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101 lines
1.6 KiB
C++
101 lines
1.6 KiB
C++
#include <iostream>
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using namespace std;
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/*
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Efficient method for finding nth term in a fibonacci number sequence.
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Uses Dvide and conquer approach
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Enter Values from 1
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Eg-
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1st term = 0;
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2nd term = 1;
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3rd term = 1;
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.
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.
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.
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.
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*/
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long long a[2][2] = {{1,1},{1,0}};//fibonacci matrix
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long long ans[2][2] = {{1,1},{1,0}};//final ans matrix
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/*
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Working Principal:
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[F(k+1) F(k)] [1 1]^k
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| | = | |
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[F(k) F(k-1)] [1 0]
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where F(k) is the kth term of the Fibonacci Sequence.
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*/
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void product(long long b[][2],long long k[][2])
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{
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/*
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Function for computing product of the two matrices b and k
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and storing them into the variable ans.
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Implementation :
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Simple matrix multiplication of two (2X2) matrices.
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*/
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long long c[2][2];//temporary stores the answer
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c[0][0] = b[0][0]*k[0][0]+b[0][1]*k[1][0];
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c[0][1] = b[0][0]*k[0][1]+b[0][1]*k[1][1];
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c[1][0] = b[1][0]*k[0][0]+b[1][1]*k[1][0];
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c[1][1] = b[1][0]*k[0][1]+b[1][1]*k[1][1];
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ans[0][0] = c[0][0];
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ans[0][1] = c[0][1];
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ans[1][0] = c[1][0];
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ans[1][1] = c[1][1];
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}
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void power_rec(long long n)
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{
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/*
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Function for calculating A^n(exponent) in a recursive fashion
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Implementation:
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A^n = { A^(n/2)*A^(n/2) if n is even
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{ A^((n-1)/2)*A^((n-1)/2)*A if n is odd
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*/
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if((n == 1)||(n==0))
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return;
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else
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{
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if((n%2) == 0)
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{
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power_rec(n/2);
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product(ans,ans);
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}
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else
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{
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power_rec((n-1)/2);
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product(ans,ans);
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product(ans,a);
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}
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}
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}
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int main()
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{
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//Main Function
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cout <<"Enter the value of n\n";
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long long n;
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cin >>n;
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if(n == 1)
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{
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cout<<"Ans: 0"<<endl;
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}
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else
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{
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power_rec(n-1);
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cout <<"Ans :"<<ans[0][1]<<endl;
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}
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return 0;
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}
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