Methods of computing square roots

Methods of computing square roots are <a href="/facts/Algorithm/fnl5NmRt">algorithms</a> for approximating the non-negative <a href="/facts/Square_root/AfrzfBdQ">square root</a> 
 
 
 
 
 
 S
 
 
 
 
 {\displaystyle {\sqrt {S}}}
 
 of a positive <a href="/facts/Real_number/R02gw5Pb">real number</a> 
 
 
 
 S
 
 
 {\displaystyle S}
 
.
Since all square roots of <a href="/facts/Natural_number/u65og8JE">natural numbers</a>, other than of <a href="/facts/Square_number/jfakASPK">perfect squares</a>, are <a href="/facts/Irrational_number/Psv19TET">irrational</a>,
square roots can usually only be computed to some finite precision: these methods typically construct a series of increasingly accurate <a href="/facts/Numerical_approximation/QT9cYa9z">approximations</a>.
Most square root computation methods are iterative: after choosing a suitable initial estimate of 
 
 
 
 
 
 S
 
 
 
 
 {\displaystyle {\sqrt {S}}}
 
, an iterative refinement is performed until some termination criterion is met.
One refinement scheme is Heron's method, a special case of <a href="/facts/Newton%27s_method/VlRhI2FL">Newton's method</a>.
If division is much more costly than multiplication, it may be preferable to compute the inverse square root instead.
Other methods are available to compute the square root digit by digit, or using Taylor series.
Rational approximations of square roots may be calculated using continued fraction expansions.
The method employed depends on the needed accuracy, and the available tools and computational power. The methods may be roughly classified as those suitable for mental calculation, those usually requiring at least paper and pencil, and those which are implemented as programs to be executed on a digital electronic computer or other computing device. Algorithms may take into account convergence (how many iterations are required to achieve a specified precision), computational complexity of individual operations (i.e. division) or iterations, and error propagation (the accuracy of the final result).
A few methods like paper-and-pencil synthetic division and series expansion, do not require a starting value. In some applications, an <a href="/facts/Integer_square_root/tkdp4lC0">integer square root</a> is required, which is the square root rounded or truncated to the nearest integer (a modified procedure may be employed in this case).

Methods of computing square roots open-in-new

Methods of computing square roots