%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%KWD: large coefficients
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Name{ Polynomial having large coefficients 3}
\Ring{\CC[x], \RR[x]}

\begin{Equations}
a 10^{-200} x^{n}+10^{100} x^{n-1}+10^{200},  \ \  (n=20,  a=1,i)
\end{Equations}

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

\begin{Characteristics}
 \item The polynomial cannot be normalized in the standard IEEE arithmetic
       without generating an overflow condition.
 \item The polynomial, even if its coefficients are representable as double,
       has a root which cannot be represented as a
       double without generating an overflow condition.
       Failure is likely for programs that do not use
       extended arithmetic.
\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 $n=20$, $a=1$, 2.3 seconds;
for $n=200$, $a=1$, 15 seconds;
for $n=500$, $a=1$, 18 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}


