chem/KPP/kpp/kpp-2.1/int/WRF_conform/TEMP/ros3w.f90

   1 MODULE KPP_ROOT_Integrator
   2
   3  USE KPP_ROOT_Parameters
   4  USE KPP_ROOT_Precision
   5  USE KPP_ROOT_JacobianSP
   6
   7   IMPLICIT NONE
   8
   9
  10
  11   INTEGER, PARAMETER :: ifun=1, ijac=2, istp=3, iacc=4, &
  12     irej=5, idec=6, isol=7, isng=8, itexit=1, ihexit=2
  13
  14
  15   ! description of the error numbers IERR
  16   CHARACTER(LEN=50), PARAMETER, DIMENSION(-8:1) :: IERR_NAMES = (/ &
  17     'Matrix is repeatedly singular                     ', & ! -8
  18     'Step size too small                               ', & ! -7
  19     'No of steps exceeds maximum bound                 ', & ! -6
  20     'Improper tolerance values                         ', & ! -5
  21     'FacMin/FacMax/FacRej must be positive             ', & ! -4
  22     'Hmin/Hmax/Hstart must be positive                 ', & ! -3
  23     'Selected Rosenbrock method not implemented        ', & ! -2
  24     'Improper value for maximal no of steps            ', & ! -1
  25     '                                                  ', & !  0 (not used)
  26     'Success                                           ' /) !  1
  27
  28 CONTAINS
  29
  30 SUBROUTINE  KPP_ROOT_INTEGRATE( TIN, TOUT, &
  31   FIX, VAR,  RCONST, ATOL, RTOL,           &
  32   ICNTRL_U, RCNTRL_U, ISTATUS_U, RSTATUS_U, IERR_U  )
  33
  34    USE KPP_ROOT_Parameters
  35 !!   USE KPP_ROOT_Global
  36    IMPLICIT NONE
  37    KPP_REAL, INTENT(INOUT), DIMENSION(NFIX) :: FIX
  38    KPP_REAL, INTENT(INOUT), DIMENSION(NVAR) :: VAR
  39    KPP_REAL, INTENT(IN), DIMENSION(NSPEC) :: ATOL, RTOL
  40    KPP_REAL, INTENT(IN), DIMENSION(NREACT) :: RCONST
  41    KPP_REAL, INTENT(IN) :: TIN  ! Start Time
  42    KPP_REAL, INTENT(IN) :: TOUT ! End Time
  43    ! Optional input parameters and statistics
  44    INTEGER,  INTENT(IN),  OPTIONAL :: ICNTRL_U(20)
  45    KPP_REAL, INTENT(IN),  OPTIONAL :: RCNTRL_U(20)
  46    INTEGER,  INTENT(OUT), OPTIONAL :: ISTATUS_U(20)
  47    KPP_REAL, INTENT(OUT), OPTIONAL :: RSTATUS_U(20)
  48    INTEGER,  INTENT(OUT), OPTIONAL :: IERR_U
  49
  50    KPP_REAL  :: STEPMIN
  51
  52
  53    INTEGER :: N_stp, N_acc, N_rej, N_sng
  54    SAVE N_stp, N_acc, N_rej, N_sng
  55    INTEGER :: i, IERR
  56    KPP_REAL :: RCNTRL(20), RSTATUS(20)
  57    INTEGER :: ICNTRL(20), ISTATUS(20)
  58
  59    ICNTRL(:)  = 0
  60    RCNTRL(:)  = 0.0_dp
  61    ISTATUS(:) = 0
  62    RSTATUS(:) = 0.0_dp
  63
  64    ! If optional parameters are given, and if they are >0,
  65    ! then they overwrite default settings.
  66    IF (PRESENT(ICNTRL_U)) THEN
  67      WHERE(ICNTRL_U(:) > 0) ICNTRL(:) = ICNTRL_U(:)
  68    END IF
  69    IF (PRESENT(RCNTRL_U)) THEN
  70      WHERE(RCNTRL_U(:) > 0) RCNTRL(:) = RCNTRL_U(:)
  71    END IF
  72
  73    CALL KPP_ROOT_Rosenbrock(VAR, FIX, RCONST, TIN,TOUT,   &
  74          ATOL,RTOL,               &
  75          RCNTRL,ICNTRL,RSTATUS,ISTATUS,IERR)
  76
  77    STEPMIN = RCNTRL(ihexit)
  78    ! if optional parameters are given for output they to return information
  79 !!$   IF (PRESENT(ISTATUS_U)) ISTATUS_U(:) = ISTATUS(:)
  80 !!$   IF (PRESENT(RSTATUS_U)) RSTATUS_U(:) = RSTATUS(:)
  81 !!$   IF (PRESENT(IERR_U))    IERR_U       = IERR
  82
  83 END SUBROUTINE  KPP_ROOT_INTEGRATE
  84
  85 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  86 SUBROUTINE  KPP_ROOT_Rosenbrock(Y, FIX, RCONST, Tstart,Tend, &
  87            AbsTol,RelTol,            &
  88            RCNTRL,ICNTRL,RSTATUS,ISTATUS,IERR)
  89 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  90 !
  91 !    Solves the system y'=F(t,y) using a Rosenbrock method defined by:
  92 !
  93 !     G = 1/(H*gamma(1)) - Jac(t0,Y0)
  94 !     T_i = t0 + Alpha(i)*H
  95 !     Y_i = Y0 + \sum_{j=1}^{i-1} A(i,j)*K_j
  96 !     G * K_i = Fun( T_i, Y_i ) + \sum_{j=1}^S C(i,j)/H * K_j +
  97 !         gamma(i)*dF/dT(t0, Y0)
  98 !     Y1 = Y0 + \sum_{j=1}^S M(j)*K_j
  99 !
 100 !    For details on Rosenbrock methods and their implementation consult:
 101 !      E. Hairer and G. Wanner
 102 !      "Solving ODEs II. Stiff and differential-algebraic problems".
 103 !      Springer series in computational mathematics, Springer-Verlag, 1996.
 104 !    The codes contained in the book inspired this implementation.
 105 !
 106 !    (C)  Adrian Sandu, August 2004
 107 !    Virginia Polytechnic Institute and State University
 108 !    Contact: sandu@cs.vt.edu
 109 !    This implementation is part of KPP - the Kinetic PreProcessor
 110 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 111 !
 112 !~~~>   INPUT ARGUMENTS:
 113 !
 114 !-     Y(NVAR)    = vector of initial conditions (at T=Tstart)
 115 !-    [Tstart,Tend]  = time range of integration
 116 !     (if Tstart>Tend the integration is performed backwards in time)
 117 !-    RelTol, AbsTol = user precribed accuracy
 118 !- SUBROUTINE  Fun( T, Y, Ydot ) = ODE function,
 119 !                       returns Ydot = Y' = F(T,Y)
 120 !- SUBROUTINE  Jac( T, Y, Jcb ) = Jacobian of the ODE function,
 121 !                       returns Jcb = dFun/dY
 122 !-    ICNTRL(1:20)    = integer inputs parameters
 123 !-    RCNTRL(1:20)    = real inputs parameters
 124 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 125 !
 126 !~~~>     OUTPUT ARGUMENTS:
 127 !
 128 !-    Y(NVAR)    -> vector of final states (at T->Tend)
 129 !-    ISTATUS(1:20)   -> integer output parameters
 130 !-    RSTATUS(1:20)   -> real output parameters
 131 !-    IERR            -> job status upon return
 132 !                        success (positive value) or
 133 !                        failure (negative value)
 134 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 135 !
 136 !~~~>     INPUT PARAMETERS:
 137 !
 138 !    Note: For input parameters equal to zero the default values of the
 139 !       corresponding variables are used.
 140 !
 141 !    ICNTRL(1) = 1: F = F(y)   Independent of T (AUTONOMOUS)
 142 !              = 0: F = F(t,y) Depends on T (NON-AUTONOMOUS)
 143 !
 144 !    ICNTRL(2) = 0: AbsTol, RelTol are NVAR-dimensional vectors
 145 !              = 1: AbsTol, RelTol are scalars
 146 !
 147 !    ICNTRL(3)  -> selection of a particular Rosenbrock method
 148 !        = 0 :  default method is Rodas3
 149 !        = 1 :  method is  Ros2
 150 !        = 2 :  method is  Ros3
 151 !        = 3 :  method is  Ros4
 152 !        = 4 :  method is  Rodas3
 153 !        = 5:   method is  Rodas4
 154 !
 155 !    ICNTRL(4)  -> maximum number of integration steps
 156 !        For ICNTRL(4)=0) the default value of 100000 is used
 157 !
 158 !    RCNTRL(1)  -> Hmin, lower bound for the integration step size
 159 !          It is strongly recommended to keep Hmin = ZERO
 160 !    RCNTRL(2)  -> Hmax, upper bound for the integration step size
 161 !    RCNTRL(3)  -> Hstart, starting value for the integration step size
 162 !
 163 !    RCNTRL(4)  -> FacMin, lower bound on step decrease factor (default=0.2)
 164 !    RCNTRL(5)  -> FacMax, upper bound on step increase factor (default=6)
 165 !    RCNTRL(6)  -> FacRej, step decrease factor after multiple rejections
 166 !            (default=0.1)
 167 !    RCNTRL(7)  -> FacSafe, by which the new step is slightly smaller
 168 !         than the predicted value  (default=0.9)
 169 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 170 !
 171 !~~~>     OUTPUT PARAMETERS:
 172 !
 173 !    Note: each call to Rosenbrock adds the current no. of fcn calls
 174 !      to previous value of ISTATUS(1), and similar for the other params.
 175 !      Set ISTATUS(1:20) = 0 before call to avoid this accumulation.
 176 !
 177 !    ISTATUS(1) = No. of function calls
 178 !    ISTATUS(2) = No. of jacobian calls
 179 !    ISTATUS(3) = No. of steps
 180 !    ISTATUS(4) = No. of accepted steps
 181 !    ISTATUS(5) = No. of rejected steps (except at the beginning)
 182 !    ISTATUS(6) = No. of LU decompositions
 183 !    ISTATUS(7) = No. of forward/backward substitutions
 184 !    ISTATUS(8) = No. of singular matrix decompositions
 185 !
 186 !    RSTATUS(1)  -> Texit, the time corresponding to the
 187 !                   computed Y upon return
 188 !    RSTATUS(2)  -> Hexit, last accepted step before exit
 189 !    For multiple restarts, use Hexit as Hstart in the following run
 190 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 191
 192   USE KPP_ROOT_Parameters
 193 !!  USE KPP_ROOT_LinearAlgebra
 194   IMPLICIT NONE
 195
 196 !~~~>  Arguments
 197    KPP_REAL, INTENT(INOUT) :: Y(NVAR)
 198    KPP_REAL, INTENT(IN), DIMENSION(NFIX) :: FIX
 199    KPP_REAL, INTENT(IN), DIMENSION(NREACT) :: RCONST
 200    KPP_REAL, INTENT(IN)   :: Tstart,Tend
 201    KPP_REAL, INTENT(IN)   :: AbsTol(NVAR),RelTol(NVAR)
 202    INTEGER, INTENT(IN)    :: ICNTRL(20)
 203    KPP_REAL, INTENT(IN)   :: RCNTRL(20)
 204    INTEGER, INTENT(INOUT) :: ISTATUS(20)
 205    KPP_REAL, INTENT(INOUT) :: RSTATUS(20)
 206    INTEGER, INTENT(OUT)   :: IERR
 207 !~~~>  The method parameters
 208    INTEGER, PARAMETER :: Smax = 6
 209    INTEGER  :: Method, ros_S
 210    KPP_REAL, DIMENSION(Smax) :: ros_M, ros_E, ros_Alpha, ros_Gamma
 211    KPP_REAL, DIMENSION(Smax*(Smax-1)/2) :: ros_A, ros_C
 212    KPP_REAL :: ros_ELO
 213    LOGICAL, DIMENSION(Smax) :: ros_NewF
 214    CHARACTER(LEN=12) :: ros_Name
 215
 216 !~~~>  Statistics on the work performed by the Rosenbrock method
 217   INTEGER :: Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng
 218
 219
 220 !~~~>  Local variables
 221    KPP_REAL :: Roundoff, FacMin, FacMax, FacRej, FacSafe
 222    KPP_REAL :: Hmin, Hmax, Hstart, Hexit
 223    KPP_REAL :: Texit
 224    INTEGER :: i, UplimTol, Max_no_steps
 225    LOGICAL :: Autonomous, VectorTol
 226 !~~~>   Parameters
 227    KPP_REAL, PARAMETER :: ZERO = 0.0_dp, ONE  = 1.0_dp
 228    KPP_REAL, PARAMETER :: DeltaMin = 1.0E-5_dp
 229
 230 !~~~>  Initialize statistics
 231    Nfun = ISTATUS(ifun)
 232    Njac = ISTATUS(ijac)
 233    Nstp = ISTATUS(istp)
 234    Nacc = ISTATUS(iacc)
 235    Nrej = ISTATUS(irej)
 236    Ndec = ISTATUS(idec)
 237    Nsol = ISTATUS(isol)
 238    Nsng = ISTATUS(isng)
 239
 240 !~~~>  Autonomous or time dependent ODE. Default is time dependent.
 241 !!$   Autonomous = .NOT.(ICNTRL(1) == 0)
 242       Autonomous=.FALSE.
 243
 244
 245
 246
 247 !~~~>  For Scalar tolerances (ICNTRL(2).NE.0)  the code uses AbsTol(1) and RelTol(1)
 248 !   For Vector tolerances (ICNTRL(2) == 0) the code uses AbsTol(1:NVAR) and RelTol(1:NVAR)
 249 !!$   IF (ICNTRL(2) == 0) THEN
 250       VectorTol = .TRUE.
 251          UplimTol  = NVAR
 252 !!$   ELSE
 253 !!$      VectorTol = .FALSE.
 254 !!$         UplimTol  = 1
 255 !!$   END IF
 256
 257 !~~~>  The particular Rosenbrock method chosen
 258 !!$   IF (ICNTRL(3) == 0) THEN
 259 !!$      Method = 4
 260 !!$   ELSEIF ( (ICNTRL(3) >= 1).AND.(ICNTRL(3) <= 5) ) THEN
 261 !!$      Method = ICNTRL(3)
 262 !!$   ELSE
 263 !!$      PRINT * , 'User-selected Rosenbrock method: ICNTRL(3)=', Method
 264 !!$      CALL KPP_ROOT_ros_ErrorMsg(-2,Tstart,ZERO,IERR)
 265 !!$      RETURN
 266 !!$   END IF
 267
 268 !~~~>   The maximum number of steps admitted
 269 !!$   IF (ICNTRL(4) == 0) THEN
 270       Max_no_steps = 10000
 271 !!$   ELSEIF (ICNTRL(4) > 0) THEN
 272 !!$      Max_no_steps=ICNTRL(4)
 273 !!$   ELSE
 274 !!$      PRINT * ,'User-selected max no. of steps: ICNTRL(4)=',ICNTRL(4)
 275 !!$      CALL KPP_ROOT_ros_ErrorMsg(-1,Tstart,ZERO,IERR)
 276 !!$      RETURN
 277 !!$   END IF
 278
 279 !~~~>  Unit roundoff (1+Roundoff>1)
 280    Roundoff = KPP_ROOT_WLAMCH('E')
 281
 282 !~~~>  Lower bound on the step size: (positive value)
 283 !!$   IF (RCNTRL(1) == ZERO) THEN
 284       Hmin = ZERO
 285 !!$   ELSEIF (RCNTRL(1) > ZERO) THEN
 286 !!$      Hmin = RCNTRL(1)
 287 !!$   ELSE
 288 !!$      PRINT * , 'User-selected Hmin: RCNTRL(1)=', RCNTRL(1)
 289 !!$      CALL KPP_ROOT_ros_ErrorMsg(-3,Tstart,ZERO,IERR)
 290 !!$      RETURN
 291 !!$   END IF
 292 !!$!~~~>  Upper bound on the step size: (positive value)
 293 !!$   IF (RCNTRL(2) == ZERO) THEN
 294       Hmax = ABS(Tend-Tstart)
 295 !!$   ELSEIF (RCNTRL(2) > ZERO) THEN
 296 !!$      Hmax = MIN(ABS(RCNTRL(2)),ABS(Tend-Tstart))
 297 !!$   ELSE
 298 !!$      PRINT * , 'User-selected Hmax: RCNTRL(2)=', RCNTRL(2)
 299 !!$      CALL KPP_ROOT_ros_ErrorMsg(-3,Tstart,ZERO,IERR)
 300 !!$      RETURN
 301 !!$   END IF
 302 !!$!~~~>  Starting step size: (positive value)
 303 !!$   IF (RCNTRL(3) == ZERO) THEN
 304       Hstart = MAX(Hmin,DeltaMin)
 305 !!$   ELSEIF (RCNTRL(3) > ZERO) THEN
 306 !!$      Hstart = MIN(ABS(RCNTRL(3)),ABS(Tend-Tstart))
 307 !!$   ELSE
 308 !!$      PRINT * , 'User-selected Hstart: RCNTRL(3)=', RCNTRL(3)
 309 !!$      CALL KPP_ROOT_ros_ErrorMsg(-3,Tstart,ZERO,IERR)
 310 !!$      RETURN
 311 !!$   END IF
 312 !~~~>  Step size can be changed s.t.  FacMin < Hnew/Hexit < FacMax
 313 !!$   IF (RCNTRL(4) == ZERO) THEN
 314       FacMin = 0.2_dp
 315 !!$   ELSEIF (RCNTRL(4) > ZERO) THEN
 316 !!$      FacMin = RCNTRL(4)
 317 !!$   ELSE
 318 !!$      PRINT * , 'User-selected FacMin: RCNTRL(4)=', RCNTRL(4)
 319 !!$      CALL KPP_ROOT_ros_ErrorMsg(-4,Tstart,ZERO,IERR)
 320 !!$      RETURN
 321 !!$   END IF
 322 !!$   IF (RCNTRL(5) == ZERO) THEN
 323       FacMax = 6.0_dp
 324 !!$   ELSEIF (RCNTRL(5) > ZERO) THEN
 325 !!$      FacMax = RCNTRL(5)
 326 !!$   ELSE
 327 !!$      PRINT * , 'User-selected FacMax: RCNTRL(5)=', RCNTRL(5)
 328 !!$      CALL KPP_ROOT_ros_ErrorMsg(-4,Tstart,ZERO,IERR)
 329 !!$      RETURN
 330 !!$   END IF
 331 !~~~>   FacRej: Factor to decrease step after 2 succesive rejections
 332 !!$   IF (RCNTRL(6) == ZERO) THEN
 333       FacRej = 0.1_dp
 334 !!$   ELSEIF (RCNTRL(6) > ZERO) THEN
 335 !!$      FacRej = RCNTRL(6)
 336 !!$   ELSE
 337 !!$      PRINT * , 'User-selected FacRej: RCNTRL(6)=', RCNTRL(6)
 338 !!$      CALL KPP_ROOT_ros_ErrorMsg(-4,Tstart,ZERO,IERR)
 339 !!$      RETURN
 340 !!$   END IF
 341 !~~~>   FacSafe: Safety Factor in the computation of new step size
 342 !!$   IF (RCNTRL(7) == ZERO) THEN
 343       FacSafe = 0.9_dp
 344 !!$   ELSEIF (RCNTRL(7) > ZERO) THEN
 345 !!$      FacSafe = RCNTRL(7)
 346 !!$   ELSE
 347 !!$      PRINT * , 'User-selected FacSafe: RCNTRL(7)=', RCNTRL(7)
 348 !!$      CALL KPP_ROOT_ros_ErrorMsg(-4,Tstart,ZERO,IERR)
 349 !!$      RETURN
 350 !!$   END IF
 351 !!$!~~~>  Check if tolerances are reasonable
 352 !!$    DO i=1,UplimTol
 353 !!$      IF ( (AbsTol(i) <= ZERO) .OR. (RelTol(i) <= 10.0_dp*Roundoff) &
 354 !!$         .OR. (RelTol(i) >= 1.0_dp) ) THEN
 355 !!$        PRINT * , ' AbsTol(',i,') = ',AbsTol(i)
 356 !!$        PRINT * , ' RelTol(',i,') = ',RelTol(i)
 357 !!$        CALL KPP_ROOT_ros_ErrorMsg(-5,Tstart,ZERO,IERR)
 358 !!$        RETURN
 359 !!$      END IF
 360 !!$    END DO
 361
 362
 363 !~~~>   Initialize the particular Rosenbrock method
 364 !!$   SELECT CASE (Method)
 365 !!$     CASE (1)
 366 !!$       CALL KPP_ROOT_Ros2(ros_S, ros_A, ros_C, ros_M, ros_E,   &
 367 !!$          ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
 368 !!$     CASE (2)
 369        CALL KPP_ROOT_Ros3(ros_S, ros_A, ros_C, ros_M, ros_E,   &
 370           ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
 371 !!$     CASE (3)
 372 !!$       CALL KPP_ROOT_Ros4(ros_S, ros_A, ros_C, ros_M, ros_E,   &
 373 !!$          ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
 374 !!$     CASE (4)
 375 !!$       CALL KPP_ROOT_Rodas3(ros_S, ros_A, ros_C, ros_M, ros_E, &
 376 !!$          ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
 377 !!$     CASE (5)
 378 !!$       CALL KPP_ROOT_Rodas4(ros_S, ros_A, ros_C, ros_M, ros_E, &
 379 !!$          ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
 380 !!$     CASE DEFAULT
 381 !!$       PRINT * , 'Unknown Rosenbrock method: ICNTRL(4)=', Method
 382 !!$       CALL KPP_ROOT_ros_ErrorMsg(-2,Tstart,ZERO,IERR)
 383 !!$       RETURN
 384 !!$   END SELECT
 385
 386 !~~~>  CALL Rosenbrock method
 387    CALL KPP_ROOT_ros_Integrator(Y,Tstart,Tend,Texit,      &
 388         AbsTol, RelTol,                          &
 389 !  Rosenbrock method coefficients
 390         ros_S, ros_M, ros_E, ros_A, ros_C,       &
 391         ros_Alpha, ros_Gamma, ros_ELO, ros_NewF, &
 392 !  Integration parameters
 393         Autonomous, VectorTol, Max_no_steps,     &
 394         Roundoff, Hmin, Hmax, Hstart, Hexit,     &
 395         FacMin, FacMax, FacRej, FacSafe,         &
 396 !  Error indicator
 397         IERR,                                    &
 398 !  Statistics on the work performed by the Rosenbrock method
 399          Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng,&
 400 !~~~>
 401          RCONST, FIX &
 402 )
 403
 404
 405 !~~~>  Collect run statistics
 406    ISTATUS(ifun) = Nfun
 407    ISTATUS(ijac) = Njac
 408    ISTATUS(istp) = Nstp
 409    ISTATUS(iacc) = Nacc
 410    ISTATUS(irej) = Nrej
 411    ISTATUS(idec) = Ndec
 412    ISTATUS(isol) = Nsol
 413    ISTATUS(isng) = Nsng
 414 !~~~> Last T and H
 415    RSTATUS(itexit) = Texit
 416    RSTATUS(ihexit) = Hexit
 417
 418 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 419 CONTAINS !  SUBROUTINES internal to Rosenbrock
 420 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 421
 422 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 423  SUBROUTINE  KPP_ROOT_ros_ErrorMsg(Code,T,H,IERR)
 424 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 425 !    Handles all error messages
 426 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 427    USE KPP_ROOT_Precision
 428
 429    KPP_REAL, INTENT(IN) :: T, H
 430    INTEGER, INTENT(IN)  :: Code
 431    INTEGER, INTENT(OUT) :: IERR
 432
 433    IERR = Code
 434    PRINT * , &
 435      'Forced exit from Rosenbrock due to the following error:'
 436    IF ((Code>=-8).AND.(Code<=-1)) THEN
 437      PRINT *, IERR_NAMES(Code)
 438    ELSE
 439      PRINT *, 'Unknown Error code: ', Code
 440    ENDIF
 441
 442    PRINT *, "T=", T, "and H=", H
 443
 444  END SUBROUTINE  KPP_ROOT_ros_ErrorMsg
 445
 446 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 447  SUBROUTINE  KPP_ROOT_ros_Integrator (Y, Tstart, Tend, T,     &
 448         AbsTol, RelTol,                          &
 449 !~~~> Rosenbrock method coefficients
 450         ros_S, ros_M, ros_E, ros_A, ros_C,       &
 451         ros_Alpha, ros_Gamma, ros_ELO, ros_NewF, &
 452 !~~~> Integration parameters
 453         Autonomous, VectorTol, Max_no_steps,     &
 454         Roundoff, Hmin, Hmax, Hstart, Hexit,     &
 455         FacMin, FacMax, FacRej, FacSafe,         &
 456 !~~~> Error indicator
 457         IERR,                                    &
 458 !~~~>   Statistics on the work performed by the Rosenbrock method
 459         Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng, &
 460 !~~~>
 461         RCONST, FIX &
 462  )
 463 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 464 !   Template for the implementation of a generic Rosenbrock method
 465 !      defined by ros_S (no of stages)
 466 !      and its coefficients ros_{A,C,M,E,Alpha,Gamma}
 467 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 468
 469   IMPLICIT NONE
 470
 471 !~~~> Input: the initial condition at Tstart; Output: the solution at T
 472    KPP_REAL, INTENT(INOUT) :: Y(NVAR)
 473 !~~~> Input: integration interval
 474    KPP_REAL, INTENT(IN) :: Tstart,Tend
 475 !~~~> Output: time at which the solution is returned (T=Tend if success)
 476    KPP_REAL, INTENT(OUT) ::  T
 477 !~~~> Input: tolerances
 478    KPP_REAL, INTENT(IN) ::  AbsTol(NVAR), RelTol(NVAR)
 479 !~~~> Input: The Rosenbrock method parameters
 480    INTEGER, INTENT(IN) ::  ros_S
 481    KPP_REAL, INTENT(IN) :: ros_M(ros_S), ros_E(ros_S),  &
 482        ros_Alpha(ros_S), ros_A(ros_S*(ros_S-1)/2), &
 483        ros_Gamma(ros_S), ros_C(ros_S*(ros_S-1)/2), ros_ELO
 484    LOGICAL, INTENT(IN) :: ros_NewF(ros_S)
 485 !~~~> Input: integration parameters
 486    LOGICAL, INTENT(IN) :: Autonomous, VectorTol
 487    KPP_REAL, INTENT(IN) :: Hstart, Hmin, Hmax
 488    INTEGER, INTENT(IN) :: Max_no_steps
 489    KPP_REAL, INTENT(IN) :: Roundoff, FacMin, FacMax, FacRej, FacSafe
 490 !~~~> Output: last accepted step
 491    KPP_REAL, INTENT(OUT) :: Hexit
 492 !~~~> Output: Error indicator
 493    INTEGER, INTENT(OUT) :: IERR
 494 !~~~> Input
 495    KPP_REAL, INTENT(IN), DIMENSION(NFIX) :: FIX
 496 !~~~> Input
 497    KPP_REAL, INTENT(IN), DIMENSION(NREACT) :: RCONST
 498
 499 !~~~>  Statistics on the work performed by the Rosenbrock method
 500   INTEGER, INTENT(INOUT)  :: Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng
 501
 502 ! ~~~~ Local variables
 503    KPP_REAL :: Ynew(NVAR), Fcn0(NVAR), Fcn(NVAR)
 504    KPP_REAL :: K(NVAR*ros_S), dFdT(NVAR)
 505 #ifdef FULL_ALGEBRA
 506    KPP_REAL :: Jac0(NVAR,NVAR), Ghimj(NVAR,NVAR)
 507 #else
 508    KPP_REAL :: Jac0(LU_NONZERO), Ghimj(LU_NONZERO)
 509 #endif
 510    KPP_REAL :: H, Hnew, HC, HG, Fac, Tau
 511    KPP_REAL :: Err, Yerr(NVAR)
 512    INTEGER :: Pivot(NVAR), Direction, ioffset, j, istage
 513    LOGICAL :: RejectLastH, RejectMoreH, Singular
 514 !~~~>  Local parameters
 515    KPP_REAL, PARAMETER :: ZERO = 0.0_dp, ONE  = 1.0_dp
 516    KPP_REAL, PARAMETER :: DeltaMin = 1.0E-5_dp
 517 !~~~>  Locally called functions
 518 !    KPP_REAL WLAMCH
 519 !    EXTERNAL WLAMCH
 520 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 521
 522
 523 !~~~>  Initial preparations
 524    T = Tstart
 525    Hexit = 0.0_dp
 526    H = MIN(Hstart,Hmax)
 527    IF (ABS(H) <= 10.0_dp*Roundoff) H = DeltaMin
 528
 529    IF (Tend  >=  Tstart) THEN
 530      Direction = +1
 531    ELSE
 532      Direction = -1
 533    END IF
 534
 535    RejectLastH=.FALSE.
 536    RejectMoreH=.FALSE.
 537
 538 !~~~> Time loop begins below
 539
 540 TimeLoop: DO WHILE ( (Direction > 0).AND.((T-Tend)+Roundoff <= ZERO) &
 541        .OR. (Direction < 0).AND.((Tend-T)+Roundoff <= ZERO) )
 542
 543    IF ( Nstp > Max_no_steps ) THEN  ! Too many steps
 544       CALL KPP_ROOT_ros_ErrorMsg(-6,T,H,IERR)
 545       RETURN
 546    END IF
 547    IF ( ((T+0.1_dp*H) == T).OR.(H <= Roundoff) ) THEN  ! Step size too small
 548       CALL KPP_ROOT_ros_ErrorMsg(-7,T,H,IERR)
 549       RETURN
 550    END IF
 551
 552 !~~~>  Limit H if necessary to avoid going beyond Tend
 553    Hexit = H
 554    H = MIN(H,ABS(Tend-T))
 555
 556 !~~~>   Compute the function at current time
 557    CALL KPP_ROOT_FunTemplate(T,Y,Fcn0, RCONST, FIX, Nfun)
 558
 559 !~~~>  Compute the function derivative with respect to T
 560    IF (.NOT.Autonomous) THEN
 561       CALL KPP_ROOT_ros_FunTimeDeriv ( T, Roundoff, Y, &
 562                 Fcn0, dFdT, RCONST, FIX, Nfun )
 563    END IF
 564
 565 !~~~>   Compute the Jacobian at current time
 566    CALL KPP_ROOT_JacTemplate(T,Y,Jac0, FIX, Njac, RCONST)
 567
 568 !~~~>  Repeat step calculation until current step accepted
 569 UntilAccepted: DO
 570
 571    CALL KPP_ROOT_ros_PrepareMatrix(H,Direction,ros_Gamma(1), &
 572           Jac0,Ghimj,Pivot,Singular, Ndec,  Nsng )
 573    IF (Singular) THEN ! More than 5 consecutive failed decompositions
 574        CALL KPP_ROOT_ros_ErrorMsg(-8,T,H,IERR)
 575        RETURN
 576    END IF
 577
 578 !~~~>   Compute the stages
 579 Stage: DO istage = 1, ros_S
 580
 581       ! Current istage offset. Current istage vector is K(ioffset+1:ioffset+NVAR)
 582        ioffset = NVAR*(istage-1)
 583
 584       ! For the 1st istage the function has been computed previously
 585        IF ( istage == 1 ) THEN
 586          CALL KPP_ROOT_WCOPY(NVAR,Fcn0,1,Fcn,1)
 587       ! istage>1 and a new function evaluation is needed at the current istage
 588        ELSEIF ( ros_NewF(istage) ) THEN
 589          CALL KPP_ROOT_WCOPY(NVAR,Y,1,Ynew,1)
 590          DO j = 1, istage-1
 591            CALL KPP_ROOT_WAXPY(NVAR,ros_A((istage-1)*(istage-2)/2+j), &
 592             K(NVAR*(j-1)+1),1,Ynew,1)
 593          END DO
 594          Tau = T + ros_Alpha(istage)*Direction*H
 595          CALL KPP_ROOT_FunTemplate(Tau,Ynew,Fcn, RCONST, FIX, Nfun)
 596        END IF ! if istage == 1 elseif ros_NewF(istage)
 597        CALL KPP_ROOT_WCOPY(NVAR,Fcn,1,K(ioffset+1),1)
 598        DO j = 1, istage-1
 599          HC = ros_C((istage-1)*(istage-2)/2+j)/(Direction*H)
 600          CALL KPP_ROOT_WAXPY(NVAR,HC,K(NVAR*(j-1)+1),1,K(ioffset+1),1)
 601        END DO
 602        IF ((.NOT. Autonomous).AND.(ros_Gamma(istage).NE.ZERO)) THEN
 603          HG = Direction*H*ros_Gamma(istage)
 604          CALL KPP_ROOT_WAXPY(NVAR,HG,dFdT,1,K(ioffset+1),1)
 605        END IF
 606        CALL KPP_ROOT_ros_Solve(Ghimj, Pivot, K(ioffset+1), Nsol)
 607
 608    END DO Stage
 609
 610
 611 !~~~>  Compute the new solution
 612    CALL KPP_ROOT_WCOPY(NVAR,Y,1,Ynew,1)
 613    DO j=1,ros_S
 614          CALL KPP_ROOT_WAXPY(NVAR,ros_M(j),K(NVAR*(j-1)+1),1,Ynew,1)
 615    END DO
 616
 617 !~~~>  Compute the error estimation
 618    CALL KPP_ROOT_WSCAL(NVAR,ZERO,Yerr,1)
 619    DO j=1,ros_S
 620         CALL KPP_ROOT_WAXPY(NVAR,ros_E(j),K(NVAR*(j-1)+1),1,Yerr,1)
 621    END DO
 622    Err = KPP_ROOT_ros_ErrorNorm ( Y, Ynew, Yerr, AbsTol, RelTol, VectorTol )
 623
 624 !~~~> New step size is bounded by FacMin <= Hnew/H <= FacMax
 625    Fac  = MIN(FacMax,MAX(FacMin,FacSafe/Err**(ONE/ros_ELO)))
 626    Hnew = H*Fac
 627
 628 !~~~>  Check the error magnitude and adjust step size
 629    Nstp = Nstp+1
 630    IF ( (Err <= ONE).OR.(H <= Hmin) ) THEN  !~~~> Accept step
 631       Nacc = Nacc+1
 632       CALL KPP_ROOT_WCOPY(NVAR,Ynew,1,Y,1)
 633       T = T + Direction*H
 634       Hnew = MAX(Hmin,MIN(Hnew,Hmax))
 635       IF (RejectLastH) THEN  ! No step size increase after a rejected step
 636          Hnew = MIN(Hnew,H)
 637       END IF
 638       RejectLastH = .FALSE.
 639       RejectMoreH = .FALSE.
 640       H = Hnew
 641       EXIT UntilAccepted ! EXIT THE LOOP: WHILE STEP NOT ACCEPTED
 642    ELSE           !~~~> Reject step
 643       IF (RejectMoreH) THEN
 644          Hnew = H*FacRej
 645       END IF
 646       RejectMoreH = RejectLastH
 647       RejectLastH = .TRUE.
 648       H = Hnew
 649       IF (Nacc >= 1) THEN
 650          Nrej = Nrej+1
 651       END IF
 652    END IF ! Err <= 1
 653
 654    END DO UntilAccepted
 655
 656    END DO TimeLoop
 657
 658 !~~~> Succesful exit
 659    IERR = 1  !~~~> The integration was successful
 660
 661   END SUBROUTINE  KPP_ROOT_ros_Integrator
 662
 663
 664 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 665   KPP_REAL FUNCTION  KPP_ROOT_ros_ErrorNorm ( Y, Ynew, Yerr, &
 666                AbsTol, RelTol, VectorTol )
 667 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 668 !~~~> Computes the "scaled norm" of the error vector Yerr
 669 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 670    IMPLICIT NONE
 671
 672 ! Input arguments
 673    KPP_REAL, INTENT(IN) :: Y(NVAR), Ynew(NVAR), &
 674           Yerr(NVAR), AbsTol(NVAR), RelTol(NVAR)
 675    LOGICAL, INTENT(IN) ::  VectorTol
 676 ! Local variables
 677    KPP_REAL :: Err, Scale, Ymax
 678    INTEGER  :: i
 679    KPP_REAL, PARAMETER :: ZERO = 0.0_dp
 680
 681    Err = ZERO
 682    DO i=1,NVAR
 683      Ymax = MAX(ABS(Y(i)),ABS(Ynew(i)))
 684      IF (VectorTol) THEN
 685        Scale = AbsTol(i)+RelTol(i)*Ymax
 686      ELSE
 687        Scale = AbsTol(1)+RelTol(1)*Ymax
 688      END IF
 689      Err = Err+(Yerr(i)/Scale)**2
 690    END DO
 691    Err  = SQRT(Err/NVAR)
 692
 693     KPP_ROOT_ros_ErrorNorm = Err
 694
 695   END FUNCTION  KPP_ROOT_ros_ErrorNorm
 696
 697
 698 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 699   SUBROUTINE KPP_ROOT_ros_FunTimeDeriv ( T, Roundoff, Y, &
 700                 Fcn0, dFdT, RCONST, FIX, Nfun )
 701 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 702 !~~~> The time partial derivative of the function by finite differences
 703 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 704    IMPLICIT NONE
 705
 706 !~~~> Input arguments
 707    KPP_REAL, INTENT(IN) :: T, Roundoff, Y(NVAR), Fcn0(NVAR)
 708    KPP_REAL, INTENT(IN) :: RCONST(NREACT), FIX(NFIX)
 709 !~~~> Output arguments
 710    KPP_REAL, INTENT(OUT) :: dFdT(NVAR)
 711 !~~~> InOut args
 712    INTEGER, INTENT(INOUT) ::Nfun
 713 !~~~> Local variables
 714    KPP_REAL :: Delta
 715    KPP_REAL, PARAMETER :: ONE = 1.0_dp, DeltaMin = 1.0E-6_dp
 716
 717    Delta = SQRT(Roundoff)*MAX(DeltaMin,ABS(T))
 718    CALL KPP_ROOT_FunTemplate(T+Delta,Y,dFdT, RCONST, FIX, Nfun)
 719    CALL KPP_ROOT_WAXPY(NVAR,(-ONE),Fcn0,1,dFdT,1)
 720    CALL KPP_ROOT_WSCAL(NVAR,(ONE/Delta),dFdT,1)
 721
 722   END SUBROUTINE  KPP_ROOT_ros_FunTimeDeriv
 723
 724
 725 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 726   SUBROUTINE  KPP_ROOT_ros_PrepareMatrix ( H, Direction, gam, &
 727              Jac0, Ghimj, Pivot, Singular, Ndec,  Nsng  )
 728 ! --- --- --- --- --- --- --- --- --- --- --- --- ---
 729 !  Prepares the LHS matrix for stage calculations
 730 !  1.  Construct Ghimj = 1/(H*ham) - Jac0
 731 !      "(Gamma H) Inverse Minus Jacobian"
 732 !  2.  Repeat LU decomposition of Ghimj until successful.
 733 !       -half the step size if LU decomposition fails and retry
 734 !       -exit after 5 consecutive fails
 735 ! --- --- --- --- --- --- --- --- --- --- --- --- ---
 736    IMPLICIT NONE
 737
 738 !~~~> Input arguments
 739 #ifdef FULL_ALGEBRA
 740    KPP_REAL, INTENT(IN) ::  Jac0(NVAR,NVAR)
 741 #else
 742    KPP_REAL, INTENT(IN) ::  Jac0(LU_NONZERO)
 743 #endif
 744    KPP_REAL, INTENT(IN) ::  gam
 745    INTEGER, INTENT(IN) ::  Direction
 746 !~~~> Output arguments
 747 #ifdef FULL_ALGEBRA
 748    KPP_REAL, INTENT(OUT) :: Ghimj(NVAR,NVAR)
 749 #else
 750    KPP_REAL, INTENT(OUT) :: Ghimj(LU_NONZERO)
 751 #endif
 752    LOGICAL, INTENT(OUT) ::  Singular
 753    INTEGER, INTENT(OUT) ::  Pivot(NVAR)
 754 !~~~> Inout arguments
 755    KPP_REAL, INTENT(INOUT) :: H   ! step size is decreased when LU fails
 756    INTEGER, INTENT(INOUT) ::  Ndec, Nsng
 757 !~~~> Local variables
 758    INTEGER  :: i, ising, Nconsecutive
 759    KPP_REAL :: ghinv
 760    KPP_REAL, PARAMETER :: ONE  = 1.0_dp, HALF = 0.5_dp
 761
 762    Nconsecutive = 0
 763    Singular = .TRUE.
 764
 765    DO WHILE (Singular)
 766
 767 !~~~>    Construct Ghimj = 1/(H*gam) - Jac0
 768 #ifdef FULL_ALGEBRA
 769      CALL KPP_ROOT_WCOPY(NVAR*NVAR,Jac0,1,Ghimj,1)
 770      CALL KPP_ROOT_WSCAL(NVAR*NVAR,(-ONE),Ghimj,1)
 771      ghinv = ONE/(Direction*H*gam)
 772      DO i=1,NVAR
 773        Ghimj(i,i) = Ghimj(i,i)+ghinv
 774      END DO
 775 #else
 776      CALL KPP_ROOT_WCOPY(LU_NONZERO,Jac0,1,Ghimj,1)
 777      CALL KPP_ROOT_WSCAL(LU_NONZERO,(-ONE),Ghimj,1)
 778      ghinv = ONE/(Direction*H*gam)
 779      DO i=1,NVAR
 780        Ghimj(LU_DIAG(i)) = Ghimj(LU_DIAG(i))+ghinv
 781      END DO
 782 #endif
 783 !~~~>    Compute LU decomposition
 784      CALL KPP_ROOT_ros_Decomp( Ghimj, Pivot, ising, Ndec )
 785      IF (ising == 0) THEN
 786 !~~~>    If successful done
 787         Singular = .FALSE.
 788      ELSE ! ising .ne. 0
 789 !~~~>    If unsuccessful half the step size; if 5 consecutive fails then return
 790         Nsng = Nsng+1
 791         Nconsecutive = Nconsecutive+1
 792         Singular = .TRUE.
 793         PRINT*,'Warning: LU Decomposition returned ising = ',ising
 794         IF (Nconsecutive <= 5) THEN ! Less than 5 consecutive failed decompositions
 795            H = H*HALF
 796         ELSE  ! More than 5 consecutive failed decompositions
 797            RETURN
 798         END IF  ! Nconsecutive
 799       END IF    ! ising
 800
 801    END DO ! WHILE Singular
 802
 803   END SUBROUTINE  KPP_ROOT_ros_PrepareMatrix
 804
 805
 806 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 807   SUBROUTINE  KPP_ROOT_ros_Decomp( A, Pivot, ising, Ndec )
 808 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 809 !  Template for the LU decomposition
 810 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 811    IMPLICIT NONE
 812 !~~~> Inout variables
 813    KPP_REAL, INTENT(INOUT) :: A(LU_NONZERO)
 814 !~~~> Output variables
 815    INTEGER, INTENT(OUT) :: Pivot(NVAR), ising
 816    INTEGER, INTENT(INOUT) :: Ndec
 817
 818 #ifdef FULL_ALGEBRA
 819    CALL  KPP_ROOT_DGETRF( NVAR, NVAR, A, NVAR, Pivot, ising )
 820 #else
 821    CALL KPP_ROOT_KppDecomp ( A, ising )
 822    Pivot(1) = 1
 823 #endif
 824    Ndec = Ndec + 1
 825
 826   END SUBROUTINE  KPP_ROOT_ros_Decomp
 827
 828
 829 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 830   SUBROUTINE  KPP_ROOT_ros_Solve( A, Pivot, b, Nsol )
 831 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 832 !  Template for the forward/backward substitution (using pre-computed LU decomposition)
 833 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 834    IMPLICIT NONE
 835 !~~~> Input variables
 836    KPP_REAL, INTENT(IN) :: A(LU_NONZERO)
 837    INTEGER, INTENT(IN) :: Pivot(NVAR)
 838 !~~~~>  InOut args
 839    INTEGER, INTENT(INOUT) :: nsol
 840 !~~~> InOut variables
 841    KPP_REAL, INTENT(INOUT) :: b(NVAR)
 842
 843
 844 #ifdef FULL_ALGEBRA
 845    CALL  KPP_ROOT_DGETRS( 'N', NVAR , 1, A, NVAR, Pivot, b, NVAR, 0 )
 846 #else
 847    CALL KPP_ROOT_KppSolve( A, b )
 848 #endif
 849
 850    Nsol = Nsol+1
 851
 852   END SUBROUTINE  KPP_ROOT_ros_Solve
 853
 854
 855
 856 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 857   SUBROUTINE  KPP_ROOT_Ros2 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
 858             ros_Gamma,ros_NewF,ros_ELO,ros_Name)
 859 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 860 ! --- AN L-STABLE METHOD, 2 stages, order 2
 861 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 862
 863   IMPLICIT NONE
 864
 865    INTEGER, PARAMETER :: S=2
 866    INTEGER, INTENT(OUT) ::  ros_S
 867    KPP_REAL, DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
 868    KPP_REAL, DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C
 869    KPP_REAL, INTENT(OUT) :: ros_ELO
 870    LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF
 871    CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
 872 !cms   DOUBLE PRECISION g
 873     KPP_REAL :: g
 874
 875     g = 1.0_dp + 1.0_dp/SQRT(2.0_dp)
 876
 877 !~~~> Name of the method
 878     ros_Name = 'ROS-2'
 879 !~~~> Number of stages
 880     ros_S = S
 881
 882 !~~~> The coefficient matrices A and C are strictly lower triangular.
 883 !   The lower triangular (subdiagonal) elements are stored in row-wise order:
 884 !   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
 885 !   The general mapping formula is:
 886 !       A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
 887 !       C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
 888
 889     ros_A(1) = (1.0_dp)/g
 890     ros_C(1) = (-2.0_dp)/g
 891 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
 892 !   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
 893     ros_NewF(1) = .TRUE.
 894     ros_NewF(2) = .TRUE.
 895 !~~~> M_i = Coefficients for new step solution
 896     ros_M(1)= (3.0_dp)/(2.0_dp*g)
 897     ros_M(2)= (1.0_dp)/(2.0_dp*g)
 898 ! E_i = Coefficients for error estimator
 899     ros_E(1) = 1.0_dp/(2.0_dp*g)
 900     ros_E(2) = 1.0_dp/(2.0_dp*g)
 901 !~~~> ros_ELO = estimator of local order - the minimum between the
 902 !    main and the embedded scheme orders plus one
 903     ros_ELO = 2.0_dp
 904 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
 905     ros_Alpha(1) = 0.0_dp
 906     ros_Alpha(2) = 1.0_dp
 907 !~~~> Gamma_i = \sum_j  gamma_{i,j}
 908     ros_Gamma(1) = g
 909     ros_Gamma(2) =-g
 910
 911  END SUBROUTINE  KPP_ROOT_Ros2
 912
 913
 914 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 915   SUBROUTINE  KPP_ROOT_Ros3 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
 916            ros_Gamma,ros_NewF,ros_ELO,ros_Name)
 917 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 918 ! --- AN L-STABLE METHOD, 3 stages, order 3, 2 function evaluations
 919 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 920
 921   IMPLICIT NONE
 922
 923    INTEGER, PARAMETER :: S=3
 924    INTEGER, INTENT(OUT) ::  ros_S
 925    KPP_REAL, DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
 926    KPP_REAL, DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C
 927    KPP_REAL, INTENT(OUT) :: ros_ELO
 928    LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF
 929    CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
 930
 931 !~~~> Name of the method
 932    ros_Name = 'ROS-3'
 933 !~~~> Number of stages
 934    ros_S = S
 935
 936 !~~~> The coefficient matrices A and C are strictly lower triangular.
 937 !   The lower triangular (subdiagonal) elements are stored in row-wise order:
 938 !   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
 939 !   The general mapping formula is:
 940 !       A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
 941 !       C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
 942
 943    ros_A(1)= 1.0_dp
 944    ros_A(2)= 1.0_dp
 945    ros_A(3)= 0.0_dp
 946
 947    ros_C(1) = -0.10156171083877702091975600115545E+01_dp
 948    ros_C(2) =  0.40759956452537699824805835358067E+01_dp
 949    ros_C(3) =  0.92076794298330791242156818474003E+01_dp
 950 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
 951 !   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
 952    ros_NewF(1) = .TRUE.
 953    ros_NewF(2) = .TRUE.
 954    ros_NewF(3) = .FALSE.
 955 !~~~> M_i = Coefficients for new step solution
 956    ros_M(1) =  0.1E+01_dp
 957    ros_M(2) =  0.61697947043828245592553615689730E+01_dp
 958    ros_M(3) = -0.42772256543218573326238373806514E+00_dp
 959 ! E_i = Coefficients for error estimator
 960    ros_E(1) =  0.5E+00_dp
 961    ros_E(2) = -0.29079558716805469821718236208017E+01_dp
 962    ros_E(3) =  0.22354069897811569627360909276199E+00_dp
 963 !~~~> ros_ELO = estimator of local order - the minimum between the
 964 !    main and the embedded scheme orders plus 1
 965    ros_ELO = 3.0_dp
 966 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
 967    ros_Alpha(1)= 0.0E+00_dp
 968    ros_Alpha(2)= 0.43586652150845899941601945119356E+00_dp
 969    ros_Alpha(3)= 0.43586652150845899941601945119356E+00_dp
 970 !~~~> Gamma_i = \sum_j  gamma_{i,j}
 971    ros_Gamma(1)= 0.43586652150845899941601945119356E+00_dp
 972    ros_Gamma(2)= 0.24291996454816804366592249683314E+00_dp
 973    ros_Gamma(3)= 0.21851380027664058511513169485832E+01_dp
 974
 975   END SUBROUTINE  KPP_ROOT_Ros3
 976
 977 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 978
 979
 980 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 981   SUBROUTINE  KPP_ROOT_Ros4 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
 982            ros_Gamma,ros_NewF,ros_ELO,ros_Name)
 983 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 984 !     L-STABLE ROSENBROCK METHOD OF ORDER 4, WITH 4 STAGES
 985 !     L-STABLE EMBEDDED ROSENBROCK METHOD OF ORDER 3
 986 !
 987 !      E. HAIRER AND G. WANNER, SOLVING ORDINARY DIFFERENTIAL
 988 !      EQUATIONS II. STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS.
 989 !      SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS,
 990 !      SPRINGER-VERLAG (1990)
 991 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 992
 993   IMPLICIT NONE
 994
 995    INTEGER, PARAMETER :: S=4
 996    INTEGER, INTENT(OUT) ::  ros_S
 997    KPP_REAL, DIMENSION(4), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
 998    KPP_REAL, DIMENSION(6), INTENT(OUT) :: ros_A, ros_C
 999    KPP_REAL, INTENT(OUT) :: ros_ELO
1000    LOGICAL, DIMENSION(4), INTENT(OUT) :: ros_NewF
1001    CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
1002 !cms   DOUBLE PRECISION g
1003    KPP_REAL :: g
1004
1005
1006 !~~~> Name of the method
1007    ros_Name = 'ROS-4'
1008 !~~~> Number of stages
1009    ros_S = S
1010
1011 !~~~> The coefficient matrices A and C are strictly lower triangular.
1012 !   The lower triangular (subdiagonal) elements are stored in row-wise order:
1013 !   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
1014 !   The general mapping formula is:
1015 !       A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
1016 !       C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
1017
1018    ros_A(1) = 0.2000000000000000E+01_dp
1019    ros_A(2) = 0.1867943637803922E+01_dp
1020    ros_A(3) = 0.2344449711399156E+00_dp
1021    ros_A(4) = ros_A(2)
1022    ros_A(5) = ros_A(3)
1023    ros_A(6) = 0.0_dp
1024
1025    ros_C(1) =-0.7137615036412310E+01_dp
1026    ros_C(2) = 0.2580708087951457E+01_dp
1027    ros_C(3) = 0.6515950076447975E+00_dp
1028    ros_C(4) =-0.2137148994382534E+01_dp
1029    ros_C(5) =-0.3214669691237626E+00_dp
1030    ros_C(6) =-0.6949742501781779E+00_dp
1031 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
1032 !   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
1033    ros_NewF(1)  = .TRUE.
1034    ros_NewF(2)  = .TRUE.
1035    ros_NewF(3)  = .TRUE.
1036    ros_NewF(4)  = .FALSE.
1037 !~~~> M_i = Coefficients for new step solution
1038    ros_M(1) = 0.2255570073418735E+01_dp
1039    ros_M(2) = 0.2870493262186792E+00_dp
1040    ros_M(3) = 0.4353179431840180E+00_dp
1041    ros_M(4) = 0.1093502252409163E+01_dp
1042 !~~~> E_i  = Coefficients for error estimator
1043    ros_E(1) =-0.2815431932141155E+00_dp
1044    ros_E(2) =-0.7276199124938920E-01_dp
1045    ros_E(3) =-0.1082196201495311E+00_dp
1046    ros_E(4) =-0.1093502252409163E+01_dp
1047 !~~~> ros_ELO  = estimator of local order - the minimum between the
1048 !    main and the embedded scheme orders plus 1
1049    ros_ELO  = 4.0_dp
1050 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
1051    ros_Alpha(1) = 0.0_dp
1052    ros_Alpha(2) = 0.1145640000000000E+01_dp
1053    ros_Alpha(3) = 0.6552168638155900E+00_dp
1054    ros_Alpha(4) = ros_Alpha(3)
1055 !~~~> Gamma_i = \sum_j  gamma_{i,j}
1056    ros_Gamma(1) = 0.5728200000000000E+00_dp
1057    ros_Gamma(2) =-0.1769193891319233E+01_dp
1058    ros_Gamma(3) = 0.7592633437920482E+00_dp
1059    ros_Gamma(4) =-0.1049021087100450E+00_dp
1060
1061   END SUBROUTINE  KPP_ROOT_Ros4
1062
1063 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1064   SUBROUTINE  KPP_ROOT_Rodas3 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
1065             ros_Gamma,ros_NewF,ros_ELO,ros_Name)
1066 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1067 ! --- A STIFFLY-STABLE METHOD, 4 stages, order 3
1068 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1069
1070   IMPLICIT NONE
1071
1072    INTEGER, PARAMETER :: S=4
1073    INTEGER, INTENT(OUT) ::  ros_S
1074    KPP_REAL, DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
1075    KPP_REAL, DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C
1076    KPP_REAL, INTENT(OUT) :: ros_ELO
1077    LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF
1078    CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
1079 !cms   DOUBLE PRECISION g
1080    KPP_REAL :: g
1081
1082 !~~~> Name of the method
1083    ros_Name = 'RODAS-3'
1084 !~~~> Number of stages
1085    ros_S = S
1086
1087 !~~~> The coefficient matrices A and C are strictly lower triangular.
1088 !   The lower triangular (subdiagonal) elements are stored in row-wise order:
1089 !   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
1090 !   The general mapping formula is:
1091 !       A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
1092 !       C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
1093
1094    ros_A(1) = 0.0E+00_dp
1095    ros_A(2) = 2.0E+00_dp
1096    ros_A(3) = 0.0E+00_dp
1097    ros_A(4) = 2.0E+00_dp
1098    ros_A(5) = 0.0E+00_dp
1099    ros_A(6) = 1.0E+00_dp
1100
1101    ros_C(1) = 4.0E+00_dp
1102    ros_C(2) = 1.0E+00_dp
1103    ros_C(3) =-1.0E+00_dp
1104    ros_C(4) = 1.0E+00_dp
1105    ros_C(5) =-1.0E+00_dp
1106    ros_C(6) =-(8.0E+00_dp/3.0E+00_dp)
1107
1108 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
1109 !   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
1110    ros_NewF(1)  = .TRUE.
1111    ros_NewF(2)  = .FALSE.
1112    ros_NewF(3)  = .TRUE.
1113    ros_NewF(4)  = .TRUE.
1114 !~~~> M_i = Coefficients for new step solution
1115    ros_M(1) = 2.0E+00_dp
1116    ros_M(2) = 0.0E+00_dp
1117    ros_M(3) = 1.0E+00_dp
1118    ros_M(4) = 1.0E+00_dp
1119 !~~~> E_i  = Coefficients for error estimator
1120    ros_E(1) = 0.0E+00_dp
1121    ros_E(2) = 0.0E+00_dp
1122    ros_E(3) = 0.0E+00_dp
1123    ros_E(4) = 1.0E+00_dp
1124 !~~~> ros_ELO  = estimator of local order - the minimum between the
1125 !    main and the embedded scheme orders plus 1
1126    ros_ELO  = 3.0E+00_dp
1127 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
1128    ros_Alpha(1) = 0.0E+00_dp
1129    ros_Alpha(2) = 0.0E+00_dp
1130    ros_Alpha(3) = 1.0E+00_dp
1131    ros_Alpha(4) = 1.0E+00_dp
1132 !~~~> Gamma_i = \sum_j  gamma_{i,j}
1133    ros_Gamma(1) = 0.5E+00_dp
1134    ros_Gamma(2) = 1.5E+00_dp
1135    ros_Gamma(3) = 0.0E+00_dp
1136    ros_Gamma(4) = 0.0E+00_dp
1137
1138   END SUBROUTINE  KPP_ROOT_Rodas3
1139
1140 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1141   SUBROUTINE  KPP_ROOT_Rodas4 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
1142              ros_Gamma,ros_NewF,ros_ELO,ros_Name)
1143 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1144 !     STIFFLY-STABLE ROSENBROCK METHOD OF ORDER 4, WITH 6 STAGES
1145 !
1146 !      E. HAIRER AND G. WANNER, SOLVING ORDINARY DIFFERENTIAL
1147 !      EQUATIONS II. STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS.
1148 !      SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS,
1149 !      SPRINGER-VERLAG (1996)
1150 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1151
1152   IMPLICIT NONE
1153
1154    INTEGER, PARAMETER :: S=6
1155    INTEGER, INTENT(OUT) ::  ros_S
1156    KPP_REAL, DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
1157    KPP_REAL, DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C
1158    KPP_REAL, INTENT(OUT) :: ros_ELO
1159    LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF
1160    CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
1161 !cms   DOUBLE PRECISION g
1162     KPP_REAL :: g
1163
1164 !~~~> Name of the method
1165     ros_Name = 'RODAS-4'
1166 !~~~> Number of stages
1167     ros_S = 6
1168
1169 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
1170     ros_Alpha(1) = 0.000_dp
1171     ros_Alpha(2) = 0.386_dp
1172     ros_Alpha(3) = 0.210_dp
1173     ros_Alpha(4) = 0.630_dp
1174     ros_Alpha(5) = 1.000_dp
1175     ros_Alpha(6) = 1.000_dp
1176
1177 !~~~> Gamma_i = \sum_j  gamma_{i,j}
1178     ros_Gamma(1) = 0.2500000000000000E+00_dp
1179     ros_Gamma(2) =-0.1043000000000000E+00_dp
1180     ros_Gamma(3) = 0.1035000000000000E+00_dp
1181     ros_Gamma(4) =-0.3620000000000023E-01_dp
1182     ros_Gamma(5) = 0.0_dp
1183     ros_Gamma(6) = 0.0_dp
1184
1185 !~~~> The coefficient matrices A and C are strictly lower triangular.
1186 !   The lower triangular (subdiagonal) elements are stored in row-wise order:
1187 !   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
1188 !   The general mapping formula is:  A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
1189 !                  C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
1190
1191     ros_A(1) = 0.1544000000000000E+01_dp
1192     ros_A(2) = 0.9466785280815826E+00_dp
1193     ros_A(3) = 0.2557011698983284E+00_dp
1194     ros_A(4) = 0.3314825187068521E+01_dp
1195     ros_A(5) = 0.2896124015972201E+01_dp
1196     ros_A(6) = 0.9986419139977817E+00_dp
1197     ros_A(7) = 0.1221224509226641E+01_dp
1198     ros_A(8) = 0.6019134481288629E+01_dp
1199     ros_A(9) = 0.1253708332932087E+02_dp
1200     ros_A(10) =-0.6878860361058950E+00_dp
1201     ros_A(11) = ros_A(7)
1202     ros_A(12) = ros_A(8)
1203     ros_A(13) = ros_A(9)
1204     ros_A(14) = ros_A(10)
1205     ros_A(15) = 1.0E+00_dp
1206
1207     ros_C(1) =-0.5668800000000000E+01_dp
1208     ros_C(2) =-0.2430093356833875E+01_dp
1209     ros_C(3) =-0.2063599157091915E+00_dp
1210     ros_C(4) =-0.1073529058151375E+00_dp
1211     ros_C(5) =-0.9594562251023355E+01_dp
1212     ros_C(6) =-0.2047028614809616E+02_dp
1213     ros_C(7) = 0.7496443313967647E+01_dp
1214     ros_C(8) =-0.1024680431464352E+02_dp
1215     ros_C(9) =-0.3399990352819905E+02_dp
1216     ros_C(10) = 0.1170890893206160E+02_dp
1217     ros_C(11) = 0.8083246795921522E+01_dp
1218     ros_C(12) =-0.7981132988064893E+01_dp
1219     ros_C(13) =-0.3152159432874371E+02_dp
1220     ros_C(14) = 0.1631930543123136E+02_dp
1221     ros_C(15) =-0.6058818238834054E+01_dp
1222
1223 !~~~> M_i = Coefficients for new step solution
1224     ros_M(1) = ros_A(7)
1225     ros_M(2) = ros_A(8)
1226     ros_M(3) = ros_A(9)
1227     ros_M(4) = ros_A(10)
1228     ros_M(5) = 1.0E+00_dp
1229     ros_M(6) = 1.0E+00_dp
1230
1231 !~~~> E_i  = Coefficients for error estimator
1232     ros_E(1) = 0.0E+00_dp
1233     ros_E(2) = 0.0E+00_dp
1234     ros_E(3) = 0.0E+00_dp
1235     ros_E(4) = 0.0E+00_dp
1236     ros_E(5) = 0.0E+00_dp
1237     ros_E(6) = 1.0E+00_dp
1238
1239 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
1240 !   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
1241     ros_NewF(1) = .TRUE.
1242     ros_NewF(2) = .TRUE.
1243     ros_NewF(3) = .TRUE.
1244     ros_NewF(4) = .TRUE.
1245     ros_NewF(5) = .TRUE.
1246     ros_NewF(6) = .TRUE.
1247
1248 !~~~> ros_ELO  = estimator of local order - the minimum between the
1249 !        main and the embedded scheme orders plus 1
1250     ros_ELO = 4.0_dp
1251
1252   END SUBROUTINE  KPP_ROOT_Rodas4
1253
1254 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1255 !   End of the set of internal Rosenbrock subroutines
1256 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1257 END SUBROUTINE  KPP_ROOT_Rosenbrock
1258 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1259
1260
1261 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1262 SUBROUTINE  KPP_ROOT_FunTemplate( T, Y, Ydot, RCONST, FIX, Nfun )
1263 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1264 !  Template for the ODE function call.
1265 !  Updates the rate coefficients (and possibly the fixed species) at each call
1266 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1267    USE KPP_ROOT_Parameters
1268 !! USE KPP_ROOT_Global
1269 !! USE KPP_ROOT_Function
1270 !! USE KPP_ROOT_Rates
1271 !~~~> Input variables
1272    KPP_REAL :: T, Y(NVAR)
1273    KPP_REAL :: RCONST(NREACT)
1274    KPP_REAL :: FIX(NFIX)
1275 !~~~> Output variables
1276    KPP_REAL :: Ydot(NVAR)
1277    INTEGER :: Nfun
1278
1279
1280 !~~~> Local variables
1281 !!   KPP_REAL :: Told
1282
1283 !!   Told = TIME
1284 !!   TIME = T
1285 !!   CALL Update_SUN()
1286 !!   CALL Update_RCONST()
1287    CALL KPP_ROOT_Fun( Y, FIX, RCONST, Ydot )
1288 !!   TIME = Told
1289
1290    Nfun = Nfun+1
1291
1292 END SUBROUTINE  KPP_ROOT_FunTemplate
1293
1294
1295 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1296 SUBROUTINE  KPP_ROOT_JacTemplate( T, Y, Jcb, FIX, Njac, RCONST )
1297 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1298 !  Template for the ODE Jacobian call.
1299 !  Updates the rate coefficients (and possibly the fixed species) at each call
1300 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1301  USE KPP_ROOT_Parameters
1302  !!USE KPP_ROOT_Global
1303  USE KPP_ROOT_Jacobian
1304 !! USE KPP_ROOT_LinearAlgebra
1305 !! USE KPP_ROOT_Rates
1306 !~~~> Input variables
1307     KPP_REAL :: T, Y(NVAR)
1308     KPP_REAL :: FIX(NFIX)
1309     KPP_REAL :: RCONST(NREACT)
1310
1311     INTEGER :: Njac
1312
1313 !~~~> Output variables
1314 #ifdef FULL_ALGEBRA
1315     KPP_REAL :: JV(LU_NONZERO), Jcb(NVAR,NVAR)
1316 #else
1317     KPP_REAL :: Jcb(LU_NONZERO)
1318 #endif
1319 !~~~> Local variables
1320     KPP_REAL :: Told
1321 #ifdef FULL_ALGEBRA
1322     INTEGER :: i, j
1323 #endif
1324
1325 !!    Told = TIME
1326 !!    TIME = T
1327 !!    CALL Update_SUN()
1328 !!    CALL Update_RCONST()
1329 #ifdef FULL_ALGEBRA
1330     CALL KPP_ROOT_Jac_SP(Y, FIX, RCONST, JV)
1331     DO j=1,NVAR
1332       DO i=1,NVAR
1333          Jcb(i,j) = 0.0d0
1334       END DO
1335     END DO
1336     DO i=1,LU_NONZERO
1337        Jcb(LU_IROW(i),LU_ICOL(i)) = JV(i)
1338     END DO
1339 #else
1340     CALL KPP_ROOT_Jac_SP( Y, FIX, RCONST, Jcb )
1341 #endif
1342 !!    TIME = Told
1343
1344     Njac = Njac+1
1345
1346 END SUBROUTINE  KPP_ROOT_JacTemplate
1347
1348 !!!END MODULE KPP_ROOT_Integrator