lsode {deSolve} | R Documentation |
Solves the initial value problem for stiff or nonstiff systems of ordinary differential equations (ODE) in the form:
dy/dt = f(t,y)
.
The R function lsode
provides an interface to the Fortran ODE
solver of the same name, written by Alan C. Hindmarsh and Andrew
H. Sherman.
The system of ODE's is written as an R function or be defined in compiled code that has been dynamically loaded.
In contrast to lsoda
, the user has to specify whether or
not the problem is stiff and choose the appropriate solution method.
lsode
is very similar to vode
, but uses a
fixed-step-interpolate method rather than the variable-coefficient
method in vode
. In addition, in vode
it is
possible to choose whether or not a copy of the Jacobian is saved for
reuse in the corrector iteration algorithm; In lsode
, a copy is
not kept.
lsode(y, times, func, parms, rtol = 1e-6, atol = 1e-6, jacfunc = NULL, jactype = "fullint", mf = NULL, verbose = FALSE, tcrit = NULL, hmin = 0, hmax = NULL, hini = 0, ynames = TRUE, maxord = NULL, bandup = NULL, banddown = NULL, maxsteps = 5000, dllname = NULL, initfunc = dllname, initpar = parms, rpar = NULL, ipar = NULL, nout = 0, outnames = 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 |
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:
yprime = func(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.
|
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 Details below 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; overruled if mf is not NULL .
|
mf |
the "method flag" passed to function lsode - overrules
jactype - provides more options than jactype - see
details.
|
verbose |
if TRUE: full output to the screen, e.g. will output the settings of vectors istate and rstate - see details. |
tcrit |
if not NULL , then lsode cannot integrate
past tcrit . The Fortran routine lsode 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 |
if FALSE: names of state variables are not passed to
function func ; this may speed up the simulation especially
for multi-D models.
|
maxord |
the maximum order to be allowed. NULL uses the default,
i.e. order 12 if implicit Adams method (meth = 1), order 5 if BDF
method (meth = 2). 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.
|
... |
additional arguments passed to func and
jacfunc allowing this to be a generic function.
|
The work is done by the Fortran subroutine lsode
, 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 lsode, from Netlib.
Before using the integrator lsode
, the user has to decide
whether or not the problem is stiff.
If the problem is nonstiff, use method flag mf
= 10, which
selects a nonstiff (Adams) method, no Jacobian used.
If the problem
is stiff, there are four standard choices which can be specified with
jactype
or mf
.
The options for jactype are
mf
= 22,
jacfunc
, corresponds to mf
= 21,
jacfunc
; the size of the bands specified by
bandup
and banddown
, corresponds to mf
= 24,
bandup
and
banddown
, corresponds to mf
= 25.
More options are available when specifying mf directly.
The
legal values of mf
are 10, 11, 12, 13, 14, 15, 20, 21, 22, 23,
24, 25.
mf
is a positive two-digit integer, mf
=
(10*METH + MITER), where
func
per df/dy value). MITER = 3 means chord
iteration with an internally generated diagonal Jacobian
approximation (using 1 extra call to func
per df/dy
evaluation). MITER = 4 means chord iteration with a user-supplied
banded Jacobian. MITER = 5 means chord iteration with an
internally generated banded Jacobian (using ML+MU+1 extra calls to
func
per df/dy evaluation).
If MITER = 1 or 4, the user must supply a subroutine jacfunc
.
Inspection of the example below shows how to specify both a banded and full jacobian.
The input parameters rtol
, and atol
determine the
error control performed by the solver. See lsoda
for details.
Models may be defined in compiled C or Fortran code, as well as
in an R-function. See package vignette "compiledCode"
for details.
The output will have the attributes istate, and rstate, two vectors with several useful elements.
If verbose
= TRUE, the settings of istate and rstate will be
written to the screen.
The following elements of istate are meaningful:
For more information, see the comments in the original code lsode.f
A matrix 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 an
additional column (the first) for the time value. There will be one
row for each element in times
unless the Fortran routine
`lsode' returns with an unrecoverable error. If y
has a names
attribute, it will be used to label the columns of the output value.
The output will have the attributes istate
, and rstate
,
two vectors with several useful elements. See details. The first
element of istate returns the conditions under which the last call to
lsode returned. Normal is istate[1] = 2
. If verbose
=
TRUE, the settings of istate and rstate will be written to the screen
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.
rk
, rk4
and euler
for
Runge-Kutta integrators.
lsoda
,
lsodes
, lsodar
, 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: ## Various ways to solve the same model. ## ======================================== ## the model, 5 state variables f1 <- function (t, y, parms) { ydot <- vector(len = 5) ydot[1] <- 0.1*y[1] -0.2*y[2] ydot[2] <- -0.3*y[1] +0.1*y[2] -0.2*y[3] ydot[3] <- -0.3*y[2] +0.1*y[3] -0.2*y[4] ydot[4] <- -0.3*y[3] +0.1*y[4] -0.2*y[5] ydot[5] <- -0.3*y[4] +0.1*y[5] return(list(ydot)) } ## the jacobian, written as a full matrix fulljac <- function (t, y, parms) { jac <- matrix(nrow = 5, ncol = 5, byrow = TRUE, data = c(0.1, -0.2, 0 , 0 , 0 , -0.3, 0.1, -0.2, 0 , 0 , 0 , -0.3, 0.1, -0.2, 0 , 0 , 0 , -0.3, 0.1, -0.2, 0 , 0 , 0 , -0.3, 0.1) ) return(jac) } ## the jacobian, written in banded form bandjac <- function (t, y, parms) { jac <- matrix(nrow = 3, ncol = 5, byrow = TRUE, data = c( 0 , -0.2, -0.2, -0.2, -0.2, 0.1, 0.1, 0.1, 0.1, 0.1, -0.3, -0.3, -0.3, -0.3, 0) ) return(jac) } ## initial conditions and output times yini <- 1:5 times <- 1:20 ## default: stiff method, internally generated, full jacobian out <- lsode(yini, times, f1, parms = 0, jactype = "fullint") ## stiff method, user-generated full jacobian out2 <- lsode(yini, times, f1, parms = 0, jactype = "fullusr", jacfunc = fulljac) ## stiff method, internally-generated banded jacobian ## one nonzero band above (up) and below(down) the diagonal out3 <- lsode(yini, times, f1, parms = 0, jactype = "bandint", bandup = 1, banddown = 1) ## stiff method, user-generated banded jacobian out4 <- lsode(yini, times, f1, parms = 0, jactype = "bandusr", jacfunc = bandjac, bandup = 1, banddown = 1) ## non-stiff method out5 <- lsode(yini, times, f1, parms = 0, mf = 10) ## ===================================== ## Example 2: ## diffusion on a 2-D grid ## partially specified jacobian ## ===================================== diffusion2D <- function(t, Y, par) { y <- matrix(nr = n, nc = n, data = Y) dY <- r*y # production ## diffusion in X-direction; boundaries = 0-concentration Flux <- -Dx * rbind(y[1,],(y[2:n,]-y[1:(n-1),]),-y[n,])/dx dY <- dY - (Flux[2:(n+1),]-Flux[1:n,])/dx ## diffusion in Y-direction Flux <- -Dy * cbind(y[,1],(y[,2:n]-y[,1:(n-1)]),-y[,n])/dy dY <- dY - (Flux[,2:(n+1)]-Flux[,1:n])/dy return(list(as.vector(dY))) } ## parameters dy <- dx <- 1 # grid size Dy <- Dx <- 1 # diffusion coeff, X- and Y-direction r <- 0.025 # production rate times <- c(0, 1) n <- 50 y <- matrix(nr = n, nc = n, 0.) pa <- par(ask = FALSE) ## initial condition for (i in 1:n) { for (j in 1:n) { dst <- (i-n/2)^2+(j-n/2)^2 y[i,j] <- max(0.,1.-1./(n*n)*(dst-n)^2) } } filled.contour(y, color.palette = terrain.colors) ## jacfunc need not be estimated exactly ## a crude approximation, with a smaller bandwidth will do. ## Here the half-bandwidth 1 is used, whereas the true ## half-bandwidths are equal to n. ## This corresponds to ignoring the y-direction coupling in the ODEs. print(system.time( for (i in 1:20) { out <- lsode(func = diffusion2D, y = as.vector(y), times = times, parms = NULL, jactype = "bandint", bandup = 1, banddown = 1) filled.contour(matrix(nr = n, nc = n, out[2,-1]), zlim = c(0,1), color.palette = terrain.colors, main = i) y <- out[2,-1] } )) par(ask = pa)