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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 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
The Symmetric Rank 1 (SR1) method is a quasi-Newton method to update the second derivative (Hessian) based on the derivatives (gradients) calculated at two points. It is a generalization to the secant method for a multidimensional problem.
Created with Gnuplot using the following input file: set terminal svg font "Bitstream Vera Sans,12" set output "Secant_method_example_code_result.svg" set xrange [0:1] set yrange [-0.5:3] set xzeroaxis linetype -1 set yzeroaxis linetype -1 set xtics axis nomirror offset 0,0.3 set ytics axis nomirror offset -0.8 set key off set border 0 plot -1.4596976941*x + 1.0000000000 with line lt rgbcolor ...
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 ...
Many methods exist to accelerate the convergence of a given sequence, i.e., to transform one sequence into a second sequence that converges more quickly to the same limit. Such techniques are in general known as "series acceleration" methods. These may reduce the computational costs of approximating the limits of the original sequences.
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.