Alternatives to Hermite.fromroots() for Hermite Polynomial Creation in NumPy

Purpose

This function constructs a Hermite polynomial series based on provided roots. Hermite polynomials are a special type of polynomial used in various scientific and engineering applications.

Breakdown

.fromroots()
This is a class method of the Hermite class. It takes a sequence of roots (roots) as input and returns a Hermite series object.
polynomial.hermite.Hermite
This refers to the Hermite class within the polynomial.hermite submodule of NumPy. This class deals with Hermite polynomial functionalities.

Arguments

roots (array_like)
This is the mandatory argument. It represents a list or array containing the roots of the desired Hermite polynomial. These roots will determine the shape of the polynomial.

Optional Arguments

symbol (str), optional
This argument lets you define the variable symbol used in the polynomial expression. The default symbol is "x".
window (None, array_like), optional
This argument defines the windowing interval to which the domain will be mapped. It's helpful for scaling and shifting the polynomial. The default value is the same as domain.
domain ([], None, array_like), optional
This argument allows you to specify the domain (interval) for the polynomial. By default, it's set to [-1, 1].

Functionality

Root Processing
The fromroots method takes the provided roots and constructs the polynomial by multiplying factors of the form (x - root_i), where i iterates through each root. This essentially creates a polynomial that passes through all the specified roots.
Series Creation
The method returns a Hermite series object that encapsulates the polynomial coefficients and other attributes like domain and window (if provided).

Use Case

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

roots = [-1, 2]  # Specifying roots
poly = hm.Hermite.fromroots(roots)

# Print the polynomial coefficients
print(poly.coef)

# Evaluate the polynomial at a specific point (x = 0)
x = 0
result = poly(x)
print(result)

This code snippet creates a Hermite polynomial with roots -1 and 2, then prints the coefficients and evaluates the polynomial at x = 0.

Example 1: Constructing a Hermite Polynomial and Evaluating

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

# Define roots
roots = np.array([0.5, 1.5])

# Create Hermite polynomial from roots
poly = hm.Hermite.fromroots(roots)

# Print coefficients
print("Coefficients:", poly.coef)

# Evaluate polynomial at x = -1
x = -1
result = poly(x)
print("Value at x = -1:", result)

This example constructs a Hermite polynomial with roots 0.5 and 1.5. It then prints the coefficients and evaluates the polynomial at x = -1.

Example 2: Specifying Domain and Window

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

# Define roots and domain
roots = np.array([-2, 1])
domain = [-3, 2]  # Custom domain

# Create Hermite polynomial with domain
poly = hm.Hermite.fromroots(roots, domain=domain)

# Print information
print("Coefficients:", poly.coef)
print("Domain:", poly.domain)

# Window (same as domain for illustration)
window = domain
print("Window:", window)

This example demonstrates specifying the domain ([-3, 2]) during polynomial creation. It also shows setting the window (optional here) to be the same as the domain.

Example 3: Using a Symbol

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

# Define roots and symbol
roots = np.array([-1, 0])
symbol = "t"  # Custom symbol

# Create Hermite polynomial with symbol
poly = hm.Hermite.fromroots(roots, symbol=symbol)

# Print polynomial representation with the symbol
print(poly)

This example defines a custom symbol "t" for the variable in the polynomial expression printed by the poly object.

Using numpy.polyval() and Hermite Coefficients

This method involves calculating the Hermite coefficients for the desired roots and then using numpy.polyval() to evaluate the polynomial at various points.

import numpy as np

def hermite_from_roots(roots):
  """
  Calculates Hermite coefficients from roots and evaluates using polyval.
  """
  n = len(roots)
  coefs = np.zeros(n + 1)  # Initialize coefficients
  coefs[0] = 1  # Constant term

  for i in range(1, n + 1):
    # Build polynomial using nested multiplication
    for root in roots:
      coefs[i] = coefs[i] * (2 * root)

  return np.polyval, coefs  # Return evaluation function and coefficients

# Example usage
roots = np.array([0.5, 1.5])
polyval, coefs = hermite_from_roots(roots)

x = -1
result = polyval(coefs, x)  # Evaluate using coefficients and polyval
print(result)

Here, we define a function hermite_from_roots that calculates the coefficients and returns a function to evaluate the polynomial using numpy.polyval.

Symbolic Libraries (Optional)

Libraries like SymPy can be used to create symbolic expressions of Hermite polynomials based on roots. These expressions can then be evaluated numerically.

# This example requires installing SymPy (pip install sympy)
import sympy as sp

def hermite_symbolic(roots):
  """
  Creates a symbolic Hermite polynomial using SymPy.
  """
  x = sp.symbols("x")  # Define symbolic variable
  poly = 1
  for root in roots:
    poly *= (x - root)
  return poly

# Example usage
roots = [0.5, 1.5]
symbolic_poly = hermite_symbolic(roots)

# You can then evaluate the symbolic expression numerically at specific points

Symbolic libraries are useful for symbolic manipulation and understanding the underlying polynomial structure, especially for complex scenarios.
If you need more flexibility in manipulating coefficients or evaluating at various points, calculating coefficients and using polyval offers more control.
For simple use cases with known roots, Hermite.fromroots() might be the most efficient choice.