Wilkinson's Polynomial - The Effect of The Basis

The Effect of The Basis

The expansion

expresses the polynomial in a particular basis, namely that of the monomials. 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 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

Every polynomial (of degree 20 or less) can be expressed in this basis:

For Wilkinson's polynomial, we find

Given the definition of the Lagrange basis polynomial ℓ₀(x), a change in the coefficient d₀ will produce no change in the roots of 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.