Differentiate a Laguerre series with multidimensional coefficients over specific axis in Python

To differentiate a Laguerre series with multidimensional coefficients, use the laguerre.lagder() method in Python. This method returns the Laguerre series coefficients differentiated m times along a specified axis. At each iteration, the result is multiplied by a scalar value.

The coefficient array represents Laguerre polynomials where [1,2,3] represents the series 1*L_0 + 2*L_1 + 3*L_2. For multidimensional arrays like [[1,2],[1,2]], it represents 1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y) if axis=0 is x and axis=1 is y.

Syntax

laguerre.lagder(c, m=1, scl=1, axis=0)

Parameters

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

Example

Let's differentiate a Laguerre series with a 2D coefficient array along different axes ?

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

# Create a multidimensional array of coefficients
c = np.arange(4).reshape(2,2)
print("Coefficient Array:")
print(c)
print(f"Shape: {c.shape}")

# Differentiate along axis 0
result_axis0 = L.lagder(c, axis=0)
print("\nDifferentiation along axis 0:")
print(result_axis0)

# Differentiate along axis 1  
result_axis1 = L.lagder(c, axis=1)
print("\nDifferentiation along axis 1:")
print(result_axis1)

Coefficient Array:
[[0 1]
 [2 3]]
Shape: (2, 2)

Differentiation along axis 0:
[[-2. -2.]]

Differentiation along axis 1:
[[-1.]
 [-3.]]

Multiple Derivatives

You can take multiple derivatives by setting the m parameter ?

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

# Create a larger coefficient array
c = np.arange(12).reshape(3,4)
print("Original coefficients:")
print(c)

# Take second derivative along axis 1
result = L.lagder(c, m=2, axis=1)
print("\nSecond derivative along axis 1:")
print(result)

Original coefficients:
[[ 0  1  2  3]
 [ 4  5  6  7]
 [ 8  9 10 11]]

Second derivative along axis 1:
[[-2. -2.]
 [-2. -2.]
 [-2. -2.]]

Using Scale Factor

The scl parameter multiplies each differentiation step ?

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

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

# Differentiate with scale factor
result = L.lagder(c, m=1, scl=2, axis=1)
print("\nDifferentiation with scl=2 along axis 1:")
print(result)

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

Differentiation with scl=2 along axis 1:
[[-2. -2.]
 [-2. -2.]]

Comparison

Conclusion

The lagder() method efficiently differentiates multidimensional Laguerre series along specified axes. Use the axis parameter to control differentiation direction and m for multiple derivatives.