| In this thesis,we consider the solution to linear system Ax=bby the preconditioned conjugate gradient method(PCG),where A is a real symmetric positive definite and strictly diagonally dominant matrix.We propose two types of preconditioned approaches.Let A = D-B be the Jacobi splitting of A,where D is the diagonal matrix consisting of the diagonal entries of A.The first kind of preconditioner we proposed is P1=D-?vvT,in which v=|B|e,e =(1,1,…,1)T,? = vTBv/||v||24 which minimizes ||cvvT-B||F.We obtain lower and upper bounds of the eigenvalues of preconditioned matrix P1-1A,which are sharper and simpler than those given in reference[10].The second class of preconditioning strategy is first to transform the original system Ax=b into the preconditioned system Ay = b,where A = I-B,B=D 2/1BD2/1,b=D 2/1b,y= D2/1x,and then consider the solution to Ay =b by PCG.The second class of preconditioner is P2=I-?vvT,in which I is the identity matrix,v=|B|e,?= vTBv/||v||24 which minimizes||cvvT-B||F We give the lower and upper bounds of the eigenvalues of preconditioned matrix P2-1A.The structure of this thesis is divided into five chapters as follows:In Chapter 1,we will give a brief introduction regarding research backgrounds,current status,research contents of such a class of systems,and the innovation of this thesis.In Chapter 2,we recall some definitions,theorems,which will be used in sequel.The Chapter 3 is a simple review of CG and PCG methods.In Chapter 4,We construct two types of precondictioners,and analyze the spectral properties and convergence rate.In Chapter 5,we implement many numerical tests.Numerical results show that using our preconditioners can improve remarkably the rate of convergence.They also show that for an M-matrix,our preconditioners performs slightly better than Jin's preconditioner proposed in reference[10]and that for an H-matrix,it works slightly better than Jacobi's preconditioner and much better than T.Chan's preconditioner in reference[39]. |
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