Wilkinson's polynomial

In numerical analysis, Wilkinson's polynomial of degree k is given by the formula

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: f(x) = prod_{i=1}^k (x - i) = (x-1)(x-2) cdots (x-k)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

which has k roots: 1, 2, ..., k.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The problem of finding the roots is ill-conditioned: A small change in one coefficient can lead to drastic changes in the roots.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Wilkinson's polynomial of degree 20 has 20 roots, but, as the graph below shows, the function becomes almost horizontal near the x-axis.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In 1984, James H. Wilkinson admitted

Related Topics:
1984 - James H. Wilkinson

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:Speaking for myself I regard it as the most traumatic experience in my career as a numerical analyst.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~