On the computation of Pfaffians
Discrete Applied Mathematics
Matrix computations (3rd ed.)
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
Eigensystem Computation for Skew-Symmetric and a Class of Symmetric Matrices
ACM Transactions on Mathematical Software (TOMS)
ACM Transactions on Mathematical Software (TOMS)
Banded Eigenvalue Solvers on Vector Machines
ACM Transactions on Mathematical Software (TOMS)
Band reduction algorithms revisited
ACM Transactions on Mathematical Software (TOMS)
Computational Discrete Mathematics
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
The Snap-Back Pivoting Method for Symmetric Banded Indefinite Matrices
SIAM Journal on Matrix Analysis and Applications
Quantum Signatures of Chaos
Optimal block-tridiagonalization of matrices for coherent charge transport
Journal of Computational Physics
Partitioned Triangular Tridiagonalization
ACM Transactions on Mathematical Software (TOMS)
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Computing the Pfaffian of a skew-symmetric matrix is a problem that arises in various fields of physics. Both computing the Pfaffian and a related problem, computing the canonical form of a skew-symmetric matrix under unitary congruence, can be solved easily once the skew-symmetric matrix has been reduced to skew-symmetric tridiagonal form. We develop efficient numerical methods for computing this tridiagonal form based on Gaussian elimination, using a skew-symmetric, blocked form of the Parlett-Reid algorithm, or based on unitary transformations, using block Householder transformations and Givens rotations, that are applicable to dense and banded matrices, respectively. We also give a complete and fully optimized implementation of these algorithms in Fortran (including a C interface), and also provide Python, Matlab and Mathematica implementations for convenience. Finally, we apply these methods to compute the topological charge of a class D nanowire, and show numerically the equivalence of definitions based on the Hamiltonian and the scattering matrix.