Multiplicative Congruence method for generating Pseudo Random Numbers
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 seedm, a, and X0 should be chosen appropriately to get a period almost equal to m.
![\[X_{i + 1} = aX_{i} \hspace{0.2cm} mod \hspace{0.2cm} m\]](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-69aad1e541ff34e8f05b8fd4ee32dbac_l3.png "Rendered by QuickLaTeX.com")
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 a storage to keep the genrated random numbers (here, vector is considered) of size noOfRandomNums.
- Initialize the 0th index of the vector with the seed value.
- For 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; } |
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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 |
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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 |
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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 |
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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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