Does anyone recognize this problem? There are $2p$ equations and $2p$ unknowns, and it feels like a classic, but I've never encountered it before:

Given $d_1, d_2, \ldots d_p$, find $a_1, a_2, \ldots a_p$ and $n_1, n_2, \ldots n_p$: $$ \left[ \begin{array}{ccc} d_1\\ d_2\\ d_3\\ \vdots\\ d_{2p} \end{array} \right] = \left[ \begin{array}{ccc} a_1, ~a_2, \cdots ~a_p\\ a_1^2, ~a_2^2, \cdots ~a_p^2\\ a_1^3, ~a_2^3, \cdots ~a_p^3\\ \vdots\\ a_1^{2p}, ~a_2^{2p}, \cdots ~a_p^{2p} \end{array} \right] \cdot \left[ \begin{array}{c} n_1\\ n_2\\ \vdots\\ n_p \end{array} \right] $$

(and where $a \succeq 0$ and $n \succeq 0$ and only real values are considered, so wherever there can be positive and negative roots for the values of $a_i$, only the positive roots need be considered).

It seems straightforward to solve with elimination: From the first row, let $a_1 = \frac{d_1-\sum_{i>1} a_i n_i}{n_1}$. Then from the second row, solve for $a_2$ (which will now involve a quadratic, etc.). Because the polynomials get larger, solving it with elimination in this manner can be quite slow (I crashed sage doing this on a fairly small problem).

Has this family of problems been characterized well in the literature and can it be solved efficiently in practice? Even a numerical solution would suffice, as long as it will converge certainly (no multi-start Levenberg-Marquardt, etc.)...