Research On The Iterative Methods And The Related Problems Of The Linear Systems

With the development of the computer technology,the iterative methods for the lin-ear system take a more and more important role in the computation in science and en-gineering.In this thesis,the iterative methods for the linear system,error bounds andpreconditioning techniques have been researched.The main results and innovations areas follows:The error bound for the USAOR method is studied.We assume that the coefficientmatrix of AX=b is symmetric,definite and consistently ordered so that the practicalerror bound is derived.Since many practical problems such as the solution to the partialdifferential equations can be turned to the iterative methods for the large linear systems,then our result is useful,the numerical example proves that the error bound for the US-AOR method is effective.We have studied the error bound for the preconditioned simultaneous displacementmethod which is presented by D.J.Evans and N.M.Missirlis.If the coefficient matrixis symmetric,definite and consistently ordered,we obtain a practical error bound for thepreconditioned simultaneous displacement method.The linear complementary problem is changed to the equivalent equations.On thebasis of the matrix splitting and the iterative method,we present the preconditioned si-multaneous displacement method for the linear complementary problem and prove theconvergence of the method.For the saddle point linear system,the MAOR-like method is proposed,and itsconvergence is obtained,which generalizes the results of G.H.Golub et al.in 2001 and in2004.The MPSD method is considered for solving the saddle point linear system and theMPSD-like method is derived.We theoretically prove the convergence of this method.The AOR method for the preconditioned linear system is studied and our resultsshow that some improvements in the convergence rate of this iterative method can be ob-tained on basis of H- matrices.We present the modified SAOR method for the preconditioned linear systems,andprove the convergence of the method on condition that the coefficient matrix is an irre- ducible diagonal dominant Z-matrix.