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A Modified Direct Eigenvalue Swapping Algorithm with Guaranteed Stability

  • Meiyue Shao
  • Rentao Xu
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Abstract

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~\texttt{SLAEXC}/\texttt{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( u lVert A rVert ) under mild assumptions, where u is the unit roundoff. We present empirical evidence to illustrate the superior stability of our algorithm over existing approaches.

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Cite

BibTeXRIS
BibTeX
@misc{shao2026modified,
  title = {{A Modified Direct Eigenvalue Swapping Algorithm with Guaranteed Stability}},
  author = {Meiyue Shao and Rentao Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30795},
  eprint = {2610.30795},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30795},
}
RIS
TY  - UNPB
TI  - A Modified Direct Eigenvalue Swapping Algorithm with Guaranteed Stability
AU  - Meiyue Shao
AU  - Rentao Xu
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.30795
UR  - https://arxiv.org/abs/2610.30795
AB  - 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~\texttt{SLAEXC}/\texttt{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( u lVert A rVert ) under mild assumptions, where u is the unit roundoff. We present empirical evidence to illustrate the superior stability of our algorithm over existing approaches.
ER  -