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April 7th, 2009, 01:16 AM   #1
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Newton's method for multivariate polynomial equation systems

I'm looking to apply Newton's method to solve a system of polynomial equations, where all polynomials are of degree two.
Since for a single polynomial equation of degree 2 Newton's method always converges (except for singular points), I was wondering whether a similar theorem holds for the multivariate case. Anyone knows?
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April 7th, 2009, 06:05 AM   #2
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Re: Newton's method for multivariate polynomial equation systems

I would expect so, since polynomials are well-behaved (in particular they have second derivatives). But I haven't seen a proof or even a claim of that.
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April 7th, 2009, 04:34 PM   #3
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Re: Newton's method for multivariate polynomial equation systems

Actually, after thinking about it some more, it shouldn't work in general for degree two - what you need should be some convex-property I reckon.
Reason why degree 2 doesn't mean much: you can emulate any polynomial by a set of polynomial equations of degree 2, e.g.
x^3 - x^2 + 1 = 0 (Newton can fail here I believe)
is equivalent to
x^2 - y = 0
xy - y + 1 = 0
Any ideas about theorems involving convex functions?
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