Finding Roots From Newton's Sums

Given non-negative real numbers  x_1, x_2,..., x_n  and the values of
           
           a_k   =   x_1^k  +  x_2^k  +  ...  +  x_n^k

for all positive EVEN integers k, can one determine the x_i's (or 
even just the sum of the x_i's) as an explicit function of the a_i's?
 
Let y_i = (x_i)^2  (i=1,2,..,n)  be the roots of the polynomial

       y^n  +  c1 y^(n-1)  +  c2 y^(n-2)  +  ...  +  cn  =  0

and let s(k) denote the sum of the kth powers of the roots.  (Note 
that s(k) equals a_{2k}.)  Clearly s(0) = n.  The other values 
of s(i) are related to the coefficients ci by Newton's Identities

                            s(1)    + 1 c1  =  0
                     s(2) + s(1) c1 + 2 c2  =  0
           s(3) + s(2) c1 + s(1) c2 + 3 c3  =  0

                                etc.

Thus, given s(1) through s(n) (which are equivalent to a_{2k} for 
k=1 to n), we can easily compute the coefficients ci.  We can then 
solve this polynomial using any of the usual methods to determine 
the n roots y_i, which determines the square roots x_i = sqrt(y_i) 
up to sign.  Since the x_i were specified to be non-negative, they 
are uniquely determined.