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.)...



Deceptive linear algebra problem
Let $$f(X) = (X - a\_1) \cdots (X - a\_p) = X^p + e\_1 X^{p-1} + \dots + e\_p.$$ Using $f(a\_i) = 0$ for each $i$, we get 
\begin{equation} \begin{bmatrix} d\_{p+1} & d\_p & \cdots & d\_1 \\ d\_{p+2} & d\_{p+1} & \cdots & d\_2 \\ \vdots & \vdots & \ddots & \vdots \\ d\_{2p} & d\_{2p-1} & \cdots & d\_p \end{bmatrix} \begin{bmatrix} 1 \\ e\_1 \\ \vdots \\ e\_p \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \end{equation}
Note that the dimension of the first matrix is $p \times (p+1)$. In the general case, it will be possible to determine $e\_1, \dots, e\_p$ uniquely. Then calculate the roots of $f(X)$ to obtain $a\_i$, and the remaining $n\_i$ part will be a simple elimination problem. 


If the dimension of the null space of the '$d\_i$ matrix' is greater than $1$, it probably means that $a\_i$ are not pairwise distinct. Then you might want to check whether system is also solvable with a smaller number of variables. 


