Editing Newton's method (section)

===Multiplicative inverses of numbers and power series===
An important application is [[Division algorithm#Newton–Raphson division|Newton–Raphson division]], which can be used to quickly find the [[Multiplicative inverse|reciprocal]] of a number {{mvar|a}}, using only multiplication and subtraction, that is to say the number {{mvar|x}} such that {{math|1={{sfrac|1|{{var|x}}}} = {{var|a}}}}. We can rephrase that as finding the zero of {{math|1={{var|f}}({{var|x}}) = {{sfrac|1|{{var|x}}}} − {{var|a}}}}. We have {{math|1={{var|{{prime|f}}}}({{var|x}}) = −{{sfrac|1|{{var|x}}{{sup|2}}}}}}.

Newton's iteration is

<math display="block">x_{n+1} = x_n-\frac{f(x_n)}{f'(x_n)} = x_n+\frac{\frac{1}{x_n}-a}{\frac{1}{x_n^2}} = x_n(2-ax_n).
</math>

Therefore, Newton's iteration needs only two multiplications and one subtraction.

This method is also very efficient to compute the multiplicative inverse of a [[power series]].