Newton's method

Summary

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 sufficient 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 x0. 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 - \frac{f(x_n)}{f'(x_n)} until a sufficiently precise value is reached. The number of correct digits roughly doubles with each st

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