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JuliaPy
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Description

Julia interface to SymPy via PyCall

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SymPy Package to bring Python's 
Sympy
functionality into 
Julia
via 
PyCall

SymPy (

http://sympy.org/
) is a Python library for symbolic mathematics.

With the excellent 

PyCall
package of 
julia
, one has access to the many features of the SymPy library from within a 
Julia
session.

This 

SymPy
package provides a light interface for the features of the SymPy library that makes working with SymPy objects a bit easier.

The documentation inludes an introduction document and a version of the SymPy tutorial translated from the Python syntax into Julia.

Installation

To use this package, both Python and its SymPy library must be installed on your system. If 

PyCall
is installed using 
Conda
(which is the default if no system 
python
is found), then the underlying SymPy library will be installed via 
Conda
when the package is first loaded. Otherwise, installing both Python and the SymPy library (which also requires mpmath) can be done by other means. In this case, the 
Anaconda
distribution is suggested, as it provides a single installation of Python that includes SymPy and many other scientific libraries that can be profitably accessed within 
Julia
via 
PyCall
. (Otherwise, install Python then download the SymPy library from https://github.com/sympy/sympy/releases and install.)

To upgrade the underlying 

sympy
library, which has new releases at a rate similar to 
Julia
, when installed with 
Conda
, the following commands are available: 
using Pkg
Pkg.add("Conda") #  if needed
using Conda
Conda.update()

The 
PyCall
interface to 
SymPy

The only point to this package is that using 

PyCall
to access SymPy is somewhat cumbersome. The following is how one would define a symbolic value 
x
, take its sine, then evaluate the symboic expression for 
x
equal 
pi
, say: 
using PyCall
sympy = pyimport("sympy")  #
x = sympy.Symbol("x")      # PyObject x
y = sympy.sin(x)           # PyObject sin(x)
z = y.subs(x, sympy.pi)    # PyObject 0
convert(Float64, z)        # 0.0

The 

sympy
object imported on the second line provides the access to much of SymPy's functionality, allowing access to functions (
sympy.sin
), properties, modules (
sympy
), and classes (
sympy.Symbol
, 
sympy.Pi
). The 
Symbol
and 
sin
operations are found within the imported 
sympy
module and, as seen, are referenced with 
Python
's dot call syntax, as implemented in 
PyCall
through a specialized 
getproperty
method.

SymPy's functionality is also found through methods bound to an object of a certain class. The 

subs
method of the 
y
object is an example. Such methods are also accessed with Python's dot-call syntax. The call above substitutes a value of 
sympy.pi
for the symbolic variable 
x
. This leaves the object as a 
PyObject
storing a number which can be brought back into 
julia
through conversion, in this case through an explicit 
convert
call.

Alternatively, 

PyCall
now has a 
*
method, so the above could also be done with: 
x = sympy.Symbol("x")
y = sympy.pi * x
z = sympy.sin(y)
convert(Float64, z.subs(x, 1))

With the 

SymPy
package this gets replaced by a more 
julia
n syntax: 
using SymPy
x = symbols("x")               # or   @vars x, Sym("x"), or  Sym(:x)
y = sin(pi*x)
y(1)                           # Does y.subs(x, 1). Use y(x=>1) to be specific as to which symbol to substitute

The object 

x
we create is of type 
Sym
, a simple proxy for the underlying 
PyObject
. The package overloads the familiar math functions so that working with symbolic expressions can use natural 
julia
idioms. The final result here is a symbolic value of 
0
, which prints as 
0
and not 
PyObject 0
. To convert it into a numeric value within 
Julia
, the 
N
function may be used, which acts like the float conversion, only there is an attempt to preserve the variable type.

(There is a subtlety, the value of 

pi
here (an 
Irrational
in 
Julia
) is converted to the symbolic 
PI
, but in general won't be if the math constant is coerced to a floating point value before it encounters a symbolic object. It is better to just use the symbolic value 
PI
, an alias for 
sympy.pi
used above.)

SymPy has a mix of function calls (as in 

sin(x)
) and method calls (as in 
y.subs(x,1)
). The function calls are from objects in the base 
sympy
module. When the 
SymPy
package is loaded, in addition to specialized methods for many generic 
Julia
functions, such as 
sin
, a priviledged set of the function calls in 
sympy
are imported as generic functions narrowed on their first argument being a symbolic object, as constructed by 
Sym
or 
symbols
. (Calling 
import_from(sympy)
will import all the function calls.)

The basic usage follows these points:

  • generic methods from 

    Julia
    and imported functions in the 
    sympy
    namespace are called through 
    fn(object)

  • SymPy methods are called through Python's dot-call syntax: 

    object.fn(...)

  • Contructors, like 

    sympy.Symbol
    , and other non-function calls from 
    sympy
    are qualified with 
    sympy.Constructor(...)
    . Such qualified calls are also useful when the first argument is not symbolic.

So, these three calls are different,

sin(1), sin(Sym(1)), sympy.sin(1)

The first involves no symbolic values. The second and third are related and return a symbolic value for 

sin(1)
. The second dispatches on the symbolic argument 
Sym(1)
, the third has no dispatch, but refers to a SymPy function from the 
sympy
object. Its argument, 
1
, is converted by 
PyCall
into a Python object for the function to process.

In the initial example, slightly rewritten, we could have written:

x = symbols("x")
y = sin(pi*x)
y.subs(x, 1)

The first line calls a provided alias for 

sympy.symbols
which is defined to allow a string (or a symbol) as an argument. The second, dispatches to 
sympy.sin
, as 
pi*x
is symbolic-- 
x
is, and multiplication promotes to a symbolic value. The third line uses the dot-call syntax of 
PyCall
to call the 
subs
method of the symbolic 
y
object.

Not illustrated above, but classes and other objects from SymPy are not brought in by default, and can be accessed using qualification, as in 

sympy.Function
(used to define symbolic functions).

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