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.
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.