Numeric and symbolic evaluation of the pfaffian of general skew-symmetric matrices

TL;DR

This paper presents two numerically stable and symbolic methods for evaluating the pfaffian of general skew-symmetric matrices, including detailed algorithms and implementations in Fortran, Python, and Mathematica.

Contribution

It introduces a tridiagonalization method and an Aitken's block diagonalization approach for pfaffian evaluation, with practical code implementations.

Findings

Tridiagonalization offers high numerical stability.

Aitken's method enables symbolic and numeric evaluation.

Provided implementations facilitate testing and application.

Abstract

Evaluation of pfaffians arises in a number of physics applications, and for some of them a direct method is preferable to using the determinantal formula. We discuss two methods for the numerical evaluation of pfaffians. The first is tridiagonalization based on Householder transformations. The main advantage of this method is its numerical stability that makes unnecessary the implementation of a pivoting strategy. The second method considered is based on Aitken's block diagonalization formula. It yields to a kind of LU (similar to Cholesky's factorization) decomposition (under congruence) of arbitrary skew-symmetric matrices that is well suited both for the numeric and symbolic evaluations of the pfaffian. Fortran subroutines (FORTRAN 77 and 90) implementing both methods are given. We also provide simple implementations in Python and Mathematica for purpose of testing, or for exploratory studies of methods that make use of pfaffians.