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{isqrt}( n ) = \lfloor \sqrt n \rfloor.

For example, $\mbox{isqrt}(27) = 5$ because $5\cdot 5=25 \le 27$ and $6\cdot 6=36 > 27$ .

Algorithm

One way of calculating $\sqrt{n}$ and $\mbox{isqrt}( n )$ is to use Newton's method to find a solution for the equation $x^{2} - n = 0$ , giving the iterative formula

{x}_{k+1} = \frac{1}{2}\left(x_k + \frac{ n }{x_k}\right), \quad k \ge 0, \quad x_0 > 0.

The sequence $\{ x_k \}$ converges quadratically to $\sqrt{n}$ as $k\to \infty$ . It can be proven that if $x_{0} = n$ is chosen as the initial guess, one can stop as soon as

| x_{k+1}-x_{k}| < 1

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

Using only integer division

For computing $\lfloor \sqrt n \rfloor$ for very large integers n, one can use the quotient of Euclidean division for both of the division operations. This has the advantage of only using integers for each intermediate value, thus making the use of floating point representations of large numbers unnecessary. It is equivalent to using the iterative formula

{x}_{k+1} = \left\lfloor \frac{1}{2}\left(x_k + \left\lfloor \frac{ n }{x_k} \right\rfloor \right) \right\rfloor, \quad k \ge 0, \quad x_0 > 0, \quad x_0 \in \mathbb{Z}.

By using the fact that

\left\lfloor \frac{1}{2}\left(x_k + \left\lfloor \frac{ n }{x_k} \right\rfloor \right) \right\rfloor = \left\lfloor \frac{1}{2}\left(x_k + \frac{ n }{x_k} \right) \right\rfloor,

one can show that this will reach $\lfloor \sqrt n \rfloor$ within a finite number of iterations.

However, $\lfloor \sqrt n \rfloor$ is not necessarily a fixed point of the above iterative formula. Indeed, it can be shown that $\lfloor \sqrt n \rfloor$ is a fixed point if and only if $n + 1$ is not a perfect square. If $n + 1$ is a perfect square, the sequence ends up in a period-two cycle between $\lfloor \sqrt n \rfloor$ and $\lfloor \sqrt n \rfloor + 1$ instead of converging.

Domain of computation

Although $\sqrt{n}$ is irrational for many $n$ , the sequence $\{ x_k \}$ contains only rational terms when $x_0$ is rational. Thus, with this method it is unnecessary to exit the field of rational numbers in order to calculate $\mbox{isqrt}( n )$ , a fact which has some theoretical advantages.

Stopping criterion

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

|x_{k+1} - x_{k}| < c\

ensures $\lfloor x_{k+1} \rfloor=\lfloor \sqrt n \rfloor$ 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.

External links

A geometric view of the square root algorithm

Number-theoretic algorithms

Primality tests	AKS test APR test Baillie–PSW Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius test Solovay–Strassen Miller–Rabin

Prime-generating	Sieve of Atkin Sieve of Eratosthenes Sieve of Sundaram Wheel factorization

Integer factorization	Continued fraction (CFRAC) Dixon's Lenstra elliptic curve (ECM) Euler's Pollard's rho p − 1 p + 1 Quadratic sieve (QS) General number field sieve (GNFS) Special number field sieve (SNFS) Rational sieve Fermat's Shanks' square forms Trial division Shor's

Multiplication	Ancient Egyptian Long Karatsuba Toom–Cook Schönhage–Strassen Fürer's

Discrete logarithm	Baby-step giant-step Pollard rho Pollard kangaroo Pohlig–Hellman Index calculus Function field sieve

Greatest common divisor	Binary Euclidean Extended Euclidean Lehmer's

Modular square root	Cipolla Pocklington's Tonelli–Shanks

Other algorithms	Chakravala Cornacchia LLL Integer square root Modular exponentiation Schoof's

Italics indicate that algorithm is for numbers of special forms

This article is issued from Wikipedia - version of the 7/3/2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.