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Toggle Asymptotic rates of convergence for iterative methods subsection. 1.1 Definitions. 1.1.1 R-convergence. 1.2 Examples. ... Exponential response formula;
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.
This convergence result is widely applied to prove the convergence of other series as well, whenever those series's terms can be bounded from above by a suitable geometric series; that proof strategy is the basis for the ratio test and root test for the convergence of infinite series. [11]
In numerical analysis, order of accuracy quantifies the rate of convergence of a numerical approximation of a differential equation to the exact solution. Consider , the exact solution to a differential equation in an appropriate normed space (, | | | |).
Additionally, the general formula can be used to estimate (leading order step size behavior of Truncation error) when neither its value nor is known a priori. Such a technique can be useful for quantifying an unknown rate of convergence.
Two classical techniques for series acceleration are Euler's transformation of series [1] and Kummer's transformation of series. [2] A variety of much more rapidly convergent and special-case tools have been developed in the 20th century, including Richardson extrapolation, introduced by Lewis Fry Richardson in the early 20th century but also known and used by Katahiro Takebe in 1722; the ...
The same formula for ... and so the rate of convergence of CGNR may be slow and the quality of the approximate solution may be sensitive to roundoff errors.
Many of these functions can be used to find their own solutions by repeatedly recycling the result back as input, but the rate of convergence can be slow, or the function can fail to converge at all, depending on the individual function. Steffensen's method accelerates this convergence, to make it quadratic.