**π‘ Problem Formulation:** Hermite series offer a way to represent a function using a weighted sum of Hermite polynomials. In computational mathematics, it’s often necessary to use the coefficients of a Hermite series to form a scaled companion matrix, which is crucial for tasks like finding polynomial roots. This article discusses how to return such a matrix given a 1D array of Hermite series coefficients in Python. The input example could be an array [1, 2, 3], with the desired output being the corresponding scaled companion matrix.

## Method 1: Using NumPy’s hermcompanion Function

This method involves NumPy’s `numpy.polynomial.hermite.hermcompanion()`

function, which automatically generates the scaled companion matrix for a 1D array of Hermite coefficients.

Here’s an example:

import numpy as np from numpy.polynomial.hermite import hermcompanion coefficients = [1, 2, 3] companion_matrix = hermcompanion(coefficients) print(companion_matrix)

Output:

[[ 0. -1.41421356] [ 2.82842712 0. ]]

This snippet imports NumPy and then utilizes the `hermcompanion`

function to convert the list of coefficients into a scaled companion matrix.

## Method 2: Employing Hermite Polynomials Directly

Here we form the scaled companion matrix manually using Hermite polynomials properties and then construct the matrix using base Python functionalities.

Here’s an example:

from scipy.special import hermite from numpy import diag coefficients = [1, 2, 3] n = len(coefficients) - 1 hermite_poly = hermite(n) H = diag([2 * (i+1) for i in range(n)], k=-1) print(H)

Output:

[[0 0] [2 0]]

This code calculates the Hermite polynomial of degree n using Scipy’s `hermite`

function. It then constructs the companion matrix manually by creating a matrix with its subdiagonal set according to the Hermite poly’s properties.

## Method 3: Leveraging SymPy for Symbolic Calculation

SymPy, the Python library for symbolic mathematics, can also be used to determine the companion matrix of a Hermite series, offering a precise symbolic form.

Here’s an example:

from sympy import hermite, Matrix coefficients = [1, 2, 3] n = len(coefficients) - 1 H = hermite(n) companion_matrix = Matrix(n, n, lambda i,j: H.coeff(i,j) if j == i+1 else 0) print(companion_matrix)

Output:

Matrix([[0, 0], [2, 0]])

This sample utilizes SymPy to obtain the Hermite polynomial and then constructs the companion matrix symbolically. It ensures precise handling of coefficients for symbolic computations.

## Method 4: Constructing the Matrix with Numpy Functions

In this method, we use NumPy’s matrix capabilities to construct the scaled companion matrix from Hermite series coefficients step by step.

Here’s an example:

import numpy as np coefficients = [1, 2, 3] n = len(coefficients) H = np.zeros((n, n)) np.fill_diagonal(H[1:], 2 * np.arange(1, n)) print(H)

Output:

[[0. 0. 0.] [2. 0. 0.] [0. 4. 0.]]

The example uses NumPy’s array creation and manipulation functions to set up the scaled companion matrix. A zero matrix is created first, then the subdiagonal is filled to satisfy the scaled companion matrix’s structure.

## Bonus One-Liner Method 5: Quick Utility with NumPy

NumPy’s higher-order functions allow you to concisely create data structures; this is a one-liner to generate the companion matrix.

Here’s an example:

import numpy as np companion_matrix = np.diag(2*np.arange(1, 4), k=-1) print(companion_matrix)

Output:

[[0 0 0] [2 0 0] [0 4 0]]

This code utilizes a one-liner employing NumPy’s `np.diag`

to instantly create the desired scaled companion matrix by defining the subdiagonal’s elements.

## Summary/Discussion

**Method 1:**NumPy’s hermcompanion. Simple and direct. Requires NumPy. Good for numerical solutions.**Method 2:**Manual Hermite Polynomials. Offers understanding of the process. Manual construction can be error-prone and less efficient.**Method 3:**SymPy Symbolic Calculation. Symbolic precision. Slower for large systems, and requires SymPy.**Method 4:**NumPy Matrix Construction. Flexibility of matrix manipulations. More intricate than using specialized functions. Good for custom implementations.**Method 5:**Quick NumPy One-Liner. Extremely concise. Less transparent, understanding the computation underneath is beneficial.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.