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

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 Aitkenʼs block diagonalization formula. It yields to a kind of LU (similar to Choleskyʼs 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 summaryProgram title:PfaffianCatalogue identifier: AEJD_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEJD_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 2281No. of bytes in distributed program, including test data, etc.: 13 226Distribution format: tar.gzProgramming language: Fortran 77 and 90Computer: Any supporting a FORTRAN compilerOperating system: Any supporting a FORTRAN compilerRAM: a few MBClassification: 4.8Nature 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. Aitkenʼs 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.