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

The Hermite series evaluation in Python NumPy allows you to compute polynomial values at specific points using the hermite.hermval() method. This function is particularly useful when working with multidimensional coefficient arrays and controlling how the evaluation is broadcast across dimensions.

Syntax

hermite.hermval(x, c, tensor=True)

Parameters

The hermval() method accepts three parameters:

• x: Points at which to evaluate the series. Can be a scalar, list, or array
• c: Coefficient array where c[n] contains coefficients for degree n terms
• tensor: Controls broadcasting behavior (default: True)

Understanding the tensor Parameter

When tensor=True, the coefficient array shape is extended with ones for each dimension of x, evaluating every coefficient column for every x element. When tensor=False, x is broadcast over the coefficient columns.

Basic Example

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

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

# Display the array
print("Coefficient Array:")
print(c)

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

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

Dimensions: 2
Datatype: int64
Shape: (2, 4)

Evaluating with tensor=True

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

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

# Evaluate at points [1, 2] with tensor=True
result_tensor = H.hermval([1, 2], c, tensor=True)
print("Result with tensor=True:")
print(result_tensor)
print("Shape:", result_tensor.shape)

Result with tensor=True:
[[ 8. 16.]
 [11. 21.]
 [14. 26.]
 [17. 31.]]
Shape: (4, 2)

Evaluating with tensor=False

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

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

# Evaluate at points [1, 2] with tensor=False
result_no_tensor = H.hermval([1, 2], c, tensor=False)
print("Result with tensor=False:")
print(result_no_tensor)
print("Shape:", result_no_tensor.shape)

Result with tensor=False:
[ 8. 16.]
Shape: (2,)

Comparison

Conclusion

The hermite.hermval() method provides flexible Hermite series evaluation with the tensor parameter controlling how multidimensional coefficients are broadcast. Use tensor=True for comprehensive evaluation across all coefficient combinations.