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Commit ccb58a6

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Merge pull request avinashkranjan#1985 from Mukesh751/master
Nth-Fibonacci-Number-Generator Added
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'''
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This algorithm calculates the Nth Fibonacci Nummber in log(N) time complexity
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The key idea behind this algorimth is to use matrix exponentiation technique (matrix multiplication + binary exponentiation)
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This algorithm uses the fact that the Fibonacci Numbers can be converted to a 2D matrix of 2x2 size, [[1, 1][1, 0]], and the Fibonacci
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Number which is to be calculated with the help of matrix multiplication will be nothing but cell, (0,0) of (Matrix)^n ,i.e, our resultant
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matrix.
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'''
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def fib(n):
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F = [[1, 1],
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[1, 0]]
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if (n == 0):
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return 0
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power(F, n - 1)
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return F[0][0]
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'''
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Matrix multiplication is calculated using Strassen Algorithm which calculates the product of Two Matrices in o(N^3) time complexity.
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Since, the size of matrix here is fixed, i.e, 2x2, so, the Time Complexity will be O(1).
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'''
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def multiply(F, M):
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x = (F[0][0] * M[0][0] +
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F[0][1] * M[1][0])
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y = (F[0][0] * M[0][1] +
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F[0][1] * M[1][1])
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z = (F[1][0] * M[0][0] +
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F[1][1] * M[1][0])
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w = (F[1][0] * M[0][1] +
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F[1][1] * M[1][1])
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F[0][0] = x
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F[0][1] = y
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F[1][0] = z
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F[1][1] = w
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'''
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Power function uses Binary Exponentiation Technique to calculate the value of x^n in O(logN) time instead of O(N) time.
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'''
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def power(F, n):
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if(n == 0 or n == 1):
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return
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M = [[1, 1],
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[1, 0]]
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power(F, n // 2)
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multiply(F, F)
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if (n % 2 != 0):
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multiply(F, M)
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# Driver Code
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if __name__ == "__main__":
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n = 5
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print(fib(n))
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'''
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I hope you have a fun time reading this algorithm.
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Problems for practice:
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https://practice.geeksforgeeks.org/problems/generalised-fibonacci-numbers1820/1
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https://practice.geeksforgeeks.org/problems/nth-fibonacci-number1335/1
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https://practice.geeksforgeeks.org/problems/37f36b318a7d99c81f17f0a54fc46b5ce06bbe8c/0
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'''
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# Nth Fibonacci Number Generator
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This script calculates Nth Fibonacci Nunber.
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Version of Python used : Python 3.10.7
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## Setup instructions
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Click on the script and run the program.
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## Output
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![Screenshot (146)](https://github.com/Mukesh751/Amazing-Python-Scripts/assets/91366697/72b62e84-62de-4b6f-a37a-21844b685157)
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## Author(s)
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Mukesh751
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