*		       PROGRAM I.4				   *

*******************************************************************************
*                              TDHPCG                                         *
* THIS PROGRAM IS THE DIFFERENTIAL EQUATION SOLVER-DHPCG WHICH                *
* USES THE FOURTH ORDER CORRECTOR-PREDICTOR METHOD.                           *
*                                                                             *
* THIS IS A MODIFICATION OF SUBROUTINE DHPCG FROM THE                         *
* SCIENTIFIC SUBROUTINE PACKAGE (SSP) PUBLISHED BY                            *
* IBM. THE ORIGINAL VERSION OF THE ROUTINE WAS WRITTEN IN                     *
* FORTRAN-4 AND WITH NO INDENTATION AS WELL AS POORLY                         *
* STRUCTURED.                                                                 *
*                                                                             *
* THE FOLLOWING VERSION IS VERY MUCH MODERNIZED TO A STRUCTURED               *
* CODE AND INDENTED IN FORTRAN-77. THE USER ARE REFERED TO THE                *
* MANUAL OF SSP FOR FURTHER IMFORMATION ABOUT THE METHODS AND                 *
* DETAILED DESCRIPTION.                                                       *
*******************************************************************************

C     ****************************************************************
C     *
C     *	 ROUTINE DHPCG-NUMERICAL INTEGRATION-DOUBLE PRECISION
C     *
C     ****************************************************************
C
      SUBROUTINE DHPCG(PRMT,PV,DERY,NDIM,IHLF,FCT,OUTP,AUX)
C

      REAL*8 PRMT(1),X,H,Z,DELT,PV(1),DERY(1),AUX(16,1)
      INTEGER NHIT
      COMMON /NHIT/NHIT

      NHIT=0
      N=1  
      IHLF1=0
      IHLF=0
      X=PRMT(1)
      H=PRMT(3)
      PRMT(5)=0.D0 
      DO I=1,NDIM  
         AUX(16,I)=0.D0 
         AUX(15,I)=DERY(I)  
         AUX(1,I)=PV(I)  
      END DO
C     ERROR RETURNS
      IF((H*PRMT(2)).LT.X*H) THEN
         IHLF=13
      ELSE IF((H*PRMT(2)).EQ.X*H) THEN
         IHLF=12
      END IF
      ISW=0
C     ****************************************************************
C
C     COMPUTATION OF DERY FOR STARTING VALUES  
C     BLOCK 1   VS OLD LABELS 4_20
C
C     ***************************************************************
      DO WHILE (ISW.NE.4)
         IF(ISW.EQ.0) THEN
            IHLF1=0
            CALL FCT(X,PV,DERY)
C           RECORDING OF STARTING VALUES
            CALL OUTP(X,PV,DERY,IHLF,NDIM,PRMT,NHIT)
            IF (NHIT.NE.0) RETURN
            IF(PRMT(5).NE.0.) THEN
               RETURN
            END IF
            DO I=1,NDIM  
               AUX(8,I)=DERY(I) 
            END DO
C           COMPUTATION OF AUX(2,I)  
            ISW=1
         ELSE  IF(ISW.EQ.1) THEN
            IF(IHLF1.NE.1) THEN
               X=X+H
               DO I=1,NDIM 
                  AUX(2,I)=PV(I)  
               END DO
            END IF
            IHLF1=0
C           INCREMENT IS TESTED BY MEANS OF BISECTION  
            IHLF=IHLF+1  
            X=X-H
            DO I=1,NDIM 
               AUX(4,I)=AUX(2,I)  
            END DO
            H=.5D0*H 
            N=1  
            ISW=2
         ELSE IF(ISW.EQ.2) THEN
            X=X+H
            CALL FCT(X,PV,DERY)
            N=2  
            DO  I=1,NDIM 
               AUX(2,I)=PV(I)  
               AUX(9,I)=DERY(I) 
            ENDDO
            ISW=3
         ELSE IF(ISW.EQ.3) THEN
C           COMPUTATION OF TEST VALUE DELT 
            DELT=0.D0
            DO  I=1,NDIM 
               DELT=DELT+AUX(15,I)*DABS(PV(I)-AUX(4,I))  
            ENDDO
            DELT=.66666666667D-1*DELT  
            IF(DELt.LE.PRMT(4)) THEN
               ISW=5
            ELSE
               IF(IHLF.GE.10) THEN
                  IHLF=11
                  X=X+H
                  ISW=0
                  IHLF1=2
               ELSE
c                  write(6,*) 'NO SATISFACTORY ACCURACY AFTER ten B' 
                  IHLF1=1
                  ISW=1
               ENDIF
            ENDIF
         ELSE IF(ISW.EQ.5) THEN
D            write(6,*) 'SATISFACT ACCURACY AFTER LESS THAN eleven B'  
            X=X+H
            CALL FCT(X,PV,DERY)
            DO I=1,NDIM 
               AUX(3,I)=PV(I)  
               AUX(10,I)=DERY(I)  
            ENDDO
            N=3  
            ISW=4
         ENDIF
C        *******************************************************************
C
C        THE FOLLOWING PART OF SUBROUTINE HPCG COMPUTES BY MEANS OF 
C        RUNGE-KUTTA METHOD STARTING VALUES FOR THE NOT SELF-STARTING 
C        PREDICTOR-CORRECTOR METHOD.
C        BLOCK 2   VS OLD LABELS 100_104 
C
C        *******************************************************************
         IF(ISW.NE.5.AND.IHLF1.EQ.0) THEN
            DO I=1,NDIM  
               Z=H*AUX(N+7,I) 
               AUX(5,I)=Z
               PV(I)=AUX(N,I)+.4D0*Z 
            ENDDO
C           Z IS AN AUXILIARY STORAGE LOCATION 
            Z=X+.4D0*H
            CALL FCT(X,PV,DERY)
            DO I=1,NDIM  
               Z=H*DERY(I)  
               AUX(6,I)=Z
               PV(I)=AUX(N,I)+.29697760925D0*AUX(5,I)+.15875964497D0*Z 
            ENDDO
            Z=X+.45573725422D0*H 
            CALL FCT(X,PV,DERY)
            DO I=1,NDIM  
               Z=H*DERY(I)  
               AUX(7,I)=Z
       PV(I)=AUX(N,I)+0.21810038823D0*AUX(5,I)-3.0509651487D0*AUX(6,I)  
     *        + 3.8328647605D0*Z  
            ENDDO
            Z=X+H
            CALL FCT(X,PV,DERY)
            DO I=1,NDIM  
          PV(I)=AUX(N,I)+.17476028223D0*AUX(5,I)-.55148066288D0*AUX(6,I)  
     *     +1.2055355994D0*AUX(7,I)+.17118478122D0*H*DERY(I)  
            ENDDO
         ENDIF
      ENDDO
C     *******************************************************************
C
C     BLOCK 3   VS OLD LABELS 21_22
C
C     *******************************************************************
      N=1  
      X=X+H
      CALL FCT(X,PV,DERY)
      X=PRMT(1)
      DO I=1,NDIM 
         AUX(11,I)=DERY(I)  
         PV(I)=AUX(1,I)+H*(.375D0*AUX(8,I)+.79166666667D0*AUX(9,I) 
     *   -.20833333333D0*AUX(10,I)+.41666666667D-1*DERY(I)) 
      ENDDO
C     *********************************************************************
C
C     BLOCK 4   VS OLD LABELS 23_30
C
C     *********************************************************************
      DO WHILE (N.LT.4)
         X=X+H
         N=N+1
         CALL FCT(X,PV,DERY)
         CALL OUTP(X,PV,DERY,IHLF,NDIM,PRMT,NHIT)
         IF (NHIT.NE.0) RETURN
         IF(PRMT(5).NE.0) THEN
            RETURN
         ELSE IF((N-4).LT.0) THEN
            DO I=1,NDIM 
               AUX(N,I)=PV(I)  
               AUX(N+7,I)=DERY(I) 
            ENDDO
            IF(N.LT.3) THEN
               DO I=1,NDIM 
                  DELT=AUX(9,I)+AUX(9,I) 
                  DELT=DELT+DELT 
               PV(I)=AUX(1,I)+.33333333333D0*H*(AUX(8,I)+DELT+AUX(10,I)) 
               ENDDO
            ELSE 
               DO I=1,NDIM 
                  DELT=AUX(9,I)+AUX(10,I)
                  DELT=DELT+DELT+DELT  
                  PV(I)=AUX(1,I)+.375D0*H*(AUX(8,I)+DELT+AUX(11,I)) 
               ENDDO
            ENDIF
         ENDIF
      ENDDO
      ISW=6
C     ******************************************************************
C
C     POSSIBLE BREAK POINT FOR LINKAGE
C     STARTING VALUES ARE COMPUTED
C     NOW START HAMMINGS MODIFIED PREDICTOR-CORRECTOR METHOD.  
C     BLOCK 5   VS OLD LABELS 200_226
C
C     ******************************************************************
      ISTEP=3  
      DO WHILE (ISW.GE.6)
         IF(ISW.EQ.6) THEN
            IF(N.EQ.8) THEN
C              N=8 CAUSES THE ROWS OF AUX TO CHANGE THEIR STORAGE LOCATIONS 
               DO N=2,7 
                  DO I=1,NDIM  
                     AUX(N-1,I)=AUX(N,I)  
                     AUX(N+6,I)=AUX(N+7,I)  
                  ENDDO
               ENDDO
               N=7  
            ENDIF
C           N LESS THAN 8 CAUSES N+1 TO GET N  
            N=N+1
C           COMPUTATION OF NEXT VECTOR PV
            DO I=1,NDIM  
               AUX(N-1,I)=PV(I)  
               AUX(N+6,I)=DERY(I) 
            ENDDO
            X=X+H
            ISW=7
         ELSE IF(ISW.EQ.7) THEN
            ISTEP=ISTEP+1  
            DO I=1,NDIM  
               DELT=AUX(N-4,I)+1.3333333333D0*H*(AUX(N+6,I)+AUX(N+6,I)
     *         -AUX(N+5,I)+AUX(N+4,I)+AUX(N+4,I))
               PV(I)=DELT-.92561983471D0*AUX(16,I) 
               AUX(16,I)=DELT 
            ENDDO
C           PREDICTOR IS NOW GENERATED IN ROW 16 OF AUX, MODIFIED PREDICTOR
C           IS GENERATED I PV.  DELT MEANS AN AUXILIARY STORAGE.  
            CALL FCT(X,PV,DERY)
C           DERIVATIVE OF MODIFIED PREDICTOR IS GENERATED IN DERY  
            DO I=1,NDIM  
               DELT=.125D0*(9.D0*AUX(N-1,I)-AUX(N-3,I)+3.D0*H*(DERY(I)
     *         +AUX(N+6,I)+AUX(N+6,I)-AUX(N+5,I)))
               AUX(16,I)=AUX(16,I)-DELT
               PV(I)=DELT+.7438016529D-1*AUX(16,I) 
            ENDDO
C           TEST WHETHER H MUST BE HALVED OR DOUBLED 
            DELT=0.D0
            DO I=1,NDIM  
               DELT=DELT+AUX(15,I)*DABS(AUX(16,I))  
            ENDDO
            ISW=8
         ELSE IF(ISW.EQ.8) THEN
            IF(DELT.LT.PRMT(4).OR.IHLF1.EQ.5) THEN
               IHLF1=0
C              H MUST NOT BE HALVED.  THAT MEANS PV(I) ARE GOOD. 
               CALL FCT(X,PV,DERY)
               CALL OUTP(X,PV,DERY,IHLF,NDIM,PRMT,NHIT)
               IF (NHIT.NE.0) RETURN
               IF(PRMT(5).NE.0.OR.IHLF.GE.11.OR.(H*(X-PRMT(2))).GE.0
     *         .OR.DABS(X-PRMT(2)).LT.0.1D0*DABS(H)) THEN
 1150             FORMAT(I5)
                  RETURN
               ELSE
                  IF(DELT.LE..2D-1*PRMT(4).AND.IHLF.GT.0.AND.N.GE.7.AND.
     *            ISTEP.GE.4) THEN
C                    H COULD BE DOUBLED IF ALL NECESSARY PRECEEDING VALUES ARE AVAILABLE
                     IMOD=ISTEP/2 
                     IF((ISTEP-IMOD-IMOD).EQ.0) THEN
                        H=H+H
                        IHLF=IHLF-1  
                        ISTEP=0  
                        DO 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  
             AUX(16,I)=8.962962963D0*(PV(I)-AUX(N-3,I))-3.361111111D0*H* 
     *                     (DERY(I)+DELT+AUX(N+4,I))  
                        ENDDO
                        ISW=6
                     ELSE
                        ISW=6
                     ENDIF
                  ELSE
                     ISW=6
                  ENDIF
               ENDIF
               ISW=6
            ELSE
               ISW=9
            ENDIF
         ELSE IF(ISW.EQ.9) THEN
C           H MUST BE HALVED 
            IHLF=IHLF+1  
            IF(IHLF.LE.10) THEN
               ISW=10
            ELSE
               ISW=8
               IHLF1=5
            ENDIF
         ELSE IF(ISW.EQ.10) THEN
            H=.5D0*H 
            ISTEP=0  
            DO I=1,NDIM  
               PV(I)=.390625D-2*(8.D1*AUX(N-1,I)+135.D0*AUX(N-2,I)
     *        + 4.D1*AUX(N-1,I)+AUX(N-4,I))-.1171875D0*(AUX(N+6,I)
     *        -  6.D0*AUX(N+5,I)-AUX(N+4,I))*H
           AUX(N-4,I)=.390625D-2*(12.D0*AUX(N-1,I)+135.D0*AUX(N-2,I)+ 
     *         08.D0*AUX(N-3,I)+AUX(N-4,I))-.234375D-1*(AUX(N+6,I)+  
     *         8.D0*AUX(N+5,I)-9.D0*AUX(N+4,I))*H  
               AUX(N-3,I)=AUX(N-2,I)  
               AUX(N+4,I)=AUX(N+5,I)  
            ENDDO
            X=X-H
            DELT=X-(H+H) 
            CALL FCT(X,PV,DERY)
            DO I=1,NDIM  
               AUX(N-2,I)=PV(I)  
               AUX(N+5,I)=DERY(I) 
               PV(I)=AUX(N-4,I)  
            ENDDO
            DELT=DELT-(H+H)  
            CALL FCT(X,PV,DERY)
            DO I=1,NDIM  
               DELT=AUX(N+5,I)+AUX(N+4,I) 
               DELT=DELT+DELT+DELT  
               AUX(16,I)=8.9629696296D0*(AUX(N-1,I)-PV(I))
     *       -  3.3611111111D0*H*(AUX(N+6,I)+DELT+DERY(I))  
               AUX(N+3,I)=DERY(I) 
            ENDDO
            ISW=7
         ENDIF
      ENDDO
      END  
C-----------------------------------------------------------------------

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