# Newton's method

## Algorithm for finding zeros of functions / From Wikipedia, the free encyclopedia

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In numerical analysis, **Newton's method**, also known as the **Newton–Raphson method**, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a real-valued function f, its derivative f′, and an initial guess `x`_{0} for a root of f. If f satisfies certain assumptions and the initial guess is close, then

$x_{1}=x_{0}-{\frac {f(x_{0})}{f'(x_{0})}}$

is a better approximation of the root than `x`_{0}. Geometrically, (`x`_{1}, 0) is the x-intercept of the tangent of the graph of f at (`x`_{0}, `f`(`x`_{0})): that is, the improved guess, `x`_{1}, is the unique root of the linear approximation of f at the initial guess, `x`_{0}. The process is repeated as

$x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}$

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 Householder's methods, and was succeeded by Halley's method. The method can also be extended to complex functions and to systems of equations.