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May 19th, 2017, 04:27 AM   #1
Joined: May 2017
From: UAE

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Lightbulb Finding roots of multivariable equation

I have a set of two equations each of two variables x and y :
f1(x,y) =a1+b1x+c1x^2+...+x^7+a2+b2y+c2y^2+...+y^7+a3xy+b3 x^2y^2+...+x^7y^9.

f2(x,y) =a4+b4x+c4x^2+...+x^7+a5+b5y+c5y^2+...+y^7+a6xy+b6 x^2y^2+...+x^8y^10.

These equations are the partial derivatives of one original equation called Performance Index (J)

I need to find the roots of both equations that equate both of them to zero simultaneously so the solution should be set of pairs points aiming to identify the local minimum and maximum points.

I have managed to find the roots by numeric analysis through Newton Raphson technique with many iterations.

The questions remain:

1) I have managed to find so far four roots that make both equations equal to zero simultaneously, since I have to find the roots one by one through Newton Raphson, when I have to stop saying that these are all possible roots.
3) How many roots we can estimate as a set of solution for above two equations. Is there any rule that we can identify the number of all possible roots?

Thanks in advance.

Last edited by skipjack; May 19th, 2017 at 08:15 AM.
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