Symbolic Mathematics with SymPy

Overview

In this article, I will explain SymPy, a Python library for symbolic mathematics. Using SymPy allows you to perform algebraic calculations, calculus, matrix operations, and more symbolically (analytically).

This article covers the following topics:

  1. Basic Operations
  2. Algebra (Expansion, Factorization, Solving Equations)
  3. Calculus (Differentiation, Integration)
  4. Linear Algebra (Matrices)

Author’s Environment

The author’s environment is as follows.

!sw_vers

ProductName:		macOS
ProductVersion:		15.5
BuildVersion:		24F74

!python -V

Python 3.12.12

Load necessary libraries. We will mainly use sympy this time.

import sympy
from sympy import symbols, expand, factor, solve, diff, integrate, sin, cos, Matrix, init_printing, pprint
import numpy as np
from pprint import pprint as py_pprint

# Settings for pretty printing of math formulas
init_printing()

print("sympy version :", sympy.__version__)

sympy version : 1.14.0

1. Basic Operations

In SymPy, by defining variables as Symbols, you can handle mathematical expressions as they are. Here, we define variables $x, y$ and create a simple expression.

x, y = symbols('x y')
expr = x + 2*y
expr

$\displaystyle x + 2 y$

2. Algebra

Expansion

We use the expand function to expand mathematical expressions. As an example, we expand $(x + 1)^2$.

expr = (x + 1)**2
expanded_expr = expand(expr)
expanded_expr

$\displaystyle x^{2} + 2 x + 1$

Factorization

We use the factor function for factorization. As an example, we factorize $x^2 + 2x + 1$.

factored_expr = factor(expanded_expr)
factored_expr

$\displaystyle \left(x + 1\right)^{2}$

Solving Equations

We use the solve function to solve equations. As an example, we solve the quadratic equation $x^2 - 3x + 2 = 0$ for $x$.

eq = x**2 - 3*x + 2
solutions = solve(eq, x)

# Output list results with pprint
py_pprint(solutions)

[1, 2]

3. Calculus

Differentiation

We use the diff function to perform differentiation. As an example, we differentiate $\sin(x)$ with respect to $x$.

diff_expr = diff(sin(x), x)
diff_expr

$\displaystyle \cos{\left(x \right)}$

Integration

We use the integrate function to perform integration. As an example, we find the indefinite integral of $\cos(x)$ with respect to $x$.

int_expr = integrate(cos(x), x)
int_expr

$\displaystyle \sin{\left(x \right)}$

Definite integration is also possible. We calculate $\int_{0}^{\pi} \sin(x) dx$.

def_int_expr = integrate(sin(x), (x, 0, sympy.pi))
def_int_expr

$\displaystyle 2$

4. Linear Algebra (Matrices)

Matrices are defined using the Matrix class. Here, we define matrix $\mathbf{A}$ as follows:

$$ \mathbf{A} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} $$

A = Matrix([[1, 2], [3, 4]])
A

$\displaystyle \left[\begin{matrix}1 & 2\\3 & 4\end{matrix}\right]$

Determinant

The determinant can be calculated with the det method.

det_A = A.det()
det_A

$\displaystyle -2$

Inverse Matrix

The inverse matrix can be calculated with the inv method.

inv_A = A.inv()
inv_A

$\displaystyle \left[\begin{matrix}-2 & 1\\\frac{3}{2} & - \frac{1}{2}\end{matrix}\right]$

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors can be calculated with the eigenvects method.

eigen_info = A.eigenvects()
display(eigen_info)

$\displaystyle \left[ \left( \frac{5}{2} - \frac{\sqrt{33}}{2}, \ 1, \ \left[ \left[\begin{matrix}- \frac{\sqrt{33}}{6} - \frac{1}{2}\\1\end{matrix}\right]\right]\right), \ \left( \frac{5}{2} + \frac{\sqrt{33}}{2}, \ 1, \ \left[ \left[\begin{matrix}- \frac{1}{2} + \frac{\sqrt{33}}{6}\\1\end{matrix}\right]\right]\right)\right]$

Conclusion

In this article, I explained the basics of symbolic mathematics using Python’s SymPy library.

Using SymPy allows for accurate execution of complex calculations within programs. The features introduced in this article are just a part of SymPy, but combining them will likely enable you to tackle a wide variety of mathematical problems.

References