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Solving initial value problems for non-stiff systems of first-order ordinary differential equations (ODEs).
The R function rk is a top-level function that provides
interfaces to a collection of common explicit one-step solvers of the
Runge-Kutta family with fixed or variable time steps.
The system of ODE's is written as an R function (which may, of
course, use .C, .Fortran,
.Call, etc., to call foreign code) or be defined in
compiled code that has been dynamically loaded. A vector of
parameters is passed to the ODEs, so the solver may be used as part of
a modeling package for ODEs, or for parameter estimation using any
appropriate modeling tool for non-linear models in R such as
optim, nls, nlm or
nlme
rk(y, times, func, parms, rtol = 1e-6, atol = 1e-6,
verbose = FALSE, tcrit = NULL, hmin = 0, hmax = NULL,
hini = hmax, ynames = TRUE, method = rkMethod("rk45dp7", ... ),
maxsteps = 5000, dllname = NULL, initfunc = dllname,
initpar = parms, rpar = NULL, ipar = NULL,
nout = 0, outnames = NULL, forcings = NULL,
initforc = NULL, fcontrol = NULL, events = NULL, ...)A matrix of class deSolve with up to as many rows as elements
in times and as many columns as elements in y plus the
number of "global" values returned in the next elements of the return
from func, plus and additional column for the time value.
There will be a row for each element in times unless the
integration routine returns with an unrecoverable error. If y
has a names attribute, it will be used to label the columns of the output value.
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 at which explicit estimates for y are
desired. The first value in times must be the initial time.
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:
func <- function(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
must be specified in the same order as the state variables y.
If func is
a string, then dllname must give the name of the shared
library (without extension) which must be loaded before
rk is called. See package vignette "compiledCode"
for more details.
vector or list of parameters used in func.
relative error tolerance, either a scalar or an array as
long as y. Only applicable to methods with variable time
step, see details.
absolute error tolerance, either a scalar or an array as
long as y. Only applicable to methods with variable time
step, see details.
if not NULL, then rk cannot integrate past
tcrit. This parameter is for compatibility with other solvers.
a logical value that, when TRUE, triggers more verbose output from the ODE solver.
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!
an optional maximum value of the integration stepsize. If
not specified, hmax is set to the maximum of hini and
the largest difference in times, to avoid that the simulation
possibly ignores short-term events. If 0, no maximal size is
specified. Note that hmin and hmax are ignored by
fixed step methods like "rk4" or "euler".
initial step size to be attempted; if 0, the initial step
size is determined automatically by solvers with flexible time step.
For fixed step methods, setting hini = 0 forces
internal time steps identically to external time steps provided by
times. Similarly, internal time steps of non-interpolating
solvers cannot be bigger than external time steps specified in times.
if FALSE: names of state variables are not passed
to function func ; this may speed up the simulation especially
for large models.
the integrator to use. This can either be a string
constant naming one of the pre-defined methods or a call to function
rkMethod specifying a user-defined method. The most
common methods are the fixed-step methods "euler", second and
fourth-order Runge Kutta ("rk2", "rk4"), or the
variable step methods Bogacki-Shampine "rk23bs",
Runge-Kutta-Fehlberg "rk34f", the fifth-order Cash-Karp
method "rk45ck" or the fifth-order Dormand-Prince method with
seven stages "rk45dp7".
As a suggestion, one may use "rk23bs" (alias "ode23") for
simple problems and "rk45dp7" (alias "ode45") for
rough problems.
average maximal number of steps per output interval
taken by the solver. This argument is defined such as to ensure
compatibility with the Livermore-solvers. rk only accepts the maximal
number of steps for the entire integration. It is calculated
as max(length(times) * maxsteps, max(diff(times)/hini + 1).
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 "compiledCode".
if not NULL, the name of the initialisation function
(which initialises values of parameters), as provided in
dllname. See package vignette "compiledCode".
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++).
only when dllname is specified: a vector with
double precision values passed to the dll-functions whose names are
specified by func and jacfunc.
only when dllname is specified: a vector with
integer values passed to the dll-functions whose names are specified
by func and jacfunc.
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 calculated in the dll - you have
to perform this check in the code. See package vignette
"compiledCode".
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.
only used if dllname is specified: 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.
See forcings or package vignette "compiledCode".
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 forcings or package vignette "compiledCode".
A list of control parameters for the forcing functions.
See forcings or vignette compiledCode.
A matrix or data frame that specifies events, i.e. when the value of a
state variable is suddenly changed. See events for more information.
Not also that if events are specified, then polynomial interpolation
is switched off and integration takes place from one external time step
to the next, with an internal step size less than or equal the difference
of two adjacent points of times.
additional arguments passed to func allowing this
to be a generic function.
Thomas Petzoldt thomas.petzoldt@tu-dresden.de
Function rk is a generalized implementation that can be used to
evaluate different solvers of the Runge-Kutta family of explicit ODE
solvers. A pre-defined set of common method parameters is in function
rkMethod which also allows to supply user-defined
Butcher tables.
The input parameters rtol, and atol determine the error
control performed by the solver. The solver will control the vector
of estimated local errors in y, according to an inequality of
the form max-norm of ( e/ewt ) rtol and atol should all be non-negative. The form of
ewt is:
where multiplication of two vectors is element-by-element.
Models can be defined in R as a user-supplied
R-function, that must be called 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.
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 second element contains output variables that are
required at each point in time. Examples are given below.
Butcher, J. C. (1987) The numerical analysis of ordinary differential equations, Runge-Kutta and general linear methods, Wiley, Chichester and New York.
Engeln-Muellges, G. and Reutter, F. (1996) Numerik Algorithmen: Entscheidungshilfe zur Auswahl und Nutzung. VDI Verlag, Duesseldorf.
Hindmarsh, Alan C. (1983) ODEPACK, A Systematized Collection of ODE Solvers; in p.55--64 of Stepleman, R.W. et al.[ed.] (1983) Scientific Computing, North-Holland, Amsterdam.
Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (2007) Numerical Recipes in C. Cambridge University Press.
For most practical cases, solvers of the Livermore family (i.e. the ODEPACK solvers, see below) are superior. Some of them are also suitable for stiff ODEs, differential algebraic equations (DAEs), or partial differential equations (PDEs).
rkMethod for a list of available Runge-Kutta
parameter sets,
rk4 and euler for special
versions without interpolation (and less overhead),
lsoda, lsode,
lsodes, lsodar, vode,
daspk for solvers of the Livermore family,
ode for a general interface to most of the ODE solvers,
ode.band for solving models with a banded
Jacobian,
ode.1D for integrating 1-D models,
ode.2D for integrating 2-D models,
ode.3D for integrating 3-D models,
diagnostics to print diagnostic messages.
## =======================================================================
## Example: Resource-producer-consumer Lotka-Volterra model
## =======================================================================
## Notes:
## - Parameters are a list, names accessible via "with" function
## - Function sigimp passed as an argument (input) to model
## (see also ode and lsoda examples)
SPCmod <- function(t, x, parms, input) {
with(as.list(c(parms, x)), {
import <- input(t)
dS <- import - b*S*P + g*C # substrate
dP <- c*S*P - d*C*P # producer
dC <- e*P*C - f*C # consumer
res <- c(dS, dP, dC)
list(res)
})
}
## The parameters
parms <- c(b = 0.001, c = 0.1, d = 0.1, e = 0.1, f = 0.1, g = 0.0)
## vector of timesteps
times <- seq(0, 200, length = 101)
## external signal with rectangle impulse
signal <- data.frame(times = times,
import = rep(0, length(times)))
signal$import[signal$times >= 10 & signal$times <= 11] <- 0.2
sigimp <- approxfun(signal$times, signal$import, rule = 2)
## Start values for steady state
xstart <- c(S = 1, P = 1, C = 1)
## Euler method
out1 <- rk(xstart, times, SPCmod, parms, hini = 0.1,
input = sigimp, method = "euler")
## classical Runge-Kutta 4th order
out2 <- rk(xstart, times, SPCmod, parms, hini = 1,
input = sigimp, method = "rk4")
## Dormand-Prince method of order 5(4)
out3 <- rk(xstart, times, SPCmod, parms, hmax = 1,
input = sigimp, method = "rk45dp7")
mf <- par("mfrow")
## deSolve plot method for comparing scenarios
plot(out1, out2, out3, which = c("S", "P", "C"),
main = c ("Substrate", "Producer", "Consumer"),
col =c("black", "red", "green"),
lty = c("solid", "dotted", "dotted"), lwd = c(1, 2, 1))
## user-specified plot function
plot (out1[,"P"], out1[,"C"], type = "l", xlab = "Producer", ylab = "Consumer")
lines(out2[,"P"], out2[,"C"], col = "red", lty = "dotted", lwd = 2)
lines(out3[,"P"], out3[,"C"], col = "green", lty = "dotted")
legend("center", legend = c("euler", "rk4", "rk45dp7"),
lty = c(1, 3, 3), lwd = c(1, 2, 1),
col = c("black", "red", "green"))
par(mfrow = mf)
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