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The staggered geometric progression () =,,,,, …, / ⌊ ⌋, …, using the floor function ⌊ ⌋ that gives the largest integer that is less than or equal to , converges R-linearly to 0 with rate 1/2, but it does not converge Q-linearly; see the second plot of the figure below. The defining Q-linear convergence limits do not exist for this ...
We then use this new value of x as x 2 and repeat the process, using x 1 and x 2 instead of x 0 and x 1. We continue this process, solving for x 3 , x 4 , etc., until we reach a sufficiently high level of precision (a sufficiently small difference between x n and x n −1 ):
The following iterates are 1.0103, 1.00093, 1.0000082, and 1.00000000065, illustrating quadratic convergence. This highlights that quadratic convergence of a Newton iteration does not mean that only few iterates are required; this only applies once the sequence of iterates is sufficiently close to the root.
This can be compared with approximately 1.618, exactly the golden ratio, for the secant method and with exactly 2 for Newton's method. So, the secant method makes less progress per iteration than Muller's method and Newton's method makes more progress. More precisely, if ξ denotes a single root of f (so f(ξ) = 0 and f'(ξ) ≠ 0), f is three ...
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]
One can also show that if a sequence converges to its limit at a rate strictly greater than 1, [] does not have a better rate of convergence. (In practice, one rarely has e.g. quadratic convergence which would mean over 30 (respectively 100) correct decimal places after 5 (respectively 7) iterations (starting with 1 correct digit); usually no ...
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 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 :