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While eagerly searching for iterative methods to approximate $f(x)=0$ (commonly known as root-finder) for the quantile function or probit, I stumbled over "Ostrowski’s square root technique". Alas, it was mentioned only, given a formula: $$x_{n+1}=x_n-\frac{f}{f'}\sqrt{\frac{m}{\displaystyle 1-\frac{f\cdot f''}{(f')^2}}}$$ where $f$ stands for $f(x_n)$ and homological regarding $f'$ and $f''$. About $m$ the hint "... $m$ is the modification when the multiplicity $m$ is $\gt 1$", but no description, no discussion.

Apparently all publications of Ostrowski or papers related to his work are guarded behind pay-walls. That is why I ask for public available descriptions of "Ostrowski’s square root technique", its convergence behaviour, the use of $m$ and/or its relation to $f$.

BTW, I miss the tag root-finder, what is the canonical correct qualifier for those algorithms?

m-stgt
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  • There are many "enhanced (or accelerated) Newton's method(s)" of the kind you show : usually they are established by manipulating Taylor expansions like for this one : https://math.stackexchange.com/a/2624492/305862 – Jean Marie Jul 16 '24 at 21:16
  • Here is a paper explaining Ostrowski's method : https://www.davidpublisher.com/Public/uploads/Contribute/64213422ecc27.pdf – Jean Marie Jul 16 '24 at 21:23
  • @JeanMarie -- TY for the links. There are plenty other root-finding methods, it's a vast field of iterations. The paper explaining Ostrowski's method explains the "Classic Ostrowski’s Method" (a multistep approach), some "Derivative-Free Ostrowski’s Method", and "Ostrowski’s Method with Memory" -- alas, "Ostrowski’s square root technique" is missing. – m-stgt Jul 16 '24 at 23:12
  • @JeanMarie -- Encore une fois moi: just saw you are retired professor, so maybe it's you who could answer my query (open for quite a while now). What root finding method is this which I found by reverse engineering of an ancient pocket calculator. With all the papers I browsed regarding root-finders I'm convinced now, it's not of higher order. My question here about "Ostrowski’s square root technique" is rather for curiosity. Merci pour votre temps. – m-stgt Jul 16 '24 at 23:34

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