# Thread: Complex equation to solve

1. ## Complex equation to solve

Hi,

I'm a network engineer working on a quite complex research about routing protocols. I managed to model my problem and I obtained an equation that I'm unfortunately unable to solve. I'm attaching an image below (trying to use Latex code resulted in a too big image according to edit rules) and the Latex code too.

<LateX Code>
\displaystyle\Biggl[\displaystyle\biggl[\frac{[1-k^{i+1}]^2-[1-k^{i}]^2}{[1-k^{N+1}]^2}\biggr] \cdot \displaystyle\biggl[\frac{P_0\tau}{1-k^{P_L}}(-k^{P_L(i+1)}ln(i+1)+E_{RX})\biggr]\Biggr]=\\
\displaystyle\Biggl[\displaystyle\biggl[P_0\tau \frac{1-(k^{P_L})^{i+1}}{1-k^{P_L}}+(i-1)E_{RX}\biggr] \cdot \displaystyle\biggl[ \frac{2[(1-k^{i+1})(k^{i+1}ln(i+1))-(1-k^i)(k^iln(i))]}{(1-k^{N+1})^2}\biggr]\Biggr]
</LateX Code>

I should solve it in k. I know it has at least a solution for k>1 (I used a graphical tool..) but I'm unable to find a closed expression of it even if I've managed in some way to simplify it (removing denominator and working on ln). I'm providing the full expression to let you start fresh.

If any of you can give a clue on how to get there I'd really appreciate that. I need to find an expression of k that holds for k>1 (with constraints if needed).

Have fun :-),

Lu

2. Without some clue of the values of the various parameters, it is hard to say how one might proceed.

There is substantial redundancy in the formulation as written. With a few caveats, it can be significantly simplified.

$P_{0} \ne 0$

$k^{P_{L}} \ne 1$

$k^{N+1} \ne 1$

After that, and a little factoring, these factors can be removed from most of it:

$\frac{P_{0}\tau}{1-k^{P_{L}}}$

$(1-K^{N+1})^{2}$

3. I already managed to eliminate those factors. The form above can be simplified in the following one:

<Latex Code>
\displaystyle\Biggl[\biggl((1-k^{i+1})^2-(1-k^{i})^2)\biggr)\cdot\biggl(-k^{P_L(i+1)}ln(i+1)+E_{RX}\biggr)\Biggr]=\\
\displaystyle\Biggl[\biggl((1-(k^{P_L})^{i+1})+(1-k^{P_L})(i-1)\frac{E_{RX}}{P_0\tau}\biggr)\cdot\biggl((1-k^{i+1})(k^{i+1}ln(i+1))-(1-k^i)(k^iln(i)\biggl)\Biggr]
</Latex Code>

$
E_{RX} > 0
$

$
2 < PL < 5
$

$
P_0 > 0
$

$
i > 0
$

I tried to simplify the logaritms but It brought me only to other complications. I can accept also a reasonable approximation rather than a full solution (if it is too complex..).