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2、,Nyg?X/x/Z%C0.8c?Z5IOka:a4?zIO;,a?zIO.4?zIOxk x0+ Kk(A,r0),?krkk2= minkb Axk2: x x0+ Kk(A,r0),(4.4): rk= b Axkxk?().?0?dIO?:)?5|?F(CG),)5|?4?(MINRES)SYMMLQ,)5|?24?(GMRES)!4?(QMR)!LSQR2?.4/294JJIIJIBackClose?zIOxk x0+ Kk(A,r0),?rk= b Axk Lk,(4.5): Lk,?kfm,?m.Lkkn;.?: Lk= Kk(A,r0); Lk= AKk(A,r0); Lk=

3、Kk(AT,r0).(4.5)Galerkin.p0?na:V?F(BiCG)!?F(CGS)-zV?F(BICGSTAB).Krylovfm?w?A:?OL?9?A,?,?31L|A?A5.d,a.?5|?)AOu.D5|?).?,?k,A(?.5|?.5/294JJIIJIBackClose4.1?F?!0?)?5|?a?Krylovfm?F.4.1.1?CG?A Rnnb Rn,A?.ux0 Rn,z Rnv(4.2),B?|(4.1)?)x = x0+ z.duA?,3Rn:kvkA=vTAvkvkA1=pvTA1v,v Rn.(4.6)d(4.2)?X?Amr0?)?Krylovfm

4、Kk(A,r0).P(4.1)?)x=x= A1b.6/294JJIIJIBackClosexk x0+ Kk(A,r0),3,ex?Z%C.e?nuZ5?A?dL.n4.1X,Ken?d:(1)xk x0+ Kk(A,r0),vkxkxkA= minkxxkA: x x0+Kk(A,r0) (?4?).(4.7)(2)xk x0+ Kk(A,r0),vkbAxkkA1= minkbAxkA1: x x0+Kk(A,r0) (4?).(4.8)(3)xk x0+ Kk(A,r0),vzT(bAxk) = 0 (=rk Kk(A,r0), z Kk(A,r0) (?).(4.9)7/294JJI

5、IJIBackClose54.1n4.1,xk x0+ Kk(A,r0)3k kAe)x?l?durk= b Axk3k kA1e?,?durk= b AxkKk(A,r0)?.?F?S“?I|Lanczos?zL.b?v1= r0/kr0k2?k?Lanczos)(2.13):AVk= VkTk+ kvk+1eTk,(4.10): VTkVk= Ik, Tk= VTkAVkXe/?n?8/294JJIIJIBackCloseTk=12223.k1k1kkk,vTk+1Vk= 0, kvk+1k2= 1.pbv1= r0/kr0k2,r0= bAx0.LLanzcos?z?,?Vk= v1,v

6、2, ,vk,?KrylovfmKk(A,r0)?|IO?,=kKk(A,r0) = spanr0,Ar0, ,Ak1r0 = R(Vk).(4.11)y3Fxk x0+Kk(A,r0)?(4.9).d(4.11)9/294JJIIJIBackClose,?uzk Rk,?VTk(r0 AVkzk) = VTk(b Axk) = 0,(4.12): xk= x0+ Vkzk.5?Tk= VTkAVkr0= kr0k2v1,d(4.12),?Tkzk= e(k)1,(4.13): = kr0k2, e(k)1L111!0?k?.5?A?%XTk?,BTk?LDLT)3,?Tk= LkDkLTk,

7、(4.14): Lk en?; Dk= diag(1,2, ,k), i 0,10/294JJIIJIBackClosei = 1,2, ,k. Tk?n?(?%XLk7kXe/G:Lk=111.k11.(4.15)|)(4.14),5|(4.13)?)Lzk= LTkD1kL1k(e(k)1),l?kxk= x0+ VkLTkD1kL1k(e(k)1).(4.16)2-ePk= p1, p2, , pk = VkLTk,zk= (1,2, ,k)T= D1kL1k(e(k)1),11/294JJIIJIBackCloseK(4.16)q?xk= x0+ePkzk.(4.17)5?Tkkekk

8、eTkk+1= Tk+1:= Lk+1Dk+1LTk+1=Lk0keTk1Dk00k+1Lk0keTk1T,: ekL1k1!0?k?.u12/294JJIIJIBackClosekePk+1= Vk+1LTk+1= Vk,vk+1Lk0keTk1T= Vk,vk+1LTkkLTke(k)k01= VkLTk,vk+1 kVkLTke(k)k= ePk,vk+1 kePke(k)k := ePk,e pk+1.dd?e pk+1= vk+1 kePke(k)k= vk+1 ke pk,(4.18)13/294JJIIJIBackClosezk+1= D1k+1L1k+1(e(k+1)1)=D1

9、k001k+1L1k0keTkL1k1(e(k+1)1)=D1kL1k(e(k)1)k+1=zkk+1,: k+1= k1k+1eTkL1ke(k)1.l?kxk+1= x0+ePk+1zk+1= x0+ePk,e pk+1zkk+1= xk+k+1e pk+1.(4.19)14/294JJIIJIBackClosed(4.19),?rk+1= b Axk+1= b Axk k+1Ae pk+1= rk k+1Ae pk+1.(4.20)?,n?,?Xen?S“:xk+1= xk+ k+1e pk+1,rk+1= rk k+1Ae pk+1,e pk+1= vk+1 ke pk.(4.21)e

10、?(4.21)13?vk+1.dr0 Kk(A,r0),xk x0+ Kk(A,r0), rk= b Axk Kk+1(A,r0),l?rkdVk+1?5L,=rk= e 1v1+ e 2v2+ + e kvk+ e k+1vk+1.15/294JJIIJIBackCloseqduVTkrk= 0,?e i= 0, i = 1,2, ,k,l?rk= e k+1vk+1.(4.22)d(4.22)|e k+1| = krkk2,”?e k+1= krkk2,pk+1= krkk2e pk+1,(4.23)Kd(4.21)(4.22),?xk+1= xk+k+1krkk2pk+1,rk+1= r

11、kk+1krkk2Apk+1,pk+1= rkkrkk2kkrk1k2pk.-k+1=k+1krkk2,k= krkk2kkrk1k2,16/294JJIIJIBackClose=?xk+1= xk+ k+1pk+1,rk+1= rk k+1Apk+1,pk+1= rk+ kpk.(4.24)e?9)(4.14)?&EOk+1k?.d,k?|pirik?5.dePk?,=?ePTkAePk= L1kVTkAVkLTk= Dk.Le pimA?(A?),=ke pTiAe pj= 0, i 6= j.ukpTiApj= 0, i 6= j.(4.25)17/294JJIIJIBackClosed

12、?,dePk?(4.23)%XKk(A,r0) = R(Vk) = R(ePk) = spanp1,p2, ,pk,(4.26)=p1,p2, ,pkKk(A,r0)?|A?,2d(4.22)vi?p?5,q?rTirj= 0, i 6= j.(4.27)=p?.?,2dKk(A,r0) = R(Vk)BkKk(A,r0) = spanr0,r1, ,rk1,(4.28)=r0,r1, ,rk1?Kk(A,r0)?|?.3(4.24)?13pTkA|(4.25),?0 = pTkApk+1= pTkArk+ kpTkApk.18/294JJIIJIBackCloseukk= pTkArkpTk

13、Apk.(4.29)23(4.24)?12rTk|(4.27),?0 = rTkrk+1= rTkrk k+1rTkApk+1.dqkk+1=rTkrkrTkApk+1.(4.30)?I)(4.14)?&EO(4.24)?Xkk+1?.,|(4.25)(4.27)?k?O.k3(4.24)?13pTk+1A,|(4.25),?pTk+1Apk+1= pTk+1Ark= rTkApk+1.(4.31)19/294JJIIJIBackClose(4.24)12?kk 1O,krk= rk1 kApk.OrTkrTk1,|(4.27),?rTkrk= krTkApk,(4.32)rTk1rk1= k

14、rTk1Apk= kpTkApk,(4.33)p(4.33)?12?|?(4.31).y(4.31)“(4.30),?k+1=rTkrkpTk+1Apk+1.(4.34)2(4.32)!(4.33)(4.29)(,kk=rTkrkrTk1rk1.(4.35)20/294JJIIJIBackClosew,O?5?O?S?g,dk?.n?,?F(CG)?5|?S“Xe.?x0,Or0= b Ax0; p1= r0; k = 0;while (krkk2/kr0k )k = k + 1;k=(rk1,rk1)(pk,Apk); xk= xk1+ kpk;rk= rk1 kApk;k=(rk,rk)(

15、rk1,rk1); pk+1= rk+ kpk;end?zS(O;),?Xe/.21/294JJIIJIBackClose4.1 (CG)?|Ax = b 0.?Oxk,?krkk2/kr0k26 ,rk= b Axk.?x0; r0= b Ax0; p1= r0; 0= rT0r0; k = 0;while (krkk2/kr0k2 )k = k + 1;zk= Apk; k= k1/zTkpk;xk= xk1+ kpk; rk= rk1 kzk;k= rTkrk; k= k/k1;pk+1= rk+ kpk;end4.1?S“OKk6 0, 022/294JJIIJIBackClose?#

16、N?. CG?MATLABSXe:%?FS-mcg.mfunction x,iter,time,res,resvec=mcg(A,b,x,max_it,tol)%:X?A,mb,x,NN?tol,S“max_it%:)x,S“giter,CPUmtime,%?res,?resvectic;r=b-A*x;p=r;rho=r*r;mr=sqrt(rho);iter=0;while (itermax_it)iter=iter+1;z=A*p; alpha=rho/(z*p);x=x+alpha*p; r=r-alpha*z;rho1=r*r; beta=rho1/rho;23/294JJIIJIB

17、ackClosep=r+beta*p;res=sqrt(rho1)/mr;resvec(iter)=res;if (res 1.(4.36)e2?eChebyshev?5.n4.2?Ck(t)d(4.36)?Chebyshev,KkXe5:(1) Ck(t)k4C0(t) = 1; C1(t) = t; Ck+1(t) = 2tCk(t) Ck1(t), k = 1,2, .(2)u|t| 1, Ck(t)kLCk(t) =12h?t +pt2 1?k+?t +pt2 1?ki.(4.37)27/294JJIIJIBackClose(3)?t 1,1,k|Ck(t)| 6 1,Ck(t(k)i

18、) = (1)i,t(k)i= cosik,i = 0,1, ,k,=Ck(t)31,1fkk+14:,3?:?/?11.(4)?s 1,kminpPk,p(s)=1maxt1,1|p(t)| =1Ck(s),(4.38):Pk=?p : pgLk?X?,?4?4Kk)p(t) =Ck(t)Ck(s).(4.39)28/294JJIIJIBackClosey(1)u|t| 6 1,- = arccost,|u?zcos + cos = 2cos + 2cos 2,kCk+1(t) + Ck1(t) = cos(k + 1) + cos(k 1)= 2coskcos = 2tCk(t).u|t|

19、 1,- = arccosh(t),|V-u?cosh() =e+ e2,29/294JJIIJIBackClosekCk+1(t) + Ck1(t) = cosh(k + 1) + cosh(k 1)=e(k+1)+ e(k+1)2+e(k1)+ e(k1)2= eke+ e2+ eke+ e2= 2 e+ e2ek+ ek2= 2coshcosh(k) = 2tCk(t).(2)u|t| 1,- = arccosh(t),kt = cosh() = e+ e/2,=?e?2 2te+ 1 = 0.30/294JJIIJIBackClose?e=2t +4t2 42= t +pt2 1.u,

20、kCk(t) = cosh(k) =12?ek+ ek?=12?e?k+?e?k?=12h?t +pt2 1?k+?t +pt2 1?ki.(3) t 1,1, |Ck(t)| 6 1w,?.?y,?Ck(t(k)i) = cos(k arccost(k)i) = cos(i) = (1)i.(4)5?s 1, p(t)kgp(s) = 1.P = 1/Ck(s) 0, pmax= max16t61|p(t)|.31/294JJIIJIBackCloseepmax pmax 0.L = Ck(t) p(t)3t(k)i(i = 0,1, ,k)?k?O?K.31,1kk:.?s 6 1,1kg

21、Ck(t) p(t)?:,l?kgCk(t) p(t)kk + 1:,g.?7kmax16t61|p(t)| =1Ck(s).e?p(t) =Ck(t)Ck(s),Kmax16t61|p(t)| = max16t61?Ck(t)Ck(s)?=max16t61|Ck(t)|Ck(s)=1Ck(s).32/294JJIIJIBackClosedd?minpPk,p(s)=1maxt1,1|p(t)| =1Ck(s).y.?4.1?a b 1(1 a b).ep Pkp(1) = 1.KminpPk,p(1)=1maxta,b|p(t)| =1Ck(w(1),(4.40):w(t) =2t a bb

22、 a, a b 1,2t a ba b, 1 a 2 n 0.y34/294JJIIJIBackCloser0UA?A?m,kr0= 1u1+ 2u2+ + nun=nXi=1iui.?,Bkkx xk2A= kA1(A)r0k2A=?A1(A)r0,(A)r0?=?nXi=1iA1(A)ui,nXj=1j(A)uj?=?nXi=1i(i)1iui,nXj=1j(j)uj?=nXi=12i2(i)1i6 max16i6n2(i)nXi=11i2i= max16i6n2(i)kx x0k2A,35/294JJIIJIBackClosel?kkx xkAkx x0kA6 minP0kmax16i6

23、n|(i)|,(4.41): P0k= Pk: (0) = 1.-a = min16i6ni= n,b = max16i6ni= 1,Kd4.1,4?4KminP0kmaxa,b|()|k?)b() =Ck?b + a 2b a?Ck?b + ab a?,36/294JJIIJIBackClose?kminP0kmaxa,b|()| =1Ck?b + ab a?,(4.42)pb?1 a 0.?Oxk,?krkk2/kr0k26 ,rk= b Axk.?x0; r0= b Ax0; z0= M1r0;40/294JJIIJIBackClosep1= z0; 0= zT0r0; k = 0;wh

24、ile (krkk2/kr0k2 )k = k + 1;uk= Apk; k= k1/uTkpk;xk= xk1+ kpk; rk= rk1 kuk;zk= M1rk; k= zTkrk; k= k/k1;pk+1= zk+ kpk;end4.2=?n?Mk,?L.?M ?,vkL?n?F.?F?,?n?F=O?(4.47)?,=zI?n?MX?|?zk.4.241/294JJIIJIBackClosez?g?A?gM1?.e?n?M = LLT?. M?A?Cholesky),=A =bLbLT+R,?M := LbLT,R = A Mv,D?,?.XJA?1?,=A = M N,KI?;S

25、“?Mv,=?.uJacobiS“, MA?|?,w,?,d?n?.?nJacobi?n.uGaussSeidelSORS“,?Men?,?,dU?n?.?uSSORS“,?M?,d?n?,SSOR?n?.42/294JJIIJIBackClose?n?F?MATLABSXe:%?n?FS-pcg.mfunction x,iter,time,res,resvec=pcg(A,b,x,M,max_it,tol)%:X?A,mb,x,?nfM,%NN?tol,S“gmax_it%:)x,S“giter,CPUmtime,%?res,?resvectic; r=b-A*x; z=Mr; p=z;rh

26、o=z*r; mr=norm(r); iter=0;while (itermax_it)iter=iter+1;u=A*p; alpha=rho/(p*u);x=x+alpha*p; r=r-alpha*u;43/294JJIIJIBackClosez=Mr; rho1=z*r;beta=rho1/rho; p=z+beta*p;res=norm(r)/mr;resvec(iter)=res;if (restol), break; endrho=rho1;endtime=toc;?n?Jk*? 0.?Oxk,?krkk2/kr0k26,rk= b Axk.?x0;Or0= b Ax0; p0=

27、 ATr0; k = 0;while (krkk2/kr0k )46/294JJIIJIBackClosek=(ATrk,ATrk)(ATpkATpk);xk+1= xk+ kpk; rk+1= rk kApk;k=(ATrk+1,ATrk+1)(ATrk,ATrk);pk+1= ATrk+1+ kpk;k = k + 1;end4.4zgS“IOng?: Apk,ATpkATrk+1.5?, CGNR4?z?kekkATA= kxk xkATA,47/294JJIIJIBackClose: x(),vx= A1b; kekkATA?mx0+ span?ATr0,(ATA)ATr0, ,(AT

28、A)k1ATr0?:= x0+ Kk(ATA,ATr0)ub Axk?2.d,CG?5E|CGNR?1A?5.2. CGNE?|Ax = b=zXe/?|,=AATy = b, x = ATy.(4.50)3/y0m,|(4.50)ACG,?Xe/:?y0,Or0= b AATy0; p0= r0; k = 0;48/294JJIIJIBackClosewhile (krkk2/kr0k )k=(rk,rk)(pk,AATpk)=(rk,rk)(ATpk,ATpk);yk+1= yk+ kpk; rk+1= rk kAATpk;k=(rk+1,rk+1)(rk,rk); pk+1= rk+1+

29、 kpk;k = k + 1;endCOATyk xk, ATpk pk,2zkfK?e?u?F(CGNE).4.4 (CGNE)x0NN? 0.?Oxk,?krkk2/kr0k26 ,rk= bAxk.?x0;Or0= b Ax0; p0= ATr0; k = 0;while (krkk2/kr0k )49/294JJIIJIBackClosek=(rk,rk)(pk,pk); xk+1= xk+ kpk;rk+1= rk kApk; k=(rk+1,rk+1)(rk,rk);pk+1= ATrk+1+ kpk;k = k + 1;end4.4zgS“IOg?: ApkATrk+1.d?,

30、CGNE4?z?ke ekkATA= kyk ykATA,50/294JJIIJIBackClose: yvy= (ATA)1b; ke ekkATA?mxk x0+ span?ATr0,(ATA)ATr0, ,(ATA)k1ATr0? x0+ Kk(ATA,ATr0)?2kxk xk2.5, CGNRCGNE?(J:3u?C?5?,?,3k?ek?.wz25?.51/294JJIIJIBackClose4.3|Ax = b,A =12323 12322 3 1232.2 31232312 Rnn,b = A11.11 Rn.?n = 1000,OCGNRCGNEA?T5|,S“310?( =

31、 1010),O?Cq)b xe x)xm?Okb x xk2= 3.4705 1010,ke x xk2= 3.4515 1010.52/294JJIIJIBackCloseOkb Ab xk2= 4.5764 109,kb Ae xk2= 4.6018 109.S“L?;,X4.3,IS“k,pI?krkk2/kr0k2,prk1k?.4.3 CGNRCGNE?A553/294JJIIJIBackClose4.224?24?8c).D5|?,3zGMRES(Generalized MinimalRESidual Method).?!0?O95n.4.2.1 GMRESXe?5|Ax = b

32、,(4.51):?A Rnnb Rn?,?x Rn?.pbX?A?.D?,?54/294JJIIJIBackCloseAT6= A.24?xk x0+ Kk(A,r0),?krkk2= minkb Axk2: x x0+ Kk(A,r0),(4.52): rk= b Axk,=xk x0+ Kk(A,r0),?rk?2?.duy3A?,U/A?Arnoldi?zL4?zK(4.52).?eArnoldi?zL(2.11):?v1,?kv1k2= 1;for j = 1 : kfor i = 1 : jhij= (Avj,vi);55/294JJIIJIBackClosee vj+1= Avjj

33、Xi=1hijvi;(4.53)hj+1,j= ke vj+1k2; vj+1= e vj+1/hj+1,j;endendJuy,(4.53)?AVk= VkHk+ kvk+1eTk= Vk+1fHk+1,k,(4.54): Vk+1= Vk,vk+1 Rn(k+1)vVTk+1Vk+1= Ik+1;?fHk+1,k=HkkeTk R(k+1)k56/294JJIIJIBackCloseHessenberg?,k= hk+1,k,Hk=h11h12h1kh21h22h2kh32h3k.hk,k1hk,k.5?Kk(A,r0) = R(Vk),?x = x0+ Vkz x0+Kk(A,r0),k

34、kb Axk2= kb Ax0 AVkzk2= kr0 AVkzk2= kVk+1e(k+1)1 Vk+1fHk+1,kzk2= ke(k+1)1fHk+1,kzk2,: = kr0k2; e(k+1)1L111!0?57/294JJIIJIBackClosek + 1?.dd,4?zK(4.52)?duzk Rk,?ke(k+1)1fHk+1,kzkk2= minke(k+1)1fHk+1,kzk2: z Rk.(4.55)?zk?,KI?xkxk= x0+ Vkzk.?K(4.55)fHk+1,k?QR)5).dufHk+1,kHessenberg?,OkGivensCGi= diagIi

35、1,cisisici,Iki,c2i+ s2i= 1,?(GkGk1G2G1)fHk+1,k=Rk0,(4.56)58/294JJIIJIBackClose: Rk?n?(fHk+1,k?g?).-G = GkGk1G2G1,tkk= G(e1), tk= (1,2, ,k)T,(4.57)KGk + 1?,?O,k1= c1,i= (1)i1s1s2si1ci, i = 2,3, ,k,k= (1)ks1s2sk.(4.58)dd=?K(4.55)?)zk= R1ktk.(4.59)59/294JJIIJIBackClosed?,d?krkk2= ke(k+1)1fHk+1,kzkk2= |

36、k|.(4.60)n?,24?(GMRES)?o(Xe:(1)-v1= r0/kr0k2,?)k?Arnoldi)(4.54).(2)|GivensCfHk+1,k?QR)(4.56),U(4.58)?tkk.(3)“)n?|Rkzk= tk,?zk.(4)Oxk= x0+ Vkzk.(5)e|k|/ 0,?x0.?Oxk Rn,?krkk2/kr0k26 ,rk= b Axk.1,Or0= b Ax0, = kr0k2?zv1= r0/, = e1= (1,0, ,0)T.k := 1.2,ArnoldiLOvk+1hi,k(i = 1,2, ,k + 1).GivensCGiu?fHi+1

37、,i?:hi,khi+1,k:=cisisicihi,khi+1,k,i = 1, ,k 1.61/294JJIIJIBackClose3,O1kgGivensCGk?cksk:ck=hk,kqh2k,k+ h2k+1,k,sk=hk+1,kqh2k,k+ h2k+1,k,(4.61)?k=hk+1,khk,k,ck=1p1 + 2k,sk= ckk?.4,k +1?fHk+1,kO1kgGivensCGk,kkk+1:=ckskskckk0=ckkskk,hk,khk+1,k:=ckskskckhk,khk+1,k=ckhk,k+ skhk+1,k0.(4.62)62/294JJIIJIBa

38、ckClose5,e|k|/ = |k+1|/ 6 ,)uzk?n?|Hk,kzk= k1,OCq)xk= x0+ Vkzk,;K,k := k + 1,=2.4.5?MATLABSXe:%GMRESS-mgmres.mfunction x,k,time,res,resvec,flag=mgmres(A,b,x,max_it,tol)tic; flag=0; r=b-A*x; beta=norm(r);%O?n=length(b); e1=zeros(n,1); e1(1)=1.0;res=norm(r)/beta; resvec(1)=res;V(:,1)=r/beta; xi=beta*e

39、1; k=0;while (k=max_it)k=k+1; w=A*V(:,k);for i=1:k%?ArnoldiL63/294JJIIJIBackCloseH(i,k)=w*V(:,i);w=w-H(i,k)*V(:,i);endH(k+1,k)=norm(w);if abs(H(k+1,k)/betatol,return;elseV(:,k+1)=w/H(k+1,k);endfor i=1:k-1,temp=c(i)*H(i,k)+s(i)*H(i+1,k);H(i+1,k)=-s(i)*H(i,k)+c(i)*H(i+1,k);H(i,k)=temp;64/294JJIIJIBack

40、Closeendc(k),s(k),H(k,k)=givens(H(k,k), H(k+1,k);%1kgGivensCxi(k+1)=-s(k)*xi(k);xi(k)=c(k)*xi(k); H(k+1,k)=0.0;res=abs(xi(k+1)/beta;resvec(k+1)=res;if (restol), flag=1; end;%time=toc;65/294JJIIJIBackClose4.4X?AmbOA =1100n121.00.00.1n 11n001n,b = A11.11?5|(4.51).w,T|?)x= (1,1, ,1)T.?n = 103,AGMRES?T|

41、,S“172?Cq)e x = x172vke x xk2= 1.1427 107.S“L?;,X4.4,IS“k,pI?krkk2/kr0k2,prk1k?.66/294JJIIJIBackClose4.4 GMRES?A5SGMRES?KkU?,?SIO(kn),?O?Sqk?,?yCq)xkv,?kU2O?/.)K?1k?-mE.?g,k?m,GMRES?)xm,?2xm-#m.-m67/294JJIIJIBackCloseGMRES,PGMRES(m),NOLXe.4.6 (GMRES(m)?A Rnn,b Rn,?m,x0#N? 0.?Oxm Rn,?kb Axmk2/kb Ax0k

42、26 .?x0;Or0= b Ax0; = kr0k2; m:= ; xm:= x0;while (m/ )r0= b Axm; 0= kr0k2; v1= r0/0;v1?)m?Arnoldi)AVm= Vm+1fHm+1,m;OfHm+1,m?QR):fHm+1,m= GTRm0;68/294JJIIJIBackCloseU(4.58)Otmm;)Rmzm= tm?zm;Oxm= x0+ Vmzm;end8c).D5|?.u4.6?m?,y3vkn?(J.3n=?y,3X?Ak?,?m,4.6o?(?5n).?u/,?moU?y.X,5|Ax :=011 0 x1x2=11:= b,69/

43、294JJIIJIBackClose|GMRES(1),?g,okxk= 0,?kAxkbk2=2,?|)v?Cq).?XJGMRES),KI?T|?().GMRES(m)?MATLABSXe:%-mGMRES-GMRES(m)-gmresm.mfunction x,out,int,time,res,resvec,flag=gmresm(A,b,x,m,max_it,tol)tic; flag=0; int=0; r= b-A*x;%O?beta=norm(r); res=norm(r)/beta; resvec(1)=res;n=length(b); %m=restrt;V(1:n,1:m+

44、1)=zeros(n,m+1);H(1:m+1,1:m)=zeros(m+1,m);e1=zeros(n,1); e1(1)=1.0;70/294JJIIJIBackClosec(1:m)=zeros(m,1); s(1:m)=zeros(m,1);for k=1:max_itV(:,1)=r/norm(r);xi=norm(r)*e1;for j=1:m%Arnoldi?E?w=A*V(:,j);for i=1:jH(i,j)=w*V(:,i);w=w-H(i,j)*V(:,i);endH(j+1,j)=norm(w);if abs(H(j+1,j)/betatol,return;71/29

45、4JJIIJIBackCloseelseV(:,j+1)=w/H(j+1,j);endfor i=1:j-1%1igGivensCtemp=c(i)*H(i,j)+s(i)*H(i+1,j);H(i+1,j)=-s(i)*H(i,j)+c(i)*H(i+1,j);H(i,j)=temp;endc(j),s(j),H(j,j)=givens(H(j,j),H(j+1,j);%1jgGivensCxi(j+1)=-s(j)*xi(j);xi(j)=c(j)*xi(j);H(j+1,j)=0.0;res=abs(xi(j+1)/beta;resvec(k-1)*m+j+1)=res;72/294JJ

46、IIJIBackCloseif(res=tol)y=H(1:j,1:j)xi(1:j);x=x+V(:,1:j)*y;break;%aSendendif (restol), flag=1; end;%time=toc;4.5GMRES(m)4.4?5|.L4.1?u?m,?(NN? = 1010)I?S“g!SS“g!oS“gCPUm(s).lL4.1L4.1 GMRES(m)m?65m?S“gSS“goS“gCPUm?104734630.26219.8273e-112014122720.14889.1166e-1130982480.13279.3534e-11406272270.12679.

47、4923e-11505192190.12449.9472e-11604262060.12039.9062e-11GMRES11721720.16518.8473e-11?w,u?,GMRES,u?74/294JJIIJIBackClose?m, GMRES(m).,c,k3GMRES(XO?S),GMRES(m),m?.d?,u?m,S“L?;,X4.5,IS“k,pI?krkk2/kr0k2,prk1k?.4.5 GMRES?A575/294JJIIJIBackClose4.2.2?nGMRES?F)|Ax = b?nE?-5,?n24?(PPGMRES).?Fu?A,?n?MI?.u?A,

48、?n?Mvk7?.d|Ax = b?nM1Ax = M1b.(4.63)pMAT?A?Cq?,?yGMRES)?n|(4.63)?|Ax = bk?.duGMRES?9?,?yM1,r?z = M1rNO,=Mz = r76/294JJIIJIBackCloseN).u?M()?()n?/?,XJacobi?n?M = D = diaga11, ,ann,GaussSeidel?n?M = D L9SOR?n?M =1(D L),:L = 0a210.an1 an,n10.77/294JJIIJIBackClose|(4.63)GMRES,?XePGMRES.4.7 (PGMRES)?A Rn

49、n,b Rn,?nfM,x0#N? 0.?Oxk Rn,?krkk2/kr0k26 ,rk= b Axk.r0= M1(b Ax0); = kr0k2;for k = 1,2,Kk(M1A,v1)Arnoldi)OVk= v1,v2, ,vk9(k + 1) k?fHk+1,k.OfHk+1,k?QR):fHk+1,k= GTRk0;U(4.58)Otkk;if |k|/ 6 78/294JJIIJIBackClose)Rkzk= tk?zk;Oxk= x0+ Vkzk;.endend|(4.63)GMRES(m)?Xe:4.8 (PGMRES(m)?A Rnn,b Rn,?m,?nfM#N?

50、 0,?x0 Rn.?Oxk Rn,?krkk2/kr0k26 ,rk= b Axk.1,Or0= M1(b Ax0), = kr0k2?zv1= r0/.79/294JJIIJIBackClose2,Km(M1A,v1)Arnoldi)OVm= v1,v2, ,vm9(m + 1) m?fHm+1,m.3,Ov4?zKmin?ke(m+1)1fHm+1,mzk2: z Rm?)zm,-xm= x0+Vmzm.exm?,.4,x0:= xm,=1.eLf,*?PGMRES?J.4.6E4.4?5|.?n?M =diag(A),APGMRES?T|,S“12?80/294JJIIJIBackCl

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