[Octave-bug-tracker] [bug #58398] Given an integer matrix, the characteristic polynomial has none integer value

[Kai Torben Ohlhus]

Update of bug #58398 (project octave):

                Severity:              3 - Normal => 2 - Minor              
                Priority:              5 - Normal => 3 - Low                
                  Status:                    None => Invalid                
             Open/Closed:                    Open => Closed                 

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Follow-up Comment #1:

Thank you for your report.  This problem seems to occur due to the nature of
numerical software.  The claim "roots(poly(A)) = eig(A)" is only approximately
valid in numerical mathematics and strongly dependent on your problem data.

Your matrix `A` may be stored with full precision, but as soon as you convert
it into a polynomial, numerical computations are performed.  Double precision
is insufficient to perform this conversion with full accuracy.  Your
polynomial is only an approximation.  This effect grows with the matrix size.


>> max (poly (A))
ans =  55572125.19238

>> min (poly (A))
ans = -3317398.57544


Your observation, that "roots(poly(A)) = eig(A)" does not hold only for your
matrix at N=100, but for other random integer matrices, is not necessarily
true.  Using random integer matrices, e.g.,


N=300;
A=randi ([-1, 1], N);
e=eig(A);
p=roots (poly (A));
figure;
plot(real(e),imag(e),'ro');hold on;
plot(real(p),imag(p),'gx');hold off;
axis equal


the difference becomes obvious at larger dimensions, as well.

Matlab R2019a does not behave better than Octave for your example code.

How to overcome this issue?

- Know about the structure of your matrix, and search in literature about
specialized numerical methods treating that structure accurately.
- Use symbolic calculation (your problem dimensions are not too large).


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