GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Direct methods for sparse matrices
Direct methods for sparse matrices
Preconditioning techniques for nonsymmetric and indefinite linear systems
Journal of Computational and Applied Mathematics - Special issue on iterative methods for the solution of linear systems
Iterative solution methods
Matrix computations (3rd ed.)
CIMGS: An Incomplete Orthogonal Factorization Preconditioner
SIAM Journal on Scientific Computing
Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems
Applied Mathematics and Computation
Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems
SIAM Journal on Matrix Analysis and Applications
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Numerical study on incomplete orthogonal factorization preconditioners
Journal of Computational and Applied Mathematics
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We present a class of new preconditioners based on the incomplete Givens orthogonalization (IGO) methods for solving large sparse systems of linear equations. In the new methods, instead of dropping entries and accepting fill-ins according to the magnitudes of values and the sparsity patterns, we adopt a diagonal compensation strategy, in which the dropped entries are re-used by adding to the main diagonal entries of the same rows of the incomplete upper-triangular factors, possibly after suitable relaxation treatments, so that certain constraints on the preconditioning matrices are further satisfied. This strategy can make the computed preconditioning matrices possess certain desired properties, e.g., having the same weighted row sums as the target matrices. Theoretical analysis shows that these modified incomplete Givens orthogonalization (MIGO) methods can preserve certain useful properties of the original matrix, and numerical results are used to verify the stability, the accuracy, and the efficiency of the MIGO methods employed to precondition the Krylov subspace iteration methods such as GMRES. Both theoretical and numerical studies show that the MIGO methods may have the potential to present high-quality preconditioners for large sparse nonsymmetric matrices.