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

To differentiate a Hermite_e series with multidimensional coefficients, use the hermite_e.hermeder() method in Python. This method allows you to compute derivatives across specific axes of multidimensional coefficient arrays.

Parameters

The hermite_e.hermeder() method accepts the following parameters:

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

Example: Differentiating Along Axis 1

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

import numpy as np
from numpy.polynomial import hermite_e as H

# 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 along axis 1
print("\nResult...\n", H.hermeder(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...
 [[1.]
 [3.]]

How It Works

When differentiating along axis 1, the method takes the derivative of each row independently. The original coefficient matrix has shape (2, 2), and after differentiation, we get a (2, 1) result. Each element in the result represents the derivative coefficient for the corresponding polynomial.

Multiple Derivatives

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

import numpy as np
from numpy.polynomial import hermite_e as H

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

# First derivative along axis 1
result1 = H.hermeder(c, m=1, axis=1)
print("\nFirst derivative:\n", result1)

# Second derivative along axis 1
result2 = H.hermeder(c, m=2, axis=1)
print("\nSecond derivative:\n", result2)

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

First derivative:
 [[1. 4.]
 [4. 10.]]

Second derivative:
 [[4.]
 [10.]]

Conclusion

The hermite_e.hermeder() method provides flexible differentiation of Hermite_e series along specified axes. Use axis=1 to differentiate across columns, which is useful for multivariable polynomial operations.