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.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?

The classical example from Wilkinson

http://en.wikipedia.org/wiki/Wilkinson's_polynomial

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.

RJF

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

http://books.google.com/books/about/Numerical_Methods_for_Roots_of_Polynomia.html?id=j0rY3D9fx-0C