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IPython Cookbook, Second Edition This is one of the 100+ free recipes of the IPython Cookbook, Second Edition, by Cyrille Rossant, a guide to numerical computing and data science in the Jupyter Notebook. The ebook and printed book are available for purchase at Packt Publishing.

▶ Text on GitHub with a CC-BY-NC-ND license
▶ Code on GitHub with a MIT license

Chapter 15 : Symbolic and Numerical Mathematics

15.2. Solving equations and inequalities

SymPy offers several ways to solve linear and nonlinear equations and systems of equations. Of course, these functions do not always succeed in finding closed-form exact solutions. In this case, we can fall back to numerical solvers and obtain approximate solutions.

How to do it...

  1. Let's define a few symbols:
from sympy import *
init_printing()
var('x y z a')

(x, y, z, a)

  1. We use the solve() function to solve equations (the right-hand side is 0 by default):
solve(x**2 - a, x)

Output

  1. We can also solve inequalities. Here, we need to use the solve_univariate_inequality() function to solve this univariate inequality in the real domain:
x = Symbol('x')
solve_univariate_inequality(x**2 > 4, x)

Output

  1. The solve() function also accepts systems of equations (here, a linear system):
solve([x + 2*y + 1, x - 3*y - 2], x, y)

Output

  1. Nonlinear systems are also handled:
solve([x**2 + y**2 - 1, x**2 - y**2 - S(1) / 2], x, y)

Output

  1. Singular linear systems can also be solved (here, there is an infinite number of solutions because the two equations are collinear):
solve([x + 2*y + 1, -x - 2*y - 1], x, y)

Output

  1. Now, let's solve a linear system using matrices containing symbolic variables:
var('a b c d u v')

Output

  1. We create the augmented matrix, which is the horizontal concatenation of the system's matrix with the linear coefficients and the right-hand side vector. This matrix corresponds to the following system in $x, y$: $ax+by=u, cx+dy=v$:
M = Matrix([[a, b, u], [c, d, v]])
M

Output

solve_linear_system(M, x, y)

Output

  1. This system needs to be nonsingular in order to have a unique solution, which is equivalent to saying that the determinant of the system's matrix needs to be nonzero (otherwise the denominators in the preceding fractions are equal to zero):
det(M[:2, :2])

Output

There's more...

Matrix support in SymPy is quite rich; we can perform a large number of operations and decompositions (see the reference guide at http://docs.sympy.org/latest/modules/matrices/matrices.html).

Here are more references about linear algebra: