Methods of computing square roots

Methods Of Computing Square Roots

Methods of computing square roots

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There are several methods for calculating the principal square root of a nonnegative real number. For the square roots of a negative or complex number, see below.

Rough estimation

Many of the methods for calculating square roots of a positive real number S require an initial seed value. If the initial value is too far from the actual square root, the calculation will be slowed down. It is therefore useful to have a rough estimate, which may be very inaccurate but easy to calculate. If S ≥ 1, let D be the number of digits to the left of the decimal point. If S < 1, let D be the negative of the number of zeros to the immediate right of the decimal point. Then the rough estimation is this:
If D is odd, D = 2n + 1, then use <math> sqrt approx 2 cdot 10^n.</math>
If D is even, D = 2n + 2, then use <math> sqrt approx 6 cdot 10^n.</math>

Two and six are used because they approximate the geometric means of the lowest and highest possible values with the given number of digits: <math>sqrt = sqrt approx 2 ,</math> and <math>sqrt = sqrt approx 6 ,.</math>

When working in the binary numeral system (as computers do internally), an alternative method is to use <math>2^</math> (here D is the number of binary digits).

== Babylonian method ==<!-- this section is linked from Babylonian method -->

Perhaps the first algorithm used for approximating <math>sqrt S</math> is known as the...
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