How to solve $1-\sum_{i=1}^Nx_i=a_nx_n^{1/n}$ given $a_n$

Question

I'm trying to solve (numerically) the following system of coupled equations:

$$1-\sum_{i=1}^Nx_i=a_nx_n^{1/n}$$ given $a_n$, I need to find the $N$ $x_n$.

I tried to write it as a matrix equation but of course, I can't isolate the vector $x$ because of the powers $1/n$.

So, is there an algorithm which would permit to solve it? Or can we isolate all the $x_n$?

Answer 1

Let $y = 1-\displaystyle\sum_{i = 1}^{N}x_i$. Then, your system of equations becomes $y = a_nx_n^{1/n}$, i.e. $x_n = \left(\dfrac{y}{a_n}\right)^n$ for $n = 1,\ldots,N$.

Substituting this back into the first equation for $y$ gives us $y = 1-\displaystyle\sum_{i = 1}^{N}\left(\dfrac{y}{a_i}\right)^i$.

Now, simply find the roots of this polynomial. Note, if all your $a_n$'s are positive, and your $x_n$'s are real, then we only care about roots $y \in (0,1)$.