5 Base 0. + 5 Base - 1 + 5 Base - 2
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Methods of computing square
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The sequence we get by computing the square root of two with the Babylonian method with different starting points
In numerical analysis, a branch of mathematics, there are several square root algorithms or methods of computing the principal square root of a non-negative real number. For the square roots of a negative or complex number, see below.
Finding {\displaystyle {\sqrt {S}}} {\sqrt {S}} is the same as solving the equation {\displaystyle f(x)=x^{2}-S=0\,\!} f(x)=x^{2}-S=0\,\! for a positive {\displaystyle x} x. Therefore, any general numerical root-finding algorithm can be used. Newton's method, for example, reduces in this case to the so-called Babylonian method:
{\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}=x_{n}-{\frac {x_{n}^{2}-S}{2x_{n}}}={\frac {1}{2}}\left(x_{n}+{\frac {S}{x_{n}}}\right)} x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}=x_{n}-{\frac {x_{n}^{2}-S}{2x_{n}}}={\frac {1}{2}}\left(x_{n}+{\frac {S}{x_{n}}}\right)
These methods generally yield approximate results, but can be made arbitrarily precise by increasing the number of calculation steps.