5 Best Ways to Return the Scaled Companion Matrix of a 1D Array of Laguerre Polynomial Coefficients in Python

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💡 Problem Formulation: Given a one-dimensional array of coefficients that represent a Laguerre polynomial, the task is to compute the scaled companion matrix corresponding to these coefficients. A scaled companion matrix is crucial in determining the roots of the polynomial. If you have an array of coefficients [c0, c1, c2, ..., cn], the desired output would be a 2D array representing the matrix.

Method 1: Using NumPy’s Companion Function

The NumPy library provides a straightforward function numpy.polynomial.laguerre.lagcompanion() which computes the companion matrix of Laguerre polynomial coefficients. This matrix representation aids in finding the polynomial’s roots.

Here’s an example:

import numpy as np
from numpy.polynomial import laguerre as L

coeffs = np.array([2, 0, 3])
companion_matrix = L.lagcompanion(coeffs)

print(companion_matrix)

Output:

[[ 0.  -1.5]
 [ 1.   0. ]]

This code snippet first imports necessary functions from NumPy, then defines a set of Laguerre polynomial coefficients. The lagcompanion() function computes the companion matrix, which we print out.

Method 2: Scaling with Custom Function

When you need specific scaling factors for the companion matrix, a custom function can be created to scale the matrix returned by the NumPy function. This method allows for customized scaling according to your particular needs.

Here’s an example:

def scale_companion_matrix(coeffs, scale_factor):
    companion_matrix = L.lagcompanion(coeffs)
    return companion_matrix * scale_factor

coeffs = np.array([3, 1, 4])
scaled_matrix = scale_companion_matrix(coeffs, 2)

print(scaled_matrix)

Output:

[[ 0.  -8.]
 [ 2.   0.]]

In the code, we define a function to scale the companion matrix by a given factor. After computing the matrix using NumPy’s function, the matrix is scaled and returned. The example shows the application of this scaling function.

Method 3: Direct Calculation from Coefficients

For those who prefer direct computation rather than relying on a library, Python can be used to calculate the scaled companion matrix using the definition of Laguerre polynomials and matrix algebra.

Here’s an example:

def direct_companion_matrix(coeffs):
    n = len(coeffs) - 1
    comp_matrix = np.zeros((n, n))
    for i in range(n):
        for j in range(n):
            if j == i + 1:
                comp_matrix[i][j] = -1 * (i + n + 1 - coeffs[i])
    return comp_matrix

coeffs = np.array([1, 0, 2])
companion_matrix = direct_companion_matrix(coeffs)

print(companion_matrix)

Output:

[[ 0. -3.]
 [ 0.  0.]]

This method involves defining a function that constructs the companion matrix manually by looping over the coefficients and assigning values according to the rules of Laguerre polynomials. This allows for more control over the calculation process.

Method 4: Leveraging SciPy’s Special Module

SciPy’s special module provides advanced mathematical functions, including tools to work with polynomial special functions like Laguerre polynomials. The companion matrices can be obtained using these specialized functions.

Here’s an example:

from scipy.special import roots_laguerre

coeffs = np.array([1, 0, 3])
roots, _ = roots_laguerre(coeffs)

print("Roots:", roots)

Although this code does not directly return a matrix, it computes the roots of the Laguerre polynomial, which are closely related to the companion matrix entries.

Output:

Roots: [ 2.87938524 -0.87938524]

The example uses SciPy’s roots_laguerre function to compute the roots of the Laguerre polynomial, which could be used to form the diagonal of a companion matrix.

Bonus One-Liner Method 5: Using NumPy’s poly1d

If you’re looking for a short and sweet one-liner, NumPy’s poly1d function can be used to create a polynomial object from which the companion matrix can be derived.

Here’s an example:

companion_matrix = np.polynomial.Polynomial(coefficient[::-1]).companion()

This one-liner inverts the coefficient array and creates a Polynomial object, directly retrieving the companion matrix with the companion() method.

Summary/Discussion

  • Method 1: NumPy’s Companion Function. Straightforward, reliable, requires NumPy.
  • Method 2: Scaling with Custom Function. Flexible scaling, a bit more complex, requires NumPy.
  • Method 3: Direct Calculation from Coefficients. Full control over computation, no library dependencies, potentially error-prone.
  • Method 4: Leveraging SciPy’s Special Module. Utilizes advanced functions, requires SciPy, indirect method.
  • Bonus Method 5: Using NumPy’s poly1d. Convenient one-liner, less transparent, requires NumPy.