runsteady {rootSolve}

Dynamically runs a system of ordinary differential equations (ODE) to steady-state

Description

Solves the steady-state condition of ordinary differential equations (ODE) in the form:

dy/dt = f(t,y)

by dynamically running till the summed absolute values of the derivatives become smaller than some predefined tolerance.

The R function runsteady makes use of the FORTRAN ODE solver DLSODE, written by Alan C. Hindmarsh and Andrew H. Sherman

The system of ODE's is written as an R function or defined in compiled code that has been dynamically loaded. The user has to specify whether or not the problem is stiff and choose the appropriate solution method (e.g. make choices about the type of the Jacobian).

Usage

runsteady(y, time = c(0, Inf), func, parms, 
          stol = 1e-8, rtol = 1e-6, atol = 1e-6,  
          jacfunc = NULL, jactype = "fullint", mf = NULL, 
          verbose = FALSE, tcrit = NULL, hmin = 0, hmax = NULL, 
          hini = 0, ynames = TRUE, maxord = NULL, bandup = NULL, 
          banddown = NULL, maxsteps = 100000, dllname = NULL, 
          initfunc = dllname, initpar = parms, rpar = NULL, 
          ipar = NULL, nout = 0, outnames = NULL, 
          forcings = NULL, initforc = NULL, fcontrol = NULL, 
          lrw = NULL, liw = NULL, times = time, ...)

Arguments

y

the initial (state) values for the ODE system. If y has a name attribute, the names will be used to label the output matrix.

time, times

The simulation time. This should be a 2-valued vector, consisting of the initial time and the end time. The last time value should be large enough to make sure that steady-state is effectively reached in this period. The simulation will stop either when times[2] has been reached or when maxsteps have been performed. (note: since version 1.7, argument time has been added, for consistency with other solvers.)

func

either an R-function that computes the values of the derivatives in the ODE system (the model definition) at time t, or a character string giving the name of a compiled function in a dynamically loaded shared library.

If func is an R-function, it must be defined as: yprime = func(t, y, parms,...). t is the current time point in the integration, y is the current estimate of the variables in the ODE system. If the initial values y has a names attribute, the names will be available inside func. parms is a vector or list of parameters; ... (optional) are any other arguments passed to the function.

The return value of func should be a list, whose first element is a vector containing the derivatives of y with respect to time, and whose next elements are global values that are required at each point in times.

The derivatives should be specified in the same order as the state variables y.

parms

vector or list of parameters used in func or jacfunc.

stol

steady-state tolerance; it is assumed that steady-state is reached if the average of absolute values of the derivatives drops below this number.

rtol

relative error tolerance of integrator, either a scalar or an array as long as y. See details.

atol

absolute error tolerance of integrator, either a scalar or an array as long as y. See details.

jacfunc

if not NULL, an R function that computes the jacobian of the system of differential equations dydot(i)/dy(j), or a string giving the name of a function or subroutine in ‘dllname’ that computes the jacobian (see Details below for more about this option). In some circumstances, supplying jacfunc can speed up the computations, if the system is stiff. The R calling sequence for jacfunc is identical to that of func.

If the jacobian is a full matrix, jacfunc should return a matrix dydot/dy, where the ith row contains the derivative of dy_i/dt with respect to y_j, or a vector containing the matrix elements by columns (the way R and Fortran store matrices).

If the jacobian is banded, jacfunc should return a matrix containing only the nonzero bands of the jacobian, rotated row-wise. See first example of lsode.

jactype

the structure of the jacobian, one of "fullint", "fullusr", "bandusr", "bandint", "sparse" - either full, banded or sparse and estimated internally or by user; overruled if mf is not NULL. If "sparse" then method lsodes is used, else lsode.

mf

the "method flag" passed to function lsode - overrules jactype - provides more options than jactype - see details.

verbose

if TRUE: full output to the screen, e.g. will output the settings of vectors *istate* and *rstate* - see details.

tcrit

if not NULL, then lsode cannot integrate past tcrit. The Fortran routine lsode overshoots its targets (times points in the vector times), and interpolates values for the desired time points. If there is a time beyond which integration should not proceed (perhaps because of a singularity), that should be provided in tcrit.

hmin

an optional minimum value of the integration stepsize. In special situations this parameter may speed up computations with the cost of precision. Don't use hmin if you don't know why!

hmax

an optional maximum value of the integration stepsize. If not specified, hmax is set to the largest difference in times, to avoid that the simulation possibly ignores short-term events. If 0, no maximal size is specified.

hini

initial step size to be attempted; if 0, the initial step size is determined by the solver.

ynames

if FALSE: names of state variables are not passed to function func ; this may speed up the simulation.

maxord

the maximum order to be allowed. NULL uses the default, i.e. order 12 if implicit Adams method (meth=1), order 5 if BDF method (meth=2) or if jacType == 'sparse'. Reduce maxord to save storage space.

bandup

number of non-zero bands above the diagonal, in case the jacobian is banded.

banddown

number of non-zero bands below the diagonal, in case the jacobian is banded.

maxsteps

maximal number of steps. The simulation will stop either when maxsteps have been performed or when times[2] has been reached.

dllname

a string giving the name of the shared library (without extension) that contains all the compiled function or subroutine definitions refered to in func and jacfunc. See package vignette.

initfunc

if not NULL, the name of the initialisation function (which initialises values of parameters), as provided in ‘dllname’. See package vignette.

initpar

only when ‘dllname’ is specified and an initialisation function initfunc is in the dll: the parameters passed to the initialiser, to initialise the common blocks (fortran) or global variables (C, C++).

rpar

only when ‘dllname’ is specified: a vector with double precision values passed to the dll-functions whose names are specified by func and jacfunc.

ipar

only when ‘dllname’ is specified: a vector with integer values passed to the dll-functions whose names are specified by func and jacfunc.

nout

only used if dllname is specified and the model is defined in compiled code: the number of output variables calculated in the compiled function func, present in the shared library. Note: it is not automatically checked whether this is indeed the number of output variables calculed in the dll - you have to perform this check in the code - See package vignette of deSolve.

outnames

only used if ‘dllname’ is specified and nout > 0: the names of output variables calculated in the compiled function func, present in the shared library.

forcings

only used if ‘dllname’ is specified: a vector with the forcing function values, or a list with the forcing function data sets, each present as a two-columned matrix, with (time,value); interpolation outside the interval [min(times), max(times)] is done by taking the value at the closest data extreme.

This feature is here for compatibility with models defined in compiled code from package deSolve; see deSolve's package vignette "compiledCode".

initforc

if not NULL, the name of the forcing function initialisation function, as provided in ‘dllname’. It MUST be present if forcings has been given a value. See deSolve's package vignette "compiledCode".

fcontrol

A list of control parameters for the forcing functions. See deSolve's package vignette "compiledCode".

lrw

Only if jactype = 'sparse', the length of the real work array rwork; due to the sparsicity, this cannot be readily predicted. If NULL, a guess will be made, and if not sufficient, lsodes will return with a message indicating the size of rwork actually required. Therefore, some experimentation may be necessary to estimate the value of lrw.

For instance, if you get the error:

 
DLSODES- RWORK length is insufficient to proceed.                               
  Length needed is .ge. LENRW (=I1), exceeds LRW (=I2)                    
  In above message,  I1 =     27627   I2 =     25932 

set lrw equal to 27627 or a higher value

liw

Only if jactype = 'sparse', the length of the integer work array iwork; due to the sparsicity, this cannot be readily predicted. If NULL, a guess will be made, and if not sufficient, lsodes will return with a message indicating the size of iwork actually required. Therefore, some experimentation may be necessary to estimate the value of liw.

...

additional arguments passed to func and jacfunc allowing this to be a generic function.

Details

The work is done by the Fortran subroutine dlsode or dlsodes (if sparse), whose documentation should be consulted for details (it is included as comments in the source file ‘src/lsodes.f’). The implementation is based on the November, 2003 version of lsode, from Netlib.

Before using runsteady, the user has to decide whether or not the problem is stiff.

If the problem is nonstiff, use method flag mf = 10, which selects a nonstiff (Adams) method, no Jacobian used..

If the problem is stiff, there are four standard choices which can be specified with jactype or mf.

The options for jactype are

• jactype = "fullint" : a full jacobian, calculated internally by lsode, corresponds to mf=22.

• jactype = "fullusr" : a full jacobian, specified by user function jacfunc, corresponds to mf=21.

• jactype = "bandusr" : a banded jacobian, specified by user function jacfunc; the size of the bands specified by bandup and banddown, corresponds to mf=24.

• jactype = "bandint" : a banded jacobian, calculated by lsode; the size of the bands specified by bandup and banddown, corresponds to mf=25.

• jactype = "sparse" : the soler lsodes is used, and the sparse jacobian is calculated by lsodes - not possible to specify jacfunc.

More options are available when specifying mf directly.

The legal values of mf are 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25.

mf is a positive two-digit integer, mf = (10*METH + MITER), where

• METH indicates the basic linear multistep method: METH = 1 means the implicit Adams method. METH = 2 means the method based on backward differentiation formulas (BDF-s).

• MITER indicates the corrector iteration method: MITER = 0 means functional iteration (no Jacobian matrix is involved). MITER = 1 means chord iteration with a user-supplied full (NEQ by NEQ) Jacobian. MITER = 2 means chord iteration with an internally generated (difference quotient) full Jacobian (using NEQ extra calls to func per df/dy value). MITER = 3 means chord iteration with an internally generated diagonal Jacobian approximation (using 1 extra call to func per df/dy evaluation). MITER = 4 means chord iteration with a user-supplied banded Jacobian. MITER = 5 means chord iteration with an internally generated banded Jacobian (using ML+MU+1 extra calls to func per df/dy evaluation).

  If MITER = 1 or 4, the user must supply a subroutine jacfunc.

Inspection of the example below shows how to specify both a banded and full jacobian.

The input parameters rtol, and atol determine the error control performed by the solver.

See stode for details.

Models may be defined in compiled C or Fortran code, as well as in an R-function. See function stode for details.

The output will have the attributes *istate*, and *rstate*, two vectors with several useful elements.

if verbose = TRUE, the settings of istate and rstate will be written to the screen.

the following elements of istate are meaningful:

• el 1 : gives the conditions under which the last call to the integrator returned. 2 if lsode was successful, -1 if excess work done, -2 means excess accuracy requested. (Tolerances too small), -3 means illegal input detected. (See printed message.), -4 means repeated error test failures. (Check all input), -5 means repeated convergence failures. (Perhaps bad Jacobian supplied or wrong choice of MF or tolerances.), -6 means error weight became zero during problem. (Solution component i vanished, and atol or atol(i) = 0.)

• el 12 : The number of steps taken for the problem so far.

• el 13 : The number of evaluations for the problem so far.,

• el 14 : The number of Jacobian evaluations and LU decompositions so far.,

• el 15 : The method order last used (successfully).,

• el 16 : The order to be attempted on the next step.,

• el 17 : if el 1 =-4,-5: the largest component in the error vector,

rstate contains the following:

• 1: The step size in t last used (successfully).

• 2: The step size to be attempted on the next step.

• 3: The current value of the independent variable which the solver has actually reached, i.e. the current internal mesh point in t.

• 4: A tolerance scale factor, greater than 1.0, computed when a request for too much accuracy was detected.

For more information, see the comments in the original code lsode.f

Value

A list containing

y

a vector with the state variable values from the last iteration during estimation of steady-state condition of the system of equations. If y has a names attribute, it will be used to label the output values.

...

the number of "global" values returned.

The output will have the attribute steady, which returns TRUE, if steady-state has been reached, the attribute precis with the precision attained at the last iteration estimated as the mean absolute rate of change (sum(abs(dy))/n), the attribute time with the simulation time reached and the attribute steps with the number of steps performed.

The output will also have the attributes istate, and rstate, two vectors with several useful elements of the dynamic simulation. See details. The first element of istate returns the conditions under which the last call to the integrator returned. Normal is istate[1] = 2. If verbose = TRUE, the settings of istate and rstate will be written to the screen.

Author(s)

Karline Soetaert <[e-mail]>

References

Alan C. Hindmarsh, "ODEPACK, A Systematized Collection of ODE Solvers," in Scientific Computing, R. S. Stepleman, et al., Eds. (North-Holland, Amsterdam, 1983), pp. 55-64.

S. C. Eisenstat, M. C. Gursky, M. H. Schultz, and A. H. Sherman, Yale Sparse Matrix Package: I. The Symmetric Codes, Int. J. Num. Meth. Eng., 18 (1982), pp. 1145-1151.

S. C. Eisenstat, M. C. Gursky, M. H. Schultz, and A. H. Sherman, Yale Sparse Matrix Package: II. The Nonsymmetric Codes, Research Report No. 114, Dept. of Computer Sciences, Yale University, 1977.

See Also

steady, for a general interface to most of the steady-state solvers

steady.band, to find the steady-state of ODE models with a banded Jacobian

steady.1D, steady.2D, steady.3D steady-state solvers for 1-D, 2-D and 3-D partial differential equations.

stode, iterative steady-state solver for ODEs with full or banded Jacobian.

stodes, iterative steady-state solver for ODEs with arbitrary sparse Jacobian.

Examples

## =======================================================================
## A simple sediment biogeochemical model
## =======================================================================

model<-function(t, y, pars) {

with (as.list(c(y, pars)),{

  Min       = r*OM
  oxicmin   = Min*(O2/(O2+ks))
  anoxicmin = Min*(1-O2/(O2+ks))* SO4/(SO4+ks2)

  dOM  = Flux - oxicmin - anoxicmin
  dO2  = -oxicmin      -2*rox*HS*(O2/(O2+ks)) + D*(BO2-O2)
  dSO4 = -0.5*anoxicmin  +rox*HS*(O2/(O2+ks)) + D*(BSO4-SO4)
  dHS  = 0.5*anoxicmin   -rox*HS*(O2/(O2+ks)) + D*(BHS-HS)

  list(c(dOM, dO2, dSO4, dHS), SumS = SO4+HS)
})
}

# parameter values
pars <- c(D = 1, Flux = 100, r = 0.1, rox = 1,
          ks = 1, ks2 = 1, BO2 = 100, BSO4 = 10000, BHS = 0)
# initial conditions
y <- c(OM = 1, O2 = 1, SO4 = 1, HS = 1)

# direct iteration
print( system.time(
  ST <- stode(y = y, func = model, parms = pars, pos = TRUE)
))

print( system.time(
  ST2 <- runsteady(y = y, func = model, parms = pars, times = c(0, 1000))
))

print( system.time(
  ST3 <- runsteady(y = y, func = model, parms = pars, times = c(0, 1000), 
    jactype = "sparse")
))

rbind("Newton Raphson" = ST$y, "Runsteady" = ST2$y, "Run sparse" = ST3$y)