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.
Read more about this topic: Wilkinson's Polynomial
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