Space efficient iterative method to Fibonacci number
Given a number n, find n-th Fibonacci Number. Note that F0 = 0, F1 = 1, F2 = 2, …..
Examples :
Input : n = 5 Output : 5 Input : n = 10 Output : 89
We have discussed below recursive solution in method 4 of Program for Fibonacci numbers.
F[2][2] = |1, 1| |1, 0| M[2][2] = |1, 1| |1, 0| F[n][n] = fib(n) | fib(n-1) ------------------ fib(n-1)| fib(n-2)
In this post an iterative method is discussed that avoids extra recursion call stack space. We have also used bitwise operators to further optimize. In the previous method, we divide the number with 2 so that at the end we get 1 and then we start the multiplication process
In this method we get the second MSB then start to multiply with FxF matrix then if bit is set then multiply again FxM matrix and so on. then we get the final result.
Approach : 1. First get the MSB of a number. 2. while (MSB > 0) multiply(F, F); if (n & MSB) multiply(F, M); and then shift MSB till MSB != 0
C++
// C++ code to find nth fibonacci #include <bits/stdc++.h> using namespace std; // get second MSB int getMSB( int n) { // consecutively set all the bits n |= n >> 1; n |= n >> 2; n |= n >> 4; n |= n >> 8; n |= n >> 16; // returns the second MSB return ((n + 1) >> 2); } // Multiply function void multiply( int F[2][2], int M[2][2]) { int x = F[0][0] * M[0][0] + F[0][1] * M[1][0]; int y = F[0][0] * M[0][1] + F[0][1] * M[1][1]; int z = F[1][0] * M[0][0] + F[1][1] * M[1][0]; int w = F[1][0] * M[0][1] + F[1][1] * M[1][1]; F[0][0] = x; F[0][1] = y; F[1][0] = z; F[1][1] = w; } // Function to calculate F[][] // raise to the power n void power( int F[2][2], int n) { // Base case if (n == 0 || n == 1) return ; // take 2D array to store number's int M[2][2] = { 1, 1, 1, 0 }; // run loop till MSB > 0 for ( int m = getMSB(n); m; m = m >> 1) { multiply(F, F); if (n & m) { multiply(F, M); } } } // To return fibonacci number int fib( int n) { int F[2][2] = { { 1, 1 }, { 1, 0 } }; if (n == 0) return 0; power(F, n - 1); return F[0][0]; } // Driver Code int main() { // Given n int n = 6; cout << fib(n) << " " ; return 0; } |
Java
// Java code to // find nth fibonacci class GFG { // get second MSB static int getMSB( int n) { // consecutively set // all the bits n |= n >> 1 ; n |= n >> 2 ; n |= n >> 4 ; n |= n >> 8 ; n |= n >> 16 ; // returns the // second MSB return ((n + 1 ) >> 2 ); } // Multiply function static void multiply( int F[][], int M[][]) { int x = F[ 0 ][ 0 ] * M[ 0 ][ 0 ] + F[ 0 ][ 1 ] * M[ 1 ][ 0 ]; int y = F[ 0 ][ 0 ] * M[ 0 ][ 1 ] + F[ 0 ][ 1 ] * M[ 1 ][ 1 ]; int z = F[ 1 ][ 0 ] * M[ 0 ][ 0 ] + F[ 1 ][ 1 ] * M[ 1 ][ 0 ]; int w = F[ 1 ][ 0 ] * M[ 0 ][ 1 ] + F[ 1 ][ 1 ] * M[ 1 ][ 1 ]; F[ 0 ][ 0 ] = x; F[ 0 ][ 1 ] = y; F[ 1 ][ 0 ] = z; F[ 1 ][ 1 ] = w; } // Function to calculate F[][] // raise to the power n static void power( int F[][], int n) { // Base case if (n == 0 || n == 1 ) return ; // take 2D array to // store number's int [][] M ={{ 1 , 1 }, { 1 , 0 }}; // run loop till MSB > 0 for ( int m = getMSB(n); m > 0 ; m = m >> 1 ) { multiply(F, F); if ((n & m) > 0 ) { multiply(F, M); } } } // To return // fibonacci number static int fib( int n) { int [][] F = {{ 1 , 1 }, { 1 , 0 }}; if (n == 0 ) return 0 ; power(F, n - 1 ); return F[ 0 ][ 0 ]; } // Driver Code public static void main(String[] args) { // Given n int n = 6 ; System.out.println(fib(n)); } } // This code is contributed // by mits |
Python3
# Python3 code to find nth fibonacci # get second MSB def getMSB(n): # consecutively set all the bits n | = n >> 1 n | = n >> 2 n | = n >> 4 n | = n >> 8 n | = n >> 16 # returns the second MSB return ((n + 1 ) >> 2 ) # Multiply function def multiply(F, M): x = F[ 0 ][ 0 ] * M[ 0 ][ 0 ] + F[ 0 ][ 1 ] * M[ 1 ][ 0 ] y = F[ 0 ][ 0 ] * M[ 0 ][ 1 ] + F[ 0 ][ 1 ] * M[ 1 ][ 1 ] z = F[ 1 ][ 0 ] * M[ 0 ][ 0 ] + F[ 1 ][ 1 ] * M[ 1 ][ 0 ] w = F[ 1 ][ 0 ] * M[ 0 ][ 1 ] + F[ 1 ][ 1 ] * M[ 1 ][ 1 ] F[ 0 ][ 0 ] = x F[ 0 ][ 1 ] = y F[ 1 ][ 0 ] = z F[ 1 ][ 1 ] = w # Function to calculate F[][] # raise to the power n def power(F, n): # Base case if (n = = 0 or n = = 1 ): return # take 2D array to store number's M = [[ 1 , 1 ], [ 1 , 0 ]] # run loop till MSB > 0 m = getMSB(n) while m: multiply(F, F) if (n & m): multiply(F, M) m = m >> 1 # To return fibonacci number def fib(n): F = [[ 1 , 1 ], [ 1 , 0 ]] if (n = = 0 ): return 0 power(F, n - 1 ) return F[ 0 ][ 0 ] # Driver Code # Given n n = 6 print (fib(n)) # This code is contributed by Mohit Kumar |
C#
// C# code to find nth fibonacci using System; class GFG { // get second MSB static int getMSB( int n) { // consecutively set // all the bits n |= n >> 1; n |= n >> 2; n |= n >> 4; n |= n >> 8; n |= n >> 16; // returns the // second MSB return ((n + 1) >> 2); } // Multiply function static void multiply( int [,]F, int [,]M) { int x = F[0,0] * M[0,0] + F[0,1] * M[1,0]; int y = F[0,0] * M[0,1] + F[0,1] * M[1,1]; int z = F[1,0] * M[0,0] + F[1,1] * M[1,0]; int w = F[1,0] * M[0,1] + F[1,1] * M[1,1]; F[0,0] = x; F[0,1] = y; F[1,0] = z; F[1,1] = w; } // Function to calculate F[][] // raise to the power n static void power( int [,]F, int n) { // Base case if (n == 0 || n == 1) return ; // take 2D array to // store number's int [,] M ={{1, 1}, {1, 0}}; // run loop till MSB > 0 for ( int m = getMSB(n); m > 0; m = m >> 1) { multiply(F, F); if ((n & m) > 0) { multiply(F, M); } } } // To return // fibonacci number static int fib( int n) { int [,] F = {{1, 1}, {1, 0}}; if (n == 0) return 0; power(F, n - 1); return F[0,0]; } // Driver Code static public void Main () { // Given n int n = 6; Console.WriteLine(fib(n)); } } // This code is contributed ajit |
PHP
<?php // PHP code to find nth fibonacci // get second MSB function getMSB( $n ) { // consecutively set all the bits $n |= $n >> 1; $n |= $n >> 2; $n |= $n >> 4; $n |= $n >> 8; $n |= $n >> 16; // returns the second MSB return (( $n + 1) >> 2); } // Multiply function function multiply(& $F , & $M ) { $x = $F [0][0] * $M [0][0] + $F [0][1] * $M [1][0]; $y = $F [0][0] * $M [0][1] + $F [0][1] * $M [1][1]; $z = $F [1][0] * $M [0][0] + $F [1][1] * $M [1][0]; $w = $F [1][0] * $M [0][1] + $F [1][1] * $M [1][1]; $F [0][0] = $x ; $F [0][1] = $y ; $F [1][0] = $z ; $F [1][1] = $w ; } // Function to calculate F[][] // raise to the power n function power(& $F , $n ) { // Base case if ( $n == 0 || $n == 1) return ; // take 2D array to store number's $M = array ( array (1, 1), array (1, 0)); // run loop till MSB > 0 for ( $m = getMSB( $n ); $m ; $m = $m >> 1) { multiply( $F , $F ); if ( $n & $m ) { multiply( $F , $M ); } } } // To return fibonacci number function fib( $n ) { $F = array ( array ( 1, 1 ), array ( 1, 0 )); if ( $n == 0) return 0; power( $F , $n - 1); return $F [0][0]; } // Driver Code // Given n $n = 6; echo fib( $n ) . " " ; // This code is contributed by ita_c ?> |
Javascript
<script> // Javascript code to find nth fibonacci // Get second MSB function getMSB(n) { // Consecutively set // all the bits n |= n >> 1; n |= n >> 2; n |= n >> 4; n |= n >> 8; n |= n >> 16; // Returns the // second MSB return ((n + 1) >> 2); } // Multiply function function multiply(F, M) { let x = F[0][0] * M[0][0] + F[0][1] * M[1][0]; let y = F[0][0] * M[0][1] + F[0][1] * M[1][1]; let z = F[1][0] * M[0][0] + F[1][1] * M[1][0]; let w = F[1][0] * M[0][1] + F[1][1] * M[1][1]; F[0][0] = x; F[0][1] = y; F[1][0] = z; F[1][1] = w; } // Function to calculate F[][] // raise to the power n function power(F, n) { // Base case if (n == 0 || n == 1) return ; // Take 2D array to // store number's let M = [ [ 1, 1 ], [ 1, 0 ] ]; // Run loop till MSB > 0 for (let m = getMSB(n); m > 0; m = m >> 1) { multiply(F, F); if ((n & m) > 0) { multiply(F, M); } } } // To return // fibonacci number function fib(n) { let F = [ [ 1, 1 ], [ 1, 0 ] ]; if (n == 0) return 0; power(F, n - 1); return F[0][0]; } // Driver code // Given n let n = 6; document.write(fib(n)); // This code is contributed by decode2207 </script> |
Output:
8
Time Complexity :- O(logn) and space complexity :- O(1).