Note to the Global GMRES for Solving the Matrix Equation AXB = F

Authors: Fatemeh Panjeh Ali Beik

Abstract:

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.

Keywords: Matrix equation, Iterative method, linear systems, block Krylov subspace method, global generalized minimum residual (Gl-GMRES).

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1070829

References:

[1] T. Ando, Generalized Schur complements, Linear Algebra Appl, 27 (1979) 173-186.
[2] R. Bouyouli, K. Jbilou, A. Messaoudi and H. Sadok, On block minimal residual methods, Appl. Math. Lett, 20 (2007) 284-289.
[3] 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.
[4] 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.
[5] 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.
[6] K. Jbilou, A. Messaoudi and H. Sadok, Global FOM et GMRES algorithms for matrix equations, Appl. Numer. Math, 31 (1999) 43- 49.
[7] P. Lancaster, Theory of Matrix, Academic Press, New York, 1969.
[8] Y.-Q. Lin, Implicitly restarted global FOM and GMRES for nonsymmetric matrix equations and Sylvester equations, Appl. Math. Comput, 167 (2005) 1004-1025.
[9] 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.
[10] 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.
[11] 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.
[12] Y. Saad, Iterative Methods for Spars Linear System, PWS Press, New York, 1995.
[13] M. Sadkane, Block Arnoldi and Davidson methods for unsymmetrical large eigenvalue problems, Numer. Math, 64 (1993) 687-706.
[14] F. Zhang, Matrix Theory, Springer-Verlag, New York, 1999.
[15] F. Zhang, The Schur Complement and its Applications, Springer, New York, 2005.