On 7/7/2014 4:25 PM, Stavros Macrakis (Σταῦρος Μακράκης) wrote:
Understood 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?
I don't know for sure, but I am skeptical that anything both simple and so useful could be said.
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

16.23992085329114 -1.542591489044811*%i

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.


On Mon, Jul 7, 2014 at 7:20 PM, Richard Fateman <fateman@berkeley.edu> wrote:
mcnamee has an enormous bibliography on this topic.  Also see