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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.
A Taylor series of f about point a may diverge, converge at only the point a, converge for all x such that | | < (the largest such R for which convergence is guaranteed is called the radius of convergence), or converge on the entire real line. Even a converging Taylor series may converge to a value different from the value of the function at ...
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
Iterative method; Rate of convergence — the speed at which a convergent sequence approaches its limit Order of accuracy — rate at which numerical solution of differential equation converges to exact solution; Series acceleration — methods to accelerate the speed of convergence of a series
The examples given in that book for the extended definition involve using the Taylor series and reduce the problem to one of finding the first non-zero term in the Taylor series after the limit. For example, one exercise from Chapter 1 of the Burden and Faires book asked us to find the rate of convergence for
Convergence proof techniques are canonical patterns of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity.. There are many types of sequences and modes of convergence, and different proof techniques may be more appropriate than others for proving each type of convergence of each type of sequence.
A comparison of the convergence of gradient descent with optimal step size (in green) and conjugate vector (in red) for minimizing a quadratic function associated with a given linear system. Conjugate gradient, assuming exact arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2).
However, the convergence of such a scheme is not guaranteed in general; moreover, the rate of convergence is usually linear, which can become too slow if the evaluation of the function is computationally expensive. [2] Anderson acceleration is a method to accelerate the convergence of the fixed-point sequence. [2]