# Solution of an Equation

I have this equation, and I want to find solution for x.

\begin{align*}&(-2 x+2 α+1/(2 σ^2))\exp[(-(x-α)^2+(x-μ)/(2 σ^2))]+\\&(-2 x+2 β+1/(2σ^2))\exp[(-(x-β)^2+(x-μ)/(2 σ^2))]=0\end{align*}

I have already used Wolfram, but it calculates the solution for $\sigma$, and other solvers say that they "Can not solve for x". Does anyone have an idea? Thank you all, in advance, for your concern.

P.S. Can I say that if this equation is equal to $0$, then only $(-2x+2 α+1/(2 σ^2))=0$ and $(-2 x+2 β+1/(2σ^2))=0$ at the same time, since exp is always $>0$.

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It looks to me that you'd be hard-pressed to find a closed-form solution, given that you have polynomials of different degrees inside and outside the exponential... – J. M. Nov 20 '11 at 15:25
@J.M. What do you think about my idea at (P.S.)? It's completely wrong? – johan paul Nov 20 '11 at 15:33
According to Maple solution for $x$ is composed of Lambert W functions – pedja Nov 20 '11 at 15:34
If you're sure everything is always real, then the P.S. ought to nail it, yes. – Henning Makholm Nov 20 '11 at 15:45
In fact, if we assume $\alpha \lt \beta$, \alpha+1/(4\sigma^2) \lt x \lt \beta+1/(4\sigma^2)(Otherwise just flip the inequalities). This will allow numeric solution to proceed easily. You could check out chapter 9 of apps.nrbook.com/c/index.html or any numerical analysis text. Like the others, I don't think you will find an algebraic solution. – Ross Millikan Nov 20 '11 at 16:33 ## 1 Answer First a trivial algebraic simplification: \begin{align}&{}\qquad (-2 x+2 α+1/(2 σ^2))\exp[(-(x-α)^2+(x-μ)/(2 σ^2))]\\ &{}\quad+(-2 x+2 β+1/(2σ^2))\exp[(-(x-β)^2+(x-μ)/(2 σ^2))] \\ \\ & = (-2 x+2 α+1/(2 σ^2))\exp[-(x-α)^2] \cdot \exp[(x-μ)/(2 σ^2)]\\ &{}\quad+(-2 x+2 β+1/(2σ^2))\exp[-(x-β)^2]\cdot\exp(x-μ)/(2 σ^2)], \end{align} so if this is0$, then you can divide both sides by$\exp[(x-\mu)/(2\sigma^2)]$, since, as a value of the exponential function, that can never be$0$. You have $$\left(-2(x-\alpha) + \frac{1}{2\sigma^2}\right) \exp(-(x-\alpha)^2) + \left(-2(x-\beta) + \frac{1}{2\sigma^2}\right) \exp(-(x-\beta)^2) = 0.$$ A trivial substitution moves some complications into one place rather than two: $$\left(-2w + \frac{1}{2\sigma^2}\right) \exp(-w^2) + \left(-2(w-\gamma) + \frac{1}{2\sigma^2}\right) \exp(-(w-\gamma)^2) = 0.$$ At this point I ponder whether some other trivial simplifications might help. E.g. you could cancel$\exp(-w^2)$from both sides, and a few other things like that. But those don't seem potentially fruitful. So, I'm thinking maybe Newton--Raphson or the like. But: This looks like something that might have come from trying to find some MLEs. So are you sure you shouldn't be trying to solve for$\alpha$,$\beta$, and$\sigma\$ instead?

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i have to solve this for x, in any way, because this part is only a calculating part of my whole exercise, in order to find a rejection algorithm. I want the solution that verifies my equation, in order to use it to find an upper bound. – johan paul Nov 20 '11 at 17:09