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 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.

The purpose of Newton's method is to find a root of a function. The idea is to start with an initial guess near a root, approximate the function by its tangent line near the guess, and then take the root of the linear approximation as a next guess at the function's root. This will typically be closer to the function's root than the previous guess, and the method can be iterated.

The best linear approximation to an arbitrary differentiable function ${\displaystyle f(x)}$ near the point ${\displaystyle x=x_{n}}$ is the tangent line to the curve, with equation

$${\displaystyle f(x)\approx f(x_{n})+f'(x_{n})(x-x_{n}).}$$