MATLAB中大型线性方程组的非定常迭代法-王泽文

第20卷第1期大　学　数　学Vol.20,№.12004年2月COLLEGEMATHEMATICSFeb.2004MATLAB王泽文,　邱淑芳(,344000)　　[　],,,,.MATLAB,GMRES.[]MATLAB;;;;[]O241.6,TP301.6　　[]B　　[]1672-1454(2004)01-0068-041　　　,.,,Ax=b,Ab,(ill-posed),A.,Matlab,.2626.00001x1x2=88.00001(1.1)(1,1),MatlabA=[2,6;2,6.00001];b=[8;8.00001];x=A\b;(1,1).,.(1.1),2625.99999x1x2=88.00002,(1.2)Ab,(10,-2)..,,A\b.MATLABbicgstab,.,,,,.,,(GeneralizedMinimumResidualMethod,GMRES)、(Quasi-MinimalResidualMethod,QMR)、(ConjugateGradientMethod,CG)　[]2002-05-27　[](2003)(PCG、CGS、BICG、LSQR).,.MATLAB,GMRESMATLAB.2　MATLABMATLAB6.5gmres,bicg,qmr,cgs,bicgstab,pcg,minres,symmlq,lsqr,x=(A,b),gmresMATLAB.,.,gmres,bicg,qmr,cgs,bicgstab;pcg;minres,symmlq;lsqr.3　GMRES(m),x0∈Rn,r0=b-Ax0,v1=r0/‖r0‖,m;,Arnoldi{vi}mi=1H-m;,‖βe1-H-my‖ym;,xm=x0+Vmym;,‖rm‖=‖b-Axm‖,r0=rm,‖rm‖X,,;,x0=xm,v1=rm/‖rm‖,).注　MATLAB,GMRES,.4　MATLABGMRES4.1　MATLABGMRES12,:1:x=gmres(A,b);　　　　　　　　　2:x=gmres(A,b,restart);3:x=gmres(A,b,restart,tol);4:x=gmres(A,b,restart,tol,maxit);5:x=gmres(A,b,restart,tol,maxit,M);6:x=gmres(A,b,restart,tol,maxit,M1,M2);7:x=gmres(A,b,restart,tol,maxit,M1,M2,x0);8:x=gmres(A,b,restart,tol,maxit,M1,M2,x0);9:[x,flag]=gmres(A,b,restart,tol,maxit,M1,M2,x0);10:[x,flag,relres]=gmres(A,b,restart,tol,maxit,M1,M2,x0);11:[x,flag,relres,iter]=gmres(A,b,restart,tol,maxit,M1,M2,x0);12:[x,flag,relres,iter,resvec]=gmres(A,b,restart,tol,maxit,M1,M2,x0).4.2　GMRESAx=b,An×n,bn.GMRESx0,x0,n(),x0.GMRESrestart(restar,n),1restart,69第1期　　　　　　　王泽文,等:MATLAB中大型线性方程组的非定常迭代法restartn,.MATLABGMRES,maxit.maxit,maxit,maxit=min{n/restart,10}.x‖b-Ax‖‖b‖(tol,0.000006),.、()maxit,.GMRES,MATLAB,gmres(25)convergedatiteration2(2)toasolutionwithrel-ativeresidual8.3e-007.GMRESmaxit,,MATLAB,、,gmres(5)stoppedatiteration30(5)withoutconvergingtothedesiredtolerance1e-006,becausethemaximumnumberofiterationswasreached.Theiteratereturned(number30(5))hasrelativeresidual0.032.4.1,GMRES:[x,flag,relres,iter,resvec]=gmres(A,b,restart,tol,maxit,M1,M2,x0).,A,b,A,b,restart,tol,maxit,x0.M1,M2()MLU,[M1,M2]=lu(M),GMRES,M1,M2.:x=gmres(A,b,restart,tol,maxit,M),M().x=gmres(A,b,restart,tol,maxit,M),[M1,M2]=lu(M);x=gmres(A,b,restart,tol,maxit,M1,M2).x;relresx;iter2,,();resvec()‖b-Ax‖,(i-1)*restart+j+1,i,jiter.flag0,1,23.,flag=0;maxit,flag=1;,,flag=2;GMRES,flag=3.注　GMRES,,:[x,flag,relres]=gmres(A,b),[x,f,rel]=gmres(A,b,r,x0).[x,f,rel]=gmres(A,b,r,x0),rrestart,x0,tol();fflag;relrelres.4.3　A=111111111n×n,0,b=111n×1,GMRESAx=b.,MATLABGMRES,m.5　MATLABbicg,qmr,cgs,bicgstab,pcg,minres,symmlq,lsqr,gmres,MATLAB.MATLAB,70大　学　数　学　　　　　　　　　　　　　　第20卷.MATLAB,,,、、,.,MATLAB,.,MATLAB,.MATLAB,MATLABGMRES,.[　　　][1]　,.[M].:,2000.[2]　,.[M].:,1999.8.[3]　BarrettR,BerryM,ChanTFetal.TemplatesfortheSolutionofLinearSystems:BuildingBlocksforIterativeMethods[M].SIAM,Philadelphia,1994.NonstationaryIterativeMethodsforLargeLinearSystemsinMATLABWANGZe-wen,QIUShu-fang(Dept.ofComputationalScience&SystemsEngineering,EastChinaInstituteofTechnology,FuzhouJiangxi344000,China)Abstract:Lotsofproblemsaredefinedbynonlinearmathematicmodelinscientificresearchandlargetechnologicaldesign.Thusextractingsolutionofthesemodelsleadtosolvingthelargelinearsystems.Therefore,whetherthelargelinearsystems,especiallyill-posedequations,canbeeffectivelysolvedisveryimportant.SomenonstationaryIterativemethodsforlargelinearsystemsinMATLABareintroducedinthispaper.WealsointroducethemathematicdescriptionoftheGMRESmethod.Keywords:MATLAB;largelinearsystems;ill-posed;nonstationaryiterativemethods;scientificcomputing71第1期　　　　　　　王泽文,等:MATLAB中大型线性方程组的非定常迭代法