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

Syntax

numpy.polynomial.hermite_e.hermegrid3d(x, y, z, c)

Parameters

The parameters are x, y, z − The three dimensional series is evaluated at the 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 and, if it isn't an ndarray, it is treated as a scalar.

The parameter c is an array of coefficients ordered so that the 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. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape.

Example

Let's create a 3D coefficient array and evaluate the Hermite_e series ?

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

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

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

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

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

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

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

Dimensions: 3
Datatype: int64
Shape: (2, 2, 4)

Result:
[[[-20. 248.]
  [-30. 404.]]

 [[-30. 436.]
  [-45. 702.]]]

How It Works

The hermegrid3d() function evaluates the 3D Hermite_e polynomial at all combinations of the input points. With input arrays [1,2] for each dimension, it creates a 2×2×2 grid of evaluation points and returns the corresponding polynomial values based on the coefficient array.

Different Input Arrays

You can use different arrays for x, y, and z coordinates ?

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

# Create coefficient array
c = np.arange(8).reshape(2,2,2)
print("Coefficient Array:")
print(c)

# Use different coordinate arrays
x_vals = [0, 1]
y_vals = [0, 1, 2]
z_vals = [1]

result = H.hermegrid3d(x_vals, y_vals, z_vals, c)
print(f"\nResult shape: {result.shape}")
print("Result:")
print(result)

Coefficient Array:
[[[0 1]
  [2 3]]

 [[4 5]
  [6 7]]]

Result shape: (2, 3, 1)
Result:
[[[6.]
  [8.]
  [14.]]]

Conclusion

The hermegrid3d() method efficiently evaluates 3D Hermite_e series on Cartesian products. It's useful for multidimensional polynomial interpolation and approximation problems in scientific computing.