Symbolica

Symbolica is a blazing fast and modern computer algebra system which aims to handle huge expressions. It can easily be incorporated into existing projects using its Python, Rust or C++ bindings. Check out the live Jupyter Notebook demo!

For documentation and more, see symbolica.io.

Quick Example

Symbolica allows you to build and manipulate mathematical expressions through matching and replacing patterns, similar to regex for text:

You are able to perform these operations from the comfort of a programming language that you (probably) already know, by using Symbolica's bindings to Python, Rust and C++:

Installation

Visit the Get Started page for detailed installation instructions.

Python

Symbolica can be installed for Python >3.5 using pip:

pip install symbolica

The installation may take some time on Mac OS and Windows, as it may have to compile Symbolica.

Rust

If you want to use Symbolica as a library in Rust, simply include it in the Cargo.toml:

[dependencies]
symbolica = "0.4"

Examples

Below we list some examples of the features of Symbolica. Check the guide for a complete overview.

Pattern matching

Variables ending with a _ are wildcards that match to any subexpression. In the following example we try to match the pattern f(w1_,w2_):

from symbolica import Expression
x, y, w1_, w2_ = Expression.vars('x','y','w1_','w2_')
f = Expression.fun('f')
e = f(3,x)*y**2+5
r = e.replace_all(f(w1_,w2_), f(w1_ - 1, w2_**2))
print(r)

which yields y^2*f(2,x^2)+5.

Solving a linear system

Solve a linear system in x and y with a parameter c:

from symbolica import Expression

x, y, c = Expression.vars('x', 'y', 'c')
f = Expression.fun('f')

x_r, y_r = Expression.solve_linear_system(
    [f(c)*x + y + c, y + c**2], [x, y])
print('x =', x_r, ', y =', y_r)

which yields x = (-c+c^2)*f(c)^-1 and y = -c^2.

Series expansion

Perform the Taylor series in x of an expression that contains a user-defined function f:

from symbolica import Expression

x, y = Expression.vars('x', 'y')
f = Expression.fun('f')

e = 2* x**2 * y + f(x)
e = e.taylor_series(x, 0, 2)

print(e)

which yields f(0)+x*der(1,f(0))+1/2*x^2*(4*y+der(2,f(0))).

Rational arithmetic

Symbolica is world-class in rational arithmetic, outperforming Mathematica, Maple, Form, Fermat, and other computer algebra packages. Simply convert an expression to a rational polynomial:

from symbolica import Expression
x, y = Expression.vars('x','y')
p = Expression.parse('(x*y^2*5+5)^2/(2*x+5)+(x+4)/(6*x^2+1)').to_rational_polynomial()
print(p)

which yields (45+13*x+50*x*y^2+152*x^2+25*x^2*y^4+300*x^3*y^2+150*x^4*y^4)/(5+2*x+30*x^2+12*x^3).

Development

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