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

Evaluation of pfaffians arises in a number of physics applications, and for some of them a direct method is preferable to using the deteminantal 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. Program summary Program title: PFAFFIAN Catalogue identifier: AEJD_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEJD_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 2281 No. of bytes in distributed program, including test data, etc.: 13 226 Distribution format: tar.gz Programming language: Fortran 77 and 90 Computer: Any supporting a FORTRAN compiler Operating system: Any supporting a FORTRAN compiler RAM: a few MB Classification: 4.8 Nature of problem: Evaluation of the pfaffian of a skew symmetric matrix. Evaluation of pfaffians arises in a number of physics applications involving fermionic mean field wave functions and their overlaps. Solution method: Householder tridiagonalization. Aitken's block diagonalization formula. Additional comments: Python and Mathematica implementations are provided in the main body of the paper. Running time: Depends on the size of the matrices. For matrices with 100 rows and columns a few milliseconds are required. (C) 2011 Elsevier B.V. All rights reserved.