dopri5: Dormand-Prince or CashCarp Runge-Kutta of Order (4)5

In deTestSet: Test Set for Differential Equations

Dormand-Prince or CashCarp Runge-Kutta of Order (4)5

Description

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.

Usage

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, ...)

Arguments

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	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.
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: 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.
parms	vector or list of parameters used in func or jacfunc.
rtol	relative error tolerance, either a scalar or an array as long as y. See details.
atol	absolute error tolerance, either a scalar or an array as long as y. See details.
verbose	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.
hmax	an optional maximum value of the integration stepsize. If not specified, hmax is set to the largest difference in times.
hini	initial step size to be attempted.
ynames	logical, if FALSE names of state variables are not passed to function func; this may speed up the simulation especially for multi-D models.
maxsteps	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!
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 vignette "compiledCode" from package deSolve.
initfunc	if not NULL, the name of the initialisation function (which initialises values of parameters), as provided in ‘dllname’. See vignette "compiledCode" from package deSolve.
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 vignette "compiledCode" from package 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. These names will be used to label the output matrix.
forcings	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".
initforc	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".
fcontrol	A list of control parameters for the forcing functions. See forcings or vignette compiledCode.
stiffness	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.

Details

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).

Value

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.

Author(s)

Karline Soetaert <[e-mail]>

References

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)

See Also

• 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.

Examples

## =======================================================================
## 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)

deTestSet documentation built on July 9, 2023, 6:10 p.m.