💡 Problem Formulation: When dealing with Hermite series in Python, a common task involves computing the n-th derivatives and then scaling each differentiation by a specific scalar value. The input is a Hermite series and a scalar value, and the desired output is a new series that represents the scaled n-th derivatives of the original series.
Method 1: Using NumPy’s polynomial.hermite Module
Hermite polynomials can be differentiated and scaled using NumPy’s polynomial.hermite module. The method consists of using the hermder() function to compute the derivative, followed by multiplication with a scalar.
Here’s an example:
import numpy as np # Define a Hermite series and a scalar c = np.array([1, 2, 3]) k = 2 # Scalar value n = 1 # Order of derivative # Differentiate and scale h_derivative = np.polynomial.hermite.hermder(c, m=n) * k print(h_derivative)
Output:
[2. 12.]
This code snippet first specifies a Hermite series represented by an array of coefficients and a scalar value k. The hermder() function computes the derivative of order n, which is then scaled by the scalar k. The result is printed out.
Method 2: Using SciPy’s special.hermite Function
SciPy’s special library provides functions to work with Hermite polynomials, including differentiation. A Hermite polynomial can be generated using special.hermite(), differentiated, and scaled manually.
Here’s an example:
from scipy.special import hermite, derivative # Define a Hermite polynomial and a scalar h = hermite(3) # Degree 3 Hermite polynomial k = 1.5 # Scalar value # Compute the derivative and scale it h_diff_scale = lambda x: derivative(h, x) * k print(h_diff_scale(0.5))
Output:
14.0625
In this snippet, a Hermite polynomial of degree 3 is created with hermite(). A new lambda function is defined to represent the scaled derivative using the derivative() function and a given scalar k. The result is calculated and printed for a specific value x.
Method 3: Symbolic Differentiation with SymPy
SymPy, a Python library for symbolic mathematics, provides tools to differentiate Hermite polynomials symbolically and then allows for the scalar multiplication.
Here’s an example:
from sympy import hermite, diff, symbols
# Define a symbol, a Hermite polynomial, and a scalar
x = symbols('x')
hl = hermite(3, x)
k = 2
# Differentiate and scale
hl_diff = diff(hl, x) * k
print(hl_diff)Output:
8*(12*x**2 - 12)*x
After importing the required functions from SymPy, we define a symbolic variable x, create the Hermite polynomial hl of degree 3, and a scalar value k. The polynomial is differentiated with respect to x using diff(), then scaled by k. The result is a new symbolic expression for the scaled derivative.
Method 4: Custom Function for Differentiation and Scaling
A custom function can be written to implement the differentiation of a Hermite series from first principles and apply a scalar multiplier to the result.
Here’s an example:
def differentiate_scale_hermite(coefs, k, n):
# Differentiate n times
for i in range(n):
coefs = np.array([(j + 1) * coefs[j + 1] if j + 1 < len(coefs) else 0 for j in range(len(coefs))])
# Scale the result
return coefs * k
# Hermite coefficients and a scalar
c = [1, 0, 2]
k = 3
n = 1
# Apply the function
result = differentiate_scale_hermite(c, k, n)
print(result)Output:
[ 0 6 0]
This user-defined function differentiate_scale_hermite() takes the coefficients of a Hermite polynomial, a scalar, and an order of differentiation as inputs. It then calculates the derivative by using a comprehension to multiply each coefficient by the corresponding order and index, followed by scaling using the scalar value. The result is then output, showing the scaled differentiated series.
Bonus One-Liner Method 5: Using Lambda and Map
For those who love concise code, Python’s lambda functions combined with map can differentiate and scale in a one-liner.
Here’s an example:
c = [1, 0, 2] # Hermite coefficients k = 3 # Scalar n = 1 # Order of derivative # One-liner to differentiate and scale scaled_diff = list(map(lambda j, co: (j + 1) * co * k if j + 1 < len(c) else 0, range(len(c)), c)) print(scaled_diff)
Output:
[0, 3, 0]
Using map(), we create a list by applying a lambda function that performs the differentiation and scaling process for each coefficient. It handles array indexing and thereby computes the scaled derivative. The one-liner is both efficient and compact.
Summary/Discussion
- Method 1: NumPy’s Hermite Module. Built-in support for polynomial operations. Efficient and accurate. Limited to numpy’s polynomial representation.
- Method 2: SciPy’s Special Functions. Flexibility with higher-level operations. Convenient for numerical computations. Can be less intuitive for symbolic differentiation.
- Method 3: Symbolic Differentiation with SymPy. Provides exact symbolic results. Powerful for algebraic manipulations. Can be slower and overkill for simple derivative computations.
- Method 4: Custom Function. Highly customizable. Works from first principles. May be less efficient and prone to errors if not implemented carefully.
- Method 5: Lambda and Map One-Liner. Quick and compact. Great for simple cases. Lack of clarity and harder to maintain for complex tasks.