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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 .
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.
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 ...
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.
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 ...
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 ...
Integrals of hyperbolic tangent, cotangent, secant, cosecant functions [ edit ] ∫ tanh x d x = ln cosh x + C {\displaystyle \int \tanh x\,dx=\ln \cosh x+C}