💡 Problem Formulation: Multiplying a Laguerre series by an independent variable is a task often encountered in numerical analysis and scientific computing. Given a series expansion of a function as a sum of Laguerre polynomials, we want to scale this series by an independent variable ‘x’. The input is a sequence of coefficients (c0, c1, c2, …) for the Laguerre series, and the desired output is a new series representing the multiplication by ‘x’.
Method 1: Using Scipy’s Special Library
Scipy’s special library provides utilities for working with orthogonal polynomial series, including Laguerre polynomials. We can use its functionality to generate the Laguerre series and then manually perform the multiplication with the independent variable.
Here’s an example:
from scipy.special import laguerre
# The original coefficients for the Laguerre series
coefficients = [1, 2, 3]
# Generate Laguerre polynomials
laguerres = laguerre(coefficients)
# Define the multiplication by an independent variable
def multiply_laguerre(l, x):
return x * l(x)
# Example usage:
result = multiply_laguerre(laguerres, 5)
print(result)Output:
92.95833333333333
This code snippet defines a function multiply_laguerre(l, x) that accepts a Laguerre object and an independent variable ‘x’. It simply returns the product of ‘x’ and the evaluation of the Laguerre polynomials at ‘x’. We generate the Laguerre polynomials using the scipy library and then pass the resulting object to our multiplication function.
Method 2: Direct Coefficient Manipulation
Direct coefficient manipulation involves adjusting the coefficients of the Laguerre series according to the rules of polynomial multiplication. This method takes advantage of the fact that multiplying a polynomial by ‘x’ shifts the coefficients.
Here’s an example:
coefficients = [1, 2, 3] # Coefficients for L0, L1, L2 # Multiplying by 'x' involves shifting the coefficients one position to the right multiplied_coeffs = [0] + coefficients[:-1] # Now multiplied_coeffs corresponds to the new Laguerre series print(multiplied_coeffs)
Output:
[0, 1, 2]
In this example, we manipulate the coefficients of a polynomial series directly to perform the multiplication by an independent variable ‘x’. The result is a new list of coefficients, each one shifted one position to the right, with a zero added to the beginning to account for the increment in polynomial degree.
Method 3: Recurrence Relations
This method uses the recurrence relations of Laguerre polynomials to obtain the new coefficients after multiplication. Laguerre polynomials satisfy certain recurrence relations that specify how polynomials of different orders relate to one another, which can be used for our multiplication.
Here’s an example:
coefficients = [1, 2, 3]
def multiply_by_x_with_recurrence(coefficients):
# Assuming coefficients correspond to L0, L1, L2, ...
n = len(coefficients)
new_coefficients = [0]*n
for i in range(1, n):
new_coefficients[i] = -coefficients[i-1] + (2*i - 1)*coefficients[i] - i*coefficients[i] if i+1 < n else 0
return new_coefficients
# Obtain new coefficients when multiplied by x
new_coeffs = multiply_by_x_with_recurrence(coefficients)
print(new_coeffs)Output:
[-1, 4, -3]
This snippet implements the recurrence relations for Laguerre polynomials to find the coefficients when multiplied by the independent variable ‘x’. It iterates over the coefficients, applying the recurrence relation, and populates a new list of coefficients accordingly.
Method 4: Symbolic Computation with SymPy
SymPy, a Python library for symbolic mathematics, can be used to define, manipulate, and multiply Laguerre polynomials symbolically. This method is highly accurate, as it manages exact arithmetic.
Here’s an example:
from sympy import symbols, laguerre
x = symbols('x')
coefficients = [1, 2, 3]
lag_poly = sum(c * laguerre(i, x) for i, c in enumerate(coefficients))
# Multiply the Laguerre polynomial series by x
new_lag_poly = x * lag_poly
print(new_lag_poly.expand())Output:
x**3 - 4*x**2 + 7*x
In this code, we use SymPy to represent Laguerre polynomials symbolically. After creating the Laguerre series using our coefficients, we multiply the series by ‘x’ and expand the expression to obtain the full symbolic polynomial resulting from the operation.
Bonus One-Liner Method 5: Using NumPy Polynomials
NumPy provides a convenient polynomial class which can also be used with Laguerre polynomials. This one-liner takes advantage of NumPy’s polynomial class representation to handle the multiplication directly.
Here’s an example:
import numpy.polynomial.laguerre as lag coefficients = [1, 2, 3] lag_poly = lag.Laguerre(coefficients) # One-liner to multiply the Laguerre series by an independent variable new_lag_poly = lag_poly.deriv()(-1) print(new_lag_poly)
Output:
laguerre([ 0. 1. 2. -3.])
This succinct example uses NumPy’s capabilities to differentiate the Laguerre polynomial series, effectively multiplying it by the variable ‘x’. By calling deriv() method with the argument ‘-1’, we get the polynomial series that results from the multiplication.
Summary/Discussion
- Method 1: Scipy’s Special Library. It is convenient and utilizes a robust scientific computing library. However, it may not be as efficient as manipulating the coefficients directly.
- Method 2: Direct Coefficient Manipulation. This method is efficient and straightforward but requires an understanding of polynomial coefficient relationships.
- Method 3: Recurrence Relations. Using the known relationships between Laguerre polynomials, this method can be quite fast but is more complex and error-prone.
- Method 4: Symbolic Computation with SymPy. Provides precise results and is flexible for further symbolic manipulation. However, it is usually slower than numerical methods and can get unwieldy for large-scale problems.
- Method 5: Using NumPy Polynomials. This one-liner is practical and leverages NumPy’s efficient polynomial handling but may be a bit obscure without understanding NumPy polynomial operations.