%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%KWD: Wilkinson, ill conditioning
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Name{Wilkinson's Polynomial}
\Ring{\RR[x]}

\begin{Equations}
   \prod_{i=1}^n(x-i), \ \ \ (n=20).
\end{Equations}

% II ORIGIN OF THE PROBLEM
Polynomial  supplied by Wilkinson. Added to the list by D. A. Bini
(Oct. 1996).


\begin{Characteristics}
  \item Coefficients are very large integers and the roots are very 
        ill-conditioned
        already for $n=20$.
  \item The condition number of the roots, for $n=20$ ranges from 420 to
        $10^{14}$.
\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 for $n=20$: 2.94 seconds;
for $n=40$: 40 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}


