A Polynomial Preconditioned Global CMRH Method for Linear Systems with Multiple Right-Hand Sides

The restarted global CMRH method (Gl-CMRH(m)) (Heyouni, 2001) is an attractive method for linear systems with multiple right-hand sides. However, Gl-CMRH(m) may converge slowly or even stagnate due to a limited Krylov subspace. To ameliorate this drawback, a polynomial preconditioned variant of Gl-CMRH(m) is presented. We give a theoretical result for the square case that assures that the number of restarts can be reduced with increasing values of the polynomial degree. Numerical experiments from real applications are used to validate the effectiveness of the proposed method.

[1]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[2]  Ronald B. Morgan,et al.  Improved seed methods for symmetric positive definite linear equations with multiple right‐hand sides , 2008, Numer. Linear Algebra Appl..

[3]  Mathematisch Instituut,et al.  A polynomial preconditioner for the GMRES algorithm , 1995 .

[4]  Gene H. Golub,et al.  Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems , 2002, SIAM J. Matrix Anal. Appl..

[5]  Gene H. Golub,et al.  Block Triangular and Skew-Hermitian Splitting Methods for Positive-Definite Linear Systems , 2005, SIAM J. Sci. Comput..

[6]  Chuanqing Gu,et al.  Skew-symmetric methods for nonsymmetric linear systems with multiple right-hand sides , 2009 .

[7]  M. Heyouni,et al.  The global Hessenberg and CMRH methods for linear systems with multiple right-hand sides , 2001, Numerical Algorithms.

[8]  H. Sadok,et al.  A block version of BiCGSTAB for linear systems with multiple right-hand sides. , 2003 .

[9]  Michael K. Ng,et al.  Galerkin Projection Methods for Solving Multiple Linear Systems , 1999, SIAM J. Sci. Comput..

[10]  R. Freund,et al.  A block QMR algorithm for non-Hermitian linear systems with multiple right-hand sides , 1997 .

[11]  Efstratios Gallopoulos,et al.  An Iterative Method for Nonsymmetric Systems with Multiple Right-Hand Sides , 1995, SIAM J. Sci. Comput..

[12]  H. Sadok,et al.  Global FOM and GMRES algorithms for matrix equations , 1999 .

[13]  Khalide Jbilou,et al.  The block Lanczos method for linear systems with multiple right-hand sides , 2004 .

[14]  Tony F. Chan,et al.  Analysis of Projection Methods for Solving Linear Systems with Multiple Right-Hand Sides , 1997, SIAM J. Sci. Comput..

[15]  D. O’Leary.  The block conjugate gradient algorithm and related methods , 1980 .

[16]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[17]  Zhenxin Yang,et al.  Global SCD algorithm for real positive definite linear systems with multiple right-hand sides , 2007, Appl. Math. Comput..

[18]  R. Mittra,et al.  A conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields , 1989 .

[19]  H. Sadok,et al.  OBLIQUE PROJECTION METHODS FOR LINEAR SYSTEMS WITH MULTIPLE RIGHT-HAND SIDES , 2005 .

[20]  Lei Du,et al.  A block IDR(s) method for nonsymmetric linear systems with multiple right-hand sides , 2011, J. Comput. Appl. Math..

[21]  Faezeh Toutounian,et al.  The block least squares method for solving nonsymmetric linear systems with multiple right-hand sides , 2006, Appl. Math. Comput..

[22]  B. Vital.  Etude de quelques methodes de resolution de problemes lineaires de grande taille sur multiprocesseur , 1990 .

[23]  Hassane Sadok,et al.  CMRH: A new method for solving nonsymmetric linear systems based on the Hessenberg reduction algorithm , 1999, Numerical Algorithms.

[24]  M. Heyouni,et al.  Matrix Krylov subspace methods for linear systems with multiple right-hand sides , 2005, Numerical Algorithms.

[25]  V. Simoncini,et al.  Convergence properties of block GMRES and matrix polynomials , 1996 .