Numerical method for square roots?

Question

All the maths questions have triggered something in my memory.

I'm sure there is a numerical means of working out a square root - but I can't remember it. Can anyone help?

Answer 1

There's a method which looks a lot like long division. To start, break the number you're dividing into pairs of digits, based around the decimal point. For instance, to take the square root of 832.491, you'd write that under the "long division" shape as:

8 32. 49 10

Having done that, start with the 8. You want to find the largest digit (we'll call it x) with a square less than or equal to 8. In this case, that's 2. So write 2 over the 8 (above the long division bar), and since 2² = 4, write 4 under the 8 and subtract.

That leaves 4. Write down the 4, and "drop down" the next pair of digits, 32. Now we have 432 as a sort of "intermediate value," and our result so far is "2."

Now, each step will be the same: we look for some digit x such that x times (20 * the result so far + x) is as large as possible without being greater than our intermediate value. For instance: 8 * (20*2 + 8) = 384, but 9 * (20*2 + 9) = 441, which is too big, so our next digit is 8. We put the 8 in place above the "32," so our "result so far" is 28. We subtract the 384 from 432, leaving 48, and we drop down the 49 for our new intermediate value: 4849.

Now we repeat: 8 * (20*28 + 8) = 4544, but 9 * (20*28 + 9) = 5121, which is too high, so 8 is our next digit, etc.

*****

The nice thing about that method is that it gives us one accurate digit at a time, but it takes a LOT of work. Newton-Raphson iteration is faster, even without a calculator, and has the advantage that, if you keep your answers in fractional form, it gives you a better rational approximation to your answer at each step.

This method is simple: start with a guess value. Then, divide the number to be "rooted" by that value, and average the result with the value. Keep repeating this until you converge on an answer.

For instance, to find the square root of 300, we might start with 16 as a first guess.

Average of 16 and 300/16: 139/8 = 17.375
Average of 17.375 and 300/17.375: 17.3205935 (approximately)
Average of 17.3205935 and 300/17.3205935: 17.3205081 (approximately)
Average of 17.3205081 and 300/17.3205081: 17.3205081 (approximately)

So 17.3205081 is the square root, to seven decimal places.

You can find out more at the link below.

Hope that helps!

Answer 2

The most common method of square root calculation by hand is known as the "Babylonian method". It involves a simple algorithm, which will bring you closer and closer to the actual square root each time it is repeated. To find r, the square root of a real number x:

1. Start with an arbitrary positive start value r (the closer to the square root of x, the better).
2. Replace r by the average of r and x / r.
3. Repeat steps 2 and 3.

--Stolen from Wikipedia

Answer 3

Any means of determining a square root is numerical. If you mean by longhand, I found a site using google search by typing:
"square root by longhand" not more than a week ago... try that.

Answer 4

You can use Newton's method, which solves f(x)=0 by the iteration

x_n+1 = x_n - f(x_n) / f'(x_n).

For computing sqrt(a), use f(x) = x^2 - a. The iteration converges quite quickly, if you choose a decent starting value x_0.

Answer 5

I have heard of another very good NUMERICAL method, using a pocket calculator with a sq.root key.

Answer 6

Newton-Rapson method of successive approximation, look it up.