Differentiate a Chebyshev series in Python

To differentiate a Chebyshev series, use the polynomial.chebder() method in NumPy. This method returns the Chebyshev series coefficients of the derivative. The coefficients are differentiated m times along the specified axis, with each iteration multiplied by a scale factor.

The argument c is an array of coefficients from low to high degree. For example, [1,2,3] represents the series 1*T_0 + 2*T_1 + 3*T_2, where T_n are Chebyshev polynomials.

Syntax

numpy.polynomial.chebyshev.chebder(c, m=1, scl=1, axis=0)

Parameters

The method accepts four parameters ?

• c ? Array of Chebyshev series coefficients
• m ? Number of derivatives taken, must be non-negative (Default: 1)
• scl ? Scale factor multiplied at each differentiation (Default: 1)
• axis ? Axis over which the derivative is taken (Default: 0)

Basic Example

Let's start with a simple example to differentiate a Chebyshev series ?

import numpy as np
from numpy.polynomial import chebyshev as C

# Create an array of Chebyshev series coefficients
c = np.array([1, 2, 3, 4])

print("Original coefficients:", c)
print("First derivative:", C.chebder(c))
print("Second derivative:", C.chebder(c, m=2))

Original coefficients: [1 2 3 4]
First derivative: [14. 12. 24.]
Second derivative: [24. 72.]

Using Scale Factor

The scale factor scl multiplies each differentiation, useful for variable transformations ?

import numpy as np
from numpy.polynomial import chebyshev as C

c = np.array([1, 2, 3, 4])

# Differentiate with scale factor of 2
result_scaled = C.chebder(c, scl=2)
print("With scale factor 2:", result_scaled)

# Compare with normal differentiation
result_normal = C.chebder(c)
print("Normal differentiation:", result_normal)

With scale factor 2: [28. 24. 48.]
Normal differentiation: [14. 12. 24.]

Multidimensional Arrays

For multidimensional coefficients, you can specify which axis to differentiate along ?

import numpy as np
from numpy.polynomial import chebyshev as C

# 2D coefficient array
c_2d = np.array([[1, 2], [3, 4], [5, 6]])
print("Original 2D coefficients:")
print(c_2d)

# Differentiate along axis 0 (default)
result_axis0 = C.chebder(c_2d, axis=0)
print("\nDerivative along axis 0:")
print(result_axis0)

# Differentiate along axis 1
result_axis1 = C.chebder(c_2d, axis=1)
print("\nDerivative along axis 1:")
print(result_axis1)

Original 2D coefficients:
[[1 2]
 [3 4]
 [5 6]]

Derivative along axis 0:
[[ 6.  8.]
 [20. 24.]]

Derivative along axis 1:
[[2.]
 [4.]
 [6.]]

Complete Example

Here's a comprehensive example showing array properties and differentiation ?

import numpy as np
from numpy.polynomial import chebyshev as C

# Create an array of Chebyshev series coefficients
c = np.array([1, 2, 3, 4])

# Display the coefficient array
print("Our coefficient Array...")
print(c)

# Check the Dimensions
print("\nDimensions of our Array...")
print(c.ndim)

# Get the Datatype
print("\nDatatype of our Array object...")
print(c.dtype)

# Get the Shape
print("\nShape of our Array object...")
print(c.shape)

# Differentiate the Chebyshev series
print("\nResult...")
print(C.chebder(c))

Our coefficient Array...
[1 2 3 4]

Dimensions of our Array...
1

Datatype of our Array object...
int64

Shape of our Array object...
(4,)

Result...
[14. 12. 24.]

Conclusion

The chebder() method efficiently computes derivatives of Chebyshev series by returning the differentiated coefficients. Use the m parameter for higher-order derivatives and scl for variable transformations.