# A simple division-free algorithm for computing Pfaffians

## TL;DR

This paper introduces a straightforward, division-free algorithm for computing Pfaffians that leverages matrix multiplication and truncation, offering efficiency especially for sparse matrices and enabling extraction of related polynomials.

## Contribution

The paper presents a novel, simple algorithm for Pfaffian computation that avoids division and adapts the Bird determinant algorithm for this purpose.

## Key findings

- Algorithm has complexity O(nM(n)) for 2n x 2n matrices.
- Efficient extraction of characteristic and Pfaffian characteristic polynomials.
- Suitable for sparse matrices with optimized dense-sparse multiplication.

## Abstract

We present a very simple algorithm for computing Pfaffians which uses no division operations. Essentially, it amounts to iterating matrix multiplication and truncation. Its complexity, for a $2n\times 2n$ matrix, is $O(nM(n))$, where $M(n)$ is the cost of matrix multiplication. In case of a sparse matrix, $M(n)$ is the cost of the dense-sparse matrix multiplication. The algorithm is an adaptation of the Bird algorithm for determinants. We show how to extract, with practically no additional work, the characteristic polynomial and the Pfaffian characteristic polynomial from these algorithms.