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I am trying to calculate the solution(s) to these equations 1, the k0,k1,k2,k3 are the unknowns, everything else is known. The problem is that I cannot find a point of intersection of all equations so I split them into combinations of each equation with another i.e.: equation1 = equation2, equation1 = equation3, etc.

The only source that could give me real solutions was Wolfram alpha. My question is, with which numerical method can I calculate the aproximate values of my unknowns? I have noticed that I have 2 equations and 3 unknowns, but Wolfram can still calculate aproximate values.

Equations

Paulius Gagelas
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1 Answers1

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When you have $n>4$ equations of the type $$f_i(k_0,k_1,k_2,k_3)=0$$ you could minimize $$\Phi=\frac 12\sum_{i=1}^n \left[f_i(k_0,k_1,k_2,k_3)\right]^2$$which means solving the four equations $$\frac{\partial\Phi}{\partial k_j}=\sum_{i=1}^n f_i(k_0,k_1,k_2,k_3)\times\frac{\partial f_i(k_0,k_1,k_2,k_3)}{\partial k_j}=0\qquad \text{with}\qquad j=1,2,3,4$$ Now, you face a square system.

Claude Leibovici
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  • Could you please provide me a source where I could see a similar example, I am trying to get my head around the RHS of the phi equation (not sure where this comes from). – Paulius Gagelas Nov 12 '18 at 10:44
  • @PauliusGagelas. $\Phi$ is the norm of the system. If you think about it, this is very "similar" to nonlinear least-square fit. – Claude Leibovici Nov 12 '18 at 10:52
  • @PauliusGagelas. Have a look at https://math.stackexchange.com/questions/46036/solving-an-overdetermined-system-of-nonlinear-equations – Claude Leibovici Nov 12 '18 at 10:54
  • Thank you very much for your information. I still wanted to ask, what is the × in the last equation? Is it a cross multiplication? Also, will I be able to use the Newton-Raphson method with the equations? (Or is there a better way?) – Paulius Gagelas Nov 12 '18 at 14:35
  • @PauliusGagelas. It is just multiplication ! – Claude Leibovici Nov 12 '18 at 14:36