Evaluate a Hermite series at points x broadcast over the columns of the coefficient in Python

To evaluate a Hermite series at points x, use the hermite.hermval() method in Python NumPy. This function allows you to evaluate Hermite polynomials with broadcasting capabilities over coefficient arrays.

Syntax

numpy.polynomial.hermite.hermval(x, c, tensor=True)

Parameters

x: If x is a list or tuple, it is converted to an ndarray, otherwise it is left unchanged and treated as a scalar. The elements must support addition and multiplication with themselves and with the elements of c.

c: An array of coefficients ordered so that the coefficients for terms of degree n are contained in c[n]. If c is multidimensional, the remaining indices enumerate multiple polynomials. In the two-dimensional case, the coefficients may be thought of as stored in the columns of c.

tensor: If True, the shape of the coefficient array is extended with ones on the right, one for each dimension of x. If False, x is broadcast over the columns of c for the evaluation. The default value is True.

Example with tensor=False

Let's create a multidimensional coefficient array and evaluate it at specific points ?

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)

# Evaluate Hermite series at points x with tensor=False
print("\nResult...")
print(H.hermval([1,2], c, tensor=False))

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...
[ 4. 13.]

Understanding tensor Parameter

The difference between tensor=True and tensor=False is important for broadcasting ?

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

c = np.arange(6).reshape(3,2)
x = [1, 2]

print("Coefficient array:")
print(c)

print("\nWith tensor=False (broadcasting):")
result_false = H.hermval(x, c, tensor=False)
print("Shape:", result_false.shape)
print("Result:", result_false)

print("\nWith tensor=True (default):")
result_true = H.hermval(x, c, tensor=True)
print("Shape:", result_true.shape) 
print("Result:", result_true)

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

With tensor=False (broadcasting):
Shape: (2,)
Result: [ 12.  23.]

With tensor=True (default):
Shape: (2, 2)
Result: [[ 8. 11.]
         [28. 43.]]

Comparison

Conclusion

Use hermite.hermval() to evaluate Hermite series efficiently. Set tensor=False for broadcasting x over coefficient columns, or tensor=True (default) when evaluating multiple polynomials at multiple points.