[size=150]The Babylonian square-root algorithm. The iterative method is called [url=https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method]the Babylonian method for finding square roots[/url], or sometimes Hero's method. It was known to the ancient Babylonians (1500 BC) and Greeks (100 AD) long before Newton invented his general procedure. Here's how it works. Suppose you are given any positive number [i]S[/i]. To find the square root of [i]S[/i], do the following:[list=1][*][b]Make an initial guess.[/b] Guess any positive number [i]x[sub]0[/sub][/i].[/*][*][b]Improve the guess.[/b] Apply the formula [i]x[sub]1[/sub][/i] = ([i]x[sub]0[/sub][/i] + [i]S[/i] / [i]x[sub]0[/sub][/i]) / 2. The number [i]x[/i]1 is a better approximation to sqrt(S).[/*][*][b]Iterate until convergence.[/b] Apply the formula [i]x[sub]n[/sub][/i]+1 = ([i]x[sub]n[/sub][/i] + [i]S[/i] / [i]x[sub]n[/sub][/i]) / 2 until the process converges. Convergence is achieved when the digits of x[sub]n[/sub]+1 and x[sub]n[/sub] agree to as many decimal places as you desire.[/*][/list]Let's use this algorithm to compute the square root of S = 20 to at least two decimal places.[list=1][*]An initial guess is [i]x[sub]0[/sub][/i] = 10.[/*][*]Apply the formula: [i]x[sub]1[/sub][/i] = (10 + 20/10)/2 = 6. The number 6 is a better approximation to sqrt(20).[/*][*]Apply the formula again to obtain [i]x[sub]2[/sub][/i] = (6 + 20/6)/2 = 4.66667. The next iterations are [i]x[sub]3[/sub][/i] = 4.47619 and [i]x[/i]4 = 4.47214.[/*][/list]Because [i]x[sub]3[/sub][/i] and [i]x[sub]4[/sub][/i] agree to two decimal places, the algorithm ends after four iterations. An estimate for sqrt(20) is 4.47214.[/size]