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Finding one root. The most widely used method for computing a root is Newton's method, which consists of the iterations of the computation of. + = ′ {\displaystyle x_ {n+1}=x_ {n}- {\frac {f (x_ {n})} {f' (x_ {n})}},} by starting from a well-chosen value. If f is a polynomial, the computation is faster when using Horner's method or evaluation ...
Root-finding algorithm. In numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function f is a number x such that f(x) = 0. As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form, root-finding algorithms provide ...
Laguerre's method. In numerical analysis, Laguerre's method is a root-finding algorithm tailored to polynomials. In other words, Laguerre's method can be used to numerically solve the equation p(x) = 0 for a given polynomial p(x). One of the most useful properties of this method is that it is, from extensive empirical study, very close to being ...
Brent's method. In numerical analysis, Brent's method is a hybrid root-finding algorithm combining the bisection method, the secant method and inverse quadratic interpolation. It has the reliability of bisection but it can be as quick as some of the less-reliable methods. The algorithm tries to use the potentially fast-converging secant method ...
The Aberth method, or Aberth–Ehrlich method or Ehrlich–Aberth method, named after Oliver Aberth [1] and Louis W. Ehrlich, [2] is a root-finding algorithm developed in 1967 for simultaneous approximation of all the roots of a univariate polynomial. This method converges cubically, an improvement over the Durand–Kerner method, another ...
Muller's method is a recursive method that generates a new approximation of a root ξ of f at each iteration using the three prior iterations. Starting with three initial values x 0, x −1 and x −2, the first iteration calculates an approximation x 1 using those three, the second iteration calculates an approximation x 2 using x 1, x 0 and x −1, the third iteration calculates an ...
Berlekamp–Rabin algorithm. In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials over the field with elements. The method was discovered by Elwyn Berlekamp in 1970 [1] as an auxiliary to the algorithm for polynomial factorization over ...
Jenkins–Traub algorithm. The Jenkins–Traub algorithm for polynomial zeros is a fast globally convergent iterative polynomial root-finding method published in 1970 by Michael A. Jenkins and Joseph F. Traub. They gave two variants, one for general polynomials with complex coefficients, commonly known as the "CPOLY" algorithm, and a more ...