💡 Problem Formulation: When dealing with polynomial series, particularly Hermite series with multidimensional coefficients, differentiating over a specific axis can be operationally complex. In the context of Python, ‘axis 1’ often refers to differentiating each array in a multidimensional array stack. For example, given an array representing the coefficients of a Hermite series, we want to obtain a new array where each element is the derivative of the original series with respect to the specified axis.
Method 1: Using NumPy’s gradient Function
NumPy’s gradient function computes the gradient of an array representing the values of the function at each point. However, to differentiate a Hermite series over a particular axis, one needs to generate the series on a grid and then differentiate. Specific to multidimensional coefficients, the gradient along ‘axis 1’ can be obtained.
Here’s an example:
import numpy as np from numpy.polynomial.hermite import hermval # Define multidimensional Hermite coefficients coeffs = np.array([[1, 0, 3], [0, 4, 0], [5, 0, 0]]) # Generate a grid of points to evaluate the series x = np.linspace(-1, 1, 100) # Evaluate the Hermite series on the grid values = hermval(x[:, None], coeffs.T) # Compute the derivative over axis 1 derivatives = np.gradient(values, axis=1) # The derivatives evaluated at the grid points print(derivatives)
The output is a multidimensional array of the evaluated derivatives of the Hermite series at each point in the grid.
This code snippet uses NumPy’s capabilities to evaluate a Hermite series using the coefficients and then employs the gradient function to compute the derivatives. The process involves evaluating the polynomial at a set of points and then computing the gradient. While straightforward for analytic differentiation, this method might not be the most computationally efficient as it involves extra steps of function evaluation.
Method 2: Manually Scaling the Coefficients
When dealing with Hermite series, differentiating them can be executed by manually scaling the coefficients. This method directly manipulates the array of coefficients to reflect differentiation over the desired axis.
Here’s an example:
import numpy as np # Define multidimensional Hermite coefficients coeffs = np.array([[1, 0, 3], [0, 4, 0], [5, 0, 0]]) # Derivate Hermite series coefficients over axis 1 n = coeffs.shape[1] # number of coefficients scaled_coeffs = np.arange(1, n) * coeffs[:, 1:] # Scaled coefficients are the derivative over axis 1 print(scaled_coeffs)
The output is an array representing the new derivatives of the Hermite series coefficients after the operation.
This snippet illustrates direct coefficient manipulation to obtain the derivative. By scaling each coefficient with its corresponding Hermite polynomial order and truncating the highest order, it effectively computes the derivative. This method is computationally efficient and avoids unnecessary evaluations of the series.
Method 3: Using SciPy’s Hermite Class
The SciPy library has a dedicated class for Hermite polynomials which includes a differentiation method. Leveraging the Hermite class, one can create an instance of the polynomial and then differentiate it.
Here’s an example:
from scipy.special import Hermite # Define multidimensional Hermite coefficients coeffs = np.array([[1, 0, 3], [0, 4, 0], [5, 0, 0]]) # Create Hermite objects and differentiate hermite_polys = [Hermite(c) for c in coeffs] differentiated_polys = [h.deriv(m=1) for h in hermite_polys] # Extract the coefficients of the differentiated polynomials diff_coeffs = np.array([p.coefficients for p in differentiated_polys]) # The new coefficients after differentiation print(diff_coeffs)
The output is the array of coefficients that correspond to the differentiated Hermite polynomials.
This approach uses the object-oriented features in SciPy to create Hermite polynomial objects and differentiate them. While this method is highly readable and leverages built-in polynomial manipulation functions, it may not be as efficient for large arrays of coefficients due to the overhead of object creation.
Method 4: Automating Differentiation with SymPy
SymPy is a Python library for symbolic mathematics which can be utilized to automate the differentiation process for Hermite polynomials. Using symbolic computation allows for precision and potentially simplifying algebraic expressions before numerical evaluation.
Here’s an example:
from sympy import symbols, diff, hermite
import numpy as np
# Define a symbolic variable
x = symbols('x')
# Define multidimensional Hermite coefficients
coeffs = np.array([[1, 0, 3], [0, 4, 0], [5, 0, 0]])
# Differentiate Hermite polynomials symbolically
diff_coeffs = [coeff*np.array([diff(hermite(n, x), x).subs(x, 0) for n in range(len(row))]) for coeff, row in zip(coeffs, coeffs)]
# Convert the symbolic coefficients to a numerical array
diff_coeffs = np.array(diff_coeffs, dtype=float)
# The numerical coefficients after differentiation
print(diff_coeffs)The output is the array with numerically evaluated coefficients of the differentiated polynomials.
This method computes the derivative symbolically, which can be very powerful in more complex analytical tasks. However, for numerical differentiation of series coefficients, it introduces the overhead of symbolic computation, making it less efficient for simple tasks.
Bonus One-Liner Method 5: Leveraging NumPy’s “polyder” for Polynomial Differentiation
NumPy provides a convenient polyder function designed for polynomial differentiation. This one-liner can be used for quickly differentiating a Hermite series’ coefficients, especially when the multidimensional coefficient array is handled carefully.
Here’s an example:
import numpy as np from numpy.polynomial.hermite import polyder # Define multidimensional Hermite coefficients coeffs = np.array([[1, 0, 3], [0, 4, 0], [5, 0, 0]]) # Differentiate Hermite series coefficients over axis 1 deriv_coeffs = np.apply_along_axis(polyder, 1, coeffs) # The resulting coefficients after differentiation print(deriv_coeffs)
The output displays the differentiated coefficients of the Hermite series.
This compact snippet succinctly computes the derivative of Hermite series coefficients by applying NumPy’s polyder function along the specified axis. It combines the efficiency of numerical libraries with convenient array manipulation functions.
Summary/Discussion
- Method 1: NumPy’s gradient. Pros: Built-in function, works directly on evaluated series. Cons: Less efficient due to series evaluation step.
- Method 2: Manually Scaling Coefficients. Pros: Highly efficient and direct manipulation. Cons: Code can be less intuitive for those unfamiliar with polynomial series.
- Method 3: SciPy’s Hermite Class. Pros: Intuitive object-oriented approach, exploits dedicated Hermite polynomial tools. Cons: Not optimal for large arrays due to object instantiation overhead.
- Method 4: Automating with SymPy. Pros: Precise and algebraically powerful. Cons: Symbolic computation overhead and could be overkill for numerical tasks.
- Bonus Method 5: NumPy’s polyder. Pros: Quick and easy one-liner. Cons: Less transparent process compared to manual coefficient handling.