,,,
蠠
:
2002
............................................................................................................................................................................. 3
1. .................................................................................................. 7
1.1. ......................................................................................................... 7
1.2. ................................................................................................... 9
1.3. .......................................................................... 10
1.4. ................................................................................................... 11
1.5. QR........................................................................................................................................ 15
1.6. ....................................................................................................................... 20
2. .......................................................................... 25
2.1. .................................................................................................................................................. 25
2.2. ....................................................................................................................... 27
2.3. .................................................................................................. 29
3. ⠠ 33
................................................................................................................................................................... 38
..................................................................................................................................................................... 39
1. ............................................................................... 40
2. ..................................................................................................... 45
. , . : , ..
, .
m´n A k£min(m,n) m b. n x0, Axb.
y n , x n , b y n j:
.
, b, e=yXb, X , n , xj: xj. , . . :
. . .
, =0
.
, , . . , , ( ) , . .
. (. 1): y=0 x=0; y=1 x=1; y=2 x=3; y=5 x=4. , , . , . :
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b=(XTX)-1XTy
, ( ) , . , . , , . t. , , , . D, , :
. (1)
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. 1. .
(1). , (1) :
, m>2, C D . .
, C D l m. , . , l m, .
1.|| Ax-b ||.
1. m´n k,
A=HRKT (2)
H m´m ; R m´n
, (3)
: R11 kxk k; K kxk.
(4)
. (5)
R11y1=g1. :
1. ||Ax-b|| , y2 .
2. . (6)
3. r
4.
. A HRKT (2) HT ( )
(7)
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(4)
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(3). , y2=0. , . .
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(8)
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s = +1, = 1, .
. .
, :
X Nxp, N>p, r X. r=p, , r<p.
,
(10)
V Nxr, , .. ; U pxr; , ; rxr, , X, . , V, U, X, (10), :
(11)
.
XX' NxN, r Nr . XX' , . , XX', V .
X'X pxp, r pr . X'X , , , X'X, U .
X'X XX' . , um X'X, vm XX', , um vm
(12)
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. (13)
X'X R-, XX' Q. (12)(13) , Q R .
.
1. X'X XX', , . , X'X.
2. X'X .
3. .
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(11) . (11) , . .
1. S, pxp r . , pxp, k, k<r, S .
2. X, Nxp r k<r. W pxp k, X .
:
(14)
k . T W S X . : T S , ST . W X , XW. , .. rk XX.
(15)
.
p=r.
S p . T p , T k<p, p k . , p , S, k .
X Nxp X=VU, V Nxp c ; pxp, U pxp.
W
k V, k , k U. WX,
(16)
(17)
pxk X p k ; (16) , X Nxp Nxk. X N p , k . k .
2. m´n. m´m Q , QA=R .
. m´m Q (9), , Q1A 2 m. (m-1)´(m1) P2 . m1 , 2 m Q1A, 3 m .
, Q2Q1A . , , n , . .
QR.
3. m´n , k<n£m. m´m Q m´n P ,
, (18)
R ´ .
. , AP, . 2, m´m Q , QAP . , QAP.
QAP, +1,...,m +1,...,n, . rankQAP>k, rankA=k. , QAP , (4). .
[R:T] (18) , R 2. .
1. [R:T] ´, R . n´n W ,
.
3, n, k, [R:T], W m, n, AT, QT 3.
3 1 .
4. m´n . m´m n´n ,
(19)
R11 ´.
, (19) , R11 .
(19) A=HRKT, R , . , R . ATA AAT (. 11).
5. m´n k. m´m U, n´n V m´n S ,
UTAV=S, A=USVT (20)
S , ; .
S . m=n=rankA.
2. n´n n. n´n U, n´n V n´n S , UTAV=S, A=USVT S .
. ATA
ATA=VDVT, (21)
V n´n, D , D . S n´n, D.
D=STS=S2, S-1DS-1=I. (22)
U=AVS-1 (23)
(21), (22), (23) V ,
UTU=S-1VTATAVS-1=S-1DS-1=I .. U . (23) V USVT=AVS-1SVT=AVVT=A .
5. A=HRKT, H, R, KT , 4. R11 (19) ´, 2 ,
(24)
´, , . (24) , R (19)
(25)
:
m´m;
n´n;
m´n;
, U V
(26)
(24) (26), A=USVT, U, S, V , 5. .
, , , U, V . s , l. , I ,
k=min(m,n), Q ´
l´l A=USVT si==si+l-1, , .
. .
:
: (x-2)2=10-6
2+10-3 2-10-3. 10-6 , 10-3.
, , . . , . , . , . .
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1)
2) ;
3) .
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1) ;
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3)
4)
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(7)
[6] .
, , . . , 1/ , . , . (7) . .
Ax=b, , : A(x+Dx)=b+Db . Db b, Dx x, . A(Dx)=Db, M m , m¹0,
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2.QR - , : , , :
QR . n3. , [8] . ( ) , . , , [9]. n2, n.
Qs , Ls . QL.
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, As s- , kij . QL . , As As-ksI (QL ). :
. , , . ks ( ), . , , n-1- . , QL , , ks.
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,
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:
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2, . 0.000, , . , , :
, . , , . H3 , . H3 1.080 A4.
, . . n n H n2 Q, . m, m , .
3.(SVD Singular Value Decomposition) . y=Xb y=USVTb. , b UTy=SVTb. .. sj, j=1,,n, S ,
, .
, sj . SVD t . sj, , t, , (1.20). sj, , t, , . , . , , t, , . , , =0, sj £t.
sj, , t, . , .
:
t | Y |
1900 | 75994575 |
1910 | 91972266 |
1920 | 105710620 |
1930 | 123203000 |
1940 | 131669275 |
1950 | 150697361 |
1960 | 179323175 |
1970 | 203211926 |
Y X =1980.
, . . . , , , Y=227780000 , Y=145210000. , . . 8 ´ 3
. 2.
: .
, 1, t t2 . , .
, , . , . :
. , , , , , . Y(1980) 212910000 214960000. , .
, . : . t 1900 1970 0.0017, a , , 0.0017a. .
, .
. , [1900, 1970] [0, 70] , , [3.5, 3.5]. 5750 10.7 . .
, .. 1. , , 0, .
, , , , .
, , . 2 , . , .
. FORTRAN. , .
, .
1. . . -.: , 1969, 368.
2. .. . -.: , 1988, 548.
3. . . -.: , 1982, 387.
4. ., . . .: , 1979, 447
5. .. . .: , 1980
6. . . .: , 1988, 350
7. . . .: , 1980, 454
8. ., . . , .: , 1976, 390
9. .., .. . -.: , 1963, 536.
10. ., ., . . .: , 1980, 279
11. .. . .: - , 1995, 76 .
1.REAL A(3,3), U(3,3), V(3,3), SIGMA(3), WORK(3),Y(3),C(3),Y0(3)
INTEGER I,IERR, J, M, N, NM
OPEN (6,FILE="SVD.OUT",STATUS="UNKNOWN",FORM="FORMATTED")
OPEN (5,FILE= "SVD.IN",STATUS="UNKNOWN",FORM="FORMATTED")
140 FORMAT(3I5)
150 FORMAT(4E15.7)
READ(5,140) NM,M,N
DO 131 I=1,M
READ(5,150) (A(I,J),J=1,N)
131 CONTINUE
READ (5,150) (Y(I),I=1,M)
CALL SVD(NM,M,N,A,SIGMA,.TRUE.,U,.TRUE.,V,IERR,WORK)
IF(IERR.NE.0) WRITE (6,2) IERR
2 FORMAT(15H TROUBLE.IERR=,I4)
WRITE(6,120)
120 FORMAT(/' ')
DO 121 I=1,M
WRITE(6,130) (A(I,J),J=1,N)
130 FORMAT(8E15.7)
121 CONTINUE
WRITE (6,160) (Y(I),I=1,N)
160 FORMAT(/' '/8E15.7)
210 FORMAT(/' ')
WRITE(6,210)
DO 3 J=1,N
WRITE(6,6) SIGMA(J)
3 CONTINUE
SMA=SIGMA(1)
SMI=SIGMA(1)
DO 211 J=2,N
IF(SIGMA(J).GT.SMA) SMA=SIGMA(J)
IF(SIGMA(J).LT.SMI.AND.SIGMA(J).GT.0.) SMI=SIGMA(J)
211 CONTINUE
OBU=SMA/SMI
230 FORMAT(/' =',E15.7)
WRITE(6,230) OBU
SIGMA1=0.
DO 30 J=1,N
IF(SIGMA(J) .GT. SIGMA1) SIGMA1=SIGMA(J)
C(J)=0.
30 CONTINUE
TAU=SIGMA1*0.1E-6
DO 60 J=1,N
IF(SIGMA(J).LE.TAU) GO TO 60
S=0.
DO 40 I=1,N
S=S+U(I,J)*Y(I)
40 CONTINUE
S=S/SIGMA(J)
DO 50 I=1,N
C(I)=C(I) + S*V(I,J)
50 CONTINUE
60 CONTINUE
write (6,560)
WRITE (6,6) (C(I),I=1,3)
DO 322 J=1,N
SS=0.
DO 321 I=1,M
321 SS=A(J,I)*C(I)+SS
322 Y0(J)=SS
write (6,570)
WRITE (6,6) (Y0(I),I=1,3)
C WRITE(6,7)
C DO 4 I=1,M
C WRITE(6,6) (U(I,J),J=1,N)
C4 CONTINUE
C WRITE(6,7)
C DO 5 I=1,N
C WRITE(6,6) (V(I,J),J=1,N)
C5 CONTINUE
6 FORMAT(3E15.7)
560 format(2x,'roots')
570 format(2x,'right')
7 FORMAT(1H )
STOP
E N D
SUBROUTINE SVD(NM,M,N,A,W,MATU,U,MATV,V,IERR,RV1)
REAL A(NM,N),W(N),U(NM,N),V(NM,N),RV1(N)
LOGICAL MATU,MATV
IERR=0
DO 100 I=1,M
DO 100 J=1,N
U(I,J)=A(I,J)
100 CONTINUE
G=0.0
SCALE=0.0
ANORM=0.0
DO 300 I=1,N
L=I+1
RV1(I)=SCALE*G
G=0.0
S=0.0
SCALE=0.0
IF(I.GT.M) GO TO 210
DO 120 K=I,M
120 SCALE=SCALE+ABS(U(K,I))
IF(SCALE.EQ.0.0) GO TO 210
DO 130 K=I,M
U(K,I)=U(K,I)/SCALE
S=S+U(K,I)**2
130 CONTINUE
F=U(I,I)
G=-SIGN(SQRT(S),F)
H=F*G-S
U(I,I)=F-G
IF(I.EQ.N) GO TO 190
DO 150 J=L,N
S=0.0
DO 140 K=I,M
140 S=S+U(K,I)*U(K,J)
F=S/H
DO 150 K=I,M
U(K,J)=U(K,J)+F*U(K,I)
150 CONTINUE
190 DO 200 K=I,M
200 U(K,I)=SCALE*U(K,I)
210 W(I)=SCALE*G
G=0.0
S=0.0
SCALE=0.0
IF(I.GT.M.OR.I.EQ.N) GO TO 290
DO 220 K=L,N
220 SCALE=SCALE+ABS(U(I,K))
IF(SCALE.EQ.0.0) GO TO 290
DO 230 K=L,N
U(I,K)=U(I,K)/SCALE
S=S+U(I,K)**2
230 CONTINUE
F=U(I,L)
G=-SIGN(SQRT(S),F)
H=F*G-S
U(I,L)=F-G
DO 240 K=L,N
240 RV1(K)=U(I,K)/H
IF(I.EQ.M) GO TO 270
DO 260 J=L,M
S=0.0
DO 250 K=L,N
250 S=S+U(J,K)*U(I,K)
DO 260 K=L,N
U(J,K)=U(J,K)+S*RV1(K)
260 CONTINUE
270 DO 280 K=L,N
280 U(I,K)=SCALE*U(I,K)
290 ANORM=AMAX1(ANORM,ABS(W(I))+ABS(RV1(I)))
300 CONTINUE
IF(.NOT.MATV) GO TO 410
DO 400 II=1,N
I=N+1-II
IF(I.EQ.N) GO TO 390
IF(G.EQ.0.0) GO TO 360
DO 320 J=L,N
320 V(J,I)=(U(I,J)/U(I,L))/G
DO 350 J=L,N
S=0.0
DO 340 K=L,N
340 S=S+U(I,K)*V(K,J)
DO 350 K=L,N
V(K,J)=V(K,J)+S*V(K,I)
350 CONTINUE
360 DO 380 J=L,N
V(I,J)=0.0
V(J,I)=0.0
380 CONTINUE
390 V(I,I)=1.0
G=RV1(I)
L=I
400 CONTINUE
410 IF(.NOT.MATU) GO TO 510
MN=N
IF(M.LT.N) MN=M
DO 500 II=1,MN
I=MN+1-II
L=I+1
G=W(I)
IF(I.EQ.N) GO TO 430
DO 420 J=L,N
420 U(I,J)=0.0
430 IF(G.EQ.0.0) GO TO 475
IF(I.EQ.MN) GO TO 460
DO 450 J=L,N
S=0.0
DO 440 K=L,M
440 S=S+U(K,I)*U(K,J)
F=(S/U(I,I))/G
DO 450 K=I,M
U(K,J)=U(K,J)+F*U(K,I)
450 CONTINUE
460 DO 470 J=I,M
470 U(J,I)=U(J,I)/G
GO TO 490
475 DO 480 J=I,M
480 U(J,I)=0.0
490 U(I,I)=U(I,I)+1.0
500 CONTINUE
510 DO 700 KK=1,N
K1=N-KK
K=K1+1
ITS=0
520 DO 530 LL=1,K
L1=K-LL
L=L1+1
IF(ABS(RV1(L))+ANORM.EQ.ANORM) GO TO 565
IF(ABS(W(L1))+ANORM.EQ.ANORM) GO TO 540
530 CONTINUE
540 C=0.0
S=1.0
DO 560 I=L,K
F=S*RV1(I)
RV1(I)=C*RV1(I)
IF(ABS(F)+ANORM.EQ.ANORM) GO TO 565
G=W(I)
H=SQRT(F*F+G*G)
W(I)=H
C=G/H
S=-F/H
IF(.NOT.MATU) GO TO 560
DO 550 J=1,M
Y=U(J,L1)
Z=U(J,I)
U(J,L1)=Y*C+Z*S
U(J,I)=-Y*S+Z*C
550 CONTINUE
560 CONTINUE
565 Z=W(K)
IF(L.EQ.K) GO TO 650
IF(ITS.EQ.30) GO TO 1000
ITS=ITS+1
X=W(L)
Y=W(K1)
G=RV1(K1)
H=RV1(K)
F=((Y-Z)*(Y+Z)+(G-H)*(G+H))/(2.0*H*Y)
G=SQRT(F*F+1.0)
F=((X-Z)*(X+Z)+H*(Y/(F+SIGN(G,F))-H))/X
C=1.0
S=1.0
DO 600 I1=L,K1
I=I1+1
G=RV1(I)
Y=W(I)
H=S*G
G=C*G
Z=SQRT(F*F+H*H)
RV1(I1)=Z
C=F/Z
S=H/Z
F=X*C+G*S
G=-X*S+G*C
H=Y*S
Y=Y*C
IF(.NOT.MATV) GO TO 575
DO 570 J=1,N
X=V(J,I1)
Z=V(J,I)
V(J,I1)=X*C+Z*S
V(J,I)=-X*S+Z*C
570 CONTINUE
575 Z=SQRT(F*F+H*H)
W(I1)=Z
IF(Z.EQ.0.0) GO TO 580
C=F/Z
S=H/Z
580 F=C*G+S*Y
X=-S*G+C*Y
IF(.NOT.MATU) GO TO 600
DO 590 J=1,M
Y=U(J,I1)
Z=U(J,I)
U(J,I1)=Y*C+Z*S
U(J,I)=-Y*S+Z*C
590 CONTINUE
600 CONTINUE
RV1(L)=0.0
RV1(K)=F
W(K)=X
GO TO 520
650 IF(Z.GE.0.0) GO TO 700
W(K)=-Z
IF(.NOT.MATV) GO TO 700
DO 690 J=1,N
690 V(J,K)=-V(J,K)
700 CONTINUE
GO TO 1001
1000 IERR=K
1001 RETURN
E N D
2.( )
3 3 3
.3200000E 02 .1400000E 02 .7400000E 02
-0.2400000E 02 -0.1000000E 02 -0.5700000E 02
-0.8000000E 01 -0.4000000E 01 -0.1700000E 02
-0.1400000E 02 0.1300000E 02 0.1000000E 01
.3200000E+02 .1400000E+02 .7400000E+02
-.2400000E+02 -.1000000E+02 -.5700000E+02
-.8000000E+01 -.4000000E+01 -.1700000E+02
-.1400000E+02 .1300000E+02 .1000000E+01
.1048255E+03
.7310871E-06
.1271749E+01
= .1433830E+09
.1215394E+01 .1821742E+01 -.1059419E+01
-.1400000E+02 .1300000E+02 .1000001E+01
, . .
[1] .
[2] , .
[3] m´n , U V , S . S (, i=1,...,k, k=min(m,n)) . . .
[4] , . P , .
[5] , . P , . mxn . , rankA=n.
[6] , .
[7] - , . .
[8] ( ) j<i1 ( ). .
[9] |i-j|>1. .
Copyright (c) 2025 Stud-Baza.ru , , , .