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

To differentiate a Hermite series with multidimensional coefficients, use the hermite.hermder() method in Python. This function allows you to compute derivatives along specific axes of multidimensional coefficient arrays.

Syntax

hermite.hermder(c, m=1, scl=1, axis=0)

Parameters

The hermder() method accepts the following parameters ?

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

Example

Let's create a multidimensional coefficient array and differentiate along different axes ?

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

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

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

# Check the dimensions
print("\nDimensions of our Array...")
print(c.ndim)

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

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

# Differentiate along axis 1
print("\nResult (axis=1)...")
print(H.hermder(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 (axis=1)...
[[2.]
 [6.]]

Differentiating Along Different Axes

You can specify different axes for differentiation ?

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

c = np.arange(6).reshape(2, 3)
print("Original coefficient array:")
print(c)

# Differentiate along axis 0 (default)
print("\nDerivative along axis 0:")
print(H.hermder(c, axis=0))

# Differentiate along axis 1
print("\nDerivative along axis 1:")
print(H.hermder(c, axis=1))

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

Derivative along axis 0:
[[6. 8. 10.]]

Derivative along axis 1:
[[ 2.  8.]
 [ 8. 20.]]

Using Multiple Derivatives and Scaling

The method also supports higher-order derivatives and scaling factors ?

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

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

# Second derivative with scaling
print("\nSecond derivative (m=2, scl=2):")
print(H.hermder(c, m=2, scl=2, axis=1))

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

Second derivative (m=2, scl=2):
[[48.]
 [96.]]

Conclusion

The hermite.hermder() method provides flexible differentiation of Hermite series with multidimensional coefficients. Use the axis parameter to control which dimension to differentiate along, and adjust m and scl for higher-order derivatives and scaling.