 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 [itex] sqrt approx 2 cdot 10^n.[/itex]
If D is even, D = 2n + 2, then use [itex] sqrt approx 6 cdot 10^n.[/itex]

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

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

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

Perhaps the first algorithm used for approximating [itex]sqrt S[/itex] is known as the...

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