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%KWD: large coefficients
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\Name{ Polynomial having large coefficients 1}
\Ring{\CC[x], \RR[x]}

\begin{Equations}
a x^{n}+10^{300}x^{14}+x^5+1, \ \  n>14, \ |a|\leq 1\ \ (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 has a coefficient with modulus close to the largest
       number representable as double in the IEEE standard.
       Overflow conditions/failure are likely for programs that do not use
       extended arithmetic. 
 \item For $n=20$ the polynomial has some root with very large modulus and
   some roots with very small modulus.
\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 $n=20$, $a=1$: 0.1 seconds;
for $n=200$, $a=10^{-300}$: 0.8 seconds;
for $n=500$, $a=10^{-300}$: 3.8 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}



