Solve Complex Equations in Python

Complex numbers seem scary, but Python can help you understand and solve equations with them. This article explores the concept of solving complex equations in Python and various approaches to solve complex equations.

Solve Complex Equations in Python

Below, are the approaches for solving complex equations in Python:

• Using Symbolic Mathematics with SymPy
• Using Numerical Solver with SciPy
• Using Numpy for Roots of Polynomials
• Using Iterative Methods (e.g., Newton’s Method)

Solve Complex Equations Using Symbolic Mathematics with SymPy

The below approach code uses the SymPy library to solve a complex equation (z**2 + 1 = 0) symbolically. It defines the variable z, sets up the equation, and utilizes SymPy’s solve function to find the solutions, printing the results. In this specific example, it finds the solutions for the equation z**2 + 1 = 0, which corresponds to complex numbers satisfying the equation.

Python3

# importing from sympy library
from sympy import symbols, Eq, solve, I

# defining the symbolic variable 'z'
z = symbols('z')

# setting up the complex equation z^2 + 1 = 0
equation = Eq(z**2 + 1, 0)

# solving the equation symbolically to find complex solutions
solutions = solve(equation, z)

# printing solutions
print("Solutions:", solutions)

Output:

Solutions: [-I, I]

Solve Complex Equations Using Numerical Solver with SciPy

This below approach code uses SciPy’s fsolve to find the root of a system of complex equations. The complex_equation_to_solve function defines the equations, and the initial guess is provided with real and imaginary parts, yielding a complex solution that is then printed.

Python3

# import necessary functions from scipy library
from scipy.optimize import fsolve
import numpy as np

# define the function representing the system of equations for the complex problem
def complex_equation_to_solve(z):
    return [z[0]**2 + z[1]**2 - 1, 2*z[0] + z[1] - 2]

# set an initial guess with real and imaginary parts
initial_guess = np.array([0.0, 1.0])

# use fsolve to find the root of the complex equation
complex_root = fsolve(complex_equation_to_solve, initial_guess)

# create a complex number from the obtained root
complex_solution = complex(complex_root[0], complex_root[1])

# print the result
print("SciPy Complex Solution:", complex_solution)

Output:

SciPy Complex Solution: (0.6+0.7999999999999999j)

Solve Complex Equations Using Numpy for Roots of Polynomials

The below approach code uses NumPy’s np.roots to find the complex roots of the polynomial equation z**2 + 1 = 0, represented by the coefficients [1, 0, 1]. The complex roots are then printed as the output.

Python3

import numpy as np

# defining the coefficients of the complex polynomial equation
coefficients = [1, 0, 1]

# finding the roots of the polynomial equation
complex_roots = np.roots(coefficients)

# printing output
print("NumPy Complex Roots:", complex_roots)

Output:

NumPy Complex Roots: [-0.+1.j  0.-1.j]

Solve Complex Equations Using Iterative Methods (e.g., Newton’s Method)

The below approach code applies Newton’s method iteratively to find the complex root of the equation z²+1=0, starting with an initial guess of 0.0+1.0j. The result is printed as the complex solution after the specified maximum number of iterations (100). We can adjust the equation and initial guess as needed for different complex problems.

Python3

import numpy as np

# defining the complex equation to solve
def complex_equation_to_solve(z):
    return z**2 + 1

# defining the derivative of the equation for Newton's method
def complex_equation_derivative(z):
    return 2 * z

# setting an initial guess with a complex number
initial_guess = complex(0.0, 1.0)

# setting the maximum number of iterations
max_iterations = 100

# Newton's method iteratively
for _ in range(max_iterations):
    initial_guess -= complex_equation_to_solve(
        initial_guess) / complex_equation_derivative(initial_guess)

complex_solution = initial_guess

# printing output
print("Newton's Method Complex Solution:", complex_solution)

Output:

Newton's Method Complex Solution: 1j

Conclusion

In conclusion, Python offers a variety of powerful tools and libraries for solving complex equations efficiently. libraries such as SymPy, SciPy, and NumPy, users can tackle mathematical problems with ease. SymPy provides symbolic mathematics capabilities, allowing users to manipulate algebraic expressions and solve equations symbolically.