💡 Problem Formulation: When working with approximations and polynomials in numerical calculations, it’s occasionally necessary to divide one Chebyshev series by another. Chebyshev series allow for efficient computations, but division can be non-trivial. This article discusses five methods to achieve this in Python, given two Chebyshev series C1 and C2, and aims to find a new series C3 that represents the division of C1 by C2.
Method 1: Using numpy.polynomial.chebyshev.Chebyshev Class
This method involves the NumPy library’s polynomial package, which provides a class specifically for Chebyshev series. The Chebyshev class offers a __div__ method for dividing two Chebyshev series objects directly, returning the quotient as a new Chebyshev object.
Here’s an example:
from numpy.polynomial.chebyshev import Chebyshev # The coefficients of the Chebyshev series are in ascending order C1_coeffs = [1, 2, 3] C2_coeffs = [1, 1] C1 = Chebyshev(C1_coeffs) C2 = Chebyshev(C2_coeffs) C3 = C1 / C2 print(C3)
The output will be a new Chebyshev series object containing the coefficients of the divided series.
In this snippet, Chebyshev series C1 is divided by series C2 using the division operator, which under the hood calls the __div__ method. The result is printed, showing the resulting Chebyshev object which represents the quotient series.
Method 2: Polynomial Long Division
Similar to how long division works for numbers, long division can be applied to polynomials, and by extension, Chebyshev series. This involves manually computing the coefficients of the quotient using a divide and conquer strategy until the remainder is of lesser degree than the divisor.
Here’s an example:
from numpy.polynomial import chebyshev
def chebyshev_long_division(C1_coeffs, C2_coeffs):
# Initialize quotient and remainder
quotient = []
remainder = list(C1_coeffs)
while len(remainder) >= len(C2_coeffs):
lead_coeff = remainder[0] / C2_coeffs[0]
quotient.append(lead_coeff)
subtracted = [coeff * lead_coeff for coeff in C2_coeffs] + [0] * (len(remainder) - len(C2_coeffs))
remainder = [rem - sub for rem, sub in zip(remainder, subtracted)][1:]
return quotient, remainder
C1_coeffs = [1, 2, 3]
C2_coeffs = [1, 1]
quotient, remainder = chebyshev_long_division(C1_coeffs, C2_coeffs)
print("Quotient:", quotient)
print("Remainder:", remainder)The output will show the calculated quotient and remainder as lists of coefficients.
This code uses a custom function to perform long division on the coefficients of two Chebyshev series. The function returns both the quotient and the remainder after division. The main loop continues until the remainder’s degree is less than that of the divisor, replicating the long division process.
Method 3: Using NumPy’s polydiv Function
NumPy’s polydiv function is meant for dividing polynomials but can be adapted for use with Chebyshev series by converting between the standard polynomial basis and the Chebyshev basis using the library’s functions.
Here’s an example:
from numpy.polynomial import chebyshev
from numpy.polynomial.polynomial import polydiv
C1_coeffs = [1, 2, 3]
C2_coeffs = [1, 1]
# Convert to standard polynomial basis
p1_coeffs = chebyshev.cheb2poly(C1_coeffs)
p2_coeffs = chebyshev.cheb2poly(C2_coeffs)
# Perform polynomial division
quotient_coeffs, remainder_coeffs = polydiv(p1_coeffs, p2_coeffs)
# Convert back to Chebyshev basis
C3_coeffs = chebyshev.poly2cheb(quotient_coeffs)
remainder_cheb_coeffs = chebyshev.poly2cheb(remainder_coeffs)
print("Quotient:", C3_coeffs)
print("Remainder:", remainder_cheb_coeffs)The output will display the quotient and remainder Chebyshev coefficients obtained after converting back from the standard polynomial basis.
This snippet converts the given Chebyshev coefficients to the standard polynomial basis, uses the NumPy polydiv function to perform division, and then converts the quotient and remainder back to the Chebyshev basis. This yields the division results in the desired Chebyshev form.
Method 4: Using scipy.interpolate.Chebyshev Series
The SciPy library extends NumPy’s functionalities and provides many additional modules for scientific computations, one of which is scipy.interpolate for interpolation purposes, including working with Chebyshev series.
Here’s an example:
from scipy.interpolate import Chebyshev C1_coeffs = [1, 2, 3] C2_coeffs = [1, 1] C1 = Chebyshev(C1_coeffs) C2 = Chebyshev(C2_coeffs) C3 = C1.__div__(C2) print(C3.coefficients)
The output will be an array displaying the coefficients of the quotient series.
Here we use the SciPy library’s Chebyshev class instead of NumPy’s. The Chebyshev objects are created from the given series coefficients and the division is accomplished using the __div__ method. The coefficients of the result are printed.
Bonus One-Liner Method 5: Chebyshev Series Division Using numpy.polynomial Package
For those who prefer concise code, the numpy.polynomial package allows for a one-liner division of Chebyshev series using a functional approach.
Here’s an example:
from numpy.polynomial.chebyshev import chebdiv
C1_coeffs = [1, 2, 3]
C2_coeffs = [1, 1]
C3_coeffs, remainder = chebdiv(C1_coeffs, C2_coeffs)
print("Quotient:", C3_coeffs)
print("Remainder:", remainder)The output will show the quotient and remainder arrays, representing the Chebyshev coefficients after division.
This concise line of code uses the chebdiv function to divide two sets of Chebyshev coefficients. It returns both the quotient and the remainder efficiently. The result is printed out for confirmation.
Summary/Discussion
- Method 1: Using the Chebyshev class from NumPy. Strengths: Officially supported and straightforward for users familiar with NumPy. Weaknesses: May not handle all edge cases smoothly.
- Method 2: Implementing polynomial long division manually. Strengths: Complete control over the division process. Weaknesses: Can be error-prone and is not as efficient as library functions.
- Method 3: Utilizing NumPy’s polydiv function. Strengths: Uses NumPy’s robust polynomial division. Weaknesses: Requires conversion between polynomial bases, which might introduce round-off errors.
- Method 4: Applying SciPy’s enhanced Chebyshev support. Strengths: SciPy provides specialized functionalities for scientific computations. Weaknesses: An additional dependency beyond NumPy that may not be necessary for all applications.
- Bonus One-Liner Method 5: Using numpy.polynomial’s chebdiv function. Strengths: Very concise, simple syntax. Weaknesses: Can obscure understanding of the underlying process for beginners.