Evaluate a 2-D Chebyshev series at points (x, y) with 3D array of coefficient in Python

In this article, we will discuss how to evaluate a 2-D Chebyshev series at points (x, y) with 3D array of coefficient in Python

Example

Input: [[[ 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]]]

Output: [[[1920.  522.]
  [ 414.  108.]]

 [[2020.  552.]
  [ 444.  117.]]

 [[2120.  582.]
  [ 474.  126.]]]

Explanation: Two dimensional Chebyshev series.

NumPy.polynomial.Chebyshev.chebgrid2d method

To perform Chebyshev differentiation, NumPy provides a function called Chebyshev.chebgrid2d which can be used to evaluate the cartesian product of the 2D Chebyshev series. This function converts the parameters x and y to arrays only if they are tuples or a list and of the same shape, otherwise, it is left unchanged and, if it is not an array, it is treated as a scalar. If c has a dimension greater than 2 the remaining indices enumerate multiple sets of coefficients.

Syntax: chebyshev.chebgrid2d(x,y, c)

Parameters:

• x,y: Input array.
• c: Array of coefficients ordered

Returns: Two dimensional Chebyshev series at points in the Cartesian product of x and y.

Example 1:

In the first example. let us consider a 3D array c of size 27 and a series of [2,2],[2,2] to evaluate against the 2D array.

Python3

import numpy as np
from numpy.polynomial import chebyshev

# co.efficient array
c = np.arange(27).reshape(3, 3, 3)

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 co.eff array with a chebyshev series
res = chebyshev.chebgrid2d([2, 2], [2, 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]]]
The shape of the array is (3, 3, 3)
The dimension of the array is 3D
The datatype of the array is int32
Resultant series ---> [[[1920. 1920.]
  [1920. 1920.]]

 [[2020. 2020.]
  [2020. 2020.]]

 [[2120. 2120.]
  [2120. 2120.]]]

Example 2:

In the first example. let us consider a 3D array c of size 27 and a series of [2,1],[2,1], to evaluate against the 2D array.

Python3

import numpy as np
from numpy.polynomial import chebyshev

# co.efficient array
c = np.arange(27).reshape(3, 3, 3)

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 co.eff array with a chebyshev series
res = chebyshev.chebgrid2d([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]]]
The shape of the array is (3, 3, 3)
The dimension of the array is 3D
The datatype of the array is int32
Resultant series ---> [[[1920.  522.]
  [ 414.  108.]]

 [[2020.  552.]
  [ 444.  117.]]

 [[2120.  582.]
  [ 474.  126.]]]