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The secant method does not require or guarantee that the root remains bracketed by sequential iterates, like the bisection method does, and hence it does not always converge. The false position method (or regula falsi) uses the same formula as the secant method.
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 ...
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
Exponential response formula; ... For example, the secant method, when converging to a regular, simple root, has an order of the golden ratio φ ≈ 1.618. [6]
The other method, namely the secant method, uses the compliance-crack length equation given by ASTM standard to calculate effective crack length from an effective compliance. Compliance at any point in Load vs displacement curve is essentially the reciprocal of the slope of the curve that ensues if the specimen is unloaded at that point.
The above formula is also used in the secant method, but the secant method always retains the last two computed points, and so, while it is slightly faster, it does ...
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.