Meaning of the progress report for numerical_irreducible_decomposition

[jzuiddam]

I am running numerical_irreducible_decomposition and am seeing the following progress report:

julia> N = numerical_irreducible_decomposition(F)
◒ Computing witness sets...    Time: 22:37:47
  Intersect with hypersurface:       19 / 48
  Tracking paths:                    6179/848952
  Current number of witness points:  
  Dimension 31:                      1
  Dimension 30:                      6
  Dimension 29:                      3103

where the number 3103 is still going up. I was wondering, what can we deduce from these values in the progress report while the program is still running? Can we deduce already that the dimension is 30? Or 31? Or do we need to wait for the full computation to finish to say anything? Earlier in the computation the report said:

julia> N = numerical_irreducible_decomposition(F)
◒ Computing witness sets...    Time: 22:02:10
  Intersect with hypersurface:       18 / 48
  Tracking paths:                    254155/294819
  Current number of witness points:  
  Dimension 35:                      1
  Dimension 34:                      1
  Dimension 33:                      558

where 558 was going up during the computation, but these lines later disappeared. So it seems that seeing a value 1 does not necessarily indicate a component of the corresponding dimension.

[PBrdng]

Hi, these numbers mean that intersecting the first 19 hypersurfaces gives you components of dimensions 31, 30 and 29. There is not much more to deduce from these values.

Would it be helpful to add an option that says "compute only for codimension < k"? This should be fairly straight forward to add.

Please, make sure to update HC.jl to the latest version. There have been some significant updates to numerical irreducible decompositions lately.

[jzuiddam]

Thanks for the answer! So if I understand correctly, the values we see under Current number of witness points: are only for the current stage in which it is intersecting with say 19 hypersurfaces, and we do not see the numbers of previous stages in which we were intersecting with fewer hypersurfaces. One thing I would like to know is simply the largest dimension among all components of this system. It seems that an option "compute only for codimension < k" would be helpful for that, as I can then guess a small k and the computation would be finished quicker. Another option could be to keep the numbers of witness points of previous stages (intersections) on the screen in the progress report so that they can be inspected during a (long) computation.

(I think I am running the latest version:)

(@v1.10) pkg> status
Status `~/.julia/environments/v1.10/Project.toml`
  [f213a82b] HomotopyContinuation v2.10.0

[PBrdng]

I can will the option "compute only for codimension < k" in the next update (I plan to work on numerical irreducible decomposition later this year, there is still some work to be done for improving it).

. Another option could be to keep the numbers of witness points of previous stages (intersections) on the screen in the progress report so that they can be inspected during a (long) computation.

This is not possible, because the data is continuously updated to avoid memory overhead.