Background
Wilkinson's polynomial arose in the study of algorithms for finding the roots of a polynomial
It is a natural question in numerical analysis to ask whether the problem of finding the roots of p from the coefficients ci is 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 x2 has a double root at x = 0. However, the polynomial x2−ε (a perturbation of size ε) has roots at ±√ε, which is much bigger than ε when 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 w(x) to illustrate this point (Wilkinson 1963).
In 1984, he described the personal impact of this discovery:
- Speaking for myself I regard it as the most traumatic experience in my career as a numerical analyst.
Wilkinson's polynomial is often used to illustrate the undesirability of naively computing eigenvalues of a 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.
Read more about this topic: Wilkinson's Polynomial
Famous quotes containing the word background:
“I had many problems in my conduct of the office being contrasted with President Kennedy’s conduct in the office, with my manner of dealing with things and his manner, with my accent and his accent, with my background and his background. He was a great public hero, and anything I did that someone didn’t approve of, they would always feel that President Kennedy wouldn’t have done that.”
—Lyndon Baines Johnson (1908–1973)
“... every experience in life enriches one’s background and should teach valuable lessons.”
—Mary Barnett Gilson (1877–?)
“Silence is the universal refuge, the sequel to all dull discourses and all foolish acts, a balm to our every chagrin, as welcome after satiety as after disappointment; that background which the painter may not daub, be he master or bungler, and which, however awkward a figure we may have made in the foreground, remains ever our inviolable asylum, where no indignity can assail, no personality can disturb us.”
—Henry David Thoreau (1817–1862)