Evaluate a 2-D Hermite_e series at points (x,y) in Python

To evaluate a 2D Hermite_e series at points (x, y), use the hermite_e.hermeval2d() method in NumPy. This method returns the values of the two-dimensional polynomial at points formed with pairs of corresponding values from x and y.

Syntax

numpy.polynomial.hermite_e.hermeval2d(x, y, c)

Parameters

The function takes three parameters:

• x, y − The two dimensional series is evaluated at points (x, y), where x and y must have the same shape. If x or y is a list or tuple, it is first converted to an ndarray
• c − Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is 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 2D coefficient array and evaluate the Hermite_e series at specific points ?

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

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

# 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 2D Hermite_e series at points (x, y), use the hermeval2d() method
print("\nResult...\n", H.hermeval2d([1,2], [1,2], c))

Our Array...
 [[0 1]
 [2 3]]

Dimensions of our Array...
2

Datatype of our Array object...
int64

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

Result...
 [ 6. 18.]

How It Works

The coefficient matrix c represents the 2D Hermite_e polynomial:

• c[0,0] = 0 (coefficient for H_e0(x) * H_e0(y))
• c[0,1] = 1 (coefficient for H_e0(x) * H_e1(y))
• c[1,0] = 2 (coefficient for H_e1(x) * H_e0(y))
• c[1,1] = 3 (coefficient for H_e1(x) * H_e1(y))

For points (1,1) and (2,2), the function evaluates the polynomial and returns the corresponding values [6., 18.].

Conclusion

The hermeval2d() method efficiently evaluates 2D Hermite_e series at multiple points simultaneously. The coefficient array structure directly maps to polynomial terms, making it straightforward to define complex 2D polynomials.