Is there a fast method to calculate $1/(x+z)$ where $z$ is a root of unity and $x$ is real.
By fast computation, I mean is there a faster method than Newton-Rhaphson method.
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Why isn't this simple algebra? $$\frac{1}{x - (\cos (2 \pi/n) + i \sin (2 \pi/n))}$$ for $n \in \mathbb{N}^+$ – David G. Stork Jan 20 '21 at 05:03
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It is simple algebra of a specific form (The imaginary part). I am interested in knowing if this specific form can be calculated faster than an arbitrary complex inversion. – Bhavesh Lakhotia Jan 20 '21 at 05:11
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What does speed have to do with anything? This takes $0.000077$ seconds on a Mac laptop. – David G. Stork Jan 20 '21 at 05:17
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I am looking to calculate such forms efficiently for a more broader problem. The computation time on your machine reflects a certain degree of precision. Say accurate upto 20 digits. The requirement could be higher than this. That’s why I am interested in speed. – Bhavesh Lakhotia Jan 20 '21 at 05:31
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1These details should go to the text of the question. As it reads out currently, it brings little to no clarity. – metamorphy Jan 20 '21 at 06:11