Talk:Bernoulli's method

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Mathematics

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Please clarify: How a sequence of real numbers can have a complex root as a limit?

@D.Lazard This is a question that deserves a longer answer. One thing to note is that the coefficients of the polynomial don't need to be real. The methods works with either case so it could find a dominant complex root (see Henrici p155). If everything is real, then finding dominant complex conjugates requires extra steps after computing the sequence. McNamee & Pan in chapter 10.2 describe these steps (they cite Jennings too) and Henrici also explains it in chapter 7.5 and it a little involved. The short version requires setting up a system of equations to solve for the angle and radius by essentially using four consecutive terms in the sequence to compute two determinants and pluging them into three equations.

I'm not sure if adding these would really benefit the article because if multiple roots are desired, then generalized approaches like QD or Aitken could compute these. Is there something that could be said to clarify things but without prompting more questions? Or would it be better to describe how to find conjugate pairs? Basilelp (talk) 04:25, 24 April 2025 (UTC)[reply]

Writing a good Wikipedia article requires to respect the WP:principle of least astonishment. The tag {{clarification needed}} means simply that the principle is not respected, since everybody may ask the same question. Here are several points, where the article deserves to be improved. It is important to say what is known for the 3 last items, since they rely on questions that evey competent reader cac ask himself.

Readers must know what is an absolute value, and not using this term may confuse readers. So, "magnitude" must be replaced everywhere with "absolute value, as I did in the lead.
It must be explicitely said that, if the coefficients are real and the root of largest absolute values is not real, then the algorithm does not work since the conjugate root has the same absolute value.
If there are several roots of maximal absolute value, does the sequence has a limit (never, sometimes of always)? If it has a limit, how is it related to the roots (in general or when the maximal roots are complex conjugate)?
Same questions in the case of multiple roots.

By the way, I'll add a sentence explaining why the given initial conditions provide nonzero c_i's. D.Lazard (talk) 14:09, 24 April 2025 (UTC)[reply]