Evaluate a 3-D Chebyshev series on the Cartesian product of x, y and z with 4d array of coefficient in Python

In this article, we will discuss how to Evaluate a 3-D Chebyshev series on the Cartesian product of x, y, and z with a 4d array of coefficients in Python and NumPy.

NumPy.polynomial.chebyshev.chebgrid3d method

Chebyshev polynomials are significant in approximation theory because the Chebyshev nodes are used as matching points for optimizing polynomial interpolation. 
To perform Chebyshev differentiation, NumPy provides a function called Chebyshev.chebgrid3d which can be used to evaluate the cartesian product of the 3D Chebyshev 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.chebyshev.chebgrid3d(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.

import numpy as np
from numpy.polynomial import chebyshev

# 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 chebyshev series
res = chebyshev.chebgrid3d([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 int32

Resultant series ---> [[[[ 240. 2480.]
   [ 392. 4008.]]

  [[ 424. 4296.]
   [ 684. 6876.]]]


 [[[ 256. 2624.]
   [ 416. 4224.]]

  [[ 448. 4512.]
   [ 720. 7200.]]]]

Example 2:

In the second example. let us consider a 4D array c of size 64.  Let us consider a 3D series [2,1],[2,1],[2,1] to evaluate against the 4D array. Import the necessary packages as shown and pass the appropriate parameters as shown below.

import numpy as np
from numpy.polynomial import chebyshev

# co.efficient array
c = np.arange(64).reshape(4, 4, 2, 2)

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

# evaluating 4d co.eff array with a 3d chebyshev series
res = chebyshev.chebgrid3d([2, 1], [2, 1], [2, 1], 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]]

 [[[32 33]   [34 35]]  [[36 37][38 39]]

  [[40 41]   [42 43]]  [[44 45][46 47]]]

 [[[48 49]   [50 51]]  [[52 53][54 55]]

  [[56 57]   [58 59]]  [[60 61][62 63]]]]

   

The shape of the array is (4, 4, 2, 2)

The dimension of the array is 4D

The datatype of the array is int32

Resultant series ---> [[[[208224. 137952.][ 21216.  14048.]]  [[ 15456.  10208.][  1504.    992.]]]

 [[[212112. 140544.][ 21648.  14336.]]  [[ 15888.  10496.] [  1552.   1024.]]]]