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Newton's method is an extremely powerful technique—in general the convergence is quadratic: as the method converges on the root, the difference between the root and the approximation is squared (the number of accurate digits roughly doubles) at each step.However, there are some difficulties with the 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 until a sufficiently precise value is reached.
Newton's method was used by 17th-century Japanese mathematician Seki Kōwa to solve single-variable equations, though the connection with calculus was missing.
instead of the more complicated sequence of polynomials used by Newton.
It is only here that the Hessian matrix of the SSE is positive and the first derivative of the SSE is close to zero.
In a robust implementation of Newton's method, it is common to place limits on the number of iterations, bound the solution to an interval known to contain the root, and combine the method with a more robust root finding method.In these situations, it may be appropriate to approximate the derivative by using the slope of a line through two nearby points on the function.Using this approximation would result in something like the secant method whose convergence is slower than that of Newton's method.Arthur Cayley in 1879 in The Newton–Fourier imaginary problem was the first to notice the difficulties in generalizing Newton's method to complex roots of polynomials with degree greater than 2 and complex initial values.This opened the way to the study of the theory of iterations of rational functions.However, the extra computations required for each step can slow down the overall performance relative to Newton's method, particularly if or its derivatives are computationally expensive to evaluate.The name "Newton's method" is derived from Isaac Newton's description of a special case of the method in De analysi per aequationes numero terminorum infinitas (written in 1669, published in 1711 by William Jones) and in De metodis fluxionum et serierum infinitarum (written in 1671, translated and published as Method of Fluxions in 1736 by John Colson).This algorithm is first in the class of Householder's methods, succeeded by Halley's method. But, in the absence of any intuition about where the zero might lie, a "guess and check" method might narrow the possibilities to a reasonably small interval by appealing to the intermediate value theorem.) The method will usually converge, provided this initial guess is close enough to the unknown zero, and that .The method can also be extended to complex functions and to systems of equations. Furthermore, for a zero of multiplicity 1, the convergence is at least quadratic (see rate of convergence) in a neighbourhood of the zero, which intuitively means that the number of correct digits roughly doubles in every step.For situations where the method fails to converge, it is because the assumptions made in this proof are not met.If the first derivative is not well behaved in the neighborhood of a particular root, the method may overshoot, and diverge from that root.