Editing Wilkinson's polynomial (section)

==The effect of the basis==
The expansion
<math display="block">p(x) = \sum_{i=0}^n c_i x^i</math>
expresses the polynomial in a particular [[basis (linear algebra)|basis]], namely that of the [[monomial]]s. If the polynomial is expressed in another basis, then the problem of finding its roots may cease to be ill-conditioned. For example, in a [[Lagrange polynomial|Lagrange form]], a small change in one (or several) coefficients need not change the roots too much.  Indeed, the basis polynomials for interpolation at the points 0, 1, 2, ..., 20 are
<math display="block"> \ell_k(x) = \prod_{i = 0,\ldots,20 \atop i \neq k} \frac{x - i}{k - i}, \qquad\text{for}\quad  k=0,\ldots,20. </math>

Every polynomial (of degree 20 or less) can be expressed in this basis:
<math display="block"> p(x) = \sum_{i=0}^{20} d_i \ell_i(x). </math>

For Wilkinson's polynomial, we find
<math display="block"> w(x) = (20!) \ell_0(x) = \sum_{i=0}^{20} d_i \ell_i(x) \quad\text{with}\quad d_0=(20!) ,\, d_1=d_2= \cdots =d_{20}=0. </math>

Given the definition of the Lagrange basis polynomial {{math|ℓ<sub>0</sub>(''x'')}}, a change in the coefficient {{math|''d''<sub>0</sub>}} will produce no change in the roots of {{math|''w''}}.  However, a perturbation in the other coefficients (all equal to zero) will slightly change the roots. Therefore, Wilkinson's polynomial is well-conditioned in this basis.