I have got a product of variables which equals a constant :
$x_1x_2...x_n=k$
I have initial values for ${x_1,..., x_n}$ and therefore $k$. My question is, how could I determine the new values of ${x_1,..., x_n}$ when $k$ changes by a some amount? e.g. $k + \Delta k$, given that I know any ${x_i}$ is constrained to values between $0.01$ and $10$.
For example, say I have $n=6$, such that $x_1=0.5$, $x_2=1.2$, $x_3=1.5$, $x_4=3$, $x_5=0.3$, $x_6=7$, then
$x_1x_2...x_6= 0.5 \cdot 1.2 \cdot 1.5 \cdot 3 \cdot 0.3 \cdot 7 = 5.67 $
I would like to know by how much the ${x_1,..., x_6}$ need to change in order to equate to $5.67 + 0.5 = 6.17$ (i.e. $k=5.67$ and $\Delta k = 0.5$)
Any ideas on how I can go about solving this problem? I've been cracking my head for a few days without knowing what could work. Thanks!
Since the constraint is a product form, it might help to think of the changes also as fractional (multiplicative) changes as opposed to additive changes. I.e.
$k \rightarrow k' = a' k$
$x_i \rightarrow x'_i = a'_i x_i$
Then $x'_1 x'_2 ... x'_n = k' \iff a'_1 a'_2 ... a'_n = a'$ and you can see you have lots of (in fact $n-1$ degrees of) freedom in the choices of $a'_i$.
In the special case that the changes are "small", we have $a' = (1 + \delta')$ and $a'_1 = (1+ \delta'_i)$ where all the deltas are small (i.e. $\ll 1$). In that case an approximate solution is $\delta'_1 + \delta'_2 + ... + \delta'_n = \delta'$, if we ignore the higher order terms (products of deltas). This may be more intuitive when rephrased as something like: a $2$% increase in $k$ can be made from $10$ separate $0.2$% increases in the $x$'s (or $5$ increases of $0.5$% plus $2$ decreases of $0.25$%, etc). This is not exact so if you want exact equality you need to adjust a little bit.