The Research Of RA Accelerated Projection Method And New SSOR Decomposition Preprocessing Iterative Method To Solve Ill-conditioned System Of Equations

In science, technology, engineering science and other fields, many problems canbe through the "discretization" or " linearization" process boils down to solving linearequations. For low-order dense linear equations, the direct method is more effective;but if the coefficient matrix is irregular large sparse matrix or has importantcharacteristics, such as symmetric positive definite, tridiagonal, diagonal dominant,etc., the iterative method would be more rapid and effective. Then we also found thatthe iterative methods will fail, when the equations are not just a sparse and oftenaccompanied with morbid, so the research of iterative methods for the ill-conditionedlinear equations will become an issue of concern. For the equations, paper make thefollowing several aspects work: firstly, conducting a preliminary study to the statediscriminant analysis method of equations and introducing the currently effectiveiterative methods; secondly, we make the research analysis and determined to thepretreatment technology and preprocessing matrix; thirdly, based on the characteristicsof coefficient matrix, constructing a new iterative algorithm or reducing the conditionnumber of it by the pretreatment technology, to make the ill-conditioned equations canbe solved.The paper was divided into five chapters. In the first chapter, the developmenthistory, research situation of the solving for linear equations were retrospect. Thebasic idea of the direct method and iterative method was introduced. The statediscriminant analysis method of linear equations and its practical significance werediscussed. The second chapter narrated the research background and solvingsignificance of morbid equations. Then, the latest research progress of iterativemethods for the ill-conditioned linear equations at home and abroad were summarizedand the significance of the research projects. The third chapter, combining with theadvantages and disadvantages of existing morbid equations iterative methods, wepresented the new iterative methods, such as the RA accelerated projection method,with the fast convergence speed, wide applicable scope. Meanwhile we make the further analysis to the algorithm convergence. The results indicate the validity andefficiency of the proposed method. The fourth chapter was on pretreatment technologyand pretreatment iteration methods for solving ill-conditioned linear equationsproblems. The research background and practical significance of pretreatment werediscussed. Then the detailed iterative formulations and theorems were introduced, andthe main related algorithms, which will serve as the prototype of the pretreatmenttechnology for the subsequent chapter. The fifth chapter narrated some applications ofthe pretreatment technology in solving ill-conditioned linear equations. On the basis ofthe present commonly used ill-conditioned linear equations iterative algorithm and thepretreatment technology, according to the different characteristics of coefficientmatrix to select the optimal preconditioning matrix, putting forward two kinds of newSSOR decomposition preprocessing iterative method to solve ill-conditioned systemof equations. The given example proves that the method is effective and feasible.Moreover, we provided the main conclusions of the paper and the late prospects of thesolving methods to ill-conditioned linear equations.