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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 secant method can be thought of as a finite-difference approximation of Newton's method, so it is considered a quasi-Newton method.
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
Linear multistep method — the other main class of methods for initial-value problems Backward differentiation formula — implicit methods of order 2 to 6; especially suitable for stiff equations; Numerov's method — fourth-order method for equations of the form ″ = (,)
The secant method increases the number of correct digits by "only" a factor of roughly 1.6 per step, but one can do twice as many steps of the secant method within a given time. Since the secant method can carry out twice as many steps in the same time as Steffensen's method, [b] in practical use the secant method actually converges faster than ...
Most of the section consists in explaining in full details how substituting variables for numerical values in the previously given formulas, and computing with these numerical values. This is not useful in Wikipedia, since one may suppose that people interested in the method know how to do such elementary operations (otherwise, they would ...
The Davidon–Fletcher–Powell (DFP) method, which is the only member of this class being published before the two methods defined by Broyden. [1]: 582 For the DFP method, =. [8]: 150 Anderson's iterative method, which uses a least squares approach to the Jacobian. [9]
multiderivative methods, which use not only the function f but also its derivatives. This class includes Hermite–Obreschkoff methods and Fehlberg methods, as well as methods like the Parker–Sochacki method [17] or Bychkov–Scherbakov method, which compute the coefficients of the Taylor series of the solution y recursively.
The method is a generalization of the secant method. Like the secant method, it is an iterative method which requires one evaluation of in each iteration and no derivatives of . The method can converge much faster though, with an order which approaches 2 provided that satisfies the regularity conditions described below.