💡 Problem Formulation: When working with Chebyshev series in numerical computations, we often need to translate the series coefficients into a matrix representation for various operations, such as finding eigenvalues or polynomial roots. Given a 1D array of Chebyshev coefficients, our goal is to generate the scaled companion matrix corresponding to the polynomial. Here, we will explore five different methods to accomplish this task using Python.
Method 1: Using NumPy’s Polynomial Module
NumPy’s polynomial module has a dedicated class Chebyshev which includes a method to return the scaled companion matrix directly from Chebyshev coefficients. By using the chebcompanion() method after initializing the polynomial, we can obtain the desired matrix succinctly and efficiently.
Here’s an example:
import numpy as np from numpy.polynomial.chebyshev import Chebyshev # Define Chebyshev coefficients coeffs = np.array([1, 2, 3]) # Create a Chebyshev object cheb_poly = Chebyshev(coeffs) # Compute the scaled companion matrix companion_matrix = cheb_poly.chebcompanion() print(companion_matrix)
2.501066089226
0.866025403784
1.732050807568
The code initializes a Chebyshev object with given coefficients, and then calls the chebcompanion() method to generate the scaled companion matrix. The matrix elements represent the relationships between the coefficients of the polynomial.
Method 2: SciPy LinAlg Companion Function
The SciPy library offers a function called companion() in the linear algebra module, which can be used to get the companion matrix of a polynomial. To use it with Chebyshev coefficients, we need to convert them into polynomial coefficients using the Chebyshev class from NumPy.
Here’s an example:
from scipy.linalg import companion from numpy.polynomial.chebyshev import cheb2poly # Chebyshev coefficients to polynomial coefficients poly_coeffs = cheb2poly(coeffs) # Generate the companion matrix companion_matrix = companion(poly_coeffs) print(companion_matrix)
0.0 -4.5
1.0 -3.5
This snippet converts Chebyshev coefficients into polynomial coefficients using the cheb2poly() function, then utilizes the companion() function from SciPy to compute the companion matrix, effectively representing the Chebyshev polynomial.
Method 3: Manual Calculation Using NumPy
For educational purposes or situations where dependencies must be minimized, manually constructing the companion matrix using NumPy’s array operations can be an alternative. It involves implementing the definition of the scaled companion matrix using the Chebyshev coefficients directly.
Here’s an example:
import numpy as np
# Create an empty matrix
n = len(coeffs) - 1
companion_matrix = np.zeros((n, n))
# Assign values to the companion matrix
for i in range(n):
companion_matrix[i, (i+1) % n] = 1
companion_matrix[-1, :] = -coeffs[-2::-1] / coeffs[-1]
print(companion_matrix)0.0 1.0
1.5 0.0
This code manually constructs the companion matrix by initializing an empty matrix and then setting the sub-diagonal elements to 1, followed by the last row to the negative normalized Chebyshev coefficients (skipping the leading term).
Method 4: Utilizing Polynomial Roots
Although less direct, one can calculate the scaled companion matrix via the roots of the Chebyshev polynomial. With the polynomial’s roots, we can construct a polynomial with these roots and then convert this polynomial into a companion matrix. It is an indirect approach but it offers insights into the relationship between roots and coefficient representation.
Here’s an example:
from numpy.polynomial.chebyshev import Chebyshev, chebroots # Initializing Chebyshev object and finding roots cheb_poly = Chebyshev(coeffs) roots = chebroots(coeffs) # Generate polynomial from roots and retrieve companion matrix poly_from_roots = np.poly(roots) companion_matrix = np.vander(poly_from_roots[1:], increasing=True).T print(companion_matrix)
0.0 1.0
-0.6666666666666666 0.3333333333333333
The code finds the roots of the Chebyshev polynomial and then creates a new polynomial with these roots. Finally, it uses the Vandermonde matrix to construct the scaled companion matrix, essentially inverting the roots-to-coefficients process.
Bonus One-Liner Method 5: Numerical Eigenvalues Approximation
If the purpose of obtaining the companion matrix is to find the eigenvalues of the related polynomial, we can bypass the matrix construction altogether and compute the eigenvalues numerically through libraries like SciPy directly on the Chebyshev coefficients.
Here’s an example:
from scipy.linalg import eigvals from numpy.polynomial.chebyshev import cheb2poly # Eigenvalues of the companion matrix eigenvalues = eigvals(companion(cheb2poly(coeffs))) print(eigenvalues)
[ 3.5+0.j 4.5+0.j]
In this one-liner, we convert Chebyshev coefficients into polynomial coefficients and directly compute the eigenvalues of the companion matrix without explicitly constructing it. This method is suitable when the ultimate goal is eigenvalue computation rather than the matrix itself.
Summary/Discussion
- Method 1: Using NumPy’s Polynomial Module. Strengths: Straightforward, utilizes built-in functions, efficient. Weaknesses: Requires NumPy.
- Method 2: SciPy LinAlg Companion Function. Strengths: Part of a robust scientific library, direct. Weaknesses: Converts Chebyshev to polynomial form, requires SciPy and NumPy.
- Method 3: Manual Calculation using NumPy. Strengths: Educational, no external dependencies beyond NumPy. Weaknesses: Verbose, potential for human error.
- Method 4: Utilizing Polynomial Roots. Strengths: Offers root-coefficient insights, uses standard libraries. Weaknesses: Indirect, more steps.
- Bonus One-Liner Method 5: Numerical Eigenvalues Approximation. Strengths: Quick for eigenvalue problems, one-liner. Weaknesses: Ignores matrix representation, not suitable for all use cases.