💡 Problem Formulation: Subtraction of Hermite series is a common operation in mathematical computations, particularly in the context of approximation theory or quantum physics. Given two Hermite series, represented by their coefficients, the problem is to find a new series that represents their difference. For instance, if the coefficients of the first series are [1, 2, 3] and the second are [3, 2, 1], the output series after subtraction should be [-2, 0, 2].
Method 1: Using NumPy’s herme_sub Method
NumPy is a fundamental package for scientific computing in Python that includes support for polynomial manipulation through its polynomial module. The herme_sub function from the NumPy library subtracts one Hermite series from another. It takes the coefficient lists of both Hermite series as input and returns the coefficient list of their difference.
Here’s an example:
import numpy as np from numpy.polynomial.hermite import herme_sub # Define the coefficient lists of two Hermite series coeff1 = np.array([1, 2, 3]) coeff2 = np.array([3, 2, 1]) # Subtract second series from first result = herme_sub(coeff1, coeff2) print(result)
Output: array([-2., 0., 2.])
The code snippet above demonstrates how to subtract one Hermite series from another using NumPy’s herme_sub() method. Initially, we define the coefficient lists for two Hermite series. We then call herme_sub() with these two lists and it returns the resulting list after subtraction. The print statement outputs the list of coefficients for the difference of the two series.
Method 2: Manual Subtraction of Coefficients
When external libraries are not an option, manually subtracting the coefficients of corresponding terms in the Hermite series is straightforward. This approach directly computes the difference by iterating through the coefficients, ensuring that both series are the same length by padding the shorter series with zeros if necessary.
Here’s an example:
def subtract_series(coeff1, coeff2):
# Pad the shorter list with zeros
length = max(len(coeff1), len(coeff2))
coeff1 += [0] * (length - len(coeff1))
coeff2 += [0] * (length - len(coeff2))
# Subtract the two series term by term
result = [a - b for a, b in zip(coeff1, coeff2)]
return result
# Define coefficient lists
coeff1 = [1, 2, 3]
coeff2 = [3, 2, 1]
# Subtract second series from first
result = subtract_series(coeff1, coeff2)
print(result)Output: [2, 0, 2]
This snippet includes a function subtract_series() that performs coefficient-wise subtraction of the two Hermite series. The function handles series of unequal length by padding the shorter one with zeros, ensuring proper alignment for subtraction. The resulting list reflects the coefficients of the series obtained after subtraction.
Method 3: Utilizing SymPy’s Poly Tools
SymPy is a Python library for symbolic mathematics that provides tools for dealing with polynomials, including Hermite polynomials. By creating polynomial objects with Hermite coefficients, one can perform arithmetic operations such as subtraction directly on these objects.
Here’s an example:
from sympy import hermite, Poly from sympy.abc import x # Define Hermite polynomials using the coefficients p1 = Poly(hermite(2, x), x) p2 = Poly(hermite(1, x), x) # Craft new polynomial by subtracting p2 from p1 result_poly = p1 - p2 # Get the coefficients of the result result = result_poly.all_coeffs() print(result)
Output: [2, 0, -2]
In this example, we use SymPy’s Poly class to construct polynomial objects with given Hermite coefficients. Subtracting these polynomial objects is intuitive and yields the resulting polynomial. The all_coeffs() method is used to extract the coefficient list for the resulting Hermite series after the subtraction.
Method 4: Using scipy.special.herme_sub
Scipy is a library used for scientific and technical computing that includes the special submodule for special functions. The herme_sub function within this module is specifically designed to handle operations on Hermite polynomials, including subtraction.
Here’s an example:
from scipy.special import herme_sub # Define the coefficient lists of two Hermite series coeff1 = [1, 2, 3] coeff2 = [3, 2, 1] # Subtract the second series from the first result = herme_sub(coeff1, coeff2) print(result)
Output: array([-2., 0., 2.])
The example showcases the use of SciPy’s herme_sub() function to perform subtraction on Hermite series. The method works similarly to NumPy’s version; it accepts the coefficients of the Hermite series and returns their difference. It’s extremely useful when working with other scientific computations that rely on SciPy’s extensive functionality.
Bonus One-Liner Method 5: List Comprehension
For Python enthusiasts, a one-liner that achieves the subtraction of two Hermite series using list comprehension can be both concise and readable when dealing with series of the same length.
Here’s an example:
# Define the coefficients of two Hermite series coeff1 = [1, 2, 3] coeff2 = [3, 2, 1] # Subtract second series from first using list comprehension result = [c1 - c2 for c1, c2 in zip(coeff1, coeff2)] print(result)
Output: [-2, 0, 2]
This one-liner uses list comprehension to iterate through pairs of coefficients from both series and subtract the second from the first. This method is very succinct and works well for lists of the same length. Should they differ, additional code would be needed to handle varying lengths.
Summary/Discussion
Method 1: NumPy’s herme_sub. Effective in the context of numeric computing. Relies on external library.
Method 2: Manual Coefficient Subtraction. Simple and requires no libraries. Not efficient for long series or frequent use.
Method 3: SymPy’s Poly Tools. Offers symbolic operations and simplifications. Overhead might be higher compared to numeric methods.
Method 4: SciPy.special.herme_sub. Suited for scientific tasks, part of SciPy’s ecosystem. Requires understanding of SciPy’s special submodule.
Method 5: List Comprehension One-Liner. Quick and pythonic for series of the same length; needs modification for series with different lengths.