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An illustration of Newton's method. 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.
Newton's method assumes the function f to have a continuous derivative. Newton's method may not converge if started too far away from a root. However, when it does converge, it is faster than the bisection method; its order of convergence is usually quadratic whereas the bisection method's is linear. Newton's method is also important because it ...
Newton's method, in its original version, has several caveats: It does not work if the Hessian is not invertible. This is clear from the very definition of Newton's method, which requires taking the inverse of the Hessian. It may not converge at all, but can enter a cycle having more than 1 point. See the Newton's method § Failure analysis.
For finding real roots of a polynomial, the common strategy is to divide the real line (or an interval of it where root are searched) into disjoint intervals until having at most one root in each interval. Such a procedure is called root isolation, and a resulting interval that contains exactly one root is an isolating interval for this root.
Anderson's iterative method, which uses a least squares approach to the Jacobian. [9] Schubert's or sparse Broyden algorithm – a modification for sparse Jacobian matrices. [10] The Pulay approach, often used in density functional theory. [11] [12] A limited memory method by Srivastava for the root finding problem which only uses a few recent ...
Finding roots in a specific region of the complex plane, typically the real roots or the real roots in a given interval (for example, when roots represents a physical quantity, only the real positive ones are interesting). For finding one root, Newton's method and other general iterative methods work generally well.
Stochastic approximation methods are a family of iterative methods typically used for root-finding problems or for optimization problems. The recursive update rules of stochastic approximation methods can be used, among other things, for solving linear systems when the collected data is corrupted by noise, or for approximating extreme values of functions which cannot be computed directly, but ...
Root-finding algorithm — algorithms for solving the equation f(x) = 0 General methods: Bisection method — simple and robust; linear convergence Lehmer–Schur algorithm — variant for complex functions; Fixed-point iteration; Newton's method — based on linear approximation around the current iterate; quadratic convergence