A Modified Direct Eigenvalue Swapping Algorithm with Guaranteed Stability

For a given matrix $A$ in the real Schur form, the eigenvalue swapping algorithm is used to swap two adjacent diagonal blocks. In LAPACK, this operation is performed by the subroutine~$\mathtt{\text{SLAEXC}}$/$\mathtt{\text{DLAEXC}}$, which is based on a direct swapping algorithm proposed by Bai and Demmel in 1993. In this work, we present a thorough error analysis of a modified direct swapping algorithm. By a carefully designed pivoted Givens QR factorization, this algorithm can achieve an error bound of the form $O(\mathit{u}\parallel A\parallel )$ under mild assumptions, where $\mathit{u}$ is the unit roundoff. We present empirical evidence to illustrate the superior stability of our algorithm over existing approaches.