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Given a real valued trigonometric polynomial,

$$ f(x) = \sum_{k=0}^{n} a_k \cos(k x + \phi_k) $$

what is the current fastest numerical method to find the roots of the polynomial for a given error? I have just come across the Durand-Kerner method, are there any others?

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I added the numerical-methods tag; it's the most important one here. Also, there is scicomp.SE site, and it really needs more questions to graduate... – user53153 Jan 15 '13 at 23:27
Thanks, shall I repost it there or perhaps get a mod to move it? – geometrikal Jan 16 '13 at 1:59
Up to you, of course... you did get an answer here after I bumped and retagged the question. You can always request migration by flagging your question. – user53153 Jan 16 '13 at 2:28
Have put a new question over at scicomp, with some changes. Thanks for the bump! – geometrikal Jan 16 '13 at 2:44
up vote 2 down vote accepted

For finding the roots of an algebraic polynomial, I swear by the Jenkins-Traub algorithm, which I've found in practice to be both faster and more robust than alternatives. The only downside is that Jenkins-Traub is rather complicated if you must implement it yourself. Codes are available online, for instance I've used this C++ implementation:

I would therefore use the standard substitution $y=e^{ix}$ to turn the trigonometric polynomial into an algebraic polynomial of degree $2n$, and then apply Jenkins-Traub.

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Thanks, that was just the start I needed. I didn't realise there had been so much work done on this. – geometrikal Jan 16 '13 at 2:08

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