The Faddeev-LeVerrier algorithm and the Pfaffian

TL;DR

This paper introduces a simple, parallelizable algorithm based on the Faddeev-LeVerrier method for efficiently computing the Pfaffian of skew-symmetric matrices, with applications in differential geometry.

Contribution

It adapts the Faddeev-LeVerrier algorithm for Pfaffian computation, providing a novel, easy-to-implement method with competitive performance.

Findings

The algorithm has computational cost $O(n^{eta+1})$ with $eta$ in [2, 2.37286).

Performance comparisons show its efficiency over existing algorithms.

It can be used to compute the Euler form of Riemannian manifolds using computer algebra.

Abstract

We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of computational cost $O(n^{\beta+1})$ where $n$ is the size of the matrix and $O(n^{\beta})$ is the cost of multiplying $n\times n$-matrices, $\beta\in[2,2.37286)$. We compare its performance to that of other algorithms and show how it can be used to compute the Euler form of a Riemannian manifold using computer algebra.