Linear Congruential Method

Generating Uniform Random Numbers

Faisal Z. Qureshi
faisal.qureshi@ontariotechu.ca

Copyright information

© Faisal Qureshi

License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

import numpy as np
import matplotlib.pyplot as plt

LCM Method

Generates the next number given the current number x and values for m, a, and c.

def lcm(x,a,c,m):
	return (a * x + c) % m

Generates n numbers starting from x

def gen(n,x0,a,c,m):
	lastx = x0
	l = [lastx]
	for i in range(1,n):
		newx = lcm(lastx,a,c,m)
		l.append(newx)
		lastx = newx
	x = np.array(l)
	return x

The shuffle technique

def shuffle(n, x0, a, c, m):
	x = np.zeros(n)
	k = 100000
	v = gen(k+1,x0,a,c,m)
	lastx = v[-1]
	v = v / float(m)
	for i in range(0,n-1):
		newx = lcm(lastx, a, c, m)
		j = int(newx * k / float(m))
		x[i] = v[j]
		lastx = newx
		newx = lcm(lastx, a, c, m)
		v[j] = newx / float(m)
		lastx = newx
	return x

Visualizations

def histogram(x, n, bins, fignum, a, c, m, s):
	hist, bins = np.histogram(x,bins = bins)
	width = 0.7* (bins[1]-bins[0])
	center = (bins[:-1]+bins[1:])/2
	plt.figure(fignum, figsize=(3,3))
	plt.title("Histogram (%s): a=%d, c=%d, m=%d" % (s, a, c, m) )
	plt.bar(center, hist, align = 'center', width = width)

Bad values for a, c, m

n = 1000000
x0 = 0
a = 200
c = 1000
m = 65000
x = gen(n,x0,a,c,m)  / float(m)
histogram(x, n, 50, 1, a, c, m, 'LCM')

n = 1000000
x0 = 0
a = 4023
c = 200000
m = 800000
x = gen(n,x0,a,c,m)  / float(m)
histogram(x, n, 50, 3, a, c, m, 'LCM')

Good values for a, c, and m

n = 1000000
x0 = 0
a = 4095
#a = 4096
c = 150889
m = 714025
x = gen(n,x0,a,c,m)  / float(m)
histogram(x, n, 50, 2, a, c, m, 'LCM')

Using shuffle method

n = 1000000
x0 = 0
a = 4096
c = 150889
m = 714025
x = shuffle(n,x0,a,c,m)
histogram(x, n, 50, 4, a, c, m, 'Shuffle')