5 Best Ways to Remove Small Trailing Coefficients from Legendre Polynomial in Python

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πŸ’‘ Problem Formulation: When working with Legendre polynomials in numerical computations, small trailing coefficients that are close to machine precision may occur due to operations like differentiation or integration. These near-zero coefficients can lead to unnecessary complexity and inaccuracy. The goal is to remove these small coefficients from the polynomial’s representation. For instance, turning P(x) = x^3 + 2x^2 + 0.000001x into P(x) = x^3 + 2x^2.

Method 1: Using NumPy’s isclose Function

A convenient method for trimming small trailing coefficients from a polynomial is employing NumPy’s isclose function, which compares each coefficient against a specified tolerance. One can iterate through the coefficients in reverse order, remove those that are ‘close’ to zero, and reconstruct the polynomial.

Here’s an example:

import numpy as np

def trim_legendre_coeffs(coeffs, tol=1e-10):
    trimmed_coeffs = []
    for coeff in reversed(coeffs):
        if np.isclose(coeff, 0, atol=tol):
            continue
        trimmed_coeffs.insert(0, coeff) # Insert at the beginning
    return np.array(trimmed_coeffs)

# Example coefficients
coeffs = np.array([1, 0.5, 0.0000003, 0.00000002])
trimmed = trim_legendre_coeffs(coeffs)
print(trimmed)

Output:

[1.  0.5]

This code snippet defines a function trim_legendre_coeffs that takes a NumPy array of coefficients and a tolerance level. For coefficients smaller than the tolerance in absolute terms, it removes those values. Here, only significant coefficients are retained, reducing polynomial complexity while preserving its shape.

Method 2: Truncating Small Coefficients Directly

This technique involves setting a threshold and directly truncating coefficients that are below it. This method is quick and does not depend on additional functions or tools.

Here’s an example:

import numpy as np

def truncate_small_coeffs(coeffs, threshold=1e-10):
    coeffs[np.abs(coeffs) < threshold] = 0
    return coeffs

coeffs = np.array([3, 1e-12, -1e-11, 5])
cleaned_coeffs = truncate_small_coeffs(coeffs)
print(cleaned_coeffs)

Output:

[3 0 0 5]

Using the function truncate_small_coeffs, this snippet sets coefficients below the threshold to zero, effectively truncating the small values. The result is a sanitized array of coefficients without the small, potentially troublesome trailing numbers.

Method 3: Using Polynomial Library’s trim Function

Python’s polynomial library features a trim function designed for trimming small coefficients. The method can be customized to handle very small numbers typical of floating-point arithmetic imprecision.

Here’s an example:

from numpy.polynomial import Polynomial

p = Polynomial([2e-20, 1, 1e-5, 2])
p_trimmed = p.trim(tol=1e-10)
print(p_trimmed)

Output:

Polynomial([1.0, 0.0, 2.0], domain=[-1,  1], window=[-1,  1])

The code utilizes the trim method of the Polynomial object to remove coefficients smaller than the provided tolerance. The remaining coefficients are used to represent the simplified polynomial.

Method 4: Filtering with List Comprehension

Python’s list comprehension offers a succinct way to filter out small coefficients. With this method, one can easily define custom logic to identify and remove undesired coefficients.

Here’s an example:

coeffs = [2e-10, -3, 1e-15, 5]
tol = 1e-10
filtered_coeffs = [coeff for coeff in coeffs if abs(coeff) > tol]
print(filtered_coeffs)

Output:

[-3, 5]

Here, the snippet uses a list comprehension to create a new list of coefficients that excludes those below the defined tolerance level. Only significant coefficients remain, thus simplifying the polynomial while maintaining its primary characteristics.

Bonus One-Liner Method 5: Using Lambda and Filter

For a quick, one-liner solution, one can use a combination of the filter function and a lambda expression to eliminate small trailing coefficients in a polynomial.

Here’s an example:

coeffs = [3e-14, 2, 3e-12, 1]
tol = 1e-10
clean_coeffs = list(filter(lambda x: abs(x) > tol, coeffs))
print(clean_coeffs)

Output:

[2, 1]

This code leverages a lambda function within the filter built-in to discard coefficients smaller than the specified tolerance, comparable to the list comprehension method but potentially more readable for some Python users.

Summary/Discussion

  • Method 1: Using NumPy’s isclose Function. Precise control over tolerance. Slightly more verbose.
  • Method 2: Truncating Directly. Quick and simple. Can introduce zeros into polynomial representation.
  • Method 3: Polynomial Library’s trim Function. Part of the standard library, reliable for polynomial operations. Requires use of Polynomial objects.
  • Method 4: Filtering with List Comprehension. Offers fine control over selection criteria. Less straightforward than library functions.
  • Method 5: Using Lambda and Filter. Very concise. Could be less clear to those unfamiliar with functional programming style.