一般反对称矩阵的pfm的数值和符号计算
Numeric and symbolic evaluation of the pfaffian of general skew-symmetric matrices
摘要：
在许多物理学应用中出现了对pf的评估，对于其中一些应用，直接方法优于使用行列式公式。我们讨论了两种方法的数值计算的pfetanans。第一种是基于Householder变换的三对角化。这种方法的主要优点是其数值稳定性，使得不必要的实施一个旋转策略。第二种方法考虑的是基于艾特肯的块对角化公式。它产生一种LU（类似于Cholesky的分解）分解（在同余下）的任意斜对称矩阵，非常适合于数值和符号评估的pfeclian。给出了实现这两种方法的FORTRAN子程序（FORTRAN 77和90）。我们还提供了Python和Mathematica中的简单实现，用于测试，或用于探索性研究使用pfuncans的方法。
英文摘要：
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 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.
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