Newton Raphson Algorithm for square root

Author: Prakash5bisht

NewtonRaphsonAlgo.cpp

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+//calculating the square root of a number using Newton Raphson method.
+
+// what is Newton Raphson method?
+
+/*
+  It is a gradient based technique used to find root of a given number.
+  according to this method,
+
+  Xn+1 = Xn - f(Xn)/f'(Xn)  - (1)    {where n is the nth term i.e 0,1,2,....}
+
+
+  we know,
+      X = (N)^1/r
+
+      where N is a number of which we want to find rth root (X).
+
+       above equation can be written as,
+              X^r - N = 0
+        =>   f(X) = X^r - N = 0  - (a)
+        =>   f'(X) = r*X^(r-1)   - (b)
+
+        now, putting nth term of equation (a) and (b) in  equation (1)
+              Xn+1 = Xn  -  (Xn^r - N) / r*Xn^(r-1)
+
+              taking L.C.M,
+                   Xn+1 = [ r*Xn^r  -  (Xn^r - N) ] / r*Xn^(r-1)
+                
+                sovling further,
+
+                   Xn+1 = [(r-1)Xn^r + N] / r*Xn^r-1   - (2)
+
+                   This is the formula to get the rth root of a positive integer N, where X is our approximation number.
+
+    now, our aim is to get square root right, so, in equation (2) put r = 2.
+
+    Xn+1 = ( Xn^2 + N ) / 2 * Xn^2  [This is our equation which we are going to use to calculate square root of a number programatically]
+    */
+
+#include<math.h>
+#include<stdlib.h>
+#include<iostream>
+using namespace std;
+
+void squareRoot(int number){
+
+   double x, root;
+
+   x = number;        // x is our approximation number(or Xn in equation (2)) 
+                      //  i.e the number which we assume can be the square root of number
+                      // which we assume to be the number itself
+   while(1){
+     
+     root = (x + number/x)/2;         //  root is our Xn+1 th term (as in equation (2))
+     if(abs(root - x) < 0.0001)      // 0.0001 is our precision value an the loop is run until we get precision upto desired decimal places
+         break;
+
+     x = root;
+   }
+
+   cout<<root;
+}
+
+int main(){
+	system("cls");
+	int n;
+
+	cout<<"enter a number: ";
+	cin>>n;
+	fflush(stdin);
+	squareRoot(n);
+	getchar();
+	return 0;
+}