Differentiate a Chebyshev series with multidimensional coefficients in Python

To differentiate a Chebyshev series with multidimensional coefficients, use the polynomial.chebder() method in NumPy. The method returns the Chebyshev series coefficients of the derivative, differentiated m times along a specified axis.

The argument c is an array of coefficients from low to high degree along each axis. For example, [1,2,3] represents the series 1*T_0 + 2*T_1 + 3*T_2 while [[1,2],[1,2]] represents 1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y) if axis=0 is x and axis=1 is y.

Syntax

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

Parameters

• c − Array of Chebyshev series coefficients. If multidimensional, different axes correspond to different variables
• m − Number of derivatives taken, must be non-negative (Default: 1)
• scl − Each differentiation is multiplied by scl. Final result is multiplied by scl**m (Default: 1)
• axis − Axis over which the derivative is taken (Default: 0)

Example

Let's differentiate a 2D Chebyshev series along different axes ?

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

# Create a multidimensional array of Chebyshev series coefficients
c = np.arange(4).reshape(2,2)

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

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

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

# Differentiate along axis 0 (default)
print("\nDerivative along axis 0...\n",C.chebder(c))

# Differentiate along axis 1
print("\nDerivative along axis 1...\n",C.chebder(c, axis=1))

Our coefficient Array...
 [[0 1]
 [2 3]]

Dimensions of our Array...
2

Shape of our Array object...
(2, 2)

Derivative along axis 0...
 [[2. 3.]]

Derivative along axis 1...
 [[1.]
 [3.]]

Multiple Derivatives

You can take higher-order derivatives by specifying the m parameter ?

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

# Create coefficients for a higher degree polynomial
c = np.arange(9).reshape(3,3)
print("Original coefficients...\n", c)

# First derivative
print("\nFirst derivative...\n", C.chebder(c, m=1))

# Second derivative  
print("\nSecond derivative...\n", C.chebder(c, m=2))

Original coefficients...
 [[0 1 2]
 [3 4 5]
 [6 7 8]]

First derivative...
 [[ 3.  4.  5.]
 [12. 14. 16.]]

Second derivative...
 [[12. 14. 16.]]

Using Scale Factor

The scl parameter multiplies each differentiation step ?

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

c = np.arange(6).reshape(2,3)
print("Original coefficients...\n", c)

# Differentiate with scale factor
print("\nWith scl=2...\n", C.chebder(c, scl=2))

# Compare with default scale
print("\nWith scl=1 (default)...\n", C.chebder(c, scl=1))

Original coefficients...
 [[0 1 2]
 [3 4 5]]

With scl=2...
 [[6. 8.]]

With scl=1 (default)...
 [[3. 4.]]

Conclusion

Use chebder() to differentiate multidimensional Chebyshev series along any axis. The m parameter controls derivative order, while scl applies scaling for variable transformations.