integer square root について

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integer square root ：ウィキペディア英語版

integer square root
In number theory, the integer square root (isqrt) of a positive integer ''n'' is the positive integer ''m'' which is the greatest integer less than or equal to the square root of ''n'',
:

\mbox( n ) = \lfloor \sqrt n \rfloor.

For example,

\mbox(27) = 5

because

5\cdot 5=25 \le 27

and

6\cdot 6=36 > 27

.
==Algorithm==
One way of calculating

\sqrt

and

\mbox( n )

is to use Newton's method to find a solution for the equation

x^ - n = 0

, giving the iterative formula
:

_ = \frac\left(x_k + \frac\right), \quad k \ge 0, \quad x_0 > 0.

\

\sqrt

k\to \infty

. It can be proven that if

x_ = n

is chosen as the initial guess, one can stop as soon as
:

| x_-x_| < 1

to ensure that

\lfloor x_ \rfloor=\lfloor \sqrt n \rfloor.

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