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In [[number theory]], the '''integer square root''' (isqrt) of a [[positive integer]] ''n'' is the positive integer ''m'' which is the [[floor and ceiling functions|greatest integer less than or equal]] to the [[square root]] of ''n'',
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: <math>\mbox{isqrt}( n ) = \lfloor \sqrt n \rfloor.</math>
 
For example, <math>\mbox{isqrt}(27) = 5</math> because <math>5\cdot 5=25 \le 27</math> and <math>6\cdot 6=36 > 27</math>.
 
==Algorithm==
One way of calculating <math>\sqrt{n}</math> and <math>\mbox{isqrt}( n )</math> is to use [[Newton's method]] to find a solution for the equation <math>x^{2} - n = 0</math>, giving the [[Recursion|recursive]] formula
 
: <math>{x}_{k+1} = \frac{1}{2}\left(x_k + \frac{ n }{x_k}\right), \quad k \ge 0, \quad x_0 > 0.</math>
 
The [[sequence]] <math>\{ x_k \}</math> [[Limit (mathematics)|converges]] [[Rate of convergence|quadratically]] to <math>\sqrt{n}</math> as <math>k\to \infty</math>. It can be proven that if <math>x_{0} = n</math> is chosen as the initial guess, one can stop as soon as
:<math>| x_{k+1}-x_{k}| < 1</math>
to ensure that <math>\lfloor x_{k+1} \rfloor=\lfloor \sqrt n \rfloor.</math>
 
==Domain of computation==
Although <math>\sqrt{n}</math> is [[irrational number|irrational]] for [[almost all]] <math>n</math>, the sequence <math>\{ x_k \}</math> contains only [[rational number|rational]] terms when <math> x_0 </math> is rational. Thus, with this method it is unnecessary to exit the [[field (mathematics)|field]] of rational numbers in order to calculate <math>\mbox{isqrt}( n )</math>, a fact which has some theoretical advantages.
 
==Stopping criterion==
One can prove that <math>c=1</math> is the largest possible number for which the stopping criterion
:<math>|x_{k+1} - x_{k}| < c\ </math>
ensures <math>\lfloor x_{k+1} \rfloor=\lfloor \sqrt n \rfloor</math>
in the algorithm above.
 
In implementations which use number formats that cannot represent all rational numbers exactly (for example, floating point), a stopping constant less than one should be used to protect against roundoff errors.
 
== See also ==
* [[Methods of computing square roots]]
 
==External links==
*[http://mathcentral.uregina.ca/RR/database/RR.09.95/grzesina1.html A geometric view of the square root algorithm]
 
{{number theoretic algorithms}}
 
[[Category:Number theoretic algorithms]]
[[Category:Number theory]]
[[Category:Root-finding algorithms]]

Latest revision as of 07:31, 11 January 2015

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