The Preconditioned Iterative Methods Of A Linear System

The thesis mainly deals with how to accelerate the rate of convergence for a given linear system of equations. As is known to all, many problems are equivalent to solving one or more large (usually sparse ) linear systems of equations as a matter of fact, the iterative method is the usual way, the convergence of an iterative method and its rate of convergence are of great significance, a divergent method does not work, while one that has a low rate of convergence is of little use, therefore a method that has a better rate of convergence must be found and the parameters in the iterative method be determined, one of the ways is to seek better iterative methods and improvement on linear systems of equations. The thesis mainly discusses preconditioners and the most common methods such as GS, AOR, and Jacobi iterative methods.Throughout the thesis,a linear system is of the form Ax = b ,with A = (a_ij)nxn x, b e R".The thesis proper contains chapter one,chapter two and chapter three.Chapter one is the summary,which briefly deals with some common iterative methods and introduces the developments in preconditioned theory in recent years, chapter two is the essence of the thesis,it puts great emphasis on preconditioned GS iterative method and preconditioned AOR iterative method, the preconditioned Jacobi iterative method discussed in the last part of this chapter mainly lays the foundation for the preconditioned 2PPJ in chapter three, which mainly applies the preconditioners constructed in the last part of chapter two to 2PPJ, here we give details for chapter two and chapter three.In Chapter two, the writer bases the constructed preconditioners P1, E1/2, and Te on the preconditioners Ps, Va proposed by A D.Gunawardena^[23] in 1991 and T.Konhno^[17] in 1997 .To extend the scope of applications, each of the three preconditioners contains as few off-diagonal entries as possible, which makes it easier to compute, E, contains one off-diagonal entry, Te contains two off-diagonal entries, and P₁ contain n-1 off-diagonal elements, all of the off-diagonal entries of P₁ are in the last row.The writer first proves a conclusion concerning P₁A in § 2.3,then the conclusion is applied to Theorem 2.3.2, the preconditioners is applied to GS iterative method in theorem...