# Implementing sqrt

I was reading some post about interview questions of 2011 and came across one that stated “find the square root of a number”.

Assuming we can’t use the sqrtf function of the standard C math library, let’s see how we can calculate the square root of a number x.

Given n, we know that its square root is a number x that holds:

$\sqrt{n} = x$

Let’s work on this equation a little. Raise both sides to the second power:

$n = x^2$

Move to the left of the equality:

$0 = x^2 - n$

If we found the roots of this last equation somehow, we would have found the square root of n. We can do this by using the Newton-Raphson iteration.

The Newton-Raphson iteration states that we can find the root of an equation using the following formula iteratively:

$x_{n+1} = x_n - \frac{f(x)}{f'(x)}$

Where f'(x) is the derivative of function f(x). We will approximate the derivative using the definition of derivative at a point (we could also note that the derivative could be trivially calculated; this method is more general).

$f'(x) = \frac{f(x+h) - f(x)}{h}$

The error of the Newton-Raphson iteration is given by:

$|x_{n+1} - x_n|$

Starting with a hardcoded seed value, we will perform this iteration in a loop until the error is less than a given value. I have chosen to iterate until the error is less than 1×10^(-10): 0.00000000001.

Let us see what a tentative “pythonesque” pseudocode for this loop could be:

Assuming we have a symbolic function type, that loop does not seem too difficult. In order to code this in C, since the equation is always the same, I will hardcode it as a plain function.

Here are the results of running our custom square root function, compared to the standard version provided with the C programming language: