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In mathematics, particularly numerical analysis, multiple methods to compute, either precisely or approximately, the principal square root of a nonnegative real number have been developed. The square root of a number is the number that, when squared, equals that number. In other words, the square root of a number $S$, denoted by ${\sqrt {S}}$, is the number $n$ such that $n^{2}=S$. Each positive number has two square roots, the positive or principal square root $n$ ($n^{2}=S$), and the negative square root which is the negative of $n$, $-n$ ($-n^{2}=S$), as the square of a negative number is positive.

Numerous methods with the sole purpose of computing the square root of a number, some of which rely on certain representations of numbers, such as IEEE floating point representation. However, the problem can also be solved with root-finding algorithms, methods that compute, for a function $f$, at what $x$ does $f(x)=0$, which is its root. The square root problem of finding ${\sqrt {S}}$ can be reduced to finding the root of the function $f(x)=x^{2}-S$. Square root-finding methods typically compute approximate results, but typically converge to the actual square root of a number as the number of iterations increase.

The square root of a negative number, which has a solution in the complex numbers, need not be computed by a dedicated method, as they can be computed by computing the square root of its absolute value ${\sqrt {|S|}}$, and then multiplying it by the imaginary unit $i$, the square root of $-1$ (so that $i^{2}=1$).

Methods

Root-finding

It is possible to reduce the square root problem of finding ${\sqrt {S}}$ to computing the root (or zero) of the function $f(x)=x^{2}-S$, that is, finding $x$ such that $f(x)=0$. Thus, root-finding algorithms can be used to approximate ${\sqrt {S}}$.

Newton's method is commonly used, the Babylonian method described below being a special case of Newton's; given an initial guess, it approximates the root of a function $f$ by using its derivative $f'(x)$ to generate a better estimate of the root. It can be applied successively to generate increasingly precise estimates. Newton's method can be described as follows, where $x_{n}$ is the $n^{\text{th}}$ estimate generated.

$$x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}$$

Babylonian method

The Babylonian method was among the first algorithms used to estimate square roots. The method generates a better estimate $x_{n+1}$ of ${\sqrt {S}}$ from the initial guess or previous estimate by taking the arithmetic mean of $S$ and ${\frac {S}{x_{n}}}$. The Babylonian method can be derived from Newton's method as follows. $${\begin{aligned}{\text{Babylonian method:}}\ x_{n+1}={\frac {1}{2}}(x_{n}+{\frac {S}{x_{n}}})\\{\frac {d}{dx}}x^{2}=2x\\x_{n+1}=x_{n}-{\frac {x_{n}^{2}-S}{2x_{n}}}&\quad {\text{Newton's method to find the square root}}\\\end{aligned}}$$

References

Works

• Hardy, Godfrey H. (1908). A Course of Pure Mathematics. Cambridge, UK: Cambridge University Press. ISBN 0521720559. {{cite book}}: ISBN / Date incompatibility (help)