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The secant method can be interpreted as a method in which the derivative is replaced by an approximation and is thus a quasi-Newton method. If we compare Newton's method with the secant method, we see that Newton's method converges faster (order 2 against order the golden ratio φ ≈ 1.6). [2]
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 ...
Brent's method is a combination of the bisection method, the secant method and inverse quadratic interpolation. At every iteration, Brent's method decides which method out of these three is likely to do best, and proceeds by doing a step according to that method. This gives a robust and fast method, which therefore enjoys considerable popularity.
As noted in the introduction, inverse quadratic interpolation is used in Brent's method. Inverse quadratic interpolation is also closely related to some other root-finding methods. Using linear interpolation instead of quadratic interpolation gives the secant method. Interpolating f instead of the inverse of f gives Muller's method.
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]
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.
The implementation of this method in the free software MPSolve is a reference for its efficiency and its accuracy. Another method with this style is the Dandelin–Gräffe method (sometimes also ascribed to Lobachevsky), which uses polynomial transformations to repeatedly and implicitly square the roots. This greatly magnifies variances in the ...
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