This method can be derived from (but predates) Newton – Raphson method.
1 Start with an arbitrary positive start value x (the closer to the
root, the better).
2 Initialize y = 1.
3. Do following until desired approximation is achieved.
a) Get the next approximation for root using average of x and y
b) Set y = n/x
n = 4 /*n itself is used for initial approximation*/ Initialize x = 4, y = 1 Next Approximation x = (x + y)/2 (= 2.500000), y = n/x (=1.600000) Next Approximation x = 2.050000, y = 1.951220 Next Approximation x = 2.000610, y = 1.999390 Next Approximation x = 2.000000, y = 2.000000 Terminate as (x - y) > e now.
If we are sure that n is a perfect square, then we can use following method. The method can go in infinite loop for non-perfect-square numbers. For example, for 3 the below while loop will never terminate.