The Study Of Iterative Methods For Large Sparse Linear Algebraic Systems

Solutions of large-scale sparse linear algebraic systems are deeply involved in variousscientific and engineering fields, such as fluid mechanics, numerical solutions of highorderdifferential equations, computational electromagnetics, optimization problems andreservoir modeling. Moreover, research of methods for solving large-scale sparse systemsof linear algebraic equations becomes one of the key issues of large-scale scientific andengineering computing and such research has important theoretic significance and practicalapplications. This doctoral dissertation deeply studies numerical characteristics ofsome special matrices as well as their solutions related to iterative solutions of large-scalesparse linear algebraic systems. In particular, convergence properties of matrix splittingmethods have been investigated and the comparison theories as well as preconditioningtechniques have been also studied. The dissertation consists of four parts with six chapters:Part 1 contributes to the upper infinity norm bounds for the inverse of strictly diagonaldominant M-matrices. Making use of the special structures of strictly diagonaldominant M-matrices and the relationships between the entries of inverse M-matricesand M-matrices, the estimate of the upper infinity norm bounds for inverse M-matricesare derived. Furthermore, the estimate of the spectral radius is also obtained.Part 2 is to study preconditioning techniques for iterative solution of saddle pointproblems. Relaxed inexact Uzawa algorithm and preconditioned Uzawa algorithm arefirstly proposed. These two proposed algorithms are the extensions of the original ones.Furthermore, the convergence of the algorithms is discussed and the numerical experimentsverify the effectiveness of these two algorithms. Next, two classes of preconditionersfor solving symmetric and nonsymmetric saddle point problems with highly singular(1,1) block matrices are established and their spectral properties are comprehensivelystudied. This kind of ill-conditioned saddle point problems can be effectively handledwith the proposed preconditioners through numerical examples. At last, according tothe special properties of the coefficient matrices of the linear systems obtain from the discretizationof the mixed time-harmonic Maxwell equations, augment-free and Schur complementfree block triangular preconditioners with parameters are proposed. Theoretical analysis illustrates that the construction and application cost of the new preconditionersis the same as that of the existing augment-free and Schur complement-free block diagonalpreconditioners, but with better clustering properties for eigenvalues, particularly withgiven relative optimal parameters. Numerical experiments depict that the performance ofthe new preconditioners is much superior to the augment-free and Schur complement-freeblock diagonal preconditioners, and also show the performance of the new preconditionersis the most efficient when the parameters are equal to relative optimal parameters.Part 3 is devoted to investigating convergence properties and comparison theoriesof matrix multisplitting methods at first, proposing nonstationary multisplitting algorithmwith K + 1 relaxed parameters and nonstationary two-stage multisplitting algorithm withK + 1 relaxed parameters. The convergence of the algorithm is discussed in detail inthe case of H-matrices as the coefficient matrices. The convergence of the above algorithmis also studied when the multisplitting matrices are derived from the incomplete LUfactorizations based on different zero patterns. Constructions of preconditioners are alsoobtained by iterative matrices. Next, according to the special structures and properties ofblock tridiagonal H-matrices, block LU precondifioners are established and the efficiencyof our preconditioners with numerical experiments is also showed.Part 4 firstly has an analysis of the convergence properties of two classes of modifiedpreconditioners with Gauss-Seidel and SOR methods having L-matrices as the coefficientmatrices and presents the comparison results. Consequently, the optimal structures of thetwo classes of preconditioners are given. And then mixed-type splitting method is studiedto solve linear equations with Z-matrices as the coefficient matrices. Finally, relaxedalternative iterative methods are proposed and the corresponding convergence propertiesares studies when the coefficient matrices are monotone matrices and Hermitian positivedefinite matrices.