Explaining NumPy's polynomial.hermite_e.hermetrim() for Trimming Coefficients

Purpose

This function helps with cleaning up polynomial representations. It essentially removes coefficients that are considered "small" in absolute value. These small coefficients are often the result of rounding errors or insignificant contributions to the overall polynomial.

Breakdown

• polynomial.hermite_e.hermetrim(c, tol=0)
  
  • c: This is the input argument representing the polynomial coefficients as a 1D NumPy array.
  • tol (optional): This argument specifies the tolerance threshold for considering a coefficient "small." Coefficients with absolute values less than tol will be trimmed. The default value for tol is zero.

Functionality

1. The function iterates through the coefficients array (c) starting from the end (highest degree term).
2. For each coefficient, it checks the absolute value against the tol value.
3. If the absolute value is less than tol, that coefficient is considered insignificant and removed from the array.
4. The trimming process continues until a non-removable coefficient (i.e., absolute value greater than or equal to tol) is encountered.

Benefits

• It can also enhance the numerical stability of calculations by eliminating the influence of rounding errors in insignificant terms.
• Trimming small coefficients can improve the efficiency of polynomial operations by reducing the number of terms involved.

Things to Consider

• hermtrim is specifically designed for Hermite polynomials, a type of polynomial related to quantum mechanics and wave functions. However, the concept of trimming small coefficients can be applied to other polynomial types as well.
• The choice of tol value depends on the specific application and the desired level of accuracy. A smaller tol removes more coefficients, potentially affecting the accuracy of the polynomial representation.

---

import numpy as np

# Sample polynomial coefficients with rounding errors
coeffs = np.array([1.2345e+00, 1.0001e-05, -2.345e-07, 1.5e-10])

# Tolerance threshold (adjust this value as needed)
tol = 1e-6

# Trim the coefficients using hermtrim
trimmed_coeffs = np.polynomial.hermite_e.hermetrim(coeffs, tol=tol)

# Print original and trimmed coefficients
print("Original coefficients:", coeffs)
print("Trimmed coefficients:", trimmed_coeffs)

In this example:

1. We import the NumPy library (import numpy as np).
2. We create a sample NumPy array (coeffs) to represent the polynomial coefficients. Here, we've introduced small values to simulate rounding errors.
3. We define the tol value (tolerance threshold) for trimming insignificant coefficients.
4. We use np.polynomial.hermite_e.hermetrim() to trim the coeffs array based on the specified tol. The trimmed coefficients are stored in trimmed_coeffs.
5. Finally, we print both the original and trimmed coefficients to see the effect of the trimming process.

---

Manual Loop

def manual_trim(coeffs, tol):
  trimmed_coeffs = []
  for coeff in coeffs:
    if abs(coeff) >= tol:
      trimmed_coeffs.append(coeff)
  return np.array(trimmed_coeffs)

# Usage:
trimmed_coeffs = manual_trim(coeffs, tol)

Thresholding with np.where

This method utilizes NumPy's np.where function to create a mask that identifies coefficients above the threshold. You can then use this mask to filter the original array.

def threshold_trim(coeffs, tol):
  mask = np.abs(coeffs) >= tol
  return coeffs[mask]

# Usage:
trimmed_coeffs = threshold_trim(coeffs, tol)

Relative Tolerance

Instead of an absolute tolerance, you can consider a relative tolerance based on the largest coefficient. This can be useful when dealing with polynomials with varying magnitudes in their coefficients.

def relative_trim(coeffs, tol):
  max_coeff = np.abs(coeffs).max()
  trimmed_coeffs = coeffs[np.abs(coeffs) >= tol * max_coeff]
  return trimmed_coeffs

# Usage:
trimmed_coeffs = relative_trim(coeffs, tol)

Choosing the right method depends on your specific needs and preferences.

• Relative tolerance adapts to varying coefficient magnitudes.
• Thresholding with np.where is concise and efficient.
• Manual loop provides fine-grained control but requires more coding.
• hermtrim offers convenience, especially for Hermite polynomials.