Numerical solution of singular Sylvester equations

Eric K. W. Chu, Liangshao Hou, Daniel B. Szyld, Jieyong Zhou*

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review


We consider the numerical solution of the large scale singular Sylvester equation of the form AX - XDT = (or AXBT - CXDT = E), where the spectra Λ (A) and Λ (D) (or, Λ (A C) and Λ (D B)) have a nonempty intersection. Using appropriate invariant subspaces, the singular Sylvester equation is rewritten as four Sylvester equations, three of which are nonsingular and one singular. When the invariant subspaces are small, so are three of the equations (including the singular one) which can be solved efficiently. The fourth is large but nonsingular with structures and may be solved using the projection method with Krylov subspaces or techniques involving hierarchical matrices. Some numerical examples for the subspace method are provided.
Original languageEnglish
Article number115426
JournalJournal of Computational and Applied Mathematics
Early online date6 Jul 2023
Publication statusPublished - 15 Jan 2024

Scopus Subject Areas

  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Krylov subspace
  • Least squares solution
  • Lyapunov equation
  • Projection method
  • Singular equation
  • Sylvester equation


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