CMRH: A new method for solving nonsymmetric linear systems based on the Hessenberg reduction algorithm. The generalized minimal residual (GMRES) methods and the quasi-minimal residual (QMR) method are two Krylov methods for solving linear systems. The main difference between these methods is the generation of the basis vectors for the Krylov subspace. The GMRES method uses the Arnoldi process while QMR uses the Lanczos algorithm for constructing a basis of the Krylov subspace. In this paper we give a new method similar to QMR but based on the Hessenberg process instead of the Lanczos process. We call the new method the CMRH method. The CMRH method is less expensive and requires slightly less storage than GMRES. Numerical experiments suggest that it has behaviour similar to GMRES.

References in zbMATH (referenced in 27 articles , 1 standard article )

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  1. Abdaoui, Ilias; Elbouyahyaoui, Lakhdar; Heyouni, Mohammed: The simpler block CMRH method for linear systems (2020)
  2. Gu, Xian-Ming; Huang, Ting-Zhu; Carpentieri, Bruno; Imakura, Akira; Zhang, Ke; Du, Lei: Efficient variants of the CMRH method for solving a sequence of multi-shifted non-Hermitian linear systems simultaneously (2020)
  3. Heyouni, Mohammed; Saberi-Movahed, Farid; Tajaddini, Azita: A tensor format for the generalized Hessenberg method for solving Sylvester tensor equations (2020)
  4. Najafi-Kalyani, Mehdi; Beik, Fatemeh Panjeh Ali; Jbilou, Khalide: On global iterative schemes based on Hessenberg process for (ill-posed) Sylvester tensor equations (2020)
  5. Ramezani, Z.; Toutounian, F.: Extended and rational Hessenberg methods for the evaluation of matrix functions (2019)
  6. Addam, Mohamed; Elbouyahyaoui, Lakhdar; Heyouni, Mohammed: On Hessenberg type methods for low-rank Lyapunov matrix equations (2018)
  7. Amini, S.; Toutounian, F.; Gachpazan, M.: The block CMRH method for solving nonsymmetric linear systems with multiple right-hand sides (2018)
  8. Gu, Xian-Ming; Huang, Ting-Zhu; Yin, Guojian; Carpentieri, Bruno; Wen, Chun; Du, Lei: Restarted Hessenberg method for solving shifted nonsymmetric linear systems (2018)
  9. Teng, Zhongming; Wang, Xuansheng: Heavy ball restarted CMRH methods for linear systems (2018)
  10. Addam, M.; Heyouni, M.; Sadok, H.: The block Hessenberg process for matrix equations (2017)
  11. Meurant, Gérard: An optimal Q-OR Krylov subspace method for solving linear systems (2017)
  12. Duintjer Tebbens, Jurjen; Meurant, Gérard: On the convergence of Q-OR and Q-MR Krylov methods for solving nonsymmetric linear systems (2016)
  13. Duminil, Sébastien; Heyouni, Mohammed; Marion, Philippe; Sadok, Hassane: Algorithms for the CMRH method for dense linear systems (2016)
  14. Bertolazzi, Enrico; Frego, Marco: Preconditioning complex symmetric linear systems (2015)
  15. Zhang, Ke; Gu, Chuanqing: A flexible CMRH algorithm for nonsymmetric linear systems (2014)
  16. Zhang, Ke; Gu, Chuanqing: Flexible global generalized Hessenberg methods for linear systems with multiple right-hand sides (2014)
  17. Duminil, Sébastien: A parallel implementation of the CMRH method for dense linear systems (2013)
  18. Pestana, Jennifer; Wathen, Andrew J.: On the choice of preconditioner for minimum residual methods for non-Hermitian matrices (2013)
  19. Alia, Ahlem; Sadok, Hassane; Souli, Mhamed: CMRH method as iterative solver for boundary element acoustic systems (2012)
  20. Sadok, Hassane; Szyld, Daniel B.: A new look at CMRH and its relation to GMRES (2012)

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