Evaluate a 2-D Hermite series on the Cartesian product of x and y with 1d 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. This 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

• 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.

Example

Let's create a 1d array of coefficients and evaluate the 2-D Hermite series ?

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

# Create a 1d array of coefficients
c = np.array([3, 5])

# 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 series on Cartesian product
print("\nResult...\n", H.hermgrid2d([1, 2], [1, 2], c))

Our Array...
 [3 5]

Dimensions of our Array...
 1

Datatype of our Array object...
 int64

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

Result...
 [ 59. 105.]

How It Works

The hermgrid2d() function evaluates the 2-D Hermite polynomial at each point in the Cartesian product of the input arrays. For a 1-D coefficient array [3, 5], it creates a 2-D coefficient matrix and computes the polynomial values at each (x, y) pair.

Multiple Coefficient Sets

You can also work with 2-D coefficient arrays for more complex polynomials ?

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

# Create a 2D array of coefficients
c = np.array([[1, 2], [3, 4]])

print("Coefficient array:")
print(c)

# Evaluate on Cartesian product
x = [0, 1]
y = [0, 1]
result = H.hermgrid2d(x, y, c)
print("\nResult for 2D coefficients:")
print(result)

Coefficient array:
[[1 2]
 [3 4]]

Result for 2D coefficients:
[[ 1.  2.]
 [ 7. 16.]]

Conclusion

Use hermite.hermgrid2d() to evaluate 2-D Hermite series on Cartesian products of coordinate arrays. The function automatically handles coefficient array reshaping and returns polynomial values at all coordinate combinations.