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Mathematica

Call Mathematica from Julia

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Mathematica.jl

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The Mathematica.jl package provides an interface for using Wolfram Mathematica™ from the Julia language. You cannot use Mathematica.jl without having purchased and installed a copy of Mathematica™ from Wolfram Research. This package is available free of charge and in no way replaces or alters any functionality of Wolfram's Mathematica product.

The package provides is a no-hassle Julia interface to Mathematica. It aims to follow Julia's philosophy of combining high-level expressiveness without sacrificing low-level optimisation.

Pkg.add("Mathematica")

Provided Mathematica is installed, its usage is as simple as:

using Mathematica
Fibonacci(1000)
#=> 43466557686937456435688527675040625802564660517371780402481729089536555417949051890403879840079255169295922593080322634775209689623239873322471161642996440906533187938298969649928516003704476137795166849228875

All of Mathematica's functions are available as both functions and macros, and splicing ($) works as you would expect:

Integrate(:(x^2), :x) # or
@Integrate(x^2, x)
#=> :(*(1//3,^(x,3)))

@Integrate(log(x), {x,0,2})
#=> :(+(-2,log(4)))

eval(ans) # or
@N($ans) # or
N(ans) # or
@N(Integrate(log(x), {x,0,2}))
#=> -0.6137056388801094

Including those that return Mathematica data:

@Plot(x^2, {x,0,2})
#=> Graphics[{{{},{},{Hue[0.67, 0.6, 0.6],Line[{{4.081632653061224e-8,1.6659725114535607e-15},...}]}}}, {:AspectRatio->Power[:GoldenRatio, -1],:Axes->true, ...}]

Mathematical data can participate in Julia functions directly, with no wrapping required. For example -

using MathLink
d = BinomialDistribution(10,0.2) #=> BinomialDistribution[10, 0.2]
probability(b::MExpr{:BinomialDistribution}) = b.args[2]
probability(d) #=> 0.2

Julia compatible data (e.g. lists, complex numbers etc.) will all be converted automatically, and you can extend the conversion to other types.

Note that Mathematica expressions are not converted to Julia expressions by default. Functions/macros with the ::Expr hint (see below) will convert their result, but for others you must use convert or MathLink.to_expr.

Log(-1) #=> Times[0 + 1im, :Pi]
convert(Expr, ans) #=> :(*(0 + 1im,Pi))
N(Log(-1)) #=> 0.0 + 3.141592653589793im

Printing and warnings are also supported:

Print("hi")
#=> hi
@Print(x^2/3)
#=>  2
#   x
#   --
#   3
Binomial(10)
#=> WARNING: Binomial::argr: Binomial called with 1 argument; 2 arguments are expected.
#=> Binomial[10]

Finally, of course:

WolframAlpha("hi") #=>
2-element Array{Any,1}:
 {{"Input",1},"Plaintext"}->"Hello."
 {{"Result",1},"Plaintext"}->"Hello, human."

Advanced Use

Typing

In the file Mathematica.jl, you'll see a listing of function and macro specifications, each in one of these formats:

Function::ReturnType # or
Function(Arg1Type, Arg2Type, ...)::ReturnType # (functions only)

For example:

Integrate::Expr
RandomReal(Number)::Float64
RandomReal(Number, Integer)::Vector{Float64}

The return type hint here is an optimisation; it allows MathLink.jl to grab the value from Mathematica without first doing a type check, and makes the function type stable - for example, RandomReal(10, 5) would return an Any array if not for this definition. The argument types allow type checking and multiple definitions.

Not many functions have type signatures yet, so providing them for the functions you want to use is an easy way to contribute.

Extending to custom datatypes

The Mathematica data expression Head[x,y,z,...] is represented in Julia as MExpr{:Head}(args = {x,y,z,...}). We can extend Mathematica.jl to support custom types by overloading MathLink.to_mma and MathLink.from_mma.

For example, we can pass a Julia Dict straight through Mathematica with just two lines of definitions:

using MathLink; import MathLink: to_mma, from_mma
d = [:a => 1, :b => 2]

to_mma(d::Dict) = MExpr{:Dict}(map(x->MExpr(:Rule, x[1], x[2]),d))
Identity(d) #=> Dict[:b->2, :a->1]
from_mma(d::MExpr{:Dict}) = Dict(map(x->x.args[1], d.args), map(x->x.args[2], d.args))
Identity(d) #=> {:b=>2,:a=>1}

Usage Issues

using Mathematica

This should work so long as either math is on the path (normally true on linux). Mathematica.jl will also look for math.exe on Windows, which should work for Mathematica versions 8 or 9 installed in default locations. If it doesn't work for you, open an issue (in particular I don't know how this will behave on Macs).

Current Limitations / Planned Features

  • Error handling: Error checking is currently reasonable, but the only way to reset the current link once an error is encountered is to restart Julia.
  • Passing native arrays and matrices is not currently supported.
  • MRefs: see the MVars section of clj-mma
  • Connect to a running session and injecting callbacks to Julia functions would be really cool, but would probably require a C extension for Mathematica.

First Commit

12/10/2013

Last Touched

9 days ago

Commits

29 commits