Evaluate a 3-D Hermite series on the Cartesian product of x, y and z with 4d array of coefficient in Python

To evaluate a 3-D Hermite series on the Cartesian product of x, y and z, use the hermite.hermgrid3d(x, y, z, c) method in Python. This method evaluates a three-dimensional Hermite polynomial at all combinations of points from the input arrays.

Parameters

The method takes four parameters:

• x, y, z − The three coordinate arrays. The series is evaluated at points in the Cartesian product of x, y, and z. If any parameter is a list or tuple, it's converted to an ndarray.
• c − A 4D array of coefficients where c[i,j,k,:] contains coefficients for terms of degree i,j,k. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape.

Example

Let's create a 4D coefficient array and evaluate the Hermite series:

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

# Create a 4d array of coefficients
c = np.arange(48).reshape(2,2,6,2)

# Display the array properties
print("Coefficient Array Shape:", c.shape)
print("Dimensions:", c.ndim)
print("Datatype:", c.dtype)

# Evaluate 3-D Hermite series on Cartesian product
result = H.hermgrid3d([1,2], [1,2], [1,2], c)
print("\nResult shape:", result.shape)
print("Result:\n", result)

Coefficient Array Shape: (2, 2, 6, 2)
Dimensions: 4
Datatype: int64

Result shape: (2, 2, 2, 2, 2)
Result:
 [[[[[  -8100.  32472.]
     [-14148.  56976.]]

    [[-14796.  59832.]
     [-25740. 104480.]]]


   [[[ -8343.  33543.]
     [-14553.  58761.]]

    [[-15201.  61617.]
     [-26415. 107455.]]]]]

How It Works

The function evaluates the 3-D Hermite polynomial:

The result has shape (2, 2, 2, 2, 2) because:

• Coefficient array shape: (2, 2, 6, 2) ? last dimension gives us 2
• Each coordinate array [1,2] has length 2
• Final shape: c.shape[3:] + x.shape + y.shape + z.shape = (2,) + (2,) + (2,) + (2,) = (2,2,2,2)

Conclusion

The hermgrid3d() function efficiently evaluates 3-D Hermite series on Cartesian products. The result shape depends on both the coefficient array dimensions and input coordinate array shapes.