`Sympy`

functionality into `Julia`

via `PyCall`

SymPy Package to bring Python's The `SymPy`

package (`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 `SymPy`

from a `julia`

session.

This `SymPy`

package provides a light interface for *some* of the
features of `SymPy`

that makes working with `SymPy`

objects a bit
easier.

The tutorial provides an overview. It is
viewable as an `IJulia`

notebook
here.

### 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
`SymPy`

(which also requires `mpmath`

) can be done by other means.
The `Anaconda`

distribution is suggested, as it provides a single
installation of `Python`

that includes `SymPy`

and many other
scientifice 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.)

`PyCall`

interface to `SymPy`

The 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 at `pi`

, say:

```
using PyCall
@pyimport sympy
x = sympy.Symbol("x")
y = sympy.sin(x)
y[:subs](x, sympy.pi) |> float
```

The `Symbol`

and `sin`

function of `SymPy`

are found within the
imported `sympy`

object. They may be referenced with `Python`

's dot
notation. However, the `subs`

method of the `y`

object is accessed
differently, using indexing notation with a symbol. The call above
substitutes a value of `sympy.pi`

for `x`

. This leaves the object as a
`PyObject`

storing a number which can be brought back into `julia`

through conversion, in this case with the `float`

function.

The above isn't so awkward, but even more cumbersome is the similarly
simple task of finding `sin(pi*x)`

. As this multiplication is done at
the python level and is not a method of `sympy`

or the `x`

object, we
need to evaluate python code. Here is one solution:

```
x = sympy.Symbol("x")
y = pyeval("k*x", k=sympy.pi, x=x)
z = sympy.sin(y)
z[:subs](x, 1) |> float
```

This gets replaced by a more `julia`

n syntax:

```
using SymPy
x = symbols("x") # or @syms x, Sym("x"), or Sym(:x)
y = sin(pi*x)
y(1) # Does subs(y, 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`

. We then overload the familiar math functions so
that working with symbolic expressions can use natural `julia`

idioms. The final result is 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`

call, only attempts to preserve the variable type.

(There is a subtlety, the value of `pi`

here 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. A similar comment applies for `e`

, where `E`

should be
used.)

However, for some tasks the `PyCall`

interface is still needed, as
only a portion of the `SymPy`

interface is exposed. To call an
underlying SymPy method, the `getindex`

method is overloaded for
`symbol`

indices so that `ex[:meth_name](...)`

dispatches to either to
SymPy's `ex.meth_name(...)`

or `meth_name(ex, ...)`

, as possible. Any
`Sym`

objects are projected down onto the underlying `PyObject`

for
use within `PyCall`

. Otherwise, to dig the `PyObject`

out of a `Sym`

object, you access its property `x`

, as in `y.x`

. To find a SymPy
method, a call like `sympy.meth_name(...)`

is possible,
e.g. `sympy.harmonic(10)`

. Any `Sym`

-type arguments must be projected
into `PyObject`

s or an error will be thrown.

## Notes

Some aspects of `SymPy`

require more modern versions of `sympy`

to be
installed. For example, the matrix functions rely on features of
`sympy`

that are not exposed in the `sympy`

installed with Ubuntu LTS
12.04.

In that particular instance, calls such as

```
x = symbols("x")
a = [x 1; 1 x]
det(a)
```

Can be replaced with

```
a[:det]()
```

Similarly for `:trace`

, `:eigenvects`

, ... . Note these are `sympy`

methods, not `Julia`

methods that have been ported. (Hence,
`:eigenvects`

and not `eigvecs`

.)