Nth fibonacci number using matrix exponentiation (#1215)

* Nth fibonacci number using matrix exponentiation
Co-authored-by: unknown <villizain6@gmail.com>

math/fibonacci_matrix_exponentiation.cpp

@@ -0,0 +1,113 @@
+/**
+ * @file 
+ * @brief This program computes the N^th Fibonacci number in modulo mod
+ * input argument .
+ *
+ * Takes O(logn) time to compute nth Fibonacci number
+ * 
+ *
+ * \author [villayatali123](https://github.com/villayatali123)
+ * \author [unknown author]()
+ * @see fibonacci.cpp, fibonacci_fast.cpp, string_fibonacci.cpp, fibonacci_large.cpp
+ */
+
+#include<iostream>
+#include<vector>
+#include <cassert>
+
+/**
+ * This function finds nth fibonacci number in a given modulus
+ * @param n nth fibonacci number
+ * @param mod  modulo number 
+ */
+uint64_t fibo(uint64_t n , uint64_t mod )
+{
+	std::vector<uint64_t> result(2,0);
+	std::vector<std::vector<uint64_t>> transition(2,std::vector<uint64_t>(2,0));
+	std::vector<std::vector<uint64_t>> Identity(2,std::vector<uint64_t>(2,0));
+	n--;
+	result[0]=1, result[1]=1;
+	Identity[0][0]=1; Identity[0][1]=0;
+	Identity[1][0]=0; Identity[1][1]=1;
+	 
+	transition[0][0]=0;
+	transition[1][0]=transition[1][1]=transition[0][1]=1;
+	
+	while(n)
+	{
+		if(n%2)
+		{
+			std::vector<std::vector<uint64_t>> res(2, std::vector<uint64_t>(2,0));
+	                for(int i=0;i<2;i++)
+			{
+				for(int j=0;j<2;j++)
+				{
+					for(int k=0;k<2;k++)
+						{
+							res[i][j]=(res[i][j]%mod+((Identity[i][k]%mod*transition[k][j]%mod))%mod)%mod;
+						}
+				}
+			}
+		       	for(int i=0;i<2;i++)
+			{
+				for(int j=0;j<2;j++)
+				{
+				Identity[i][j]=res[i][j];
+				}
+	    		}
+			n--;
+		}
+		else{
+			std::vector<std::vector<uint64_t>> res1(2, std::vector<uint64_t>(2,0));
+			for(int i=0;i<2;i++)
+			{
+				for(int j=0;j<2;j++)
+				{
+					for(int k=0;k<2;k++)
+					{
+						res1[i][j]=(res1[i][j]%mod+((transition[i][k]%mod*transition[k][j]%mod))%mod)%mod;
+					}
+				}
+			}
+			for(int i=0;i<2;i++)
+			{
+				for(int j=0;j<2;j++)
+				{
+					transition[i][j]=res1[i][j];
+				}
+			} 
+			n=n/2;
+			}
+	}
+	return ((result[0]%mod*Identity[0][0]%mod)%mod+(result[1]%mod*Identity[1][0]%mod)%mod)%mod;
+}
+
+/**
+ * Function to test above algorithm
+ */
+void test()
+{
+    assert(fibo(6, 1000000007 ) == 8);
+    std::cout << "test case:1 passed\n";
+    assert(fibo(5, 1000000007  ) == 5);
+    std::cout << "test case:2 passed\n";
+    assert(fibo(10 , 1000000007) == 55);
+    std::cout << "test case:3 passed\n";
+    assert(fibo(500 , 100) == 25);
+    std::cout << "test case:3 passed\n";
+    assert(fibo(500 , 10000) == 4125);
+    std::cout << "test case:3 passed\n";
+    std::cout << "--All tests passed--\n";
+}
+
+/**
+ * Main function
+ */
+int main()
+{
+	test();
+	uint64_t mod=1000000007;
+	std::cout<<"Enter the value of N: ";
+	uint64_t n=0; std::cin>>n; 
+	std::cout<<n<<"th Fibonacci number in modulo " << mod << ": "<< fibo( n , mod) << std::endl;
+}