Kahan 2x2 determinant computation reduces numerical imprecision.

[TotoGaz]

Related to issue GEOS-DEV/GEOS#1440: some tests testTensorOpsInverseTwoArgs and testTensorOpsInverseOneArg are failing due to large numerical imprecision.

Using the Kahan method to compute 2x2 determinants brings more precision.
3x3 case still untouched.

Warning, I sometimes get compilation warnings like

../coreComponents/LvArray/unitTests/../src/fixedSizeSquareMatrixOpsImpl.hpp(100): warning: calling a constexpr __host__ function("fma") from a __host__ __device__ function("determinant") is not allowed. The experimental flag '--expt-relaxed-constexpr' can be used to allow this.

which I find surprising since this fma function is supposed to be device friendly.
fma also seems pretty standard so one should not need any fallback.
So I need to double-check.

Any opinion?

[corbett5]

If we want to use fma we should put it in LvArray::math and wrap it such that it uses std::fma on host and CUDA's fma on device.

Also this modification makes calculating the determinant significantly more costly. I'm not sure you can answer this question but maybe the kind of matrices that pop up in the unit tests aren't representative of the normal use case.

[corbett5]

I think matrix[ 0 ][ 0 ] is more readable.

[TotoGaz]

OK, I will change this.

[rrsettgast]

Also this modification makes calculating the determinant significantly more costly. I'm not sure you can answer this question but maybe the kind of matrices that pop up in the unit tests aren't representative of the normal use case.

It is 3 flop vs 2 flop. Look at the second variant I listed in https://godbolt.org/z/4bGdjYbd5

I don't think this is a big deal...for 3d it may be a bit more costly...but not much. I don't think we will often run into this sort of round off problem, but then again, if it doesn't cost much to "fix", then I kind of like this approach.

[corbett5]

Is the fma required to obtain adequate precision? I'd prefer to do auto const e = -b * c + w and let the compiler do it's thing.

[TotoGaz]

Sure, I'll check too.

[rrsettgast]

https://godbolt.org/z/ao9GMhPsq

I think you need the fma

[TotoGaz]

Also this modification makes calculating the determinant significantly more costly. I'm not sure you can answer this question but maybe the kind of matrices that pop up in the unit tests aren't representative of the normal use case.

I could spot 3x3 dets in the code (FEM stuff), but only 2x2 in the unit tests.
I will double check.

EDIT: no 2x2 dets in the code, only in tests.