# 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
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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++

 `// CPP code to find nth fibonacci ` `#include ` `using` `namespace` `std; ` ` `  `// get second MSB ` `int` `getMSB(``int` `n) ` `{ ` `    ``// consectutively 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 numebr ` `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) ` `{ ` `    ``// consectutively 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 numebr ` `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 `

/div>

## C#

 `// C# code to find nth fibonacci  ` `using` `System; ` ` `  `class` `GFG { ` `     `  `// get second MSB  ` `static` `int` `getMSB(``int` `n)  ` `{ ` `     `  `    ``// consectutively 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 numebr  ` `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

> 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 numebr
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
?>

Output:

```8
```

Time Complexity :- O(logn) and space complexity :- O(1).

## tags:

Mathematical Fibonacci Mathematical Fibonacci