Differentiate a Legendre series with multidimensional coefficients over axis 1 in Python

To differentiate a Legendre series with multidimensional coefficients, use the polynomial.legendre.legder() method in Python. This method 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 function accepts the following parameters:

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

Example: Differentiating over Axis 1

Let's create a multidimensional array and differentiate along axis 1 ?

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

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

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

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

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

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

# Differentiate the Legendre series along axis 1
print("\nResult (differentiated along axis 1)...\n",L.legder(c, axis=1))

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

Dimensions of our Array...
2

Datatype of our Array object...
int64

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

Result (differentiated along axis 1)...
 [[1.]
 [3.]]

How It Works

When differentiating along axis 1, the function treats each row as a separate Legendre series. For a 2D array with shape (2,2), each row represents coefficients of a Legendre polynomial. The derivative of a Legendre series [a?, a?] along axis 1 gives [a?] scaled appropriately.

Example with Multiple Derivatives

You can also take multiple derivatives by specifying the m parameter ?

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

# Create a larger coefficient array
c = np.arange(6).reshape(2,3)
print("Original array:\n", c)

# First derivative along axis 1
print("\nFirst derivative (m=1):\n", L.legder(c, m=1, axis=1))

# Second derivative along axis 1  
print("\nSecond derivative (m=2):\n", L.legder(c, m=2, axis=1))

Original array:
 [[0 1 2]
 [3 4 5]]

First derivative (m=1):
 [[1. 4.]
 [4. 10.]]

Second derivative (m=2):
 [[4.]
 [10.]]

Conclusion

The legder() method efficiently differentiates Legendre series along any specified axis. Use axis=1 to differentiate each row independently, which is useful for handling multiple polynomial series simultaneously.