Interoperability between Symbolics.jl and MathLink.jl

[Shuvomoy]

Dear All,

Is interoperability possible between Symbolics.jl and MathLink.jl? I would like to create a symbolic object in Julia, do part of the calculation using Symbolics.jl and then pass some intermediate symbolic object to Mathematica or Wolfram Engine via MathLink.jl for further manipulation, and then convert it back to Symbolics variables.

For example, consider the following example, where I multiply two matrices in Symbolics.jl and then compute the eigenvalues of the new matrix in MathLink.jl.

using Symbolics, MathLink


@variables α β
a = Symbolics.variables(:a, 1:2, 1:2) 
a[1,2] = a[2,1] 
# will create 
# 2×2 Matrix{Num}:
# a₁ˏ₁  a₂ˏ₁
# a₂ˏ₁  a₂ˏ₂

b = Symbolics.variables(:b, 1:2, 1:2)
b[1,1] = α
b[2, 2] = β
b[2, 1] = 0
b[1, 2] = 0

# at this stage b is
# 2×2 Matrix{Num}:
# α  0
# 0  β

c = a*b 
# c is equal to 
# 2×2 Matrix{Any}:
# a₁ˏ₁*α  a₂ˏ₁*β
# a₂ˏ₁*α  a₂ˏ₂*β

To compute the eigenvalues of c in Mathematica, I convert c into a MathLink.WExpr object manually and call the Wolfram function Eigenvalues via weval as follows:

weval(W`
Eigenvalues[{{a₁ˏ₁*\[Alpha], a₂ˏ₁*\[Beta]}, {a₂ˏ₁*\[Alpha], a₂ˏ₂*\[Beta]}}]
`)     

which gives the correct eigenvalues, but now I have to painstackingly bring back the MathLink.WExpr into Symbolics variables. I have to do these steps manually, and I was wondering if there is any way of automating the process? I found the Julia package https://github.com/eswagel/SymbolicsMathLink.jl which I believe has similar use cases, unfortunately it does not work even for simplest of examples and seem to have many bugs.

Any tips or suggestions regarding how to automate this conversion process between Symbolics.jl and MathLink.jl will be much appreciated.

(Of course, I am talking about this specific example just to illustrate what my intention is. My actual use case is somewhat involved and requires to use some special functions in Mathematica that is not currently possible in Symbolics.jl.)

[fremling]

Hi Shuvomoy,

It is a verry interesting problem indeed. I'm myself not familiar with the inner workings of the symbolic package so the only think i can say at face value is that there is not support within MathLink for it. And neither should there be, as this should be built in a different package.

That being said, i playd around with is eer so slightly and it looks like the sumbolics package stores its data in asimilar hirearchy to that we do in the MathLink package. Therefore the best way formward as i see it would be to write a converter that can swap between the two formats.

You mention the https://github.com/eswagel/SymbolicsMathLink.jl package with sound like a good candidate for this. It's sad to hear that it is not doing in this task correctly.

Maybe we can get it moving bu putting in some specific requests there ;)

\Mikael

[Shuvomoy]

Thanks for your very detailed comments Mikael. Yes, I think SymbolicsMathLink is a good candidate for this, I have put couple of issues there.

[eswagel]

Hi all, just wanted to make sure to respond in this thread since we discussed separately. SymbolicsMathLink.jl is fully functional. Although it doesn't cover literally every Mathematica function, it's relatively robust for most applications, and can easily be extended. I've found it very helpful (in my unbiased opinion) in some of my work and hope others do too!

[fremling]

Great to hear! I also added a shutout to the Symbolics and SymbolicsMathLink packages at the end of the readme.