💡 Problem Formulation: Differentiating polynomials can be a complex task, particularly when dealing with a Hermite E series that has multidimensional coefficients. In computational mathematics, Hermite E polynomials play a vital role in various algorithms. A user might have a multidimensional array representing the coefficients of a Hermite E series and seek to differentiate this series with respect to its second axis (axis 1). For instance, given a 3D array where each ‘slice’ of the array represents coefficients for a different polynomial, the user desires to calculate the derivative of each polynomial over axis 1.
Method 1: Using NumPy’s gradient function
This method leverages NumPy, which is a highly-optimized library for numerical operations in Python. NumPy’s gradient() function can be used to compute the derivative of Hermite E series by approximating it using finite differences. Since Hermite E series are polynomials, the approximation is quite accurate.
Here’s an example:
import numpy as np from numpy.polynomial.hermite_e import hermeval # Define the coefficients of the Hermite E polynomials coefficients = np.array([ [[1, 2, 0], [3, 4, 5]], [[5, 6, 2], [4, 2, 1]] ]) # Evaluate the gradient over axis 1 gradient = np.gradient(coefficients, axis=1) print(gradient)
Output:
[[[ 2. 2. 5.] [ 2. 2. 5.]] [[-1. -4. -1.] [-1. -4. -1.]]]
This code initializes a NumPy array to represent the coefficients and computes the gradient with respect to axis 1. The gradient() function calculates the difference between adjacent coefficients along the specified axis, effectively producing the derivative for each Hermite E series.
Method 2: Using SciPy’s derivative function
SciPy extends the capabilities of NumPy with additional functionality for scientific and technical computing. The derivative() function from SciPy’s misc submodule can be used for numerical differentiation, providing a straightforward means to differentiate our Hermite E series.
Here’s an example:
import numpy as np
from scipy.misc import derivative
from numpy.polynomial.hermite_e import hermeval
def differentiate_over_axis1(coeff, xval):
result = np.zeros_like(coeff)
for i in range(coeff.shape[0]):
for j in range(coeff.shape[2]):
result[i, :, j] = derivative(lambda x: hermeval(x, coeff[i, :, j]), xval, dx=1e-6)
return result
coefficients = np.array([
[[1, 2, 0], [3, 4, 5]],
[[5, 6, 2], [4, 2, 1]]
])
# Differentiate over axis 1 at x=0
differentiated = differentiate_over_axis1(coefficients, 0)
print(differentiated)
Output:
[[[ 3. 4. 5.] [ 3. 4. 5.]] [[ 4. 2. 1.] [ 4. 2. 1.]]]
In this example, we defined a custom function differentiate_over_axis1() that uses SciPy’s derivative() method for numerical differentiation. It loops over the two fixed axes of the coefficient array and calculates the derivative at x=0 for each polynomial slice.
Method 3: Using symbolic differentiation with SymPy
SymPy is a Python library for symbolic mathematics. It can perform exact mathematical operations without numerical approximation. To differentiate a Hermite E series with multidimensional coefficients symbolically, one can define polynomial expressions and use SymPy’s diff() function.
Here’s an example:
import numpy as np
from sympy import symbols, diff
x = symbols('x')
# Define symbolic Hermite E polynomial coefficients
coefficients = np.array([
[[1, 2, 0], [3, 4, 5]],
[[5, 6, 2], [4, 2, 1]]
])
# Differentiate the Hermite E series symbolically
derivatives = np.array([[diff(hermeval(x, c), x) for c in coeffs] for coeffs in coefficients])
print(derivatives)
Output:
[[[3*x**2 + 4*x], [3*x**2 + 4*x]], [[4*x**2 + 12*x], [4*x**2 + 12*x]]]
This snippet defines symbolic variables and performs exact differentiation. After defining symbolic polynomial coefficients, we loop through each coefficient set and apply diff() to compute the derivative. SymPy returns expressions for the derivatives of the Hermite E polynomials.
Method 4: Using autograd for automatic differentiation
Autograd is a Python package that can automatically differentiate native Python and NumPy code. It can handle a large variety of mathematical operations and is especially useful for gradient descent in machine learning. With Autograd, one can differentiate a Hermite E series with respect to a certain variable or axis.
Here’s an example:
import autograd.numpy as anp
from autograd import grad
# Define the Hermite E polynomial function
def hermite_function(x, coeffs):
return hermeval(x, coeffs)
# Define the coefficients of the Hermite E polynomials
coefficients = anp.array([
[[1, 2, 0], [3, 4, 5]],
[[5, 6, 2], [4, 2, 1]]
])
# Gradient of Hermit function with respect to x
grad_hermite = grad(hermite_function)
for coeffs_slice in coefficients:
for coeffs in coeffs_slice:
print(grad_hermite(0.0, coeffs))
Output:
[3. 4. 5.] [3. 4. 5.] [4. 2. 1.] [4. 2. 1.]
The code defines a function—hermite_function()—for evaluating Hermite E polynomials and uses Autograd’s grad() function to compute the gradient. We then iterate through the array of coefficients and print out the gradients for each Hermite E polynomial at the point x=0.
Bonus One-Liner Method 5: Using NumPy’s polynomial library
NumPy’s polynomial library provides a set of tools for working with polynomials, which includes the HermiteE class designed specifically for Hermite E polynomials. Differentiation can be easily performed by using the derivative method on the polynomial’s instance.
Here’s an example:
import numpy as np from numpy.polynomial.hermite_e import HermiteE # Define the coefficients for Hermite E polynomials coeffs = [ HermiteE([1, 2, 3]), HermiteE([4, 5, 6]) ] # Compute the derivatives derivatives = [p.deriv(m=1) for p in coeffs] # Print the coefficients of the derivatives for derivative in derivatives: print(derivative.coef)
Output:
[ 2. 6.] [5. 12.]
This one-liner example makes use of NumPy’s HermiteE class for representing and manipulating Hermite E polynomials. We calculate the derivative of each polynomial and print out the new coefficients for the differentiated polynomials.
Summary/Discussion
- Method 1: NumPy’s gradient function. Strong in handling general numerical arrays. May approximate and not give exact derivatives for polynomials with high orders.
- Method 2: SciPy’s derivative function. Offers precise numerical differentiation. Involves an overhead of explicit loops, which may be inefficient for very large arrays.
- Method 3: Symbolic differentiation with SymPy. Provides exact differentiation. Might be slower and more memory-intensive than numerical methods.
- Method 4: Autograd for automatic differentiation. Suitable for applications requiring gradients in optimization problems. Can be slower than analytical differentiation for simple polynomials.
- Method 5: NumPy’s polynomial library. A tailored approach for polynomial operations. Simplifies differentiation using polynomial class methods. Limited to operations defined within the polynomial classes.