      SUBROUTINE DSTODE (NEQ, Y, YH, NYH, YH1, EWT, SAVF, ACOR,
     1   WM, IWM, F, JAC, PJAC, SLVS)
C***BEGIN PROLOGUE  DSTODE
C***SUBSIDIARY
C***PURPOSE  Performs one step of an ODEPACK integration.
C***TYPE      DOUBLE PRECISION (SSTODE-S, DSTODE-D)
C***AUTHOR  Hindmarsh, Alan C., (LLNL)
C***DESCRIPTION
C
C  DSTODE performs one step of the integration of an initial value
C  problem for a system of ordinary differential equations.
C  Note:  DSTODE is independent of the value of the iteration method
C  indicator MITER, when this is .ne. 0, and hence is independent
C  of the type of chord method used, or the Jacobian structure.
C  Communication with DSTODE is done with the following variables:
C
C  NEQ    = integer array containing problem size in NEQ(1), and
C           passed as the NEQ argument in all calls to F and JAC.
C  Y      = an array of length .ge. N used as the Y argument in
C           all calls to F and JAC.
C  YH     = an NYH by LMAX array containing the dependent variables
C           and their approximate scaled derivatives, where
C           LMAX = MAXORD + 1.  YH(i,j+1) contains the approximate
C           j-th derivative of y(i), scaled by h**j/factorial(j)
C           (j = 0,1,...,NQ).  on entry for the first step, the first
C           two columns of YH must be set from the initial values.
C  NYH    = a constant integer .ge. N, the first dimension of YH.
C  YH1    = a one-dimensional array occupying the same space as YH.
C  EWT    = an array of length N containing multiplicative weights
C           for local error measurements.  Local errors in Y(i) are
C           compared to 1.0/EWT(i) in various error tests.
C  SAVF   = an array of working storage, of length N.
C           Also used for input of YH(*,MAXORD+2) when JSTART = -1
C           and MAXORD .lt. the current order NQ.
C  ACOR   = a work array of length N, used for the accumulated
C           corrections.  On a successful return, ACOR(i) contains
C           the estimated one-step local error in Y(i).
C  WM,IWM = real and integer work arrays associated with matrix
C           operations in chord iteration (MITER .ne. 0).
C  PJAC   = name of routine to evaluate and preprocess Jacobian matrix
C           and P = I - h*el0*JAC, if a chord method is being used.
C  SLVS   = name of routine to solve linear system in chord iteration.
C  CCMAX  = maximum relative change in h*el0 before PJAC is called.
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 absolute value of the step size h to be used.
C  HMXI   = inverse of the maximum absolute value of h to be used.
C           HMXI = 0.0 is allowed and corresponds to an infinite hmax.
C           HMIN and HMXI may be changed at any time, but will not
C           take effect until the next change of h is considered.
C  TN     = the independent variable. TN is updated on each step taken.
C  JSTART = an integer used for input only, with the following
C           values and meanings:
C                0  perform the first step.
C            .gt.0  take a new step continuing from the last.
C               -1  take the next step with a new value of H, MAXORD,
C                     N, METH, MITER, and/or matrix parameters.
C               -2  take the next step with a new value of H,
C                     but with other inputs unchanged.
C           On return, JSTART is set to 1 to facilitate continuation.
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               -2  corrector convergence could not be achieved.
C               -3  fatal error in PJAC or SLVS.
C           A return with KFLAG = -1 or -2 means either
C           abs(H) = HMIN or 10 consecutive failures occurred.
C           On a return with KFLAG negative, the values of TN and
C           the YH array are as of the beginning of the last
C           step, and H is the last step size attempted.
C  MAXORD = the maximum order of integration method to be allowed.
C  MAXCOR = the maximum number of corrector iterations allowed.
C  MSBP   = maximum number of steps between PJAC calls (MITER .gt. 0).
C  MXNCF  = maximum number of convergence failures allowed.
C  METH/MITER = the method flags.  See description in driver.
C  N      = the number of first-order differential equations.
C  The values of CCMAX, H, HMIN, HMXI, TN, JSTART, KFLAG, MAXORD,
C  MAXCOR, MSBP, MXNCF, METH, MITER, and N are communicated via COMMON.
C
C***SEE ALSO  DLSODE
C***ROUTINES CALLED  DCFODE, DVNORM
C***COMMON BLOCKS    DLS001
C***REVISION HISTORY  (YYMMDD)
C   791129  DATE WRITTEN
C   890501  Modified prologue to SLATEC/LDOC format.  (FNF)
C   890503  Minor cosmetic changes.  (FNF)
C   930809  Renamed to allow single/double precision versions. (ACH)
C   010418  Reduced size of Common block /DLS001/. (ACH)
C   031105  Restored 'own' variables to Common block /DLS001/, to
C           enable interrupt/restart feature. (ACH)
C***END PROLOGUE  DSTODE
C**End
      EXTERNAL F, JAC, PJAC, SLVS
      INTEGER NEQ, NYH, IWM
      DOUBLE PRECISION Y, YH, YH1, EWT, SAVF, ACOR, WM
      DIMENSION NEQ(*), Y(*), YH(NYH,*), YH1(*), EWT(*), SAVF(*),
     1   ACOR(*), WM(*), IWM(*)
      INTEGER ILLIN, INIT, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
     1   MXSTEP, MXHNIL, NHNIL, NTREP, NSLAST, CNYH,
     2   IALTH, IPUP, LMAX, MEO, NQNYH, NSLP
      INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, METH, MITER,
     1   MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
      INTEGER I, I1, IREDO, IRET, J, JB, M, NCF, NEWQ
      DOUBLE PRECISION CONIT, CRATE, EL, ELCO, HOLD, RMAX, TESCO,
     2   CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
      DOUBLE PRECISION DCON, DDN, DEL, DELP, DSM, DUP, EXDN, EXSM, EXUP,
     1   R, RH, RHDN, RHSM, RHUP, TOLD, DVNORM
      COMMON /DLS001/ CONIT, CRATE, EL(13), ELCO(13,12),
     1   HOLD, RMAX, TESCO(3,12),
     2   CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
     2   ILLIN, INIT, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
     3   MXSTEP, MXHNIL, NHNIL, NTREP, NSLAST, CNYH,
     3   IALTH, IPUP, LMAX, MEO, NQNYH, NSLP,
     4   ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, METH, MITER,
     5   MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
C
C***FIRST EXECUTABLE STATEMENT  DSTODE
      KFLAG = 0
      TOLD = TN
      NCF = 0
      IERPJ = 0
      IERSL = 0
      JCUR = 0
      ICF = 0
      DELP = 0.0D0
      IF (JSTART .GT. 0) GO TO 200
      IF (JSTART .EQ. -1) GO TO 100
      IF (JSTART .EQ. -2) GO TO 160
C-----------------------------------------------------------------------
C On the first call, the order is set to 1, and other variables are
C initialized.  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 to 2
C for the next increase.
C-----------------------------------------------------------------------
      LMAX = MAXORD + 1
      NQ = 1
      L = 2
      IALTH = 2
      RMAX = 10000.0D0
      RC = 0.0D0
      EL0 = 1.0D0
      CRATE = 0.7D0
      HOLD = H
      MEO = METH
      NSLP = 0
      IPUP = MITER
      IRET = 3
      GO TO 140
C-----------------------------------------------------------------------
C The following block handles preliminaries needed when JSTART = -1.
C IPUP is set to MITER to force a matrix update.
C If an order increase is about to be considered (IALTH = 1),
C IALTH is reset to 2 to postpone consideration one more step.
C If the caller has changed METH, DCFODE is called to reset
C the coefficients of the method.
C If the caller has changed MAXORD to a value less than the current
C order NQ, NQ is reduced to MAXORD, and a new H chosen accordingly.
C If H is to be changed, YH must be rescaled.
C If H or METH is being changed, IALTH is reset to L = NQ + 1
C to prevent further changes in H for that many steps.
C-----------------------------------------------------------------------
 100  IPUP = MITER
      LMAX = MAXORD + 1
      IF (IALTH .EQ. 1) IALTH = 2
      IF (METH .EQ. MEO) GO TO 110
      CALL DCFODE (METH, ELCO, TESCO)
      MEO = METH
      IF (NQ .GT. MAXORD) GO TO 120
      IALTH = L
      IRET = 1
      GO TO 150
 110  IF (NQ .LE. MAXORD) GO TO 160
 120  NQ = MAXORD
      L = LMAX
      DO 125 I = 1,L
 125    EL(I) = ELCO(I,NQ)
      NQNYH = NQ*NYH
      RC = RC*EL(1)/EL0
      EL0 = EL(1)
      CONIT = 0.5D0/(NQ+2)
      DDN = DVNORM (N, SAVF, EWT)/TESCO(1,L)
      EXDN = 1.0D0/L
      RHDN = 1.0D0/(1.3D0*DDN**EXDN + 0.0000013D0)
      RH = MIN(RHDN,1.0D0)
      IREDO = 3
      IF (H .EQ. HOLD) GO TO 170
      RH = MIN(RH,ABS(H/HOLD))
      H = HOLD
      GO TO 175
C-----------------------------------------------------------------------
C DCFODE is called to get all the integration coefficients for the
C current METH.  Then the EL vector and related constants are reset
C whenever the order NQ is changed, or at the start of the problem.
C-----------------------------------------------------------------------
 140  CALL DCFODE (METH, ELCO, TESCO)
 150  DO 155 I = 1,L
 155    EL(I) = ELCO(I,NQ)
      NQNYH = NQ*NYH
      RC = RC*EL(1)/EL0
      EL0 = EL(1)
      CONIT = 0.5D0/(NQ+2)
      GO TO (160, 170, 200), IRET
C-----------------------------------------------------------------------
C If H is being changed, the H ratio RH is checked against
C RMAX, HMIN, and HMXI, and the YH array rescaled.  IALTH is set to
C L = NQ + 1 to prevent a change of H for that many steps, unless
C forced by a convergence or error test failure.
C-----------------------------------------------------------------------
 160  IF (H .EQ. HOLD) GO TO 200
      RH = H/HOLD
      H = HOLD
      IREDO = 3
      GO TO 175
 170  RH = MAX(RH,HMIN/ABS(H))
 175  RH = MIN(RH,RMAX)
      RH = RH/MAX(1.0D0,ABS(H)*HMXI*RH)
      R = 1.0D0
      DO 180 J = 2,L
        R = R*RH
        DO 180 I = 1,N
 180      YH(I,J) = YH(I,J)*R
      H = H*RH
      RC = RC*RH
      IALTH = L
      IF (IREDO .EQ. 0) GO TO 690
C-----------------------------------------------------------------------
C This section computes the predicted values by effectively
C multiplying the YH 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 CCMAX, IPUP is set to MITER
C to force PJAC to be called, if a Jacobian is involved.
C In any case, PJAC is called at least every MSBP steps.
C-----------------------------------------------------------------------
 200  IF (ABS(RC-1.0D0) .GT. CCMAX) IPUP = MITER
      IF (NST .GE. NSLP+MSBP) IPUP = MITER
      TN = TN + H
      I1 = NQNYH + 1
      DO 215 JB = 1,NQ
        I1 = I1 - NYH
Cdir$ ivdep
        DO 210 I = I1,NQNYH
 210      YH1(I) = YH1(I) + YH1(I+NYH)
 215    CONTINUE
C-----------------------------------------------------------------------
C Up to MAXCOR corrector iterations are taken.  A convergence test is
C made on the R.M.S. norm of each correction, weighted by the error
C weight vector EWT.  The sum of the corrections is accumulated in the
C vector ACOR(i).  The YH array is not altered in the corrector loop.
C-----------------------------------------------------------------------
 220  M = 0
      DO 230 I = 1,N
 230    Y(I) = YH(I,1)
      CALL F (NEQ, TN, Y, SAVF)
      NFE = NFE + 1
      IF (IPUP .LE. 0) GO TO 250
C-----------------------------------------------------------------------
C If indicated, the matrix P = I - h*el(1)*J is reevaluated and
C preprocessed before starting the corrector iteration.  IPUP is set
C to 0 as an indicator that this has been done.
C-----------------------------------------------------------------------
      CALL PJAC (NEQ, Y, YH, NYH, EWT, ACOR, SAVF, WM, IWM, F, JAC)
      IPUP = 0
      RC = 1.0D0
      NSLP = NST
      CRATE = 0.7D0
      IF (IERPJ .NE. 0) GO TO 430
 250  DO 260 I = 1,N
 260    ACOR(I) = 0.0D0
 270  IF (MITER .NE. 0) GO TO 350
C-----------------------------------------------------------------------
C In the case of functional iteration, update Y directly from
C the result of the last function evaluation.
C-----------------------------------------------------------------------
      DO 290 I = 1,N
        SAVF(I) = H*SAVF(I) - YH(I,2)
 290    Y(I) = SAVF(I) - ACOR(I)
      DEL = DVNORM (N, Y, EWT)
      DO 300 I = 1,N
        Y(I) = YH(I,1) + EL(1)*SAVF(I)
 300    ACOR(I) = SAVF(I)
      GO TO 400
C-----------------------------------------------------------------------
C In the case of the chord method, compute the corrector error,
C and solve the linear system with that as right-hand side and
C P as coefficient matrix.
C-----------------------------------------------------------------------
 350  DO 360 I = 1,N
 360    Y(I) = H*SAVF(I) - (YH(I,2) + ACOR(I))
      CALL SLVS (WM, IWM, Y, SAVF)
      IF (IERSL .LT. 0) GO TO 430
      IF (IERSL .GT. 0) GO TO 410
      DEL = DVNORM (N, Y, EWT)
      DO 380 I = 1,N
        ACOR(I) = ACOR(I) + Y(I)
 380    Y(I) = YH(I,1) + EL(1)*ACOR(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-----------------------------------------------------------------------
 400  IF (M .NE. 0) CRATE = MAX(0.2D0*CRATE,DEL/DELP)
      DCON = DEL*MIN(1.0D0,1.5D0*CRATE)/(TESCO(2,NQ)*CONIT)
      IF (DCON .LE. 1.0D0) GO TO 450
      M = M + 1
      IF (M .EQ. MAXCOR) GO TO 410
      IF (M .GE. 2 .AND. DEL .GT. 2.0D0*DELP) GO TO 410
      DELP = DEL
      CALL F (NEQ, TN, Y, SAVF)
      NFE = NFE + 1
      GO TO 270
C-----------------------------------------------------------------------
C The corrector iteration failed to converge.
C If MITER .ne. 0 and the Jacobian is out of date, PJAC is called for
C the next try.  Otherwise the YH array is retracted to its values
C before prediction, and H is reduced, if possible.  If H cannot be
C reduced or MXNCF failures have occurred, exit with KFLAG = -2.
C-----------------------------------------------------------------------
 410  IF (MITER .EQ. 0 .OR. JCUR .EQ. 1) GO TO 430
      ICF = 1
      IPUP = MITER
      GO TO 220
 430  ICF = 2
      NCF = NCF + 1
      RMAX = 2.0D0
      TN = TOLD
      I1 = NQNYH + 1
      DO 445 JB = 1,NQ
        I1 = I1 - NYH
Cdir$ ivdep
        DO 440 I = I1,NQNYH
 440      YH1(I) = YH1(I) - YH1(I+NYH)
 445    CONTINUE
      IF (IERPJ .LT. 0 .OR. IERSL .LT. 0) GO TO 680
      IF (ABS(H) .LE. HMIN*1.00001D0) GO TO 670
      IF (NCF .EQ. MXNCF) GO TO 670
      RH = 0.25D0
      IPUP = MITER
      IREDO = 1
      GO TO 170
C-----------------------------------------------------------------------
C The corrector has converged.  JCUR is set to 0
C to signal that the Jacobian involved may need updating later.
C The local error test is made and control passes to statement 500
C if it fails.
C-----------------------------------------------------------------------
 450  JCUR = 0
      IF (M .EQ. 0) DSM = DEL/TESCO(2,NQ)
      IF (M .GT. 0) DSM = DVNORM (N, ACOR, EWT)/TESCO(2,NQ)
      IF (DSM .GT. 1.0D0) GO TO 500
C-----------------------------------------------------------------------
C After a successful step, update the YH array.
C Consider changing H if IALTH = 1.  Otherwise decrease IALTH by 1.
C If IALTH is then 1 and NQ .lt. MAXORD, then ACOR 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, IALTH is set to 3 to prevent
C testing for that many steps.
C-----------------------------------------------------------------------
      KFLAG = 0
      IREDO = 0
      NST = NST + 1
      HU = H
      NQU = NQ
      DO 470 J = 1,L
        DO 470 I = 1,N
 470      YH(I,J) = YH(I,J) + EL(J)*ACOR(I)
      IALTH = IALTH - 1
      IF (IALTH .EQ. 0) GO TO 520
      IF (IALTH .GT. 1) GO TO 700
      IF (L .EQ. LMAX) GO TO 700
      DO 490 I = 1,N
 490    YH(I,LMAX) = ACOR(I)
      GO TO 700
C-----------------------------------------------------------------------
C The error test failed.  KFLAG keeps track of multiple failures.
C Restore TN and the YH 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.  After 2 or more failures, H is forced to decrease
C by a factor of 0.2 or less.
C-----------------------------------------------------------------------
 500  KFLAG = KFLAG - 1
      TN = TOLD
      I1 = NQNYH + 1
      DO 515 JB = 1,NQ
        I1 = I1 - NYH
Cdir$ ivdep
        DO 510 I = I1,NQNYH
 510      YH1(I) = YH1(I) - YH1(I+NYH)
 515    CONTINUE
      RMAX = 2.0D0
      IF (ABS(H) .LE. HMIN*1.00001D0) GO TO 660
      IF (KFLAG .LE. -3) GO TO 640
      IREDO = 2
      RHUP = 0.0D0
      GO TO 540
C-----------------------------------------------------------------------
C Regardless of the success or failure of the step, factors
C RHDN, RHSM, and RHUP are computed, by which H could be multiplied
C at order NQ - 1, order NQ, or order NQ + 1, respectively.
C In the case of failure, RHUP = 0.0 to avoid an order increase.
C The largest 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  RHUP = 0.0D0
      IF (L .EQ. LMAX) GO TO 540
      DO 530 I = 1,N
 530    SAVF(I) = ACOR(I) - YH(I,LMAX)
      DUP = DVNORM (N, SAVF, EWT)/TESCO(3,NQ)
      EXUP = 1.0D0/(L+1)
      RHUP = 1.0D0/(1.4D0*DUP**EXUP + 0.0000014D0)
 540  EXSM = 1.0D0/L
      RHSM = 1.0D0/(1.2D0*DSM**EXSM + 0.0000012D0)
      RHDN = 0.0D0
      IF (NQ .EQ. 1) GO TO 560
      DDN = DVNORM (N, YH(1,L), EWT)/TESCO(1,NQ)
      EXDN = 1.0D0/NQ
      RHDN = 1.0D0/(1.3D0*DDN**EXDN + 0.0000013D0)
 560  IF (RHSM .GE. RHUP) GO TO 570
      IF (RHUP .GT. RHDN) GO TO 590
      GO TO 580
 570  IF (RHSM .LT. RHDN) GO TO 580
      NEWQ = NQ
      RH = RHSM
      GO TO 620
 580  NEWQ = NQ - 1
      RH = RHDN
      IF (KFLAG .LT. 0 .AND. RH .GT. 1.0D0) RH = 1.0D0
      GO TO 620
 590  NEWQ = L
      RH = RHUP
      IF (RH .LT. 1.1D0) GO TO 610
      R = EL(L)/L
      DO 600 I = 1,N
 600    YH(I,NEWQ+1) = ACOR(I)*R
      GO TO 630
 610  IALTH = 3
      GO TO 700
 620  IF ((KFLAG .EQ. 0) .AND. (RH .LT. 1.1D0)) GO TO 610
      IF (KFLAG .LE. -2) RH = MIN(RH,0.2D0)
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 YH 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 150
C-----------------------------------------------------------------------
C Control reaches this section if 3 or more failures have occured.
C If 10 failures have occurred, exit with KFLAG = -1.
C It is assumed that the derivatives that have accumulated in the
C YH 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 until it succeeds or H reaches HMIN.
C-----------------------------------------------------------------------
 640  IF (KFLAG .EQ. -10) GO TO 660
      RH = 0.1D0
      RH = MAX(HMIN/ABS(H),RH)
      H = H*RH
      DO 645 I = 1,N
 645    Y(I) = YH(I,1)
      CALL F (NEQ, TN, Y, SAVF)
      NFE = NFE + 1
      DO 650 I = 1,N
 650    YH(I,2) = H*SAVF(I)
      IPUP = MITER
      IALTH = 5
      IF (NQ .EQ. 1) GO TO 200
      NQ = 1
      L = 2
      IRET = 3
      GO TO 150
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 720
 670  KFLAG = -2
      GO TO 720
 680  KFLAG = -3
      GO TO 720
 690  RMAX = 10.0D0
 700  R = 1.0D0/TESCO(2,NQU)
      DO 710 I = 1,N
 710    ACOR(I) = ACOR(I)*R
 720  HOLD = H
      JSTART = 1
      RETURN
C----------------------- END OF SUBROUTINE DSTODE ----------------------
      END
