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

To evaluate a 3-D Hermite_e series on the Cartesian product of x, y and z, use the hermite_e.hermegrid3d(x, y, z, c) method in Python. This method returns the values of the three-dimensional polynomial at points in the Cartesian product of x, y and z.

Parameters

The method accepts the following parameters:

• x, y, z: The three-dimensional series is evaluated at points in the Cartesian product of x, y, and z. If x, y, or z is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged.
• c: A 4D array of coefficients ordered so that the coefficients for terms of degree i, j, k are contained in c[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_e series ?

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

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

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

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

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

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

# Evaluate 3-D Hermite_e series on Cartesian product
result = H.hermegrid3d([1, 2], [1, 2], [1, 2], c)
print("\nResult...\n", result)

Our Array...
 [[[[ 0  1]
   [ 2  3]
   [ 4  5]
   [ 6  7]
   [ 8  9]
   [10 11]]

  [[12 13]
   [14 15]
   [16 17]
   [18 19]
   [20 21]
   [22 23]]]


 [[[24 25]
   [26 27]
   [28 29]
   [30 31]
   [32 33]
   [34 35]]

  [[36 37]
   [38 39]
   [40 41]
   [42 43]
   [44 45]
   [46 47]]]]

Dimensions of our Array...
 4

Datatype of our Array object...
 int64

Shape of our Array object...
 (2, 2, 6, 2)

Result...
 [[[[ 424. -1848.]
   [ 684. -2952.]]

  [[ 732. -3132.]
   [1170. -4968.]]]


 [[[ 440. -1908.]
   [ 708. -3042.]]

  [[ 756. -3222.]
   [1206. -5103.]]]]

Understanding the Output

The output shape is determined by the formula c.shape[3:] + x.shape + y.shape + z.shape. In our example:

• Coefficient array shape: (2, 2, 6, 2)
• Input arrays x, y, z each have shape (2,)
• Result shape: (2,) + (2,) + (2,) + (2,) = (2, 2, 2, 2)

Conclusion

The hermegrid3d() method efficiently evaluates 3-D Hermite_e series on Cartesian product grids. The coefficient array's fourth dimension allows for multiple polynomial evaluations simultaneously.