Note to the Global GMRES for Solving the Matrix Equation AXB = F
Authors: Fatemeh Panjeh Ali Beik
In the present work, we propose a new projection method for solving the matrix equation AXB = F. For implementing our new method, generalized forms of block Krylov subspace and global Arnoldi process are presented. The new method can be considered as an extended form of the well-known global generalized minimum residual (Gl-GMRES) method for solving multiple linear systems and it will be called as the extended Gl-GMRES (EGl- GMRES). Some new theoretical results have been established for proposed method by employing Schur complement. Finally, some numerical results are given to illustrate the efficiency of our new method.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1070829Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1712
 T. Ando, Generalized Schur complements, Linear Algebra Appl, 27 (1979) 173-186.
 R. Bouyouli, K. Jbilou, A. Messaoudi and H. Sadok, On block minimal residual methods, Appl. Math. Lett, 20 (2007) 284-289.
 F. Ding, P. X. Liu and J. Ding, Iterative solutions of the generalized Sylvester matrix equation by using hierarchical identification principle, Appl. Math. Comput, 197 (2008) 41-50.
 A. El Guennouni, K. Jbilou and H. Sadok, A block version of BiCGSTAB for linear systems with multiple right-hand sides, Trans. Numer. Anal, 16 (2003) 129-142.
 G. X. Huang, F. Yin, K. Guo, An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB =C, J. Comput. Appl. Math, 212 (2008) 231-244.
 K. Jbilou, A. Messaoudi and H. Sadok, Global FOM et GMRES algorithms for matrix equations, Appl. Numer. Math, 31 (1999) 43- 49.
 P. Lancaster, Theory of Matrix, Academic Press, New York, 1969.
 Y.-Q. Lin, Implicitly restarted global FOM and GMRES for nonsymmetric matrix equations and Sylvester equations, Appl. Math. Comput, 167 (2005) 1004-1025.
 M. Mohseni Moghadam and F. Panjeh Ali Beik, A new weighted global full orthogonalization method for solving nonsymmetric linear systems with multiple right-hand sides, Int. Elect. J. Pure Appl. Math, 2(2)(2010) 47-67.
 M. Mohseni Moghadam and F. Panjeh Ali Beik, On a new weighted global GMRES for solving nonsymmetric linear system with multiple right-hand sides. Int. Journal of contemp. Math. Sciences, 5 (2010) 2237-2255.
 F. Panjeh Ali Beik, Extending global full orthogonalization method for solving the matrix equation AXB = F, International Journal of Mathematical and Computer Sciences, 7(4) (2011) 196-199.
 Y. Saad, Iterative Methods for Spars Linear System, PWS Press, New York, 1995.
 M. Sadkane, Block Arnoldi and Davidson methods for unsymmetrical large eigenvalue problems, Numer. Math, 64 (1993) 687-706.
 F. Zhang, Matrix Theory, Springer-Verlag, New York, 1999.
 F. Zhang, The Schur Complement and its Applications, Springer, New York, 2005.