Java Program for n-th Fibonacci numbers

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.

Method 1 ( Use recursion ) 

Java

// Fibonacci Series using Recursion
class Fibonacci {
    static int fib(int n)
    {
        if (n==0||n==1)
            return 0;
      else if(n==2)
            return 1;
        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 Musharraf Hassan */

Output:

34

Method 2 ( Use Dynamic Programming ) 

Java

// Fibonacci Series using Dynamic Programming

class Fibonacci {
    static int fib(int n)
    {
        /* Declare an array to store Fibonacci numbers. */
        int f[] = new int[n + 1];
        int i;

        /* 0th and 1st number of the series are 0 and 1*/
        f[0] = 0;

        if (n > 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
and improved by MichaelJoshuaRamos */

Output:

34

Method 3 ( Use Dynamic Programming with Space Optimization) 

Java

// Java program for Fibonacci Series using Space
// Optimized Method
class 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

Output:

34

Method 4 (Divide and Conquer) 

Java

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 */

Output:

34

Method 5 (Divide and Conquer) 

Java

// Fibonacci Series using  Optimized Method
class 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 */

Output:

34

Please refer complete article on Program for Fibonacci numbers for more details!