# Approximate a solution of a system of non linear equations

I have a system of non-linear equations of the form

$$A x_1^B \exp \bigg(\frac{- C}{x_1} \bigg) = k_1$$ $$A x_2^B \exp \bigg(\frac{- C}{x_2} \bigg) = k_2$$ $$A x_3^B \exp \bigg(\frac{- C}{x_3} \bigg) = k_3$$

where [x1, x2, x3] and [k1, k2, k3] are known. The couple of constants [A, B, C] is the unknown. The solution of this non-linear system of equations is given here: Solving a system of non linear equations

We must now ensure that B < 0 at all times. How would you find one couple [A', B', C'] that best approach the solution of the system, with B' < 0.

Thanks a lot,

• I highly suggest you edit this to use TeX/MathJax, or people will probably ignore. – The Count Jul 5 '16 at 13:35

If you take logs, you get linear equations. The first becomes $$\log A +B \log x_1-\frac C{x_1}=\log k_1$$ Your variables are now $\log A, B,$ and $C$ and you should have a unique solution. Whether $B \lt 0$ depends on the constants.
• What do you mean best approximation? Should you leave $A'=A, C'=C,$ and set $B'=-\epsilon?$ That is about the closest point in $A,B,C$ space. You could sum the squares of the errors in the equations (maybe with some weights that suit you) and minimize that subject to $B \lt 0$ – Ross Millikan Jul 5 '16 at 15:12