  We use Newton's method to find all roots of several polynomials in one
complex variable of degree up to and exceeding one million and show that the
method, applied to appropriately chosen starting points, can be turned into an
algorithm that can be applied routinely to find all roots without deflation and
with the inherent numerical stability of Newton's method.
  We specify an algorithm that provably terminates and finds all roots of any
polynomial of arbitrary degree, provided all roots are distinct and exact
computation is available. It is known that Newton's method is inherently
stable, so computing errors do not accumulate; we provide an exact bound on how
much numerical precision is sufficient.
