Treffer: Skew-Symmetric Differential qd Algorithm: Skew-symmetric differential \(qd\) algorithm

Differential qd (dqd) algorithm with shifts is probably the fastest known algorithm which computes eigenvalues of symmetric tridiagonal matrices with high relative accuracy. In this paper we will construct a similar algorithm for computing eigenvalues of skew-symmetric matrices, which is based on implicit usage of both the QR and the symplectic QR factorizations. If we apply this algorithm to tridiagonal skew-symmetric matrices, we obtain the skew-symmetric dqd algorithm. This algorithm also enjoys high relative stability. However, incorporation of shifts is much harder then in the symmetric case, and yet to be implemented. Finally, the standard algorithm for computing the eigenvalues of tridiagonal skew-symmetric matrices can also be interpreted in the context of the skew-symmetric dqd algorithm.