Evaluate a Hermite series at tuple of points x in Python

To evaluate a Hermite series at points x, use the hermite.hermval() method in Python NumPy. This function evaluates a Hermite polynomial series at given points using the coefficients provided.

Syntax

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

Parameters

The function accepts three 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.
• tensor − If True (default), 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.

Example

Let's create a simple Hermite series evaluation ?

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

# Create an array of coefficients
c = np.array([1, 2, 3])

# 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)

# Here, x is a tuple
x = (5, 10, 15)

# To evaluate a Hermite series at points x, use the hermite.hermval() method
print("\nResult...\n", H.hermval(x, c))

Our Array...
 [1 2 3]

Dimensions of our Array...
 1

Datatype of our Array object...
 int64

Shape of our Array object...
 (3,)

Result...
 [ 315. 1235. 2755.]

How It Works

The Hermite series is evaluated using the formula: H(x) = c[0] + c[1]*H?(x) + c[2]*H?(x) + ... where H?, H?, etc. are Hermite polynomials of increasing degree.

Example with Different Input Types

You can also evaluate the series with scalar input ?

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

# Same coefficients
c = np.array([1, 2, 3])

# Scalar input
x_scalar = 5
result_scalar = H.hermval(x_scalar, c)
print("Result for scalar x=5:", result_scalar)

# Array input
x_array = np.array([1, 2, 3])
result_array = H.hermval(x_array, c)
print("Result for array x=[1,2,3]:", result_array)

Result for scalar x=5: 315.0
Result for array x=[1,2,3]: [15. 47. 99.]

Conclusion

The hermite.hermval() function efficiently evaluates Hermite polynomial series at single points, tuples, or arrays. It's particularly useful for mathematical computations involving Hermite polynomials in scientific applications.