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In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f.
The false position method can be faster than the bisection method and will never diverge like the secant method. However, it may fail to converge in some naive implementations due to roundoff errors that may lead to a wrong sign for f(c). Typically, this may occur if the derivative of f is large in the neighborhood of the root.
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.
For instance, if by any chance two of the function values f n−2, f n−1 and f n coincide, the algorithm fails completely. Thus, inverse quadratic interpolation is seldom used as a stand-alone algorithm. The order of this convergence is approximately 1.84 as can be proved by secant method analysis.
Sidi's method reduces to the secant method if we take k = 1. In this case the polynomial p n , 1 ( x ) {\displaystyle p_{n,1}(x)} is the linear approximation of f {\displaystyle f} around α {\displaystyle \alpha } which is used in the n th iteration of the secant method.
This three step procedure guarantees that the minmax properties of the bisection method are enjoyed by the estimate as well as the superlinear convergence of the secant method. And, is observed to outperform both bisection and interpolation based methods under smooth and non-smooth functions. [15]
The Davidon–Fletcher–Powell formula (or DFP; named after William C. Davidon, Roger Fletcher, and Michael J. D. Powell) finds the solution to the secant equation that is closest to the current estimate and satisfies the curvature condition. It was the first quasi-Newton method to generalize the secant method to a
In numerical analysis, the ITP method (Interpolate Truncate and Project method) is the first root-finding algorithm that achieves the superlinear convergence of the secant method [1] while retaining the optimal [2] worst-case performance of the bisection method. [3]