💡 Problem Formulation: Integrating a Hermite E series over axis 0 in Python refers to the process of computing the integral of the probabilist’s Hermite polynomials along the first axis of a multidimensional array or a sequence. For a given n-dimensional input array hermite_array, the desired output is a (n-1)-dimensional array that represents the integrated values over axis 0.
Method 1: Using NumPy’s apply_along_axis
NumPy’s apply_along_axis function allows users to apply a specified function along a particular axis of an array. For the Hermite E series integration, we can use this function to apply the integration routine over axis 0 by importing the HermiteE class from numpy.polynomial.hermite_e and defining an integration function.
Here’s an example:
import numpy as np
from scipy.integrate import quad
from numpy.polynomial.hermite_e import HermiteE
# Define the Hermite E polynomial series
coeffs = np.array([1, 2, 3])
herm_e = HermiteE(coeffs)
# Integration function
def integrate_hermite(x):
return quad(lambda t: herm_e(t), -np.inf, np.inf)[0]
# Input array of Hermite E series coefficients
input_array = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
integrated = np.apply_along_axis(integrate_hermite, 0, input_array)
print(integrated)Output:
[Value1, Value2, ...]
The code snippet demonstrates how to define and integrate a Hermite E series using NumPy’s apply_along_axis. We create a Hermite E object herm_e from a series of coefficients and define the integrate_hermite function to perform numerical integration using SciPy’s quad function. The apply_along_axis method is called over axis 0 to carry out the integration for each series in the input array.
Method 2: Using a Custom Vectorization with NumPy
A custom approach involves creating a manually vectorized function that computes the integral for each Hermite E series, allowing vector operations over the series coefficients. This leverages NumPy’s broadcasting and vectorization capabilities to integrate over axis 0 efficiently.
Here’s an example:
import numpy as np
from scipy.integrate import quad
from numpy.polynomial.hermite_e import HermiteE
# Vectorized integration function
def vectorized_integrate_hermite(arr):
results = np.zeros_like(arr[0], dtype=float)
for i, coeffs in enumerate(arr):
herm_e = HermiteE(coeffs)
results[i] = quad(lambda t: herm_e(t), -np.inf, np.inf)[0]
return results
# Input array of Hermite E series coefficients
input_array = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
integrated = vectorized_integrate_hermite(input_array)
print(integrated)Output:
[Value1, Value2, ...]
The code defines a function vectorized_integrate_hermite that takes an array of Hermite E series coefficients, initializes a results array, and iterates over each set of coefficients to compute the integral using a lambda function. The integration results are stored in the corresponding index of the results array.
Method 3: Using SciPy’s integrate Functions with Mapping
SciPy’s integrate submodule offers a variety of integration functions such as quad. By mapping these integration functions onto each sub-array of Hermite E coefficients within our multidimensional array, we integrate over axis 0.
Here’s an example:
import numpy as np from scipy.integrate import quad from numpy.polynomial.hermite_e import HermiteE # Input array of Hermite E series coefficients and integrator input_array = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) integrated = np.array(list(map(lambda coeffs: quad(HermiteE(coeffs), -np.inf, np.inf)[0], input_array))) print(integrated)
Output:
[Value1, Value2, ...]
This snippet uses Python’s built-in map function to apply the SciPy quad integration method to each sub-array in the input array. The HermiteE function is used to calculate the polynomial values, and quad does the actual integration.
Method 4: Exploiting SymPy for Symbolic Integration
For a symbolic approach to integrating a Hermite E series, SymPy, Python’s symbolic mathematics library, offers powerful integration capabilities. Using SymPy allows the integration to be computed symbolically, which is precise and may be useful for analysis or further symbolic manipulation.
Here’s an example:
import numpy as np
import sympy as sp
# Define the symbolic variable and Hermite E polynomial
x = sp.symbols('x')
hermite_poly = sp.hermite(3, x) # Example for the 3rd Hermite E polynomial
# Perform symbolic integration over the entire real line
integrated = sp.integrate(hermite_poly, (x, -sp.oo, sp.oo))
print(integrated)Output:
Symbolic value of the integral (may be zero for some polynomials)
The code defines a symbolic variable x and a third-degree Hermite E polynomial using SymPy’s hermite function. sp.integrate is then used to compute the symbolic integral over the entire real line. The result can be a symbolic expression or a numeric value, depending on the properties of the polynomial.
Bonus One-Liner Method 5: Using NumPy and SciPy’s Quad with a Lambda One-Liner
For quick integrations, NumPy and SciPy can be combined into a single, efficient one-liner that utilizes a lambda function along with the quad function for on-the-fly Hermite E series evaluation and integration.
Here’s an example:
import numpy as np from scipy.integrate import quad from numpy.polynomial.hermite_e import HermiteE # One-liner for integrating Hermite E series coefficients array input_array = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) integrated = np.array([quad(lambda t: HermiteE(c)(t), -np.inf, np.inf)[0] for c in input_array]) print(integrated)
Output:
[Value1, Value2, ...]
The one-liner uses a list comprehension to iterate over each array of coefficients, creating a Hermite E polynomial object and integrating it immediately using quad. This compact method is both elegant and practical for simple use cases.
Summary/Discussion
- Method 1: NumPy’s apply_along_axis. Best for clean code and readability. Less efficient for large arrays due to Python-level loop overhead.
- Method 2: Custom Vectorization with NumPy. Offers performance improvements by exploiting NumPy’s vectorization but requires manual setup of the vectorized function.
- Method 3: SciPy’s integrate Functions with Mapping. Straightforward and Pythonic, but potentially less efficient due to the use of Python’s map function.
- Method 4: SymPy’s Symbolic Integration. Provides exact, algebraic results which can be useful for certain types of analysis or where numerical precision is critical.
- Bonus Method 5: NumPy and SciPy’s Quad with Lambda One-Liner. Quick and concise method ideal for simple tasks or interactive exploration.