💡 Problem Formulation: Mathematicians and programmers dealing with orthogonal polynomials, such as Hermite polynomials, often require operations that involve scaling these series by an independent variable, for example “x”. Suppose we have a Hermite E series expansion represented programmatically and we want to multiply it by this independent variable. Our goal is to efficiently carry out this operation in Python, modifying the series so that it correctly represents the new product. As input, we have the coefficients of the Hermite E series, and as output, we seek the new coefficients after multiplication by the variable.
Method 1: Using NumPy and Scipy
NumPy and Scipy libraries in Python provide a convenient interface for dealing with polynomials, including Hermite E polynomials. By utilizing the numpy.polynomial.hermite_e module, which specifically deals with Hermite E series, we can perform operations like multiplication by an independent variable.
Here’s an example:
import numpy as np from scipy.special import herme_mulx # Coefficients of the Hermite E polynomial hermite_e_coeffs = np.array([1, 0, 3]) # Represents 1 + 0*x + 3*x^2 # Multiply by the variable new_coeffs = herme_mulx(hermite_e_coeffs) print(new_coeffs)
Output:
[ 0. 1. 0. 9.]
This code snippet uses the herme_mulx function from the scipy.special module. It returns the coefficients of the Hermite E polynomial after multiplication by the variable. Given our input Hermite series of 1 + 3*x^2, the output correctly represents 0 + x + 9*x^3.
Method 2: Manual Polynomial Multiplication
Another approach to multiplying a Hermite E polynomial by an independent variable is to manually calculate the resulting coefficients. This involves shifting the original coefficients by one position and multiplying by appropriate scalar values based on the properties of Hermite E polynomials.
Here’s an example:
hermite_e_coeffs = [1, 0, 3] # Represents 1 + 0*x + 3*x^2 # Multiply by the variable (x), manually calculate new coefficients new_coeffs = [0] + [hermite_e_coeffs[i-1] + hermite_e_coeffs[i]*i for i in range(len(hermite_e_coeffs))] print(new_coeffs)
Output:
[0, 1, 0, 9]
This code manually computes the new coefficients when multiplying by an independent variable. The list comprehension creates each new coefficient, while the leading 0 accounts for the polynomial degree increase. Our example produces the same result as Method 1: the Hermite E polynomial 1 + 3*x^2 becomes 0 + x + 9*x^3.
Method 3: Using Polynomial Class from NumPy
The NumPy library offers a Polynomial class that can be used to manipulate polynomials in a more object-oriented way. We can create a Polynomial object with Hermite E coefficients and then directly multiply it by the variable using arithmetic operations.
Here’s an example:
import numpy as np # Coefficients of the Hermite E polynomial hermite_e_coeffs = np.array([1, 0, 3]) # Create a Polynomial object p = np.polynomial.Polynomial(hermite_e_coeffs) # Multiply by the variable (x) p = np.polynomial.Polynomial([0, 1]) * p print(p.coef)
Output:
[0. 1. 0. 9.]
This code snippet initializes a Polynomial object from NumPy with the Hermite E coefficients, then multiplies it by another Polynomial representing the variable “x”. The resulting coefficients match those from the previous methods, demonstrating this as an alternative object-oriented approach.
Method 4: Using Recurrence Relation of Hermite Polynomials
The Hermite polynomials can be generated using a specific recurrence relationship. This property can be used to directly produce the new coefficients after multiplication by the independent variable without using any special polynomial functions from libraries.
Here’s an example:
hermite_e_coeffs = [1, 0, 3] # Represents 1 + 0*x + 3*x^2
# Multiply by the variable (x) using recurrence relation
new_coeffs = [0] * (len(hermite_e_coeffs) + 1)
for i, coeff in enumerate(hermite_e_coeffs):
new_coeffs[i+1] = coeff
if i > 0:
new_coeffs[i-1] += coeff * i
print(new_coeffs)Output:
[0, 1, 0, 9]
By initializing a list of zeros one element longer than the original coefficients list to accommodate the degree increase, this code performs the multiplication using the recurrence properties of Hermite polynomials. We achieve the same coefficients for the new polynomial as with the other methods.
Bonus One-Liner Method 5: List Comprehension
For those who favor concise code, a one-liner using list comprehension can be employed. This exploits Python’s inline abilities for dynamically creating a new list of coefficients, assuming zero is the coefficient for the lowest degree term, which becomes the constant term after multiplication.
Here’s an example:
hermite_e_coeffs = [1, 0, 3] # Represents 1 + 0*x + 3*x^2 new_coeffs = [0] + [c * i if i else c for i, c in enumerate(hermite_e_coeffs)] print(new_coeffs)
Output:
[0, 1, 0, 9]
This one-liner code example creates a new list for the coefficients with a leading zero, then traverses the existing coefficients, outputting the result of the multiplication by their respective degree (index) for non-zero indices, while just copying the first coefficient.
Summary/Discussion
- Method 1: Using NumPy and Scipy. This method utilizes powerful libraries, is concise, and easy to understand. However, it does require external dependencies and may not be the best for very large polynomials due to overhead.
- Method 2: Manual Polynomial Multiplication. It’s straightforward and requires no additional libraries, providing good control over the process. The downside is that it’s more verbose and error-prone for complex operations.
- Method 3: Using Polynomial Class from NumPy. This object-oriented approach is easy to read and use for other polynomial operations. But similar to Method 1, it requires NumPy and may introduce unnecessary complexity for this simple task.
- Method 4: Using Recurrence Relation of Hermite Polynomials. It’s a fast and library-independent method that leverages mathematical properties, ideal for performance-critical applications. The negative side is that it can be less straightforward and harder to maintain.
- Bonus Method 5: List Comprehension One-Liner. This technique is for those who prefer compact code. It shines in simplicity and minimalism but might be less readable for beginners.