Evaluate a 3-D Hermite series on the Cartesian product of x, y and z 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 returns the values of the three-dimensional polynomial at points in the Cartesian product of x, y, and z coordinates.

Syntax

numpy.polynomial.hermite.hermgrid3d(x, y, z, c)

Parameters

The parameters are:

• 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. If it isn't an ndarray, it is treated as a scalar.
• c − An array of coefficients ordered so that coefficients for terms of degree i,j are contained in c[i,j]. If c has dimension greater than two, the remaining indices enumerate multiple sets of coefficients. If c has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D.

Return Value

The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape.

Example

Let's create a 3D array of coefficients and evaluate the Hermite series ?

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

# Create a 3D array of coefficients
c = np.arange(16).reshape(2,2,4)

# 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 series on Cartesian product
print("\nResult...\n", H.hermgrid3d([1,2], [1,2], [1,2], c))

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

 [[ 8  9 10 11]
  [12 13 14 15]]]

Dimensions of our Array...
3

Datatype of our Array object...
int64

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

Result...
 [[[   18.  5616.]
  [   38.  9832.]]

 [[   46. 10304.]
  [   90. 17960.]]]

How It Works

The hermgrid3d() function evaluates the 3D Hermite series at each point in the Cartesian product of the input arrays. For coordinates [1,2] in each dimension, it computes 2×2×2 = 8 total combinations and evaluates the polynomial at each point using the coefficient array.

Conclusion

The hermite.hermgrid3d() method efficiently evaluates 3D Hermite series on Cartesian products. It's particularly useful for multidimensional polynomial approximations and scientific computing applications.