*(English)*Zbl 1091.65041

Authorâ€™s abstract: For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices.

Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as generalized minimal residual (GMRES) methods and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to efficient and high-quality preconditioning matrices for some typical matrices from the real-world applications.

##### MSC:

65F35 | Matrix norms, conditioning, scaling (numerical linear algebra) |

65F10 | Iterative methods for linear systems |

65F50 | Sparse matrices (numerical linear algebra) |