Newton's method

In <a href="/facts/Numerical_analysis/QT9cYa9z">numerical analysis</a>, the Newton–Raphson method, also known simply as Newton's method, named after <a href="/facts/Isaac_Newton/4L7gTncN">Isaac Newton</a> and <a href="/facts/Joseph_Raphson/oLHlrvGR">Joseph Raphson</a>, is a <a href="/facts/Root-finding_algorithm/IxTDkhBq">root-finding algorithm</a> which produces successively better <a href="/facts/Numerical_analysis/QT9cYa9z">approximations</a> to the <a href="/facts/Root_of_a_function/mlK3dhoT">roots</a> (or zeroes) of a <a href="/facts/Real_number/R02gw5Pb">real</a>-valued <a href="/facts/Function_(mathematics)/CdReEKCh">function</a>. The most basic version starts with a <a href="/facts/Real-valued_function/jetqlJpK">real-valued function</a> f, its <a href="/facts/Derivative/7Ito70Dh">derivative</a> f′, and an initial guess x0 for a <a href="/facts/Zero_of_a_function/mlK3dhoT">root</a> of f. If f satisfies certain assumptions and the initial guess is close, then

 
 
 
 
 x
 
 1
 
 
 =
 
 x
 
 0
 
 
 −
 
 
 
 f
 (
 
 x
 
 0
 
 
 )
 
 
 
 f
 ′
 
 (
 
 x
 
 0
 
 
 )
 
 
 
 
 
 {\displaystyle x_{1}=x_{0}-{\frac {f(x_{0})}{f'(x_{0})}}}

is a better approximation of the root than x0. Geometrically, (x1, 0) is the <a href="/facts/X-intercept/mlK3dhoT">x-intercept</a> of the <a href="/facts/Tangent/sF0jH3EI">tangent</a> of the <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> of f at (x0, f(x0)): that is, the improved guess, x1, is the unique root of the <a href="/facts/Linear_approximation/SDBAneFx">linear approximation</a> of f at the initial guess, x0. The process is repeated as

 
 
 
 
 x
 
 n
 +
 1
 
 
 =
 
 x
 
 n
 
 
 −
 
 
 
 f
 (
 
 x
 
 n
 
 
 )
 
 
 
 f
 ′
 
 (
 
 x
 
 n
 
 
 )
 
 
 
 
 
 {\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}}

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 <a href="/facts/Householder%27s_method/koUW3umF">Householder's methods</a>, and was succeeded by <a href="/facts/Halley%27s_method/t0tcGcHA">Halley's method</a>. The method can also be extended to <a href="/facts/Complex-valued_function/hgCoQlH0">complex functions</a> and to <a href="/facts/Systems_of_equations/KzP6TJ4u">systems of equations</a>.

Newton's method open-in-new

Newton's method