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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 .
Muller's method is a root-finding algorithm, a numerical method for solving equations of the form f(x) = 0.It was first presented by David E. Muller in 1956.. Muller's method proceeds according to a third-order recurrence relation similar to the second-order recurrence relation of the secant method.
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.
Broyden suggested using the most recent estimate of the Jacobian matrix, J n−1, and then improving upon it by requiring that the new form is a solution to the most recent secant equation, and that there is minimal modification to J n−1:
If the result of the secant method, s, lies strictly between b k and m, then it becomes the next iterate (b k+1 = s), otherwise the midpoint is used (b k+1 = m). Then, the value of the new contrapoint is chosen such that f(a k+1) and f(b k+1) have opposite signs. If f(a k) and f(b k+1) have opposite signs, then the contrapoint remains the same ...
Download QR code; Print/export Download as PDF; Printable version; ... It may be useful to some readers to see the secant method applied in a numerical example. An ...
The simplest form of the formula for Steffensen's method occurs when it is used to find a zero of a real function; that is, to find the real value that satisfies () =.Near the solution , the derivative of the function, ′, is supposed to approximately satisfy < ′ <; this condition ensures that is an adequate correction-function for , for finding its own solution, although it is not required ...
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.