What is the strategy for estimating square roots?

Answer 1

A simple strategy is the Newton Raphson method. See link for details.

Suppose to wish to find the square root of a number k.

Define f(x) = x2 - k

Let f'(x) = 2*x

Now start with some estimate for sqrt(k), say x0.

Then calculate x1 = x0 - f(x0)/f'(x0) as the next estimate.

Repeat, using xn+1 = xn - f(xn)/f'(xn).

The estimates will soon be very close to sqrt(k)

To illustrate:

Try sqrt(50) - which is approx 7.071068.

Even though you know the answer, start with a pretty feeble first estimate:

x0 = 10

Then x1 = 10 - (102 - 50)/(2*10) = 7.5

x3 is accurate to 4 decimal places and x4 to 11 dp.