lsodar {deSolve} | 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.
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, ...)
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 |
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 lsodar() 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.
|
jacfunc |
if not NULL , an R function, that computes the
Jacobian of the system of differential equations dydot(i)/dy(j), or
a string giving the name of a function or subroutine in
‘dllname’ that computes the Jacobian (see vignette
"compiledCode" for more about this option).
In some circumstances, supplying jacfunc can speed up the computations, if the system is
stiff. The R calling sequence for jacfunc is identical to
that of func .
If the Jacobian is a full matrix, jacfunc should return a
matrix dydot/dy, where the ith row contains the derivative of
dy_i/dt with respect to y_j, or a vector containing the
matrix elements by columns (the way R and FORTRAN store matrices).
If the Jacobian is banded, jacfunc should return a matrix
containing only the nonzero bands of the Jacobian, rotated
row-wise. See first example of lsode .
|
jactype |
the structure of the Jacobian, one of
"fullint" , "fullusr" , "bandusr" or
"bandint" - either full or banded and estimated internally or
by user.
|
rootfunc |
if not NULL , an R function that computes the
function whose root has to be estimated or a string giving the name
of a function or subroutine in ‘dllname’ that computes the root
function. The R calling sequence for rootfunc is identical
to that of func . rootfunc should return a vector with
the function values whose root is sought.
|
verbose |
a logical value that, when TRUE , will
print the diagnostiscs of the integration - see details.
|
nroot |
only used if ‘dllname’ is specified: the number of
constraint functions whose roots are desired during the integration;
if rootfunc is an R-function, the solver estimates the number
of roots.
|
tcrit |
if not NULL , then lsodar cannot integrate
past tcrit . The FORTRAN routine lsodar overshoots its
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 .
|
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. |
ynames |
logical, if FALSE : names of state variables are not
passed to function func ; this may speed up the simulation
especially for large models.
|
maxordn |
the maximum order to be allowed in case the method is
non-stiff. Should be <= 12. Reduce maxord to save storage space.
|
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 func and
jacfunc . See package vignette "compiledCode" .
|
initfunc |
if not NULL , the name of the initialisation function
(which initialises values of parameters), as provided in
‘dllname’. See package vignette "compiledCode" .
|
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 package vignette
"compiledCode" .
|
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 .
|
... |
additional arguments passed to func and
jacfunc allowing this to be a generic function.
|
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.
It finds the root of at least one of a set of constraint functions g(i) 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.
The form of the Jacobian can be specified by jactype
which can take the following values:
jacfunc
,
jacfunc
; the size of the bands specified by
bandup
and banddown
,
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 `lsoda'
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 <k.soetaert@nioo.knaw.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.
Kathie L. Hiebert and Lawrence F. Shampine, Implicitly Defined Output Points for Solutions of ODEs, Sandia Report SAND80-0180, February 1980.
Netlib: http://www.netlib.org
rk
, 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 <- NULL ny <- length(y) 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)