Integer square root

In number theory (a branch of mathematics), 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,

<math>\mbox{isqrt}( n ) = \lfloor \sqrt n \rfloor.<math>

Algorithm

To calculate √n, and in particular, isqrt(n), one can use Newton's method for the equation x² – n = 0, which gives us the recursive formula

<math>{x}_{k+1} = \frac{1}{2}\left(x_k + \frac{ n }{x}_{k}\right), \ k \ge 0.<math>

One can choose for example x₀ = n as initial guess.

The sequence {x _k} converges quadratically to √n as k→∞. It can be proved (the proof is not trivial) that one can stop as soon as

| x_{k + 1} − x_k | < 1

to ensure that <math>\lfloor x_{k+1} \rfloor=\lfloor \sqrt n \rfloor.<math>

Domain of computation

Let us note that even if √n is irrational for most n, the sequence {x _k} contains only rational terms. Thus, with Newton's method one never needs to exit the field of rational numbers in order to calculate isqrt(n), a fact which has some theoretical advantages in number theory.

Stopping criterion

One can prove that <math>c=1<math> is the largest possible number for which the stopping criterion

| x_k+1 − x_k | < c

ensures <math>\lfloor x_{k+1} \rfloor=\lfloor \sqrt n \rfloor<math> in the algorithm above.

Since actual computer calculations involve roundoff errors, one needs to use c < 1 in the stopping criterion, e.g.,

| x_k+1 − x_k | < 0.5.

See also

• Wikisource: integer square root