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Solving initial value problems for systems of first-order ordinary differential equations (ODEs) using Euler's method or the classical Runge-Kutta 4th order integration.
euler(y, times, func, parms, verbose = FALSE, ynames = TRUE,
dllname = NULL, initfunc = dllname, initpar = parms,
rpar = NULL, ipar = NULL, nout = 0, outnames = NULL,
forcings = NULL, initforc = NULL, fcontrol = NULL, ...)rk4(y, times, func, parms, verbose = FALSE, ynames = TRUE,
dllname = NULL, initfunc = dllname, initpar = parms,
rpar = NULL, ipar = NULL, nout = 0, outnames = NULL,
forcings = NULL, initforc = NULL, fcontrol = NULL, ...)
euler.1D(y, times, func, parms, nspec = NULL, dimens = NULL,
names = NULL, verbose = FALSE, ynames = TRUE,
dllname = NULL, initfunc = dllname, initpar = parms,
rpar = NULL, ipar = NULL, nout = 0, outnames = NULL,
forcings = NULL, initforc = NULL, fcontrol = NULL, ...)
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 rk4 is called. See package vignette
"compiledCode" for more details.
vector or list of parameters used in func.
for 1D models only: the number of species (components)
in the model. If NULL, then dimens should be specified.
for 1D models only: the number of boxes in the
model. If NULL, then nspec should be specified.
for 1D models only: the names of the components; used for plotting.
a logical value that, when TRUE, triggers more
verbose output from the ODE solver.
if FALSE: names of state variables are not passed
to function func ; this may speed up the simulation especially
for large models.
a string giving the name of the shared library
(without extension) that contains all the compiled function or
subroutine definitions refered to in func.
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.
additional arguments passed to func allowing this
to be a generic function.
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.
rk4 and euler are special versions of the two fixed step
solvers with less overhead and less functionality (e.g. no interpolation
and no events) compared to the generic Runge-Kutta codes called by
ode resp. rk.
If you need different internal and external time steps or want to use events,
please use:
rk(y, times, func, parms, method = "rk4") or
rk(y, times, func, parms, method = "euler").
See help pages of rk and rkMethod
for details.
Function euler.1D essentially calls function euler but
contains additional code to support plotting of 1D models, see
ode.1D and plot.1D for details.
rkMethod for a list of available Runge-Kutta
parameter sets,
rk for the more general Runge-Code,
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,
dede for integrating models with delay
differential equations,
diagnostics to print diagnostic messages.
# NOT RUN {
## =======================================================================
## Example: Analytical and numerical solutions of logistic growth
## =======================================================================
## the derivative of the logistic
logist <- function(t, x, parms) {
with(as.list(parms), {
dx <- r * x[1] * (1 - x[1]/K)
list(dx)
})
}
time <- 0:100
N0 <- 0.1; r <- 0.5; K <- 100
parms <- c(r = r, K = K)
x <- c(N = N0)
## analytical solution
plot(time, K/(1 + (K/N0-1) * exp(-r*time)), ylim = c(0, 120),
type = "l", col = "red", lwd = 2)
## reasonable numerical solution with rk4
time <- seq(0, 100, 2)
out <- as.data.frame(rk4(x, time, logist, parms))
points(out$time, out$N, pch = 16, col = "blue", cex = 0.5)
## same time step with euler, systematic under-estimation
time <- seq(0, 100, 2)
out <- as.data.frame(euler(x, time, logist, parms))
points(out$time, out$N, pch = 1)
## unstable result
time <- seq(0, 100, 4)
out <- as.data.frame(euler(x, time, logist, parms))
points(out$time, out$N, pch = 8, cex = 0.5)
## method with automatic time step
out <- as.data.frame(lsoda(x, time, logist, parms))
points(out$time, out$N, pch = 1, col = "green")
legend("bottomright",
c("analytical","rk4, h=2", "euler, h=2",
"euler, h=4", "lsoda"),
lty = c(1, NA, NA, NA, NA), lwd = c(2, 1, 1, 1, 1),
pch = c(NA, 16, 1, 8, 1),
col = c("red", "blue", "black", "black", "green"))
# }
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