Square root of a number... Accurate up to n precision

Question

I could not understand the above method. Can someone please explain? I have done some code but its is limited to some hard coded precision and seems to consume too much resource of computer.

R = 0.00001
INPUT N
WHILE R*R != N
        R = R + 0.00001
ENDWHILE
PRINT R

What is the Algorithm or C++ code for square root of a number upto n precision? n can be taken from user if required.

Answer 1

There are algorithms that are much better suited to computer evaluation. I learned the one in the question in the 1960's, as a way of manually calculating a square root digit-by-digit using a process rather like long division.

The objective, during calculation of the nth digit of the result, is to find the largest prefix string such that the square is less than or equal to the first 2n digits of the input.

The key underlying idea is that (a+b)^2 = a^2 + b^2 + 2ab. In the algorithm, a is the partial result so far, and b is the new digit. It accounts for factors of 100 in the square and 10 in the root by moving two places in the input for one generated digit in the result.

Let p be the partial result before appending digit d. We have already subtracted p^2 from the input. We need to also subtract d^2 + 2pd, to maintain subtraction of the square of the new partial result. Equivalently, subtract d(2p+d). We keep p already doubled, append d, and multiply by d. Before going on to the next step, we need to double d as well.