Roots of simultaneous power sum equations (numerically or otherwise)

Question

I'm a physicist, and I've come across a problem in my research where I need to solve a set of equations looking like (e.g. in 3D)

$$r_1 + r_2 + r_3 = k_1$$ $$r_1^2 + r_2^2 + r_3^2 = k_2$$ $$r_1^3 + r_2^3 + r_3^3 = k_3$$

Where the $\{k_n\}$ are known and the $\{r_n\}$ are the roots I need to solve for. The roots can be complex, but will always appear in conjugate pairs (by the way I construct these things in the first place).

Ideally I need to generalise this to the nth case (where there will always be $n$ unknowns and $n$ equations following the pattern above). Numerical solutions are fine, as long as they're reasonably cheap.

If it's the case that there isn't always a unique solution to these things, then I may have to rethink my strategy for the problem as a whole.

Thanks in advance.

Answer 1

the elementary symmetric polynomials may be computed from the power sums. with these you can then construct a single polynomial on one unknown which has all the $r_j$ as roots.

rather than write the solutions explicitly, the scheme is easier to remember in the form suited to recursive evaluation. define:

$$ e_1 = k_1 \\ 2e_2 = e_1k_1 - k_2 \\ 3e_3 = e_2k_1-e_1k_2 +k_3 \\ \cdots $$ then (with $e_0=1$) the polynomial $$ P(x) = \sum_{k=0}^n (-1)^k e_k x^{n-k} $$ has roots $r1, r_2,...$

Answer 2

To expand on the answer by David Holden, you can use Newton's identities to find the elementary symmetric functions $e_i$ of the unknowns $r_i$ from their given power sums $p_i=k_i$. Then the $r_i$ are the roots of the polynomial $$ X^n-e_1X^{n-1}+e_2X^{n-2}-e_3X^{n-3}+\cdots+(-1)^ne_nX^0. $$ After that it is just general polynomial root finding, which is hard but for which you can probably find library routines.