Counting real roots of integer polynomials

[Gordon Royle <gordon.royle@uwa.edu.au>]

Hi,

I have a large number (hundreds of millions) of integer polynomials of moderate degree (about 15-20) and coefficients (less than 6 digits each). They may have complex roots, but I just want to collect data about certain real roots.

In particular, I want to know

(1) How many roots the polynomial has in the open interval (1,2)
(2) The multiplicity (possibly 0) of 2 as a root

For other reasons, I am sure that x=1 is a simple root of each of these polynomials.


At the moment I do the following:

- count the roots in the closed interval [1,2] with “polrootsreal” with argument [1,2]
- count the roots at 2 by using polrootsreal again with argument [2,2]
- subtract the second value from the first value and take off another 1

I am not sure if I should do something else instead - maybe factor the polynomial to find the multiplicity of the root at 2, and save one call to polrootsreal? 

Being primarily an end-user of polynomial routines, I do not have a feel for the relative costs for factoring versus solving etc; though I could probably try some simple experiments there may be a more sophisticated way of addressing my problem that I would just never come across.

Any advice will be gratefully received.

Thanks

Gordon