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SymPy - Symbolic Mathematics in Python

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SymPy - Symbolic Mathematics in Python

Overview

SymPy is a Python library for symbolic mathematics that enables exact computation using mathematical symbols rather than numerical approximations. This skill provides comprehensive guidance for performing symbolic algebra, calculus, linear algebra, equation solving, physics calculations, and code generation using SymPy.

Installation

Tested against SymPy 1.14.0 (stable; April 2025). Requires Python 3.9+.

bash
# Install SymPy using uv
uv pip install "sympy>=1.14"

# Optional: for lambdify and plotting examples
uv pip install numpy scipy matplotlib

Check your version:

python
import sympy
print(sympy.__version__)

When to Use This Skill

Use this skill when:

  • Solving equations symbolically (algebraic, differential, systems of equations)
  • Performing calculus operations (derivatives, integrals, limits, series)
  • Manipulating and simplifying algebraic expressions
  • Working with matrices and linear algebra symbolically
  • Doing physics calculations (mechanics, quantum mechanics, vector analysis)
  • Number theory computations (primes, factorization, modular arithmetic)
  • Geometric calculations (2D/3D geometry, analytic geometry)
  • Converting mathematical expressions to executable code (Python, C, Fortran)
  • Generating LaTeX or other formatted mathematical output
  • Needing exact mathematical results (e.g., sqrt(2) not 1.414...)

Core Capabilities

Seven capability areas are documented in references/core_capabilities.md:

  1. Symbolic computation basics — symbols, expressions, simplification, substitution.
  2. Calculus — differentiation, integration, limits, series.
  3. Equation solving — solve, solveset, linear and nonlinear systems, ODEs.
  4. Matrices and linear algebra — see references/matrices-linear-algebra.md.
  5. Physics and mechanics — see references/physics-mechanics.md.
  6. Advanced mathematics — see references/advanced-topics.md.
  7. Code generation and output — see references/code-generation-printing.md.

Deeper treatment of the first three is in references/core-capabilities.md.

Working with SymPy: Best Practices

1. Always Define Symbols First

python
from sympy import symbols
x, y, z = symbols('x y z')
# Now x, y, z can be used in expressions

2. Use Assumptions for Better Simplification

python
x = symbols('x', positive=True, real=True)
sqrt(x**2)  # Returns x (not Abs(x)) due to positive assumption

Common assumptions: real, positive, negative, integer, rational, complex, even, odd

3. Use Exact Arithmetic

python
from sympy import Rational, S
# Correct (exact):
expr = Rational(1, 2) * x
expr = S(1)/2 * x

# Incorrect (floating-point):
expr = 0.5 * x  # Creates approximate value

4. Numerical Evaluation When Needed

python
from sympy import pi, sqrt
result = sqrt(8) + pi
result.evalf()    # 5.96371554103586
result.evalf(50)  # 50 digits of precision

5. Convert to NumPy for Performance

python
# Slow for many evaluations:
for x_val in range(1000):
    result = expr.subs(x, x_val).evalf()

# Fast:
f = lambdify(x, expr, 'numpy')
results = f(np.arange(1000))

6. Use Appropriate Solvers

  • solveset: Algebraic equations (primary)
  • linsolve: Linear systems
  • nonlinsolve: Nonlinear systems
  • dsolve: Differential equations
  • solve: General purpose (legacy, but flexible)

Reference Files Structure

This skill uses modular reference files for different capabilities:

  1. core-capabilities.md: Symbols, algebra, calculus, simplification, equation solving

    • Load when: Basic symbolic computation, calculus, or solving equations

  2. matrices-linear-algebra.md: Matrix operations, eigenvalues, linear systems

    • Load when: Working with matrices or linear algebra problems

  3. physics-mechanics.md: Classical mechanics, quantum mechanics, vectors, units

    • Load when: Physics calculations or mechanics problems

  4. advanced-topics.md: Geometry, number theory, combinatorics, logic, statistics

    • Load when: Advanced mathematical topics beyond basic algebra and calculus

  5. code-generation-printing.md: Lambdify, codegen, LaTeX output, printing

    • Load when: Converting expressions to code or generating formatted output

Common Use Case Patterns

Pattern 1: Solve and Verify

python
from sympy import symbols, solve, simplify
x = symbols('x')

# Solve equation
equation = x**2 - 5*x + 6
solutions = solve(equation, x)  # [2, 3]

# Verify solutions
for sol in solutions:
    result = simplify(equation.subs(x, sol))
    assert result == 0

Pattern 2: Symbolic to Numeric Pipeline

python
# 1. Define symbolic problem
x, y = symbols('x y')
expr = sin(x) + cos(y)

# 2. Manipulate symbolically
simplified = simplify(expr)
derivative = diff(simplified, x)

# 3. Convert to numerical function
f = lambdify((x, y), derivative, 'numpy')

# 4. Evaluate numerically
results = f(x_data, y_data)

Pattern 3: Document Mathematical Results

python
# Compute result symbolically
integral_expr = Integral(x**2, (x, 0, 1))
result = integral_expr.doit()

# Generate documentation
print(f"LaTeX: {latex(integral_expr)} = {latex(result)}")
print(f"Pretty: {pretty(integral_expr)} = {pretty(result)}")
print(f"Numerical: {result.evalf()}")

Integration with Scientific Workflows

With NumPy

python
import numpy as np
from sympy import symbols, lambdify

x = symbols('x')
expr = x**2 + 2*x + 1

f = lambdify(x, expr, 'numpy')
x_array = np.linspace(-5, 5, 100)
y_array = f(x_array)

With Matplotlib

python
import matplotlib.pyplot as plt
import numpy as np
from sympy import symbols, lambdify, sin

x = symbols('x')
expr = sin(x) / x

f = lambdify(x, expr, 'numpy')
x_vals = np.linspace(-10, 10, 1000)
y_vals = f(x_vals)

plt.plot(x_vals, y_vals)
plt.show()

With SciPy

python
from scipy.optimize import fsolve
from sympy import symbols, lambdify

# Define equation symbolically
x = symbols('x')
equation = x**3 - 2*x - 5

# Convert to numerical function
f = lambdify(x, equation, 'numpy')

# Solve numerically with initial guess
solution = fsolve(f, 2)

Quick Reference: Most Common Functions

python
# Symbols
from sympy import symbols, Symbol
x, y = symbols('x y')

# Basic operations
from sympy import simplify, expand, factor, collect, cancel
from sympy import sqrt, exp, log, sin, cos, tan, pi, E, I, oo

# Calculus
from sympy import diff, integrate, limit, series, Derivative, Integral

# Solving
from sympy import solve, solveset, linsolve, nonlinsolve, dsolve

# Matrices
from sympy import Matrix, eye, zeros, ones, diag

# Logic and sets
from sympy import And, Or, Not, Implies, FiniteSet, Interval, Union

# Output
from sympy import latex, pprint, lambdify, init_printing

# Utilities
from sympy import evalf, N, nsimplify

Getting Started Examples

Example 1: Solve Quadratic Equation

python
from sympy import symbols, solve, sqrt
x = symbols('x')
solution = solve(x**2 - 5*x + 6, x)
# [2, 3]

Example 2: Calculate Derivative

python
from sympy import symbols, diff, sin
x = symbols('x')
f = sin(x**2)
df_dx = diff(f, x)
# 2*x*cos(x**2)

Example 3: Evaluate Integral

python
from sympy import symbols, integrate, exp
x = symbols('x')
integral = integrate(x * exp(-x**2), (x, 0, oo))
# 1/2

Example 4: Matrix Eigenvalues

python
from sympy import Matrix
M = Matrix([[1, 2], [2, 1]])
eigenvals = M.eigenvals()
# {3: 1, -1: 1}

Example 5: Generate Python Function

python
from sympy import symbols, lambdify
import numpy as np
x = symbols('x')
expr = x**2 + 2*x + 1
f = lambdify(x, expr, 'numpy')
f(np.array([1, 2, 3]))
# array([ 4,  9, 16])

Troubleshooting Common Issues

  1. "NameError: name 'x' is not defined"

    • Solution: Always define symbols using symbols() before use

  2. Unexpected numerical results

    • Issue: Using floating-point numbers like 0.5 instead of Rational(1, 2)
    • Solution: Use Rational() or S() for exact arithmetic

  3. Slow performance in loops

    • Issue: Using subs() and evalf() repeatedly
    • Solution: Use lambdify() to create a fast numerical function

  4. "Can't solve this equation"

    • Try different solvers: solve, solveset, nsolve (numerical)
    • Check if the equation is solvable algebraically
    • Use numerical methods if no closed-form solution exists

  5. Simplification not working as expected

    • Try different simplification functions: simplify, factor, expand, trigsimp
    • Add assumptions to symbols (e.g., positive=True)
    • Use simplify(expr, force=True) for aggressive simplification

Additional Resources