PDF课件第2章PPT.pdf
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1、1/137JJIIJIBackClose5“1? ?CK2/137JJIIJIBackCloseS?z?(q)C3“kA?/ . QR)u5|AO?K?).?uQR)?QRuO?A?A?.d?, Krylovfm?zu?E5|?S“.K?gK?E5|S“?A?O?:u:.?0?C!QR)!5|?z9Krylovfm?59?z,?0?K?e.3/137JJIIJIBackClose2.1?C2.1.1 HouseholderC2.1?u Rnvkuk2= 1,n?H = I 2uuT(2.1)Householder?(?), uHouseholder.e?n?Householder?5.n2.1
2、?H(2.1)?Householder?,K(1) det(H) = 1.(2) HT= H, HTH = I, H1= H, H2= I.(3) H=kp?A?11,1-?,A?A?u.?1n 1-?,A?A?ku?.4/137JJIIJIBackClose(4)?x,u RnvuTx = 0, R.KH(x + u) = x u.y(1)|?$B/?ydet(H) =1.,I0uT1I 2u01I 2uuT0uT1=I0uT1I2uuT1=I2u0 1 2uTu?1?,?det(H) = 1 2uTu = 1.(2)|H?,N?yo?.(3)duHu = (I 2uuT)u = u,?1H
3、?A?,A-?1, uA?A?.,?,u?5/137JJIIJIBackCloseu?nx,kHx = x,=1H?A?,A-?n1,u?A?A?.duA?A-L“-,?A?11?“-O?1n 1,?n.5?A?“-Ln,?11?“-Of1n 1.(4)?OH(x + u) = (I 2uuT)(x + u) = x u.y.?6/137JJIIJIBackClosee?nHouseholder?,5.n2.2?Hnmn m?Householder?,KH =ImOO Hnmn?Householder?.y ?unmn m ?,KkHnm= Inm 2unmuTnm,7/137JJIIJIBa
4、ckCloseH =ImOO Inm 2unmuTnm=ImOO Inm 2OOO unmuTnm=ImOO Inm 20unmh0 uTnmi= In 2unuTn,: un=0unm Rn.duuTnun= uTnmunm= 1,Hn?Householder?.y.?8/137JJIIJIBackClosen2.3?x = (x1,x2, ,xn)T Rn(n 1)?,K3u Rnvkuk2= 1,?(2.1)?Householder?vHx = e1,(2.2): = kxk2.?(2.2)?u3?e(?.y d2.1, H = I 2uuT,uHx = x 2(uTx)u.(2.2),
5、7Lv2(uTx)u = x e1.dukuk2= 1,?u =x e1kx e1k2.qH?,(2.2),7Lkkxk2= kHxk2= ke1k2= | ke1k2= |,9/137JJIIJIBackClose= = kxk2.N?y,X?Hv(2.2).,kHx = x 2(x e1)(x e1)Tkx e1k22x= x 2(x e1)Txkx e1k22(x e1)= x 2kxk22 2eT1xkxk22 2eT1x + 2(x e1)= x 22 2eT1x2 2eT1x + 2(x e1)= x (x e1) = e1.d?,du?L,?u3?e(?.y.?52.1dn2.3
6、,UXe5?E(H? u :10/137JJIIJIBackClose(1)Ov = x kxk2e1.(2)Ou = v/kvk2.O?9kxk2c?K.XJ?v = x kxk2e1,yOv1= x1 kxk2?K,v1,x1OLv,x?11.3x1 0,UOv1U?k?i?”.3/Ue?O,=v1= x1 kxk2=x21 kxk22x1+ kxk2=(x22+ + x2n)x1+ kxk2,3x1 0O,;C?/.11/137JJIIJIBackClose5?H = I 2uuT= I 2vTvvvT= I vvT,: = 2/(vTv).?vk7w/u,Iv=.3SO, v8z111
7、?(ICv := v/v1=).?:rv?n1?3x?n1z0? ,?v?11I?.3O,y7L?K.3O,XJx,L,$Uy?y.?)K,x/kxk“Ox5?Ev,?u3?5?vc? = 1/kxk,?vv? z?.u?,?Xe?.12/137JJIIJIBackClose2.1?OHouseholder?H = I vvTvv1= 1?v.1,nx.O = kxk,x := x/.2, v := 1,x(2 : n)T,O = x22+ + x2n.3,u = 0,ex1 0, := 0,K := 2,O.4,u 0,O :=px21+ .ex16 0, v1= x1,Kv1= /(x1+
8、 ).5,O := 2v21/( + v21), v := v/v1.12.1?$?4n.?,?MATLABSXe:%HouseholderCS-r_house.mfunction v,beta=r_house(x)%?OHouseholder?H=I-beta*v*v13/137JJIIJIBackClosevv(1)=1?vbeta.n=length(x);eta=norm(x,inf); x=x/eta;sigma=x(2:n)*x(2:n);v=1; x(2:n);if sigma=0if x(1)=0beta=0;elsebeta=2;endelsealpha=(x(1)2+sigm
9、a)0.5;14/137JJIIJIBackCloseif x(1) x=2,5,7,1; v,beta=r_house(x); H=eye(length(x)-beta*v*v; y=H*x=?I?(J.15/137JJIIJIBackClose52.23AHouseholderC?z?,I?/,|A(?-?.?OHouseholder?H = I vvT Rmm?A Rmn?,SOHIw?,?Xe?5O:HA = (I vvT)A = A v(ATv)T= A vwT,: w = ATv.,Ue?OHA :(1)Ow = ATv.(2)OB = A vwT.?B=?HA.?O?I?$4mn
10、.16/137JJIIJIBackClose52.33E/, HouseholderCXe/?H(w) = I 2wwH,(2.3): w CnvwHw = 1.AuHouseholder?,EHouseholder?j?.,EHouseholderC?HouseholderCA?(1) (4)5?,kXe5:(5)?x,y Cn, x 6= y.K3HouseholderCH(w)?H(w)x = y?7xHx = yHy,xHy = yHx.3,I?w?w = eix ykx yk2, R.17/137JJIIJIBackClosed52.3,e?x = (x1,x2, ,xn)T6= 0
11、, y = kxk2e1, =1,?x1= 0,x1|x1|,?x16= 0,(2.4)KvxHx = yHy, xHy = yHx.dH(w)x = kxk2e1,: w?w =x + e1kx + e1k2, = kxk2.?x,?H(w)?.u?,ke?.2.2x Cn,?Ow,?Hx = e1,H = I wwH, kwk2=2 .18/137JJIIJIBackClosefunction w, = c house(x)w = x; = kxk2;if = 0w(1) =2;(endif w(1) 6= 0 = w(1)/|w(1)|;else = 1;endw = (/)w;w(1)
12、 = w(1) + 1;w = w/pw(1); = ;19/137JJIIJIBackClose52.42.2aquMATLAB?,?N/.d,GMATLAB-?.,aquMATLAB?5 i?.52.52.1?HouseholderC?,?2.2QUuqUuE.:-?,NSK,X?6|?O,l/.DkA(?E5|.dKrylovfm(XGMRES),I?E?HouseholderC.2.2?MATLABSXe:%E?HouseholderCS-c_house.mfunction w,gamma=c_house(x)20/137JJIIJIBackClose%Ex,?Owgamma%vH*x
13、=gamma*e1,H=I-w*w, |w|=sqrt(2).w=x; gamma=norm(x);if gamma=0w(1)=sqrt(2); return;endif w(1)=0tau=1;elsetau=conj(w(1)/abs(w(1);endw=(tau/gamma)*w; w(1)=w(1)+1;w=w/sqrt(w(1); gamma=-conj(tau)*gamma;21/137JJIIJIBackClose2.2Ex = (2 + i,5 3i,7,1 + 2i)T,?EHouse-holder?H?Hx = kxk2e1.) 3MATLAB-Ig$1Xe“: x=2+
14、i,5-3*i,7,1+2*i; w,gama=c_house(x); H=eye(length(x)-w*w; y=H*x=?I?(J.2.1.2 GivensCHouseholderCeZ?z0.?XJ?,z0,KGivensCk?. GivensC(?)Xe.22/137JJIIJIBackClose2.2?csvc2+ s2= 1,n?Gik(c,s) =1.cs.sc.1(i)(k)(i 6= k)(2.5)GivensC(?),?=?.Ny, Givens?kXe5.n2.4?Gd(2.5)?Givens?.K(1) GTG = I,=Gik(c,s)?.(2) det(G) =
15、1.(3) Gik(c,s)1= Gik(c,s)T= Gik(c,s).(4)u?x Rn, GivensCy = Gik(c,s)xUCx?1i, k.23/137JJIIJIBackClosey(1) (3)N?y.y(4).?x =(x1,x2, ,xn)T, y = (y1,y2, ,yn)T,Kdy = Gik(c,s)x,?yi= cxi+ sxk,yk= sxi+ cxk,yj= xj,(j 6= i,k).y.?dn2.4?(4)Juy,?x2i+ x2k6= 0,?c =xipx2i+ x2k,s =xkpx2i+ x2k,(2.6)yi=px2i+ x2k 0, yk=
16、0.dd,x1i1k,Ue?Ocs?Gik(c,s)x?1k0.24/137JJIIJIBackClose2.3 (OGivensC)ab,?Onc,s,?css cab=0.function c,s, = r givens(a,b)if b = 0c = 1; s = 0; = a;(endif a = 0c = 0; s = 1; = b;(endif |b| |a|t = a/b; s = 1/1 + t2; c = st; = |b|/s;25/137JJIIJIBackCloseelset = b/a; c = 1/1 + t2; s = ct; = |a|/c;end12.3I5g
17、!1g1gm$.XJO,K?g$.?,?MATLABSXe:%GivenCS-r_givens.mfunction c,s,eta=r_givens(a,b)%Oc,svc s;-s c*a;b=eta;0if b=0, c=1; s=0; eta=a; endif a=0, c=0; s=1; eta=b; endif abs(b)abs(a)t=a/b; s=1/sqrt(1+t2); c=s*t;eta=abs(b)/s;26/137JJIIJIBackCloseelset=b/a; c=1/sqrt(1+t2); s=c*t;eta=abs(a)/c;enden?(LGivensCkH
18、ouseholderC?U.n2.5?x = (x1,x2, ,xn)T6= 0,K3kGivens?G,?Gx = kxk2e1.y(1)?x16= 0,g?EG12: c =x1px21+ x22, s =x2px21+ x22,=G12x =?qx21+ x22,0,x3, ,xn?T.27/137JJIIJIBackCloseG13: c =px21+ x22px21+ x22+ x23, s =x3px21+ x22+ x23,=G13(G12x) =?qx21+ x22+ x23,0,0,x4, ,xn?T.G1n: c =qx21+ + x2n1px21+ + x2n, s =x
19、npx21+ + x2n,=G1n(G1,n1G13G12x) =?qx21+ + x2n,0, ,0?T.-G = G1nG1,n1G13G12,KkGx = kxk2e1.(2)?x1= = xi1= 0, xi6= 0 (1 n)?,ek?Q Rmmn?R Rmn?A = QR,KQRA?QR).2.2.1 HouseholderCQR)?!0?XHouseholderC?QR).34/137JJIIJIBackClosen2.6?A Rmn(m n)?,K3kHouseholder?Q,?A = QeR = QRO,(2.9): R Rnn?n?.y ?A1:= A.A = (aij)
20、 Rmn?,?A?11?a16= 0.u?EHouseholder?H1 Rmm,?H1a1= 1e1,p1= ka1k2, e1 Rm.KkA2= H1A1=10 A(2)22.A(2)22 R(m1)(n1)?.A(2)2211?m 1a2?Em 1?Householder?fH2,?fH2a2= 2e1,35/137JJIIJIBackClosep2= ka2k2, e1 Rm1.-H2= diag(1,fH2),KkA3= H2A2=1 02 00 .00 =A(3)11A(3)12OA(3)222m 2,2n 2: A(3)11n?.-EL,b?1?k 1,?HouseholderC
21、H1,H2,Hk1,?Ak= Hk1H2H1A1=A(k)11A(k)12OA(k)22k 1m k + 1,k 1n k + 1: A(k)11n?.36/137JJIIJIBackClosebA(k)22=?ak,ak+1, ,an?.1kkak,?EfHk= Imk+1 kvkvTk R(mk+1)(mk+1),?fHkak= ke1,pk= kakk2, e1 Rmk+1.-Hk=diag(Ik1,fHk),KAk+1= HkAk=A(k)11A(k)12OfHkA(k)22=A(k+1)11A(k+1)12OA(k+1)22km k,kn k: A(k+1)11n?.?,lk = 1
22、u,Ag?1ngHouseholderC,A?zn?eR,=37/137JJIIJIBackCloseeR = An+1= HnAn=A(n+1)11Onm n,: A(n+1)11n?.yPR = A(n+1)11, QT= HnHn1H1,KA = QRO,: R Rnnn?.e-Q =?Q1,Q2?,pQ1 Rmn,KkA = Q1R.y.?38/137JJIIJIBackClose52.7eA Rnn?,Kdn2.6, AkQR)A = QR,: Q Rnn?; R Rnn?n?.eOA?QR)?;K.5,3?QR)?A2I,5QR.7Qw/,?nHouseholder?Hk(k =1
23、,2, ,n),?zHk,I?vkk=.5?vkkXe/:vk=?1,v(k)k+1, ,v(k)n?T Rmk+1,?vk?12?mk +1,=vk(2 : mk +1)3A?1k?e? ,?A?n?5R?n?.n?,?Xe.39/137JJIIJIBackClose2.5 (HouseholderCQR)for k = 1 : nif k n),OHouseholder?H1, H2, , Hnv:XJQ = H1H2Hn,KQTA = Rn?/?,A?n?R?n?CX,1kHouseholder?1k + 1?1muA(k + 1 : m,k), k m? .NO,T?$2n2(m n/
24、3)flop.40/137JJIIJIBackClose?MATLABSXe:function A,d=house_qr(A)%HouseholderCQR),wO;?Qm,n=size(A);for k=1:nif kmv,beta=r_house(A(k:m,k);I=eye(m-k+1);A(k:m,k:n)=(I-beta*v*v)*A(k:m,k:n);d(k)=beta;A(k+1:m,k)=v(2:m-k+1);endend41/137JJIIJIBackClose2.3|2.5?A?1QR),A =76 96 85 3526 55 26 2051 14 82 2670 15 2
25、5 6290 26 93 48.) du2.5vkwO;?Q,?IShouse qr.m?U:function Q,R=Qhouse_qr(A)%Householder QR),wO;?Qm,n=size(A); Q=eye(m);for k=1:nif k A=76,96,85,35;26,55,26,20; .51,14,82,26;70,15,25,62;90,26,93,48; Q,R=Qhouse_qr(A)=?A?QR),vkA QRk2= 4.3169 1014.43/137JJIIJIBackClose2.2.2 GivensCQR)GivensCy?A?QR).kA?/,ke
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