  The DLG root-squaring iterations, due to Dandelin 1826 and rediscovered by
Lobachevsky 1834 and Graeffe 1837, have been the main approach to root-finding
for a univariate polynomial p(x) in the 19th century and beyond, but not so
nowadays because these iterations are prone to severe numerical stability
problems. Trying to avoid these problems we have found simple but novel
reduction of the iterations applied for Newton's inverse ratio -p'(x)/p(x) to
approximation of the power sums of the zeros of p(x) and its reverse
polynomial. The resulting polynomial root-finders can be devised and performed
independently of DLG iterations, based on Newton's identities or Cauchy
integrals. In the former case the computation involve a set of leading or
tailing coefficients of an input polynomial. In the latter case we must scale
the variable and increase the arithmetic computational cost to ensure numerical
stability. Nevertheless the cost is still manageable, at least for fast
root-refinement, and the algorithms can be applied to a black box polynomial
p(x)--given by a black box for the evaluation of the ratio p'(x)/p(x) rather
than by its coefficients. This enables important computational benefits,
including efficient recursive as well as concurrent approximation of a set of
zeros of p(x) or even all of its zeros, acceleration where an input polynomial
can be evaluated fast, and extension to approximation of the eigenvalues of a
matrix or a polynomial matrix, being efficient if the matrix can be inverted
fast, e.g., is data sparse. We also recall our recent fast algorithms for
approximation of the root radii, that is, the distances to the roots from the
origin or any complex value to the zeros of p(x), and apply it for fast black
box initialization of polynomial root-finding by means of functional
iterations, e.g., Newton's, Ehrlich's, and Weierstrass's.
