epsilon-prepare failed in this case

[F-Feng]

Dear epsilon supporter,

I got a problem when using epsilon-prepare to transform a matrix in Mathematica format, I got a message like this one: "singularity is not a number. @ (10,4)", could you give a test on my case please?

PS: I have attached the matrix in Mathematica format in the attachment. (I gzip the file since it is not allowed to upload a .m file on the website.)

Thanks very much!
Feng
mat.m.gz

[vsht]

Hi Feng,

as you know, I'm not a developer of this package, but I think the problem you are facing is due to the presence of terms like 1/(s + 2 x - 8) which may give a different singularity depending on the value of s, which epsilon obviously cannot know. So it refuses to handle such terms. We can isolate the problem like this

mat = Get["mat.m"];
Put[mat[[10 ;; 10]], "mat2.m"]

and applying epsilon-prepare to mat2.m shows that problem comes from mat[[10]][[4]], which is
-((2 (-2 + 3 ep))/(-8 + s + 2 x))

If we multiply the whole matrix with (s + 2 x - 8)

Map[Factor[(s + 2 x - 8)*#] &, mat];
Put[%, "mat2.m"]

the problem reappears in the matrix element mat[[11]][[3]] which is proportional to (-16 s + s^2 + 4 x + 2 s x)

Doing

Map[Factor[(-16 s + s^2 + 4 x + 2 s x)(s + 2 x - 8)*#] &, mat];
Put[%, "mat2.m"]

finally leaves us

polynomials of degree 4 unsupported. @ (19,1)

So far my attempt to understand the problem. It seems that Fuchsia can deal with multivariate dependence of the matrices, cf. this discussion of the developers

gituliar/fuchsia#11

Perhaps the author of epsilon could also comment on this, as he definitely has more experience with differential equations than me.

Cheers,
Vladyslav

[F-Feng]

Hi Vladyslav,

Thanks very much for the helpful comments, I am going to have a try with your suggestions.

And I also tried Fuchsia, but it was running for such a long time to give a fuchsified result,
maybe the DE matrix is too large, and jordan_decomposition seems taking much longer time.

Anyway, Thanks again!

Best regards!
Feng

[mprausa]

Hi,

sorry for the late response for this issue.

Vladyslav is right that epsilon-prepare can not deal with singularities at non-numeric positions.
Epsilon was designed to handle two-scale problems, i.e. with only one dimensionless kinematic variable.

In principle epsilon might do its job even if the system has singularities at symbolic positions. One can use the option --symbols to add additional symbols to the Fermat computer algebra system. But note that this is highly experimental and was never tested. Unfortunately, in that case epsilon-prepare can not be used to generate the input files and this has to be done by other means (It sould not be too hard to write a Mathematica code to produce working input files).

Best,
Mario

[F-Feng]

Dear Mario,

Thanks very much for your reply, I will have a try on your suggestions.

Currently, I am using DESS developed by Smirnov etc. , and I just need the DE matrix
to be fuchsified only at x=0 (the singular point), so I am trying to write a small code
for this purpose according to Lee's Algorithm 1, and to reduce running time on matrix operations,
I use the Fermat intensively, now I can get the fuchsified result at tolerable time.

Anyway, Thanks again!

Best regards!
Feng