Talk:Bairstow's method

Convergence of Bairstow's method

Copied from User talk:Hermitian. -- Jitse Niesen (talk) 15:11, 14 January 2006 (UTC)[reply]

[…] Is the method really quadratically convergent, even in the case of double roots, as the article implies? Do you have a reference for this? I would guess that it is only linearly convergent in this case, just as Newton's method. Cheers, Jitse Niesen (talk) 23:00, 13 January 2006 (UTC)[reply]

Bairstow's method loses its quadratic rate of convergence when trying to find a quadratic factor of multiplicity greater than one, although there are a couple of modifications to it which fix this. I do not believe it loses its quadratic convergence to a single quadratic factor based on whether that quadratic factor has distinct or identical roots, as it is searching for the quadratic coefficients, and not for the roots directly. I've never seen a reference which stated the latter specifically, but there is lots of literature on the quadratic multiplicity difficulty, and one would conjecture that if double roots also caused a problem, they would be mentioned in that.Hermitian 02:44, 14 January 2006 (UTC)[reply]

I was refering to the case of a quadratic factor of multiplicity greater than one. The article says

"Like Newton's Method, Bairstow's Algorithm will converge quadratically providing the initial guess is close enough to the zero. The algorithm can be quite slow to converge to quadratic factors of multiplicity higher than 1."

I read this as saying that the algorithm will also converge quadratically, albeit slowly, in the case of quadratic factors of multiplicity higher than one. What do you think about reformulating this as

"Bairstow's algorithm inherits the quadratic convergence of Newton's method, except in the case of quadratic factors of multiplicity higher than 1, when convergence can be rather slow.."

or something similar. […] -- Jitse Niesen (talk) 03:47, 14 January 2006 (UTC)[reply]

I intended the comment about multiplicity to be an exception to the first line about quadratic convergence. You are correct that this is less than clear. Your suggested phrasing is fine. […] Hermitian 05:05, 14 January 2006 (UTC)[reply]

Thank you for your answer. I changed Bairstow's method as indicated above, and copied this section to Talk:Bairstow's method for future reference. I hope that you continue to contribute to Wikipedia; you clearly know this stuff and it is essential for Wikipedia to attract more experts. Cheers, Jitse Niesen (talk) 15:11, 14 January 2006 (UTC)[reply]

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Example

In the section "Example", it says 'the method produced a quadratic factor that contains the roots -1/3 and -3'. I have checked that this method does indeed produce these roots for this example. It's not obvious from the table that these are the roots. — Preceding unsigned comment added by 82.0.88.79 (talk) 17:19, 7 November 2011 (UTC)[reply]