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
Famous quotes containing the words effect and/or basis:
“The attention of those who frequent the camp-meetings at Eastham is said to be divided between the preaching of the Methodists and the preaching of the billows on the back side of the Cape, for they all stream over here in the course of their stay. I trust that in this case the loudest voice carries it. With what effect may we suppose the ocean to say, “My hearers!” to the multitude on the bank. On that side some John N. Maffit; on this, the Reverend Poluphloisboios Thalassa.”
—Henry David Thoreau (1817–1862)
““The Love that dare not speak its name” in this century is such a great affection of an elder for a younger man as there was between David and Jonathan, such as Plato made the very basis of his philosophy, and such as you find in the sonnets of Michelangelo and Shakespeare. It is that deep, spiritual affection that is as pure as it is perfect.... It is in this century misunderstood ... and on account of it I am placed where I am now.”
—Oscar Wilde (1854–1900)