Spectral Analysis Of Iterative Matrices

The solution of linear systems of algebraic equations and computing matrices eigenvaluesarises in many problems of computational Science and Engineering computingproblem. So, the research of solutions for the two questions becomes one of the keyissues of scientific and engineering computing and has important theoretic significanceand valuable practical applications. This doctoral dissertation has a comprehensive studyon eigenvalues of some special matrices as well as iterative methods of linear algebraicsystems. In particular, comparison theories of some iterative methods are presented, andrelationships of eigenvalues among some iterative methods with Jacobi iterative methodhave also been studied and preconditioning techniques for the stationary iterative methodsare also investigated and some comparison theories are obtained.Stein-Rosenberg Theorem and its generalized forms are studied. Comparison relationshipsof the spectral radius between some iterative matrices of MPSD iterative methodand the Jacobi iterative matrices are obtained. We show that the spectral radius of theJacobi iterative method is smaller than others under some assumptions. So, the SteinRosenbergTheorem and other results in references are perfected.p-cyclic matrices are investigated. Relationships of eigenvalues between MPSD iterativematrices and Jacobi iterative matrices are obtained at first. The relationships ofeigenvalues between GMPSD iterative matrices and Jacobi iterative matrices are establishedlater and the relationships of eigenvalue between GUSAOR iterative matrices andJacobi iterative matrices are investigated in the end, The results in corresponding referencesare improved and perfected.Preconditioning techniques for stationary iterative methods are studied.(1) The Upper Jacobi and Upper Gauss-Seidel iterative methods are proposed. In thespecial preconditioner, the Upper Jacobi and Upper Gauss-Seidel iterative methods arestudied, the comparison relationships of preconditioned Upper Jacobi and Upper GaussSeideliterative methods with the origional ones, as well as comparison relationships ofpreconditioned Upper Jacobi method with preconditioned Upper Gauss-Seidel iterativemethod are obtained.(2) The precodtioned AOR iterative method under the percondtioners in [106, 107]is investigated and some comparison relationships are obtained.(3) The precodtioned Gauss-Seidel iterative method under the percondtioners in[107, 111] is studied and some comparison relationships are obtained.