Convert a Hermite_e series to a polynomial in Python

To convert a Hermite_e series to a polynomial, use the hermite_e.herme2poly() method in NumPy. This function converts an array representing the coefficients of a Hermite_e series (ordered from lowest degree to highest) to an array of coefficients of the equivalent polynomial in the standard basis.

Syntax

numpy.polynomial.hermite_e.herme2poly(c)

Parameters

The parameter c is a 1-D array containing the Hermite_e series coefficients, ordered from lowest order term to highest.

Example

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

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

# Create an array of Hermite_e coefficients
c = np.array([1, 2, 3, 4, 5])

# Display the array
print("Hermite_e coefficients:", c)

# Check array properties
print("Dimensions:", c.ndim)
print("Datatype:", c.dtype)
print("Shape:", c.shape)

# Convert Hermite_e series to polynomial
result = H.herme2poly(c)
print("Polynomial coefficients:", result)

Hermite_e coefficients: [1 2 3 4 5]
Dimensions: 1
Datatype: int64
Shape: (5,)
Polynomial coefficients: [ 13. -10. -27.   4.   5.]

How It Works

The function transforms the Hermite_e series representation into standard polynomial form. The input coefficients [1, 2, 3, 4, 5] represent:

• 1 × He?(x) + 2 × He?(x) + 3 × He?(x) + 4 × He?(x) + 5 × He?(x)

The output [13, -10, -27, 4, 5] represents the polynomial:

• 13 + (-10)x + (-27)x² + 4x³ + 5x?

Practical Example

Here's a simpler example with fewer coefficients ?

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

# Simple Hermite_e series: 1 + 2*He_1(x)
coeffs = np.array([1, 2])
polynomial = H.herme2poly(coeffs)

print("Hermite_e coefficients:", coeffs)
print("Polynomial coefficients:", polynomial)
print("This represents: {} + {}x".format(polynomial[0], polynomial[1]))

Hermite_e coefficients: [1 2]
Polynomial coefficients: [1. 2.]
This represents: 1.0 + 2.0x

Conclusion

The herme2poly() function efficiently converts Hermite_e series coefficients to standard polynomial coefficients. This transformation is useful in mathematical computations involving orthogonal polynomials and numerical analysis.