rk {deSolve} | R Documentation |
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 step 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). 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-06, atol = 1e-06, tcrit = NULL, verbose = FALSE, hmin = 0, hmax = NULL, hini = hmax, method = rkMethod("rk45dp7", ...), maxsteps = 5000, ...)
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 |
times at which explicit estimates for y are desired. The first value in times must be the initial time. |
func |
an R-function that computes the values of the
derivatives in the ODE system (the model definition) at time t.
The R-function func 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, and 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 . |
parms |
any parameters used in func |
rtol |
relative error tolerance, either a scalar or an array as
long as y . Only applicable to methods with variable time step, see details. |
atol |
absolute error tolerance, either a scalar or an array as
long as y . Only applicable to methods with variable time step, see details. |
tcrit |
if not NULL , then rk cannot integrate past
tcrit . The solver routines may overshoot their 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 . |
verbose |
a logical value that, when TRUE, triggers more verbose output from the ODE solver. |
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 |
method |
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", "rk2", "rk4"
or the variable step methods "rk23bs", "rk34f", "rk45f" or "rk45dp7". |
maxsteps |
maximal number of steps during one call to the solver |
... |
additional arguments passed to func allowing this to be a generic function |
The Runge-Kutta solvers are primarily provided for didactic reasons.
For most practical cases, solvers of the Livermore family
(lsoda
, lsode
, lsodes
,
lsodar
, vode
, daspk
)
are superior because of higher efficiency and faster implementation
(FORTRAN and C).
In addition to this, some of the Livermore solvers are also suitable for stiff
ODEs, differential algebraic equations (DAEs), or partial differential equations
(PDEs).
Function rk
is a generalized implementation that can be used to evaluate
different solvers of the Runge-Kutta family. 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 )
<= 1, where ewt is a vector of positive error
weights. The values of rtol
and atol
should all be
non-negative.
The form of ewt is:
rtol * abs(y) + atol
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.
An example is given below:
model<-function(t,Y,parameters)
{
with (as.list(parameters),{
dy1 = -k1*Y[1] + k2*Y[2]*Y[3]
dy3 = k3*Y[2]*Y[2]
dy2 = -dy1 - dy3
list(c(dy1,dy2,dy3))
})
}
A matrix 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 solver
returns with an unrecoverable error. If y
has a names
attribute, it will be used to label the columns of the output value.
The output will have the attributes istate
, and rstate
,
two vectors with several useful elements.
See details.
The output will have the attributes *istate*, and *rstate*,
two vectors with several useful elements, whose interpretation is
compatible with lsoda
:
el 1: |
0 for normal return, -2 means excess accuracy requested. |
el 12: |
The number of steps taken for the problem so far. |
el 13: |
The number of function evaluations for the problem so far. |
el 15: |
The order of the method. |
normal-bracket143bracket-normal
Thomas Petzoldt thomas.petzoldt@tu-dresden.de
Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (2007) Numerical Recipes in C. Cambridge University Press.
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.
rkMethod
, ode
,
lsoda
,
lsode
,
lsodes
,
lsodar
,
vode
,
daspk
######################################### ## Example: Lotka-volterra model ######################################### ## Note: ## parameters are a list, names accessible via "with" statement ## (see also ode and lsoda examples) lvmodel <- function(t, x, parms) { S <- x[1] # substrate P <- x[2] # producer K <- x[3] # consumer with(parms,{ import <- approx(signal$times, signal$import, t)$y dS <- import - b * S * P + g * K dP <- c * S * P - d * K * P dK <- e * P * K - f * K res<-c(dS, dP, dK) list(res) }) } ## vector of timesteps times <- seq(0, 100, length=101) ## external signal with rectangle impulse signal <- as.data.frame(list(times = times, import = rep(0,length(times)))) signal$import[signal$times >= 10 & signal$times <=11] <- 0.2 ## Parameters for steady state conditions parms <- list(b=0.0, c=0.1, d=0.1, e=0.1, f=0.1, g=0.0) ## Start values for steady state y<-xstart <- c(S=1, P=1, K=1) ## Euler method out1 <- as.data.frame(rk(xstart, times, lvmodel, parms, hini = 0.1, method="euler")) ## classical Runge-Kutta 4th order out2 <- as.data.frame(rk(xstart, times, lvmodel, parms, hini = 1, method="rk4")) ## Dormand-Prince method of order 5(4) out3 <- as.data.frame(rk(xstart, times, lvmodel, parms, hmax=1, method = "rk45dp7")) mf <- par(mfrow=c(2,2)) plot (out1$time, out1$S, type="l", ylab="Substrate") lines(out2$time, out2$S, col="red", lty="dotted", lwd=2) lines(out3$time, out3$S, col="green", lty="dotted") plot (out1$time, out1$P, type="l", ylab="Producer") lines(out2$time, out2$P, col="red", lty="dotted") lines(out3$time, out3$P, col="green", lty="dotted") plot (out1$time, out1$K, type="l", ylab="Consumer") lines(out2$time, out2$K, col="red", lty="dotted", lwd=2) lines(out3$time, out3$K, col="green", lty="dotted") plot (out1$P, out1$K, type="l", xlab="Producer",ylab="Consumer") lines(out2$P, out2$K, col="red", lty="dotted", lwd=2) lines(out3$P, out3$K, 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)