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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
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 Steffensen's method, when both algorithms succeed: The secant method achieves a factor of about (1.6) 2 ≈ 2.6 times as many digits for every two steps (two function ...
In numerical analysis, Aitken's delta-squared process or Aitken extrapolation is a series acceleration method used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926. [1] It is most useful for accelerating the convergence of a sequence that is converging linearly.
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 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.
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 ...
If an equation can be put into the form f(x) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x 1 in the basin of attraction of x, and let x n+1 = f(x n) for n ≥ 1, and the sequence {x n} n ≥ 1 will converge to the solution x.
Is there a fixed order of convergence for repeated roots with the secant method? For instance, with the Newton-Raphson method, R=2 (quadratic) for simple roots and R=1 for repeated roots. For the Secant Method, R=1.618.... for simple roots, but what about repeated/complex roots? Computer Guru 21:40, 26 May 2008 (UTC)