Differentiate a Chebyshev series with multidimensional coefficients in Python

In this article, we will cover how to differentiate the Chebyshev series with multidimensional coefficients in Python using NumPy.

Example

Input: [[1 2 3 4 5]

 [3 4 2 6 7]]

Output: [[3. 4. 2. 6. 7.]]

Explanation: Chebyshev series of the derivative.

chebyshev.chebder method

To evaluate a Chebyshev series at points x with a multidimensional coefficient array, NumPy provides a function called chebyshev.chebder(). This method is used to generate the Chebyshev series and this method is available in the NumPy module in python, it returns a multi-dimensional coefficient array, Below is the syntax of the Chebyshev method.

Syntax: chebyshev.chebder(x, m, axis)

Parameter:

• x: array
• m: Number of derivatives taken, must be non-negative. (Default: 1)
• axis: Axis over which the derivative is taken. (Default: 1).

Return: Chebyshev series.

Example 1:

In this example, we are creating a coefficient multi-dimensional array of 5 x 2 and, displaying the shape and dimensions of an array. Also, we are using chebyshev.chebder() method to differentiate a Chebyshev series.

Python3

# import the numpy module
import numpy

# import chebyshev
from numpy.polynomial import chebyshev

# create array of coefficients with 5 elements each
coefficients_data = numpy.array([[1, 2, 3, 4, 5],
                                 [3, 4, 2, 6, 7]])

# Display the coefficients
print(coefficients_data)

# get the shape
print(f"\nShape of an array: {coefficients_data.shape}")

# get the dimensions
print(f"Dimension: {coefficients_data.ndim}")

# using  chebyshev.chebder() method to differentiate
# a chebyshev series.
print("\nChebyshev series", chebyshev.chebder(coefficients_data))

Output:

[[1 2 3 4 5]
 [3 4 2 6 7]]

Shape of an array: (2, 5)
Dimension: 2

Chebyshev series [[3. 4. 2. 6. 7.]]

Example 2:

In this example, we are creating a coefficient multi-dimensional array of 5 x 3 and, displaying the shape and dimensions of an array. Also, we are using the number of derivatives=2, and the axis over which the derivative is taken is 1.

Python3

# import the numpy module
import numpy

# import chebyshev
from numpy.polynomial import chebyshev

# create array of coefficients with 5 elements each
coefficients_data = numpy.array(
    [[1, 2, 3, 4, 5], [3, 4, 2, 6, 7], [43, 45, 2, 6, 7]])

# Display the coefficients
print(coefficients_data)

# get the shape
print(f"\nShape of an array: {coefficients_data.shape}")

# get the dimensions
print(f"Dimension: {coefficients_data.ndim}")

# using  chebyshev.chebder() method to differentiate a
# chebyshev series.
print("\nChebyshev series", chebyshev.chebder(coefficients_data,
                                              m=2, axis=1))

Output:

[[ 1  2  3  4  5]
 [ 3  4  2  6  7]
 [43 45  2  6  7]]

Shape of an array: (3, 5)
Dimension: 2

Chebyshev series [[172.  96. 240.]
 [232. 144. 336.]
 [232. 144. 336.]]