On 7/7/2014 4:25 PM, Stavros Macrakis
(Σταῦρος Μακράκης) wrote:
I don't know for sure, but I am skeptical that anything both simple
and so useful could be said.
that there's a vast literature on all this.
But what simple and correct statements can we make to help
users understand what allroots does?
For example, could we say that for polynomials of degree
< 20 and some characteristic of the coefficients, or of the
roots, abs(calcroot-trueroot) < xxx?
The classical example from Wilkinson
is something like the polynomial product(x-n, n,0,25) whose
roots are obvioiusly 0,1,2,...25.
Allroots finds stuff like
perturbing one of the coefficients by a trivial amount blows the
accuracy to smithereens.
Actually, Wilkinson's polynomial stopped at n=20, but allroots
almost works at that point.
See wikipedia for more info.