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Does Monotone (linear) convergence of iterates imply monotone (linear) convergence of function values?
I am considering a proof that would require a certain connection between convergence of iterates and corresponding function values: Consider an algorithm with iterates $\left\{{\mathbf{x}}^k\right\}_{...
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Can Carlsons's iterative algorithm for $\arctan x$ be inverted to get one for $\tan x$?
In the article An algorithm for computing logarithms and arctangents, by B. C. Carlson, the following iterative algorithm for arctangents is given:
The algorithm uses that $2^n\tan(2^{-n}\arctan(x))=\...
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