Volume 46, pp. 460-473, 2017.

The block Hessenberg process for matrix equations

M. Addam, M. Heyouni, and H. Sadok


In the present paper, we first introduce a block variant of the Hessenberg process and discuss its properties. Then, we show how to apply the block Hessenberg process in order to solve linear systems with multiple right-hand sides. More precisely, we define the block CMRH method for solving linear systems that share the same coefficient matrix. We also show how to apply this process for solving discrete Sylvester matrix equations. Finally, numerical comparisons are provided in order to compare the proposed new algorithms with other existing methods.

Full Text (PDF) [325 KB]

Key words

Block Krylov subspace methods, Hessenberg process, Arnoldi process, CMRH, GMRES, low-rank matrix equations.

AMS subject classifications

65F10, 65F30

Links to the cited ETNA articles

[16]Vol. 33 (2008-2009), pp. 207-220 L. Elbouyahyaoui, A. Messaoudi, and H. Sadok: Algebraic properties of the block GMRES and block Arnoldi methods
[18]Vol. 16 (2003), pp. 129-142 A. El Guennouni, K. Jbilou, and H. Sadok: A block version of BiCGSTAB for linear systems with multiple right-hand sides
[28]Vol. 30 (2008), pp. 1-9 Hualei Liu and Baojiang Zhong: Simpler Block GMRES for nonsymmetric systems with multiple right-hand sides

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