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|
C
C ________________________________________________________
C | |
C | MINIMIZE A FUNCTION USING THE FLETCHER-REEVES FORM |
C | OF THE CONJUGATE GRADIENT METHOD |
C | WITH (OR WITHOUT) PRECONDITIONING |
C | |
C | INPUT: |
C | |
C | X --ARRAY CONTAINING STARTING GUESS |
C | |
C | STEP --STARTING GUESS FOR MINIMIZER IN DIREC- |
C | TION OF NEGATIVE GRADIENT DURING FIRST |
C | ITERATION (E. G. STEP=1) WHEN STEP=0, |
C | THE PROGRAM SELECTS A STARTING GUESS |
C | |
C | T --COMPUTING TOLERANCE (ITERATIONS STOP |
C | WHEN MAX-NORM OF GRADIENT .LE. T) |
C | |
C | LIMIT --MAXIMUM NUMBER OF ITERATIONS |
C | |
C | N --NUMBER OF UNKNOWNS |
C | |
C | M --NUMBER OF ITERATIONS UNTIL THE SEARCH |
C | DIRECTIONS ARE RENORMALIZED ALONG THE |
C | NEGATIVE GRADIENT (TYPICALLY, M = N) |
C | |
C | VALUE --NAME OF COST EVALUATION FUNC. ROUTINE |
C | (EXTERNAL IN MAIN PROGRAM) |
C | VALUE(X) IS VALUE OF COST AT X |
C | |
C | GRAD --NAME OF GRADIENT EVALUATION SUBROUTINE |
C | (EXTERNAL IN MAIN PROGRAM) |
C | GRAD(G,X) PUTS IN G THE GRADIENT AT X |
C | |
C | BOTH --NAME SUBROUTINE TO EVALUATE BOTH COST |
C | AND ITS GRADIENT (EXTERNAL IN MAIN |
C | PROGRAM) BOTH(V,G,X) PUTS THE VALUE IN |
C | V AND THE GRADIENT IN G FOR THE POINT X|
C | |
C | PRE --NAME OF PRECONDITIONING SUBROUTINE |
C | (EXTERNAL IN MAIN PROGRAM) |
C | PRE(Y,Z) APPLIES THE PRECONDITIONER TO |
C | Z, STORING THE RESULT IN Y. |
C | IF PRECONDITIONING NOT USED SET Y = Z |
C | |
C | H --WORK ARRAY (LENGTH AT LEAST 3N) |
C | |
C | OUTPUT: |
C | |
C | X --MINIMIZER |
C | |
C | E --MAX-NORM OF GRADIENT |
C | |
C | IT --NUMBER OF ITERATIONS PERFORMED |
C | |
C | STEP --STEP SIZE ALONG SEARCH DIRECTION FOR |
C | FINAL ITERATION |
C | |
C | BUILTIN FUNCTIONS: DABS,DEXP,IDINT,DLOG,DSQRT,DMAX1,|
C | DMIN1,DSIGN |
C | PACKAGE ROUTINES: CUB,FD,FV,FVD,INS |
C |________________________________________________________|
C
SUBROUTINE CG(X,E,IT,STEP,T,LIMIT,N,M,VALUE,GRAD,BOTH,PRE,H)
INTEGER I,IT,J,K,L,LIMIT,M,N,NA,NB,NC,ND
REAL*8 H(N,1),X(1),Y(50),Z(50),A1,A2,A3,A4,A5,A6,A7,A8,A,B,C,C0,C1
REAL*8 D,D0,DA,DB,E,F,F0,F1,FA,FB,FC,G,L3,P,Q,R,S,STEP,T,V,W
REAL*8 FV,FD,VALUE
EXTERNAL BOTH,GRAD,PRE,VALUE
DATA A1/.1D0/,A2/.9D0/,A3/5.D0/,A4/.2D0/,A5/10.D0/,A6/.9D0/
DATA A7/.3D0/
A8 = A3 + .01D0
IT = 0
CALL BOTH(F,H(1,3),X)
E = 0.
DO 10 I = 1,N
10 IF ( DABS(H(I,3)) .GT. E ) E = DABS(H(I,3))
IF ( E .LE. T ) RETURN
L3 = 1./DLOG(A3)
CALL PRE(H(1,2),H(1,3))
A = STEP
IF ( A .GT. 0. ) GOTO 30
DO 20 I = 1,N
20 IF ( DABS(X(I)) .GT. A ) A = DABS(X(I))
A = .01*A/E
IF ( A .EQ. 0. ) A = 1.
30 G = 0.
DO 40 I = 1,N
40 G = G + H(I,2)*H(I,3)
IF ( G .LT. 0. ) GOTO 620
50 L = 0
DO 60 I = 1,N
60 H(I,1) = -H(I,2)
D = -G
70 FA = FV(A,X,H,N,VALUE)
C0 = A
F0 = FA
J = 2
Y(1) = 0.
Z(1) = F
Y(2) = A
Z(2) = FA
V = A1*D
W = A2*D
IQ = 0
IF ( FA .LE. F ) GOTO 80
C = A
B = 0.
A = 0.
FC = FA
FB = F
FA = F
GOTO 90
80 C = 0.
B = 0.
FC = F
FB = F
IQ = 1
90 NA = 0
NB = 0
NC = 0
ND = 0
Q = (D+(F-F0)/C0)/C0
IF ( Q .LT. 0. ) GOTO 110
Q = A
100 ND = ND + 1
IF ( ND .GT. 25 ) GOTO 610
Q = A3*Q
P = FV(Q,X,H,N,VALUE)
CALL INS(Q,P,A,B,C,FA,FB,FC,J,Y,Z)
IF ( P-F .LT. W*Q ) GOTO 100
GOTO 260
110 Q = .5*D/Q
IF ( Q .LT. .01*C0 ) Q = .01*C0
P = FV(Q,X,H,N,VALUE)
IF ( P .LE. F0 ) GOTO 120
F1 = F0
C1 = C0
F0 = P
C0 = Q
GOTO 130
120 F1 = P
C1 = Q
130 CALL INS(Q,P,A,B,C,FA,FB,FC,J,Y,Z)
135 IF ( A .EQ. 0. ) GOTO 140
IF ( FA-F .GE. V*A ) GOTO 160
IF ( FA-F .LT. W*A ) GOTO 210
GOTO 280
140 Q = C0
IF ( C1 .LT. Q ) Q = C1
150 NA = NA + 1
IF ( NA .GT. 25 ) GOTO 630
Q = A4*Q
P = FV(Q,X,H,N,VALUE)
CALL INS(Q,P,A,B,C,FA,FB,FC,J,Y,Z)
IF ( P-F .GE. V*Q ) GOTO 150
GOTO 250
160 IF ( C0 .GT. C1 ) GOTO 200
IF ( F0-F .GT. V*C0 ) GOTO 180
IF ( F0-F .GE. W*C0 ) GOTO 320
IF ( C1 .LE. A5*C0 ) GOTO 320
R = DLOG(C1/C0)
S = -IDINT(R*L3+.999)
R = .999*DEXP(R/S)
Q = C1
170 Q = Q*R
IF ( Q .LT. C0 ) GOTO 320
P = FV(Q,X,H,N,VALUE)
CALL INS(Q,P,A,B,C,FA,FB,FC,J,Y,Z)
NA = NA + 1
IF ( P-F .GT. V*Q ) GOTO 170
GOTO 320
180 Q = C0
190 NA = NA + 1
IF ( NA .GT. 25 ) GOTO 630
Q = A4*Q
P = FV(Q,X,H,N,VALUE)
CALL INS(Q,P,A,B,C,FA,FB,FC,J,Y,Z)
IF ( P-F .GE. V*Q ) GOTO 190
GOTO 250
200 Q = A
GOTO 190
210 IF ( C0 .LT. C1 ) GOTO 290
IF ( F0-F .GE. V*C0 ) GOTO 230
IF ( F0-F .GE. W*C0 ) GOTO 250
Q = C0
220 ND = ND + 1
IF ( ND .GT. 25 ) GOTO 610
Q = A3*Q
P = FV(Q,X,H,N,VALUE)
CALL INS(Q,P,A,B,C,FA,FB,FC,J,Y,Z)
IF ( P-F .LT. W*Q ) GOTO 220
GOTO 250
230 IF ( C0 .LE. A5*C1 ) GOTO 250
R = DLOG(C0/C1)
S = IDINT(R*L3+.999)
R = 1.001*DEXP(R/S)
Q = A
240 Q = Q*R
IF ( Q .GT. C0 ) GOTO 250
ND = ND + 1
P = FV(Q,X,H,N,VALUE)
CALL INS(Q,P,A,B,C,FA,FB,FC,J,Y,Z)
IF ( P-F .LT. W*Q ) GOTO 240
250 IF ( IQ .EQ. 1 ) GOTO 320
260 IF ( B .EQ. 0. ) GOTO 280
IF ( C .EQ. 0. ) GOTO 270
V = C - A
W = A - B
R = 1./V
S = 1./W
P = FC - FA
Q = FB - FA
E = P*R + Q*S
IF ( DSIGN(E,C-B) .NE. E ) GOTO 320
IF ( E .EQ. 0. ) GOTO 320
Q = (P*R)*W - (Q*S)*V
Q = A - .5*Q/E
P = FV(Q,X,H,N,VALUE)
CALL INS(Q,P,A,B,C,FA,FB,FC,J,Y,Z)
GOTO 320
270 R = 1./A
S = 1./B
P = R*(FA-F) - D
Q = S*(FB-F) - D
E = A - B
V = (R*P-S*Q)/E
W = (A*Q*S-B*P*R)/E
V = W*W-3.*V*D
IF ( V .LT. 0. ) V = 0.
V = DSQRT(V)
IF ( W+V .EQ. 0. ) GOTO 320
Q = -D/(W+V)
IF ( Q .LE. 0. ) GOTO 320
P = FV(Q,X,H,N,VALUE)
CALL INS(Q,P,A,B,C,FA,FB,FC,J,Y,Z)
GOTO 320
280 IF ( IQ .EQ. 1 ) GOTO 320
Q = (D+(F-FA)/A)/A
IF ( Q .GE. 0. ) GOTO 320
Q = .5*D/Q
P = FV(Q,X,H,N,VALUE)
CALL INS(Q,P,A,B,C,FA,FB,FC,J,Y,Z)
GOTO 320
290 IF ( F0-F .GT. V*C0 ) GOTO 300
IF ( F0-F .GT. W*C0 ) GOTO 320
300 Q = A
310 ND = ND + 1
IF ( ND .GT. 25 ) GOTO 610
Q = A3*Q
P = FV(Q,X,H,N,VALUE)
CALL INS(Q,P,A,B,C,FA,FB,FC,J,Y,Z)
IF ( P-F .LT. W*Q ) GOTO 310
GOTO 250
320 DA = FD(A,X,H,N,GRAD)
IF ( DA .GT. A6*G ) GOTO 410
IF ( DA .GE. 0. ) GOTO 560
R = A
Q = 0.
DO 330 I = 1,J
IF ( Y(I) .GT. A ) GOTO 370
IF ( Y(I) .LE. Q ) GOTO 330
IF ( Y(I) .EQ. A ) GOTO 330
Q = Y(I)
330 CONTINUE
IF ( A .LE. A8*Q ) GOTO 560
Q = A
340 ND = ND + 1
IF ( ND .GT. 25 ) GOTO 610
Q = A3*Q
P = FV(Q,X,H,N,VALUE)
F1 = FA
CALL INS(Q,P,A,B,C,FA,FB,FC,J,Y,Z)
IF ( P .LT. F1 ) GOTO 340
IF ( A .GT. R ) GOTO 360
DO 350 I = 1,N
350 H(I,2) = X(I) + A*H(I,1)
GOTO 560
360 DA = FD(A,X,H,N,GRAD)
IF ( DA .GT. A6*G ) GOTO 410
GOTO 560
370 Q = Y(I)
DO 380 K = I,J
IF ( Y(K) .LE. A ) GOTO 380
IF ( Y(K) .LT. Q ) Q = Y(K)
380 CONTINUE
IF ( Q .LE. A5*A ) GOTO 560
F0 = DLOG(Q/A)
S = IDINT(F0*L3+.999)
F0 = 1.001*DEXP(F0/S)
S = A
390 S = S*F0
IF ( S .GE. Q ) GOTO 320
P = FV(S,X,H,N,VALUE)
F1 = FA
CALL INS(S,P,A,B,C,FA,FB,FC,J,Y,Z)
IF ( P .LT. F1 ) GOTO 390
IF ( A .GT. R ) GOTO 320
DO 400 I = 1,N
400 H(I,2) = X(I) + A*H(I,1)
GOTO 560
410 B = 0.
K = 1
I = K
420 I = I + 1
IF ( I .GT. J ) GOTO 430
IF ( Y(I) .GE. A ) GOTO 420
IF ( Y(I) .LT. B ) GOTO 420
B = Y(I)
K = I
GOTO 420
430 FB = Z(K)
DB = D
IF ( B .NE. 0. ) DB = FD(B,X,H,N,GRAD)
440 W = 2.*DABS(B-A)
CALL CUB(C,A,B,FA,FB,DA,DB)
NC = 1
GOTO 480
450 W = .5*W
IF ( W .LT. DABS(C0-C) ) GOTO 550
IF ( C0 .LT. C ) GOTO 460
IF ( D0 .GE. D ) GOTO 470
GOTO 550
460 IF ( D0 .GT. D ) GOTO 550
470 CALL CUB(C,C,C0,F,F0,D,D0)
NC = NC + 1
IF ( NC .GT. 30 ) GOTO 600
480 R = DMAX1(A,B)
S = DMIN1(A,B)
IF ( C .GT. R ) GOTO 490
IF ( C .GT. S ) GOTO 500
C = S + (S-C)
S = .5*(A+B)
IF ( C .GT. S ) C = S
GOTO 500
490 C = R - (C-R)
S = .5*(A+B)
IF ( C .LT. S ) C = S
500 C0 = A
F0 = FA
D0 = DA
CALL FVD(F,D,C,X,H,N,BOTH)
IF ( F .LT. FA ) GOTO 510
B = C
FB = F
DB = D
GOTO 450
510 IF ( C .LT. A ) GOTO 540
IF ( D .LT. 0. ) GOTO 530
520 B = A
FB = FA
DB = DA
530 A = C
FA = F
DA = D
IF ( D .GT. A6*G ) GOTO 450
GOTO 560
540 IF ( D .LT. 0. ) GOTO 520
GOTO 530
550 C = .5*(A+B)
NB = NB + 1
W = DABS(B-A)
GOTO 500
560 E = 0.
DO 570 I = 1,N
IF ( DABS(H(I,3)) .GT. E ) E = DABS(H(I,3))
570 X(I) = H(I,2)
IT = IT + 1
IF ( E .LE. T ) GOTO 660
IF ( IT .GE. LIMIT ) GOTO 660
F = FA
D = DA
A = A7*A
CALL PRE(H(1,2),H(1,3))
R = 0.
DO 580 I = 1,N
580 R = R + H(I,2)*H(I,3)
IF ( R .LT. 0. ) GOTO 620
S = R/G
G = R
L = L + 1
IF ( L .GE. M ) GOTO 50
D = 0.
DO 590 I = 1,N
H(I,1) = -H(I,2) + S*H(I,1)
590 D = D + H(I,1)*H(I,3)
GOTO 70
600 IF ( D .LT. G ) GOTO 560
WRITE(6,*) 'UNABLE TO OBTAIN DESCENT DIRECTION'
STOP
610 WRITE(6,*) 'THE FUNCTION DECREASES WITH NO MINIMUM'
STOP
620 WRITE(6,*) 'PRECONDITIONER NOT POSITIVE DEFINITE'
STOP
630 Q = Q*A3**25
ND = 0
640 ND = ND + 1
IF ( ND .GT. 25 ) GOTO 650
Q = A3*Q
P = FV(Q,X,H,N,VALUE)
CALL INS(Q,P,A,B,C,FA,FB,FC,J,Y,Z)
IF ( P-F .GT. V*Q ) GOTO 640
GOTO 135
650 WRITE(6,*) 'UNABLE TO SATISFY ARMIJO CONDITION'
RETURN
660 STEP = A
RETURN
END
DOUBLE PRECISION FUNCTION FV(A,X,H,N,VALUE)
REAL*8 H(N,1),X(1),A,VALUE
EXTERNAL VALUE
DO 10 I = 1 , N
10 H(I,2) = X(I) + A*H(I,1)
FV = VALUE(H(1,2))
RETURN
END
DOUBLE PRECISION FUNCTION FD(A,X,H,N,GRAD)
REAL*8 H(N,1),X(1),A,D
EXTERNAL GRAD
DO 10 I = 1 , N
10 H(I,2) = X(I) + A*H(I,1)
CALL GRAD(H(1,3),H(1,2))
D = 0.
DO 20 I = 1,N
20 D = D + H(I,1)*H(I,3)
FD = D
RETURN
END
SUBROUTINE FVD(V,D,A,X,H,N,BOTH)
REAL*8 H(N,1),X(1),A,D,V
EXTERNAL BOTH
DO 10 I = 1 , N
10 H(I,2) = X(I) + A*H(I,1)
CALL BOTH(V,H(1,3),H(1,2))
D = 0.
DO 20 I = 1,N
20 D = D + H(I,1)*H(I,3)
RETURN
END
SUBROUTINE CUB(X,A,B,C,D,E,F)
REAL*8 A,B,C,D,E,F,G,V,W,X,Y,Z
G = B - A
IF ( G .EQ. 0. ) GOTO 50
V = E + F - 3*(D-C)/G
W = V*V-E*F
IF ( W .LT. 0. ) W = 0.
W = DSIGN(DSQRT(W),G)
Y = E + V
Z = F + V
IF ( DSIGN(Y,G) .NE. Y ) GOTO 30
IF ( DSIGN(Z,G) .NE. Z ) GOTO 20
IF ( Z .EQ. 0. ) GOTO 20
10 X = B - G*F/(Z+W)
RETURN
20 IF ( C .LT. D ) X = A
IF ( C .GE. D ) X = B
RETURN
30 IF ( DSIGN(Z,G) .NE. Z ) GOTO 40
IF ( DABS(E) .GT. DABS(F) ) GOTO 10
40 X = A + G*E/(Y-W)
RETURN
50 X = A
RETURN
END
SUBROUTINE INS(S,F,A,B,C,FA,FB,FC,J,Y,Z)
REAL*8 A,B,C,F,FA,FB,FC,S,Y(1),Z(1)
INTEGER J
J = J + 1
Y(J) = S
Z(J) = F
IF ( F .LE. FA ) GOTO 20
IF ( F .LE. FB ) GOTO 10
IF ( F .GT. FC ) RETURN
C = S
FC = F
RETURN
10 C = B
B = S
FC = FB
FB = F
RETURN
20 C = B
B = A
A = S
FC = FB
FB = FA
FA = F
RETURN
END
|
