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:
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 X0 should 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 0th index 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++ 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 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 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# 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 |
<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);
// Print 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
Time Complexity: O(N), where N is the total number of random numbers we need to generate.
Auxiliary Space: O(1)
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.