Introduction to SymPy

SymPy is a powerful Python library for symbolic mathematics. It provides tools for executing algebraic computations, calculus, equation solving, and more, entirely in Python. Its symbolic nature makes it a go-to library for researchers, educators, and developers working on mathematical-based projects.

Why Use SymPy?

SymPy is built to cater to users who need flexibility and precision in mathematical computations. Its APIs are intuitive, and it integrates seamlessly with Python, making it an excellent choice for both standalone mathematical applications and integration into larger projects.

Basic Usage Examples of SymPy

Below are well-abbreviated examples showcasing dozens of essential SymPy APIs for various mathematical tasks.

1. Symbol Declaration

  from sympy import symbols

  x, y = symbols('x y')
  print(x + y)  # Output: x + y

2. Simplification

  from sympy import simplify

  expr = simplify("(x**2 + 2*x + 1) / (x + 1)")
  print(expr)  # Output: x + 1

3. Solving Equations

  from sympy import Eq, solve

  equation = Eq(x**2 - 2, 0)
  roots = solve(equation, x)
  print(roots)  # Output: [-sqrt(2), sqrt(2)]

4. Calculus Operations

Differentiation and Integration are straightforward in SymPy.

Differentiation

  from sympy import diff

  derivative = diff(x**3, x)
  print(derivative)  # Output: 3*x**2

Integration

  from sympy import integrate

  integral = integrate(x**3, x)
  print(integral)  # Output: x**4 / 4

5. Matrix Operations

  from sympy import Matrix

  matrix = Matrix([[1, 2], [3, 4]])
  determinant = matrix.det()
  inverse = matrix.inv()
  print(f"Determinant: {determinant}, Inverse: {inverse}")

6. Series Expansion

  from sympy import series, exp

  expansion = series(exp(x), x, 0, 5)
  print(expansion)  # Output: 1 + x + x**2/2 + x**3/6 + x**4/24 + O(x**5)

7. Plotting

  from sympy.plotting import plot

  plot(x**2, (x, -10, 10))

8. Solving Systems of Equations

  from sympy import solve

  solutions = solve([x + y - 2, x - y - 0], (x, y))
  print(solutions)  # Output: {x: 1, y: 1}

9. Boolean Algebra

  from sympy import Or, And, Not

  expr = And(Or(x, y), Not(x))
  print(expr.simplify())  # Output: And(Not(x), y)

SymPy Application Example: Loan Repayment Calculator

Below is an example of a practical application that uses SymPy to calculate loan repayments symbolically.

  from sympy import symbols, Eq, solve

  # Symbolic variables
  principal, rate, time, repayment = symbols('principal rate time repayment')

  # Equation for monthly repayment
  equation = Eq(repayment, principal * rate * (1 + rate)**time / ((1 + rate)**time - 1))

  # Solve for repayment when other values are known
  values = {principal: 100000, rate: 0.05 / 12, time: 12 * 30}  # $100,000 loan for 30 years at 5% annual rate
  monthly_repayment = solve(equation.subs(values), repayment)
  print(monthly_repayment)  # Output: [536.82...] (approx.)

This example demonstrates the use of symbolic computation to derive and solve equations for practical financial calculations.

Conclusion

SymPy is a versatile library that serves as a cornerstone for symbolic computation in Python. Its APIs cover a broad spectrum of mathematical domains, making it invaluable for academic, financial, engineering, or scientific applications.