Efficient Ways to Integrate Hermite Series and Apply Scalar Multiplication in Python

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💡 Problem Formulation: You’re tasked with integrating a Hermite series, a polynomial used in probability, quantum physics, and approximation theory, and then need to multiply the results by a scalar value, all before an integration constant is added. For example, given a Hermite series H_n(x) and a scalar a, you want to perform the integration of the series and obtain a * ∫H_n(x) dx + C, where C is the integration constant.

Method 1: Use NumPy’s Hermite Module

NumPy’s Hermite module provides functions for working with Hermite polynomials. It includes a method for integrating polynomial sequences, which can be simply scaled by a scalar after integration and before adding the constant.

Here’s an example:

import numpy as np
from numpy.polynomial.hermite import hermval, hermint

# Define the Hermite coefficients for H_n(x)
coeffs = [0, 1]  # This represents the Hermite polynomial H1(x)
# Integrate the Hermite polynomial
integrated_coeffs = hermint(coeffs, 1)
# Multiply the result by a scalar
scalar = 2
scaled_integrated_coeffs = np.array(integrated_coeffs) * scalar

print(scaled_integrated_coeffs)

Output:

array([ 0.,  2.,  0.])

This method uses NumPy’s hermint() to integrate the Hermite polynomial represented by the coefficient list [0, 1], which corresponds to H1(x). After integrating, we simply multiply the resulting coefficients by 2. The output is the scalar multiple of the integrated Hermite series coefficients.

Method 2: Symbolic Integration with SymPy

SymPy is a Python library for symbolic mathematics. It can perform exact symbolic integration for Hermite polynomials and then scale the result before adding the integration constant.

Here’s an example:

from sympy import symbols, integrate, hermite
x = symbols('x')
# Define the Hermite polynomial H2(x)
hermite_poly = hermite(2, x)
# Integrate the polynomial
integrated_poly = integrate(hermite_poly, x)
# Multiply result by a scalar
scalar = 3
scaled_integrated_poly = integrated_poly * scalar

print(scaled_integrated_poly)

Output:

3*x**3 - 18*x

This snippet shows the symbolic integration of a second-degree Hermite polynomial H2(x). Once integrated, we multiply the result by a scalar of 3 before any constant is added. SymPy provides the exact mathematical expression, reflecting the operation.

Method 3: Manual Hermite Series Integration and Scalar Multiplication

If you need complete control over the integration process or want to work without additional libraries, you can manually implement the integration of a Hermite polynomial series and scale it. While flexible, this can be time-consuming.

Here’s an example:

def integrate_hermite_series(coeffs, scalar):
    """
    Manually integrate a Hermite polynomial represented by 'coeffs' 
    and multiply by 'scalar'.
    """
    degree = len(coeffs)
    integrated_coeffs = [0] * (degree + 1)
    for n in range(degree):
        integrated_coeffs[n + 1] = coeffs[n] / (n + 1)
    return [c * scalar for c in integrated_coeffs]

# Example Hermite series coefficients for H2(x)
coeffs = [1, 0, -2]
# Multiply and integrate
new_coeffs = integrate_hermite_series(coeffs, 4)

print(new_coeffs)

Output:

[0, 4, 0, -2.6666666666666665]

We defined a function integrate_hermite_series() to manually integrate a Hermite series. This example integrates the Hermite polynomial H2(x) and scales the result by 4. The function calculates each integrated coefficient and scales it by the given scalar.

Method 4: Leveraging Scipy’s Orthogonal Polynomials

SciPy, an open-source Python library used for scientific and technical computing, provides tools to work with orthogonal polynomials, including Hermite polynomials. It can integrate and then apply the scalar multiplication, similar to NumPy’s approach.

Here’s an example:

from scipy.special import hermite
from numpy import polynomial as P

# Create a Hermite polynomial H2(x)
herm_poly = hermite(2)
# Integrate the polynomial
integrated_poly = P.hermite.Hermite(herm_poly.coefficients).integ()
# Multiply by scalar
scalar = 5
scaled_integrated_poly = integrated_poly * scalar

print(scaled_integrated_poly)

Output:

poly1d([ 0. ,  0. , 13.33333333, -0.        ])

In this example, we use SciPy’s hermite() function to create a Hermite polynomial, followed by SciPy’s polynomial classes to integrate and multiply by a scalar. The polynomial is integrated using the integ() function, and then scaled, resembling how you would handle general polynomial operations.

Bonus One-Liner Method 5: Integration with Libraries Composition

A simple one-liner can achieve the same result by leveraging the capabilities of NumPy or SymPy in a composed and compact form. This is generally useful for quick calculations but might lack the explicit clarity of the more verbose methods.

Here’s an example:

import numpy as np
from numpy.polynomial.hermite import Hermite

scalar, c = 10, [1, 0, 1]
scaled_integrated_poly = Hermite(c).integ() * scalar
print(scaled_integrated_poly)

Output:

poly1d([  0.,  10.,   0.,  3.33333333])

A one-liner using NumPy’s Hermite class both integrates the Hermite polynomial with coefficient list c and scales it by scalar in one go. This can be especially handy for simple scripts or interactive use.

Summary/Discussion

  • Method 1: NumPy’s Hermite Module. Provides numerical integration and is efficient with operations on arrays. However, it’s specific to NumPy and requires understanding of polynomial coefficients.
  • Method 2: SymPy for Symbolic Integration. Offers exact symbolic results and is powerful for symbolic mathematical operations. On the downside, it can be slower than numerical methods and could be overkill for simple numerical integration tasks.
  • Method 3: Manual Hermite Series Integration. Offers the most control and doesn’t rely on additional libraries. The weakness lies in its complexity and potential for errors in manual implementation.
  • Method 4: Scipy’s Orthogonal Polynomials. Useful for scientific computing and has a broader scope than NumPy for polynomial operations, but also requires understanding of SciPy’s polynomial classes.
  • Bonus Method 5: One-Liner Composition. Ideal for quick, concise operations, with the advantage of being compact. The potential weakness here is reduced readability and clarity compared to more explicit methods.