%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%KWD: large clusters, small clusters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Name{ Polynomial with large and small clustered roots. }
\Ring{ \RR[x]}

\begin{Equations}
 (x^{12}-(10^{20} x-1)^4)(1+(10^{20}+x)^4 x^8)
\end{Equations}

% II ORIGIN OF THE PROBLEM
Polynomial supplied by D. A. Bini (Oct. 1996).

\begin{Characteristics}
 \item The polynomial  has 4 roots of modulus $10^{-20}$ matching in the 
first 60 digits, and 4 zeros of modulus $10^{20}$ matching in the first 
60 digits.
\end{Characteristics}


% IV THE KIND OF PROBLEM WE ARE SOLVING
\begin{Problem}
Newton-isolate all the roots.
\end{Problem}

% V LIST OF SOLUTIONS 
\begin{Solution}
 Algorithm based on simultaneous iterations (Aberth's method)
with cluster analysis (see \cite{Bini97abrt}).

 Program in Fortran 90 based on the multiprecision package
MPFUN, by D. Bailey.

 Machine used: IBM  Risc 6000.
Time for Newton-isolating all the roots: 9.75 seconds.

\Signature
{Dario Andrea Bini, Dipartimento di Matematica, via Buonarroti 2,
 56127 Pisa, ITALY}                    % <- Name(s) and Adress(es)
{bini@dm.unipi.it}                   % <- email and/or www adress
{October 15, 1996}               % <- Date of the experimentation.

\end{Solution}



