Editing Newton's method (section)

{{Short description|Algorithm for finding zeros of functions}}
{{About|Newton's method for finding roots|Newton's method for finding minima|Newton's method in optimization}}
{{Use dmy dates|date=January 2020}}
[[File:Methode newton.png|thumb|An illustration of Newton's method.]]
In [[numerical analysis]], the '''Newton–Raphson method''', also known simply as '''Newton's method''', named after [[Isaac Newton]] and [[Joseph Raphson]], is a [[root-finding algorithm]] which produces successively better [[Numerical analysis|approximations]] to the [[root of a function|roots]] (or zeroes) of a [[Real number|real]]-valued [[function (mathematics)|function]]. The most basic version starts with a [[real-valued function]]  {{mvar|f}}, its [[derivative]] {{mvar|{{prime|f}}}}, and an initial guess {{math|{{var|x}}{{sub|0}}}} for a [[Zero of a function|root]] of {{mvar|f}}. If {{mvar|f}} satisfies certain assumptions and the initial guess is close, then

<math display="block">x_{1} = x_0 - \frac{f(x_0)}{f'(x_0)}</math>

is a better approximation of the root than {{math|{{var|x}}{{sub|0}}}}. Geometrically, {{math|({{var|x}}{{sub|1}}, 0)}} is the [[x-intercept]] of the [[tangent]] of the [[graph of a function|graph]] of {{mvar|f}} at {{math|({{var|x}}{{sub|0}}, {{var|f}}({{var|x}}{{sub|0}}))}}: that is, the improved guess, {{math|{{var|x}}{{sub|1}}}}, is the unique root of the [[linear approximation]] of {{mvar|f}}  at the initial guess, {{math|{{var|x}}{{sub|0}}}}. The process is repeated as

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

until a sufficiently precise value is reached. The number of correct digits roughly doubles with each step. This algorithm is first in the class of [[Householder's method]]s, and was succeeded by [[Halley's method]]. The method can also be extended to [[Complex-valued function|complex functions]] and to [[systems of equations]].