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How can methods like Newton-Raphson or quasi Newton methods calculate multiple roots if it is given a set of equations.
I mean for every variable there is an equation but every variable has multiple roots.
Is it by bifurcation and homotopy?

Edit: For a set of equations like $( x^2+y=0 ; y^5-x=0 )$. There are multiple roots for $x$ and multiple roots for $y$. How can methods like Newton Raphson get all these roots. I mean there are only two equations with two unknowns so how can it get multiple roots like 10 roots in this case. Does it use homotopy and bifurcation?

Gerry Myerson
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Salma El-Sokkary
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    It might clarify your question if you gave an example. – Gerry Myerson Aug 28 '17 at 09:41
  • For a set of equations like( x^2+y=0 ; y^5-x=0 )There are multiple roots for X and multiple roots for y.. how methods like Newton raphson get all these roots.. I mean there are only two equations with two unknowns so how can it get multiple roots like 7 roots in this case.. does it use homotopy and bifurcation? – Salma El-Sokkary Aug 28 '17 at 09:43
  • The example should be edited into the body of the question, please. Readers should not need to consult the comments to understand the question. – Gerry Myerson Aug 28 '17 at 09:44
  • Ok; I have added it, Thank you – Salma El-Sokkary Aug 28 '17 at 09:48
  • Yes, edited..How can these roots be found? Especially for software packages that depends on Newton methods, and quasi Newton methods,, does it uses bifurcation and homotopy or these are only used for graphical solutions? And there is another way that it is done by in Newton methods? – Salma El-Sokkary Aug 28 '17 at 09:56
  • Here's an earlier question that might be useful: https://math.stackexchange.com/questions/466809/solving-a-set-of-equations-with-newton-raphson Here's a paper where homotopy is given as an alternative to Newton-Raphson: https://link.springer.com/article/10.1007/s00190-009-0346-x Here's a paper that mentions both homotopy methods and Newton-Raphson: http://waset.org/publications/1301/automatic-iterative-methods-for-the-multivariate-solution-of-nonlinear-algebraic-equations. – Gerry Myerson Aug 29 '17 at 05:44
  • Here's a paper entitled, Newton based homotopy optimization method for solving global optimization problem: http://eprints.utm.my/33234/ and here's one called Solving the nonlinear equations by the Newton-homotopy continuation method with adjustable auxiliary homotopy function: http://dl.acm.org/citation.cfm?id=2615030 If you type $$\rm newton\ raphson\ multivariate\ homotopy$$ into Google, I expect you'll get even more. – Gerry Myerson Aug 29 '17 at 05:50

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