## Pfaffian

swMATH ID: | 10808 |

Software Authors: | González-Ballestero, C.; Robledo, L.M.; Bertsch, G.F. |

Description: | Numeric and symbolic evaluation of the Pfaffian of general skew-symmetric matrices. The 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 one is tridiagonalization based on Householder transformations. The main advantage of this method is its numerical stability that makes the implementation of a pivoting strategy unnecessary. The second method considered is based on Aitken’s block diagonalization formula. It yields 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. |

Homepage: | http://cpc.cs.qub.ac.uk/summaries/AEJD_v1_0.html |

Keywords: | skew symmetric matrices; Pfaffian; LU-decomposition; numerical examples; tridiagonalization; Householder transformations; pivoting strategy; Aitken’s block diagonalization; Cholesky’s factorization; { t FORTRAN 77}; { t FORTRAN 90} |

Related Software: | Algorithm 923; SageMath; Mathematica; BLAS |

Cited in: | 6 Publications |

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### Cited by 13 Authors

### Cited in 5 Serials

2 | Computer Physics Communications |

1 | Journal of Multivariate Analysis |

1 | Linear Algebra and its Applications |

1 | Journal of Mathematical Sciences (New York) |

1 | Progress of Theoretical Physics |

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