Differentiate a Legendre series and set the derivatives in Python

To differentiate a Legendre series in Python, use the numpy.polynomial.legendre.legder() method. This function returns the Legendre series coefficients differentiated m times along the specified axis.

Syntax

numpy.polynomial.legendre.legder(c, m=1, scl=1, axis=0)

Parameters

The method accepts the following parameters:

• c: Array of Legendre series coefficients. If multidimensional, different axes correspond to different variables
• m: Number of derivatives taken (must be non-negative, default: 1)
• scl: Scalar multiplier for each differentiation (default: 1)
• axis: Axis over which the derivative is taken (default: 0)

Example

Let's differentiate a Legendre series with different derivative orders:

import numpy as np
from numpy.polynomial import legendre as L

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

# Display the original coefficients
print("Original coefficients:", c)

# First derivative
print("First derivative:", L.legder(c, 1))

# Second derivative  
print("Second derivative:", L.legder(c, 2))

# Third derivative
print("Third derivative:", L.legder(c, 3))

Original coefficients: [1 2 3 4]
First derivative: [ 6. 20. 42.]
Second derivative: [ 60. 252.]
Third derivative: [840.]

Using Scale Parameter

The scl parameter multiplies each differentiation by a scalar value:

import numpy as np
from numpy.polynomial import legendre as L

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

# Differentiate with scale factor of 2
result_scaled = L.legder(c, m=2, scl=2)
print("Second derivative with scale=2:", result_scaled)

# Compare with normal second derivative
result_normal = L.legder(c, m=2)
print("Second derivative without scaling:", result_normal)
print("Scale factor applied:", result_scaled / result_normal)

Second derivative with scale=2: [ 240. 1008.]
Second derivative without scaling: [ 60. 252.]
Scale factor applied: [4. 4.]

Multidimensional Arrays

For multidimensional coefficient arrays, specify the axis for differentiation:

import numpy as np
from numpy.polynomial import legendre as L

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

# Differentiate along axis 0
result_axis0 = L.legder(c_2d, axis=0)
print("Derivative along axis 0:\n", result_axis0)

# Differentiate along axis 1
result_axis1 = L.legder(c_2d, axis=1)
print("Derivative along axis 1:\n", result_axis1)

2D coefficients:
 [[1 2 3]
 [4 5 6]]
Derivative along axis 0:
 [[ 9. 15. 21.]]
Derivative along axis 1:
 [[ 6. 20.]
 [15. 50.]]

Conclusion

Use numpy.polynomial.legendre.legder() to differentiate Legendre series efficiently. The method supports multiple derivatives, scaling factors, and multidimensional arrays for complex polynomial operations.