The Implementation of the formulas from the paper "Basis-free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras"

Here we present implementation in Python programming language of the formulas and algorithms in Geometric Algebra from the paper "Basis-free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras".

The Implementation

The implementation requires galgebra package for symbolic computations and clifford package for numeric computations.

We implement:

In coefficients_recursive.py

• The recursive formulas for the characteristic polynomial coefficients from the paper "On computing the determinant, other characteristic polynomial coefficients, and inverse in Clifford algebras of arbitrary dimension" (Theorem 4).

In coefficients_explicit_n5.py

• The explicit formulas for the characteristic polynomial coefficients in the case n=5 from the paper "Basis-free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras" (Theorem 5.1).

In coefficients_explicit_genralform.py

• Our method to obtain a general form of the basis-free formulas for all characteristic polynomial coefficients in the cases n≤ 6 from the paper "Basis-free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras" (Sections 7 and 8).

Examples

We provide examples of application of the implementation in the Jupyter notebooks. The examples could be opened on a free Jupyter notebook environment Google Colaboratory that runs in the cloud:

Example___coefficients_recursive.ipynb

Example___coefficients_explicit_n5.ipynb

Example___coefficients_explicit_genralform.ipynb

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