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In practical numerical computations, asymptotic rates and orders of convergence follow two common conventions for two types of sequences: the first for sequences of iterations of an iterative numerical method and the second for sequences of successively more accurate numerical discretizations of a target.
Broyden's method is a generalization of the secant method to more than one dimension. The following graph shows the function f in red and the last secant line in bold blue. In the graph, the x intercept of the secant line seems to be a good approximation of the root of f.
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 ...
The factor 1 / 2 used above looks arbitrary, but it guarantees superlinear convergence (asymptotically, the algorithm will perform two regular steps after any modified step, and has order of convergence 1.442). There are other ways to pick the rescaling which give even better superlinear convergence rates. [11]
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 ...
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 above equation is underdetermined when k is greater than one. 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 :
The rate of convergence of the Gauss–Newton algorithm can approach quadratic. [7] The algorithm may converge slowly or not at all if the initial guess is far from the minimum or the matrix J r T J r {\displaystyle \mathbf {J_{r}^{\operatorname {T} }J_{r}} } is ill-conditioned .