Metric intersection

[xgarnaud]

Dear all,

Thank you very much for this library.

I was trying to understand how metric are computed and handled, and I think there might be an issue with metric intersection.
If I understand correctly when I intersect two metrics M_0$ and M_1 to get M, for any vector v I should have
v^T M_0 v < v^T M v
and
v^T M_1 v < v^T M v
or at least numerically something very close to this.

I tried to check this numerically. Usually this test passes, but when I use e.g.

M0 << 0.686582, 0.310726, -0.440491, 0.310726, 0.765968, 0.284695, -0.440491,
      0.284695, 0.657285;
M1 << 0.914328, -0.689839, -0.270308, -0.689839, 1.29997, 0.048584, -0.270308,
      0.048584, 1.15847;

I find that v^T M_1 v > v^T M v for the last two eigenvectors of M1:
v^T M_1 v = 1.14424 and v^T M v = 1.06664
and
v^T M_1 v = 1.88177 and v^T M v = 1.58967

The code I used is here:
https://gist.github.com/xgarnaud/0b6c5e54ad3398319292fb95b7c7eaf3

Have you ever seen such a behavior (it seems when it happens M_0 has a small eigenvalue, but it can work fine even if it has one)? Do you have an idea where it could comme from?
Best regards