Evaluate a Hermite_e series at points x and the shape of the coefficient array extended for each dimension of x in Python

To evaluate a Hermite_e series at points x, use the hermite_e.hermeval() method in Python NumPy. This function allows you to evaluate Hermite_e polynomials at specific points and control how the coefficient array is handled for different dimensions.

Parameters

The hermeval() method accepts three key parameters:

• x: The evaluation points. If x is a list or tuple, it is converted to an ndarray. The elements must support addition and multiplication operations.
• c: Array of coefficients where coefficients for terms of degree n are in c[n]. For multidimensional arrays, remaining indices enumerate multiple polynomials.
• tensor: Boolean parameter controlling shape extension. If True (default), the coefficient array shape is extended for each dimension of x. If False, x is broadcast over the columns of c.

Example with Tensor=True

Let's create a multidimensional coefficient array and evaluate the Hermite_e series ?

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

# Create a multidimensional array of coefficients
c = np.arange(8).reshape(2,4)

# Display the array
print("Our Array...")
print(c)

# Check the Dimensions
print("\nDimensions of our Array...")
print(c.ndim)

# Get the Datatype
print("\nDatatype of our Array object...")
print(c.dtype)

# Get the Shape
print("\nShape of our Array object...")
print(c.shape)

# Evaluate Hermite_e series at points x with tensor=True
print("\nResult with tensor=True...")
result = H.hermeval([1,2], c, tensor=True)
print(result)

Our Array...
[[0 1 2 3]
 [4 5 6 7]]

Dimensions of our Array...
2

Datatype of our Array object...
int64

Shape of our Array object...
(2, 4)

Result with tensor=True...
[[ 4.  8.]
 [ 6. 11.]
 [ 8. 14.]
 [10. 17.]]

Comparison: Tensor=True vs Tensor=False

The tensor parameter changes how the evaluation is performed ?

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

c = np.arange(8).reshape(2,4)

print("Coefficient array:")
print(c)
print("\nShape:", c.shape)

# With tensor=True (default)
result_true = H.hermeval([1,2], c, tensor=True)
print("\nWith tensor=True:")
print("Shape:", result_true.shape)
print("Result:")
print(result_true)

# With tensor=False
result_false = H.hermeval([1,2], c, tensor=False)
print("\nWith tensor=False:")
print("Shape:", result_false.shape)
print("Result:")
print(result_false)

Coefficient array:
[[0 1 2 3]
 [4 5 6 7]]

Shape: (2, 4)

With tensor=True:
Shape: (4, 2)
Result:
[[ 4.  8.]
 [ 6. 11.]
 [ 8. 14.]
 [10. 17.]]

With tensor=False:
Shape: (2, 2)
Result:
[[ 4.  6.]
 [30. 41.]]

How It Works

When tensor=True, each column of coefficients is evaluated for every element of x, creating a result where each polynomial is evaluated at each point. When tensor=False, the evaluation points are broadcast over the columns, resulting in a different output shape and values.

Conclusion

The hermite_e.hermeval() method provides flexible evaluation of Hermite_e series. Use tensor=True when you need each polynomial evaluated at each point, and tensor=False for broadcasting behavior with multidimensional coefficient arrays.