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I am attempting to solve five variables from a system of equations. Let the variables be $x_1,x_2,x_3,x_4,x_5$. Let the problem have the form:

$\exp(x_1)+x_1^6=x_3+x_4+x_5 \tag{1}$ $\exp(x_2)+x_2^6=x_3+x_4+x_5 \tag{2}$ $f_1(x_1,x_2,x_3,x_4,x_5)=0 \tag{3}$ $f_2(x_1,x_2,x_3,x_4,x_5)=0 \tag{4}$ $f_3(x_1,x_2,x_3,x_4,x_5)=0 \tag{5}$

Here $f_1,f_2,f_3$ are some generally complicated functions. I am looking for the most efficient way to solve this problem.

One can see that I can solve for the first two variables simply from the first two equations and represent them as $x_1=g_1(x_3,x_4,x_5)$ and $x_2=g_2(x_3,x_4,x_5)$. If $g_1$ and $g_2$ were analytic, I could simply plug the first two variables to the last three equations and solve for a much simpler three equation problem, which would typically improve both speed and accuracy. However, the issue is that here $g_1$ and $g_2$ cannot be derived analytically and hence the relationship is implicit.

I have tried a nested solution approach, where for each guess value for the last three variables I numerically solve for the first two from the first two equations. However, this does not appear to improve speed much over simply giving a standard solver a full problem with 5 equations and unknowns.

Could someone suggest an alternative solution approach?

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  • $\begingroup$ What can you do with $f$? Only evalute it? $\endgroup$ – user3680510 Feb 17 at 13:58
  • $\begingroup$ @user3680510 $f$ is essentially a known but complicated polynomial expression, it could be something like $f_1(x_1,x_2,x_3,x_4,x_5)=x_1x_2x_3+exp(x_1)+x_2+x_3x_4x_5$. I think specifying it is not necessary to answer the question as this needs to be feeded to a numerical solver anyway. $\endgroup$ – fesman Feb 17 at 14:39
  • $\begingroup$ With a good nonlinear programming solver, calling $g_1(x_1,x_2,x_3,x_4)=\exp(x_1)+x_1^6=x_3+x_4+x_5$ and $g_2(x_1,x_2,x_3,x_4) = \exp(x_2)+x_2^6=x_3+x_4+x_5$ we can proceed as $$ \min_X (\|g_1\|+\|g_2\|+\|f_1\|+\|f_2\|+\|f_3\|) $$ choosing a differential evolution approach. $\endgroup$ – Cesareo Feb 23 at 16:54

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