PDF课件第7章PPT.pdf
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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?
10、x = (x1,x2, ,xn)T.?i0?|xi0| = max16i6n|xi|,KdAx = x?14/252JJIIJIBackClosenXj=1ai0jxj= xi0= ( ai0i0)xi0=nXj=1,j6=i0ai0jxj= | ai0i0| =?nXj=1,j6=i0ai0jxjxi0?6 Ri0,= Gi0nSi=1Gi.y.?n7.5|?CX?A?,e0?,?|A/CX?A?n,?wc?2.n7.6?A Cnn(n 1)?A?,K u,ij=?z |z C,|z aii|z ajj| RiRj, i 6= j; i,j = 1,2, ,n?,ij(i 6= j)A?Ca
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)?=?
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