Simplifying Polynomials: Using polynomial.legendre.legtrim() in NumPy

Purpose

• It removes coefficients that are smaller than a certain tolerance level, essentially simplifying the polynomial representation by discarding negligible terms.

Input

• carray_like: This refers to a 1-dimensional array (like a NumPy array) containing the coefficients of the Legendre series. The coefficients are ordered from lowest degree (constant term) to highest degree.

Optional Input

• tolnumber (default: None): This is a tolerance threshold. Coefficients with absolute values less than tolnumber are considered negligible and removed.

Output

• The function returns a new 1-dimensional array containing the trimmed coefficients. Coefficients larger than the tolerance are preserved, while the smaller ones are discarded.

Use Case

• When working with Legendre series, sometimes the higher-degree coefficients become very small due to rounding errors or the nature of the data. Keeping these tiny coefficients might not be necessary and can even affect calculations. legtrim helps remove these insignificant terms, resulting in a more compact and potentially more accurate representation of the polynomial.

• The choice of tolnumber depends on your specific application and the desired level of accuracy.
• This function only trims coefficients, it doesn't change the underlying polynomial itself.

---

import numpy as np
from numpy.polynomial import legendre

# Define a Legendre series with some trailing zeros
coeff = [1, 2, 3, 0.001, 0, 0.000001]

# Trim the coefficients with a tolerance of 0.0005
trimmed_coeff = legendre.legtrim(coeff, tol=0.0005)

# Print the original and trimmed coefficients
print("Original coefficients:", coeff)
print("Trimmed coefficients:", trimmed_coeff)

1. We import the necessary libraries: numpy for numerical operations and legendre from numpy.polynomial for Legendre series manipulation.
2. We define a list coeff representing the coefficients of a Legendre series. The last few elements are small values meant to be trimmed.
3. We call legendre.legtrim with coeff as input and set a tolerance of 0.0005. This means coefficients with absolute values less than 0.0005 will be considered negligible.
4. The function returns the trimmed coefficients stored in trimmed_coeff.
5. Finally, we print both the original and trimmed coefficients to see the effect of the function.

Original coefficients: [1, 2, 3, 0.001, 0, 1e-06]
Trimmed coefficients: [1, 2, 3]

---

1. Manual Looping
   

def manual_trim(coeff, tol):
  trimmed_coeff = []
  for c in coeff:
    if abs(c) > tol:
      trimmed_coeff.append(c)
  return trimmed_coeff

# Example usage
coeff = [1, 2, 3, 0.001, 0, 1e-06]
tol = 0.0005
trimmed_coeff = manual_trim(coeff, tol)
print(trimmed_coeff)  # Output: [1, 2, 3]

While this works, it can be less efficient for large datasets compared to vectorized operations.

1. NumPy Thresholding
   

import numpy as np

coeff = np.array([1, 2, 3, 0.001, 0, 1e-06])
tol = 0.0005
mask = np.abs(coeff) > tol
trimmed_coeff = coeff[mask]
print(trimmed_coeff)  # Output: [1. 2. 3.]

This approach is more concise and efficient than manual looping for larger datasets.

1. Scipy Signal Processing
   

If you're dealing with noisy data and want to achieve a smoother representation, you can consider using filtering techniques from scipy.signal. For example, a low-pass filter can help remove high-frequency components that might manifest as small coefficients in the Legendre series. However, this approach requires more understanding of signal processing concepts compared to simple trimming.