Program for Fibonacci numbers

28 Jun, 2022
The Fibonacci numbers are the numbers in the following integer sequence.0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ........
In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation 
Fn = Fn-1 + Fn-2
with seed values 
F0 = 0 and F1 = 1.
Given a number n, print n-th Fibonacci Number. 
Examples: 
Input  : n = 2
Output : 1

Input  : n = 9
Output : 34
Write a function int fib(int n) that returns Fn. For example, if n = 0, then fib() should return 0. If n = 1, then it should return 1. For n > 1, it should return Fn-1 + Fn-2
For n = 9
Output:34
Following are different methods to get the nth Fibonacci number. 
Method 1 (Use recursion) A simple method that is a direct recursive implementation mathematical recurrence relation is given above.
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//Fibonacci Series using Recursion#include<bits/stdc++.h>using namespace std; int fib(int n){    if (n <= 1)        return n;    return fib(n-1) + fib(n-2);} int main (){    int n = 9;    cout << fib(n);    getchar();    return 0;} // This code is contributed// by Akanksha Rai
//Fibonacci Series using Recursion#include<stdio.h>int fib(int n){   if (n <= 1)      return n;   return fib(n-1) + fib(n-2);} int main (){  int n = 9;  printf("%d", fib(n));  getchar();  return 0;}
//Fibonacci Series using Recursionclass fibonacci{    static int fib(int n)    {    if (n <= 1)       return n;    return fib(n-1) + fib(n-2);    }          public static void main (String args[])    {    int n = 9;    System.out.println(fib(n));    }}/* This code is contributed by Rajat Mishra */
#This code gives nth number in fibonacci series - {0,1,1,2,3,5,......}.def fibonacci(n, second_last, last):    if n-1 == 0:        return second_last    else:        new_last = second_last + last        second_last = last        return fibonacci(n-1, second_last, new_last)  if __name__ == "__main__":    print(fibonacci(10, 0, 1))  # This code is contributed by Manan Tyagi.
// C# program for Fibonacci Series// using Recursionusing System; public class GFG{    public static int Fib(int n)    {        if (n <= 1)        {            return n;        }        else        {            return Fib(n - 1) + Fib(n - 2);        }    }             // driver code    public static void Main(string[] args)    {        int n = 9;        Console.Write(Fib(n));    }} // This code is contributed by Sam007
<?php// Fibonacci Series// using Recursion // function returns// the Fibonacci numberfunction fib($n){    if ($n <= 1)        return $n;    return fib($n - 1) +           fib($n - 2);} // Driver Code$n = 9;echo fib($n); // This code is contributed by aj_36?>
<script>//Fibonacci Series using Recursion    let n = 9;         // function returns the Fibonacci number    function fib(n) {    if (n <= 1)        return n;    return fib(n-1) + fib(n-2);}      //function call    document.write(fib(n));//This code is contributed by Surbhi Tyagi</script>
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Time Complexity: T(n) = T(n) which is linear. 
If the original recursion tree were to be implemented then this would have been the tree but now for n times the recursion function is called
Original tree for recursion
                          fib(5)   
                     /                \
               fib(4)                fib(3)   
             /        \              /       \ 
         fib(3)      fib(2)         fib(2)   fib(1)
        /    \       /    \        /      \
  fib(2)   fib(1)  fib(1) fib(0) fib(1) fib(0)
  /     \
fib(1) fib(0)
Optimized tree for recursion for code above
    fib(5) 
    fib(4)
    fib(3)
    fib(2)
    fib(1)
Extra Space: O(n) if we consider the function call stack size, otherwise O(1).
Method 2 (Use Dynamic Programming) We can avoid the repeated work done in method 1 by storing the Fibonacci numbers calculated so far. 
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// C++ program for Fibonacci Series // using Dynamic Programming#include<bits/stdc++.h>using namespace std; class GFG{     public:int fib(int n){         // Declare an array to store    // Fibonacci numbers.    // 1 extra to handle    // case, n = 0    int f[n + 2];    int i;     // 0th and 1st number of the    // series are 0 and 1    f[0] = 0;    f[1] = 1;     for(i = 2; i <= n; i++)    {                //Add the previous 2 numbers       // in the series and store it       f[i] = f[i - 1] + f[i - 2];    }    return f[n];    }}; // Driver codeint main (){    GFG g;    int n = 9;         cout << g.fib(n);    return 0;} // This code is contributed by SoumikMondal
//Fibonacci Series using Dynamic Programming#include<stdio.h> int fib(int n){  /* Declare an array to store Fibonacci numbers. */  int f[n+2];   // 1 extra to handle case, n = 0  int i;   /* 0th and 1st number of the series are 0 and 1*/  f[0] = 0;  f[1] = 1;   for (i = 2; i <= n; i++)  {      /* Add the previous 2 numbers in the series         and store it */      f[i] = f[i-1] + f[i-2];  }   return f[n];} int main (){  int n = 9;  printf("%d", fib(n));  getchar();  return 0;}
// Fibonacci Series using Dynamic Programmingclass fibonacci{   static int fib(int n)    {    /* Declare an array to store Fibonacci numbers. */    int f[] = new int[n+2]; // 1 extra to handle case, n = 0    int i;          /* 0th and 1st number of the series are 0 and 1*/    f[0] = 0;    f[1] = 1;         for (i = 2; i <= n; i++)    {       /* Add the previous 2 numbers in the series         and store it */        f[i] = f[i-1] + f[i-2];    }          return f[n];    }          public static void main (String args[])    {        int n = 9;        System.out.println(fib(n));    }}/* This code is contributed by Rajat Mishra */
# Fibonacci Series using Dynamic Programmingdef fibonacci(n):         # Taking 1st two fibonacci numbers as 0 and 1    f = [0, 1]              for i in range(2, n+1):        f.append(f[i-1] + f[i-2])    return f[n]     print(fibonacci(9))
// C# program for Fibonacci Series// using Dynamic Programmingusing System;class fibonacci {     static int fib(int n)    {                 // Declare an array to        // store Fibonacci numbers.        // 1 extra to handle        // case, n = 0        int []f = new int[n + 2];        int i;                 /* 0th and 1st number of the           series are 0 and 1 */        f[0] = 0;        f[1] = 1;                 for (i = 2; i <= n; i++)        {            /* Add the previous 2 numbers               in the series and store it */            f[i] = f[i - 1] + f[i - 2];        }                 return f[n];    }         // Driver Code    public static void Main ()    {        int n = 9;        Console.WriteLine(fib(n));    }} // This code is contributed by anuj_67.
<?php//Fibonacci Series using Dynamic// Programming function fib( $n){         /* Declare an array to store    Fibonacci numbers. */         // 1 extra to handle case,    // n = 0    $f = array();    $i;         /* 0th and 1st number of the    series are 0 and 1*/    $f[0] = 0;    $f[1] = 1;         for ($i = 2; $i <= $n; $i++)    {                 /* Add the previous 2        numbers in the series        and store it */        $f[$i] = $f[$i-1] + $f[$i-2];    }         return $f[$n];} $n = 9;echo fib($n); // This code is contributed by// anuj_67.?>
<script> // Fibonacci Series using Dynamic Programming     function  fib(n)    {        /* Declare an array to store Fibonacci numbers. */        let f = new Array(n+2); // 1 extra to handle case, n = 0        let i;        /* 0th and 1st number of the series are 0 and 1*/        f[0] = 0;        f[1] = 1;        for (i = 2; i <= n; i++)        {            /* Add the previous 2 numbers in the series            and store it */            f[i] = f[i-1] + f[i-2];        }        return f[n];    }    let n=9;    document.write(fib(n));         // This code is contributed by avanitrachhadiya2155     </script>
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Method 3 (Space Optimized Method 2) We can optimize the space used in method 2 by storing the previous two numbers only because that is all we need to get the next Fibonacci number in series. 
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// Fibonacci Series using Space Optimized Method#include<bits/stdc++.h>using namespace std; int fib(int n){    int a = 0, b = 1, c, i;    if( n == 0)        return a;    for(i = 2; i <= n; i++)    {       c = a + b;       a = b;       b = c;    }    return b;} // Driver codeint main(){    int n = 9;         cout << fib(n);    return 0;} // This code is contributed by Code_Mech
// Fibonacci Series using Space Optimized Method#include<stdio.h>int fib(int n){  int a = 0, b = 1, c, i;  if( n == 0)    return a;  for (i = 2; i <= n; i++)  {     c = a + b;     a = b;     b = c;  }  return b;} int main (){  int n = 9;  printf("%d", fib(n));  getchar();  return 0;}
// Java program for Fibonacci Series using Space// Optimized Methodclass fibonacci{    static int fib(int n)    {        int a = 0, b = 1, c;        if (n == 0)            return a;        for (int i = 2; i <= n; i++)        {            c = a + b;            a = b;            b = c;        }        return b;    }     public static void main (String args[])    {        int n = 9;        System.out.println(fib(n));    }} // This code is contributed by Mihir Joshi
# Function for nth fibonacci number - Space Optimisation# Taking 1st two fibonacci numbers as 0 and 1 def fibonacci(n):    a = 0    b = 1    if n < 0:        print("Incorrect input")    elif n == 0:        return a    elif n == 1:        return b    else:        for i in range(2,n+1):            c = a + b            a = b            b = c        return b # Driver Program print(fibonacci(9)) #This code is contributed by Saket Modi
// C# program for Fibonacci Series// using Space Optimized Methodusing System; namespace Fib{    public class GFG    {        static int Fib(int n)        {            int a = 0, b = 1, c = 0;                         // To return the first Fibonacci number            if (n == 0) return a;                 for (int i = 2; i <= n; i++)            {                c = a + b;                a = b;                b = c;            }                 return b;        }             // Driver function    public static void Main(string[] args)        {                         int n = 9;            Console.Write("{0} ", Fib(n));        }    }} // This code is contributed by Sam007.
<?php// PHP program for Fibonacci Series// using Space Optimized Method function fib( $n){    $a = 0;    $b = 1;    $c;    $i;    if( $n == 0)        return $a;    for($i = 2; $i <= $n; $i++)    {        $c = $a + $b;        $a = $b;        $b = $c;    }    return $b;} // Driver Code$n = 9;echo fib($n); // This code is contributed by anuj_67.?>
<script> // Javascript program for Fibonacci Series using Space Optimized Method function fib(n){    let a = 0, b = 1, c, i;    if( n == 0)        return a;    for(i = 2; i <= n; i++)    {    c = a + b;    a = b;    b = c;    }    return b;} // Driver code     let n = 9;         document.write(fib(n)); // This code is contributed by Mayank Tyagi </script>
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Time Complexity:O(n) Extra Space: O(1)
Method 4 (Using power of the matrix {{1, 1}, {1, 0}}) This another O(n) which relies on the fact that if we n times multiply the matrix M = {{1,1},{1,0}} to itself (in other words calculate power(M, n)), then we get the (n+1)th Fibonacci number as the element at row and column (0, 0) in the resultant matrix.The matrix representation gives the following closed expression for the Fibonacci numbers:  
 
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#include<bits/stdc++.h>using namespace std; // Helper function that multiplies 2// matrices F and M of size 2*2, and// puts the multiplication result// back to F[][]void multiply(int F[2][2], int M[2][2]); // Helper function that calculates F[][]// raise to the power n and puts the// result in F[][]// Note that this function is designed// only for fib() and won't work as// general power functionvoid power(int F[2][2], int n); 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];} 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;} void power(int F[2][2], int n){    int i;    int M[2][2] = { { 1, 1 }, { 1, 0 } };         // n - 1 times multiply the    // matrix to {{1,0},{0,1}}    for(i = 2; i <= n; i++)        multiply(F, M);} // Driver codeint main(){    int n = 9;         cout << " " <<  fib(n);         return 0;} // This code is contributed by shivanisinghss2110
#include <stdio.h> /* Helper function that multiplies 2 matrices F and M of size 2*2, and  puts the multiplication result back to F[][] */void multiply(int F[2][2], int M[2][2]); /* Helper function that calculates F[][] raise to the power n and puts the  result in F[][]  Note that this function is designed only for fib() and won't work as general  power function */void power(int F[2][2], int n); 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];} 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;} void power(int F[2][2], int n){  int i;  int M[2][2] = {{1,1},{1,0}};   // n - 1 times multiply the matrix to {{1,0},{0,1}}  for (i = 2; i <= n; i++)      multiply(F, M);} /* Driver program to test above function */int main(){  int n = 9;  printf("%d", fib(n));  getchar();  return 0;}
class fibonacci{         static int fib(int n)    {    int F[][] = new int[][]{{1,1},{1,0}};    if (n == 0)        return 0;    power(F, n-1);            return F[0][0];    }           /* Helper function that multiplies 2 matrices F and M of size 2*2, and     puts the multiplication result back to F[][] */    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;    }     /* Helper function that calculates F[][] raise to the power n and puts the    result in F[][]    Note that this function is designed only for fib() and won't work as general    power function */    static void power(int F[][], int n)    {    int i;    int M[][] = new int[][]{{1,1},{1,0}};         // n - 1 times multiply the matrix to {{1,0},{0,1}}    for (i = 2; i <= n; i++)        multiply(F, M);    }          /* Driver program to test above function */    public static void main (String args[])    {    int n = 9;    System.out.println(fib(n));    }}/* This code is contributed by Rajat Mishra */
# Helper function that multiplies# 2 matrices F and M of size 2*2,# and puts the multiplication# result back to F[][] # Helper function that calculates# F[][] raise to the power n and# puts the result in F[][]# Note that this function is# designed only for fib() and# won't work as general# power functiondef fib(n):    F = [[1, 1],         [1, 0]]    if (n == 0):        return 0    power(F, n - 1)         return F[0][0] 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 def power(F, n):     M = [[1, 1],         [1, 0]]     # n - 1 times multiply the    # matrix to {{1,0},{0,1}}    for i in range(2, n + 1):        multiply(F, M) # Driver Codeif __name__ == "__main__":    n = 9    print(fib(n)) # This code is contributed# by ChitraNayal
using System; class GFG {         static int fib(int n)    {        int [,]F = new int[,] {{1, 1},                               {1, 0} };        if (n == 0)            return 0;        power(F, n-1);                 return F[0,0];    }         /* Helper function that multiplies 2    matrices F and M of size 2*2, and puts    the multiplication result back to F[][] */    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;    }     /* Helper function that calculates F[][]    raise to the power n and puts the result    in F[][] Note that this function is designed    only for fib() and won't work as general    power function */    static void power(int [,]F, int n)    {        int i;        int [,]M = new int[,]{{1, 1},                              {1, 0} };                 // n - 1 times multiply the matrix to        // {{1,0},{0,1}}        for (i = 2; i <= n; i++)            multiply(F, M);    }         /* Driver program to test above function */    public static void Main ()    {        int n = 9;        Console.WriteLine(fib(n));    }} // This code is contributed by anuj_67.
<?php function fib($n){    $F = array(array(1, 1),               array(1, 0));    if ($n == 0)        return 0;    power($F, $n - 1);         return $F[0][0];} function multiply(&$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] = $w;} function power(&$F, $n){    $M = array(array(1, 1),               array(1, 0));         // n - 1 times multiply the    // matrix to {{1,0},{0,1}}    for ($i = 2; $i <= $n; $i++)        multiply($F, $M);} // Driver Code$n = 9;echo fib($n); // This code is contributed// by ChitraNayal?>
<script> // Note that this function is designed// only for fib() and won't work as// general power function   function fib( n){     var F = [ [ 1, 1 ], [ 1, 0 ] ];          if (n == 0)        return 0;              power(F, n - 1);          return F[0][0];}// Helper function that multiplies 2// matrices F and M of size 2*2, and// puts the multiplication result// back to F[][]  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;}// Helper function that calculates F[][]// raise to the power n and puts the// result in F[][]  function power( F, n){    var i;     var M = [[ 1, 1 ], [ 1, 0 ]];          // n - 1 times multiply the    // matrix to {{1,0},{0,1}}    for(i = 2; i <= n; i++)        multiply(F, M);}  // Driver code     var   n = 9;           document.write (" " +  fib(n)); //This code is contributed by sweetyty </script>
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Time Complexity: O(n) Extra Space: O(1)  
Method 5 (Optimized Method 4) The method 4 can be optimized to work in O(Logn) time complexity. We can do recursive multiplication to get power(M, n) in the previous method (Similar to the optimization done in this post)
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// Fibonacci Series using Optimized Method#include <bits/stdc++.h>using namespace std; void multiply(int F[2][2], int M[2][2]);void power(int F[2][2], int n); // Function that returns nth Fibonacci numberint fib(int n){    int F[2][2] = {{1, 1}, {1, 0}};    if (n == 0)        return 0;    power(F, n - 1);     return F[0][0];} // Optimized version of power() in method 4void power(int F[2][2], int n){    if(n == 0 || n == 1)       return;    int M[2][2] = {{1, 1}, {1, 0}};         power(F, n / 2);    multiply(F, F);         if (n % 2 != 0)        multiply(F, M);} 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;} // Driver codeint main(){    int n = 9;         cout << fib(9);    getchar();         return 0;} // This code is contributed by Nidhi_biet
#include <stdio.h> void multiply(int F[2][2], int M[2][2]); void power(int F[2][2], int n); /* function that returns nth 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];} /* Optimized version of power() in method 4 */void power(int F[2][2], int n){  if( n == 0 || n == 1)      return;  int M[2][2] = {{1,1},{1,0}};   power(F, n/2);  multiply(F, F);   if (n%2 != 0)     multiply(F, M);} 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;} /* Driver program to test above function */int main(){  int n = 9;  printf("%d", fib(9));  getchar();  return 0;}
//Fibonacci Series using Optimized Methodclass fibonacci{    /* function that returns nth Fibonacci number */    static int fib(int n)    {    int F[][] = new int[][]{{1,1},{1,0}};    if (n == 0)        return 0;    power(F, n-1);          return F[0][0];    }          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;    }          /* Optimized version of power() in method 4 */    static void power(int F[][], int n)    {    if( n == 0 || n == 1)      return;    int M[][] = new int[][]{{1,1},{1,0}};          power(F, n/2);    multiply(F, F);          if (n%2 != 0)       multiply(F, M);    }         /* Driver program to test above function */    public static void main (String args[])    {         int n = 9;     System.out.println(fib(n));    }}/* This code is contributed by Rajat Mishra */
# Fibonacci Series using# Optimized Method # function that returns nth# Fibonacci numberdef fib(n):         F = [[1, 1],         [1, 0]]    if (n == 0):        return 0    power(F, n - 1)             return F[0][0]     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         # Optimized version of# power() in method 4def power(F, n):     if( n == 0 or n == 1):        return;    M = [[1, 1],         [1, 0]];             power(F, n // 2)    multiply(F, F)             if (n % 2 != 0):        multiply(F, M)     # Driver Codeif __name__ == "__main__":    n = 9    print(fib(n)) # This code is contributed# by ChitraNayal
// Fibonacci Series using// Optimized Methodusing System; class GFG{/* function that returnsnth Fibonacci number */static int fib(int n){int[,] F = new int[,]{{1, 1},                      {1, 0}};if (n == 0)    return 0;power(F, n - 1); return F[0, 0];} 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;} /* Optimized version ofpower() in method 4 */static void power(int[,] F, int n){if( n == 0 || n == 1)return;int[,] M = new int[,]{{1, 1},                      {1, 0}}; power(F, n / 2);multiply(F, F); if (n % 2 != 0)multiply(F, M);} // Driver Codepublic static void Main (){    int n = 9;    Console.Write(fib(n));}} // This code is contributed// by ChitraNayal
<script> // Fibonacci Series using Optimized Method // Function that returns nth Fibonacci numberfunction fib(n){    var F = [ [ 1, 1 ], [ 1, 0 ] ];    if (n == 0)        return 0;             power(F, n - 1);     return F[0][0];} function multiply(F, M){    var x = F[0][0] * M[0][0] + F[0][1] * M[1][0];    var y = F[0][0] * M[0][1] + F[0][1] * M[1][1];    var z = F[1][0] * M[0][0] + F[1][1] * M[1][0];    var 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;} // Optimized version of power() in method 4 */function power(F, n){    if (n == 0 || n == 1)        return;             var M = [ [ 1, 1 ], [ 1, 0 ] ];     power(F, n / 2);    multiply(F, F);     if (n % 2 != 0)        multiply(F, M);} // Driver codevar n = 9; document.write(fib(n)); // This code is contributed by gauravrajput1 </script>
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Time Complexity: O(Logn) Extra Space: O(Logn) if we consider the function call stack size, otherwise O(1).
Method 6 (O(Log n) Time)Below is one more interesting recurrence formula that can be used to find n’th Fibonacci Number in O(Log n) time.  
If n is even then k = n/2:
F(n) = [2*F(k-1) + F(k)]*F(k)

If n is odd then k = (n + 1)/2
F(n) = F(k)*F(k) + F(k-1)*F(k-1)
How does this formula work? The formula can be derived from above matrix equation.  
Taking determinant on both sides, we get 

(-1)n = Fn+1Fn-1 - Fn2 
 
Moreover, since AnAm = An+m for any square matrix A, 
the following identities can be derived (they are obtained 
from two different coefficients of the matrix product)

FmFn + Fm-1Fn-1 = Fm+n-1         ---------------------------(1)

By putting n = n+1 in equation(1),
FmFn+1 + Fm-1Fn = Fm+n             --------------------------(2)

Putting m = n in equation(1).
F2n-1 = Fn2 + Fn-12
Putting m = n in equation(2)

F2n = (Fn-1 + Fn+1)Fn = (2Fn-1 + Fn)Fn (Source: Wiki)   --------
( By putting Fn+1 = Fn + Fn-1 )
To get the formula to be proved, we simply need to do the following 
If n is even, we can put k = n/2 
If n is odd, we can put k = (n+1)/2
Below is the implementation of above idea.  
C++
Java
Python3
C#
PHP
Javascript
// C++ Program to find n'th fibonacci Number in// with O(Log n) arithmetic operations#include <bits/stdc++.h>using namespace std; const int MAX = 1000; // Create an array for memoizationint f[MAX] = {0}; // Returns n'th fibonacci number using table f[]int fib(int n){    // Base cases    if (n == 0)        return 0;    if (n == 1 || n == 2)        return (f[n] = 1);     // If fib(n) is already computed    if (f[n])        return f[n];     int k = (n & 1)? (n+1)/2 : n/2;     // Applying above formula [Note value n&1 is 1    // if n is odd, else 0.    f[n] = (n & 1)? (fib(k)*fib(k) + fib(k-1)*fib(k-1))           : (2*fib(k-1) + fib(k))*fib(k);     return f[n];} /* Driver program to test above function */int main(){    int n = 9;    printf("%d ", fib(n));    return 0;}
// Java Program to find n'th fibonacci// Number with O(Log n) arithmetic operationsimport java.util.*; class GFG {         static int MAX = 1000;    static int f[];         // Returns n'th fibonacci number using    // table f[]    public static int fib(int n)    {        // Base cases        if (n == 0)            return 0;                     if (n == 1 || n == 2)            return (f[n] = 1);              // If fib(n) is already computed        if (f[n] != 0)            return f[n];              int k = (n & 1) == 1? (n + 1) / 2                            : n / 2;              // Applying above formula [Note value        // n&1 is 1 if n is odd, else 0.        f[n] = (n & 1) == 1? (fib(k) * fib(k) +                        fib(k - 1) * fib(k - 1))                       : (2 * fib(k - 1) + fib(k))                       * fib(k);              return f[n];    }         /* Driver program to test above function */    public static void main(String[] args)    {        int n = 9;        f= new int[MAX];        System.out.println(fib(n));    }}     // This code is contributed by Arnav Kr. Mandal.
# Python3 Program to find n'th fibonacci Number in# with O(Log n) arithmetic operationsMAX = 1000 # Create an array for memoizationf = [0] * MAX # Returns n'th fibonacci number using table f[]def fib(n) :    # Base cases    if (n == 0) :        return 0    if (n == 1 or n == 2) :        f[n] = 1        return (f[n])     # If fib(n) is already computed    if (f[n]) :        return f[n]     if( n & 1) :        k = (n + 1) // 2    else :        k = n // 2     # Applying above formula [Note value n&1 is 1    # if n is odd, else 0.    if((n & 1) ) :        f[n] = (fib(k) * fib(k) + fib(k-1) * fib(k-1))    else :        f[n] = (2*fib(k-1) + fib(k))*fib(k)     return f[n]  # Driver coden = 9print(fib(n))  # This code is contributed by Nikita Tiwari.
// C# Program to find n'th// fibonacci Number with// O(Log n) arithmetic operationsusing System; class GFG{ static int MAX = 1000;static int[] f; // Returns n'th fibonacci// number using table f[]public static int fib(int n){    // Base cases    if (n == 0)        return 0;             if (n == 1 || n == 2)        return (f[n] = 1);     // If fib(n) is already    // computed    if (f[n] != 0)        return f[n];     int k = (n & 1) == 1 ? (n + 1) / 2                         : n / 2;     // Applying above formula    // [Note value n&1 is 1 if    // n is odd, else 0.    f[n] = (n & 1) == 1 ? (fib(k) * fib(k) +                           fib(k - 1) * fib(k - 1))                        : (2 * fib(k - 1) + fib(k)) *                                            fib(k);     return f[n];} // Driver Codestatic void Main(){    int n = 9;    f = new int[MAX];    Console.WriteLine(fib(n));}} // This code is contributed by mits
<?php// PHP Program to find n'th// fibonacci Number in with// O(Log n) arithmetic operations $MAX = 1000; // Returns n'th fibonacci// number using table f[]function fib($n){    global $MAX;         // Create an array for memoization    $f = array_fill(0, $MAX, NULL);         // Base cases    if ($n == 0)        return 0;    if ($n == 1 || $n == 2)        return ($f[$n] = 1);     // If fib(n) is already computed    if ($f[$n])        return $f[$n];     $k = ($n & 1) ? ($n + 1) / 2 : $n / 2;     // Applying above formula    // [Note value n&1 is 1 if    // n is odd, else 0.    $f[$n] = ($n & 1) ? (fib($k) * fib($k) +                         fib($k - 1) * fib($k - 1)) :                    (2 * fib($k - 1) + fib($k)) * fib($k);     return $f[$n];} // Driver Code$n = 9;echo fib($n); // This code is contributed// by ChitraNayal?>
<script>      // JavaScript Program to find n'th fibonacci Number in      // with O(Log n) arithmetic operations       const MAX = 1000;       // Create an array for memoization      var f = [...Array(MAX)];      f.fill(0);       // Returns n'th fibonacci number using table f[]      function fib(n) {        // Base cases        if (n == 0) return 0;        if (n == 1 || n == 2) return (f[n] = 1);         // If fib(n) is already computed        if (f[n]) return f[n];         var k = n & 1 ? (n + 1) / 2 : n / 2;         // Applying above formula [Note value n&1 is 1        // if n is odd, else 0.        f[n] =          n & 1            ? fib(k) * fib(k) + fib(k - 1) * fib(k - 1)            : (2 * fib(k - 1) + fib(k)) * fib(k);         return f[n];      }       /* Driver program to test above function */       var n = 9;      document.write(fib(n));             // This code is contributed by rdtank.    </script>
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Time complexity of this solution is O(Log n) as we divide the problem to half in every recursive call.
Method 7 
Another approach(Using formula) :In this method we directly implement the formula for nth term in the fibonacci series. Fn = {[(√5 + 1)/2] ^ n} / √5 
Reference: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibFormula.html 
C++
C
Java
Python3
C#
PHP
Javascript
// C++ Program to find n'th fibonacci Number#include<iostream>#include<cmath> int fib(int n) {  double phi = (1 + sqrt(5)) / 2;  return round(pow(phi, n) / sqrt(5));} // Driver Codeint main (){  int n = 9;  std::cout << fib(n) << std::endl;  return 0;}//This code is contributed by Lokesh Mohanty.
// C Program to find n'th fibonacci Number#include<stdio.h>#include<math.h>int fib(int n) {  double phi = (1 + sqrt(5)) / 2;  return round(pow(phi, n) / sqrt(5));}int main (){  int n = 9;  printf("%d", fib(n));  return 0;}
// Java Program to find n'th fibonacci Numberimport java.util.*; class GFG { static int fib(int n) {double phi = (1 + Math.sqrt(5)) / 2;return (int) Math.round(Math.pow(phi, n)                        / Math.sqrt(5));} // Driver Codepublic static void main(String[] args) {        int n = 9;    System.out.println(fib(n));    }}// This code is contributed by PrinciRaj1992
# Python3 program to find n'th# fibonacci Numberimport math def fibo(n):    phi = (1 + math.sqrt(5)) / 2     return round(pow(phi, n) / math.sqrt(5))     # Driver code   if __name__ == '__main__':         n = 9         print(fibo(n))     # This code is contributed by prasun_parate
// C# Program to find n'th fibonacci Numberusing System; public class GFG{    static int fib(int n)    {    double phi = (1 + Math.Sqrt(5)) / 2;    return (int) Math.Round(Math.Pow(phi, n)                            / Math.Sqrt(5));    }         // Driver code    public static void Main()    {        int n = 9;        Console.WriteLine(fib(n));    }} // This code is contributed by 29AjayKumar
<?php// PHP Program to find n'th// fibonacci Number function fib($n){    $phi = (1 + sqrt(5)) / 2;    return round(pow($phi, $n) / sqrt(5));} // Driver Code$n = 9;echo fib($n) ; // This code is contributed by Ryuga?>
<script>    // Javascript Program to find n'th fibonacci Number    function fib(n) {      let phi = (1 + Math.sqrt(5)) / 2;      return Math.round(Math.pow(phi, n) / Math.sqrt(5));    }         let n = 9;      document.write(fib(n)); // This code is contributed by mukesh07.</script>
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Time Complexity: O(logn), this is because calculating phi^n takes logn timeSpace Complexity: O(1)
Method 8
DP using memoization(Top down approach)
We can avoid the repeated work done in method 1 by storing the Fibonacci numbers calculated so far. We just need to store all the values in an array.
C++
Java
Python3
C#
Javascript
#include <bits/stdc++.h>using namespace std;int dp[10];int fib(int n){    if (n <= 1)        return n;     // temporary variables to store    //  values of fib(n-1) & fib(n-2)    int first, second;     if (dp[n - 1] != -1)        first = dp[n - 1];    else        first = fib(n - 1);     if (dp[n - 2] != -1)        second = dp[n - 2];    else        second = fib(n - 2);     // memoization    return dp[n] = first + second;} // Driver Codeint main(){    int n = 9;     memset(dp, -1, sizeof(dp));     cout << fib(n);    getchar();    return 0;     // This code is contributed by Bhavneet Singh}
import java.util.*; class GFG{ // Initialize array of dpstatic int[] dp = new int[10]; static int fib(int n){    if (n <= 1)        return n;             // Temporary variables to store    // values of fib(n-1) & fib(n-2)    int first, second;         if (dp[n - 1] != -1)        first = dp[n - 1];    else        first = fib(n - 1);     if (dp[n - 2] != -1)        second = dp[n - 2];    else        second = fib(n - 2);     // Memoization    return dp[n] = first + second;} // Driver Codepublic static void main(String[] args){    int n = 9;     Arrays.fill(dp, -1);     System.out.print(fib(n));}} // This code is contributed by sujitmeshram
# Initialize array of dpdp = [-1 for i in range(10)] def fib(n):    if (n <= 1):        return n;    global dp;         # Temporary variables to store    # values of fib(n-1) & fib(n-2)    first = 0;    second = 0;     if (dp[n - 1] != -1):        first = dp[n - 1];    else:        first = fib(n - 1);    if (dp[n - 2] != -1):        second = dp[n - 2];    else:        second = fib(n - 2);    dp[n] = first + second;     # Memoization    return dp[n] ; # Driver Codeif __name__ == '__main__':    n = 9;    print(fib(n)); # This code contributed by Rajput-Ji
using System;class GFG {         // Initialize array of dp    static int[] dp = new int[10];    static int fib(int n)    {        if (n <= 1)            return n;                      // Temporary variables to store        // values of fib(n-1) & fib(n-2)        int first, second;                  if (dp[n - 1] != -1)            first = dp[n - 1];        else            first = fib(n - 1);              if (dp[n - 2] != -1)            second = dp[n - 2];        else            second = fib(n - 2);              // Memoization        return dp[n] = first + second;    }   // Driver code  static void Main()  {    int n = 9;    Array.Fill(dp, -1);    Console.Write(fib(n));  }} // This code is contributed by divyeshrabadiya07.
<script>// Initialize array of dpdp = Array.from({length: 10}, (_, i) => -1); function fib(n){    if (n <= 1)        return n;             // Temporary variables to store    // values of fib(n-1) & fib(n-2)    var first, second;         if (dp[n - 1] != -1)        first = dp[n - 1];    else        first = fib(n - 1);     if (dp[n - 2] != -1)        second = dp[n - 2];    else        second = fib(n - 2);     // Memoization    return dp[n] = first + second;} // Driver Codevar n = 9;document.write(fib(n)); // This code is contributed by Amit Katiyar</script>
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https://www.youtube.com/watch?v=LwZRsM7qhrI This method is contributed by Chirag Agarwal.
Related Articles: Large Fibonacci Numbers in JavaPlease write comments if you find the above codes/algorithms incorrect, or find other ways to solve the same problem.
References: http://en.wikipedia.org/wiki/Fibonacci_number http://www.ics.uci.edu/~eppstein/161/960109.html
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