Evaluate a 2-D Hermite_e series on the Cartesian product of x and y with 3d array of coefficient in Python

To evaluate a 2-D Hermite_e series on the Cartesian product of x and y, use the hermite_e.hermegrid2d() method in Python. This method returns the values of the two-dimensional polynomial at points in the Cartesian product of x and y coordinates.

Syntax

numpy.polynomial.hermite_e.hermegrid2d(x, y, c)

Parameters

The parameters are:

• x, y: The two dimensional series is evaluated at points in the Cartesian product of x and y. If x or y is a list or tuple, it is first converted to an ndarray
• c: 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

Example

Let's create a 3D array of coefficients 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(24).reshape(2, 2, 6)

# 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 2-D Hermite_e series on Cartesian product
print("\nResult...\n", H.hermegrid2d([1, 2], [1, 2], c))

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]]]

Dimensions of our Array...
 3

Datatype of our Array object...
 int64

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

Result...
 [[[ 36.  60.]
  [ 66. 108.]]

 [[ 40.  66.]
  [ 72. 117.]]

 [[ 44.  72.]
  [ 78. 126.]]

 [[ 48.  78.]
  [ 84. 135.]]

 [[ 52.  84.]
  [ 90. 144.]]

 [[ 56.  90.]
  [ 96. 153.]]]

How It Works

The method evaluates the 2-D Hermite_e polynomial series at each point (x_i, y_j) where x_i comes from the first array [1, 2] and y_j comes from the second array [1, 2]. The 3D coefficient array has shape (2, 2, 6), so it contains 6 different coefficient matrices of size 2×2. The result has shape (6, 2, 2) corresponding to the 6 coefficient sets evaluated at the 2×2 Cartesian product points.

Conclusion

The hermegrid2d() method efficiently evaluates 2-D Hermite_e series on Cartesian products. It handles multiple coefficient sets when the coefficient array has more than 2 dimensions, making it useful for batch polynomial evaluations.