Divide one Hermite series by another in Python using NumPy

The Hermite series is a mathematical technique used to represent infinite series of Hermite polynomials. Hermite polynomials are orthogonal polynomials that solve the Hermite differential equation. NumPy provides functions to work with Hermite series, including division operations.

What is a Hermite Series?

A Hermite series is represented by the equation:

f(x) = ?n=0^? cn Hn(x)

Where:

• Hn(x) is the nth Hermite polynomial

• cn is the nth coefficient in the expansion

Creating Hermite Series

First, let's create Hermite series using NumPy's polynomial.hermite.poly2herm() function ?

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

c = np.array([1, 2, 3, 4, 5])
print("Original Array:")
print(c)
print("\nConverted to Hermite series:")
print(H.poly2herm(c))

Original Array:
[1 2 3 4 5]

Converted to Hermite series:
[6.25   4.     4.5    0.5    0.3125]

Dividing Hermite Series

You can divide one Hermite series by another using NumPy's divide() function ?

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

# Create two Hermite series
h1 = np.array([1, 2, 3, 4, 5])
h2 = np.array([1, 2, 3, 4, 5])

print("First Hermite series coefficients:", h1)
print("Second Hermite series coefficients:", h2)

# Divide the series
result = np.divide(h1, h2)
print("\nResult of division:", result)

First Hermite series coefficients: [1 2 3 4 5]
Second Hermite series coefficients: [1 2 3 4 5]

Result of division: [1. 1. 1. 1. 1.]

Dividing Different Hermite Series

Here's an example with different coefficient arrays ?

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

h1 = np.array([1, 2, 3, 4, 5])
h2 = np.array([2, 4, 6, 8, 10])

print("First series:", h1)
print("Second series:", h2)

# Convert to Hermite representation
hermite1 = H.poly2herm(h1)
hermite2 = H.poly2herm(h2)

print("\nHermite representation of first series:", hermite1)
print("Hermite representation of second series:", hermite2)

# Divide the original coefficient arrays
division_result = np.divide(h1, h2)
print("\nDivision result:", division_result)

First series: [1 2 3 4 5]
Second series: [ 2  4  6  8 10]

Hermite representation of first series: [6.25   4.     4.5    0.5    0.3125]
Hermite representation of second series: [12.5    8.     9.     1.     0.625]

Division result: [0.5 0.5 0.5 0.5 0.5]

Proper Hermite Division

For actual Hermite polynomial division, use the hermdiv() function ?

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

# Define two Hermite series
dividend = np.array([1, 2, 3])
divisor = np.array([1, 1])

print("Dividend coefficients:", dividend)
print("Divisor coefficients:", divisor)

# Perform Hermite polynomial division
quotient, remainder = H.hermdiv(dividend, divisor)

print("\nQuotient:", quotient)
print("Remainder:", remainder)

Dividend coefficients: [1 2 3]
Divisor coefficients: [1 1]

Quotient: [-2.  6.]
Remainder: [3.]

Key Points

• np.divide() performs element-wise division of coefficient arrays

• H.hermdiv() performs proper polynomial division in Hermite basis

• H.poly2herm() converts standard polynomial to Hermite representation

• Arrays must have compatible dimensions for division operations

Conclusion

NumPy provides multiple ways to work with Hermite series division. Use np.divide() for coefficient-wise division or hermdiv() for proper polynomial division. Always ensure arrays have compatible dimensions to avoid errors.