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1、1/252JJIIJIBackClose5“1 ?A?K?2/252JJIIJIBackCloseNSK?),X?K!-5K?,8(,?A?A?K. n?A Cnn?A?A?,vXe? Cx Cn:p() = det(A I) = 0,(7.1)Ax = x(A I)x = 0.(7.2)(7.1)?A?A?, In? ?, det(AI)L?A I?1?,?ng“,?n?:JO().d,lO?*:5w,A?5?A?,7Lk?.3SA,?A?A?S“.?g,A?A?S?45?.?a?K?,?O?.3/252JJIIJIBackCloseNK?I,k?SKO?A?.?,?KKO?A?A?.?0?

2、A8c3O?A?K?.7.1?A?Ol3O, I?O?A?3E? .X,3S“,I?S“?(S“?Jacobi?)?A?3 ?S.qX,5X-,I?k?A?3E?.?!?0?(J.k?Hermite?A?LK.duHermite?A Cnn?A?,?nA?gS4/252JJIIJIBackClose:1 2 n.(7.3)7.1?A CnnHermite?,?x Cn,R(x) =(Ax,x)(x,x)x?Rayleigh.e?n?Hermite?A?Rayleigh?44?5.n7.1?AH= A Cnn,Kmax06=xCn(Ax,x)(x,x)= max(A),min06=xCn(Ax,

3、x)(x,x)= min(A).(7.4)y ?A?A?1 2 n,K3j?Q?5/252JJIIJIBackCloseA = Qdiag(1,2, ,n)QH,?(Ax,x)(x,x)=(Qdiag(1,2, ,n)QHx,x)(QHx,QHx)=(diag(1,2, ,n)QHx,QHx)(QHx,QHx).-y = QHx = (y1,y2, ,yn)T,K(Ax,x)(x,x)=(diag(1,2, ,n)y,y)(y,y)=1y21+ 2y22+ + ny2ny21+ y22+ + y2n.dun(y21+y22+y2n) 6 1y21+2y22+ny2n6 1(y21+y22+y2

4、n),6/252JJIIJIBackClose?n6(Ax,x)(x,x)6 1, 0 6= x Cn.?x1xnOAu1n?A?,K(Ax1,x1)(x1,x1)= 1,(Axn,xn)(xn,xn)= n,d(.y.?n7.2 AH= A Cnn?A?X(7.3),K1 6k 6 n,kk= maxVkmin06=xVk(Ax,x)(x,x),(7.5): VkCn?kfm.y ?A?AuA?1,2, ,n?A?gp1,p2,pn,IO?,?EfmWk=spanpk,pk+1, ,pn,7/252JJIIJIBackCloseodim(Wk) = n k + 1.duVk+ Wk Cn,|

5、fm?n dim(Vk+ Wk) = dim(Vk) + dim(Wk) dim(Vk Wk)= n + 1 dim(Vk Wk),=dim(VkWk) 1.u3x0 VkWk Wkvkx0k2= 1,kx0= ckpk+ ck+1pk+1+ + cnpn, (c2k+ c2k+1+ + c2n= 1),(Ax0,x0) = kc2k+ k+1c2k+1+ + nc2n6 k.dmin06=xVk(Ax,x)(x,x)6(Ax0,x0)(x0,x0)6 k.8/252JJIIJIBackClosedVk?5,?maxVkmin06=xVk(Ax,x)(x,x)6 k.,?,?kfmVk= sp

6、anp1,p2, ,pk,?x Vkvkxk2= 1,kx = c1p1+ c2p2+ + ckpk, (c21+ c22+ + c2k= 1),(Ax,x) = 1c21+ 2c22+ + kc2k k.dx Vk?5,?min06=xVk(Ax,x)(x,x) k.l?kmaxVkmin06=xVk(Ax,x)(x,x) k.n,(7.5).y.?9/252JJIIJIBackClosen7.3?n?Hermite?AB = A + E?A?g1 2 n,1 2 n.Kkn6 k k6 1,(7.6): 1nOHermite?E?A?A?.y e?xvkxk2= 1.3(7.5)?Vk,|

7、(7.4),?k minxVk(Bx,x) minxVk(Ax,x) + minxVk(Ex,x) minxVk(Ax,x) + minxCn(Ex,x) = minxVk(Ax,x) + n,10/252JJIIJIBackClosek minxVk(Ax,x) minxVk(Bx,x) + minxVk(Ex,x) minxVk(Bx,x) + minxCn(Ex,x) = minxVk(Bx,x) + (1).duVk Rn?,k maxVkminxVk(Ax,x) + n= k+ n,k maxVkminxVk(Bx,x) 1= k 1,=(7.6).y.?57.1n7.35?K.Ew

8、du?3?A?)? /?0,?BSO?.?E?z?eijk|eij| 6 , 0,KB?A?(B) = (A) +11/252JJIIJIBackCloseA?A?(A)mv(7.6),=1 n.d,k| 6 (E) 6 kEkF6 n.L,?A?k?6,3A?A?E?6?.n7.4 (n)?An?, B = QTAQ,Q Rn(n1)vQTQ = In1.2?AB?A?O1 2 n1 2 n1,Kk1 1 2 2 n1 n.(7.7)AO/,3n7.4?Q = e1, ,ei1,ei+1, ,en,=?Xe(.12/252JJIIJIBackClose7.1?Bn?A?n 1?f?,bAB?

9、A?O1 2 n1 2 n1,Kk1 1 2 2 n1 n.EA7.1,Bke.7.2?An?, BA?k?f?(1 6 k 6 n 1),bAB?A?O1 2 n1 2 k,Kki i nk+i,i = 1,2, ,k.(7.8)13/252JJIIJIBackClose?n?A?3E?,e?Gerschgorin?(X?X?).7.2?A = (aij) Cnn,-Ri=nPj=1,j6=i|aij|,KGi= z|z C : |z aii| 6 Ri,i = 1,2,n(7.9)A?1iX?.n7.5?A Cnn?A?,K nSi=1Gi,=A?A?3?nX?8S.y ?A?AuA?A?

11、ssini (kZ)/.15/252JJIIJIBackClosey ?A?AuA?A?x = (x1,x2, ,xn)T.?i06= j0v|xi0| |xj0| |xk|(k 6= i0,j0),ey i0j0.(1)XJxj0= 0,Kxi06= 0, xk= 0 (k 6= i0).dAx = x?xi0=nXk=1ai0kxk= ai0i0 xi0 = ai0i0.? ai0i0? aj0j0?= 0 Ri0Rj0.(2)XJxj06= 0,K?xi0| |xj0? 0,2dAx = x?( aii)xi=Xk6=iaikxk, (i = 1,2, ,n).?i = i0,?| ai

12、0i0|xi0| Xk6=i0|ai0k|xk| |xj0|Ri0.16/252JJIIJIBackClose?i = j0,?| aj0j0|xj0| Xk6=j0|aj0k|xk| |xi0?Rj0.d| ai0i0| aj0j0| Ri0Rj0.n(1)(2)=? i0j0.y.?7.3?A = (aij) Cnn(n 1)v|aii|ajj| RiRj(i 6= j),Kdet(A) 6= 0.y ?A?A?,o7kij? ij,=| aii| ajj| RiRj.XJ = 0,Kk|aii|ajj| RiRj,K?g,? 6= 0.l?det(A) 6= 0.y.?17/252JJI

13、IJIBackClose7.27.2.1L?A?5A?S“.5U?A?A?A?.:,NOy,:,k?56u?A?.u?/: A?A?U?|1| |2| |n|,AA?1,2, ,n5.d,?x(0)1,2, ,n5L,=x(0)= 11+ 22+ + nn,(7.10)1, 2, ,n?.S?x(k)= Akx(0),K18/252JJIIJIBackClosex(k)= Akx(0)= 1Ak1+ 2Ak2+ + nAkn= 1k11+ 2k22+ + nknn= k1h11+ 2?21?k2+ + n?n1?kni.dd,e16= 0,Kk ,k?i1?k 0, (i = 2, ,n),?

14、k,7kx(k) k111,=x(k)Cqw1A?A?;?x(k)x(k1)x(k)ix(k1)ik11(1)ik111(1)i= 1.19/252JJIIJIBackCloseu|S?x(k)QU?A?1,qA?A?1.3SO,?|1| 1, k1 ;?|1| 1,k1 0,?Ox(k)U?O/0/e0yu),?zx(k)8z?n?,=x(k)?,kx(k)k= 1.u,AU?A?1A?A?1?,8BXe.7.1 ()1,?A,v(0),?,S“gN.Pm0v(0)U?, x(0)= v(0)/m0.k := 0.2,Ov(k+1)= Ax(k).Pmk+1v(k+1)U?, x(k+1)=

15、 v(k+1)/mk+1.3,e|mk+1 mk| ,CqA?mk+1C20/252JJIIJIBackCloseqA?x(k+1);K,=4.4,ek N,k := k + 1,=2;KO”&E,.7.1.?(?,N?y.MATLABSXe:function lam,v,k=mypower(A,x,tol,N)%O?A?A?A?%:An?,x,%tol,NS“g%:lamU?A?,%vA?A?,kS“gif nargin4, N=1000; endif nargin3, tol=1e-6; end21/252JJIIJIBackClosem=0; k=0;while(kN)v=A*x;m1,

16、t=max(abs(v);m1=v(t); x=v/m1;err=abs(m1-m);if err|2| |n|,AA?1,2, ,n5.S?x(k)d7.1?),Kklimkx(k)=1max1:= 01,limkmk= 1,(7.11): 0118z?; max11?.23/252JJIIJIBackClosey d7.1?23x(k)=v(k)mk=Ax(k1)mk=A2x(k2)mkmk1= =Akx(0)mkmk1m1.dux(k)?1,=maxx(k) = 1,?mkmk1m1= maxAkx(0).l?x(k)=Akx(0)maxAkx(0)=k1h11+nPi=2i?i1?ki

17、imaxnk1h11+nPi=2i?i1?kiio=11+nPi=2i?i1?kimaxn11+nPi=2i?i1?kio.24/252JJIIJIBackCloselimkx(k)=11max11=1max1= 01.qv(k)= Ax(k1)=Akx(0)mk1m1=Akx(0)maxAk1x(0)=k1h11+nPi=2i?i1?kiik11maxn11+nPi=2i?i1?k1io,5?mkv(k)?,=kmk= maxv(k) = 1maxn11+nPi=2i?i1?kiomaxn11+nPi=2i?i1?k1io,l?limkmk= 1.y.?25/252JJIIJIBackClo

18、se7.2.2?En7.83n7.7?e,(7.1)5?.y ?k, Akx(0)?U?1j,Kmk 1= maxv(k) 1=maxAkx(0)maxAk1x(0) 1=?1k11+ 2k22+ + nknn?j?1k111+ 2k122+ + nk1nn?j 1=?2k12(2 1)2+ + nk1n(n 1)n?j?1k111+ 2k122+ + nk1nn?j.26/252JJIIJIBackCloseukmk 1=?21?k1h2(2 1)2+nPi=3i?i2?k1(i 1)iijh11+nPi=2i?i1?k1iij=?21?k1Mk,Mk M,: M.,?k ,k|mk+1 1

19、|mk 1|=?Mk+1(2/1)kMk(2/1)k1?21?,y?5.y.?n7.8L,?|2/1|?k,|2/1|?,?,?d?Cu1,27/252JJIIJIBackClose?.d,?: ,-B = A I,: .Jd?B?,?.?A?A?1,2, ,n,A?A?1,2, ,n,K?B?A?1,2, ,n, B?A?A?A?.b?: ?, B?A?1 E?A?,J?8?max26i6n|i |1 |?21?.(7.12)?/J?.dmk 1 ,mk+ 1,?x(k)E,uA?A?01.?: .3SO,du?A?k28/252JJIIJIBackClose?,?3(J,?y1 E,?B

20、(= AI)?A?,?: Jy?.?,3,?: N?,d,?: ?U?A?A?IO.?: E?MATLABSXe:function lam,v,k=mopower(A,x,alpha,tol,N)%?: ?A?A?A?%:An?,x,tol,%NS“g,alpha?: %:lamU?A?,%vA?A?,kS“gif nargin5, N=1000; endif nargin4, tol=1e-6; end29/252JJIIJIBackClosem=0; k=0;A=A-alpha*eye(length(x);while(kN)v=A*x;m1,t=max(abs(v);m1=v(t); x=

21、v/m1;err=abs(m1-m);if err |2| |n1| |n|,K1nA1U?A?,dU,7kmk 1n,x(k) 0n,31/252JJIIJIBackClose|n/n1|.?x(0),?ES?x(k+1)= A1x(k),k = 0,1,2,(7.13)UO=.?(7.13)O,kA1,?L?.SO,)|?,=Ax(k+1)= x(k),k = 0,1,2, ,(7.14)x(k+1).“O,SO?v(k)= x(k)/max(x(k),Ax(k+1)= v(k),k = 0,1,2, ,(7.15): max(x(k)x(k)?.u?CqA?,?A?JpA?.d,?: 5

22、S“32/252JJIIJIBackCloseL,u(7.15)Av(k)= x(k)/max(x(k),(A I)x(k+1)= v(k),k = 0,1,2, .(7.16)?OXe.7.2 ()1,?x(0),Cq,?,S“gN.Pm0 x(0)U?, v(0)= x(0)/m0.k := 0.2,)|(A I)x(k+1)= v(k)?x(k+1).3,Pmk+1x(k+1)U?, v(k+1)= x(k+1)/mk+1.4,e|m1k+1 m1k| ,K := m1k+1+ ,x(k+1),;K,=5.33/252JJIIJIBackClose5,ek N,k := k + 1,=2

23、,KO”&E,.57.2 (1)7.2O?C?A?9A?A?.e? = 0,KA?U?A?.(2)kA?U?CqA?7.2?,2TA?A?1(z.(3)!O,k?LU)?A I)en?Ln?U,?3S“LzI)n?|=.?MATLABSXe:function lam,v,k=mvpower(A,x,alpha,tol,N)34/252JJIIJIBackClose%O?alpha?C?A?A?A?%:An?,x,tol,%NS“,alpha,%:lamalpha?C?A?,%vA?A?,kS“gif nargin5, N=500; endif nargin4, tol=1e-5; endm=0

24、.5; k=0;A=A-alpha*eye(length(x);L,U,P=lu(A);while (kN)m1,t=max(abs(x);m1=x(t); v=x/m1;35/252JJIIJIBackClosez=L(P*v); x=Uz;err=abs(1/m1-1/m);if err=tol, break; endk=k+1; m=m1;endlam=alpha+1/m;7.3|S,7.1?A?C101,99, 20?A?A?A?.) 5?d?O?101,99,2,0.?MAT-LABS,?Mex73.m,?NN? = 106,3-I$1T,?4CqA?: 100.7462; 99.2

25、107; 1.7893;0.2538,?A?A?.36/252JJIIJIBackClose7.3 JacobiJacobiu)?A?A?A?.?nXe:(1) n?A?,A?A?5?.(2)q?k?A?.(3)en?A?,K3?Q,?QTAQ = D,D?,?1,2, ,nA?A?, Q?1i?iA?A?.Jacobiu?n,X?C?zA,=?A?,l?A?A?.37/252JJIIJIBackClose7.3.1?=?qCk?e120?L?CGivensC,Jacobi?.7.3?1 6 i j 6 n,K?Gij=1.cossin1.1sincos.1ij(7.17)ij(i,j)?=?

26、,GivensC?.w, G = Gij?,=GTG = I.ux Rn,38/252JJIIJIBackClosed5Cy = Gx?y?yi= xicos + xjsin,yj= xisin + xjcos,yk= xk, k 6= i,j,(7.18)=Gijx,UC1i,j.d?G = Gij(?Cy = Gx=C,GivensC.(7.18)N?y,?Gijke?5.n7.9?x Rn?1jxj6= 0, 1 6 i j 6 n.e-c = cos =xiqx2i+ x2j,s = sin =xjqx2i+ x2j,(7.19)39/252JJIIJIBackCloseKy = Gi

27、jx?yi=qx2i+ x2j,yj= 0,yk= xk,k 6= i,j.(7.20)n7.9L,GivensC?,C?.e?GivensC?.=?Gij?A = (aij)nn?qC,?PA1,=A1= GijAGTij= (a(1)ij).w,AT1= (GijAGTij)T= GijAGTij= A1,40/252JJIIJIBackClose=A1E?.?O,?a(1)ii= aiicos2 + ajjsin2 + 2aijcossin,a(1)jj= aiisin2 + ajjcos2 2aijcossin,a(1)ij= a(1)ji= aij(cos2 sin2) (aii a

28、jj)cossin,a(1)il= a(1)li= ailcos + ajlsin, l 6= i,j,a(1)jl= a(1)lj= ailsin + ajlcos, l 6= i,j,a(1)lm= a(1)ml= aml, m,l 6= i,j.(7.21)Jw, ALGij?qC?, A1?A?,k1i1,1j11i?,1j?u)?Cz,?A?.d(7.21)?,eaij6= 0,K?41/252JJIIJIBackClose?,?a(1)ij= a(1)ji= 0.,-aij(cos2 sin2) (aii ajj)cossin = 0,)?cot2 =aii ajj2aij=1 t

29、an22tan,4 0,d d2+ 1, d 0,e?a(k)ij? , i,j = 1,2, ,n, i 6= j,(7.30)45/252JJIIJIBackCloseKAk?,l?(O. Ak?A?A?.JacobiA?kA?.,dAk= GkAk1GTk= GkGk1Ak2GTk1GTk= = GkGk1G1AGT1GTk1GTk,ePQk= GT1GTk1GTk,(7.31)KAk= QTkAQk.(7.32): Qk?.eAk?,?=A?A?,1i?a(k)iiA?A?Qk?1i?. Qk46/252JJIIJIBackClose?OA?=qC?1.e-Q0= I,KQk= Qk1

30、GTk.(7.33)eGk= Gij,?Qk?OXe:q(k)li= q(k1)lic + q(k1)ljs, l = 1,2, ,n,q(k)lj= q(k1)lis + q(k1)ljc, l = 1,2, ,n,q(k)km= q(k1)km, k,m 6= i,j.(7.34),?1i,j?u)Cz?,?C.eIOA?,Kd.?,?Jacobi?OXe.7.3 (Jacobi)1,?A, Q = I,x,?,S“gN.k := 1.47/252JJIIJIBackClose2,3? =?airjr?= max16i,j6ni6=j|aij|,i := ir, j := jr.3,U(7

31、.23) (7.27)Od,t,c,s?A1?a(1)lm, l,m = 1,2, ,n.4,#Q?:qli:= qlic + qljs,qlj:= qlis + qljc,l = 1,2, ,n.5,e ,A1?Q?,;K,=6.6,ek N,k := k + 1,=2;KO”&E,.48/252JJIIJIBackClose7.3,?Jacobi?MATLABSXe:%JacobiS-Jacobi_eig.mfunction lambda,Q=Jacobi_eig(A,tol)%Jacobi?A?A?A?%:An?,tolNN?%:lambda,A?A?,%Q?,?A?nA?if narg

32、in=0)t=-d+sqrt(d2+1);elset=-d-sqrt(d2+1);endc=1/sqrt(t2+1); s=c*t; G=c s; -s c;A(p q,:)=G*A(p q,:);A(:,p q)=A(:,p q)*G;Q(:,p q)=Q(:,p q)*G;w1,p=max(abs(A-diag(diag(A);w2,q=max(w1); p=p(q);if (abs(A(p,q)1n(n 1)S(Ak).=?qCz0,?Ak+1,dS(Ak+1) = S(Ak) 2?a(k)ij?26 S(Ak) 2n(n 1)S(Ak)=h1 2n(n 1)iS(Ak) 6h1 2n(

33、n 1)ik+1S(A).du1 2n(n 1)?n?A?SchurIO/.w,?SchurIO/,N?A?.d,F?ES“(qC),FU%C?A?SchurIO/.X,u?A Rnn,-57/252JJIIJIBackCloseA1:= A,?ES“:Ak= QkRk,Ak+1= RkQk,k = 1,2, ,(7.39): Qk?; Rkn?.y,3e,d(7.39)?)?S?Ak/%C0uA?SchurIO/.,?,(7.39)?S“vk?.zS“?$?(?O(n3);?(6uA?l).d,p?S“,7L?U/?zS“?$,Jp.1?,kr?A?qCHessenberg/,?2?qC?1

34、S“.eXg0?QR5?A?A?.58/252JJIIJIBackClose7.4.1z?Hessenberg?3QR?A?, Householder?k:A?qC,rAzHessenberg?;?n?).k?rAzHessenberg?.?A1= A = (a(1)ij)n?,?x = (0,a(1)21, ,a(1)n1)T,Pa1= sgn(x2)kxk2,Kdn2.2n2.3?EHouseholder?H1=1000.0,?H1x = a1e2.59/252JJIIJIBackCloseH1A1?11?H1a(1)11a(1)21.a(1)n1 = H1x + H1a(1)110.0

35、= a1e2+a(1)110.0 =a(1)11a1.0.H1m?UCT?11?,uA2= H1A1H1=a(1)11a(2)12a(2)1na1a(2)22a(2)2n0a(2)32a(2)3n.0a(2)n2a(2)nn.2?x = (0,0,a(2)32, ,a(2)n2)T,Pa2= sgn(x3)kxk2,?EH2H2=1000010000.00,60/252JJIIJIBackClose?H2x = a2e3.H2A2?11?A2?11?,?H2A2?12?CH2x + H2a(2)12a(2)220.0= a2e3+a(2)12a(2)220.0=a(2)12a(2)22a2.0

36、.?H2m?UCT?11?12?,uA3= H2A2H2=a10a200.00.61/252JJIIJIBackClose?e?,Ln 2gC?,A1zHessenberg?An1,=An1= Hn2H2H1A1H1H2Hn2=a1a2a3.an1.XJA1?,KAn1E?,dAn1n?:An1=a1a1a2a2a3.an1an1.62/252JJIIJIBackClose|HouseholderC?zAHessenberg/?,o(Xe.7.4 (Hessenbergz)1,A = (aij), k := 1.2,On k?Householder?fHk,fHkak+1,kak+2,k.an

37、,k=0.0,A := HkAHk,Hk= diag(Ik,fHk).3,ek A=-1 2 3 5; 2 -3 8 1; 3 8 -2 7; 5 1 7 6; A,Q=mhessen(A)65/252JJIIJIBackCloseA =-1.0000-6.16440.00000.0000-6.164411.73681.83800.00000.00001.8380-6.59295.99380.00000.00005.9938-4.1439Q =1.000000000.3244-0.0418-0.945000.48670.86400.128900.8111-0.50170.30075,Hesse

38、nberg)?,?ye?(J.66/252JJIIJIBackClosen7.12?A RnnkXeHessenberg):UTAU = H,VTAV = G,(7.40): U = u1,u2, ,unV = v1,v2, ,vnn?; H = (hij)G = (gij)Hessenberg?.eu1= v1,H?g?hi+1,i6= 0(i = 1,2, ,n),K3?11?D,?U = V D,H = DGD,(7.41)=ui= vi, |hij| = |gij|, i,j = 1,2, ,n.y ?n8B.un = 1(w,.b?n = m(,=ui= ivi, i = 1,2,

39、,m,(7.42)67/252JJIIJIBackClose: 1= 1, i= 11, i = 2, ,m.ey3m+111?um+1= m+1vm+1.d(7.40),?AU = UH, AV = V G.O?1m?,?Aum= h1mu1+ + hmmum+ hm+1,mum+1,(7.43)Avm= g1mv1+ + gmmvm+ gm+1,mvm+1.(7.44)O3(7.43)(7.44)uTivTi(i = 1,2, ,m),?him= uTiAum, i = 1,2, ,m,(7.45)gim= vTiAvm, i = 1,2, ,m.(7.46)68/252JJIIJIBac

40、kClosed(7.42)!(7.45)(7.46),?him= imgim, i = 1,2, ,m.(7.47)(7.47)“(7.43),|(7.42)(7.44),?hm+1,mum+1= Aum 1mg1mu1 mmgmmum= m(Avm 21g1mv1 2mgmmvm)= m(Avm g1mv1 gmmvm)= mgm+1,mvm+1.(7.48)?,?|hm+1,m| = |gm+1,m|.duhm+1,m6= 0,?(7.48)%Xum+1= m+1vm+1,: m+1= 11.y.?69/252JJIIJIBackClose57.3Hessenberg?H = (hij),

41、XJg?,=hi+1,i6= 0, i = 1,2, ,n1,K?.n7.12L,XJQTAQ = H?Hessenberg?,KQH?dQ?11?(p3?Ke?).7.4.2Hessenberg?QR)uHessenberg?H =h(1)11h(1)12h(1)13h(1)1nh(1)21h(1)22h(1)23h(1)2nh(1)32h(1)33h(1)2n.h(1)n,n1h(1)nn,Ln 1gGivensCzn?,l?H?QR).N:70/252JJIIJIBackClose(1)PH1= H.?h(1)216= 0 (K?1e),?Givens?G21=c1s1s1c11.1,:

42、c1=h(1)11r1,s1=h(1)21r1,r1=q(h(1)11)2+ (h(1)21)2.KG21H1=r1h(2)12h(2)13h(2)1n0h(2)22h(2)23h(2)2nh(2)32h(2)33h(2)3n.h(2)n,n1h(2)nn:= H2.71/252JJIIJIBackClose(2)?h(1)326= 0 (K?1e),2?Givens?G32=1c2s2s2c21.1,:c2=h(2)22r2,s2=h(2)32r2,r2=q(h(2)22)2+ (h(2)32)2.KG32H2=r1h(3)12h(3)13h(3)1,n1h(3)1n0r2h(3)23h(3

43、)2,n1h(3)2n0h(3)33h(3)3,n1h(3)3nh(3)43h(3)4,n1h(3)4n.h(3)n,n1h(3)nn:= H3.72/252JJIIJIBackClose(3)b?L?1?k 1,kHk= Gk,k1Hk1=r1h(k)1,k1h(k)1kh(k)1,n1h(k)1n.rk1h(k)k1,kh(k)k1,n1h(k)k1,nh(k)kkh(k)k,n1h(k)knh(k)k+1,kh(k)k+1,n1h(k)k+1,n.h(k)n,n1h(k)nn.?h(k)k+1,k6= 0,?Givens?Gk+1,k=1.1ckskskck1.1,73/252JJIIJ

44、IBackClose:ck=h(k)kkrk,sk=h(k)k+1,krk,rk=q(h(k)kk)2+ (h(k)k+1,k)2.uGk+1,kHk=r1h(k+1)1,kh(k+1)1,k+1h(k+1)1,n1h(k+1)1n.rkh(k+1)k,k+1h(k+1)k,n1h(k+1)knh(k+1)k+1,k+1h(k+1)k+1,n1h(k+1)k+1,nh(k+1)k+2,k+1h(k+1)k+2,n1h(k+1)k+2,n.h(k+1)n,n1h(k+1)nn:= Hk+1.74/252JJIIJIBackClosed,n 1gGivensC,=?Gn,n1G32G21H =r1

45、h(n)12h(n)13h(n)1nr2h(n)23h(n)2nr3h(n)3n.rn= R.Gk,k1(k = 2, ,n)?,?H = GT21GT32GTn,n1R = QR,: Q = GT21GT32GTn,n1E?.?L?$?4n2,?QR)?$O(n3)?.?5?,yfH = RQ = QTHQEHessen-berg?,uUS“e?,?H?qu75/252JJIIJIBackClosen?n?(?1 12 2?),l?H?A?A?A?.Hessenberg?QR)?MATLABSXe:function A=hessen_qrtran(A,m)%?Sn?Hessenberg?A,

46、GivensC%?m?f?1QR),2qC,%?C?Hessenberg?A.Q=eye(m);for i=1:m-1xi=A(i,i); xk=A(i+1,i);if xk=0d=sqrt(xi2+xk2);c=xi/d; s=xk/d;76/252JJIIJIBackCloseG=c, s; -s, c;A(i:i+1,i:m)=G*A(i:i+1,i:m);Q(1:m,i:i+1)=Q(1:m,i:i+1)*G;endendA(1:m,1:m)=A(1:m,1:m)*Q;7.6|SHessenberg?A?1QRC,A =2357842359083620071300069.77/252J

47、JIIJIBackClose) 3MATLAB-I: A=2 3 5 7 8; 4 2 3 5 9; 0 8 3 6 2;.0 0 7 1 3; 0 0 0 6 9; A=hessen_qrtran(A,5)A =4.80005.36827.666410.5793-4.94457.33212.72384.95350.3480-6.832707.22181.17452.7581-3.2311006.02987.7534-4.9785000-1.52830.548378/252JJIIJIBackClose7.4.3?QR?!0?A?QR.-A1= A,A1QR):A1= Q1R1,?-A2= R

48、1Q1,2A2QR):A2= Q2R2,-A3= R2Q2,?e?S?Ak,?)LVXe:A1= A,Ak= QkRk,Ak+1= RkQk,k = 1,2, .(7.49)Ny, Ak+1Akq,?Akk?A?.79/252JJIIJIBackClose3e, Ak?un?(n?).eun?,KTn?A?A?;eun?,K?A?A?A?.du?A?, Ak?,?3QR?SA,kAzq?Hessenberg?,2A?.?OLXe.7.5 (?QR)1,Hessenberg?A Rnn.2,PA1:= A.uk = 1,2, ,k(1) Ak= QkRk(QR).(2) Ak+1= QTkAkQ

49、k= RkQk(?qC).80/252JJIIJIBackClose?QR?MATLABSXe:function iter,D=qr_eig(A,tol,N)%?QRn?A?A?%:A?,tol,NS“g%:iterS“g,DA?A?%N:mhessen.m,hessen_qrtran.m,eig-=u1,2?if nargin3, N=500; endif nargin2, tol=1e-5; endn=size(A,1); D=zeros(n,1);i=n; m=n; iter=0;%zA=mhessen(A);%z?AHessenberg?while (iter=N)%?QR?1S“it

50、er=iter+1;81/252JJIIJIBackCloseif m=2la=eig(A(1:m,1:m); D(1:m)=la;break;end%Hessenberg?QR)?qCA=hessen_qrtarn(A,m);%e?S?for k=m-1:-1:1if abs(A(k+1,k)tolif m-k A=3 2 3 4 5 6 7;11 1 2 3 4 5 6;2 8 9 1 2 3 4; .-4 2 9 11 13 15 8; -1 -2 -3 -1 -1 -1 -1; .83/252JJIIJIBackClose3 2 3 4 13 15 8; -2 -2 -3 -4 -5

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