; cin >> seed; cout << "\n"; for int j = 0 ; j < 20 ; j++) cout << random() << " "; }