Geodesic
Weighted versions of Gl-FOM and Gl-GMRES for solving general coupled linear matrix equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 10, pp. 1606-1618 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper, two algorithms called weighted Gl-FOM (WGl-FOM) and weighted Gl-GMRES (WGl-GMRES) are proposed for solving the general coupled linear matrix equations. In order to accelerate the speed of convergence, a new inner product is used. Invoking the new inner product and a new matrix product, the weighted global Arnoldi algorithm is introduced which will be utilized for employing the WGl-FOM and WGl-GMRES algorithms to solve the linear coupled linear matrix equations. After introducing the weighted methods, some relations that link Gl-FOM (Gl-GMRES) to its weighted version are established. Numerical experiments are presented to illustrate the effectiveness of the new algorithms in comparison with Gl-FOM and Gl-GMRES algorithms for solving the linear coupled linear matrix equations. 
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     title = {Weighted versions of {Gl-FOM} and {Gl-GMRES} for solving general coupled linear matrix equations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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Fatemeh Panjeh Ali Beik; Davod Khojasteh Salkuyeh. Weighted versions of Gl-FOM and Gl-GMRES for solving general coupled linear matrix equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 10, pp. 1606-1618. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_10_a2/

[1] A. Bouhamidi, K. Jbilou, “A note on the numerical approximate solutions for generalized Sylvester matrix equations with applications”, Appl. Math. Comput., 206 (2008), 687–694 | DOI | MR | Zbl

[2] X. W. Chang, J. S. Wang, “The symmetric solution of the matrix equations $AX+YA=C$, $AXA^{\mathrm{T}}+BYB^{\mathrm{T}}=C$ and $(A^{\mathrm{T}}XA, B^{\mathrm{T}}XB)=(C, D)$”, Linear Algebra Appl., 179 (1993), 171–189 | DOI | MR | Zbl

[3] K. Jbilou, A. Messaudi, H. Sadok, “Global FOM and GMRES algorithms for matrix equations”, Appl. Numer. Math., 31 (1999), 49–63 | DOI | MR | Zbl

[4] K. Jbilou, A. J. Riquet, “Projection methods for large Lyapunov matrix equations”, Linear Algebra Appl., 415 (2006), 344–358 | DOI | MR | Zbl

[5] D. K. Salkuyeh, F. Toutounian, “New approaches for solving large Sylvester equations”, Appl. Math. Comput., 173 (2006), 9–18 | DOI | MR | Zbl

[6] Q. W. Wang, J. H. Sun, S. Z. Li, “Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra”, Linear Algebra Appl., 353 (2002), 169–182 | DOI | MR | Zbl

[7] J. J. Zhang, “A note on the iterative solutions of general coupled matrix equation”, Appl. Math. Comput., 217 (2011), 9380–8386 | DOI | MR

[8] B. Zhou, G. R. Duan, “On the generalized Sylvester mapping and matrix equation”, Syst. Control Lett., 57:3 (2008), 200–208 | DOI | MR | Zbl

[9] F. P. A. Beik, D. K. Salkuyeh, “On the global Krylov subspace methods for solving general coupled matrix equations”, Comput. Math. Appl., 62 (2011), 4605–4613 | DOI | MR | Zbl

[10] M. Dehghan, M. Hajarian, “The general coupled matrix equations over generalized bisymmetric matrices”, Linear Algebra Appl., 432 (2010), 1531–1552 | DOI | MR | Zbl

[11] F. Ding, P. X. Liu, J. Ding, “Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle”, Appl. Math. Comput., 197 (2008), 41–50 | DOI | MR | Zbl

[12] Y. Saad, Iterative Methods for Sparse linear Systems, PWS, New York, 1995

[13] Y. Saad, M. H. Schultz, “GMRES: A generalized minimal residual method for solving nonsymmetric linear systems”, SIAM J. Sci. Stat. Comput., 7 (1986), 856–869 | DOI | MR | Zbl

[14] Y. Q. Lin, “Implicitly restarted global FOM and GMRES for nonsymmetric equations and Sylvester equations”, Appl. Math. Comput., 167 (2005), 1004–1025 | DOI | MR | Zbl

[15] A. Essai, “Weighted FOM and GMRES for solving nonsymmetric linear systems”, Numer. Algorithms, 18 (1998), 277–292 | DOI | MR | Zbl

[16] Y.-F. Jing, T. Z. Huang, “Restarted weighted full orthogonalization method for shifted linear systems”, Comput. Math. Appl., 57 (2009), 1583–1591 | DOI | MR | Zbl

[17] M. Heyouni, A. Essai, “Matrix Krylov subspace methods for linear systems with multiple right-hand sides”, Numer. Algorithms, 40 (2005), 137–156 | DOI | MR | Zbl

[18] R. Bouyouli, K. Jbilou, R. Sadaka, H. Sadok, “Convergence properties of some block Krylov subspace methods”, J. Comput. Appl. Math., 196 (2006), 498–511 | DOI | MR | Zbl

[19] Matrix Market, , August, 2005 http://math.nist.gov/MatrixMarket