The Pfaffian of a matrix, like the determinant, is a polynomial in the matrix elements. It is most frequently used in particle physics where the matrix is even-ordered and skew-symmetric, and the determinant is the square of the Pfaffian. In these problems, the Pfaffian is used to define a particular choice of sign for the root of the determinant. The methods in this paper first reduce the given matrix to an equivalent tridiagonal form, from which the Pfaffian can be computed directly.

The first section of the paper summarizes several published methods for the reduction of skew-symmetric matrices to tridiagonal form. The second section describes the routines that make up Algorithm 923, and a final section describes the results of tests with those routines. The routines are written in Fortran95, Mathematica, MATLAB, and Python. The Fortran routines are for complex or real matrices in single or double precision. The author also provides C-language interface definitions.

The first two sections of the paper include interesting background results on skew-symmetric real and complex matrices, as well as the Pfaffian, supported by an extensive bibliography. The author shows how methods for real skew-symmetric matrices can be adapted for complex matrices. He also describes in detail one application of the Pfaffian to a problem in particle physics. The author mentions three appendices, which are available online.