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 single-variable function f defined for a real variable x, the function's derivative f ′, and an initial guess x0 for a root of f. If the function satisfies necessary assumptions and the initial guess is close, then a better approximation x1 is x 1 = x 0 − f ( x 0 ) f ′ ( x 0 ) . {\displaystyle x_{1}=x_{0}-{\frac {f(x_{0})}{f'(x_{0})}}\,.} Geometrically, (x1, 0) is the intersection of the x-axis and the tangent of the graph of f at (x0, f (x0)): that is, the improved guess is the unique root of the linear approximation at the initial point. 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 accurate value is reached. This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions and to systems of equations.


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