Extracting Square Roots
| Field | Mathematics |
| Went Obsolete | 1970s |
| Made Obsolete By | Scientific Calculator |
| Knowledge Assumed | Understanding of a method of extracting a square root, such as the Babylonian Method |
| When useful | If you ever find yourself without a calculator or computer capable of computing a square root but need one computed |
A commonly used algorithm for approximating \sqrt S (and perhaps the best) is known as the “Babylonian method”[1] and can be derived from (but predates) Newton's method. This is a quadratically convergent algorithm, which means that the number of correct digits of the approximation roughly doubles with each iteration. It proceeds as follows:
1. Start with an arbitrary positive start value x0 (the closer to the root, the better). 2. Let xn+1 be the average of xn and S / xn (using the arithmetic mean to approximate the geometric mean). 3. Repeat steps 2 and 3, until the desired accuracy is achieved.
It can also be represented as:
x_0 \approx \sqrt{S}. x_{n+1} = \frac{1}{2} \left(x_n + \frac{S}{x_n}\right), \sqrt S = \lim_{n \to \infty} x_n.
Attach:http://upload.wikimedia.org/wikipedia/commons/e/e4/Riemann_surface_sqrt.jpg Δ | Image of a square root]
