Convergence Analysis On Higher-order 2PPJ Iterative Methods

A great quantity of problems in science and technology require to be solved by constructing the counterpart linear equations.In general,the iterative method is employed for solving large sparse equations effectively.Consider the iterative method,it's significant to investigate its convergence and convergence rate.Therefore,based on the 2PPJ iteration format,this paper examines the necessary and sufficient condition for convergence of the high-order 2PPJ iteration method and the rate of accelerating the convergence of the high-order 2PPJ iteration through the preconditions P=I+Q.The details are displayed as follows.Chapter 1 mainly introduce the Jacob iterative method,spectral radius of matrix A,Z-matrix,L-matrix,M-matrix,(q,r)consistently ordered matrix and the relevant basic lemmas such as Perron-Frobenius and interpolated iteration.In chapter 2,it is discussed when the Jacobi eigenvalue of coefficient matrix A is?j=aj±aj(j=1,2,…,n,aj ? R,i=(?))or ?j=bji(j=1,2,…,n,bj?R,i=(?)),a high-order 2PPJ iteration matrix G?,?=(1-?S)I+?T?,T?=[(1-?)l+?J]J2P+1 is proposed.The order value P can be classified into several types based on iP(P?N),further,combining with the extrapolated iterative lemma,the relevant conditions of Jacobi iteration and 2PPJ iteration convergence are generalized to obtain the necessary and suficient condition for the convergence of higher-order 2PPJ iteration.Finally,the conclusions are verified by three examples.In chapter 3,based on the preconditions proposed by Gunawardena and Niki at all,the P=I+Q preconditions matrix is constructed in this paper.When the coefficient matrix A is strictly anti-optimal Z-matrix,A=(aij)is strictly antioptimal M-matrix preconditions,and ?(J)??(J)<1,under the condition of 0<?<1,0???1,it is known that the same order value P yields ?[G?,?(J)]??[G?,?(J)]<1;while the value of P are different,which deduces that the higher value of P accompanies with the faster of the convergence.