What method to find square root?

Answer 1

Assuming the number whose square root you require is not a perfect square, one way to find its square root is through iteration. This entails making a guess at the answer and then improving on it. Repeating the procedure should lead to a better estimate at each stage. One such is the Newton-Raphson method.

To start with, if you want to find the square root of 7, define f(x) = x2 - 7.

Then finding the square root of 7 is equivalent to solving f(x) = 0.

Let f'(x) = 2x. This is the derivative of f(x) but you do not need to know that to use the N-R method.

Start with x0 as the first guess.

Then let xn+1 = xn - f(xn)/f'(xn) for n = 0, 1, 2, …

Provided you made a reasonable choice for the starting point, the iteration will very quickly converge to the true answer. Even if your first guess is not so good:

Suppose you start with x0 = 5 (a pretty poor choice since 52 is 25, which is nowhere near 7).

Even so, x3 = 2.646178944, which is the square root of approx 7.002. Finally, remember that the negative value is also a square root.