A Transformation Approach That Makes Spai, Psai and Rsai Procedures Efficient for Large Double Irregular Nonsymmetric Sparse Linear Systems

It has been known that the sparse approximate inverse preconditioning procedures SPAI and PSAI(tol) are costly to construct preconditioners for a large sparse nonsymmetric linear system with the coefficient matrix having at least one relatively dense column. This is also true for SPAI and the recently proposed sparse approximate inverse preconditioning procedure RSAI(tol) procedure when the matrix has at least one relatively dense row. The matrix is called double irregular sparse if it has at least one relatively dense column and row, and it is double regular sparse if all the columns and rows of it are sparse. Double irregular sparse linear systems have a wide range of applications, and 24.4% of the nonsymmetric matrices in the Florida University collection are double irregular sparse. Making use of the Sherman-Morrison-Woodbury formula twice, we propose an approach that transforms a double irregular sparse problem into some double regular sparse ones with the same coefficient matrix, for which SPAI, PSAI(tol) and RSAI(tol) are efficient to construct preconditioners for the transformed double regular linear systems. We consider some theoretical and practical issues on such transformation approach, and develop a practical algorithm that first preconditions the transformed systems and then solves them by Krylov iterative solvers. Numerical experiments confirm the very sharp superiority of our transformation algorithm to SPAI, PSAI(tol) and RSAI(tol) applied to the double irregular sparse linear problem directly.