<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
Wilkinson	B-Algorithm
's	I-Algorithm
polynomial	I-Algorithm
is	O
a	O
specific	O
polynomial	O
which	O
was	O
used	O
by	O
James	O
H	O
.	O
Wilkinson	O
in	O
1963	O
to	O
illustrate	O
a	O
difficulty	O
when	O
finding	O
the	O
root	O
of	O
a	O
polynomial	O
:	O
the	O
location	O
of	O
the	O
roots	O
can	O
be	O
very	O
sensitive	O
to	O
perturbations	O
in	O
the	O
coefficients	O
of	O
the	O
polynomial	O
.	O
</s>
<s>
Sometimes	O
,	O
the	O
term	O
Wilkinson	B-Algorithm
's	I-Algorithm
polynomial	I-Algorithm
is	O
also	O
used	O
to	O
refer	O
to	O
some	O
other	O
polynomials	O
appearing	O
in	O
Wilkinson	O
's	O
discussion	O
.	O
</s>
<s>
It	O
is	O
a	O
natural	O
question	O
in	O
numerical	B-General_Concept
analysis	I-General_Concept
to	O
ask	O
whether	O
the	O
problem	O
of	O
finding	O
the	O
roots	O
of	O
p	O
from	O
the	O
coefficients	O
ci	O
is	O
well-conditioned	B-Algorithm
.	O
</s>
<s>
The	O
problem	O
is	O
ill-conditioned	B-Algorithm
when	O
the	O
polynomial	O
has	O
a	O
multiple	O
root	O
.	O
</s>
<s>
It	O
is	O
therefore	O
natural	O
to	O
expect	O
that	O
ill-conditioning	B-Algorithm
also	O
occurs	O
when	O
the	O
polynomial	O
has	O
zeros	O
which	O
are	O
very	O
close	O
.	O
</s>
<s>
However	O
,	O
the	O
problem	O
may	O
also	O
be	O
extremely	O
ill-conditioned	B-Algorithm
for	O
polynomials	O
with	O
well-separated	O
zeros	O
.	O
</s>
<s>
Speaking	O
for	O
myself	O
I	O
regard	O
it	O
as	O
the	O
most	O
traumatic	O
experience	O
in	O
my	O
career	O
as	O
a	O
numerical	B-General_Concept
analyst	I-General_Concept
.	O
</s>
<s>
Wilkinson	B-Algorithm
's	I-Algorithm
polynomial	I-Algorithm
is	O
often	O
used	O
to	O
illustrate	O
the	O
undesirability	O
of	O
naively	O
computing	O
eigenvalues	O
of	O
a	O
matrix	O
by	O
first	O
calculating	O
the	O
coefficients	O
of	O
the	O
matrix	O
's	O
characteristic	O
polynomial	O
and	O
then	O
finding	O
its	O
roots	O
,	O
since	O
using	O
the	O
coefficients	O
as	O
an	O
intermediate	O
step	O
may	O
introduce	O
an	O
extreme	O
ill-conditioning	B-Algorithm
even	O
if	O
the	O
original	O
problem	O
was	O
well	O
conditioned	O
.	O
</s>
<s>
However	O
,	O
the	O
polynomial	O
is	O
still	O
very	O
ill-conditioned	B-Algorithm
.	O
</s>
<s>
Wilkinson	O
chose	O
the	O
perturbation	O
of	O
2−23	O
because	O
his	O
Pilot	B-Device
ACE	I-Device
computer	O
had	O
30-bit	O
floating	B-Algorithm
point	I-Algorithm
significands	B-Algorithm
,	O
so	O
for	O
numbers	O
around	O
210	O
,	O
2−23	O
was	O
an	O
error	O
in	O
the	O
first	O
bit	O
position	O
not	O
represented	O
in	O
the	O
computer	O
.	O
</s>
<s>
The	O
two	O
real	O
numbers	O
,	O
210	O
and	O
210	O
2−23	O
,	O
are	O
represented	O
by	O
the	O
same	O
floating	B-Algorithm
point	I-Algorithm
number	I-Algorithm
,	O
which	O
means	O
that	O
2−23	O
is	O
the	O
unavoidable	O
error	O
in	O
representing	O
a	O
real	O
coefficient	O
close	O
to	O
210	O
by	O
a	O
floating	B-Algorithm
point	I-Algorithm
number	I-Algorithm
on	O
that	O
computer	O
.	O
</s>
<s>
The	O
perturbation	O
analysis	O
shows	O
that	O
30-bit	O
coefficient	O
precision	B-Architecture
is	O
insufficient	O
for	O
separating	O
the	O
roots	O
of	O
Wilkinson	B-Algorithm
's	I-Algorithm
polynomial	I-Algorithm
.	O
</s>
<s>
In	O
the	O
example	O
of	O
Wilkinson	B-Algorithm
's	I-Algorithm
polynomial	I-Algorithm
of	O
degree	O
20	O
,	O
the	O
roots	O
are	O
given	O
by	O
αj	O
=	O
j	O
for	O
j	O
=	O
1	O
,...,	O
20	O
,	O
and	O
c(x )	O
is	O
equal	O
to	O
x19	O
.	O
</s>
<s>
Similarly	O
,	O
the	O
other	O
small	O
roots	O
of	O
Wilkinson	B-Algorithm
's	I-Algorithm
polynomial	I-Algorithm
are	O
insensitive	O
to	O
changes	O
int	O
.	O
</s>
<s>
At	O
first	O
sight	O
the	O
roots	O
α1	O
=	O
1	O
and	O
α20	O
=	O
20	O
of	O
Wilkinson	B-Algorithm
's	I-Algorithm
polynomial	I-Algorithm
appear	O
to	O
be	O
similar	O
,	O
as	O
they	O
are	O
on	O
opposite	O
ends	O
of	O
a	O
symmetric	O
line	O
of	O
roots	O
,	O
and	O
have	O
the	O
same	O
set	O
of	O
distances	O
1	O
,	O
2	O
,	O
3	O
,	O
...	O
,	O
19	O
from	O
other	O
roots	O
.	O
</s>
<s>
If	O
the	O
polynomial	O
is	O
expressed	O
in	O
another	O
basis	O
,	O
then	O
the	O
problem	O
of	O
finding	O
its	O
roots	O
may	O
cease	O
to	O
be	O
ill-conditioned	B-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
Wilkinson	B-Algorithm
's	I-Algorithm
polynomial	I-Algorithm
is	O
well-conditioned	B-Algorithm
in	O
this	O
basis	O
.	O
</s>
