Evaluate 3-D Hermite series on the Cartesian product of x, y and z using NumPy in Python

In this article, we will discuss how to Evaluate a 3-D Hermite series on the Cartesian product of x, y, and z  in Python and NumPy.

NumPy.polynomial.hermite.hermgrid3d method

Hermite polynomials are significant in approximation theory because the Hermite nodes are used as matching points for optimizing polynomial interpolation. To perform Hermite differentiation, NumPy provides a function called hermite.hermgrid3d which can be used to evaluate the cartesian product of the 3D Hermite series. This function converts the parameters x, y, and z to array only if they are tuples or a list, otherwise, it is left unchanged and, if it is not an array, it is treated as a scalar.

Syntax: polynomial.hermite.hermgrid3d(x, y, z, c)

Parameters:

• x,y,z: array_like
• c: array of coefficients

Returns: Two dimensional polynomials at points as cartesian products of x and y.

Example 1:

In the first example. let us consider a 4D array c of size 32.  Let us consider a 3D series [1,2],[1,2],[1,2] to evaluate against the 4D array. Import the necessary packages as shown and pass the appropriate parameters as shown below.

Python3

import numpy as np
from numpy.polynomial import hermite

# co.efficient array
c = np.arange(32).reshape(2, 2, 4, 2)

print(f'The co.efficient array is {c}')
print(f'The shape of the array is {c.shape}')
print(f'The dimension of the array is {c.ndim}D')
print(f'The datatype of the array is {c.dtype}')

# evaluating 4d co.eff array with a 3d hermite series
res = hermite.hermgrid3d([1, 2], [1, 2], [1, 2], c)

# resultant array
print(f'Resultant series ---> {res}')

Output:

The co.efficient array is [[[[ 0  1]
   [ 2  3]
   [ 4  5]
   [ 6  7]]

  [[ 8  9]
   [10 11]
   [12 13]
   [14 15]]]


 [[[16 17]
   [18 19]
   [20 21]
   [22 23]]

  [[24 25]
   [26 27]
   [28 29]
   [30 31]]]]
The shape of the array is (2, 2, 4, 2)
The dimension of the array is 4D
The datatype of the array is int64
Resultant series ---> [[[[3.6000e+01 1.1232e+04]
   [7.6000e+01 1.9664e+04]]

  [[9.2000e+01 2.0608e+04]
   [1.8000e+02 3.5920e+04]]]


 [[[4.5000e+01 1.1763e+04]
   [9.1000e+01 2.0549e+04]]

  [[1.0700e+02 2.1493e+04]
   [2.0500e+02 3.7395e+04]]]]

Example 2:

In this example, we are using a 1-D array to evaluate a 3-D Hermite series on the Cartesian product series.

Python3

# import packages
import numpy as np
from numpy.polynomial import hermite

# array of coefficients
c = np.array([2,2,3])
print(c)

# shape of the array is
print("Shape of the array is : ",c.shape)

# dimension of the array
print("The dimension of the array is : ",c.ndim)

# Datatype of the array
print("Datatype of our Array is : ",c.dtype)

#evaluating hermite series
print(hermite.hermgrid3d([1,2],[3,4],[5,6],c))

Output:

[2 2 3]
Shape of the array is :  (3,)
The dimension of the array is :  1
Datatype of our Array is :  int64
[4604. 5460.]