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Usage Guide

This guide explains how to use faxtored_coeffs_txt.py and root_field.py with safe coefficient input.

1. Generate a coefficients file from a factorized polynomial

The repository includes faxtored_coeffs_txt.py, which expands a factorized polynomial into a coefficient list and writes it to coeffs.txt.

Use the script

  1. Open faxtored_coeffs_txt.py.
  2. Set the factorized polynomial expression in the factored = ... line.
  3. Run the script:
cd /Users/ralf/Projects/Python/Param_Poly_Root
python3 faxtored_coeffs_txt.py

This writes the expanded coefficients into coeffs.txt in the current folder.

Example factorization

In faxtored_coeffs_txt.py, you can use expressions such as:

x = sp.symbols('x')
factored = (x - 5)**10 * (x**2 - 2*x + 2)**25 # use cluster_tolerance 1e-22
# factored = (x - 1)**40 * (x - 2)**30 * (x - 3)**20 
# # use cluster_tolerance 1e-10

Then run the script to generate the coefficients file.

2. Use root_field.py with a coefficients file

Once coeffs.txt exists, run:

python3 root_field.py --coeffs-file coeffs.txt

If you have a different filename:

python3 root_field.py --coeffs-file my_coeffs.txt

3. Use root_field.py with standard input

If you prefer to pipe coefficients directly into the script:

echo "1 -100 4875 -154550 ..." | python3 root_field.py

or:

cat coeffs.txt | python3 root_field.py

This avoids placing a long list directly in the shell command.

4. Advanced options for precision and clustering

Precision control (--dps)

Increase arbitrary-precision decimal places for higher multiplicities or ill-conditioned polynomials:

python3 root_field.py --coeffs-file coeffs.txt --dps 800

Clustering tolerance (--cluster-tol)

Adjust how roots are merged into clusters. Increase this when many m=1 clusters should be grouped into higher multiplicities:

python3 root_field.py --coeffs-file coeffs.txt --cluster-tol 1e-22

Important: dps and cluster-tol are interdependent. The noise floor is roughly 10^(-dps / m_max), so cluster-tol should remain above that floor to avoid splitting true multiple roots.

Combined usage

python3 root_field.py --coeffs-file coeffs.txt --dps 800 --cluster-tol 1e-22

The script prints diagnostic hints if many m=1 clusters are found, suggesting better tuning.

5. Notes

  • coeffs.txt should contain a single line with coefficients separated by spaces or commas.
  • Coefficients must be provided in descending degree order, starting with the coefficient of x^n.
  • root_field.py supports real and complex coefficients.
  • Use the file-based input path when coefficients are large or too long for a shell command.

6. Example workflow

cd /Users/ralf/Projects/Python/Param_Poly_Root
python3 factored_coeffs_txt.py
python3 root_field.py --coeffs-file coeffs.txt

This workflow uses a factorized polynomial definition to generate safe coefficient input and then analyses the roots with root_field.py.