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C     ALGORITHM 540R (REMARK ON ALG.540), COLLECTED ALGORITHMS FROM ACM.
C      THIS WORK PUBLISHED IN TRANSACTIONS ON MATHEMATICAL SOFTWARE,
C      VOL. 18, NO. 3, SEPTEMBER, 1992, PP. 343-344.
C
C     USED FOR:
C      PROGRAM RUNPDE
C      FOR TESTING AND DEBUGGING UNDER FORTRAN    
C      CALL PDETEST()     
C      END

C     MODEL 1: VECER
's PDEC     MODEL 2: ZHANG's PDE

C ------------------------------------------------------------------------------

      SUBROUTINE PDETEST()
CC     NOT USED BY R
CC     FOR TESTING AND DEBUGGING UNDER FORTRAN         
      IMPLICIT REAL*8 (A-H, O-Z)
      PARAMETER(MNP=10)
      DIMENSION PRICE(MNP+1), XBYS(MNP+1)
C      
      PARAMETER (MMF=12, MMX=1000)
      PARAMETER (MNPDE=1, MKORD=4, MNINT=MMX, MNCC=2, MMAXDER=5)

C     WORKING ARRAYS: 
      DIMENSION WORK
     *   (MKORD+MNPDE*(4+9*MNPDE)+(MKORD+(MNINT-1)*(MKORD-MNCC))*
     *   (3*MKORD+2+MNPDE*(3*(MKORD-1)*MNPDE+MMAXDER+4)))
      DIMENSION IWORK((MNPDE+1)*(MKORD+(MNINT-1)*(MKORD-MNCC)))
      DIMENSION XBKPT(MNINT+1)    
   
C PDE PARAMETERS:   
      NP = MNP
      MF = MMF
      MX = MMX
      NPDE = MNPDE
      KORD = MKORD
      NINT = MNINT
      NCC = MNCC
      MAXDER = MMAXDER
      
C OPTION SETTINGS:     
      SIGMA   =    0.30D0
      TIME    =    1.00D0
      RR      =    0.09D0
      XS      =  100.00D0
       XSMIN  =   90.00D0
       XSMAX  =  110.00D0
      SS      =  100.00D0    
      DELTA = (XSMAX-XSMIN)/NP
      DO I = 1, NP+1
         XBYS(I) = (XSMIN +(I-1)*DELTA)/XS
      ENDDO
      
C SET TIME POINTS:
C   T0    =  INITIAL VALUE OF T, THE INDEPENDENT VARIABLE
C   TOUT  =  VALUE OF T AT WHICH OUTPUT IS DESIRED NEXT
C   DT    =  INITIAL STEP SIZE IN T
C   EPS   =  RELATIVE TIME ERROR BOUND 
      T0   = 0.0D0
      TOUT = 1.0D0
      EPS  = 1.0D-08
      DT   = 1.0D-10

C FURTHER PARAMETERS:
C   NINT=1000 - NUMBER OF SUBINTERVALS (XLEFT,XRIGHT) IS TO BE DIVIDED 
C   KORD=4    - ORDER OF THE PIECEWISE POLYNOMIAL SPACE TO BE USED
C   NCC=2     - NUMBER OF CONTINUITY CONDITIONS TO BE IMPOSED 
C   MF=12     - METHOD FLAG
C               ADAMS METHODS - GENERALIZATIONS OF CRANK-NICOLSON AND
C               CHORD METHOD WITH FINITE DIFFERENCES JACOBIAN
C   INDEX     - INTEGER USED ON INPUT TO INDICATE TYPE OF CALL
C   WORK      - WORKING ARRAY
C   IWORK     - SIZE OF WORKING ARRAY
    
C ASIAN CALL (1) AND PUT(2) VALUE:
      Z = -1
      DO IP = 1, 2
         Z = -Z   
C        PDE PARAMETERS:
         MODSEL = 1
         SIGMAT = SIGMA * DSQRT(TIME)
         RRT    = RR*TIME   
         XM     = 5.0D0 * SIGMAT 
         WRITE (*,*)
         WRITE (*,*) " PDE - ASIAN OPTION SETTINGS"     
         WRITE (*,*) " MF  KORD  NCC : ", MF, KORD, NCC         
         WRITE (*,*) " SIGMA*TIME    : ", SIGMAT
         WRITE (*,*) " R*TIME        : ", RRT
         WRITE (*,*) " XM            : ", XM
         WRITE (*,*) " (XMIN,XMAX)/S : ", XSMIN/SS, XSMAX/SS 
         CALL ASIANVAL(
     &        Z, SS, XS, XSMIN, XSMAX, TIME, RR, SIGMA, 
     &        T0, TOUT, EPS, DT, PRICE, NP, MODSEL, 
     &        MF, NPDE, KORD, MX, NCC, MAXDER, 
     &        XBYS, XBKPT, WORK, IWORK)
C        OUTPUT U - NUMERICAL SOLUTION:   
         WRITE (*,*) " XLEFT  XRIGHT : ", XBKPT(1), XBKPT(NINT+1)
         WRITE (*,*) " EPS DT MX     : ", EPS, DT, MX
         WRITE (*,*) " ERROR CODE:   : ", INDEX
         WRITE(*,*)
         WRITE(*,*) " U - NUMERICAL SOLUTION FOR DIFF STRIKES:"
         WRITE(*,*) "      X          XI           PRICE     "
         DO I = 1, NP+1 
            XI = XBYS(I)*EXP(-RRT) - (1.0-EXP(-RRT))/RRT   
            WRITE(*,9) XS*XBYS(I), SS*XI, SS*PRICE(I), SS*(PRICE(I)-XI)
         ENDDO 
      ENDDO        
    9 FORMAT(F10.3, 4F15.7) 
      
      RETURN
      END
      
      
C ------------------------------------------------------------------------------

    
      SUBROUTINE ASIANVAL(
     &  ZZ, SS1, XS1, XSMIN, XSMAX, TIME1, RR1, SIGMA1, 
     &  T0, TOUT, EPS, DT, PRICEBYS, NP, MODSEL,
     &  MF1, NPDE1, KORD1, MX1, NCC1, MAXDER1, 
     &  XBYS, XBKPT, WORK, IWORK)
      
      IMPLICIT REAL*8 (A-H, O-Z)
      PARAMETER(MKORD=4, MDERV=0)   
      DIMENSION WORK
     *   (KORD1+NPDE1*(4+9*NPDE1)+(KORD1+(MX1-1)*(KORD1-NCC1))*
     *   (3*KORD1+2+NPDE1*(3*(KORD1-1)*NPDE1+MAXDER1+4)))
      DIMENSION IWORK((NPDE1+1)*(KORD1+(MX1-1)*(KORD1-NCC1)))
      DIMENSION XBKPT(MX1+1)      
      DIMENSION USOL(1,1,MDERV+1), SCRTCH(MKORD*(MDERV+1))
      DIMENSION XBYS(NP), PRICEBYS(NP)
        
      COMMON /SIZES/  NINT,KORD,NCC,NPDE,NCPTS,NEQN,IQUAD
      COMMON /GEAR0/  HUSED, NQUSED, NS, NF, NJ
      COMMON /GEAR1/  T,DTC,DTMN,DTMX,EPSC,UROUND,N,MFC,KFLAG,JSTART
      COMMON /GEAR9/  EPSJ,R0,ML,MU,MW,NM1,N0ML,N0W
      COMMON /OPTION/ NOGAUS,MAXDER
      COMMON /ISTART/ IW1, IW2, IW3, IDUM(15)
      COMMON /PARAMS/ PI
      COMMON /ASIAN1/ SIGMAT, RRT, XM, Z, MODEL
      COMMON /ASIAN2/ SIGMA, TIME, RR, XS, SS, ETA, XL, XR

C FOR COMMON BLOCKS:
      SIGMA = SIGMA1
      TIME = TIME1
      RR = RR1
      XS = XS1
      SS = SS1

C FOR COMMON BLOCKS:    
      MF = MF1
      NPDE = NPDE1
      KORD = KORD1
      MX = MX1
      NCC = NCC1
      MAXDER = MAXDER1
      NINT = MX1
      MODEL = MODSEL
      PI   = 4.0D0 * DATAN(1.0D0)  

C CALCULATE FOR BOTH, FOR A CALL Z=+1 OR FOR A PUT Z=-1:       
      Z = ZZ
         
C WORKSPACE SETTINGS:          
      IWORK(1) = KORD+NPDE*(4+9*NPDE)+(KORD+(MX-1)*
     *   (KORD-NCC))*(3*KORD+2+NPDE*(3*(KORD-1)*NPDE+MAXDER+4))
      IWORK(2) = (NPDE+1)*(KORD+(NINT-1)*(KORD-NCC)) 
      DO I = 1, IWORK(1)
         WORK(I)=0.0
      ENDDO
                        
C OPTION SETTINGS:          
      SIGMAT = SIGMA * DSQRT(TIME)
      RRT    = RR*TIME   
      XM     = 5.0D0 * SIGMAT 
      XL     = -XM
      XR     = +XM
      ETA    = (SIGMA**2)*(TIME**3)/6.0D0
             
C SET SPACE POINTS:  
      NX = MX
      DX = 2.0D0 * XM / NX
      DO I = 1, NX + 1
         XBKPT(I) = -XM + (I-1)*DX
      ENDDO
         
C SOLVE PDE:
      INDEX = 1
      CALL PDECOL(T0, TOUT, DT, XBKPT, EPS,  
     &     NX, KORD, NCC, NPDE, MF, INDEX, WORK, IWORK)

C OUTPUT U - NUMERICAL SOLUTION:     
      DO I = 1, NP+1
         XI = XBYS(I)*DEXP(-RRT) - (1.0D0-DEXP(-RRT))/RRT         
         CALL VALUES(XI, USOL, SCRTCH, 1, 1, 1, 0, WORK)
         PRICEBYS(I) = USOL(1,1,1)
      ENDDO 
         
      RETURN
      END
     
C ------------------------------------------------------------------------------

      SUBROUTINE F(T, X, U, UX, UXX, FVAL, NPDE)
      
      IMPLICIT REAL*8 (A-H, O-Z)
      DIMENSION U(NPDE), UX(NPDE), UXX(NPDE), FVAL(NPDE)
      COMMON /GEAR0/ HUSED, NQUSED, NS, NF, NJ
      COMMON /PARAMS/ PI
      COMMON /ASIAN1/ SIGMAT, RRT, XM, Z, MODEL
      COMMON /ASIAN2/ SIGMA, TIME, RR, XS, SS, ETA, XL, XR
      
      IF (MODEL.EQ.1) THEN
         FR = (1.0D0-DEXP(-RR*T))/RRT     
         FVAL(1) = (0.5D0*SIGMAT*SIGMAT) * ((X+FR)**2) * UXX(1)
      ENDIF
      
      IF (MODEL.EQ.2) THEN
         RT = (1.0D0-DEXP(-RR*T))/RR
         PF = (X*SIGMA*SIGMA)/(4.0D0*DSQRT(PI*ETA)) 
         FVAL(1) = (0.5D0*SIGMA*SIGMA) * ((X+RT)**2) * UXX(1)
         FVAL(1) = FVAL(1) + PF*DEXP(-0.25D0*X*X/ETA)*(X+2.0D0*RT)
      ENDIF
      
      RETURN
      
      END
      
C ------------------------------------------------------------------------------
      
      SUBROUTINE BNDRY(T, X, U, UX, DBDU, DBDUX, DZDT, NPDE)

      IMPLICIT REAL*8 (A-H, O-Z)
      DIMENSION U(NPDE), UX(NPDE), DZDT(NPDE)
      DIMENSION DBDU(NPDE,NPDE), DBDUX(NPDE,NPDE)
      COMMON /ASIAN1/ SIGMAT, RRT, XM, Z, MODEL
      COMMON /ASIAN2/ SIGMA, TIME, RR, XS, SS, ETA, XL, XR

C LEFT/RIGHT BOUNDARY MODEL 1:     
      IF (MODEL.EQ.1) THEN
         IF (X.LE.-XM) THEN
            DBDU (1,1) = (-Z*X + DABS(X) ) / 2.0D0
            DBDUX(1,1) = 0.0D0
            DZDT (1)   = 0.0D0
            RETURN
         ENDIF
         IF (X.LE.XM) THEN
           DBDU (1,1) = (-Z*X + DABS(X) ) / 2.0D0
           DBDUX(1,1) = 0.0D0
           DZDT (1)   = 0.0D0
           RETURN
         ENDIF   
      ENDIF
 
C LEFT/RIGHT BOUNDARY MODEL 2:      
      IF (MODEL.EQ.2) THEN
         EPS = 1.0D-20
         IF (X.LE.XL ) THEN
            DBDU (1,1) = EPS
            DBDUX(1,1) = 0.0D0
            DZDT (1)   = 0.0D0
            RETURN
         ENDIF   
         IF (X.GE.XR ) THEN
            DBDU (1,1) = EPS
            DBDUX(1,1) = 0.0D0
            DZDT (1)   = 0.0D0
            RETURN
         ENDIF
      ENDIF
   
      RETURN
      
      END
      
C ------------------------------------------------------------------------------
       
      SUBROUTINE UINIT(X, U, NPDE)
      
      IMPLICIT REAL*8 (A-H, O-Z)
      DIMENSION U(NPDE) 
      COMMON /ASIAN1/ SIGMAT, RRT, XM, Z, MODEL
      COMMON /ASIAN2/ SIGMA, TIME, RR, XS, SS, ETA, XL, XR
      
C NOTE : Z=+1 FOR A CALL AND Z-1 FOR A PUT  
    
      IF (MODEL.EQ.1) THEN
        U(1) = ( (-Z*X) + DABS(-X) ) / 2.0D0
      ENDIF
      
      IF (MODEL.EQ.2) THEN
        U(1) = 0.0D0
      ENDIF
    
      RETURN
      END
      
C ------------------------------------------------------------------------------
       
      SUBROUTINE DERIVF(T, X, U, UX, UXX, DFDU, DFDUX, DFDUXX, NPDE)
      
      IMPLICIT REAL*8 (A-H, O-Z)
      DIMENSION U(NPDE), UX(NPDE), UXX(NPDE)
      DIMENSION DFDU(NPDE,NPDE), DFDUX(NPDE,NPDE), DFDUXX(NPDE,NPDE)
      COMMON /ASIAN1/ SIGMAT, RRT, XM, Z, MODEL
      COMMON /ASIAN2/ SIGMA, TIME, RR, XS, SS, ETA, XL, XR

C
C IF THE USER DESIRES TO USE THE MF = 11 OR 21 OPTION IN ORDER TO SAVE
C ABOUT 10-20 PERCENT IN EXECUTION TIME (SEE BELOW), THEN THE USER MUST
C PROVIDE THE FOLLOWING SUBROUTINE WHICH PROVIDES INFORMATION ABOUT THE
C DERIVATIVES OF THE FUNCTION F ABOVE. THIS PROVIDES FOR MORE EFFICIENT
C JACOBIAN MATRIX GENERATION.  ON MOST COMPUTER SYSTEMS, THE USER WILL
C BE REQUIRED TO SUPPLY THIS SUBROUTINE AS A DUMMY SUBROUTINE IF THE
C OPTIONS MF = 12 OR 22 ARE USED (SEE BELOW).
C

C        THE PACKAGE PROVIDES VALUES OF THE INPUT VARIABLES T, X, U, UX,
C        AND UXX, AND THE USER SHOULD CONSTRUCT THIS ROUTINE TO PROVIDE
C        THE FOLLOWING CORRESPONDING VALUES OF THE OUTPUT ARRAYS
C        DFDU, DFDUX, AND DFDUXX FOR K,J = 1 TO NPDE...
C           DFDU(K,J) = PARTIAL DERIVATIVE OF THE K-TH COMPONENT OF THE
C                       PDE DEFINING FUNCTION  F  WITH RESPECT TO THE
C                       VARIABLE U(J).
C          DFDUX(K,J) = PARTIAL DERIVATIVE OF THE K-TH COMPONENT OF THE
C                       PDE DEFINING FUNCTION  F  WITH RESPECT TO THE
C                       VARIABLE UX(J).
C         DFDUXX(K,J) = PARTIAL DERIVATIVE OF THE K-TH COMPONENT OF THE
C                       PDE DEFINING FUNCTION  F  WITH RESPECT TO THE
C                       VARIABLE UXX(J).
C        NOTE... THE INCOMING VALUE OF  X  WILL BE A COLLOCATION POINT
C        VALUE.  

      PI = 4.0 * DATAN(1.0D0)
      
      IF (MODEL.EQ.1) THEN
        RT = (1.0D0-EXP(-RRT*T))/RRT
        DFDU(1,1)   = 0.0D0
        DFDUX(1,1)  = 0.0D0
        DFDUXX(1,1) = (SIGMAT**2) * ( X + RT )
      ENDIF
      
      IF (MODEL.EQ.1) THEN
         RT = (1.0D0-DEXP(-RR*T))/RR 
         F1 = (X*SIGMA*SIGMA)/(4.0D0*DSQRT(PI*ETA))    
         F2 = DEXP(-0.25D0*X*X/ETA)
         F3 = (X+2.0D0*RT)    
         DF1 = F1 / X
         DF2 = -2.0D0 * X * F2 / (4.0D0*ETA)
         DF3 = 1.0D0
         DFDUXX(1,1) = (SIGMA**2) * ( X + RT )
         DFDUX(1,1)  = 0.0D0
         DFDU(1,1)   = DF1*F2*F3 + F1*DF2*F3 + F1*F2*DF3
      ENDIF
      
      RETURN
      END

      
C ##############################################################################      
 
     
C      ALGORITHM 540R (REMARK ON ALG.540), COLLECTED ALGORITHMS FROM ACM.
C      THIS WORK PUBLISHED IN TRANSACTIONS ON MATHEMATICAL SOFTWARE,
C      VOL. 18, NO. 3, SEPTEMBER, 1992, PP. 343-344.
C
C
      SUBROUTINE PDECOL(T0, TOUT, DT, XBKPT, EPS, NINT, KORD, 
     *   NCC, NPDE, MF, INDEX, WORK, IWORK)
     
      IMPLICIT REAL*8 (A-H, O-Z)
C
C
C-------------------------------------------------------------------------------
C
C THIS IS THE MARCH 24, 1978 VERSION OF PDECOL.
C
C THIS PACKAGE WAS CONSTRUCTED SO AS TO CONFORM TO AS MANY ANSI-FORTRAN
C RULES AS WAS CONVENIENTLY POSSIBLE.  THE FORTRAN USED VIOLATES ANSI
C STANDARDS IN THE TWO WAYS LISTED BELOW....
C
C   1. SUBSCRIPTS OF THE GENERAL FORM C*V1 + V2 + V3 ARE USED
C      (POSSIBLY IN A PERMUTED ORDER), WHERE C IS AN INTEGER CONSTANT
C      AND V1, V2, AND V3 ARE INTEGER VARIABLES.
C
C   2. ARRAY NAMES APPEAR SINGLY IN DATA STATEMENTS IN THE ROUTINES
C      BSPLVN AND COSET.
C
C MACHINE DEPENDENT FEATURES......
C
C THIS VERSION OF PDECOL WAS DESIGNED FOR USE ON CDC MACHINES WITH
C A WORD LENGTH OF 60 BITS.  WE DO NOT RECOMMEND THE USE OF PDECOL WITH
C WORD LENGTHS OF LESS THAN 48 BITS.  THE MOST IMPORTANT MACHINE
C AND WORD LENGTH DEPENDENT CONSTANTS ARE DEFINED IN THE BLOCK DATA
C AND IN SUBROUTINES COLPNT AND COSET.  THE USER SHOULD CHECK THESE
C CAREFULLY FOR APPROPRIATENESS FOR HIS LOCAL SITUATION.  THE FORTRAN
C FUNCTIONS USED BY EACH ROUTINE ARE LISTED IN THE COMMENTS TO
C FACILITATE CONVERSION TO DOUBLE PRECISION.
C
C-------------------------------------------------------------------------------
C
C PDECOL IS THE DRIVER ROUTINE FOR A SOPHISTICATED PACKAGE OF
C SUBROUTINES WHICH IS DESIGNED TO SOLVE THE GENERAL SYSTEM OF
C NPDE NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS OF AT MOST SECOND
C ORDER ON THE INTERVAL (XLEFT,XRIGHT) FOR T .GT. T0 WHICH IS OF THE
C FORM....
C
C      DU/DT  =  F( T, X, U, UX, UXX )
C
C WHERE
C
C          U  =  (  U(1),  U(2), ... ,  U(NPDE) )
C         UX  =  ( UX(1), UX(2), ... , UX(NPDE) )
C        UXX  =  (UXX(1),UXX(2), ... ,UXX(NPDE) ) .
C
C EACH U(K) IS A FUNCTION OF THE SCALAR QUANTITIES T AND X.
C UX(K) REPRESENTS THE FIRST PARTIAL DERIVATIVE OF U(K) WITH RESPECT
C TO THE VARIABLE X,  UXX(K) REPRESENTS THE SECOND PARTIAL DERIVATIVE
C OF U(K) WITH RESPECT TO THE VARIABLE X, AND DU/DT IS THE VECTOR OF
C PARTIAL DERIVATIVES OF U WITH RESPECT TO THE TIME VARIABLE T.
C F  REPRESENTS AN ARBITRARY VECTOR VALUED FUNCTION WHOSE NPDE
C COMPONENTS DEFINE THE RESPECTIVE PARTIAL DIFFERENTIAL EQUATIONS OF
C THE PDE SYSTEM.  SEE SUBROUTINE F DESCRIPTION BELOW.
C
C BOUNDARY CONDITIONS
C
C   DEPENDING ON THE TYPE OF PDE(S), 0, 1, OR 2 BOUNDARY CONDITIONS
C   ARE REQUIRED FOR EACH PDE IN THE SYSTEM. THESE ARE IMPOSED AT XLEFT
C   AND/OR XRIGHT AND EACH MUST BE OF THE FORM....
C
C        B(U,UX)  =  Z(T)
C
C   WHERE  B  AND  Z  ARE ARBITRARY VECTOR VALUED FUNCTIONS WITH
C   NPDE COMPONENTS AND  U, UX, AND T  ARE AS ABOVE.  THESE BOUNDARY
C   CONDITIONS MUST BE CONSISTENT WITH THE INITIAL CONDITIONS WHICH ARE
C   DESCRIBED NEXT.
C
C INITIAL CONDITIONS
C
C   EACH SOLUTION COMPONENT  U(K)  IS ASSUMED TO BE A KNOWN (USER
C   PROVIDED) FUNCTION OF  X  AT THE INITIAL TIME T = T0.  THE
C   INITIAL CONDITION FUNCTIONS MUST BE CONSISTENT WITH THE BOUNDARY
C   CONDITIONS ABOVE, I.E. THE INITIAL CONDITION FUNCTIONS MUST
C   SATISFY THE BOUNDARY CONDITIONS FOR T = T0.  SEE SUBROUTINE UINIT
C   DESCRIPTION BELOW.
C
C-------------------------------------------------------------------------------
C
C REQUIRED USER SUPPLIED SUBROUTINES
C
C THE USER IS REQUIRED TO CONSTRUCT THREE SUBPROGRAMS AND A MAIN
C PROGRAM WHICH DEFINE THE PDE PROBLEM WHOSE SOLUTION IS TO BE
C ATTEMPTED.  THE THREE SUBPROGRAMS ARE...
C
C 1)  SUBROUTINE F( T, X, U, UX, UXX, FVAL, NPDE )
C     DIMENSION U(NPDE), UX(NPDE), UXX(NPDE), FVAL(NPDE)
C        THIS ROUTINE DEFINES THE DESIRED PARTIAL DIFFERENTIAL
C        EQUATIONS TO BE SOLVED.  THE PACKAGE PROVIDES VALUES OF THE
C        INPUT SCALARS T AND X AND INPUT ARRAYS (LENGTH NPDE) U, UX,
C        AND UXX, AND THE USER MUST CONSTRUCT THIS ROUTINE TO COMPUTE
C        THE OUTPUT ARRAY FVAL (LENGTH NPDE) WHICH CONTAINS THE
C        CORRESPONDING VALUES OF THE RIGHT HAND SIDES OF THE DESIRED
C        PARTIAL DIFFERENTIAL EQUATIONS, I.E.
C
C        FVAL(K) = THE VALUE OF THE RIGHT HAND SIDE OF THE K-TH PDE IN
C                  THE PDE SYSTEM ABOVE, FOR K = 1 TO NPDE.
C
C        THE INCOMING VALUE OF THE SCALAR QUANTITY X WILL BE A
C        COLLOCATION POINT VALUE (SEE INITAL AND COLPNT) AND THE
C        INCOMING VALUES IN THE ARRAYS U, UX AND UXX CORRESPOND TO THIS
C        POINT X AND TIME T.
C     RETURN
C     END
C
C 2)  SUBROUTINE BNDRY( T, X, U, UX, DBDU, DBDUX, DZDT, NPDE )
C     DIMENSION U(NPDE), UX(NPDE), DZDT(NPDE)
C     DIMENSION DBDU(NPDE,NPDE), DBDUX(NPDE,NPDE)
C        THIS ROUTINE IS USED TO PROVIDE THE PDE PACKAGE WITH NEEDED
C        INFORMATION ABOUT THE BOUNDARY CONDITION FUNCTIONS B AND Z
C        ABOVE.  THE PACKAGE PROVIDES VALUES OF THE INPUT VARIABLES
C        T, X, U, AND UX, AND THE USER IS TO DEFINE THE CORRESPONDING
C        OUTPUT VALUES OF THE DERIVATIVES OF THE FUNCTIONS B AND Z
C        WHERE....
C           DBDU(K,J) = PARTIAL DERIVATIVE OF THE K-TH COMPONENT OF THE
C                       VECTOR FUNCTION B(U,UX) ABOVE WITH RESPECT TO
C                       THE J-TH VARIABLE U(J).
C          DBDUX(K,J) = PARTIAL DERIVATIVE OF THE K-TH COMPONENT OF THE
C                       VECTOR FUNCTION B(U,UX) ABOVE WITH RESPECT TO
C                       THE J-TH VARIABLE UX(J).
C             DZDT(K) = DERIVATIVE OF THE K-TH COMPONENT OF THE VECTOR
C                       FUNCTION Z(T) ABOVE WITH RESPECT TO THE
C                       VARIABLE T.
C        NOTE... THE INCOMING VALUE OF X WILL BE EITHER XLEFT OR XRIGHT.
C        IF NO BOUNDARY CONDITION IS DESIRED FOR SAY THE K-TH PDE AT
C        ONE OR BOTH OF THE ENDPOINTS XLEFT OR XRIGHT, THEN DBDU(K,K)
C        AND DBDUX(K,K) SHOULD BOTH BE SET TO ZERO WHEN BNDRY IS
C        CALLED FOR THAT POINT.  WE REFER TO THIS AS A NULL BOUNDARY
C        CONDITION.  THIS ROUTINE CAN BE STRUCTURED AS FOLLOWS...
C        THE COMMON BLOCK /ENDPT/ IS NOT A PART OF PDECOL AND
C        MUST BE SUPPLIED AND DEFINED BY THE USER.
C     COMMON /ENDPT/ XLEFT
C     IF( X .NE. XLEFT ) GO TO 10
C        HERE DEFINE AND SET PROPER VALUES FOR DBDU(K,J), DBDUX(K,J),
C        AND DZDT(K) FOR K,J = 1 TO NPDE FOR THE LEFT BOUNDARY POINT
C        X = XLEFT.
C     RETURN
C  10 CONTINUE
C        HERE DEFINE AND SET PROPER VALUES FOR DBDU(K,J), DBDUX(K,J),
C        AND DZDT(K) FOR K,J = 1 TO NPDE FOR THE RIGHT BOUNDARY POINT
C        X = XRIGHT.
C     RETURN
C     END
C
C 3)  SUBROUTINE UINIT( X, U, NPDE )
C     DIMENSION U(NPDE)
C        THIS ROUTINE IS USED TO PROVIDE THE PDE PACKAGE WITH THE
C        NEEDED INITIAL CONDITION FUNCTION VALUES.  THE PACKAGE
C        PROVIDES A VALUE OF THE INPUT VARIABLE X, AND THE USER IS TO
C        DEFINE THE PROPER INITIAL VALUES (AT T = T0) FOR ALL OF THE
C        PDE COMPONENTS, I.E.
C           U(K) = DESIRED INITIAL VALUE OF PDE COMPONENT U(K) AT
C                  X AND T = T0 FOR K = 1 TO NPDE.
C        NOTE... THE INCOMING VALUE OF X WILL BE A COLLOCATION POINT
C        VALUE.  THE INITIAL CONDITIONS AND BOUNDARY CONDITIONS
C        MUST BE CONSISTENT (SEE ABOVE).
C     RETURN
C     END
C
C-------------------------------------------------------------------------------
C
C OPTIONAL USER SUPPLIED SUBROUTINE
C
C IF THE USER DESIRES TO USE THE MF = 11 OR 21 OPTION IN ORDER TO SAVE
C ABOUT 10-20 PERCENT IN EXECUTION TIME (SEE BELOW), THEN THE USER MUST
C PROVIDE THE FOLLOWING SUBROUTINE WHICH PROVIDES INFORMATION ABOUT THE
C DERIVATIVES OF THE FUNCTION F ABOVE. THIS PROVIDES FOR MORE EFFICIENT
C JACOBIAN MATRIX GENERATION.  ON MOST COMPUTER SYSTEMS, THE USER WILL
C BE REQUIRED TO SUPPLY THIS SUBROUTINE AS A DUMMY SUBROUTINE IF THE
C OPTIONS MF = 12 OR 22 ARE USED (SEE BELOW).
C
C 1)  SUBROUTINE DERIVF( T, X, U, UX, UXX, DFDU, DFDUX, DFDUXX, NPDE )
C     DIMENSION U(NPDE), UX(NPDE), UXX(NPDE)
C     DIMENSION DFDU(NPDE,NPDE), DFDUX(NPDE,NPDE), DFDUXX(NPDE,NPDE)
C        THE PACKAGE PROVIDES VALUES OF THE INPUT VARIABLES T, X, U, UX,
C        AND UXX, AND THE USER SHOULD CONSTRUCT THIS ROUTINE TO PROVIDE
C        THE FOLLOWING CORRESPONDING VALUES OF THE OUTPUT ARRAYS
C        DFDU, DFDUX, AND DFDUXX FOR K,J = 1 TO NPDE...
C           DFDU(K,J) = PARTIAL DERIVATIVE OF THE K-TH COMPONENT OF THE
C                       PDE DEFINING FUNCTION  F  WITH RESPECT TO THE
C                       VARIABLE U(J).
C          DFDUX(K,J) = PARTIAL DERIVATIVE OF THE K-TH COMPONENT OF THE
C                       PDE DEFINING FUNCTION  F  WITH RESPECT TO THE
C                       VARIABLE UX(J).
C         DFDUXX(K,J) = PARTIAL DERIVATIVE OF THE K-TH COMPONENT OF THE
C                       PDE DEFINING FUNCTION  F  WITH RESPECT TO THE
C                       VARIABLE UXX(J).
C        NOTE... THE INCOMING VALUE OF  X  WILL BE A COLLOCATION POINT
C        VALUE.
C     RETURN
C     END
C
C-------------------------------------------------------------------------------
C
C METHODS USED
C
C   THE PACKAGE PDECOL IS BASED ON THE METHOD OF LINES AND USES A
C   FINITE ELEMENT COLLOCATION PROCEDURE (WITH PIECEWISE POLYNOMIALS
C   AS THE TRIAL SPACE) FOR THE DISCRETIZATION OF THE SPATIAL VARIABLE
C   X.  THE COLLOCATION PROCEDURE REDUCES THE PDE SYSTEM TO A SEMI-
C   DISCRETE SYSTEM WHICH THEN DEPENDS ONLY ON THE TIME VARIABLE T.
C   THE TIME INTEGRATION IS THEN ACCOMPLISHED BY USE OF SLIGHTLY
C   MODIFIED STANDARD TECHNIQUES (SEE REFS. 1,2).
C
C   PIECEWISE POLYNOMIALS
C
C   THE USER IS REQUIRED TO SELECT THE PIECEWISE POLYNOMIAL SPACE
C   WHICH IS TO BE USED TO COMPUTE HIS APPROXIMATE SOLUTION. FIRST, THE
C   ORDER, KORD, OF THE POLYNOMIALS TO BE USED MUST BE SPECIFIED
C   (KORD = POLYNOMIAL DEGREE + 1).  NEXT, THE NUMBER OF PIECES
C   (INTERVALS), NINT, INTO WHICH THE SPATIAL DOMAIN (XLEFT,XRIGHT) IS
C   TO BE DIVIDED, IS CHOSEN.  THE NINT + 1 DISTINCT BREAKPOINTS OF
C   THE DOMAIN MUST BE DEFINED AND SET INTO THE ARRAY XBKPT IN
C   STRICTLY INCREASING ORDER, I.E.
C   XLEFT=XBKPT(1) .LT. XBKPT(2) .LT. ... .LT. XBKPT(NINT+1)=XRIGHT.
C   THE APPROXIMATE SOLUTION AT ANY TIME T WILL BE A POLYNOMIAL OF
C   ORDER KORD OVER EACH SUBINTERVAL (XBKPT(I),XBKPT(I+1)).  THE
C   NUMBER OF CONTINUITY CONDITIONS, NCC, TO BE IMPOSED ACROSS ALL OF
C   THE BREAKPOINTS IS THE LAST PIECE OF USER SUPPLIED DATA WHICH IS
C   REQUIRED TO UNIQUELY DETERMINE THE DESIRED PIECEWISE POLYNOMIAL
C   SPACE.  FOR EXAMPLE, NCC = 2 WOULD REQUIRE THAT THE APPROXIMATE
C   SOLUTION (MADE UP OF THE SEPARATE POLYNOMIAL PIECES) AND ITS FIRST
C   SPATIAL DERIVATIVE BE CONTINUOUS AT THE BREAKPOINTS AND HENCE ON
C   THE ENTIRE DOMAIN (XLEFT,XRIGHT).  NCC = 3 WOULD REQUIRE THAT THE
C   APPROXIMATE SOLUTION AND ITS FIRST AND SECOND SPATIAL DERIVATIVES
C   BE CONTINUOUS AT THE BREAKPOINTS, ETC. THE DIMENSION OF THIS LINEAR
C   SPACE IS KNOWN AND FINITE AND IS NCPTS = KORD*NINT - NCC*(NINT-1).
C   THE WELL-KNOWN B-SPLINE BASIS (SEE REF. 3) FOR THIS SPACE IS USED
C   BY PDECOL AND IT CONSISTS OF NCPTS KNOWN PIECEWISE POLYNOMIAL
C   FUNCTIONS BF(I,X), FOR I=1 TO NCPTS, WHICH DO NOT DEPEND ON THE
C   TIME VARIABLE T. WE WISH TO EMPHASIZE THAT THE PIECEWISE POLYNOMIAL
C   SPACE USED IN PDECOL (WHICH IS SELECTED BY THE USER) WILL DETERMINE
C   THE MAGNITUDE OF THE SPATIAL DISCRETIZATION ERRORS IN THE COMPUTED
C   APPROXIMATE SOLUTION.  THE PACKAGE HAS NO CONTROL OVER ERRORS
C   INTRODUCED BY THE USERS CHOICE OF THIS SPACE.  SEE INPUT PARAMETERS
C   BELOW.
C
C   COLLOCATION OVER PIECEWISE POLYNOMIALS
C
C   THE BASIC ASSUMPTION MADE IS THAT THE APPROXIMATE SOLUTION
C   SATISFIES
C                       NCPTS
C             U(T,X)  =  SUM  C(I,T) * BF(I,X)
C                        I=1
C
C   WHERE THE UNKNOWN COEFFICIENTS C DEPEND ONLY ON THE TIME T AND
C   THE KNOWN BASIS FUNCTIONS DEPEND ONLY ON X (WE HAVE ASSUMED THAT
C   NPDE = 1 FOR CONVENIENCE).  SO, AT ANY GIVEN TIME T THE APPROX-
C   IMATE SOLUTION IS A PIECEWISE POLYNOMIAL IN THE USER CHOSEN SPACE.
C   THE SEMI-DISCRETE EQUATIONS (ACTUALLY ORDINARY DIFFERENTIAL
C   EQUATIONS) WHICH DETERMINE THE COEFFICIENTS C ARE OBTAINED BY
C   REQUIRING THAT THE ABOVE APPROXIMATE U(T,X) SATISFY THE PDE AND
C   BOUNDARY CONDITIONS EXACTLY AT A SET OF NCPTS COLLOCATION POINTS
C   (SEE COLPNT).  THUS, PDECOL ACTUALLY COMPUTES THE BASIS FUNCTION
C   COEFFICIENTS RATHER THAN SPECIFIC APPROXIMATE SOLUTION VALUES.
C
C   REFERENCES
C
C   1. MADSEN, N.K. AND R.F. SINCOVEC, PDECOL - COLLOCATION SOFTWARE
C        FOR PARTIAL DIFFERENTIAL EQUATIONS, ACM-TOMS, VOL.  , NO.  ,
C   2. SINCOVEC, R.F. AND N.K. MADSEN, SOFTWARE FOR NONLINEAR PARTIAL
C        DIFFERENTIAL EQUATIONS, ACM-TOMS, VOL. 1, NO. 3,
C        SEPTEMBER 1975, PP. 232-260.
C   3. HINDMARSH, A.C., PRELIMINARY DOCUMENTATION OF GEARIB.. SOLUTION
C        OF IMPLICIT SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS WITH
C        BANDED JACOBIANS, LAWRENCE LIVERMORE LAB, UCID-30130, FEBRUARY
C        1976.
C   4. DEBOOR, C., PACKAGE FOR CALCULATING WITH B-SPLINES, SIAM J.
C        NUMER. ANAL., VOL. 14, NO. 3, JUNE 1977, PP. 441-472.
C
C-------------------------------------------------------------------------------
C
C USE OF PDECOL
C
C PDECOL IS CALLED ONCE FOR EACH DESIRED OUTPUT VALUE (TOUT) OF THE
C TIME T, AND IT IN TURN MAKES REPEATED CALLS TO THE CORE INTEGRATOR,
C STIFIB, WHICH ADVANCES THE TIME BY TAKING SINGLE STEPS UNTIL
C T .GE. TOUT.  INTERPOLATION TO THE EXACT TIME TOUT IS THEN DONE.
C SEE TOUT BELOW.
C
C
C SUMMARY OF SUGGESTED INPUT VALUES
C
C   IT IS OF COURSE IMPOSSIBLE TO SUGGEST INPUT PARAMETER VALUES WHICH
C   ARE APPROPRIATE FOR ALL PROBLEMS. THE FOLLOWING SUGGESTIONS ARE TO
C   BE USED ONLY IF YOU HAVE NO IDEA OF BETTER VALUES FOR YOUR PROBLEM.
C
C   DT    = 1.E-10
C   XBKPT = CHOOSE  NINT+1  EQUALLY SPACED VALUES SUCH THAT XBKPT(1) =
C           XLEFT AND  XBKPT(NINT+1) = XRIGHT.
C   EPS   = 1.E-4
C   NINT  = ENOUGH SO THAT ANY FINE STRUCTURE OF THE PROBLEM MAY BE
C           RESOLVED.
C   KORD  = 4
C   NCC   = 2
C   MF    = 22
C   INDEX = 1 (ON FIRST CALL ONLY, THEN 0 THEREAFTER).
C
C
C THE INPUT PARAMETERS ARE..
C   T0    =  THE INITIAL VALUE OF T, THE INDEPENDENT VARIABLE
C              (USED ONLY ON FIRST CALL).
C   TOUT  =  THE VALUE OF T AT WHICH OUTPUT IS DESIRED NEXT.  SINCE
C              THE PACKAGE CHOOSES ITS OWN TIME STEP SIZES, THE
C              INTEGRATION WILL NORMALLY GO SLIGHTLY BEYOND TOUT
C              AND THE PACKAGE WILL INTERPOLATE TO T = TOUT.
C   DT    =  THE INITIAL STEP SIZE IN T, IF INDEX = 1, OR, THE
C              MAXIMUM STEP SIZE ALLOWED (MUST BE .GT. 0), IF INDEX = 3.
C              USED FOR INPUT ONLY WHEN INDEX = 1 OR 3. SEE BELOW.
C   XBKPT =  THE ARRAY OF PIECEWISE POLYNOMIAL BREAKPOINTS.
C              THE NINT+1 VALUES MUST BE STRICTLY INCREASING WITH
C              XBKPT(1) = XLEFT AND XBKPT(NINT+1) = XRIGHT (USED ONLY
C              ON FIRST CALL).
C   EPS   =  THE RELATIVE TIME ERROR BOUND  (USED ONLY ON THE
C              FIRST CALL, UNLESS INDEX = 4).  SINGLE STEP ERROR
C              ESTIMATES DIVIDED BY CMAX(I) WILL BE KEPT LESS THAN
C              EPS IN ROOT-MEAN-SQUARE NORM. THE VECTOR CMAX OF WEIGHTS
C              IS COMPUTED IN PDECOL.  INITIALLY CMAX(I) IS SET TO
C              DABS(C(I)), WITH A DEFAULT VALUE OF 1 IF DABS(C(I)) .LT. 1.
C              THEREAFTER, CMAX(I) IS THE LARGEST VALUE
C              OF DABS(C(I)) SEEN SO FAR, OR THE INITIAL CMAX(I) IF
C              THAT IS LARGER.  TO ALTER EITHER OF THESE, CHANGE THE
C              APPROPRIATE STATEMENTS IN THE DO-LOOPS ENDING AT
C              STATEMENTS 50 AND 130 BELOW.  THE USER SHOULD EXERCISE
C              SOME DISCRETION IN CHOOSING EPS.  IN GENERAL, THE
C              OVERALL RUNNING TIME FOR A PROBLEM WILL BE GREATER IF
C              EPS IS CHOSEN SMALLER. THERE IS USUALLY LITTLE REASON TO
C              CHOOSE EPS MUCH SMALLER THAN THE ERRORS WHICH ARE BEING
C              INTRODUCED BY THE USERS CHOICE OF THE POLYNOMIAL SPACE.
C              SEE RELATED COMMENTS CONCERNING CMAX BELOW STATEMENT 40.
C   NINT  =  THE NUMBER OF SUBINTERVALS INTO WHICH THE SPATIAL DOMAIN
C              (XLEFT,XRIGHT) IS TO BE DIVIDED (MUST BE .GE. 1)
C              (USED ONLY ON FIRST CALL).
C   KORD  =  THE ORDER OF THE PIECEWISE POLYNOMIAL SPACE TO BE USED.
C              ITS VALUE MUST BE GREATER THAN 2 AND LESS THAN 21.  FOR
C              FIRST ATTEMPTS WE RECOMMEND KORD = 4.  IF THE SOLUTION
C              IS SMOOTH AND MUCH ACCURACY IS DESIRED, HIGHER VALUES
C              MAY PROVE TO BE MORE EFFICIENT.  WE HAVE SELDOM USED
C              VALUES OF KORD IN EXCESS OF 8 OR 9, THOUGH THEY ARE
C              AVAILABLE FOR USE IN PDECOL (USED ONLY ON FIRST CALL).
C   NCC   =  THE NUMBER OF CONTINUITY CONDITIONS TO BE IMPOSED ON THE
C              APPROXIMATE SOLUTION AT THE BREAKPOINTS IN XBKPT.
C              NCC MUST BE GREATER THAN 1 AND LESS THAN KORD.  WE
C              RECOMMEND THE USE OF NCC = 2
C              SINCE THEORY PREDICTS THAT DRAMATICALLY MORE
C              ACCURATE RESULTS CAN OFTEN BE OBTAINED USING THIS CHOICE
C              (USED ONLY ON FIRST CALL).
C   NPDE  =  THE NUMBER OF PARTIAL DIFFERENTIAL EQUATIONS IN THE SYSTEM
C              TO BE SOLVED (USED ONLY ON FIRST CALL).
C   MF    =  THE METHOD FLAG  (USED ONLY ON FIRST CALL, UNLESS
C              INDEX = 4).  ALLOWED VALUES ARE 11, 12, 21, 22.
C              FOR FIRST ATTEMPTS WE RECOMMEND THE USE OF MF = 22.
C              MF HAS TWO DECIMAL DIGITS, METH AND MITER
C              (MF = 10*METH + MITER).
C              METH IS THE BASIC METHOD INDICATOR..
C                METH = 1  MEANS THE ADAMS METHODS (GENERALIZATIONS OF
C                          CRANK-NICOLSON).
C                METH = 2  MEANS THE BACKWARD DIFFERENTIATION
C                          FORMULAS (BDF), OR STIFF METHODS OF GEAR.
C              MITER IS THE ITERATION METHOD INDICATOR
C              AND DETERMINES HOW THE JACOBIAN MATRIX IS
C              TO BE COMPUTED..
C                MITER = 1 MEANS CHORD METHOD WITH ANALYTIC JACOBIAN.
C                          FOR THIS USER SUPPLIES SUBROUTINE DERIVF.
C                          SEE DESCRIPTION ABOVE.
C                MITER = 2 MEANS CHORD METHOD WITH JACOBIAN CALCULATED
C                          INTERNALLY BY FINITE DIFFERENCES.  SEE
C                          SUBROUTINES PSETIB AND DIFFF.
C   INDEX =  INTEGER USED ON INPUT TO INDICATE TYPE OF CALL,
C              WITH THE FOLLOWING VALUES AND MEANINGS..
C                 1    THIS IS THE FIRST CALL FOR THIS PROBLEM.
C                 0    THIS IS NOT THE FIRST CALL FOR THIS PROBLEM,
C                      AND INTEGRATION IS TO CONTINUE.
C                 2    SAME AS 0 EXCEPT THAT TOUT IS TO BE HIT
C                      EXACTLY (NO INTERPOLATION IS DONE).  SEE NOTE
C                      BELOW.  ASSUMES TOUT .GE. THE CURRENT T.
C                      IF TOUT IS .LT. THE CURRENT TIME, THEN TOUT IS
C                      RESET TO THE CURRENT TIME AND CONTROL IS
C                      RETURNED TO THE USER.  A CALL TO VALUES WILL
C                      PRODUCE ANSWERS FOR THE NEW VALUE OF TOUT.
C                 3    SAME AS 0 EXCEPT CONTROL RETURNS TO CALLING
C                      PROGRAM AFTER ONE STEP.  TOUT IS IGNORED AND
C                      DT MUST BE SET .GT. 0 TO A MAXIMUM ALLOWED
C                      DT VALUE. SEE ABOVE.
C                 4    THIS IS NOT THE FIRST CALL FOR THE PROBLEM,
C                      AND THE USER HAS RESET EPS AND/OR MF.
C              SINCE THE NORMAL OUTPUT VALUE OF INDEX IS 0,
C              IT NEED NOT BE RESET FOR NORMAL CONTINUATION.
C
C NOTE.. THE PACKAGE MUST HAVE TAKEN AT LEAST ONE SUCCESSFUL TIME
C STEP BEFORE A CALL WITH INDEX = 2 OR 4 IS ALLOWED.
C AFTER THE INITIAL CALL, IF A NORMAL RETURN OCCURRED AND A NORMAL
C CONTINUATION IS DESIRED, SIMPLY RESET TOUT AND CALL AGAIN.
C ALL OTHER PARAMETERS WILL BE Y FOR THE NEXT CALL.
C A CHANGE OF PARAMETERS WITH INDEX = 4 CAN BE MADE AFTER
C EITHER A SUCCESSFUL OR AN UNSUCCESSFUL RETURN PROVIDED AT LEAST
C ONE SUCCESSFUL TIME STEP HAS BEEN MADE.
C
C   WORK  =  FLOATING POINT WORKING ARRAY FOR PDECOL. WE RECOMMEND
C              THAT IT BE INITIALIZED TO ZERO BEFORE THE FIRST CALL
C              TO PDECOL.  ITS TOTAL LENGTH MUST BE AT LEAST
C
C              KORD + 4*NPDE + 9*NPDE**2 + NCPTS*(3*KORD + 2) +
C              NPDE*NCPTS*(3*ML + MAXDER + 7)
C
C              WHERE ML AND MAXDER ARE DEFINED BELOW (SEE STORAGE
C              ALLOCATION).
C
C   IWORK =  INTEGER WORKING ARRAY FOR PDECOL.  THE FIRST TWO
C              LOCATIONS MUST BE DEFINED AS FOLLOWS...
C              IWORK(1) = LENGTH OF USERS ARRAY WORK
C              IWORK(2) = LENGTH OF USERS ARRAY IWORK
C              THE TOTAL LENGTH OF IWORK MUST BE AT LEAST
C              NCPTS*(NPDE + 1).
C OUTPUT
C
C THE SOLUTION VALUES ARE NOT RETURNED DIRECTLY TO THE USER BY PDECOL.
C THE METHODS USED IN PDECOL COMPUTE BASIS FUNCTION COEFFICIENTS, SO
C THE USER (AFTER A RETURN FROM PDECOL) MUST CALL THE PACKAGE ROUTINE
C VALUES TO OBTAIN HIS APPROXIMATE SOLUTION VALUES AT ANY DESIRED SPACE
C POINTS X AT THE TIME T = TOUT.  SEE THE COMMENTS IN SUBROUTINE VALUES
C FOR DETAILS ON HOW TO PROPERLY MAKE THE CALL.
C
C EXECUTION ERROR MESSAGES WILL BE PRINTED BY PDECOL ON LOGICAL UNIT
C LOUT WHICH IS THE ONLY VARIABLE IN THE COMMON BLOCK /IOUNIT/.  A
C DEFAULT OF LOUT = 3 IS SET IN THE BLOCK DATA.
C
C THE COMMON BLOCK /GEAR0/ CONTAINS THE VARIABLES DTUSED, NQUSED,
C NSTEP, NFE, AND NJE AND CAN BE ACCESSED EXTERNALLY BY THE USER IF
C DESIRED.  RESPECTIVELY, IT CONTAINS THE STEP SIZE LAST USED (SUCCESS-
C FULLY), THE ORDER LAST USED (SUCCESSFULLY), THE NUMBER OF STEPS TAKEN
C SO FAR, THE NUMBER OF RESIDUAL EVALUATIONS (RES CALLS) SO FAR,
C AND THE NUMBER OF MATRIX EVALUATIONS (PSETIB CALLS) SO FAR.
C DIFFUN CALLS ARE COUNTED IN WITH RESIDUAL EVALUATIONS.
C
C THE OUTPUT PARAMETERS ARE..
C   DT    =  THE STEP SIZE USED LAST, WHETHER SUCCESSFULLY OR NOT.
C   TOUT  =  THE OUTPUT VALUE OF T.  IF INTEGRATION WAS SUCCESSFUL,
C              AND THE INPUT VALUE OF INDEX WAS NOT 3, TOUT IS
C              UNCHANGED FROM ITS INPUT VALUE.  OTHERWISE, TOUT
C              IS THE CURRENT VALUE OF T TO WHICH THE INTEGRATION
C              HAS BEEN COMPLETED.
C   INDEX =  INTEGER USED ON OUTPUT TO INDICATE RESULTS,
C              WITH THE FOLLOWING VALUES AND MEANINGS..
C         0    INTEGRATION WAS COMPLETED TO TOUT OR BEYOND.
C        -1    THE INTEGRATION WAS HALTED AFTER FAILING TO PASS THE
C              ERROR TEST EVEN AFTER REDUCING DT BY A FACTOR OF
C              1.E10 FROM ITS INITIAL VALUE.
C        -2    AFTER SOME INITIAL SUCCESS, THE INTEGRATION WAS
C              HALTED EITHER BY REPEATED ERROR TEST FAILURES OR BY
C              A TEST ON EPS.  TOO MUCH ACCURACY HAS BEEN REQUESTED.
C        -3    THE INTEGRATION WAS HALTED AFTER FAILING TO ACHIEVE
C              CORRECTOR CONVERGENCE EVEN AFTER REDUCING DT BY A
C              FACTOR OF 1.E10 FROM ITS INITIAL VALUE.
C        -4    SINGULAR MATRIX ENCOUNTERED.  PROBABLY DUE TO STORAGE
C              OVERWRITES.
C        -5    INDEX WAS 4 ON INPUT, BUT THE DESIRED CHANGES OF
C              PARAMETERS WERE NOT IMPLEMENTED BECAUSE TOUT
C              WAS NOT BEYOND T.  INTERPOLATION TO T = TOUT WAS
C              PERFORMED AS ON A NORMAL RETURN.  TO TRY AGAIN,
C              SIMPLY CALL AGAIN WITH INDEX = 4 AND A NEW TOUT.
C        -6    ILLEGAL INDEX VALUE.
C        -7    ILLEGAL EPS VALUE.
C        -8    AN ATTEMPT TO INTEGRATE IN THE WRONG DIRECTION.  THE
C              SIGN OF DT IS WRONG RELATIVE TO T0 AND TOUT.
C        -9    DT .EQ. 0.0.
C       -10    ILLEGAL NINT VALUE.
C       -11    ILLEGAL KORD VALUE.
C       -12    ILLEGAL NCC VALUE.
C       -13    ILLEGAL NPDE VALUE.
C       -14    ILLEGAL MF VALUE.
C       -15    ILLEGAL BREAKPOINTS - NOT STRICTLY INCREASING.
C       -16    INSUFFICIENT STORAGE FOR WORK OR IWORK.
C
C-------------------------------------------------------------------------------
C
C SUMMARY OF ALL PACKAGE ROUTINES
C
C PDECOL - STORAGE ALLOCATION, ERROR CHECKING, INITIALIZATION, REPEATED
C          CALLS TO STIFIB TO ADVANCE THE TIME.
C
C INTERP - INTERPOLATES COMPUTED BASIS FUNCTION COEFFICIENTS TO THE
C          DESIRED OUTPUT TIMES, TOUT, FOR USE BY VALUES.
C
C INITAL - INITIALIZATION, GENERATION AND STORAGE OF PIECEWISE POLY-
C          NOMIAL SPACE BASIS FUNCTION VALUES AND DERIVATIVES, DET-
C          ERMINES THE BASIS FUNCTION COEFFICINTS OF THE PIECEWISE
C          POLYNOMIALS WHICH INTERPOLATE THE USERS INITIAL CONDITIONS.
C
C COLPNT - GENERATION OF REQUIRED COLLOCATION POINTS.
C
C BSPLVD - B-SPLINE PACKAGE ROUTINES WHICH ALLOW FOR EVALUATION OF
C BSPLVN   ANY B-SPLINE BASIS FUNCTION OR DERIVATIVE VALUE.
C INTERV
C
C VALUES - GENERATION AT ANY POINT(S) OF VALUES OF THE COMPUTED
C          APPROXIMATE SOLUTION AND ITS DERIVATIVES WHICH ARE
C          PIECEWISE POLYNOMIALS.  THE SUBROUTINE IS CALLED ONLY BY
C          THE USER.
C
C STIFIB - CORE INTEGRATOR, TAKES SINGLE TIME STEPS TO ADVANCE THE
C          TIME.  ASSEMBLES AND SOLVES THE PROPER NONLINEAR EQUATIONS
C          WHICH ARE RELATED TO USE OF ADAMS OR GEAR TYPE INTEGRATION
C          FORMULAS.  CHOOSES PROPER STEP SIZE AND INTEGRATION FORMULA
C          ORDER TO MAINTAIN A DESIRED ACCURACY.  DESIGNED FOR ODE
C          PROBLEMS OF THE FORM A * (DY/DT) = G(T,Y).
C
C COSET  - GENERATES INTEGRATION FORMULA AND ERROR CONTROL COEFFICIENTS.
C
C RES    - COMPUTES RESIDUAL VECTORS USED IN SOLVING THE NONLINEAR
C          EQUATIONS BY A MODIFIED NEWTON METHOD.
C
C DIFFUN - COMPUTES A**-1 * G(T,Y) WHERE A AND G ARE AS ABOVE (STIFIB).
C
C ADDA   - ADDS THE A MATRIX TO A GIVEN MATRIX IN BAND FORM.
C
C EVAL   - EVALUATES THE COMPUTED PIECEWISE POLYNOMIAL SOLUTION AND
C          DERIVATIVES AT COLLOCATION POINTS.
C
C GFUN   - EVALUATES THE FUNCTION G(T,Y) BY CALLING EVAL AND THE USER
C          SUBROUTINES F AND BNDRY.
C
C PSETIB - GENERATES PROPER JACOBIAN MATRICES REQUIRED BY THE MODIFIED
C          NEWTON METHOD.
C
C DIFFF  - PERFORMS SAME ROLE AS THE USER ROUTINE DERIVF. COMPUTES
C          DERIVATIVE APPROXIMATIONS BY USE OF FINITE DIFFERENCES.
C
C DECB   - PERFORM AN LU DECOMPOSTION AND FORWARD AND BACKWARD
C SOLB     SUBSTITUTION FOR SOLVING BANDED SYSTEMS OF LINEAR EQUATIONS.
C
C-----------------------------------------------------------------------
C
C
C STORAGE ALLOCATION
C
C SINCE PDECOL IS A DYNAMICALLY DIMENSIONED PROGRAM, MOST OF ITS
C WORKING STORAGE IS PROVIDED BY THE USER IN THE ARRAYS WORK AND IWORK.
C THE FOLLOWING GIVES A LIST OF THE ARRAYS WHICH MAKE UP THE CONTENTS
C WORK AND IWORK, THEIR LENGTHS, AND THEIR USES.  WHEN MORE THAN ONE
C NAME IS GIVEN, IT INDICATES THAT DIFFERENT NAMES ARE USED FOR THE
C SAME ARRAY IN DIFFERENT PARTS OF THE PROGRAM.  THE DIFFERENT NAMES
C OCCUR BECAUSE PDECOL IS AN AMALGAMATION OF SEVERAL OTHER CODES
C WRITTEN BY DIFFERENT PEOPLE AND WE HAVE TRIED TO LEAVE THE SEPARATE
C PARTS AS UNCHANGED FROM THEIR ORIGINAL VERSIONS AS POSSIBLE.
C
C
C   NAMES       LENGTH        USE
C ---------     ------------  -------------------------------------
C
C BC            4*NPDE**2     BOUNDARY CONDITION INFORMATION.
C WORK
C
C A             3*KORD*NCPTS  BASIS FUNCTION VALUES AT COLLOCATION POINT
C WORK(IW1)
C
C XT            NCPTS + KORD  BREAKPOINT SEQUENCE FOR GENERATION OF BASI
C WORK(IW2)                   FUNCTION VALUES.
C
C XC            NCPTS         COLLOCATION POINTS.
C WORK(IW3)
C
C CMAX          NPDE*NCPTS    VALUES USED IN ESTIMATING TIME
C YMAX                        INTEGRATION ERRORS.
C WORK(IW4)
C
C ERROR         NPDE*NCPTS    TIME INTEGRATION ERRORS.
C WORK(IW5)
C
C SAVE1         NPDE*NCPTS    WORKING STORAGE FOR THE TIME INTEGRATION
C WORK(IW6)                   METHOD.
C
C SAVE2         NPDE*NCPTS    WORKING STORAGE FOR THE TIME INTEGRATION
C WORK(IW7)                   METHOD.
C
C SAVE3         NPDE*NCPTS    WORKING STORAGE FOR THE TIME INTEGRATION
C WORK(IW8)                   METHOD.
C
C UVAL          3*NPDE        WORKING STORAGE FOR VALUES OF U, UX, AND
C WORK(IW9)                   UXX AT ONE POINT.
C
C C             NPDE*NCPTS*   CURRENT BASIS FUNCTION COEFFICIENT VALUES
C Y               (MAXDER+1)  AND THEIR SCALED TIME DERIVATIVES.
C WORK(IW10)
C
C DFDU          NPDE**2       WORKING STORAGE USED TO COMPUTE THE
C WORK(IW11)                  JACOBIAN MATRIX.
C
C DFDUX         NPDE**2       WORKING STORAGE USED TO COMPUTE THE
C WORK(IW12)                  JACOBIAN MATRIX.
C
C DFDUXX        NPDE**2       WORKING STORAGE USED TO COMPUTE THE
C WORK(IW13)                  JACOBIAN MATRIX.
C
C DBDU          NPDE**2       BOUNDARY CONDITION INFORMATION.
C WORK(IW14)
C
C DBDUX         NPDE**2       BOUNDARY CONDITION INFORMATION.
C WORK(IW15)
C
C DZDT          NPDE          BOUNDARY CONDITION INFORMATION.
C WORK(IW16)
C
C PW            NPDE*NCPTS*   STORAGE AND PROCESSING OF THE JACOBIAN
C WORK(IW17)      (3*ML+1)    MATRIX.
C
C ILEFT         NCPTS         POINTERS TO BREAKPOINT SEQUENCE FOR
C IWORK                       GENERATION OF BASIS FUNCTION VALUES.
C
C IPIV          NPDE*NCPTS    PIVOT INFORMATION FOR THE LU DECOMPOSED
C IWORK(IW18)                 JACOBIAN MATRIX PW.
C
C WHERE...
C
C      NCPTS = KORD*NINT - NCC*(NINT-1)
C         ML = NPDE*(KORD+IQUAD-1) - 1
C      IQUAD = 1 IF KORD = 3 AND A NULL BOUNDARY CONDITION EXISTS
C      IQUAD = 0 OTHERWISE
C     MAXDER = 5 UNLESS OTHERWISE SET BY THE USER INTO /OPTION/.
C
C THE COMMON BLOCK /OPTION/ CONTAINS THE TWO VARIABLES NOGAUS AND
C MAXDER.  NOGAUS IS SET .EQ. 0 IN THE BLOCK DATA.  IT CAN BE CHANGED
C TO BE SET .EQ. 1 IF THE GAUSS-LEGENDRE COLLOCATION POINTS ARE NOT
C DESIRED WHEN NCC = 2 (SEE ABOVE AND COLPNT).  MAXDER IS SET
C .EQ. 5 IN THE BLOCK DATA AND ITS VALUE REPRESENTS THE
C MAXIMUM ORDER OF TIME INTEGRATION FORMULA ALLOWED.  ITS VALUE
C AFFECTS THE STORAGE REQUIRED IN WORK AND MAY BE CHANGED IF
C DESIRED.  SEE COSET FOR RESTRICTIONS.  THESE CHANGES MAY BE MADE BY
C THE USER BY ACCESSING /OPTION/ IN HIS CALLING PROGRAM (BEFORE THE
C FIRST CALL TO PDECOL) OR BY CHANGING THE DATA STATEMENT IN
C THE BLOCK DATA.
C
C-----------------------------------------------------------------------
C
C
C COMMUNICATION
C
C EACH SUBROUTINE IN THE PACKAGE CONTAINS A COMMUNICATION SUMMARY
C AS INDICATED BELOW.
C
C PACKAGE ROUTINES CALLED..   EVAL,INITAL,INTERP,STIFIB
C USER ROUTINES CALLED..      BNDRY
C CALLED BY..                 USERS MAIN PROGRAM
C FORTRAN FUNCTIONS USED..    ABS,DMAX1,FLOAT,DSQRT
C-----------------------------------------------------------------------
      SAVE
      COMMON /GEAR0/ DTUSED,NQUSED,NSTEP,NFE,NJE
      COMMON /GEAR1/ T,DTC,DTMN,DTMX,EPSC,UROUND,N,MFC,KFLAG,JSTART
      COMMON /GEAR9/ EPSJ,R0,ML,MU,MW,NM1,N0ML,N0W
      COMMON /OPTION/ NOGAUS,MAXDER
      COMMON /SIZES/ NIN,KOR,NC,NPD,NCPTS,NEQN,IQUAD
      COMMON /ISTART/ IW1,IW2,IW3,IW4,IW5,IW6,IW7,IW8,IW9,IW10,
     *                 IW11,IW12,IW13,IW14,IW15,IW16,IW17,IW18
      COMMON /IOUNIT/ LOUT
      DIMENSION WORK(KORD+NPDE*(4+9*NPDE)+(KORD+(NINT-1)*(KORD-NCC))*
     *   (3*KORD+2+NPDE*(3*(KORD-1)*NPDE+MAXDER+4))),
     *   IWORK((NPDE+1)*(KORD+(NINT-1)*(KORD-NCC))), XBKPT(NINT+1)
      IF (INDEX .EQ. 0) GO TO 60
      IF (INDEX .EQ. 2) GO TO 70
      IF (INDEX .EQ. 4) GO TO 80
      IF (INDEX .EQ. 3) GO TO 90
C-----------------------------------------------------------------------
C SEVERAL CHECKS ARE MADE HERE TO DETERMINE IF THE INPUT PARAMETERS
C HAVE LEGAL VALUES.  ERROR CHECKS ARE MADE ON INDEX, EPS, (T0-TOUT)*DT,
C DT, NINT, KORD, NCC, NPDE, MF, WHETHER THE BREAKPOINT SEQUENCE IS
C STRICTLY INCREASING, AND WHETHER THERE IS SUFFICIENT STORAGE
C PROVIDED FOR WORK AND IWORK.  PROBLEM DEPENDENT PARAMETERS ARE
C CALCULATED AND PLACED IN COMMON.
C-----------------------------------------------------------------------
      IERID = -6
      IF (INDEX .NE. 1) GO TO 320
      IERID = IERID - 1
      IF (EPS .LE. 0.) GO TO 320
      IERID = IERID - 1
      IF ((T0-TOUT)*DT .GT. 0.) GO TO 320
      IERID = IERID - 1
      IF (DT .EQ. 0.0) GO TO 320
      IERID = IERID - 1
      NIN = NINT
      IF (NIN .LT. 1) GO TO 320
      IERID = IERID - 1
      KOR = KORD
      IF (KOR .LT. 3  .OR.  KOR .GT. 20) GO TO 320
      IERID = IERID - 1
      NC = NCC
      IF (NCC .LT. 2  .OR.  NCC .GE. KOR) GO TO 320
      IERID = IERID - 1
      NPD = NPDE
      NPDE2 = NPD*NPD
      IF (NPDE .LT. 1) GO TO 320
      IERID = IERID - 1
      IF (MF.NE.22.AND.MF.NE.21.AND.MF.NE.12.AND.MF.NE.11) GO TO 320
      IERID = IERID - 1
      DO 10 K=1,NIN
        IF(XBKPT(K) .GE. XBKPT(K+1)) GO TO 320
   10 CONTINUE
      NCPTS = KOR + (NIN - 1) * (KOR - NCC)
      NEQN = NPDE * NCPTS
      ML = (KOR-1)*NPDE - 1
      MU = ML
      MW = ML + ML + 1
      N0W = NEQN*MW
      IWSAVE = IWORK(1)
      IISAVE = IWORK(2)
      IW1 = 4*NPDE2 + 1
      IW2 = IW1 + 3*KORD*NCPTS
      IW3 = IW2 + NCPTS + KORD
      IW4 = IW3 + NCPTS
      IW5 = IW4 + NEQN
      IW6 = IW5 + NEQN
      IW7 = IW6 + NEQN
      IW8 = IW7 + NEQN
      IW9 = IW8 + NEQN
      IW10 = IW9 + 3*NPDE
      IW11 = IW10 + NEQN*(MAXDER+1)
      IW12 = IW11 + NPDE2
      IW13 = IW12 + NPDE2
      IW14 = IW13 + NPDE2
      IW15 = IW14 + NPDE2
      IW16 = IW15 + NPDE2
      IW17 = IW16 + NPDE
      IW18 = NCPTS + 1
      IERID = IERID - 1
      IWSTOR = IW17 + NEQN*(3*ML+1) - 1
      IISTOR = IW18 + NEQN - 1
      IF ( IWSAVE .LT. IWSTOR  .OR.  IISAVE .LT. IISTOR ) GO TO 335
C-----------------------------------------------------------------------
C PERFORM INITIALIZATION TASKS.  IF KORD .EQ. 3 THEN CALCULATE THE BAND-
C WIDTH OF THE ASSOCIATED MATRIX PROBLEM BY DETERMINING THE TYPE OF
C BOUNDARY CONDITIONS, THEN CHECK FOR SUFFICIENT STORAGE AGAIN.
C-----------------------------------------------------------------------
      CALL INITAL(KOR,WORK(IW1),WORK(IW6),XBKPT,WORK(IW2),WORK(IW3),
     *            WORK(IW17),IWORK(IW18),IWORK)
      IF(IQUAD .NE. 0) GO TO 280
      IF( KOR .NE. 3 ) GO TO 40
      CALL EVAL(1,NPDE,WORK(IW6),WORK(IW9),WORK(IW1),IWORK)
      CALL BNDRY(T0,WORK(IW3),WORK(IW9),WORK(IW9+NPDE),WORK(IW14),
     *           WORK(IW15),WORK(IW16),NPDE)
      DO 20 K=1,NPDE
        I = K + NPDE*(K-1) - 1
        IF(WORK(IW14+I) .EQ. 0.0 .AND. WORK(IW15+I) .EQ. 0.0)
     *     IQUAD = 1
   20 CONTINUE
      CALL EVAL(NCPTS,NPDE,WORK(IW6),WORK(IW9),WORK(IW1),IWORK)
      CALL BNDRY(T0,WORK(IW3+NCPTS-1),WORK(IW9),WORK(IW9+NPDE),
     *           WORK(IW14),WORK(IW15),WORK(IW16),NPDE)
      DO 30 K=1,NPDE
        I = K + NPDE*(K-1) - 1
        IF(WORK(IW14+I) .EQ. 0.0 .AND. WORK(IW15+I) .EQ. 0.0)
     *     IQUAD = 1
   30 CONTINUE
      ML = ML + IQUAD*NPDE
      MU = ML
      MW = ML + ML + 1
      N0W = NEQN*MW
   40 CONTINUE
      IWSTOR = IW17 + NEQN*(3*ML+1) - 1
      IF ( IWSAVE .LT. IWSTOR ) GO TO 335
C-----------------------------------------------------------------------
C IF INITIAL VALUES OF CMAX OTHER THAN THOSE SET BELOW ARE DESIRED,
C THEY SHOULD BE SET HERE.  ALL CMAX(I) MUST BE POSITIVE.
C HAVING PROPER VALUES OF CMAX FOR THE PROBLEM BEING SOLVED IS AS
C IMPORTANT AS CHOOSING EPS (SEE ABOVE), SINCE ERRORS ARE
C MEASURED RELATIVE TO CMAX.  IF VALUES FOR DTMN OR DTMX, THE
C BOUNDS ON DABS(DT), OTHER THAN THOSE BELOW ARE DESIRED, THEY
C SHOULD BE SET BELOW.
C-----------------------------------------------------------------------
      DO 50 I = 1,NEQN
        I1 = I - 1
        WORK(IW4+I1) = DABS(WORK(IW6+I1))
        IF (WORK(IW4+I1) .LT. 1.) WORK(IW4+I1) = 1.
   50   WORK(IW10+I1) = WORK(IW6+I1)
      N = NEQN
      T = T0
      DTC = DT
      DTMN = DABS(DT)
      DTUSED = 0.
      EPSC = EPS
      MFC = MF
      JSTART = 0
      EPSJ =  DSQRT(UROUND)
      NM1 = NEQN - 1
      N0ML = NEQN*ML
      NHCUT = 0
      KFLAG = 0
      TOUTP = T0
      IF ( T0 .EQ. TOUT ) GO TO 360
   60 DTMX = DABS(TOUT-TOUTP)*10.
      GO TO 140
C
   70 DTMX = DABS(TOUT-TOUTP)*10.
      IF ((T-TOUT)*DTC .GE. 0.) GO TO 340
      GO TO 150
C
   80 IF ((T-TOUT)*DTC .GE. 0.) GO TO 300
      JSTART = -1
      EPSC = EPS
      MFC = MF
      GO TO 100
C
   90 DTMX = DT
  100 IF ((T+DTC) .EQ. T) WRITE(LOUT,110)
  110 FORMAT(36H WARNING..  T + DT = T ON NEXT STEP.)
C-----------------------------------------------------------------------
C TAKE A TIME STEP BY CALLING THE INTEGRATOR.
C-----------------------------------------------------------------------
      CALL STIFIB (NEQN,WORK(IW10),WORK(IW4),WORK(IW5),WORK(IW6),
     *       WORK(IW7),WORK(IW8),WORK(IW17),IWORK(IW18),WORK,IWORK)
C
      KGO = 1 - KFLAG
      GO TO (120, 160, 220, 260, 280), KGO
C KFLAG  =   0,  -1,  -2,  -3   -4
C
  120 CONTINUE
C-----------------------------------------------------------------------
C NORMAL RETURN FROM INTEGRATOR.
C
C THE WEIGHTS CMAX(I) ARE UPDATED.  IF DIFFERENT VALUES ARE DESIRED,
C THEY SHOULD BE SET HERE.  A TEST IS MADE FOR EPS BEING TOO SMALL
C FOR THE MACHINE PRECISION.
C
C ANY OTHER TESTS OR CALCULATIONS THAT ARE REQUIRED AFTER EVERY
C STEP SHOULD BE INSERTED HERE.
C
C IF INDEX = 3, SAVE1 IS SET TO THE CURRENT C VALUES ON RETURN.
C IF INDEX = 2, DT IS CONTROLLED TO HIT TOUT (WITHIN ROUNDOFF
C ERROR), AND THEN THE CURRENT C VALUES ARE PUT IN SAVE1 ON RETURN.
C FOR ANY OTHER VALUE OF INDEX, CONTROL RETURNS TO THE INTEGRATOR
C UNLESS TOUT HAS BEEN REACHED.  THEN INTERPOLATED VALUES OF C ARE
C COMPUTED AND STORED IN SAVE1 ON RETURN.
C IF INTERPOLATION IS NOT DESIRED, THE CALL TO INTERP SHOULD BE
C REMOVED AND CONTROL TRANSFERRED TO STATEMENT 340 INSTEAD OF 360.
C-----------------------------------------------------------------------
      D = 0.
      DO 130 I = 1,NEQN
        I1 = I - 1
        AYI = DABS(WORK(IW10+I1))
        WORK(IW4+I1) = DMAX1(WORK(IW4+I1), AYI)
  130   D = D + (AYI/WORK(IW4+I1))**2
      D = D*(UROUND/EPS)**2
      IF (D .GT.  FLOAT(NEQN)) GO TO 240
      IF (INDEX .EQ. 3) GO TO 340
      IF (INDEX .EQ. 2) GO TO 150
  140 IF ((T-TOUT)*DTC .LT. 0.) GO TO 100
      CALL INTERP(TOUT,WORK(IW10),NEQN,WORK(IW6))
      GO TO 360
C
  150 IF (((T+DTC)-TOUT)*DTC .LE. 0.) GO TO 100
      IF (DABS(T-TOUT) .LE. 100.*UROUND*DTMX) GO TO 340
      IF ((T-TOUT)*DTC .GE. 0.) GO TO 340
      DTC = (TOUT - T)*(1. - 4.*UROUND)
      JSTART = -1
      GO TO 100
C-----------------------------------------------------------------------
C ON AN ERROR RETURN FROM INTEGRATOR, AN IMMEDIATE RETURN OCCURS IF
C KFLAG = -2 OR -4, AND RECOVERY ATTEMPTS ARE MADE OTHERWISE.
C TO RECOVER, DT AND DTMN ARE REDUCED BY A FACTOR OF .1 UP TO 10
C TIMES BEFORE GIVING UP.
C-----------------------------------------------------------------------
  160 WRITE (LOUT,170) T
  170 FORMAT(//35H KFLAG = -1 FROM INTEGRATOR AT T = ,E16.8/
     *   41H  ERROR TEST FAILED WITH DABS(DT) = DTMIN/)
  180 IF (NHCUT .EQ. 10) GO TO 200
      NHCUT = NHCUT + 1
      DTMN = .1*DTMN
      DTC = .1*DTC
      WRITE (LOUT,190) DTC
  190 FORMAT(25H  DT HAS BEEN REDUCED TO ,E16.8,
     *   26H  AND STEP WILL BE RETRIED//)
      JSTART = -1
      GO TO 100
C
  200 WRITE (LOUT,210)
  210 FORMAT(//44H PROBLEM APPEARS UNSOLVABLE WITH GIVEN INPUT//)
      GO TO 340
C
  220 WRITE (LOUT,230) T,DTC
  230 FORMAT(//35H KFLAG = -2 FROM INTEGRATOR AT T = ,E16.8,6H  DT =,
     *  E16.8/52H  THE REQUESTED ERROR IS SMALLER THAN CAN BE HANDLED//)
      GO TO 340
C
  240 WRITE (LOUT,250) T
  250 FORMAT(//37H INTEGRATION HALTED BY DRIVER AT T = ,E16.8/
     *   56H  EPS TOO SMALL TO BE ATTAINED FOR THE MACHINE PRECISION/)
      KFLAG = -2
      GO TO 340
C
  260 WRITE (LOUT,270) T
  270 FORMAT(//35H KFLAG = -3 FROM INTEGRATOR AT T = ,E16.8/
     *   45H  CORRECTOR CONVERGENCE COULD NOT BE ACHIEVED/)
      GO TO 180
C
  280 WRITE (LOUT,290)
  290 FORMAT(//28H SINGULAR MATRIX ENCOUNTERED,
     *         35H PROBABLY DUE TO STORAGE OVERWRITES//)
      KFLAG = -4
      GO TO 340
C
  300 WRITE(LOUT,310) T,TOUT,DTC
  310 FORMAT(//45H INDEX = -1 ON INPUT WITH (T-TOUT)*DT .GE. 0./
     *   4H T =,E16.8,9H   TOUT =,E16.8,8H   DTC =,E16.8/
     *   44H INTERPOLATION WAS DONE AS ON NORMAL RETURN./
     *   41H DESIRED PARAMETER CHANGES WERE NOT MADE.)
      CALL INTERP(TOUT,WORK(IW10),NEQN,WORK(IW6))
      INDEX = -5
      RETURN
C
  320 WRITE(LOUT,330) IERID
  330 FORMAT(//24H ILLEGAL INPUT...INDEX= ,I3//)
      INDEX = IERID
      RETURN
C
  335 WRITE(LOUT,336) IWSTOR,IWSAVE,IISTOR,IISAVE
  336 FORMAT(//21H INSUFFICIENT STORAGE/24H  WORK MUST BE OF LENGTH,
     * I10,5X,12HYOU PROVIDED,I10/24H IWORK MUST BE OF LENGTH,I10,5X,
     * 12HYOU PROVIDED,I10//)
      INDEX = IERID
      RETURN
C
  340 TOUT = T
      DO 350 I = 1,NEQN
        I1 = I - 1
  350   WORK(IW6+I1) = WORK(IW10+I1)
  360 INDEX = KFLAG
      TOUTP = TOUT
      DT = DTUSED
      IF (KFLAG .NE. 0) DT = DTC
      RETURN
      END
C
C
C ##############################################################################
C
C
      SUBROUTINE VALUES(X,USOL,SCTCH,NDIM1,NDIM2,NPTS,NDERV,WORK)
      IMPLICIT REAL*8 (A-H, O-Z)
C-------------------------------------------------------------------------------
C SUBROUTINE VALUES COMPUTES THE SOLUTION U AND THE FIRST NDERV
C DERIVATIVES OF U AT THE NPTS POINTS X AND AT TIME TOUT AND RETURNS
C THEM IN THE ARRAY USOL.  THIS ROUTINE MUST BE USED TO OBTAIN
C SOLUTION VALUES SINCE PDECOL DOES NOT RETURN ANY SOLUTION VALUES
C TO THE USER.  SEE PDECOL.
C
C THE CALLING PARAMETERS ARE...
C   X     =  AN ARBITRARY VECTOR OF SPATIAL POINTS OF LENGTH NPTS AT
C            WHICH THE SOLUTION AND THE FIRST NDERV DERIVATIVE VALUES
C            ARE TO BE CALCULATED.  IF X .LT. XLEFT ( X .GT. XRIGHT )
C            THEN THE PIECEWISE POLYNOMIAL OVER THE LEFTMOST ( RIGHT-
C            MOST ) INTERVAL IS EVALUATED TO CALCULATE THE SOLUTION
C            VALUES AT THIS UNUSUAL VALUE OF  X.  SEE PDECOL.
C
C   USOL  =  AN ARRAY WHICH CONTAINS THE SOLUTION AND THE FIRST
C            NDERV DERIVATIVES OF THE SOLUTION AT ALL THE POINTS IN
C            THE INPUT VECTOR X.  IN PARTICULAR, USOL(J,I,K) CONTAINS
C            THE VALUE OF THE (K-1)-ST DERIVATIVE OF THE J-TH PDE
C            COMPONENT AT THE I-TH POINT OF THE X VECTOR FOR
C            J = 1 TO NPDE, I = 1 TO NPTS, AND K = 1 TO NDERV+1.
C
C   SCTCH =  A USER SUPPLIED WORKING STORAGE ARRAY OF LENGTH AT LEAST
C            KORD*(NDERV+1).  SEE BELOW AND PDECOL FOR DEFINITIONS OF
C            THESE PARAMETERS.
C
C   NDIM1 =  THE FIRST DIMENSION OF THE OUTPUT ARRAY USOL IN THE CALLING
C            PROGRAM.  NDIM1 MUST BE .GE. NPDE.
C
C   NDIM2 =  THE SECOND DIMENSION OF THE OUTPUT ARRAY USOL IN THE
C            CALLING PROGRAM.  NDIM2 MUST BE .GE. NPTS.
C
C   NPTS  =  THE NUMBER OF POINTS IN THE X VECTOR.
C
C   NDERV =  THE NUMBER OF DERIVATIVE VALUES OF THE SOLUTION THAT ARE
C            TO BE CALCULATED.  NDERV SHOULD BE LESS THAN KORD SINCE
C            THE KORD-TH DERIVATIVE OF A POLYNOMIAL OF DEGREE KORD-1
C            IS EQUAL TO ZERO.  SEE PDECOL.
C
C   WORK  =  THE USERS WORKING STORAGE ARRAY WHICH IS USED IN THIS CASE
C            TO PROVIDE THE CURRENT BASIS FUNCTION COEFFICIENTS AND THE
C            PIECEWISE POLYNOMIAL BREAKPOINT SEQUENCE.
C
C PACKAGE ROUTINES CALLED..  BSPLVD,INTERV
C USER ROUTINES CALLED..     NONE
C CALLED BY..                USERS MAIN PROGRAM
C FORTRAN FUNCTIONS USED..   NONE
C
C-----------------------------------------------------------------------
      SAVE ILEFT, MFLAG
      COMMON /SIZES/ NINT,KORD,NCC,NPDE,NCPTS,NEQN,IQUAD
      COMMON /ISTART/ IW1,IW2,IW3,IW4,IW5,IW6,IDUM(12)
      COMMON /OPTION/ NOGAUS,MAXDER
      DIMENSION USOL(NDIM1,NDIM2,NDERV+1),X(NPTS),SCTCH(KORD*(NDERV+1)),
     *          WORK(KORD+NPDE*(4+9*NPDE)+(KORD+(NINT-1)*(KORD-NCC))*
     *               (3*KORD+2+NPDE*(3*(KORD-1)*NPDE+MAXDER+4)))
      DATA ILEFT/0/, MFLAG/0/
      NDERV1 = NDERV + 1
      DO 20 IPTS=1,NPTS
        CALL INTERV(WORK(IW2),NCPTS,X(IPTS),ILEFT,MFLAG)
        CALL BSPLVD(WORK(IW2),KORD,X(IPTS),ILEFT,SCTCH,NDERV1)
        IK = ILEFT - KORD
        DO 10 M=1,NDERV1
          I1 = (M-1)*KORD
          DO 10 K=1,NPDE
            USOL(K,IPTS,M) = 0.
            DO 10 I=1,KORD
              I2 = (I+IK-1)*NPDE + IW6 - 1
              USOL(K,IPTS,M) = USOL(K,IPTS,M) + WORK(I2+K) * SCTCH(I+I1)
   10   CONTINUE
   20 CONTINUE
      RETURN
      END
C
C
C ##############################################################################
C
C
      BLOCK DATA
      IMPLICIT REAL*8 (A-H, O-Z)
C-------------------------------------------------------------------------------
C IN THE FOLLOWING DATA STATEMENT, SET..
C    LOUT   = THE LOGICAL UNIT NUMBER FOR THE OUTPUT OF MESSAGES DURING
C             THE INTEGRATION.
C    NOGAUS = SET .EQ. 1 IF THE GAUSS-LEGENDRE COLLOCATION POINTS ARE
C             NOT DESIRED WHEN NCC = 2 (SEE PDECOL AND COLPNT).
C    MAXDER = SET .EQ. 5.  ITS VALUE REPRESENTS THE MAXIMUM ORDER OF
C             THE TIME INTEGRATION ALLOWED.  ITS VALUE AFFECTS THE STOR-
C             AGE REQUIRED IN   WORK   AND MAY BE CHANGED IF DESIRED
C             (SEE COSET FOR RESTRICTIONS).
C    UROUND = THE UNIT ROUNDOFF OF THE MACHINE, I.E. THE SMALLEST
C             POSITIVE U SUCH THAT 1. + U .NE. 1. ON THE MACHINE.
C-------------------------------------------------------------------------------
      COMMON /GEAR1/ DUM(5),UROUND,IDUM(4)
      COMMON /OPTION/ NOGAUS,MAXDER
      COMMON /IOUNIT/ LOUT
C***
C*** UROUND SET TO SINGLE PRECISION FOR A SUN SPARC2
C***
C***     DATA LOUT,NOGAUS,MAXDER,UROUND/6, 0, 5, 5.960464D-08/
C
      DATA LOUT,NOGAUS,MAXDER,UROUND/66, 0, 5, 2.22D-16/     
      END
C
C
C ##############################################################################
C
C
      SUBROUTINE INITAL(K,A,RHS,X,XT,XC,PW,IPIV,ILEFT)
      IMPLICIT REAL*8 (A-H, O-Z)
C-------------------------------------------------------------------------------
C INITAL IS CALLED ONLY ONCE BY PDECOL TO PERFORM INITIALIZATION TASKS.
C THESE TASKS INCLUDE - 1) DEFINING THE PIECEWISE POLYNOMIAL SPACE
C BREAKPOINT SEQUENCE, 2) CALLING THE SUBROUTINE COLPNT TO DEFINE THE
C REQUIRED COLLOCATION POINTS, 3) DEFING THE PIECEWISE POLYNOMIAL SPACE
C BASIS FUNCTION VALUES (PLUS FIRST AND SECOND DERIVATIVE VALUES) AT
C THE COLLOCATION POINTS, AND 4) DEFINING THE INITIAL BASIS FUNCTION
C COEFFICIENTS WHICH DETERMINE THE PIECEWISE POLYNOMIAL WHICH
C INTERPOLATES THE USER SUPPLIED (UINIT) INITIAL CONDITION FUNCTION(S)
C AT THE COLLOCATION POINTS.
C
C K     = ORDER OF PIECEWISE POLYNOMIAL SPACE.
C A     = BASIS FUNCTION VALUES GENERATED BY INITAL.
C RHS   = TEMPORARY STORAGE USED TO RETURN INITIAL CONDITION COEFFICIENT
C         VALUES.
C X     = USER DEFINED PIECEWISE POLYNOMIAL BREAKPOINTS.
C XT    = PIECEWISE POLYNOMIAL BREAKPOINT SEQUENCE GENERATED BY INITAL.
C XC    = COLLOCATION POINTS GENERATED BY INITAL.
C PW    = STORAGE FOR BAND MATRIX USED TO GENERATE INITIAL
C         COEFFICIENT VALUES.
C IPIV  = PIVOT INFORMATION FOR LINEAR EQUATION SOLVER DECB-SOLB.
C ILEFT = POINTERS TO BREAKPOINT SEQUENCE GENERATED BY INITAL.
C
C PACKAGE ROUTINES CALLED..  BSPLVD,COLPNT,DECB,INTERV,SOLB
C USER ROUTINES CALLED..     UINIT
C CALLED BY..                PDECOL
C FORTRAN FUNCTIONS USED..   MAX0,MIN0
C-------------------------------------------------------------------------------
      COMMON /SIZES/ NINT,KORD,NCC,NPDE,NCPTS,NEQN,IER
      COMMON /GEAR9/ EPSJ,R0,ML,MU,IDUM(3),N0W
      DIMENSION A(K,3,NCPTS),RHS(NEQN),X(NINT+1),XT(NCPTS+KORD),
     *          XC(NCPTS),PW(NEQN*(3*ML+1)),
     *          IPIV(NEQN),ILEFT(NCPTS)
      MFLAG = -2
      IER = 0
C-----------------------------------------------------------------------
C SET UP THE PIECEWISE POLYNOMIAL SPACE BREAKPOINT SEQUENCE.
C-----------------------------------------------------------------------
      KRPT = KORD - NCC
      DO 10 I=1,KORD
        XT(NCPTS+I) = X(NINT+1)
   10   XT(I) = X(1)
      DO 20 I=2,NINT
        I1 = (I-2)*KRPT + KORD
        DO 20 J=1,KRPT
   20     XT(I1+J) = X(I)
C-----------------------------------------------------------------------
C SET UP COLLOCATION POINTS ARRAY XC.
C-----------------------------------------------------------------------
      CALL COLPNT(X, XC, XT)
C-----------------------------------------------------------------------
C GENERATE THE ILEFT ARRAY.  STORE THE BASIS FUNCTION VALUES IN THE
C ARRAY A.  THE ARRAY A IS DIMENSIONED A(KORD,3,NCPTS) AND A(K,J,I)
C CONTAINS THE VALUE OF THE (J-1)-ST DERIVATIVE (J = 1,2,3) OF THE K-TH
C NONZERO BASIS FUNCTION (K = 1, ... ,KORD) AT THE I-TH COLLOCATION
C POINT (I = 1, ... ,NCPTS).  SET UP RHS FOR INTERPOLATING THE INITIAL
C CONDITIONS AT THE COLLOCATION POINTS.  SET THE INTERPOLATION MATRIX
C INTO THE BANDED MATRIX PW.
C-----------------------------------------------------------------------
      DO 30 I=1,N0W
   30   PW(I) = 0.
      DO 40 I=1,NCPTS
        CALL INTERV(XT,NCPTS,XC(I),ILEFT(I),MFLAG)
        CALL BSPLVD(XT,KORD,XC(I),ILEFT(I),A(1,1,I),3)
        I1 = NPDE * (I-1)
        CALL UINIT(XC(I),RHS(I1+1),NPDE)
        ICOL = ILEFT(I) - I - 1
        JL = MAX0(1,I+2-NCPTS)
        JU = MIN0(KORD,KORD+I-2)
        DO 40 J=JL,JU
          J1 = I1 + NEQN * (NPDE * (ICOL + J) - 1)
          DO 40 JJ=1,NPDE
   40       PW(JJ+J1) = A(J,1,I)
C-----------------------------------------------------------------------
C LU DECOMPOSE THE MATRIX PW.
C-----------------------------------------------------------------------
      CALL DECB (NEQN,NEQN,ML,MU,PW,IPIV,IER)
      IF ( IER .NE. 0 ) RETURN
C-----------------------------------------------------------------------
C SOLVE THE LINEAR SYSTEM   PW*Z = RHS.  THIS GIVES THE BASIS FUNCTION
C COEFFICIENTS FOR THE INITIAL CONDITIONS.
C-----------------------------------------------------------------------
      CALL SOLB (NEQN,NEQN,ML,MU,PW,RHS,IPIV)
      RETURN
      END
C
C
C ##############################################################################
C
C
      SUBROUTINE COLPNT(X, XC, XT)
      IMPLICIT REAL*8 (A-H, O-Z)
C-------------------------------------------------------------------------------
C COLPNT IS CALLED ONLY ONCE BY INITAL TO DEFINE THE REQUIRED COLLOCA-
C TION POINTS WHICH ARE TO BE USED WITH THE USER SELECTED PIECEWISE
C POLYNOMIAL SPACE.  THE COLLOCATION POINTS ARE CHOSEN SUCH THAT THEY
C ARE EITHER THE POINTS AT WHICH THE PIECEWISE POLYNOMIAL SPACE BASIS
C FUNCTIONS ATTAIN THEIR UNIQUE MAXIMUM VALUES, OR, THE GAUSS-LEGENDRE
C QUADRATURE POINTS WITHIN EACH PIECEWISE POLYNOMIAL SPACE SUBINTERVAL,
C DEPENDING UPON THE SPACE BEING USED AND THE DESIRE OF THE USER.
C
C X  = USER DEFINED PIECEWISE POLYNOMIAL BREAKPOINTS.
C XC = COLLOCATION POINTS DEFINED BY COLPNT.
C XT = PIECEWISE POLYNOMIAL BREAKPOINT SEQUENCE.
C
C PACKAGE ROUTINES CALLED..  BSPLVD,INTERV
C USER ROUTINES CALLED..     NONE
C CALLED BY..                INITAL
C FORTRAN FUNCTIONS USED..   NONE
C-------------------------------------------------------------------------------
      SAVE ILEFT
      COMMON /SIZES/ NINT,KORD,NCC,NPDE,NCPTS,NEQN,IQUAD
      COMMON /OPTION/ NOGAUS,MAXDER
      DIMENSION RHO(40),X(NINT+1),XC(NCPTS),XT(NCPTS+KORD)
      DATA ILEFT/0/
C-----------------------------------------------------------------------
C IF THE VARIABLE NOGAUS IN THE COMMON BLOCK /OPTION/ IS SET .EQ. 1,
C THE USE OF THE GAUSS-LEGENDRE POINTS IS PROHIBITED FOR ALL CASES.
C NOGAUS IS CURRENTLY SET .EQ. 0 BY A DATA STATEMENT IN THE BLOCK DATA.
C THE USER MAY CHANGE THIS AS DESIRED.
C-----------------------------------------------------------------------
      IF ( NCC .NE. 2  .OR. NOGAUS .EQ. 1 ) GO TO 200
C-----------------------------------------------------------------------
C COMPUTE THE COLLOCATION POINTS TO BE AT THE GAUSS-LEGENDRE POINTS IN
C EACH PIECEWISE POLYNOMIAL SPACE SUBINTERVAL.  THE ARRAY RHO IS SET TO
C CONTAIN THE GAUSS-LEGENDRE POINTS FOR THE STANDARD INTERVAL (-1,1).
C-----------------------------------------------------------------------
      IPTS = KORD - 2
      GO TO (10,20,30,40,50,60,70,80,90,100,110,120,130,140,150,160,170,
     *    180),IPTS
   10 RHO(1) = 0.
      GO TO 190
   20 RHO(2) = .577350269189626D-00
      RHO(1) = - RHO(2)
      GO TO 190
   30 RHO(3) = .774596669241483D-00
      RHO(1) = - RHO(3)
      RHO(2) = 0.
      GO TO 190
   40 RHO(3) = .339981043584856D-00
      RHO(2) = - RHO(3)
      RHO(4) = .861136311594053D-00
      RHO(1) = - RHO(4)
      GO TO 190
   50 RHO(4) = .538469310105683D-00
      RHO(2) = - RHO(4)
      RHO(5) = .906179845938664D-00
      RHO(1) = - RHO(5)
      RHO(3) = 0.
      GO TO 190
   60 RHO(4) = .238619186083197D-00
      RHO(3) = - RHO(4)
      RHO(5) = .661209386466265D-00
      RHO(2) = - RHO(5)
      RHO(6) = .932469514203152D-00
      RHO(1) = - RHO(6)
      GO TO 190
   70 RHO(5) = .405845151377397D-00
      RHO(3) = - RHO(5)
      RHO(6) = .741531185599394D-00
      RHO(2) = - RHO(6)
      RHO(7) = .949107912342759D-00
      RHO(1) = - RHO(7)
      RHO(4) = 0.
      GO TO 190
   80 RHO(5) = .183434642495650D-00
      RHO(4) = - RHO(5)
      RHO(6) = .525532409916329D-00
      RHO(3) = - RHO(6)
      RHO(7) = .796666477413627D-00
      RHO(2) = - RHO(7)
      RHO(8) = .960289856497536D-00
      RHO(1) = - RHO(8)
      GO TO 190
   90 RHO( 5) = .0
      RHO( 6) = .324253423403809D-00
      RHO( 7) = .613371432700590D-00
      RHO( 8) = .836031107326636D-00
      RHO( 9) = .968160239507626D-00
      DO 95 I=1,4
   95   RHO(I) = -RHO(10-I)
      GO TO 190
  100 RHO( 6) = .148874338981631D-00
      RHO( 7) = .433395394129247D-00
      RHO( 8) = .679409568299024D-00
      RHO( 9) = .865063366688984D-00
      RHO(10) = .973906528517172D-00
      DO 105 I=1,5
  105   RHO(I) = -RHO(11-I)
      GO TO 190
  110 RHO( 6) = .0
      RHO( 7) = .269543155952345D-00
      RHO( 8) = .519096129206812D-00
      RHO( 9) = .730152005574049D-00
      RHO(10) = .887062599768095D-00
      RHO(11) = .978228658146057D-00
      DO 115 I=1,5
  115   RHO(I) = -RHO(12-I)
      GO TO 190
  120 RHO( 7) = .125233408511469D-00
      RHO( 8) = .367831498998180D-00
      RHO( 9) = .587317954286617D-00
      RHO(10) = .769902674194305D-00
      RHO(11) = .904117256370475D-00
      RHO(12) = .981560634246719D-00
      DO 125 I=1,6
  125   RHO(I) = -RHO(13-I)
      GO TO 190
  130 RHO( 7) = .0
      RHO( 8) = .230458315955135D-00
      RHO( 9) = .448492751036447D-00
      RHO(10) = .642349339440340D-00
      RHO(11) = .801578090733310D-00
      RHO(12) = .917598399222978D-00
      RHO(13) = .984183054718588D-00
      DO 135 I=1,6
  135   RHO(I) = -RHO(14-I)
      GO TO 190
  140 RHO( 8) = .108054948707344D-00
      RHO( 9) = .319112368927890D-00
      RHO(10) = .515248636358154D-00
      RHO(11) = .687292904811685D-00
      RHO(12) = .827201315069765D-00
      RHO(13) = .928434883663574D-00
      RHO(14) = .986283808696812D-00
      DO 145 I=1,7
  145   RHO(I) = -RHO(15-I)
      GO TO 190
  150 RHO( 8) = .0
      RHO( 9) = .201194093997435D-00
      RHO(10) = .394151347077563D-00
      RHO(11) = .570972172608539D-00
      RHO(12) = .724417731360170D-00
      RHO(13) = .848206583410427D-00
      RHO(14) = .937273392400706D-00
      RHO(15) = .987992518020485D-00
      DO 155 I = 1,7
  155   RHO(I) = -RHO(16-I)
      GO TO 190
  160 RHO( 9) = .950125098376374D-01
      RHO(10) = .281603550779259D-00
      RHO(11) = .458016777657227D-00
      RHO(12) = .617876244402644D-00
      RHO(13) = .755404408355003D-00
      RHO(14) = .865631202387832D-00
      RHO(15) = .944575023073233D-00
      RHO(16) = .989400934991650D-00
      DO 165 I=1,8
  165   RHO(I) = -RHO(17-I)
      GO TO 190
  170 RHO( 9) = .0
      RHO(10) = .178484181495848D-00
      RHO(11) = .351231763453876D-00
      RHO(12) = .512690537086477D-00
      RHO(13) = .657671159216691D-00
      RHO(14) = .781514003896801D-00
      RHO(15) = .880239153726986D-00
      RHO(16) = .950675521768768D-00
      RHO(17) = .990575475314417D-00
      DO 175 I=1,8
  175   RHO(I) = -RHO(18-I)
      GO TO 190
  180 RHO(10) = .847750130417353D-01
      RHO(11) = .251886225691506D-00
      RHO(12) = .411751161462843D-00
      RHO(13) = .559770831073948D-00
      RHO(14) = .691687043060353D-00
      RHO(15) = .803704958972523D-00
      RHO(16) = .892602466497556D-00
      RHO(17) = .955823949571398D-00
      RHO(18) = .991565168420931D-00
      DO 185 I=1,9
  185   RHO(I) = -RHO(19-I)
C-----------------------------------------------------------------------
C COMPUTE THE GAUSS-LEGENDRE COLLOCATION POINTS IN EACH SUBINTERVAL.
C-----------------------------------------------------------------------
  190 DO 195 I=1,NINT
        FAC = ( X(I+1) - X(I) ) * .5
        DO 195 J = 1,IPTS
          KNOT = IPTS * (I-1) + J + 1
  195     XC(KNOT) = X(I) + FAC * ( RHO(J) + 1. )
      XC(1) = X(1)
      XC(NCPTS) = X(NINT+1)
      RETURN
C-----------------------------------------------------------------------
C COMPUTE THE COLLOCATION POINTS TO BE AT THE POINTS WHERE THE BASIS
C FUNCTIONS ATTAIN THEIR MAXIMA.  A BISECTION METHOD IS USED TO FIND
C THE POINTS TO MACHINE PRECISION.  THIS PROCESS COULD BE SPEEDED UP
C BY USING A SECANT METHOD IF DESIRED.
C-----------------------------------------------------------------------
  200 ITOP = NCPTS - 1
      MFLAG = -2
      XC(1) = X(1)
      XC(NCPTS) = X(NINT+1)
      DO 240 I=2,ITOP
        XOLD = 1.E+20
        XL = XT(I)
        XR = XT(I+KORD)
  210   XNEW = .5 * (XL + XR)
        IF( XOLD .EQ. XNEW ) GO TO 240
        CALL INTERV(XT,NCPTS,XNEW,ILEFT,MFLAG)
        CALL BSPLVD(XT,KORD,XNEW,ILEFT,RHO,2)
        DO 220 J=1,KORD
          IF( I .EQ. J + ILEFT - KORD ) GO TO 230
  220   CONTINUE
  230   XVAL = RHO(KORD+J)
        IF( XVAL .EQ. 0.0 ) XR = XNEW
        IF( XVAL .GT. 0.0 ) XL = XNEW
        IF( XVAL .LT. 0.0 ) XR = XNEW
        XOLD = XNEW
        GO TO 210
  240 XC(I) = XR
      RETURN
      END
C
C
C ##############################################################################
C
C
      SUBROUTINE BSPLVD ( XT, K, X, ILEFT, VNIKX, NDERIV )
      IMPLICIT REAL*8 (A-H, O-Z)
C-------------------------------------------------------------------------------
C THIS SUBROUTINE IS PART OF THE B-SPLINE PACKAGE FOR THE STABLE
C EVALUATION OF ANY B-SPLINE BASIS FUNCTION OR DERIVATIVE VALUE.
C SEE REFERENCE BELOW.
C
C CALCULATES THE VALUE AND THE FIRST NDERIV-1 DERIVATIVES OF ALL
C B-SPLINES WHICH DO NOT VANISH AT X.  THE ROUTINE FILLS THE TWO-
C DIMENSIONAL ARRAY VNIKX(J,IDERIV), J=IDERIV, ... ,K WITH NONZERO
C VALUES OF B-SPLINES OF ORDER K+1-IDERIV, IDERIV=NDERIV, ... ,1, BY
C REPEATED CALLS TO BSPLVN.
C
C XT     = PIECEWISE POLYNOMIAL BREAKPOINT SEQUENCE.
C K      = ORDER OF THE PIECEWISE POLYNOMIAL SPACE.
C X      = POINT AT WHICH THE B-SPLINE IS TO BE EVALUATED.
C ILEFT  = POINTER TO THE BREAKPOINT SEQUENCE.
C VNIKX  = TABLE OF B-SPLINE VALUES AND DERIVATIVES.
C NDERIV = DETERMINES NUMBER OF DERIVATIVES TO BE GENERATED.
C
C REFERENCE
C
C    DEBOOR, C., PACKAGE FOR CALCULATING WITH B-SPLINES, SIAM J.
C      NUMER. ANAL., VOL. 14, NO. 3, JUNE 1977, PP. 441-472.
C
C PACKAGE ROUTINES CALLED..  BSPLVN
C USER ROUTINES CALLED..     NONE
C CALLED BY..                COLPNT,INITAL,VALUES
C FORTRAN FUNCTIONS USED..   FLOAT,MAX0
C-------------------------------------------------------------------------------
      COMMON /SIZES/ NINT,KORD,NCC,NPDE,NCPTS,NEQN,IQUAD
      DIMENSION XT(NCPTS+KORD),VNIKX(K,NDERIV)
      DIMENSION A(20,20)
      KO = K + 1 - NDERIV
      CALL BSPLVN(XT,KO,1,X,ILEFT,VNIKX(NDERIV,NDERIV))
      IF (NDERIV .LE. 1) GO TO 120
      IDERIV = NDERIV
      DO 20 I=2,NDERIV
        IDERVM = IDERIV-1
        DO 10 J=IDERIV,K
   10     VNIKX(J-1,IDERVM) = VNIKX(J,IDERIV)
        IDERIV = IDERVM
        CALL BSPLVN(XT,0,2,X,ILEFT,VNIKX(IDERIV,IDERIV))
   20   CONTINUE
      DO 40 I=1,K
        DO 30 J=1,K
   30     A(I,J) = 0.
   40   A(I,I) = 1.
      KMD = K
      DO 110 M=2,NDERIV
        KMD = KMD - 1
        FKMD =  FLOAT(KMD)
        I = ILEFT
        J = K
   50   JM1 = J-1
        IPKMD = I + KMD
        DIFF = XT(IPKMD) -XT(I)
        IF (JM1 .EQ. 0) GO TO 80
        IF (DIFF .EQ. 0.) GO TO 70
        DO 60 L=1,J
   60     A(L,J) = (A(L,J) - A(L,J-1))/DIFF*FKMD
   70   J = JM1
        I = I - 1
        GO TO 50
   80   IF (DIFF .EQ. 0.) GO TO 90
        A(1,1) = A(1,1)/DIFF*FKMD
   90   DO 110 I=1,K
          V = 0.
          JLOW = MAX0(I,M)
          DO 100 J=JLOW,K
  100       V = A(I,J)*VNIKX(J,M) + V
  110     VNIKX(I,M) = V
  120 RETURN
      END
C
C
C ##############################################################################
C
C
      SUBROUTINE BSPLVN ( XT, JHIGH, INDEX, X, ILEFT, VNIKX )
      IMPLICIT REAL*8 (A-H, O-Z)
C-------------------------------------------------------------------------------
C THIS SUBROUTINE IS PART OF THE B-SPLINE PACKAGE FOR THE STABLE
C EVALUATION OF ANY B-SPLINE BASIS FUNCTION OR DERIVATIVE VALUE.
C SEE REFERENCE BELOW.
C
C CALCULATES THE VALUE OF ALL POSSIBLY NONZERO B-SPLINES AT THE
C POINT X OF ORDER MAX(JHIGH,(J+1)(INDEX-1)) FOR THE BREAKPOINT SEQ-
C UENCE XT.  ASSUMING THAT XT(ILEFT) .LE. X .LE. XT(ILEFT+1), THE ROUT-
C INE RETURNS THE B-SPLINE VALUES IN THE ONE DIMENSIONAL ARRAY VNIKX.
C
C FOR DEFINITIONS OF CALLING ARGUMENTS SEE ABOVE AND BSPLVD.
C
C REFERENCE
C
C    DEBOOR, C., PACKAGE FOR CALCULATING WITH B-SPLINES, SIAM J.
C      NUMER. ANAL., VOL. 14, NO. 3, JUNE 1977, PP. 441-472.
C
C PACKAGE ROUTINES CALLED..  NONE
C USER ROUTINES CALLED..     NONE
C CALLED BY..                BSPLVD
C FORTRAN FUNCTIONS USED..   NONE
C-------------------------------------------------------------------------------
      SAVE J,DELTAM,DELTAP
      COMMON /SIZES/ NINT,KORD,NCC,NPDE,NCPTS,NEQN,IQUAD
      DIMENSION DELTAM(20),DELTAP(20)
      DIMENSION XT(NCPTS+KORD),VNIKX(*)
      DATA J/1/,DELTAM/20*0.D-00/,DELTAP/20*0.D-00/
      GO TO (10,20),INDEX
   10 J = 1
      VNIKX(1) = 1.
      IF (J .GE. JHIGH) GO TO 40
   20 IPJ = ILEFT+J
      DELTAP(J) = XT(IPJ) - X
      IMJP1 = ILEFT-J+1
      DELTAM(J) = X - XT(IMJP1)
      VMPREV = 0.
      JP1 = J+1
      DO 30 L=1,J
        JP1ML = JP1-L
        VM = VNIKX(L)/(DELTAP(L) + DELTAM(JP1ML))
        VNIKX(L) = VM*DELTAP(L) + VMPREV
   30   VMPREV = VM*DELTAM(JP1ML)
      VNIKX(JP1) = VMPREV
      J = JP1
      IF (J .LT. JHIGH) GO TO 20
   40 RETURN
      END
C
C
C ##############################################################################
C
C
      SUBROUTINE INTERV ( XT, LXT, X, ILEFT, MFLAG )
      IMPLICIT REAL*8 (A-H, O-Z)
C-------------------------------------------------------------------------------
C THIS SUBROUTINE IS PART OF THE B-SPLINE PACKAGE FOR THE STABLE
C EVALUATION OF ANY B-SPLINE BASIS FUNCTION OR DERIVATIVE VALUE.
C SEE REFERENCE BELOW.
C
C COMPUTES LARGEST ILEFT IN (1,LXT) SUCH THAT XT(ILEFT) .LE. X.  THE
C PROGRAM STARTS THE SEARCH FOR ILEFT WITH THE VALUE OF ILEFT THAT WAS
C RETURNED AT THE PREVIOUS CALL (AND WAS SAVED IN THE LOCAL VARIABLE
C ILO) TO MINIMIZE THE WORK IN THE COMMON CASE THAT THE VALUE OF X ON
C THIS CALL IS CLOSE TO THE VALUE OF X ON THE PREVIOUS CALL.  SHOULD
C THIS ASSUMPTION NOT BE VALID, THEN THE PROGRAM LOCATES ILO AND IHI
C SUCH THAT XT(ILO) .LE. X .LT. XT(IHI) AND, ONCE THEY ARE FOUND USES
C BISECTION TO FIND THE CORRECT VALUE FOR ILEFT. MFLAG IS AN ERROR FLAG.
C
C FOR DEFINITIONS OF CALLING ARGUMENTS SEE ABOVE AND BSPLVD.
C
C REFERENCE
C
C    DEBOOR, C., PACKAGE FOR CALCULATING WITH B-SPLINES, SIAM J.
C      NUMER. ANAL., VOL. 14, NO. 3, JUNE 1977, PP. 441-472.
C
C PACKAGE ROUTINES CALLED..  NONE
C USER ROUTINES CALLED..     NONE
C CALLED BY..                COLPNT,INITAL,VALUES
C FORTRAN FUNCTIONS USED..   NONE
C-------------------------------------------------------------------------------
      SAVE ILO
      DIMENSION XT(LXT)
      IF(MFLAG.EQ.-2) ILO = 1
      IHI = ILO + 1
      IF (IHI .LT. LXT) GO TO 20
      IF (X .GE. XT(LXT)) GO TO 110
      IF (LXT .LE. 1) GO TO 90
      ILO = LXT - 1
      GO TO 21
   20 IF (X .GE. XT(IHI)) GO TO 40
   21 IF (X .GE. XT(ILO)) GO TO 100
C-----------------------------------------------------------------------
C NOW X .LT. XT(IHI).  FIND LOWER BOUND.
C-----------------------------------------------------------------------
CC   30 ISTEP = 1
      ISTEP = 1
   31 IHI = ILO
      ILO = IHI - ISTEP
      IF (ILO .LE. 1) GO TO 35
      IF (X .GE. XT(ILO)) GO TO 50
      ISTEP = ISTEP*2
      GO TO 31
   35 ILO = 1
      IF (X .LT. XT(1)) GO TO 90
      GO TO 50
C-----------------------------------------------------------------------
C NOW X .GE. XT(ILO).  FIND UPPER BOUND.
C-----------------------------------------------------------------------
   40 ISTEP = 1
   41 ILO = IHI
      IHI = ILO + ISTEP
      IF (IHI .GE. LXT) GO TO 45
      IF (X .LT. XT(IHI)) GO TO 50
      ISTEP = ISTEP*2
      GO TO 41
   45 IF (X .GE. XT(LXT)) GO TO 110
      IHI = LXT
C-----------------------------------------------------------------------
C NOW XT(ILO) .LE. X .LT. XT(IHI).  NARROW THE INTERVAL.
C-----------------------------------------------------------------------
   50 MIDDLE = (ILO + IHI)/2
      IF (MIDDLE .EQ. ILO) GO TO 100
C-----------------------------------------------------------------------
C NOTE..  IT IS ASSUMED THAT MIDDLE = ILO IN CASE IHI = ILO+1.
C-----------------------------------------------------------------------
      IF (X .LT. XT(MIDDLE)) GO TO 53
      ILO = MIDDLE
      GO TO 50
   53 IHI = MIDDLE
      GO TO 50
C-----------------------------------------------------------------------
C SET OUTPUT AND RETURN.
C-----------------------------------------------------------------------
   90 MFLAG = -1
      ILEFT = 1
      RETURN
  100 MFLAG = 0
      ILEFT = ILO
      RETURN
  110 MFLAG = 1
      ILEFT = LXT
      RETURN
      END
C
C
C ##############################################################################
C
C
      SUBROUTINE STIFIB (N0,Y,YMAX,ERROR,SAVE1,SAVE2,SAVE3,
     *                   PW,IPIV,WORK,IWORK)
      IMPLICIT REAL*8 (A-H, O-Z)
C-------------------------------------------------------------------------------
C STIFIB PERFORMS ONE STEP OF THE INTEGRATION OF AN INITIAL VALUE
C PROBLEM FOR A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS OF THE FORM,
C   A(Y,T)*(DY/DT) = G(Y,T),   WHERE   Y = (Y(1),Y(2), ... ,Y(N)).
C STIFIB IS FOR USE WHEN THE MATRICES A AND DG/DY HAVE BANDED OR NEARLY
C BANDED FORM.  THE DEPENDENCE OF A(Y,T) ON Y IS ASSUMED TO BE WEAK.
C
C REFERENCE
C
C   HINDMARSH, A.C., PRELIMINARY DOCUMENTATION OF GEARIB.. SOLUTION
C     OF IMPLICIT SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS WITH
C     BANDED JACOBIANS, LAWRENCE LIVERMORE LAB, UCID-30130, FEBRUARY
C     1976.
C
C COMMUNICATION WITH STIFIB IS DONE WITH THE FOLLOWING VARIABLES..
C
C   Y       AN N0 BY LMAX ARRAY CONTAINING THE DEPENDENT VARIABLES
C             AND THEIR SCALED DERIVATIVES.  LMAX IS 13 FOR THE ADAMS
C             METHODS AND 6 FOR THE GEAR METHODS.  LMAX - 1 = MAXDER
C             IS THE MAXIMUM ORDER AVAILABLE.  SEE SUBROUTINE COSET.
C             Y(I,J+1) CONTAINS THE J-TH DERIVATIVE OF Y(I), SCALED BY
C             H**J/FACTORIAL(J)  (J = 0,1,...,NQ).
C   N0      A CONSTANT INTEGER .GE. N, USED FOR DIMENSIONING PURPOSES.
C   T       THE INDEPENDENT VARIABLE. T IS UPDATED ON EACH STEP TAKEN.
C   H       THE STEP SIZE TO BE ATTEMPTED ON THE NEXT STEP.
C             H IS ALTERED BY THE ERROR CONTROL ALGORITHM DURING THE
C             PROBLEM.  H CAN BE EITHER POSITIVE OR NEGATIVE, BUT ITS
C             SIGN MUST REMAIN CONSTANT THROUGHOUT THE PROBLEM.
C   HMIN,   THE MINIMUM AND MAXIMUM ABSOLUTE VALUE OF THE STEP SIZE
C    HMAX     TO BE USED FOR THE STEP.  THESE MAY BE CHANGED AT ANY
C             TIME, BUT WILL NOT TAKE EFFECT UNTIL THE NEXT H CHANGE.
C   EPS     THE RELATIVE ERROR BOUND.  SEE DESCRIPTION IN PDECOL.
C   UROUND  THE UNIT ROUNDOFF OF THE MACHINE.
C   N       THE NUMBER OF FIRST-ORDER DIFFERENTIAL EQUATIONS.
C   MF      THE METHOD FLAG.  SEE DESCRIPTION IN PDECOL.
C   KFLAG   A COMPLETION CODE WITH THE FOLLOWING MEANINGS..
C                     0  THE STEP WAS SUCCESFUL.
C                    -1  THE REQUESTED ERROR COULD NOT BE ACHIEVED
C                          WITH DABS(H) = HMIN.
C                    -2  THE REQUESTED ERROR IS SMALLER THAN CAN
C                          BE HANDLED FOR THIS PROBLEM.
C                    -3  CORRECTOR CONVERGENCE COULD NOT BE
C                          ACHIEVED FOR DABS(H) = HMIN.
C                    -4  SINGULAR A-MATRIX ENCOUNTERED.
C             ON A RETURN WITH KFLAG NEGATIVE, THE VALUES OF T AND
C             THE Y ARRAY ARE AS OF THE BEGINNING OF THE LAST
C             STEP, AND H IS THE LAST STEP SIZE ATTEMPTED.
C   JSTART  AN INTEGER USED ON INPUT AND OUTPUT.
C             ON INPUT, IT HAS THE FOLLOWING VALUES AND MEANINGS..
C                     0  PERFORM THE FIRST STEP.
C                 .GT.0  TAKE A NEW STEP CONTINUING FROM THE LAST.
C                 .LT.0  TAKE THE NEXT STEP WITH A NEW VALUE OF
C                          H, EPS, N, AND/OR MF.
C             ON EXIT, JSTART IS NQ, THE CURRENT ORDER OF THE METHOD.
C   YMAX    AN ARRAY OF N ELEMENTS WITH WHICH THE ESTIMATED LOCAL
C             ERRORS IN Y ARE COMPARED.
C   ERROR   AN ARRAY OF N ELEMENTS.  ERROR(I)/TQ(2) IS THE ESTIMATED
C             ONE-STEP ERROR IN Y(I).
C   SAVE1,SAVE2,SAVE3   THREE WORKING STORAGE ARRAYS, EACH OF LENGTH N.
C   PW      A BLOCK OF LOCATIONS USED FOR THE CHORD ITERATION
C             MATRIX.  SEE DESCRIPTION IN PDECOL.
C   IPIV    AN INTEGER ARRAY OF LENGTH N FOR PIVOT INFORMATION.
C   ML,MU   THE LOWER AND UPPER HALF BANDWIDTHS, RESPECTIVELY, OF
C             THE CHORD ITERATION MATRIX.  SEE DESCRIPTION IN PDECOL.
C   WORK,IWORK   WORKING ARRAYS WHICH ARE USED TO PASS APPROPRIATE
C                  ARRAYS TO OTHER SUBROUTINES.
C
C PACKAGE ROUTINES CALLED..  COSET,DIFFUN,PSETIB,RES,SOLB
C USER ROUTINES CALLED..     NONE
C CALLED BY..                PDECOL
C FORTRAN FUNCTIONS USED..   ABS,DMAX1,DMIN1,FLOAT
C-------------------------------------------------------------------------------
C     SAVE EL,OLDL0,TQ,IER,NQ,L,METH,MITER
      SAVE BND,CON,CRATE,D,D1,E,EDN,ENQ1,ENQ2,ENQ3,EPSOLD,
     *     EUP,FN,HOLD,OLDL0,PR1,PR2,PR3,R1,RC,RH,RMAX,TOLD,
     *     I,IDOUB,IER,IREDO,IRET,IWEVAL,J,J1,J2,L,LMAX,M,MEO,METH,
     *     MFOLD,MIO,MITER,NEWQ,NOLD,NQ,NSTEPJ
      COMMON /SIZES/ NINT,KORD,NCC,NPDE,NCPTS,NEQN,IQUAD
      COMMON /ISTART/ IW1,IW2,IW3,IW4,IW5,IW6,IW7,IW8,IW9,IW10,IW11,
     *                IW12,IW13,IW14,IW15,IW16,IW17,IW18
      COMMON /GEAR1/ T,H,HMIN,HMAX,EPS,UROUND,N,MF,KFLAG,JSTART
      COMMON /GEAR9/ EPSJ,R0,ML,MU,MW,NM1,N0ML,N0W
      COMMON /GEAR0/ HUSED,NQUSED,NSTEP,NFE,NJE
      COMMON /OPTION/ NOGAUS,MAXDER
      DIMENSION Y(NEQN,MAXDER+1),YMAX(NEQN),ERROR(NEQN),SAVE1(NEQN),
     *          SAVE2(NEQN),
     *          SAVE3(NEQN),PW(NEQN*(3*ML+1)),IPIV(NEQN),
     *          IWORK((NPDE+1)*NCPTS),
     *          WORK(KORD+NPDE*(4+9*NPDE)+(KORD+(NINT-1)*(KORD-NCC))*
     *               (3*KORD+2+NPDE*(3*(KORD-1)*NPDE+MAXDER+4)))
      DIMENSION EL(13),TQ(4)
      DATA EL(2)/1./, OLDL0/1./, TQ(1)/0./, IER/0/
      KFLAG = 0
      TOLD = T
      IF (JSTART .GT. 0) GO TO 200
      IF (JSTART .NE. 0) GO TO 120
C-----------------------------------------------------------------------
C ON THE FIRST CALL, THE ORDER IS SET TO 1 AND THE INITIAL YDOT IS
C CALCULATED.  RMAX IS THE MAXIMUM RATIO BY WHICH H CAN BE INCREASED
C IN A SINGLE STEP.  IT IS INITIALLY 1.E4 TO COMPENSATE FOR THE SMALL
C INITIAL H, BUT THEN IS NORMALLY EQUAL TO 10.  IF A FAILURE
C OCCURS (IN CORRECTOR CONVERGENCE OR ERROR TEST), RMAX IS SET AT 2
C FOR THE NEXT INCREASE.
C-----------------------------------------------------------------------
      NQ = 1
      IER = 0
      CALL DIFFUN (N, T, Y, SAVE1, IER, PW, IPIV, WORK, IWORK)
      IF ( IER .NE. 0 ) GO TO 685
      DO 110 I = 1,N
  110   Y(I,2) = H*SAVE1(I)
      METH = MF/10
      MITER = MF - 10*METH
      L = 2
      IDOUB = 3
      RMAX = 1.E+04
      RC = 0.
      CRATE = 1.
      EPSOLD = EPS
      HOLD = H
      MFOLD = MF
      NOLD = N
      NSTEP = 0
      NSTEPJ = 0
      NFE = 0
      NJE = 1
      IRET = 3
      GO TO 130
C-----------------------------------------------------------------------
C IF THE CALLER HAS CHANGED METH, COSET IS CALLED TO SET
C THE COEFFICIENTS OF THE METHOD.  IF THE CALLER HAS CHANGED
C N, EPS, OR METH, THE CONSTANTS E, EDN, EUP, AND BND MUST BE RESET.
C E IS A COMPARISON FOR ERRORS OF THE CURRENT ORDER NQ. EUP IS
C TO TEST FOR INCREASING THE ORDER, EDN FOR DECREASING THE ORDER.
C BND IS USED TO TEST FOR CONVERGENCE OF THE CORRECTOR ITERATES.
C IF THE CALLER HAS CHANGED H, Y MUST BE RESCALED.
C IF H OR METH HAS BEEN CHANGED, IDOUB IS RESET TO L + 1 TO PREVENT
C FURTHER CHANGES IN H FOR THAT MANY STEPS.
C-----------------------------------------------------------------------
  120 IF (MF .EQ. MFOLD) GO TO 150
      MEO = METH
      MIO = MITER
      METH = MF/10
      MITER = MF - 10*METH
      MFOLD = MF
      IF (MITER .NE. MIO) IWEVAL = MITER
      IF (METH .EQ. MEO) GO TO 150
      IDOUB = L + 1
      IRET = 1
  130 CALL COSET (METH, NQ, EL, TQ)
      LMAX = MAXDER + 1
      RC = RC*EL(1)/OLDL0
      OLDL0 = EL(1)
  140 FN =  FLOAT(N)
      EDN = FN*(TQ(1)*EPS)**2
      E   = FN*(TQ(2)*EPS)**2
      EUP = FN*(TQ(3)*EPS)**2
      BND = FN*(TQ(4)*EPS)**2
      GO TO (160, 170, 200), IRET
  150 IF ((EPS .EQ. EPSOLD) .AND. (N .EQ. NOLD)) GO TO 160
      EPSOLD = EPS
      NOLD = N
      IRET = 1
      GO TO 140
  160 IF (H .EQ. HOLD) GO TO 200
      RH = H/HOLD
      H = HOLD
      IREDO = 3
      GO TO 175
  170 RH = DMAX1(RH,HMIN/ DABS(H))
  175 RH = DMIN1(RH,HMAX/ DABS(H),RMAX)
      R1 = 1.
      DO 180 J = 2,L
        R1 = R1*RH
        DO 180 I = 1,N
  180     Y(I,J) = Y(I,J)*R1
      H = H*RH
      RC = RC*RH
      IDOUB = L + 1
      IF (IREDO .EQ. 0) GO TO 690
C-----------------------------------------------------------------------
C THIS SECTION COMPUTES THE PREDICTED VALUES BY EFFECTIVELY
C MULTIPLYING THE Y ARRAY BY THE PASCAL TRIANGLE MATRIX.
C RC IS THE RATIO OF NEW TO OLD VALUES OF THE COEFFICIENT  H*EL(1).
C WHEN RC DIFFERS FROM 1 BY MORE THAN 30 PERCENT, OR THE CALLER HAS
C CHANGED MITER, IWEVAL IS SET TO MITER TO FORCE PW TO BE UPDATED.
C IN ANY CASE, PW IS UPDATED AT LEAST EVERY 40-TH STEP.
C PW IS THE CHORD ITERATION MATRIX A - H*EL(1)*(DG/DY).
C-----------------------------------------------------------------------
  200 IF ( DABS(RC-1.) .GT. 0.3) IWEVAL = MITER
      IF (NSTEP .GE. NSTEPJ+40) IWEVAL = MITER
      T = T + H
      DO 210 J1 = 1,NQ
        DO 210 J2 = J1,NQ
          J = (NQ + J1) - J2
          DO 210 I = 1,N
  210       Y(I,J) = Y(I,J) + Y(I,J+1)
C-----------------------------------------------------------------------
C UP TO 3 CORRECTOR ITERATIONS ARE TAKEN.  A CONVERGENCE TEST IS
C MADE ON THE R.M.S. NORM OF EACH CORRECTION, USING BND, WHICH
C IS DEPENDENT ON EPS.  THE SUM OF THE CORRECTIONS IS ACCUMULATED
C IN THE VECTOR ERROR(I).  THE Y ARRAY IS NOT ALTERED IN THE CORRECTOR
C LOOP.  THE UPDATED Y VECTOR IS STORED TEMPORARILY IN SAVE1.
C THE UPDATED H*YDOT IS STORED IN SAVE2.
C-----------------------------------------------------------------------
  220 DO 230 I = 1,N
        SAVE2(I) = Y(I,2)
  230   ERROR(I) = 0.
      M = 0
      CALL RES (T, H, Y, SAVE2, SAVE3, NPDE, NCPTS, WORK(IW1), IWORK,
     *  WORK, WORK(IW14), WORK(IW15), WORK(IW16), WORK(IW3), WORK(IW9))
      NFE = NFE + 1
      IF (IWEVAL .LE. 0) GO TO 350
C-----------------------------------------------------------------------
C IF INDICATED, THE MATRIX PW IS REEVALUATED BEFORE STARTING THE
C CORRECTOR ITERATION.  IWEVAL IS SET TO 0 AS AN INDICATOR
C THAT THIS HAS BEEN DONE.  PW IS COMPUTED AND PROCESSED IN PSETIB.
C-----------------------------------------------------------------------
      IWEVAL = 0
      RC = 1.0D0
      NJE = NJE + 1
      NSTEPJ = NSTEP
      CON = -H*EL(1)
      CALL PSETIB (Y, PW, N0, CON, MITER, IER, WORK(IW1), IWORK,
     *     WORK(IW3),WORK(IW9),SAVE2,IPIV,YMAX,WORK(IW11),WORK(IW12),
     *     WORK(IW13),WORK(IW16),WORK(IW14),WORK(IW15),WORK,NPDE)
      IF (IER .NE. 0) GO TO 420
C-----------------------------------------------------------------------
C COMPUTE THE CORRECTOR ERROR, R SUB M, AND SOLVE THE LINEAR SYSTEM
C WITH THAT AS RIGHT-HAND SIDE AND PW AS COEFFICIENT MATRIX,
C USING THE LU DECOMPOSITION OF PW.
C-----------------------------------------------------------------------
  350 CALL SOLB (N0, N, ML, MU, PW, SAVE3, IPIV)
cc  370 D = 0.0D0
      D = 0.0D0
      DO 380 I = 1,N
        ERROR(I) = ERROR(I) + SAVE3(I)
        D = D + (SAVE3(I)/YMAX(I))**2
        SAVE1(I) = Y(I,1) + EL(1)*ERROR(I)
  380   SAVE2(I) = Y(I,2) + ERROR(I)
C-----------------------------------------------------------------------
C TEST FOR CONVERGENCE.  IF M.GT.0, AN ESTIMATE OF THE CONVERGENCE
C RATE CONSTANT IS STORED IN CRATE, AND THIS IS USED IN THE TEST.
C-----------------------------------------------------------------------
CC 400 IF (M .NE. 0) CRATE = DMAX1(.9*CRATE,D/D1)
      IF (M .NE. 0) CRATE = DMAX1(.9*CRATE,D/D1)
      IF ((D*DMIN1(1.D0,2.0D0*CRATE)) .LE. BND) GO TO 450
      D1 = D
      M = M + 1
      IF (M .EQ. 3) GO TO 410
      CALL RES(T, H, SAVE1, SAVE2, SAVE3, NPDE, NCPTS, WORK(IW1), IWORK,
     *  WORK, WORK(IW14), WORK(IW15), WORK(IW16), WORK(IW3), WORK(IW9))
      GO TO 350
C-----------------------------------------------------------------------
C THE CORRECTOR ITERATION FAILED TO CONVERGE IN 3 TRIES.
C IF THE MATRIX PW IS NOT UP TO DATE, IT IS REEVALUATED FOR THE
C NEXT TRY.  OTHERWISE THE Y ARRAY IS RETRACTED TO ITS VALUES
C BEFORE PREDICTION, AND H IS REDUCED, IF POSSIBLE.  IF NOT, A
C NO-CONVERGENCE EXIT IS TAKEN.
C-----------------------------------------------------------------------
  410 NFE = NFE + 2
      IF (IWEVAL .EQ. -1) GO TO 440
  420 T = TOLD
      RMAX = 2.
      DO 430 J1 = 1,NQ
        DO 430 J2 = J1,NQ
          J = (NQ + J1) - J2
          DO 430 I = 1,N
  430       Y(I,J) = Y(I,J) - Y(I,J+1)
      IF ( DABS(H) .LE. HMIN*1.00001) GO TO 680
      RH = .25
      IREDO = 1
      GO TO 170
  440 IWEVAL = MITER
      GO TO 220
C-----------------------------------------------------------------------
C THE CORRECTOR HAS CONVERGED.  IWEVAL IS SET TO -1 TO SIGNAL
C THAT PW MAY NEED UPDATING ON SUBSEQUENT STEPS.  THE ERROR TEST
C IS MADE AND CONTROL PASSES TO STATEMENT 500 IF IT FAILS.
C-----------------------------------------------------------------------
 450  IWEVAL = -1
      NFE = NFE + M
      D = 0.
      DO 460 I = 1,N
  460   D = D + (ERROR(I)/YMAX(I))**2
      IF (D .GT. E) GO TO 500
C-----------------------------------------------------------------------
C AFTER A SUCCESSFUL STEP, UPDATE THE Y ARRAY.
C CONSIDER CHANGING H IF IDOUB = 1.  OTHERWISE DECREASE IDOUB BY 1.
C IF IDOUB IS THEN 1 AND NQ .LT. MAXDER, THEN ERROR IS SAVED FOR
C USE IN A POSSIBLE ORDER INCREASE ON THE NEXT STEP.
C IF A CHANGE IN H IS CONSIDERED, AN INCREASE OR DECREASE IN ORDER
C BY ONE IS CONSIDERED ALSO.  A CHANGE IN H IS MADE ONLY IF IT IS BY A
C FACTOR OF AT LEAST 1.1.  IF NOT, IDOUB IS SET TO 10 TO PREVENT
C TESTING FOR THAT MANY STEPS.
C-----------------------------------------------------------------------
      KFLAG = 0
      IREDO = 0
      NSTEP = NSTEP + 1
      HUSED = H
      NQUSED = NQ
      DO 470 J = 1,L
        DO 470 I = 1,N
  470     Y(I,J) = Y(I,J) + EL(J)*ERROR(I)
      IF (IDOUB .EQ. 1) GO TO 520
      IDOUB = IDOUB - 1
      IF (IDOUB .GT. 1) GO TO 700
      IF (L .EQ. LMAX) GO TO 700
      DO 490 I = 1,N
  490   Y(I,LMAX) = ERROR(I)
      GO TO 700
C-----------------------------------------------------------------------
C THE ERROR TEST FAILED.  KFLAG KEEPS TRACK OF MULTIPLE FAILURES.
C RESTORE T AND THE Y ARRAY TO THEIR PREVIOUS VALUES, AND PREPARE
C TO TRY THE STEP AGAIN.  COMPUTE THE OPTIMUM STEP SIZE FOR THIS OR
C ONE LOWER ORDER.
C-----------------------------------------------------------------------
  500 KFLAG = KFLAG - 1
      T = TOLD
      DO 510 J1 = 1,NQ
        DO 510 J2 = J1,NQ
          J = (NQ + J1) - J2
          DO 510 I = 1,N
  510       Y(I,J) = Y(I,J) - Y(I,J+1)
      RMAX = 2.
      IF ( DABS(H) .LE. HMIN*1.00001) GO TO 660
      IF (KFLAG .LE. -3) GO TO 640
      IREDO = 2
      PR3 = 1.E+20
      GO TO 540
C-----------------------------------------------------------------------
C REGARDLESS OF THE SUCCESS OR FAILURE OF THE STEP, FACTORS
C PR1, PR2, AND PR3 ARE COMPUTED, BY WHICH H COULD BE DIVIDED
C AT ORDER NQ - 1, ORDER NQ, OR ORDER NQ + 1, RESPECTIVELY.
C IN THE CASE OF FAILURE, PR3 = 1.E20 TO AVOID AN ORDER INCREASE.
C THE SMALLEST OF THESE IS DETERMINED AND THE NEW ORDER CHOSEN
C ACCORDINGLY.  IF THE ORDER IS TO BE INCREASED, WE COMPUTE ONE
C ADDITIONAL SCALED DERIVATIVE.
C-----------------------------------------------------------------------
  520 PR3 = 1.E+20
      IF (L .EQ. LMAX) GO TO 540
      D1 = 0.
      DO 530 I = 1,N
  530   D1 = D1 + ((ERROR(I) - Y(I,LMAX))/YMAX(I))**2
      ENQ3 = .5/ FLOAT(L+1)
      PR3 = ((D1/EUP)**ENQ3)*1.4 + 1.4D-06
  540 ENQ2 = .5/ FLOAT(L)
      PR2 = ((D/E)**ENQ2)*1.2 + 1.2D-06
      PR1 = 1.E+20
      IF (NQ .EQ. 1) GO TO 560
      D = 0.
      DO 550 I = 1,N
  550   D = D + (Y(I,L)/YMAX(I))**2
      ENQ1 = .5/ FLOAT(NQ)
      PR1 = ((D/EDN)**ENQ1)*1.3 + 1.3D-06
  560 IF (PR2 .LE. PR3) GO TO 570
      IF (PR3 .LT. PR1) GO TO 590
      GO TO 580
  570 IF (PR2 .GT. PR1) GO TO 580
      NEWQ = NQ
      RH = 1./PR2
      GO TO 620
  580 NEWQ = NQ - 1
      RH = 1./PR1
      GO TO 620
  590 NEWQ = L
      RH = 1./PR3
      IF (RH .LT. 1.1) GO TO 610
      DO 600 I = 1,N
  600   Y(I,NEWQ+1) = ERROR(I)*EL(L)/ FLOAT(L)
      GO TO 630
  610 IDOUB = 10
      GO TO 700
  620 IF ((KFLAG .EQ. 0) .AND. (RH .LT. 1.1)) GO TO 610
C-----------------------------------------------------------------------
C IF THERE IS A CHANGE OF ORDER, RESET NQ, L, AND THE COEFFICIENTS.
C IN ANY CASE H IS RESET ACCORDING TO RH AND THE Y ARRAY IS RESCALED.
C THEN EXIT FROM 690 IF THE STEP WAS OK, OR REDO THE STEP OTHERWISE.
C-----------------------------------------------------------------------
      IF (NEWQ .EQ. NQ) GO TO 170
  630 NQ = NEWQ
      L = NQ + 1
      IRET = 2
      GO TO 130
C-----------------------------------------------------------------------
C CONTROL REACHES THIS SECTION IF 3 OR MORE FAILURES HAVE OCCURED.
C IT IS ASSUMED THAT THE DERIVATIVES THAT HAVE ACCUMULATED IN THE
C Y ARRAY HAVE ERRORS OF THE WRONG ORDER.  HENCE THE FIRST
C DERIVATIVE IS RECOMPUTED, AND THE ORDER IS SET TO 1.  THEN
C H IS REDUCED BY A FACTOR OF 10, AND THE STEP IS RETRIED.
C AFTER A TOTAL OF 7 FAILURES, AN EXIT IS TAKEN WITH KFLAG = -2.
C-----------------------------------------------------------------------
  640 IF (KFLAG .EQ. -7) GO TO 670
      RH = .1
      RH = DMAX1(HMIN/ DABS(H),RH)
      H = H*RH
      IER = 0
      CALL DIFFUN (N, T, Y, SAVE1, IER, PW, IPIV, WORK, IWORK)
      IF ( IER .NE. 0 ) GO TO 685
      NJE = NJE + 1
      DO 650 I = 1,N
  650   Y(I,2) = H*SAVE1(I)
      IWEVAL = MITER
      IDOUB = 10
      IF (NQ .EQ. 1) GO TO 200
      NQ = 1
      L = 2
      IRET = 3
      GO TO 130
C-----------------------------------------------------------------------
C ALL RETURNS ARE MADE THROUGH THIS SECTION.  H IS SAVED IN HOLD
C TO ALLOW THE CALLER TO CHANGE H ON THE NEXT STEP.
C-----------------------------------------------------------------------
  660 KFLAG = -1
      GO TO 700
  670 KFLAG = -2
      GO TO 700
  680 KFLAG = -3
      GO TO 700
  685 KFLAG = -4
      GO TO 700
  690 RMAX = 10.
  700 HOLD = H
      JSTART = NQ
      RETURN
      END
C
C
C ##############################################################################
C
C
      SUBROUTINE GFUN ( T,C,UDOT,NPDE,NCPTS,A,BC,DBDU,DBDUX,DZDT,
     *                 XC,UVAL,ILEFT )
      IMPLICIT REAL*8 (A-H, O-Z)
C-------------------------------------------------------------------------------
C CALLING ARGUMENTS ARE DEFINED BELOW AND IN PDECOL.
C
C SUBROUTINE GFUN COMPUTES THE FUNCTION UDOT=G(C,T), THE RIGHT-
C HAND SIDE OF THE SEMI-DISCRETE APPROXIMATION TO THE ORIGINAL
C SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS AND UPDATES THE BOUNDARY
C CONDITION INFORMATION.
C
C PACKAGE ROUTINES CALLED..  EVAL
C USER ROUTINES CALLED..     BNDRY,F
C CALLED BY..                DIFFUN,PSETIB,RES
C FORTRAN FUNCTIONS USED..   NONE
C-------------------------------------------------------------------------------
      COMMON /SIZES/ NINT,KORD,NCC,NPD,NCPT,NEQN,IQUAD
      DIMENSION C(NPDE,NCPTS),UDOT(NPDE,NCPTS)
      DIMENSION A(NCPTS*3*KORD),BC(NPDE,NPDE,4),
     *          XC(NCPTS),UVAL(NPDE,3),ILEFT(NCPTS)
      DIMENSION DZDT(NPDE),DBDU(NPDE,NPDE),DBDUX(NPDE,NPDE)
      DO 10 K=1,4
        DO 10 J=1,NPDE
          DO 10 I=1,NPDE
            BC(I,J,K) = 0.0
   10 CONTINUE
C-----------------------------------------------------------------------
C UPDATE THE LEFT BOUNDARY VALUES.  SAVE LEFT BOUNDARY CONDITION
C INFORMATION IN THE FIRST 2*NPDE*NPDE LOCATIONS OF BC.
C
C NOTE.. UVAL(K,1) = U(K), UVAL(K,2) = UX(K), AND UVAL(K,3) = UXX(K).
C-----------------------------------------------------------------------
      CALL EVAL(1,NPDE,C,UVAL,A,ILEFT)
      CALL BNDRY(T,XC(1),UVAL,UVAL(1,2),DBDU,DBDUX,DZDT,NPDE)
      CALL F(T,XC(1),UVAL,UVAL(1,2),UVAL(1,3),UDOT,NPDE)
      ILIM = KORD + 2
      DO 30 K=1,NPDE
        BC(K,K,1) = 1.
        IF( DBDU(K,K) .EQ. 0.0  .AND.  DBDUX(K,K) .EQ. 0.0 ) GO TO 30
        UDOT(K,1) = DZDT(K)
        DO 20 J=1,NPDE
          BC(K,J,2) = A(ILIM) * DBDUX(K,J)
          BC(K,J,1) = DBDU(K,J) - BC(K,J,2)
   20   CONTINUE
   30 CONTINUE
C-----------------------------------------------------------------------
C MAIN LOOP TO FORM RIGHT SIDE OF ODES AT THE COLLOCATION POINTS.
C-----------------------------------------------------------------------
      ILIM = NCPTS - 1
      DO 40 I=2,ILIM
        CALL EVAL(I,NPDE,C,UVAL,A,ILEFT)
        CALL F(T,XC(I),UVAL,UVAL(1,2),UVAL(1,3),UDOT(1,I),NPDE)
   40 CONTINUE
C-----------------------------------------------------------------------
C UPDATE THE RIGHT BOUNDARY VALUES.  SAVE THE RIGHT BOUNDARY CONDITION
C INFORMATION IN THE LAST 2*NPDE*NPDE LOCATIONS IN BC.
C-----------------------------------------------------------------------
      CALL EVAL(NCPTS,NPDE,C,UVAL,A,ILEFT)
      CALL F(T,XC(NCPTS),UVAL,UVAL(1,2),UVAL(1,3),UDOT(1,NCPTS),NPDE)
      CALL BNDRY(T,XC(NCPTS),UVAL,UVAL(1,2),DBDU,DBDUX,DZDT,NPDE)
      ILIM = NCPTS * 3 * KORD - KORD - 1
      DO 60 K=1,NPDE
        BC(K,K,4) = 1.
        IF( DBDU(K,K) .EQ. 0.0  .AND.  DBDUX(K,K) .EQ. 0.0 ) GO TO 60
        UDOT(K,NCPTS) = DZDT(K)
        DO 50 J=1,NPDE
          BC(K,J,3) = A(ILIM) * DBDUX(K,J)
          BC(K,J,4) = DBDU(K,J) - BC(K,J,3)
   50   CONTINUE
   60 CONTINUE
      RETURN
      END
C
C
C ##############################################################################
C
C
      SUBROUTINE EVAL( ICPT,NPDE,C,UVAL,A,ILEFT )
      IMPLICIT REAL*8 (A-H, O-Z)
C-------------------------------------------------------------------------------
C CALLING ARGUMENTS ARE DEFINED BELOW AND IN PDECOL.
C
C SUBROUTINE EVAL EVALUATES U(K), UX(K), AND UXX(K), K=1 TO NPDE,
C AT THE COLLOCATION POINT WITH INDEX ICPT USING THE VALUES OF
C THE BASIS FUNCTION COEFFICIENTS IN C AND THE BASIS FUNCTION VALUES
C STORED IN A.  THE RESULTS ARE STORED IN UVAL AS FOLLOWS..
C UVAL(K,1) = U(K), UVAL(K,2) = UX(K), AND UVAL(K,3) = UXX(K).
C
C PACKAGE ROUTINES CALLED..  NONE
C USER ROUTINES CALLED..     NONE
C CALLED BY..                GFUN,PDECOL,PSETIB
C FORTRAN FUNCTIONS USED..   NONE
C-------------------------------------------------------------------------------
      COMMON /SIZES/ NINT,KORD,NCC,NPD,NCPTS,NEQN,IQUAD
      DIMENSION C(NPDE,NCPTS),UVAL(NPDE,3),A(NCPTS*3*KORD),ILEFT(NCPTS)
      IK = ILEFT(ICPT) - KORD
      IC = 3*KORD*(ICPT-1)
      DO 10 M=1,3
        ICC = IC + KORD*(M-1)
        DO 10 J=1,NPDE
          UVAL(J,M) = 0.
          DO 10 I=1,KORD
            UVAL(J,M) = UVAL(J,M) + C(J,I+IK)*A(I+ICC)
   10 CONTINUE
      RETURN
      END
C
C
C ##############################################################################
C
C     
      SUBROUTINE DIFFUN( N, T, Y, YDOT, IER, PW, IPIV, WORK, IWORK )
      IMPLICIT REAL*8 (A-H, O-Z)
C-------------------------------------------------------------------------------
C CALLING ARGUMENTS ARE DEFINED BELOW AND IN PDECOL.
C
C THIS ROUTINE COMPUTES YDOT = A(Y,T)**-1 * G(Y,T) BY USE OF
C THE ROUTINES GFUN, ADDA, DECB, AND SOLB.
C
C PACKAGE ROUTINES CALLED..  ADDA,DECB,GFUN,SOLB
C USER ROUTINES CALLED..     NONE
C CALLED BY..                STIFIB
C FORTRAN FUNCTIONS USED..   NONE
C-------------------------------------------------------------------------------
      COMMON /GEAR9/ EPSJ,R0,ML,MU,MW,NM1,N0ML,N0W
      COMMON /SIZES/ NINT,KORD,NCC,NPDE,NCPTS,NEQN,IQUAD
      COMMON /ISTART/ IW1,IW2,IW3,IDUM(5),IW9,IW10,IW11,IW12,IW13,IW14,
     *                IW15,IW16,IW17,IW18
      COMMON /OPTION/ NOGAUS,MAXDER
      DIMENSION Y(NEQN),YDOT(NEQN),PW(NEQN*(3*ML+1)),
     *          IPIV(NEQN),IWORK((NPDE+1)*NCPTS),
     *          WORK(KORD+NPDE*(4+9*NPDE)+(KORD+(NINT-1)*(KORD-NCC))*
     *               (3*KORD+2+NPDE*(3*(KORD-1)*NPDE+MAXDER+4)))
      CALL GFUN (T, Y, YDOT, NPDE, NCPTS, WORK(IW1), WORK, WORK(IW14),
     *           WORK(IW15), WORK(IW16), WORK(IW3), WORK(IW9), IWORK)
      DO 10 I = 1,N0W
 10     PW(I) = 0.
      N0 = NM1 + 1
      CALL ADDA (PW, N0, WORK(IW1), IWORK, WORK, NPDE)
      CALL DECB (N0, N, ML, MU, PW, IPIV, IER)
      IF ( IER .NE. 0 ) RETURN
      CALL SOLB (N0, N, ML, MU, PW, YDOT, IPIV)
      RETURN
      END
C
C
C ##############################################################################
C
C      
      SUBROUTINE ADDA( PW,N0,A,ILEFT,BC,NPDE )
      IMPLICIT REAL*8 (A-H, O-Z)
C-------------------------------------------------------------------------------
C CALLING ARGUMENTS ARE DEFINED BELOW AND IN PDECOL AND STIFIB.
C
C SUBROUTINE ADDA ADDS THE MATRIX  A  TO THE MATRIX STORED IN  PW  IN
C BAND FORM.  PW IS STORED BY DIAGONALS WITH THE LOWERMOST DIAGONAL
C STORED IN THE FIRST COLUMN OF THE ARRAY.
C
C PACKAGE ROUTINES CALLED..  NONE
C USER ROUTINES CALLED..     NONE
C CALLED BY..                DIFFUN,PSETIB
C FORTRAN FUNCTIONS USED..   NONE
C-------------------------------------------------------------------------------
      COMMON /SIZES/ NINT,KORD,NCC,NPD,NCPTS,NEQN,IQUAD
      COMMON /GEAR9/ EPSJ,R0,ML,MU,MW,NM1,N0ML,N0W
      DIMENSION PW(NEQN,3*ML+1),A(3*KORD*NCPTS),
     *          ILEFT(NCPTS),BC(NPDE,NPDE,4)
C-----------------------------------------------------------------------
C ADD THE BOUNDARY CONDITION PORTIONS OF THE A MATRIX TO PW ( THE FIRST
C AND LAST BLOCK ROWS).
C-----------------------------------------------------------------------
      ICOL = (ILEFT(1) + IQUAD - 1) * NPDE
      DO 10 I=1,NPDE
        IBOT = NEQN - NPDE + I
        DO 10 J=1,NPDE
          IND = ICOL + J - I
          PW(I,IND) = PW(I,IND) + BC(I,J,1)
          PW(I,IND+NPDE) = PW(I,IND+NPDE) + BC(I,J,2)
          PW(IBOT,IND-NPDE) = PW(IBOT,IND-NPDE) + BC(I,J,3)
          PW(IBOT,IND) = PW(IBOT,IND) + BC(I,J,4)
   10 CONTINUE
C-----------------------------------------------------------------------
C UPDATE THE REMAINING ROWS OF PW BY ADDING THE APPROPRIATE VALUES
C IN A TO PW.
C-----------------------------------------------------------------------
      IND = NCPTS - 1
      DO 20 I=2,IND
        I1 = (I-1) * NPDE
        I2 = (I-1) * KORD * 3
        ICOL = ILEFT(I) - I + IQUAD - 1
        DO 20 J=1,KORD
          J1 = (ICOL+J) * NPDE
          J2 = I2 + J
          DO 20 JJ=1,NPDE
   20       PW(I1+JJ,J1) = PW(I1+JJ,J1) + A(J2)
      RETURN
      END
C
C
C ##############################################################################
C
C      
      SUBROUTINE RES( T,H,C,V,R,NPDE,NCPTS,A,ILEFT,BC,DBDU,DBDUX,DZDT,
     *               XC,UVAL )
      IMPLICIT REAL*8 (A-H, O-Z)
C-------------------------------------------------------------------------------
C CALLING ARGUMENTS ARE DEFINED BELOW AND IN PDECOL.
C
C SUBROUTINE RES COMPUTES THE RESIDUAL VECTOR R = H*G(C,T) - A(C,T)*V
C WHERE H IS THE CURRENT TIME STEP SIZE, G IS A VECTOR, A IS A
C MATRIX, V IS A VECTOR, AND T IS THE CURRENT TIME.
C
C PACKAGE ROUTINES CALLED..  GFUN
C USER ROUTINES CALLED..     NONE
C CALLED BY..                STIFIB
C FORTRAN FUNCTIONS USED..   NONE
C-------------------------------------------------------------------------------
      SAVE
      COMMON /SIZES/ NINT,KORD,NCC,NPD,NCPT,NEQN,IQUAD
      DIMENSION C(NPDE,NCPTS),R(NPDE,NCPTS),V(NPDE,NCPTS)
      DIMENSION A(3*KORD*NCPTS),ILEFT(NCPTS),BC(NPDE,NPDE,4),XC(NCPTS),
     *          UVAL(3*NPDE)
      DIMENSION DBDU(NPDE,NPDE),DBDUX(NPDE,NPDE),DZDT(NPDE)
C-----------------------------------------------------------------------
C FORM G(C,T) AND STORE IN R.
C-----------------------------------------------------------------------
      CALL GFUN(T,C,R,NPDE,NCPTS,A,BC,DBDU,DBDUX,DZDT,XC,UVAL,ILEFT)
C-----------------------------------------------------------------------
C FORM THE FIRST AND LAST BLOCK ROWS OF THE RESIDUAL VECTOR
C WHICH ARE DEPENDENT ON THE BOUNDARY CONDITIONS.
C-----------------------------------------------------------------------
      ILIM = NCPTS - 1
      DO 20 I=1,NPDE
        SUM1 = 0.0
        SUM2 = 0.0
        DO 10 J=1,NPDE
          SUM1 = SUM1 + BC(I,J,1) * V(J,1) + BC(I,J,2) * V(J,2)
          SUM2 = SUM2 + BC(I,J,3) * V(J,ILIM) + BC(I,J,4) * V(J,NCPTS)
   10   CONTINUE
        R(I,1) = H * R(I,1) - SUM1
        R(I,NCPTS) = H * R(I,NCPTS) - SUM2
   20 CONTINUE
C-----------------------------------------------------------------------
C FORM THE REMAINING COMPONENTS OF THE RESIDUAL VECTOR.
C-----------------------------------------------------------------------
      DO 50 ICPTS=2,ILIM
        I2 = (ICPTS-1) * KORD * 3
        ICOL = ILEFT(ICPTS) - KORD
        DO 40 JJ=1,NPDE
          SUM1 = 0.
          DO 30 J=1,KORD
            SUM1 = SUM1 + A(I2+J) * V(JJ,ICOL+J)
   30       CONTINUE
          R(JJ,ICPTS) = H*R(JJ,ICPTS) - SUM1
   40     CONTINUE
   50   CONTINUE
      RETURN
      END
C
C
C ##############################################################################
C
C
      SUBROUTINE PSETIB( C,PW,N0,CON,MITER,IER,A,ILEFT,XC,UVAL,
     *    SAVE2,IPIV,CMAX,DFDU,DFDUX,DFDUXX,DZDT,DBDU,DBDUX,BC,NPDE )
      IMPLICIT REAL*8 (A-H, O-Z)
C-----------------------------------------------------------------------
C CALLING ARGUMENTS ARE DEFINED BELOW AND IN PDECOL AND STIFIB.
C
C PSETIB IS CALLED BY STIFIB TO COMPUTE AND PROCESS THE MATRIX
C PW = A - H*EL(1)*(DG/DC), WHERE A AND DG/DC ARE TREATED IN BAND
C FORM.  DG/DC IS COMPUTED, EITHER WITH THE AID OF THE USER-SUPPLIED
C ROUTINE DERIVF IF MITER = 1, OR BY FINITE DIFFERENCING WITH THE AID
C OF THE PACKAGE-SUPPLIED ROUTINE DIFFF IF MITER = 2.  FINALLY,
C PW IS SUBJECTED TO LU DECOMPOSITION IN PREPARATION FOR LATER
C SOLUTION OF LINEAR SYSTEMS WITH PW AS COEFFICIENT MATRIX.
C SEE SUBROUTINES DECB AND SOLB.
C
C IN ADDITION TO VARIABLES DESCRIBED PREVIOUSLY, COMMUNICATION
C WITH PSETIB USES THE FOLLOWING..
C   EPSJ    = DSQRT(UROUND), USED IN THE NUMERICAL JACOBIAN INCREMENTS.
C   MW      = ML + MU + 1.
C   NM1     = N0 - 1.
C   N0ML    = N0*ML.
C   N0W     = N0*MW.
C
C PACKAGE ROUTINES CALLED..  ADDA,DECB,DIFFF,EVAL,GFUN
C USER ROUTINES CALLED..     BNDRY,DERIVF
C CALLED BY..                STIFIB
C FORTRAN FUNCTIONS USED..   ABS,FLOAT,MAX0,MIN0,DSQRT
C-----------------------------------------------------------------------
      COMMON /SIZES/ NINT,KORD,NCC,NPD,NCPTS,NEQN,IQUAD
      COMMON /GEAR1/ T,H,DUMMY(3),UROUND,N,IDUMMY(3)
      COMMON /GEAR9/ EPSJ,R0,ML,MU,MW,NM1,N0ML,N0W
      DIMENSION PW(NEQN,3*ML+1),C(NEQN),CMAX(NEQN)
      DIMENSION A(3*KORD*NCPTS),ILEFT(NCPTS),BC(4*NPDE*NPDE),
     *          XC(NCPTS),UVAL(NPDE,3),SAVE2(NEQN),IPIV(NEQN)
      DIMENSION DFDU(NPDE,NPDE),DFDUX(NPDE,NPDE),DFDUXX(NPDE,NPDE)
      DIMENSION DZDT(NPDE),DBDU(NPDE,NPDE),DBDUX(NPDE,NPDE)
      DO 10 I=1,NEQN
        DO 5 J=1,MW
5         PW(I,J)=0.0E0
10    CONTINUE
      IF ( MITER .EQ. 1 ) GO TO 25
      CALL GFUN (T, C, SAVE2, NPDE, NCPTS,A,BC,DBDU,DBDUX,DZDT,XC,
     *           UVAL,ILEFT)
      D = 0.
      DO 20 I = 1,N
   20   D = D + SAVE2(I)**2
      R0 =  DABS(H)* DSQRT(D/FLOAT(N0))*1.E+03*UROUND
C-----------------------------------------------------------------------
C COMPUTE BLOCK ROWS OF JACOBIAN.
C-----------------------------------------------------------------------
   25 DO 30 I=1,NCPTS
        I1 = (I-1)*NPDE
        I2 = (I-1)*KORD*3
        CALL EVAL(I,NPDE,C,UVAL,A,ILEFT)
        IF ( MITER .EQ. 1 )
     *      CALL DERIVF(T,XC(I),UVAL,UVAL(1,2),UVAL(1,3),
     *                  DFDU,DFDUX,DFDUXX,NPDE)
        IF ( MITER .EQ. 2 )
     *       CALL DIFFF(T,XC(I),I,UVAL,UVAL(1,2),UVAL(1,3),
     *                  DFDU,DFDUX,DFDUXX,NPDE,CMAX,SAVE2)
        ICOL = ILEFT(I) - I + IQUAD - 1
        KLOW = MAX0(1,I+2-NCPTS)
        KUP = MIN0(KORD,KORD+I-2)
        DO 30 KBLK=KLOW,KUP
          J1 = (ICOL+KBLK)*NPDE
          J2 = I2 + KBLK
          J3 = J2 + KORD
          J4 = J3 + KORD
          DO 30 L=1,NPDE
            DO 30 K=1,NPDE
              PW(I1+K,J1-K+L) = DFDU(K,L)*A(J2) + DFDUX(K,L)*A(J3)
     *                        + DFDUXX(K,L)*A(J4)
   30 CONTINUE
C-----------------------------------------------------------------------
C MODIFY THE LAST AND THE FIRST BLOCK ROWS FOR THE BOUNDARY CONDITIONS.
C CURRENT INFORMATION FOR THE RIGHT BOUNDARY CONDITION IS ALREADY IN
C THE ARRAYS DBDU, DBDUX AS A RESULT OF A PREVIOUS CALL TO GFUN.
C-----------------------------------------------------------------------
      IROW = NEQN - NPDE
      DO 50 K=1,NPDE
        IROW = IROW + 1
        IF(DBDU(K,K) .EQ. 0.0  .AND.  DBDUX(K,K) .EQ. 0.0) GO TO 50
        DO 40 J=1,MW
          PW(IROW,J) = 0.0
   40   CONTINUE
   50 CONTINUE
      CALL EVAL(1,NPDE,C,UVAL,A,ILEFT)
      CALL BNDRY(T,XC(1),UVAL,UVAL(1,2),DBDU,DBDUX,DZDT,NPDE)
      DO 70 K=1,NPDE
        IF(DBDU(K,K) .EQ. 0.0  .AND.  DBDUX(K,K) .EQ. 0.0) GO TO 70
        DO 60 J=1,MW
          PW(K,J) = 0.0
   60   CONTINUE
   70 CONTINUE
      DO 80 I = 1,N0
        DO 85 J=1,MW
   85   PW(I,J)=PW(I,J)*CON
   80 CONTINUE
C-----------------------------------------------------------------------
C ADD MATRIX A(C,T) TO PW.
C-----------------------------------------------------------------------
      CALL ADDA (PW, N0, A, ILEFT, BC, NPDE)
C-----------------------------------------------------------------------
C DO LU DECOMPOSITION ON PW.
C-----------------------------------------------------------------------
      CALL DECB (N0, N, ML, MU, PW, IPIV, IER)
      RETURN
      END
C
C
C ##############################################################################
C
C
      SUBROUTINE DIFFF( T,X,IPT,U,UX,UXX,DFDU,DFDUX,DFDUXX,NPDE,CMAX,
     *                 SAVE2)
      IMPLICIT REAL*8 (A-H, O-Z)
C-----------------------------------------------------------------------
C CALLING ARGUMENTS ARE DEFINED BELOW AND IN PDECOL.
C
C SUBROUTINE DIFFF IS USED IF MITER=2 TO PROVIDE FINITE DIFFERENCE
C APPROXIMATIONS FOR THE PARTIAL DERIVATIVES OF THE K-TH USER DEFINED
C FUNCTION IN THE F ROUTINE WITH RESPECT TO THE VARIABLES U, UX, AND
C UXX.  THESE PARTIALS WITH RESPECT TO U, UX, AND UXX ARE COMPUTED,
C STORED, AND RETURNED IN THE NPDE BY NPDE ARRAYS DFDU, DFDUX, AND
C DFDUXX, RESPECTIVELY, AT COLLOCATION POINT NUMBER IPT.
C
C PACKAGE ROUTINES CALLED..  NONE
C USER ROUTINES CALLED..     F
C CALLED BY..                PSETIB
C FORTRAN FUNCTIONS USED..   DMAX1
C-----------------------------------------------------------------------
      COMMON /GEAR9/ EPSJ,R0,ML,MU,MW,NM1,N0ML,N0W
      COMMON /SIZES/ NINT,KORD,NCC,NPD,NCPTS,NEQN,IQUAD
      DIMENSION U(NPDE),UX(NPDE),UXX(NPDE),DFDU(NPDE,NPDE),
     *   DFDUX(NPDE,NPDE),DFDUXX(NPDE,NPDE),CMAX(NEQN),SAVE2(NEQN)
      ID = (IPT-1) * NPDE
      DO 40 J=1,NPDE
        UJ = U(J)
        R = EPSJ * CMAX(J)
        R = DMAX1(R,R0)
        U(J) = U(J) + R
        RINV = 1. / R
        CALL F(T,X,U,UX,UXX,DFDU(1,J),NPDE)
        DO 10 I=1,NPDE
   10     DFDU(I,J) = ( DFDU(I,J) - SAVE2(I+ID) ) * RINV
        U(J) = UJ
        UJ = UX(J)
        UX(J) = UX(J) + R
        CALL F(T,X,U,UX,UXX,DFDUX(1,J),NPDE)
        DO 20 I=1,NPDE
   20     DFDUX(I,J) = ( DFDUX(I,J) - SAVE2(I+ID) ) * RINV
        UX(J) = UJ
        UJ = UXX(J)
        UXX(J) = UXX(J) + R
        CALL F(T,X,U,UX,UXX,DFDUXX(1,J),NPDE)
        DO 30 I=1,NPDE
   30     DFDUXX(I,J) = ( DFDUXX(I,J) - SAVE2(I+ID) ) * RINV
        UXX(J) = UJ
   40 CONTINUE
      RETURN
      END
C
C
C ##############################################################################
C
C
      SUBROUTINE INTERP ( TOUT, Y, N0, Y0 )
      IMPLICIT REAL*8 (A-H, O-Z)
C-----------------------------------------------------------------------
C CALLING ARGUMENTS ARE DEFINED BELOW AND IN STIFIB
C
C SUBROUTINE INTERP COMPUTES INTERPOLATED VALUES OF THE DEPENDENT
C VARIABLE Y AND STORES THEM IN Y0.  THE INTERPOLATION IS TO THE
C POINT T = TOUT, AND USES THE NORDSIECK HISTORY ARRAY Y, AS FOLLOWS..
C                             NQ
C                  Y0(I)  =  SUM  Y(I,J+1)*S**J ,
C                            J=0
C WHERE S = -(T-TOUT)/H.
C
C PACKAGE ROUTINES CALLED..  NONE
C USER ROUTINES CALLED..     NONE
C CALLED BY..                PDECOL
C FORTRAN FUNCTIONS USED..   NONE
C-----------------------------------------------------------------------
      COMMON /SIZES/ NINT,KORD,NCC,NPD,NCPTS,NEQN,IQUAD
      COMMON /OPTION/ NOGAUS,MAXDER
      COMMON /GEAR1/ T,H,DUMMY(4),N,IDUMMY(2),JSTART
      DIMENSION Y0(NEQN),Y(NEQN,MAXDER+1)
      
      DO 10 I = 1,N
   10   Y0(I) = Y(I,1)
      L = JSTART + 1
      S = (TOUT - T)/H
      S1 = 1.
      DO 30 J = 2,L
        S1 = S1*S
        DO 20 I = 1,N
   20     Y0(I) = Y0(I) + S1*Y(I,J)
   30 CONTINUE
      RETURN
      END
C
C
C ##############################################################################
C
C
      SUBROUTINE COSET ( METH, NQ, EL, TQ )
      IMPLICIT REAL*8 (A-H, O-Z)
C-----------------------------------------------------------------------
C COSET IS CALLED BY THE INTEGRATOR AND SETS COEFFICIENTS USED THERE.
C THE VECTOR EL, OF LENGTH NQ + 1, DETERMINES THE BASIC METHOD.
C THE VECTOR TQ, OF LENGTH 4, IS INVOLVED IN ADJUSTING THE STEP SIZE
C IN RELATION TO TRUNCATION ERROR.  ITS VALUES ARE GIVEN BY THE
C PERTST ARRAY.
C
C THE VECTORS EL AND TQ DEPEND ON METH AND NQ.
C THE MAXIMUM ORDER, MAXDER, OF THE METHODS AVAILABLE IS CURRENTLY
C 12 FOR THE ADAMS METHODS AND 5 FOR THE BDF METHODS.  MAXDER DEFAULTS
C TO 5 UNLESS THE USER SETS MAXDER TO SOME OTHER LEGITIMATE VALUE
C THROUGH THE COMMON BLOCK /OPTION/.  SEE PDECOL FOR ADDITIONAL DETAILS.
C LMAX = MAXDER + 1 IS THE NUMBER OF COLUMNS IN THE Y ARRAY (SEE STIFIB
C AND THE VARIABLE C, Y, OR WORK(IW10) IN PDECOL.
C
C THE COEFFICIENTS IN PERTST NEED BE GIVEN TO ONLY ABOUT
C ONE PERCENT ACCURACY.  THE ORDER IN WHICH THE GROUPS APPEAR BELOW
C IS..  COEFFICIENTS FOR ORDER NQ - 1, COEFFICIENTS FOR ORDER NQ,
C COEFFICIENTS FOR ORDER NQ + 1.  WITHIN EACH GROUP ARE THE
C COEFFICIENTS FOR THE ADAMS METHODS, FOLLOWED BY THOSE FOR THE
C BDF METHODS.
C
C REFERENCE
C
C   GEAR, C.W., NUMERICAL INITIAL VALUE PROBLEMS IN ORDINARY
C     DIFFERENTIAL EQUATIONS, PRENTICE-HALL, ENGLEWOOD CLIFFS,
C     N. J., 1971.
C
C PACKAGE ROUTINES CALLED..  NONE
C USER ROUTINES CALLED..     NONE
C CALLED BY..                STIFIB
C FORTRAN FUNCTIONS USED..   FLOAT
C-----------------------------------------------------------------------
      DIMENSION PERTST(12,2,3),EL(13),TQ(4)
      DATA  PERTST / 1.,1.,2.,1.,.3158,.07407,.01391,.002182,
     *                 .0002945,.00003492,.000003692,.0000003524,
     *               1.,1.,.5,.1667,.04167,1.,1.,1.,1.,1.,1.,1.,
     *               2.,12.,24.,37.89,53.33,70.08,87.97,106.9,
     *                 126.7,147.4,168.8,191.0,
     *               2.0,4.5,7.333,10.42,13.7,1.,1.,1.,1.,1.,1.,1.,
     *               12.0,24.0,37.89,53.33,70.08,87.97,106.9,
     *                 126.7,147.4,168.8,191.0,1.,
     *               3.0,6.0,9.167,12.5,1.,1.,1.,1.,1.,1.,1.,1. /
C
      GO TO (1,2),METH
    1 GO TO (101,102,103,104,105,106,107,108,109,110,111,112),NQ
    2 GO TO (201,202,203,204,205),NQ
C-----------------------------------------------------------------------
C THE FOLLOWING COEFFICIENTS SHOULD BE DEFINED TO MACHINE ACCURACY.
C FOR A GIVEN ORDER NQ, THEY CAN BE CALCULATED BY USE OF THE
C GENERATING POLYNOMIAL L(T), WHOSE COEFFICIENTS ARE EL(I)..
C      L(T) = EL(1) + EL(2)*T + ... + EL(NQ+1)*T**NQ.
C FOR THE IMPLICIT ADAMS METHODS, L(T) IS GIVEN BY
C      DL/DT = (T+1)*(T+2)* ... *(T+NQ-1)/K,    L(-1) = 0,
C WHERE                 K = FACTORIAL(NQ-1).
C FOR THE BDF METHODS,
C      L(T) = (T+1)*(T+2)* ... *(T+NQ)/K,
C WHERE         K = FACTORIAL(NQ)*(1 + 1/2 + ... + 1/NQ).
C
C THE ORDER IN WHICH THE GROUPS APPEAR BELOW IS..
C IMPLICIT ADAMS METHODS OF ORDERS 1 TO 12,
C BDF METHODS OF ORDERS 1 TO 5.
C-----------------------------------------------------------------------
 101  EL(1) = 1.0D-00
      GO TO 900
 102  EL(1) = 0.5D-00
      EL(3) = 0.5D-00
      GO TO 900
 103  EL(1) = 4.1666666666667D-01
      EL(3) = 0.75D-00
      EL(4) = 1.6666666666667D-01
      GO TO 900
 104  EL(1) = 0.375D-00
      EL(3) = 9.1666666666667D-01
      EL(4) = 3.3333333333333D-01
      EL(5) = 4.1666666666667D-02
      GO TO 900
 105  EL(1) = 3.4861111111111D-01
      EL(3) = 1.0416666666667D-00
      EL(4) = 4.8611111111111D-01
      EL(5) = 1.0416666666667D-01
      EL(6) = 8.3333333333333D-03
      GO TO 900
 106  EL(1) = 3.2986111111111D-01
      EL(3) = 1.1416666666667D-00
      EL(4) = 0.625D-00
      EL(5) = 1.7708333333333D-01
      EL(6) = 0.025D-00
      EL(7) = 1.3888888888889D-03
      GO TO 900
 107  EL(1) = 3.1559193121693D-01
      EL(3) = 1.225D-00
      EL(4) = 7.5185185185185D-01
      EL(5) = 2.5520833333333D-01
      EL(6) = 4.8611111111111D-02
      EL(7) = 4.8611111111111D-03
      EL(8) = 1.9841269841270D-04
      GO TO 900
 108  EL(1) = 3.0422453703704D-01
      EL(3) = 1.2964285714286D-00
      EL(4) = 8.6851851851852D-01
      EL(5) = 3.3576388888889D-01
      EL(6) = 7.7777777777778D-02
      EL(7) = 1.0648148148148D-02
      EL(8) = 7.9365079365079D-04
      EL(9) = 2.4801587301587D-05
      GO TO 900
 109  EL(1) = 2.9486800044092D-01
      EL(3) = 1.3589285714286D-00
      EL(4) = 9.7655423280423D-01
      EL(5) = 0.4171875D-00
      EL(6) = 1.1135416666667D-01
      EL(7) = 0.01875D-00
      EL(8) = 1.9345238095238D-03
      EL(9) = 1.1160714285714D-04
      EL(10)= 2.7557319223986D-06
      GO TO 900
 110  EL(1) = 2.8697544642857D-01
      EL(3) = 1.4144841269841D-00
      EL(4) = 1.0772156084656D-00
      EL(5) = 4.9856701940035D-01
      EL(6) = 0.1484375D-00
      EL(7) = 2.9060570987654D-02
      EL(8) = 3.7202380952381D-03
      EL(9) = 2.9968584656085D-04
      EL(10)= 1.3778659611993D-05
      EL(11)= 2.7557319223986D-07
      GO TO 900
 111  EL(1) = 2.8018959644394D-01
      EL(3) = 1.4644841269841D-00
      EL(4) = 1.1715145502646D-00
      EL(5) = 5.7935819003527D-01
      EL(6) = 1.8832286155203D-01
      EL(7) = 4.1430362654321D-02
      EL(8) = 6.2111441798942D-03
      EL(9) = 6.2520667989418D-04
      EL(10)= 4.0417401528513D-05
      EL(11)= 1.5156525573192D-06
      EL(12)= 2.5052108385442D-08
      GO TO 900
 112  EL(1) = 2.7426554003160D-01
      EL(3) = 1.5099386724387D-00
      EL(4) = 1.2602711640212D-00
      EL(5) = 6.5923418209877D-01
      EL(6) = 2.3045800264550D-01
      EL(7) = 5.5697246105232D-02
      EL(8) = 9.4394841269841D-03
      EL(9) = 1.1192749669312D-03
      EL(10)= 9.0939153439153D-05
      EL(11)= 4.8225308641975D-06
      EL(12)= 1.5031265031265D-07
      EL(13)= 2.0876756987868D-09
      GO TO 900
 201  EL(1) = 1.0D-00
      GO TO 900
 202  EL(1) = 6.6666666666667D-01
      EL(3) = 3.3333333333333D-01
      GO TO 900
 203  EL(1) = 5.4545454545455D-01
      EL(3) = EL(1)
      EL(4) = 9.0909090909091D-02
      GO TO 900
 204  EL(1) = 0.48D-00
      EL(3) = 0.7D-00
      EL(4) = 0.2D-00
      EL(5) = 0.02D-00
      GO TO 900
 205  EL(1) = 4.3795620437956D-01
      EL(3) = 8.2116788321168D-01
      EL(4) = 3.1021897810219D-01
      EL(5) = 5.4744525547445D-02
      EL(6) = 3.6496350364964D-03
C
 900  DO 910 K = 1,3
 910    TQ(K) = PERTST(NQ,METH,K)
      TQ(4) = .5D-00*TQ(2)/ FLOAT(NQ+2)
      RETURN
      END
C
C
C ##############################################################################
C
C
      SUBROUTINE DECB ( NDIM, N, ML, MU, B, IPIV, IER )
      IMPLICIT REAL*8 (A-H, O-Z)
C-----------------------------------------------------------------------
C SUBROUTINES DECB AND SOLB FORM A TWO SUBROUTINE PACKAGE FOR THE
C DIRECT SOLUTION OF A SYSTEM OF LINEAR EQUATIONS IN WHICH THE
C COEFFICIENT MATRIX IS REAL AND BANDED.
C
C    LU DECOMPOSITION OF BAND MATRIX A..  L*U = P*A , WHERE P IS A
C       PERMUTATION MATRIX, L IS A UNIT LOWER TRIANGULAR MATRIX,
C       AND U IS AN UPPER TRIANGULAR MATRIX.
C    N     =  ORDER OF MATRIX.
C    B     =  N BY (2*ML+MU+1) ARRAY CONTAINING THE MATRIX A ON INPUT
C             AND ITS FACTORED FORM ON OUTPUT.
C             ON INPUT, B(I,K) (1.LE.I.LE.N) CONTAINS THE K-TH
C             DIAGONAL OF A, OR A(I,J) IS STORED IN B(I,J-I+ML+1).
C             ON OUTPUT, B CONTAINS THE L AND U FACTORS, WITH
C             U IN COLUMNS 1 TO ML+MU+1, AND L IN COLUMNS
C             ML+MU+2 TO 2*ML+MU+1.
C    ML,MU =  WIDTHS OF THE LOWER AND UPPER PARTS OF THE BAND, NOT
C             COUNTING THE MAIN DIAGONAL. TOTAL BANDWIDTH IS ML+MU+1.
C    NDIM  =  THE FIRST DIMENSION (COLUMN LENGTH) OF THE ARRAY B.
C             NDIM MUST BE .GE. N.
C    IPIV  =  ARRAY OF LENGTH N CONTAINING PIVOT INFORMATION.
C    IER   =  ERROR INDICATOR..
C          =  0  IF NO ERROR,
C          =  K  IF THE K-TH PIVOT CHOSEN WAS ZERO (A IS SINGULAR).
C    THE INPUT ARGUMENTS ARE  NDIM, N, ML, MU, B.
C    THE OUTPUT ARGUMENTS ARE  B, IPIV, IER.
C
C PACKAGE ROUTINES CALLED..  NONE
C USER ROUTINES CALLED..     NONE
C CALLED BY..                DIFFUN,INITAL,PSETIB
C FORTRAN FUNCTIONS USED..   ABS,MIN0
C-----------------------------------------------------------------------
      DIMENSION B(NDIM,2*ML+MU+1),IPIV(N)
      IER = 0
      IF (N .EQ. 1) GO TO 92
      LL = ML + MU + 1
      N1 = N - 1
      IF (ML .EQ. 0) GO TO 32
      DO 30 I = 1,ML
        II = MU + I
        K = ML + 1 - I
        DO 10 J = 1,II
   10     B(I,J) = B(I,J+K)
        K = II + 1
        DO 20 J = K,LL
   20     B(I,J) = 0.
   30   CONTINUE
   32 LR = ML
      DO 90 NR = 1,N1
        NP = NR + 1
        IF (LR .NE. N) LR = LR + 1
        MX = NR
        XM =  DABS(B(NR,1))
        IF (ML .EQ. 0) GO TO 42
        DO 40 I = NP,LR
          IF ( DABS(B(I,1)) .LE. XM) GO TO 40
          MX = I
          XM =  DABS(B(I,1))
   40     CONTINUE
   42   IPIV(NR) = MX
        IF (MX .EQ. NR) GO TO 60
        DO 50 I = 1,LL
          XX = B(NR,I)
          B(NR,I) = B(MX,I)
   50     B(MX,I) = XX
   60   XM = B(NR,1)
        IF (XM .EQ. 0.) GO TO 100
        B(NR,1) = 1./XM
        IF (ML .EQ. 0) GO TO 90
        XM = -B(NR,1)
        KK = MIN0(N-NR,LL-1)
        DO 80 I = NP,LR
          J = LL + I - NR
          XX = B(I,1)*XM
          B(NR,J) = XX
          DO 70 II = 1,KK
   70       B(I,II) = B(I,II+1) + XX*B(NR,II+1)
   80     B(I,LL) = 0.
   90   CONTINUE
   92 NR = N
      IF (B(N,1) .EQ. 0.) GO TO 100
      B(N,1) = 1./B(N,1)
      RETURN
  100 IER = NR
      RETURN
      END
C
C
C ##############################################################################
C
C
      SUBROUTINE SOLB ( NDIM, N, ML, MU, B, Y, IPIV )
      IMPLICIT REAL*8 (A-H, O-Z)
C-----------------------------------------------------------------------
C SUBROUTINES DECB AND SOLB FORM A TWO SUBROUTINE PACKAGE FOR THE
C DIRECT SOLUTION OF A SYSTEM OF LINEAR EQUATIONS IN WHICH THE
C COEFFICIENT MATRIX IS REAL AND BANDED.
C
C    SOLUTION OF  A*X = C  GIVEN LU DECOMPOSITION OF A FROM DECB.
C    Y  =  RIGHT-HAND VECTOR C, OF LENGTH N, ON INPUT,
C       =  SOLUTION VECTOR X ON OUTPUT.
C    ALL THE ARGUMENTS ARE INPUT ARGUMENTS.
C    THE OUTPUT ARGUMENT IS  Y.
C
C PACKAGE ROUTINES CALLED..  NONE
C USER ROUTINES CALLED..     NONE
C CALLED BY..                DIFFUN,INITAL,STIFIB
C FORTRAN FUNCTIONS USED..   MIN0
C-----------------------------------------------------------------------
      DIMENSION B(NDIM,2*ML+MU+1),Y(N),IPIV(N)
      IF (N .EQ. 1) GO TO 60
      N1 = N - 1
      LL = ML + MU + 1
      IF (ML .EQ. 0) GO TO 32
      DO 30 NR = 1,N1
        IF (IPIV(NR) .EQ. NR) GO TO 10
        J = IPIV(NR)
        XX = Y(NR)
        Y(NR) = Y(J)
        Y(J) = XX
   10   KK = MIN0(N-NR,ML)
        DO 20 I = 1,KK
   20     Y(NR+I) = Y(NR+I) + Y(NR)*B(NR,LL+I)
   30   CONTINUE
   32 LL = LL - 1
      Y(N) = Y(N)*B(N,1)
      KK = 0
      DO 50 NB = 1,N1
        NR = N - NB
        IF (KK .NE. LL) KK = KK + 1
        DP = 0.
        IF (LL .EQ. 0) GO TO 50
        DO 40 I = 1,KK
   40     DP = DP + B(NR,I+1)*Y(NR+I)
   50   Y(NR) = (Y(NR) - DP)*B(NR,1)
      RETURN
   60 Y(1) = Y(1)*B(1,1)
      RETURN
      END

      
C ------------------------------------------------------------------------------


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