Convert a polynomial to Hermite series in Python

To convert a polynomial to a Hermite series, use the hermite.poly2herm() method in NumPy. This function converts an array representing polynomial coefficients (ordered from lowest to highest degree) to the coefficients of an equivalent Hermite series.

Syntax

numpy.polynomial.hermite.poly2herm(pol)

Parameters:

• pol − 1-D array containing polynomial coefficients ordered from lowest to highest degree

Returns: 1-D array containing the coefficients of the equivalent Hermite series.

Example

Let's convert a polynomial with coefficients [1, 2, 3, 4, 5] to its Hermite series representation ?

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

# Create polynomial coefficients array
coefficients = np.array([1, 2, 3, 4, 5])

# Display the original polynomial coefficients
print("Polynomial coefficients:")
print(coefficients)

# Display array properties
print(f"\nDimensions: {coefficients.ndim}")
print(f"Datatype: {coefficients.dtype}")
print(f"Shape: {coefficients.shape}")

# Convert polynomial to Hermite series
hermite_coeffs = H.poly2herm(coefficients)
print(f"\nHermite series coefficients:")
print(hermite_coeffs)

Polynomial coefficients:
[1 2 3 4 5]

Dimensions: 1
Datatype: int64
Shape: (5,)

Hermite series coefficients:
[6.25 4.   4.5  0.5  0.3125]

Understanding the Conversion

The original polynomial 1 + 2x + 3x² + 4x³ + 5x? is converted to its equivalent representation using Hermite polynomials. Each coefficient in the result corresponds to the weight of the respective Hermite polynomial in the series.

Working with Different Polynomial Degrees

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

# Simple quadratic polynomial: 1 + 2x + x²
quadratic = np.array([1, 2, 1])
print("Quadratic polynomial coefficients:", quadratic)
print("Hermite series:", H.poly2herm(quadratic))

# Linear polynomial: 3 + 4x  
linear = np.array([3, 4])
print("\nLinear polynomial coefficients:", linear)
print("Hermite series:", H.poly2herm(linear))

Quadratic polynomial coefficients: [1 2 1]
Hermite series: [2.5 2.  0.5]

Linear polynomial coefficients: [3 4]
Hermite series: [3. 4.]

Conclusion

The hermite.poly2herm() function efficiently converts standard polynomial coefficients to Hermite series representation. This conversion is useful in numerical analysis and physics applications where Hermite polynomials provide better computational properties.