Re: Fast test for integer-rooted polynomial

On Tue, Apr 9, 2019, 6:10 PM Gordon Royle <gordon.royle@uwa.edu.au> wrote:

Dear Pari/GP users

This is a very simple task, but I want to do it billions of times, and so I want it to be as fast as possible.

I have written a suitable Pari/GP routine to do it, but just naively, and I wonder if there is a clever way to do it.

INPUT: Billions of symmetric matrices of smallish order (say at most 16 x 16) each of which has small integer entries.

OUTPUT: The input matrices that have only integer eigenvalues.

From the nature of the problem, I know that any integer eigenvalues lie in {0,1,2, …, n} where n is the order of the matrix.

What I do currently:

- get Pari to compute the characteristic polynomial (charpoly())

- factor the characteristic polynomial

- check that the factors are all linear

Here is an example:

a=

[7,0,-1,-1,-1,-1,-1,-1,-1;0,7,-1,-1,-1,-1,-1,-1,-1;-1,-1,7,0,-1,-1,-1,-1,-1;-1,-1,0,7,-1,-1,-1,-1,-1;-1,-1,-1,-1,7,0,-1,-1,-1;-1,-1,-1,-1,0,4,0,0,0;-1,-1,-1,-1,-1,0,7,-1,-1;-1,-1,-1,-1,-1,0,-1,7,-1;-1,-1,-1,-1,-1,0,-1,-1,7]

p=charpoly(a)

f=factor(p)

print(f)

produces

[x - 9, 2; x - 8, 3; x - 7, 2; x - 4, 1; x, 1]

showing that the eigenvalues are just 9, 9, 8, 8, 8, 7, 7, 4, 0 which are integers as required.

This works fine, but of course, faster is better, which leads me to my questions:

Q1: Is there perhaps a better way to compute the char poly, such as the other methods supplied by Pari, perhaps using the flag=4 option to take advantage of the integer entries?

Q2: Is there perhaps a better way to check that the polynomial is a product of linear factors without actually factorising it? For example, for each i in {0,1,2, …, n}, check if p(i) = 0 and then divide by (x-i) if so. Repeat.

Q3: Or is there a way to check the eigenvalues without even computing the characteristic polynomial? (I looked at qfjacobi but it returns floating point numbers for the eigenvalues)

Thanks in advance for any suggestions

Gordon

Why not compute the characteristic polynomial and then evaluate at 16 points and compute multiplicaties via the derivative?