Newton's method

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 x₀ for a root of f. If f satisfies certain assumptions and the initial guess is close, then

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

is a better approximation of the root than x₀. Geometrically, (x₁, 0) is the x-intercept of the tangent of the graph of f at (x₀, f(x₀)): that is, the improved guess, x₁, is the unique root of the linear approximation of f at the initial guess, x₀. The process is repeated as

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

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