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A sparse matrix is called double irregular sparse if it has at least onerelatively dense column and row, and it is double regular sparse if all thecolumns and rows of it are sparse. The sparse approximate inversepreconditioning procedures SPAI, PSAI($tol$) and RSAI($tol$) are costly andeven impractical to construct preconditioners for a large sparse nonsymmetriclinear system with the coefficient matrix being double irregular sparse, butthey are efficient for double regular sparse problems. Double irregular sparselinear systems have a wide range of applications, and 4.4\% of the nonsymmetricmatrices in the Florida University collection are double irregular sparse. Forthis class of problems, we propose a transformation approach, which consists offour steps: (i) transform a given double irregular sparse problem into a smallnumber of double regular sparse ones with the same coefficient matrix$\hat{A}$, (ii) use SPAI, PSAI($tol$) and RSAI($tol$) to construct sparseapproximate inverses $M$ of $\hat{A}$, (iii) solve the preconditioned doubleregular sparse linear systems by Krylov solvers, and (iv) recover anapproximate solution of the original problem with a prescribed accuracy fromthose of the double regular sparse ones. A number of theoretical and practicalissues are considered on the transformation approach. Numerical experiments ona number of real-world problems confirm the very sharp superiority of thetransformation approach to the standard approach that preconditions theoriginal double irregular sparse problem by SPAI, PSAI($tol$) or RSAI($tol$)and solves the resulting preconditioned system by Krylov solvers.

A Transformation Approach that Makes SPAI, PSAI and RSAI Procedures Efficient for Large Double Irregular Nonsymmetric Sparse Linear Systems