Multiplicative Congruence method for generating Pseudo Random Numbers

Last Updated : 12 May, 2021

Multiplicative Congruential Method (Lehmer Method) is a type of linear congruential generator for generating pseudorandom numbers in a specific range. This method can be defined as:

$X_{i + 1} = aX_{i} \hspace{0.2cm} mod \hspace{0.2cm} m$

where,
X, the sequence of pseudo-random numbers
m ( > 0), the modulus
a (0, m), the multiplier
X₀[0, m), initial value of the sequence – termed as seed

m, a, and X₀should be chosen appropriately to get a period almost equal to m.

Approach:

Choose the seed value ( X₀), modulus parameter ( m ), and multiplier term ( a ).
Initialize the required amount of random numbers to generate (say, an integer variable noOfRandomNums).
Define storage to keep the generated random numbers (here, the vector is considered) of size noOfRandomNums.
Initialize the 0^thindex of the vector with the seed value.
For the rest of the indexes follow the Multiplicative Congruential Method to generate the random numbers.

randomNums[i] = (randomNums[i – 1] * a) % m

Finally, return the random numbers.
Below is the implementation of the above approach:

C++

// C++ implementation of the
// above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to generate random numbers
void multiplicativeCongruentialMethod(
    int Xo, int m, int a,
    vector<int>& randomNums,
    int noOfRandomNums)
{
 
    // Initialize the seed state
    randomNums[0] = Xo;
 
    // Traverse to generate required
    // numbers of random numbers
    for (int i = 1; i < noOfRandomNums; i++) {
 
        // Follow the multiplicative
        // congruential method
        randomNums[i]
            = (randomNums[i - 1] * a) % m;
    }
}
 
// Driver Code
int main()
{
    int Xo = 3; // seed value
    int m = 15; // modulus parameter
    int a = 7; // multiplier term
 
    // Number of Random numbers
    // to be generated
    int noOfRandomNums = 10;
 
    // To store random numbers
    vector<int> randomNums(noOfRandomNums);
 
    // Function Call
    multiplicativeCongruentialMethod(
        Xo, m, a, randomNums,
        noOfRandomNums);
 
    // Print the generated random numbers
    for (int i = 0; i < noOfRandomNums; i++) {
        cout << randomNums[i] << " ";
    }
    return 0;
}

Java

// Java implementation of the above approach
import java.util.*;
 
class GFG{
 
// Function to generate random numbers
static void multiplicativeCongruentialMethod(
    int Xo, int m, int a,
    int[] randomNums,
    int noOfRandomNums)
{
     
    // Initialize the seed state
    randomNums[0] = Xo;
     
    // Traverse to generate required
    // numbers of random numbers
    for(int i = 1; i < noOfRandomNums; i++)
    {
         
        // Follow the multiplicative
        // congruential method
        randomNums[i] = (randomNums[i - 1] * a) % m;
    }
}
 
// Driver code
public static void main(String[] args)
{
     
    // Seed value
    int Xo = 3;
     
    // Modulus parameter
    int m = 15;
     
    // Multiplier term
    int a = 7;
     
    // Number of Random numbers
    // to be generated
    int noOfRandomNums = 10;
     
    // To store random numbers
    int[] randomNums = new int[noOfRandomNums];
     
    // Function Call
    multiplicativeCongruentialMethod(Xo, m, a,
                                     randomNums,
                                     noOfRandomNums);
     
    // Print the generated random numbers
    for(int i = 0; i < noOfRandomNums; i++)
    {
        System.out.print(randomNums[i] + " ");
    }
}
}
 
// This code is contributed by offbeat

Python3

# Python3 implementation of the
# above approach
 
# Function to generate random numbers
def multiplicativeCongruentialMethod(Xo, m, a,
                                     randomNums,
                                     noOfRandomNums):
 
    # Initialize the seed state
    randomNums[0] = Xo
 
    # Traverse to generate required
    # numbers of random numbers
    for i in range(1, noOfRandomNums):
         
        # Follow the linear congruential method
        randomNums[i] = (randomNums[i - 1] * a) % m
 
# Driver Code
if __name__ == '__main__':
     
    # Seed value
    Xo = 3
     
    # Modulus parameter
    m = 15
     
    # Multiplier term
    a = 7
 
    # Number of Random numbers
    # to be generated
    noOfRandomNums = 10
 
    # To store random numbers
    randomNums = [0] * (noOfRandomNums)
 
    # Function Call
    multiplicativeCongruentialMethod(Xo, m, a,
                                     randomNums,
                                     noOfRandomNums)
 
    # Print the generated random numbers
    for i in randomNums:
        print(i, end = " ")
 
# This code is contributed by mohit kumar 29

C#

// C# implementation of the above approach
using System;
 
class GFG{
 
// Function to generate random numbers
static void multiplicativeCongruentialMethod(
    int Xo, int m, int a,
    int[] randomNums,
    int noOfRandomNums)
{
     
    // Initialize the seed state
    randomNums[0] = Xo;
     
    // Traverse to generate required
    // numbers of random numbers
    for(int i = 1; i < noOfRandomNums; i++)
    {
         
        // Follow the multiplicative
        // congruential method
        randomNums[i] = (randomNums[i - 1] * a) % m;
    }
}
 
// Driver code
public static void Main(String[] args)
{
     
    // Seed value
    int Xo = 3;
     
    // Modulus parameter
    int m = 15;
     
    // Multiplier term
    int a = 7;
     
    // Number of Random numbers
    // to be generated
    int noOfRandomNums = 10;
     
    // To store random numbers
    int[] randomNums = new int[noOfRandomNums];
     
    // Function call
    multiplicativeCongruentialMethod(Xo, m, a,
                                     randomNums,
                                     noOfRandomNums);
     
    // Print the generated random numbers
    for(int i = 0; i < noOfRandomNums; i++)
    {
        Console.Write(randomNums[i] + " ");
    }
}
}
 
// This code is contributed by sapnasingh4991

Javascript

<script>
 
// Javascript program to implement
// the above approach
 
// Function to generate random numbers
function multiplicativeCongruentialMethod(
    Xo, m, a,
    randomNums, noOfRandomNums)
{
       
    // Initialize the seed state
    randomNums[0] = Xo;
       
    // Traverse to generate required
    // numbers of random numbers
    for(let i = 1; i < noOfRandomNums; i++)
    {
           
        // Follow the multiplicative
        // congruential method
        randomNums[i] = (randomNums[i - 1] * a) % m;
    }
}
 
    // Driver Code
           
    // Seed value
    let Xo = 3;
       
    // Modulus parameter
    let m = 15;
       
    // Multiplier term
    let a = 7;
       
    // Number of Random numbers
    // to be generated
    let noOfRandomNums = 10;
       
    // To store random numbers
    let randomNums = new Array(noOfRandomNums).fill(0);
       
    // Function Call
    multiplicativeCongruentialMethod(Xo, m, a,
                                     randomNums,
                                     noOfRandomNums);
       
    // Prlet the generated random numbers
    for(let i = 0; i < noOfRandomNums; i++)
    {
        document.write(randomNums[i] + " ");
    }
 
</script>

Output:

3 6 12 9 3 6 12 9 3 6

The literal meaning of pseudo is false. These random numbers are called pseudo because some known arithmetic procedure is utilized to generate. Even the generated sequence forms a pattern hence the generated number seems to be random but may not be truly random.

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