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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.
The following is an example of a possible implementation of Newton's method in the Python (version 3.x) programming language for finding a root of a function f which has derivative f_prime. The initial guess will be x 0 = 1 and the function will be f(x) = x 2 − 2 so that f ′ (x) = 2x. Each new iteration of Newton's method will be denoted by x1.
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 ...
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:
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.
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 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 ...