Solves the initial value problem for systems of ordinary differential equations (ODE) in the form: $$dy/dt = f(t,y)$$
The R function dopri5 provides an interface to the Fortran ODE
solver DOPRI5, written by E. Hairer and G. Wanner.
It implements the explicit Runge-Kutta method of order 4(5) due to Dormand & Prince with stepsize control and dense output
The R function cashkarp provides an interface to the Fortran ODE
solver CASHCARP, written by J. Cash and F. Mazzia.
It implements the explicit Runge-Kutta method of order 4(5) due to Cash-Carp, with stepsize control and dense output
The system of ODE's is written as an R function or can be defined in compiled code that has been dynamically loaded.
dopri5 (y, times, func, parms, rtol = 1e-6, atol = 1e-6,
verbose = FALSE, hmax = NULL, hini = hmax, ynames = TRUE,
maxsteps = 10000, dllname = NULL, initfunc = dllname,
initpar=parms, rpar = NULL, ipar = NULL, nout = 0,
outnames = NULL, forcings = NULL, initforc = NULL, fcontrol = NULL, ...)cashkarp (y, times, func, parms, rtol = 1e-6, atol = 1e-6,
verbose = FALSE, hmax = NULL, hini = hmax, ynames = TRUE,
maxsteps = 10000, dllname = NULL, initfunc = dllname, initpar = parms,
rpar = NULL, ipar = NULL, nout = 0, outnames = NULL, forcings = NULL,
initforc = NULL, fcontrol = NULL, stiffness = 2, ...)
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 FORTRAN routine `lsoda'
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.
time sequence for which output is wanted; the first
value of times must be the initial time; if only one step is
to be taken; set times = NULL.
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
should 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
lsode() is called. See package vignette "compiledCode"
for more details.
vector or list of parameters used in func or
jacfunc.
relative error tolerance, either a
scalar or an array as long as y. See details.
absolute error tolerance, either a scalar or an array as
long as y. See details.
if TRUE: full output to the screen, e.g. will
print the diagnostiscs of the integration - if the method becomes
stiff it will rpint a message.
an optional maximum value of the integration stepsize. If
not specified, hmax is set to the largest difference in
times.
initial step size to be attempted.
logical, if FALSE names of state variables are not
passed to function func; this may speed up the simulation especially
for multi-D models.
maximal number of steps taken by the solver, for the entire integration. This is different from the settings of this argument in the solvers from package deSolve!
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 vignette "compiledCode"
from package deSolve.
if not NULL, the name of the initialisation function
(which initialises values of parameters), as provided in
dllname. See vignette "compiledCode"
from package deSolve.
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 calculed in the dll - you have
to perform this check in the code - See vignette "compiledCode"
from package deSolve.
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.
These names will be used to label the output matrix.
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.
How the stiffness of the solution should be estimated.
Default = stiffness based on eigenvalue approximation;
when = stiffness = 0: no stiffness estimate; when = stiffness = 1
or -1: all stiffness estimates calculated ; when = stiffness = 2 or -2: stiffness based on
eigenvalue approximation; when = stiffness = 3 or -3: stiffness
based on error estimate; when = stiffness = 4 or -4: stiffness
based on conditioning. Positive values of stiffness will cause the
integration to stop; negative values will continue anyway.
additional arguments passed to func and
jacfunc allowing this to be a generic function.
Karline Soetaert <karline.soetaert@nioz.nl>
The work is done by the FORTRAN subroutine dop853, whose
documentation should be consulted for details. The implementation
is based on the Fortran 77 version fromOctober 11, 2009.
The input parameters rtol, and atol determine the
error control performed by the solver, which roughly keeps the
local error of y(i) below rtol(i)*abs(y(i))+atol(i).
The diagnostics of the integration can be printed to screen
by calling diagnostics. If verbose = TRUE,
the diagnostics will written to the screen at the end of the integration.
See vignette("deSolve") from the deSolve package for an
explanation of each element in the vectors
containing the diagnostic properties and how to directly access them.
Models may be defined in compiled C or FORTRAN code, as well as
in an R-function. See package vignette "compiledCode" from package
deSolve for details.
Information about linking forcing functions to compiled code is in
forcings (from package deSolve).
E. Hairer, S.P. Norsett AND G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series In Computational Mathematics, SPRINGER-VERLAG (1993)
ode for a general interface to most of the ODE solvers
from package deSolve,
ode.1D for integrating 1-D models,
ode.2D for integrating 2-D models,
ode.3D for integrating 3-D models,
mebdfi for integrating DAE models,
bimd for blended implicit methods,
gamd for the generalised adams method
diagnostics to print diagnostic messages.
## =======================================================================
## Example :
## The Arenstorff orbit model
## =======================================================================
Arenstorff <- function(t, y, parms) {
D1 <- ((y[1]+mu)^2+y[2]^2)^(3/2)
D2 <- ((y[1]-(1-mu))^2+y[2]^2)^(3/2)
dy1 <- y[3]
dy2 <- y[4]
dy3 <- y[1] + 2*y[4]-(1-mu)*(y[1]+mu)/D1 -mu*(y[1]-(1-mu))/D2
dy4 <- y[2] - 2*y[3]-(1-mu)*y[2]/D1 - mu*y[2]/D2
list(c(dy1,dy2,dy3,dy4))
}
#-----------------------------
# parameters, initial values and times
#-----------------------------
mu <- 0.012277471
yini <- c(x = 0.994, y = 0, dx = 0,
dy = -2.00158510637908252240537862224)
times <- seq(0, 18, 0.01)
#-----------------------------
# solve the model
#-----------------------------
#out <- dopri5 (times=times, y=yini, func = Arenstorff, parms=NULL )
out <- cashkarp (times = times, y = yini, func = Arenstorff, parms = NULL )
plot(out[,c("x", "y")], type = "l", lwd = 2, main = "Arenstorff")
#-----------------------------
# First and last value should be the same
#-----------------------------
times <- c(0, 17.0652165601579625588917206249)
Test <- dopri5 (times = times, y = yini, func = Arenstorff, parms = NULL)
diagnostics(Test)
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