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I've gotten into a problem I haven't really worked with before in my numerics classes.

I have a system of four nonlinear equations with six parameters.

Newtons method, Boydens method etc. all include the inverse of the jacobian, but if the system is underdetermined this is not defined as far as I understand.

My only other generic idea is to reduce the number of parameters and then solve a system of four nonlinear equations with four parameters and iterate the last two, but this seems to get out of hand as well.

Is there any straightforward way to solve this kind of problems, or am I right in being a bit stuck?

1233023
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  • Are the equations polynomial? Then there are methods to solve non-linear systems of polynomial equations, see the book of B. Sturmfels. But it is quite different from systems of linear equations. Could you write the system here with variables $x_1,\ldots ,x_6$? – Dietrich Burde May 11 '18 at 13:39
  • The equations are trigonometric, so expanding them as polynomials would be possible – 1233023 May 11 '18 at 13:43
  • I'm doing my thesis, so I'm not sure if I should get such direct help with solving a specific problem. – 1233023 May 11 '18 at 13:51
  • Use a pseudo-inverse of the Jacobian, this should move you in the direction of admissible points, if there is a piece of it close-by to the initial point. It sometimes helps to replace an angle by the point on the unit circle and translate expressions in that angle via trigonometric identities. – Lutz Lehmann May 23 '18 at 07:14

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Just consider two of the variables as free parameters and solve for the remaining four.

Unless there are other constraints, you get a double infinity of solutions.