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On Hessenberg type methods for low-rank Lyapunov matrix equations

Volume 45 / 2018

Mohamed Addam, Lakhdar Elbouyahyaoui, Mohammed Heyouni Applicationes Mathematicae 45 (2018), 255-273 MSC: Primary 65F10; Secondary 65F30. DOI: 10.4064/am2344-12-2017 Published online: 21 September 2018


The Hessenberg process can be seen as an alternative to the well-known Arnoldi and Lanczos processes. Block and global versions of the Hessenberg process were used for solving linear systems with multiple right-hand sides and large Sylvester matrix equations. In this work, we describe two new Hessenberg based methods for obtaining approximate solutions to low-rank Lyapunov matrix equations. The first one is based on a Ruhe variant of the block Hessenberg process and belongs to the class of block Krylov solvers. The second one is a matrix Krylov solver and uses the global Hessenberg process. The numerical comparisons we have made show that solving the Lyapunov equation via the global Hessenberg process gives better results compared to the use of the block Hessenberg process.


  • Mohamed AddamLaboratoire LSIA, Équipe EMMA
    ENSA Al-Hoceima
    Université Mohammed Premier
    Al-Hoceima, Maroc
  • Lakhdar ElbouyahyaouiCentre Régional des Métiers de l’Éducation
    et de la Formation
    Fes, Maroc
  • Mohammed HeyouniLaboratoire LSIA, Équipe EMMA
    ENSA Oujda
    Université Mohammed Premier
    Oujda, Maroc

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