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

C. Gonzalez-Ballestero
L. M. Robledo
G. F. Bertsch
Journal / Anthology

Computer Physics Communications
Year: 2011
Volume: 182
Issue: 10
Page range: 2213-2218

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 Aitkenís block diagonalization formula. It yields to a kind of LU (similar to Choleskyís factorization) decomposition (under congruence) of arbitrary skewsymmetric 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.