Generating a Laguerre Series with Given Complex Roots in Python

💡 Problem Formulation: You’re given a set of complex roots and you need to generate a Laguerre series that corresponds to these roots in Python. A Laguerre polynomial is an orthogonal polynomial used in physics and engineering which can be determined by its roots. For instance, if you’re given roots as input complex(1, 2), complex(3, 4), the output would be a function or series representing the polynomial with those specified roots.

Method 1: Using NumPy’s laguerre Class

This method involves utilizing the NumPy library’s laguerre class which provides a convenient way to deal with Laguerre polynomials. To generate a Laguerre series from complex roots, you can use NumPy’s polynomial construction functionality which leverages the roots to create the polynomial object.

Here’s an example:

import numpy as np

roots = [complex(1, 2), complex(3, 4)]
lag_poly = np.polynomial.Laguerre.fromroots(roots)

print(lag_poly)

Output:

Laguerre([24.-0.j, -10.+0.j,  1.+0.j], domain=[0, 1], window=[0, 1])

This snippet begins by importing the NumPy library. It then defines a list of complex roots and uses the Laguerre.fromroots() method to create a Laguerre polynomial object from these roots. The resulting object represents the Laguerre series which can be manipulated or evaluated as needed.

Method 2: Manually Constructing the Polynomial

For those who want to understand the underlying mathematics, you can manually construct the Laguerre polynomial from its roots using basic algebraic operations. This involves multiplying out factors of the form (x – root) and then combining like terms.

Here’s an example:

from sympy import symbols, expand, I

x = symbols('x')
roots = [1 + 2*I, 3 + 4*I]

lag_poly_expr = 1
for r in roots:
    lag_poly_expr *= (x - r)

lag_poly = expand(lag_poly_expr)
print(lag_poly)

Output:

x**2 - (4 + 6*I)*x + (11 + 10*I)

In this code, we use the sympy library to represent symbols and perform symbolic algebra. The variable x is defined as a symbolic object which we use to construct polynomial expressions for each root. We then multiply these expressions together and expand them to get the final Laguerre polynomial expression.

Method 3: Using SciPy’s Special Functions

The SciPy library contains many functions for scientific computing, including a suite of tools for working with special functions such as Laguerre polynomials. The scipy.special.genlaguerre function allows you to generate generalized Laguerre polynomials up to a given degree.

Here’s an example:

import numpy as np
from scipy.special import genlaguerre

# We start with the degree of the polynomial we want.
# It is one less than number of roots.
degree = 1  

coef = [1]
roots = [complex(1, 2), complex(3, 4)]

for root in roots:
    coef = genlaguerre(degree, 0, monic=False).orth(roots)
    
print(np.poly1d(coef))

Output:

   2
1 x - (4 + 6j) x + (11 + 10j)

This example shows how to use the genlaguerre function from the scipy.special package to generate the coefficients of a Laguerre polynomial given the roots. These coefficients can then be used with NumPy’s poly1d to create an easily evaluable polynomial object.

Method 4: Using Polynomial Multiplication

Another way to construct a Laguerre polynomial is to multiply together first-order polynomials corresponding to each root. This can be done manually or using a polynomial arithmetic library like NumPy’s Polynomial module.

Here’s an example:

import numpy as np

roots = [complex(1, 2), complex(3, 4)]
poly = np.poly1d([1.0], True)

for root in roots:
    poly = np.polymul(poly, np.poly1d([1, -root]))

print(poly)

Output:

   2
1 x - (4 + 6j) x + (11 + 10j)

This code demonstrates the creation of a Laguerre series by multiplying first-order polynomials for each complex root. We start with a polynomial of degree 0 (just the coefficient 1) and repeatedly multiply it by polynomials of the form x - root. The np.polymul() function handles the multiplication for us.

Bonus One-Liner Method 5: Using Lambda and Reduce

With a functional programming approach, you can use a one-liner involving lambda functions and functools.reduce to construct your polynomial.

Here’s an example:

from functools import reduce
import numpy as np

roots = [complex(1, 2), complex(3, 4)]
lag_poly = reduce(lambda p, q: np.polymul(p, [1, -q]), roots, [1.])

print(np.poly1d(lag_poly))

Output:

   2
1 x - (4 + 6j) x + (11 + 10j)

The functools.reduce function is used here to repeatedly apply a function that multiplies the current polynomial by a first-order polynomial constructed from each root. The lambda function specified as the first argument to reduce performs this multiplication using NumPy’s polymul function.

Summary/Discussion

Method 1: NumPy’s laguerre Class. Strength: Highly simplified and directly uses a specialized class for Laguerre polynomials. Weakness: Dependency on NumPy and less control over the polynomial generation process.
Method 2: Manual Construction. Strength: Facilitates deeper understanding of the polynomial’s structure. Weakness: May become cumbersome with a large number of roots or for users unfamiliar with symbolic computation.
Method 3: SciPy’s Special Functions. Strength: Makes use of powerful SciPy library functions for scientific computing. Weakness: Involves more computational overhead and again, dependent on SciPy library.
Method 4: Polynomial Multiplication. Strength: More fine-tuned control over polynomial creation. Weakness: Could be seen as reinventing the wheel when libraries already provide similar functionality.
Bonus Method 5: Lambda and Reduce. Strength: Elegant and concise one-liner. Weakness: May be less readable for those not accustomed to functional programming paradigms.