Non-deterministic determinant computation with false results

[Chaman-veteran]

Steps To Reproduce

Trying to compute:

matrix([[1.00000000000000, 0.0900540649890900, 0.00811952576623298, 0.000732079098000832], [0.0900540649890900, 0.00811952576623298, 0.000732079098000832, 0.0000660062941063044], [0.00811952576623298, 0.000732079098000832, 0.0000660062941063044, 5.95131164589146e-6], [0.000732079098000832, 0.0000660062941063044, 5.95131164589146e-6, SR.temp_var()]]).det()

Either gives the correct result ((1.0587911840678754e-22)*symbol229 + 2.5920047522169004e-29) or wrong ones. I've seen 2.5920047522169004e-29 a lot but sometimes I got 0.0 and something starting with a 4. At first I thought that it was linked to the name of the used variables but it seems not :/

Thus, I've tried to work around this with:

R.<x> = PolynomialRing(RR); matrix([[1.00000000000000, 0.0900540649890900, 0.00811952576623298, 0.000732079098000832], [0.0900540649890900, 0.00811952576623298, 0.000732079098000832, 0.0000660062941063044], [0.00811952576623298, 0.000732079098000832, 0.0000660062941063044, 5.95131164589146e-6], [0.000732079098000832, 0.0000660062941063044, 5.95131164589146e-6, x]]).det()

Which gives a wrong result (-1.05879118406788e-21*x^2 + 2.73168125489512e-20*x, how can we have a degree two here? Am I missing smthing?)

Lastly, as, AFAIK, it should return a degree 1 polynomial, I tried to work around this with:

matrix([[1.00000000000000, 0.0900540649890900, 0.00811952576623298, 0.000732079098000832], [0.0900540649890900, 0.00811952576623298, 0.000732079098000832, 0.0000660062941063044], [0.00811952576623298, 0.000732079098000832, 0.0000660062941063044, 5.95131164589146e-6], [0.000732079098000832, 0.0000660062941063044, 5.95131164589146e-6, i]]).det()

Which gives me 1.02544561451684e-16*I...

Expected Behavior

All tested computations should result in the same output

Actual Behavior

Only the computation with symbolic computing seem to work, though, only in few cases only.

Additional Information

No response

Environment

• OS: Linux archlinux 6.17.6-arch1-1
• Sage Version: 10.7
• Python Version: 3.13.7

Checklist

• I have searched the existing issues for a bug report that matches the one I want to file, without success.
• I have read the documentation and troubleshoot guide

[user202729]

not a bug, floating point is inaccurate.

There's RBF for rigorous error bounds.

An alternative solution is to implement the naive (O(n!)) algorithm which will eliminate the x^2 term. I don't think it's worth it though.