runsteady {rootSolve} | R Documentation |
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).
runsteady(y, times=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, ...)
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.
|
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.
|
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" or "bandint" - either full or
banded and estimated internally or by user; overruled if mf
is not NULL.
|
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). 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.
|
... |
additional arguments passed to func and jacfunc
allowing this to be a generic function.
|
The work is done by the Fortran subroutine dlsode
,
whose documentation should be consulted for details (it is included as
comments in the source file ‘src/lsode.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
lsode
, corresponds to mf
=22.
jacfunc
, corresponds to mf
=21.
jacfunc
; the size of the bands specified by bandup
and
banddown
, corresponds to mf
=24.
lsode
;
the size of the bands specified by bandup
and banddown
,
corresponds to mf
=25.
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
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:
rstate contains the following:
For more information, see the comments in the original code lsode.f
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.
Karline Soetaert <k.soetaert@nioo.knaw.nl>
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.
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.
## ======================================================================= ## 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)))) rbind("Newton Raphson"=ST$y,"Runsteady"=ST2$y)