lsodar | R Documentation |
Solving initial value problems for stiff or non-stiff systems of first-order ordinary differential equations (ODEs) and including root-finding.
The R function lsodar
provides an interface to the FORTRAN ODE
solver of the same name, written by Alan C. Hindmarsh and Linda
R. Petzold.
The system of ODE's is written as an R function or be defined in
compiled code that has been dynamically loaded. - see description of
lsoda
for details.
lsodar
differs from lsode
in two respects.
It switches automatically between stiff and nonstiff methods (similar as lsoda).
It finds the root of at least one of a set of constraint functions g(i) of the independent and dependent variables.
Two uses of lsodar
are:
To stop the simulation when a certain condition is met
To trigger events, i.e. sudden changes in one of the state variables when a certain condition is met.
when a particular condition is met.
lsodar(y, times, func, parms, rtol = 1e-6, atol = 1e-6,
jacfunc = NULL, jactype = "fullint", rootfunc = NULL,
verbose = FALSE, nroot = 0, tcrit = NULL, hmin = 0,
hmax = NULL, hini = 0, ynames = TRUE, maxordn = 12,
maxords = 5, bandup = NULL, banddown = NULL, maxsteps = 5000,
dllname = NULL, initfunc = dllname, initpar = parms,
rpar = NULL, ipar = NULL, nout = 0, outnames = NULL, forcings=NULL,
initforc = NULL, fcontrol=NULL, events=NULL, lags = NULL, ...)
y |
the initial (state) values for the ODE system. If |
times |
times at which explicit estimates for |
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 The return value of If |
parms |
vector or list of parameters used in |
rtol |
relative error tolerance, either a scalar or an array as
long as |
atol |
absolute error tolerance, either a scalar or an array as
long as |
jacfunc |
if not In some circumstances, supplying
If the Jacobian is a full matrix, If the Jacobian is banded, |
jactype |
the structure of the Jacobian, one of
|
rootfunc |
if not |
verbose |
a logical value that, when |
nroot |
only used if ‘dllname’ is specified: the number of
constraint functions whose roots are desired during the integration;
if |
tcrit |
if not |
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 |
hmax |
an optional maximum value of the integration stepsize. If
not specified, |
hini |
initial step size to be attempted; if 0, the initial step size is determined by the solver. |
ynames |
logical, if |
maxordn |
the maximum order to be allowed in case the method is
non-stiff. Should be <= 12. Reduce |
maxords |
the maximum order to be allowed in case the method is stiff. Should be <= 5. Reduce maxord to save storage space. |
bandup |
number of non-zero bands above the diagonal, in case the Jacobian is banded. |
banddown |
number of non-zero bands below the diagonal, in case the Jacobian is banded. |
maxsteps |
maximal number of steps per output interval taken by the solver. |
dllname |
a string giving the name of the shared library
(without extension) that contains all the compiled function or
subroutine definitions refered to in |
initfunc |
if not |
initpar |
only when ‘dllname’ is specified and an
initialisation function |
rpar |
only when ‘dllname’ is specified: a vector with
double precision values passed to the dll-functions whose names are
specified by |
ipar |
only when ‘dllname’ is specified: a vector with
integer values passed to the dll-functions whose names are specified
by |
nout |
only used if |
outnames |
only used if ‘dllname’ is specified and
|
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( See forcings or package vignette |
initforc |
if not |
fcontrol |
A list of control parameters for the forcing functions.
See forcings or vignette |
events |
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. |
lags |
A list that specifies timelags, i.e. the number of steps that has to be kept. To be used for delay differential equations. See timelags, dede for more information. |
... |
additional arguments passed to |
The work is done by the FORTRAN subroutine lsodar
, whose
documentation should be consulted for details (it is included as
comments in the source file ‘src/opkdmain.f’). The
implementation is based on the November, 2003 version of lsodar, from
Netlib.
lsodar
switches automatically between stiff and nonstiff
methods (similar as lsoda
). This means that the user does not
have to determine whether the problem is stiff or not, and the solver
will automatically choose the appropriate method. It always starts
with the nonstiff method.
lsodar
can find the root of at least one of a set of constraint functions
rootfunc
of the independent and dependent variables. It then returns the
solution at the root if that occurs sooner than the specified stop
condition, and otherwise returns the solution according the specified
stop condition.
Caution: Because of numerical errors in the function
rootfun
due to roundoff and integration error, lsodar
may
return false roots, or return the same root at two or more
nearly equal values of time
.
The form of the Jacobian can be specified by jactype
which can take the following values:
a full Jacobian, calculated internally by lsodar, the default,
a full Jacobian, specified by user
function jacfunc
,
a banded Jacobian, specified by user
function jacfunc
; the size of the bands specified by
bandup
and banddown
,
banded Jacobian, calculated by lsodar;
the size of the bands specified by bandup
and
banddown
.
If jactype
= "fullusr" or "bandusr" then the user must supply a
subroutine jacfunc
.
The input parameters rtol
, and atol
determine the
error control performed by the solver. See lsoda
for details.
The output will have the attribute iroot, if a root was found iroot is a vector, its length equal to the number of constraint functions it will have a value of 1 for the constraint function whose root that has been found and 0 otherwise.
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") 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"
for details.
More information about models defined in compiled code is in the package vignette ("compiledCode"); information about linking forcing functions to compiled code is in forcings.
Examples in both C and FORTRAN are in the ‘dynload’ subdirectory
of the deSolve
package directory.
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 ‘lsodar’
returns with an unrecoverable error. If y
has a names
attribute, it will be used to label the columns of the output value.
If a root has been found, the output will have the attribute
iroot
, an integer indicating which root has been found.
Karline Soetaert <karline.soetaert@nioz.nl>
Alan C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (Eds.), North-Holland, Amsterdam, 1983, pp. 55-64.
Linda R. Petzold, Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations, Siam J. Sci. Stat. Comput. 4 (1983), pp. 136-148. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1137/0904010")}
Kathie L. Hiebert and Lawrence F. Shampine, Implicitly Defined Output Points for Solutions of ODEs, Sandia Report SAND80-0180, February 1980.
Netlib: https://netlib.org
roots
for more examples on roots and events
rk
, rkMethod
, rk4
and euler
for
Runge-Kutta integrators.
lsoda
, lsode
,
lsodes
, vode
,
daspk
for other 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 1:
## from lsodar source code
## =======================================================================
Fun <- function (t, y, parms) {
ydot <- vector(len = 3)
ydot[1] <- -.04*y[1] + 1.e4*y[2]*y[3]
ydot[3] <- 3.e7*y[2]*y[2]
ydot[2] <- -ydot[1] - ydot[3]
return(list(ydot, ytot = sum(y)))
}
rootFun <- function (t, y, parms) {
yroot <- vector(len = 2)
yroot[1] <- y[1] - 1.e-4
yroot[2] <- y[3] - 1.e-2
return(yroot)
}
y <- c(1, 0, 0)
times <- c(0, 0.4*10^(0:8))
out <- lsodar(y = y, times = times, fun = Fun, rootfun = rootFun,
rtol = 1e-4, atol = c(1e-6, 1e-10, 1e-6), parms = NULL)
print(paste("root is found for eqn", which(attributes(out)$iroot == 1)))
print(out[nrow(out),])
diagnostics(out)
## =======================================================================
## Example 2:
## using lsodar to estimate steady-state conditions
## =======================================================================
## Bacteria (Bac) are growing on a substrate (Sub)
model <- function(t, state, pars) {
with (as.list(c(state, pars)), {
## substrate uptake death respiration
dBact <- gmax*eff*Sub/(Sub+ks)*Bact - dB*Bact - rB*Bact
dSub <- -gmax *Sub/(Sub+ks)*Bact + dB*Bact + input
return(list(c(dBact,dSub)))
})
}
## root is the condition where sum of |rates of change|
## is very small
rootfun <- function (t, state, pars) {
dstate <- unlist(model(t, state, pars)) # rate of change vector
return(sum(abs(dstate)) - 1e-10)
}
pars <- list(Bini = 0.1, Sini = 100, gmax = 0.5, eff = 0.5,
ks = 0.5, rB = 0.01, dB = 0.01, input = 0.1)
tout <- c(0, 1e10)
state <- c(Bact = pars$Bini, Sub = pars$Sini)
out <- lsodar(state, tout, model, pars, rootfun = rootfun)
print(out)
## =======================================================================
## Example 3:
## using lsodar to trigger an event
## =======================================================================
## a state variable is decaying at a first-order rate.
## when it reaches the value 0.1, a random amount is added.
derivfun <- function (t,y,parms)
list (-0.05 * y)
rootfun <- function (t,y,parms)
return(y - 0.1)
eventfun <- function(t,y,parms)
return(y + runif(1))
yini <- 0.8
times <- 0:200
out <- lsodar(func=derivfun, y = yini, times=times,
rootfunc = rootfun, events = list(func=eventfun, root = TRUE))
plot(out, type = "l", lwd = 2, main = "lsodar with event")
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