Editing Wilkinson's polynomial (section)

==Background==
Wilkinson's polynomial arose in the study of algorithms for finding the roots of a polynomial
<math display="block"> p(x) = \sum_{i=0}^n c_i x^i. </math>
It is a natural question in numerical analysis to ask whether the problem of finding the roots of {{math|''p''}} from the coefficients {{math|''c''<sub>''i''</sub>}} is [[condition number|well-conditioned]].  That is, we hope that a small change in the coefficients will lead to a small change in the roots.  Unfortunately, this is not the case here.

The problem is ill-conditioned when the polynomial has a [[multiple root]]. For instance, the polynomial {{math|''x''<sup>2</sup>}} has a double root at&nbsp;{{math|1=''x'' = 0}}. However, the polynomial {{math|''x''<sup>2</sup> − ''ε''}} (a perturbation of size&nbsp;''ε'') has roots at {{math|±√''ε''}}, which is much bigger than {{math|''ε''}} when {{math|''ε''}} is small.

It is therefore natural to expect that ill-conditioning also occurs when the polynomial has zeros which are very close. However, the problem may also be extremely ill-conditioned for polynomials with well-separated zeros. Wilkinson used the polynomial {{math|''w''(''x'')}} to illustrate this point (Wilkinson 1963).

In 1984, he described the personal impact of this discovery:

<blockquote>''Speaking for myself I regard it as the most traumatic experience in my career as a numerical analyst.''<ref>
{{Cite book|last=Wilkinson |first=James H. |authorlink=James H. Wilkinson |editor=Gene H. Golub |title= Studies in Numerical Analysis |year=1984 |publisher=Mathematical Association of America |isbn=978-0-88385-126-5 |pages=3 |chapter=The perfidious polynomial}}</ref></blockquote>

Wilkinson's polynomial is often used to illustrate the undesirability of naively computing [[eigenvalue]]s of a [[matrix (mathematics)|matrix]] by first calculating the coefficients of the matrix's [[characteristic polynomial]] and then finding its roots, since using the coefficients as an intermediate step may introduce an extreme ill-conditioning even if the original problem was well-conditioned.<ref name=TrefethenBau>{{Citation|first1=Lloyd N. |last1=Trefethen |first2= David|last2= Bau|title=Numerical Linear Algebra|publisher=SIAM|year=1997}}</ref>