This is one of the 100 recipes of the IPython Cookbook, the definitive guide to high-performance scientific computing and data science in Python.

15.1. Diving into symbolic computing with SymPy

SymPy is a pure Python package for symbolic mathematics.

First, we import SymPy, and enable rich display LaTeX-based printing in the IPython notebook (using the MathJax Javascript library).

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from sympy import *
init_printing()

With NumPy and the other packages we have been using so far, we were dealing with numbers and numerical arrays. With SymPy, we deal with symbolic variables. It's a radically different shift of paradigm, which mathematicians may be more familiar with.

To deal with symbolic variables, we need to declare them.

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var('x y')

The var function creates symbols and injects them into the namespace. This function should only be used in interactive mode. In a Python module, it is better to use the symbol function which returns the symbols.

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x, y = symbols('x y')

We can create mathematical expressions with these symbols.

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expr1 = (x + 1)**2
expr2 = x**2 + 2*x + 1

Are these expressions equal?

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expr1 == expr2

These expressions are mathematically equal, but not syntactically identical. To test whether they are equal, we can ask SymPy to simplify the difference algebraically.

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simplify(expr1-expr2)

A very common operation with symbolic expressions is substitution of a symbol by another symbol, expression, or a number.

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expr1.subs(x, expr1)
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expr1.subs(x, pi)

A rational number cannot be written simply as "1/2" as this Python expression evaluates to 0. A possibility is to use a SymPy object for 1, for example using the function S.

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expr1.subs(x, S(1)/2)

Exactly-represented numbers can be evaluated numerically with evalf:

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_.evalf()

We can transform this symbolic function into an actual Python function that can be evaluated on NumPy arrays, using the lambdify function.

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f = lambdify(x, expr1)
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import numpy as np
f(np.linspace(-2., 2., 5))

You'll find all the explanations, figures, references, and much more in the book (to be released later this summer).

IPython Cookbook, by Cyrille Rossant, Packt Publishing, 2014 (500 pages).