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In numerical analysis, Halley's method is a root-finding algorithm used for functions of one real...

In numerical analysis, Halley's method is a root-finding algorithm used for functions of one real variable with a continuous second derivative, i.e. C functions. It is named after its inventor Edmond Halley who also discovered Halley's Comet. The algorithm is second in the class of Householder's methods, right after the Newton's method. Like the latter it produces iteratively a sequence of approximations to the root, their rate of convergence to the root is cubic. Multidimensional versions of this method exist. Like any root-finding method, Halley's method is a numerical algorithm for solving the nonlinear equation ƒ(x) = 0. In this case, the function ƒ has to be a function of one real variable. The method consists of a sequence of iterations: beginning with an initial guess x0. If ƒ is a thrice continuously differentiable function and a is a zero of ƒ but not of its derivative, then, in a neighborhood of a, the iterates xn satisfy: This means that the iterates converge to the zero if the initial guess is sufficiently close, and that the convergence is cubic. Consider the function Any root of ƒ which is not a root of its derivative is a root of g; and any root of g is a root of ƒ.