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]