Evaluate a 3-D Hermite_e series at points (x,y,z) in Python

To evaluate a 3-D Hermite_e series at points (x, y, z), use the numpy.polynomial.hermite_e.hermeval3d() method. This method evaluates a three-dimensional Hermite_e polynomial series and returns values at the specified coordinate points.

Syntax

numpy.polynomial.hermite_e.hermeval3d(x, y, z, c)

Parameters

x, y, z ? The three-dimensional series is evaluated at points (x, y, z). These arrays must have the same shape. If any parameter is a list or tuple, it is converted to an ndarray.

c ? Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in c[i,j,k]. If c has dimension greater than 3, the remaining indices enumerate multiple sets of coefficients.

Example

Let's create a 3-D 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 3d array of coefficients
c = np.arange(24).reshape(2, 2, 6)

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

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

# Evaluate 3D Hermite_e series at points (1,2), (1,2), (1,2)
result = H.hermeval3d([1, 2], [1, 2], [1, 2], c)
print(f"\nEvaluation at points ([1,2], [1,2], [1,2]):")
print(result)

Coefficient 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: 3
Shape: (2, 2, 6)
Datatype: int64

Evaluation at points ([1,2], [1,2], [1,2]):
[ 212. -2484.]

Different Point Combinations

You can also evaluate at different coordinate combinations −

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)

# Evaluate at multiple point combinations
x = [0, 1]
y = [0, 1] 
z = [0, 1]

result = H.hermeval3d(x, y, z, c)
print(f"\nEvaluation at points ({x}, {y}, {z}):")
print(result)

# Single point evaluation
single_result = H.hermeval3d(1, 1, 1, c)
print(f"\nEvaluation at single point (1, 1, 1): {single_result}")

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

 [[4 5]
  [6 7]]]

Evaluation at points ([0, 1], [0, 1], [0, 1]):
[0. 6.]

Evaluation at single point (1, 1, 1): 6.0

Conclusion

The hermeval3d() method provides an efficient way to evaluate 3-D Hermite_e polynomial series at specified coordinate points. The coefficient array structure determines the polynomial terms, and the method handles both single points and arrays of coordinates.