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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.
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.
English: Example of finding the root of the function represented by the red line (cos(x) − x 3) using the secant method, with the increasingly accurate approximations represented by the blue straight lines.
Quasi-Newton methods are a generalization of the secant method to find the root of the first derivative for multidimensional problems. In multiple dimensions the secant equation is under-determined, and quasi-Newton methods differ in how they constrain the solution, typically by adding a simple low-rank update to the current estimate of the ...
The idea to combine the bisection method with the secant method goes back to Dekker (1969).. Suppose that we want to solve the equation f(x) = 0.As with the bisection method, we need to initialize Dekker's method with two points, say a 0 and b 0, such that f(a 0) and f(b 0) have opposite signs.
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 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
Limited-memory BFGS method — truncated, matrix-free variant of BFGS method suitable for large problems; Steffensen's method — uses divided differences instead of the derivative; Secant method — based on linear interpolation at last two iterates; False position method — secant method with ideas from the bisection method