Modified incomplete orthogonal factorization methods using Givens rotations

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.