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

To evaluate a 2-D Hermite series on the Cartesian product of x and y, use the hermite.hermgrid2d(x, y, c) method in Python. The method returns the values of the two-dimensional polynomial at points in the Cartesian product of x and y.

Syntax

numpy.polynomial.hermite.hermgrid2d(x, y, c)

Parameters

The parameters are:

• x, y ? The two dimensional series is evaluated at the 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, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar.
• c ? 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 two dimensions, ones are implicitly appended to its shape to make it 2-D.

Return Value

The shape of the result will be c.shape[2:] + x.shape. This means the result contains evaluations for each coefficient set and each point in the Cartesian product.

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(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)

# To evaluate a 2-D Hermite series on the Cartesian product of x and y
print("\nResult...\n",H.hermgrid2d([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...
 [[[108. 192.]
  [204. 360.]]

 [[117. 207.]
  [219. 385.]]

 [[126. 222.]
  [234. 410.]]

 [[135. 237.]
  [249. 435.]]

 [[144. 252.]
  [264. 460.]]

 [[153. 267.]
  [279. 485.]]]

How It Works

The function evaluates the Hermite polynomial at each point in the Cartesian product of x and y. Since our coefficient array has shape (2, 2, 6), the result has shape (6, 2, 2) - representing 6 coefficient sets evaluated at 2×2 = 4 points (Cartesian product of [1,2] and [1,2]).

Conclusion

Use hermite.hermgrid2d() to evaluate 2-D Hermite series on Cartesian products. The result shape depends on the coefficient array dimensions and input point arrays.