`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.

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()
```

`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 @syms 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 `@syms`

, `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 issued:

```
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, as is `@syms`

, to define symbolic functions).

04/23/2013

1 day ago

661 commits