Efficient numerical computation of the Pfaffian for dense and banded skew-symmetric matrices

TL;DR

This paper introduces efficient numerical algorithms for computing the Pfaffian of dense and banded skew-symmetric matrices, with applications in physics, and provides optimized implementations in multiple programming languages.

Contribution

It develops new numerical methods based on Gauss transformations and unitary transformations for tridiagonalization of skew-symmetric matrices, applicable to dense and banded cases.

Findings

Algorithms successfully compute the Pfaffian for large matrices.

Implementation in Fortran, Python, Matlab, and Mathematica is fully optimized.

Application to topological charge calculation in nanowires demonstrates practical utility.

Abstract

Computing the Pfaffian of a skew-symmetric matrix is a problem that arises in various fields of physics. Both computing the Pfaffian and a related problem, computing the canonical form of a skew-symmetric matrix under unitary congruence, can be solved easily once the skew-symmetric matrix has been reduced to skew-symmetric tridiagonal form. We develop efficient numerical methods for computing this tridiagonal form based on Gauss transformations, using a skew-symmetric, blocked form of the Parlett-Reid algorithm, or based on unitary transformations, using block Householder transformations and Givens rotations, that are applicable to dense and banded matrices, respectively. We also give a complete and fully optimized implementation of these algorithms in Fortran, and also provide Python, Matlab and Mathematica implementations for convenience. Finally, we apply these methods to compute the topological charge of a class D nanowire, and show numerically the equivalence of definitions based on the Hamiltonian and the scattering matrix.