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In asymptotic analysis in general, one sequence () that converges to a limit is said to asymptotically converge to with a faster order of convergence than another sequence () that converges to in a shared metric space with distance metric | |, such as the real numbers or complex numbers with the ordinary absolute difference metrics, if
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.
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 convergence rate of the bisection method could possibly be improved by using a different solution estimate. The regula falsi method calculates the new solution estimate as the x-intercept of the line segment joining the endpoints of the function on the current bracketing interval. Essentially, the root is being approximated by replacing the ...
The rate of convergence is distinguished from the number of iterations required to reach a given accuracy. For example, the function f(x) = x 20 − 1 has a root at 1. Since f ′(1) ≠ 0 and f is smooth, it is known that any Newton iteration convergent to 1 will converge quadratically. However, if initialized at 0.5, the first few iterates of ...
In this example, Aitken's method is applied to a sublinearly converging series and accelerates convergence considerably. The convergence is still sublinear, but much faster than the original convergence: the first A [ X ] {\textstyle A[X]} value, whose computation required the first three X {\textstyle X} values, is closer to the limit than the ...
Halley's method is a numerical algorithm for solving the nonlinear equation f(x) = 0.In this case, the function f has to be a function of one real variable. The method consists of a sequence of iterations:
This is generally true but not for this particular article: there are two graphical examples. The first image explains clearly how the method works. I agree that the second one is unclear, but explaining better the image and/or replacing it by a cleared one would be much more useful than adding the disputed section.