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1985-11-29
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365 lines
C
C ..................................................................
C
C SUBROUTINE DHPCG
C
C PURPOSE
C TO SOLVE A SYSTEM OF FIRST ORDER ORDINARY GENERAL
C DIFFERENTIAL EQUATIONS WITH GIVEN INITIAL VALUES.
C
C USAGE
C CALL DHPCG (PRMT,Y,DERY,NDIM,IHLF,FCT,OUTP,AUX)
C PARAMETERS FCT AND OUTP REQUIRE AN EXTERNAL STATEMENT.
C
C DESCRIPTION OF PARAMETERS
C PRMT - DOUBLE PRECISION INPUT AND OUTPUT VECTOR WITH
C DIMENSION GREATER THAN OR EQUAL TO 5, WHICH
C SPECIFIES THE PARAMETERS OF THE INTERVAL AND OF
C ACCURACY AND WHICH SERVES FOR COMMUNICATION BETWEEN
C OUTPUT SUBROUTINE (FURNISHED BY THE USER) AND
C SUBROUTINE DHPCG. EXCEPT PRMT(5) THE COMPONENTS
C ARE NOT DESTROYED BY SUBROUTINE DHPCG AND THEY ARE
C PRMT(1)- LOWER BOUND OF THE INTERVAL (INPUT),
C PRMT(2)- UPPER BOUND OF THE INTERVAL (INPUT),
C PRMT(3)- INITIAL INCREMENT OF THE INDEPENDENT VARIABLE
C (INPUT),
C PRMT(4)- UPPER ERROR BOUND (INPUT). IF ABSOLUTE ERROR IS
C GREATER THAN PRMT(4), INCREMENT GETS HALVED.
C IF INCREMENT IS LESS THAN PRMT(3) AND ABSOLUTE
C ERROR LESS THAN PRMT(4)/50, INCREMENT GETS DOUBLED.
C THE USER MAY CHANGE PRMT(4) BY MEANS OF HIS
C OUTPUT SUBROUTINE.
C PRMT(5)- NO INPUT PARAMETER. SUBROUTINE DHPCG INITIALIZES
C PRMT(5)=0. IF THE USER WANTS TO TERMINATE
C SUBROUTINE DHPCG AT ANY OUTPUT POINT, HE HAS TO
C CHANGE PRMT(5) TO NON-ZERO BY MEANS OF SUBROUTINE
C OUTP. FURTHER COMPONENTS OF VECTOR PRMT ARE
C FEASIBLE IF ITS DIMENSION IS DEFINED GREATER
C THAN 5. HOWEVER SUBROUTINE DHPCG DOES NOT REQUIRE
C AND CHANGE THEM. NEVERTHELESS THEY MAY BE USEFUL
C FOR HANDING RESULT VALUES TO THE MAIN PROGRAM
C (CALLING DHPCG) WHICH ARE OBTAINED BY SPECIAL
C MANIPULATIONS WITH OUTPUT DATA IN SUBROUTINE OUTP.
C Y - DOUBLE PRECISION INPUT VECTOR OF INITIAL VALUES
C (DESTROYED). LATERON Y IS THE RESULTING VECTOR OF
C DEPENDENT VARIABLES COMPUTED AT INTERMEDIATE
C POINTS X.
C DERY - DOUBLE PRECISION INPUT VECTOR OF ERROR WEIGHTS
C (DESTROYED). THE SUM OF ITS COMPONENTS MUST BE
C EQUAL TO 1. LATERON DERY IS THE VECTOR OF
C DERIVATIVES, WHICH BELONG TO FUNCTION VALUES Y AT
C INTERMEDIATE POINTS X.
C NDIM - AN INPUT VALUE, WHICH SPECIFIES THE NUMBER OF
C EQUATIONS IN THE SYSTEM.
C IHLF - AN OUTPUT VALUE, WHICH SPECIFIES THE NUMBER OF
C BISECTIONS OF THE INITIAL INCREMENT. IF IHLF GETS
C GREATER THAN 10, SUBROUTINE DHPCG RETURNS WITH
C ERROR MESSAGE IHLF=11 INTO MAIN PROGRAM.
C ERROR MESSAGE IHLF=12 OR IHLF=13 APPEARS IN CASE
C PRMT(3)=0 OR IN CASE SIGN(PRMT(3)).NE.SIGN(PRMT(2)-
C PRMT(1)) RESPECTIVELY.
C FCT - THE NAME OF AN EXTERNAL SUBROUTINE USED. IT
C COMPUTES THE RIGHT HAND SIDES DERY OF THE SYSTEM
C TO GIVEN VALUES OF X AND Y. ITS PARAMETER LIST
C MUST BE X,Y,DERY. THE SUBROUTINE SHOULD NOT
C DESTROY X AND Y.
C OUTP - THE NAME OF AN EXTERNAL OUTPUT SUBROUTINE USED.
C ITS PARAMETER LIST MUST BE X,Y,DERY,IHLF,NDIM,PRMT.
C NONE OF THESE PARAMETERS (EXCEPT, IF NECESSARY,
C PRMT(4),PRMT(5),...) SHOULD BE CHANGED BY
C SUBROUTINE OUTP. IF PRMT(5) IS CHANGED TO NON-ZERO,
C SUBROUTINE DHPCG IS TERMINATED.
C AUX - DOUBLE PRECISION AUXILIARY STORAGE ARRAY WITH 16
C ROWS AND NDIM COLUMNS.
C
C REMARKS
C THE PROCEDURE TERMINATES AND RETURNS TO CALLING PROGRAM, IF
C (1) MORE THAN 10 BISECTIONS OF THE INITIAL INCREMENT ARE
C NECESSARY TO GET SATISFACTORY ACCURACY (ERROR MESSAGE
C IHLF=11),
C (2) INITIAL INCREMENT IS EQUAL TO 0 OR HAS WRONG SIGN
C (ERROR MESSAGES IHLF=12 OR IHLF=13),
C (3) THE WHOLE INTEGRATION INTERVAL IS WORKED THROUGH,
C (4) SUBROUTINE OUTP HAS CHANGED PRMT(5) TO NON-ZERO.
C
C SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
C THE EXTERNAL SUBROUTINES FCT(X,Y,DERY) AND
C OUTP(X,Y,DERY,IHLF,NDIM,PRMT) MUST BE FURNISHED BY THE USER.
C
C METHOD
C EVALUATION IS DONE BY MEANS OF HAMMINGS MODIFIED PREDICTOR-
C CORRECTOR METHOD. IT IS A FOURTH ORDER METHOD, USING 4
C PRECEEDING POINTS FOR COMPUTATION OF A NEW VECTOR Y OF THE
C DEPENDENT VARIABLES.
C FOURTH ORDER RUNGE-KUTTA METHOD SUGGESTED BY RALSTON IS
C USED FOR ADJUSTMENT OF THE INITIAL INCREMENT AND FOR
C COMPUTATION OF STARTING VALUES.
C SUBROUTINE DHPCG AUTOMATICALLY ADJUSTS THE INCREMENT DURING
C THE WHOLE COMPUTATION BY HALVING OR DOUBLING.
C TO GET FULL FLEXIBILITY IN OUTPUT, AN OUTPUT SUBROUTINE
C MUST BE CODED BY THE USER.
C FOR REFERENCE, SEE
C (1) RALSTON/WILF, MATHEMATICAL METHODS FOR DIGITAL
C COMPUTERS, WILEY, NEW YORK/LONDON, 1960, PP.95-109.
C (2) RALSTON, RUNGE-KUTTA METHODS WITH MINIMUM ERROR BOUNDS,
C MTAC, VOL.16, ISS.80 (1962), PP.431-437.
C
C ..................................................................
C
SUBROUTINE DHPCG(PRMT,Y,DERY,NDIM,IHLF,FCT,OUTP,AUX)
C
C
DIMENSION PRMT(1),Y(1),DERY(1),AUX(16,1)
DOUBLE PRECISION Y,DERY,AUX,PRMT,X,H,Z,DELT
N=1
IHLF=0
X=PRMT(1)
H=PRMT(3)
PRMT(5)=0.D0
DO 1 I=1,NDIM
AUX(16,I)=0.D0
AUX(15,I)=DERY(I)
1 AUX(1,I)=Y(I)
IF(H*(PRMT(2)-X))3,2,4
C
C ERROR RETURNS
2 IHLF=12
GOTO 4
3 IHLF=13
C
C COMPUTATION OF DERY FOR STARTING VALUES
4 CALL FCT(X,Y,DERY)
C
C RECORDING OF STARTING VALUES
CALL OUTP(X,Y,DERY,IHLF,NDIM,PRMT)
IF(PRMT(5))6,5,6
5 IF(IHLF)7,7,6
6 RETURN
7 DO 8 I=1,NDIM
8 AUX(8,I)=DERY(I)
C
C COMPUTATION OF AUX(2,I)
ISW=1
GOTO 100
C
9 X=X+H
DO 10 I=1,NDIM
10 AUX(2,I)=Y(I)
C
C INCREMENT H IS TESTED BY MEANS OF BISECTION
11 IHLF=IHLF+1
X=X-H
DO 12 I=1,NDIM
12 AUX(4,I)=AUX(2,I)
H=.5D0*H
N=1
ISW=2
GOTO 100
C
13 X=X+H
CALL FCT(X,Y,DERY)
N=2
DO 14 I=1,NDIM
AUX(2,I)=Y(I)
14 AUX(9,I)=DERY(I)
ISW=3
GOTO 100
C
C COMPUTATION OF TEST VALUE DELT
15 DELT=0.D0
DO 16 I=1,NDIM
16 DELT=DELT+AUX(15,I)*DABS(Y(I)-AUX(4,I))
DELT=.066666666666666667D0*DELT
IF(DELT-PRMT(4))19,19,17
17 IF(IHLF-10)11,18,18
C
C NO SATISFACTORY ACCURACY AFTER 10 BISECTIONS. ERROR MESSAGE.
18 IHLF=11
X=X+H
GOTO 4
C
C THERE IS SATISFACTORY ACCURACY AFTER LESS THAN 11 BISECTIONS.
19 X=X+H
CALL FCT(X,Y,DERY)
DO 20 I=1,NDIM
AUX(3,I)=Y(I)
20 AUX(10,I)=DERY(I)
N=3
ISW=4
GOTO 100
C
21 N=1
X=X+H
CALL FCT(X,Y,DERY)
X=PRMT(1)
DO 22 I=1,NDIM
AUX(11,I)=DERY(I)
220Y(I)=AUX(1,I)+H*(.375D0*AUX(8,I)+.7916666666666667D0*AUX(9,I)
1-.20833333333333333D0*AUX(10,I)+.041666666666666667D0*DERY(I))
23 X=X+H
N=N+1
CALL FCT(X,Y,DERY)
CALL OUTP(X,Y,DERY,IHLF,NDIM,PRMT)
IF(PRMT(5))6,24,6
24 IF(N-4)25,200,200
25 DO 26 I=1,NDIM
AUX(N,I)=Y(I)
26 AUX(N+7,I)=DERY(I)
IF(N-3)27,29,200
C
27 DO 28 I=1,NDIM
DELT=AUX(9,I)+AUX(9,I)
DELT=DELT+DELT
28 Y(I)=AUX(1,I)+.33333333333333333D0*H*(AUX(8,I)+DELT+AUX(10,I))
GOTO 23
C
29 DO 30 I=1,NDIM
DELT=AUX(9,I)+AUX(10,I)
DELT=DELT+DELT+DELT
30 Y(I)=AUX(1,I)+.375D0*H*(AUX(8,I)+DELT+AUX(11,I))
GOTO 23
C
C THE FOLLOWING PART OF SUBROUTINE DHPCG COMPUTES BY MEANS OF
C RUNGE-KUTTA METHOD STARTING VALUES FOR THE NOT SELF-STARTING
C PREDICTOR-CORRECTOR METHOD.
100 DO 101 I=1,NDIM
Z=H*AUX(N+7,I)
AUX(5,I)=Z
101 Y(I)=AUX(N,I)+.4D0*Z
C Z IS AN AUXILIARY STORAGE LOCATION
C
Z=X+.4D0*H
CALL FCT(Z,Y,DERY)
DO 102 I=1,NDIM
Z=H*DERY(I)
AUX(6,I)=Z
102 Y(I)=AUX(N,I)+.29697760924775360D0*AUX(5,I)+.15875964497103583D0*Z
C
Z=X+.45573725421878943D0*H
CALL FCT(Z,Y,DERY)
DO 103 I=1,NDIM
Z=H*DERY(I)
AUX(7,I)=Z
103 Y(I)=AUX(N,I)+.21810038822592047D0*AUX(5,I)-3.0509651486929308D0*
1AUX(6,I)+3.8328647604670103D0*Z
C
Z=X+H
CALL FCT(Z,Y,DERY)
DO 104 I=1,NDIM
1040Y(I)=AUX(N,I)+.17476028226269037D0*AUX(5,I)-.55148066287873294D0*
1AUX(6,I)+1.2055355993965235D0*AUX(7,I)+.17118478121951903D0*
2H*DERY(I)
GOTO(9,13,15,21),ISW
C
C POSSIBLE BREAK-POINT FOR LINKAGE
C
C STARTING VALUES ARE COMPUTED.
C NOW START HAMMINGS MODIFIED PREDICTOR-CORRECTOR METHOD.
200 ISTEP=3
201 IF(N-8)204,202,204
C
C N=8 CAUSES THE ROWS OF AUX TO CHANGE THEIR STORAGE LOCATIONS
202 DO 203 N=2,7
DO 203 I=1,NDIM
AUX(N-1,I)=AUX(N,I)
203 AUX(N+6,I)=AUX(N+7,I)
N=7
C
C N LESS THAN 8 CAUSES N+1 TO GET N
204 N=N+1
C
C COMPUTATION OF NEXT VECTOR Y
DO 205 I=1,NDIM
AUX(N-1,I)=Y(I)
205 AUX(N+6,I)=DERY(I)
X=X+H
206 ISTEP=ISTEP+1
DO 207 I=1,NDIM
0DELT=AUX(N-4,I)+1.3333333333333333D0*H*(AUX(N+6,I)+AUX(N+6,I)-
1AUX(N+5,I)+AUX(N+4,I)+AUX(N+4,I))
Y(I)=DELT-.9256198347107438D0*AUX(16,I)
207 AUX(16,I)=DELT
C PREDICTOR IS NOW GENERATED IN ROW 16 OF AUX, MODIFIED PREDICTOR
C IS GENERATED IN Y. DELT MEANS AN AUXILIARY STORAGE.
C
CALL FCT(X,Y,DERY)
C DERIVATIVE OF MODIFIED PREDICTOR IS GENERATED IN DERY
C
DO 208 I=1,NDIM
0DELT=.125D0*(9.D0*AUX(N-1,I)-AUX(N-3,I)+3.D0*H*(DERY(I)+AUX(N+6,I)
1+AUX(N+6,I)-AUX(N+5,I)))
AUX(16,I)=AUX(16,I)-DELT
208 Y(I)=DELT+.07438016528925620D0*AUX(16,I)
C
C TEST WHETHER H MUST BE HALVED OR DOUBLED
DELT=0.D0
DO 209 I=1,NDIM
209 DELT=DELT+AUX(15,I)*DABS(AUX(16,I))
IF(DELT-PRMT(4))210,222,222
C
C H MUST NOT BE HALVED. THAT MEANS Y(I) ARE GOOD.
210 CALL FCT(X,Y,DERY)
CALL OUTP(X,Y,DERY,IHLF,NDIM,PRMT)
IF(PRMT(5))212,211,212
211 IF(IHLF-11)213,212,212
212 RETURN
213 IF(H*(X-PRMT(2)))214,212,212
214 IF(DABS(X-PRMT(2))-.1D0*DABS(H))212,215,215
215 IF(DELT-.02D0*PRMT(4))216,216,201
C
C
C H COULD BE DOUBLED IF ALL NECESSARY PRECEEDING VALUES ARE
C AVAILABLE
216 IF(IHLF)201,201,217
217 IF(N-7)201,218,218
218 IF(ISTEP-4)201,219,219
219 IMOD=ISTEP/2
IF(ISTEP-IMOD-IMOD)201,220,201
220 H=H+H
IHLF=IHLF-1
ISTEP=0
DO 221 I=1,NDIM
AUX(N-1,I)=AUX(N-2,I)
AUX(N-2,I)=AUX(N-4,I)
AUX(N-3,I)=AUX(N-6,I)
AUX(N+6,I)=AUX(N+5,I)
AUX(N+5,I)=AUX(N+3,I)
AUX(N+4,I)=AUX(N+1,I)
DELT=AUX(N+6,I)+AUX(N+5,I)
DELT=DELT+DELT+DELT
2210AUX(16,I)=8.962962962962963D0*(Y(I)-AUX(N-3,I))
1-3.3611111111111111D0*H*(DERY(I)+DELT+AUX(N+4,I))
GOTO 201
C
C
C H MUST BE HALVED
222 IHLF=IHLF+1
IF(IHLF-10)223,223,210
223 H=.5D0*H
ISTEP=0
DO 224 I=1,NDIM
0Y(I)=.390625D-2*(8.D1*AUX(N-1,I)+135.D0*AUX(N-2,I)+4.D1*AUX(N-3,I)
1+AUX(N-4,I))-.1171875D0*(AUX(N+6,I)-6.D0*AUX(N+5,I)-AUX(N+4,I))*H
0AUX(N-4,I)=.390625D-2*(12.D0*AUX(N-1,I)+135.D0*AUX(N-2,I)+
1108.D0*AUX(N-3,I)+AUX(N-4,I))-.0234375D0*(AUX(N+6,I)+
218.D0*AUX(N+5,I)-9.D0*AUX(N+4,I))*H
AUX(N-3,I)=AUX(N-2,I)
224 AUX(N+4,I)=AUX(N+5,I)
X=X-H
DELT=X-(H+H)
CALL FCT(DELT,Y,DERY)
DO 225 I=1,NDIM
AUX(N-2,I)=Y(I)
AUX(N+5,I)=DERY(I)
225 Y(I)=AUX(N-4,I)
DELT=DELT-(H+H)
CALL FCT(DELT,Y,DERY)
DO 226 I=1,NDIM
DELT=AUX(N+5,I)+AUX(N+4,I)
DELT=DELT+DELT+DELT
0AUX(16,I)=8.962962962962963D0*(AUX(N-1,I)-Y(I))
1-3.3611111111111111D0*H*(AUX(N+6,I)+DELT+DERY(I))
226 AUX(N+3,I)=DERY(I)
GOTO 206
END