lsodes {deSolve} | R Documentation |
Solves the initial value problem for stiff systems of ordinary differential equations (ODE) in the form:
dy/dt = f(t,y)
and where the Jacobian matrix df/dy has an arbitrary sparse structure.
The R function lsodes
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.
lsodes(y, times, func, parms, rtol = 1e-6, atol = 1e-6, jacvec = NULL, sparsetype = "sparseint", nnz = NULL, inz = NULL, verbose = FALSE, tcrit = NULL, hmin = 0, hmax = NULL, hini = 0, ynames = TRUE, maxord = NULL, maxsteps = 5000, lrw = NULL, liw = NULL, 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 |
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
lsodes() 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.
|
jacvec |
if not NULL , an R function that computes a
column of 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 column of the
Jacobian (see vignette "compiledCode" for more about this option).
The R calling sequence for jacvec is identical to that of
func , but with extra parameter j , denoting the column
number. Thus, jacvec should be called as: jacvec =
func(t, y, j, parms) and jacvec should return a vector
containing column j of the Jacobian, i.e. its i-th value is
dydot(i)/dy(j). If this function is absent, lsodes will
generate the Jacobian by differences.
|
sparsetype |
the sparsity structure of the Jacobian, one of "sparseint" or "sparseusr", sparsity estimated internally by lsodes or givenby user. |
nnz |
the number of nonzero elements in the sparse Jacobian (if this is unknown, use an estimate). |
inz |
(row,column) indices to the nonzero elements in the sparse
Jacobian. Necessary if sparsetype = "sparseusr"; else
ignored.
|
verbose |
if TRUE : full output to the screen, e.g. will
print the diagnostiscs of the integration - see details.
|
tcrit |
if not NULL , then lsodes cannot integrate
past tcrit . The FORTRAN routine lsodes 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 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.
|
maxsteps |
maximal number of steps per output interval taken by the solver. |
lrw |
the length of the real work array rwork; due to the
sparsicity, this cannot be readily predicted. If NULL , a
guess will be made, and if not sufficient, lsodes will return
with a message indicating the size of rwork actually required.
Therefore, some experimentation may be necessary to estimate the
value of lrw .
|
liw |
the length of the integer work array iwork; due to the
sparsicity, this cannot be readily predicted. If NULL , a guess will
be made, and if not sufficient, lsodes will return with a
message indicating the size of iwork actually required. Therefore,
some experimentation may be necessary to estimate the value of
liw .
|
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 lsodes
, 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 lsodes, from Netlib.
lsodes
is applied for stiff problems, where the Jacobian has a
sparse structure.
There are four choices depending on whether jacvec
and
inz
is specified.
If function jacvec
is present, then it should return the j-th
column of the Jacobian matrix.
If matrix inz
is present, then it should contain indices (row,
column) to the nonzero elements in the Jacobian matrix.
If jacvec
and inz
are present, then the Jacobian is
fully specified by the user.
If jacvec
is present, but not nnz
then the structure of
the Jacobian will be obtained from NEQ + 1 calls to jacvec
.
If nnz
is present, but not jacvec
then the Jacobian will
be estimated internally, by differences.
If neither nnz
nor jacvec
is present, then the Jacobian
will be generated internally by differences, its structure (indices to
nonzero elements) will be obtained from NEQ + 1 initial calls to
func
.
If nnz
is not specified, it is advisable to provide an estimate
of the number of non-zero elements in the Jacobian (inz
).
The input parameters rtol
, and atol
determine the
error control performed by the solver. See lsoda
for details.
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.
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.
S. C. Eisenstat, M. C. Gursky, M. H. Schultz, and A. H. Sherman, Yale Sparse Matrix Package: I. The Symmetric Codes, Int. J. Num. Meth. Eng., 18 (1982), pp. 1145-1151.
S. C. Eisenstat, M. C. Gursky, M. H. Schultz, and A. H. Sherman, Yale Sparse Matrix Package: II. The Nonsymmetric Codes, Research Report No. 114, Dept. of Computer Sciences, Yale University, 1977.
rk
, rk4
and euler
for
Runge-Kutta integrators.
lsoda
, lsode
,
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.
## Various ways to solve the same model. ## ======================================================================= ## The example from lsodes source code ## A chemical model ## ======================================================================= n <- 12 y <- rep(1, n) dy <- rep(0, n) times <- c(0, 0.1*(10^(0:4))) rtol = 1.0e-4 atol = 1.0e-6 parms <- c(rk1 = 0.1, rk2 = 10.0, rk3 = 50.0, rk4 = 2.5, rk5 = 0.1, rk6 = 10.0, rk7 = 50.0, rk8 = 2.5, rk9 = 50.0, rk10 = 5.0, rk11 = 50.0, rk12 = 50.0,rk13 = 50.0, rk14 = 30.0, rk15 = 100.0,rk16 = 2.5, rk17 = 100.0,rk18 = 2.5, rk19 = 50.0, rk20 = 50.0) # chemistry <- function (time,Y,pars) { with (as.list(pars), { dy[1] <- -rk1 *Y[1] dy[2] <- rk1 *Y[1] + rk11*rk14*Y[4] + rk19*rk14*Y[5] - rk3 *Y[2]*Y[3] - rk15*Y[2]*Y[12] - rk2*Y[2] dy[3] <- rk2 *Y[2] - rk5 *Y[3] - rk3*Y[2]*Y[3] - rk7*Y[10]*Y[3] + rk11*rk14*Y[4] + rk12*rk14*Y[6] dy[4] <- rk3 *Y[2]*Y[3] - rk11*rk14*Y[4] - rk4*Y[4] dy[5] <- rk15*Y[2]*Y[12] - rk19*rk14*Y[5] - rk16*Y[5] dy[6] <- rk7 *Y[10]*Y[3] - rk12*rk14*Y[6] - rk8*Y[6] dy[7] <- rk17*Y[10]*Y[12] - rk20*rk14*Y[7] - rk18*Y[7] dy[8] <- rk9 *Y[10] - rk13*rk14*Y[8] - rk10*Y[8] dy[9] <- rk4 *Y[4] + rk16*Y[5] + rk8*Y[6] + rk18*Y[7] dy[10] <- rk5 *Y[3] + rk12*rk14*Y[6] + rk20*rk14*Y[7] + rk13*rk14*Y[8] - rk7 *Y[10]*Y[3] - rk17*Y[10]*Y[12] - rk6 *Y[10] - rk9*Y[10] dy[11] <- rk10*Y[8] dy[12] <- rk6 *Y[10] + rk19*rk14*Y[5] + rk20*rk14*Y[7] - rk15*Y[2]*Y[12] - rk17*Y[10]*Y[12] return(list(dy)) }) } ## ======================================================================= ## application 1. lsodes estimates the structure of the Jacobian ## and calculates the Jacobian by differences ## ======================================================================= out <- lsodes(func = chemistry, y = y, parms = parms, times = times, atol = atol, rtol = rtol, verbose = TRUE) ## ======================================================================= ## application 2. the structure of the Jacobian is input ## lsodes calculates the Jacobian by differences ## this is not so efficient... ## ======================================================================= ## elements of Jacobian that are not zero nonzero <- matrix(nc = 2, byrow = TRUE, data = c( 1, 1, 2, 1, # influence of sp1 on rate of change of others 2, 2, 3, 2, 4, 2, 5, 2, 12, 2, 2, 3, 3, 3, 4, 3, 6, 3, 10, 3, 2, 4, 3, 4, 4, 4, 9, 4, # d (dyi)/dy4 2, 5, 5, 5, 9, 5, 12, 5, 3, 6, 6, 6, 9, 6, 10, 6, 7, 7, 9, 7, 10, 7, 12, 7, 8, 8, 10, 8, 11, 8, 3,10, 6,10, 7,10, 10,10, 12,10, 2,12, 5,12, 7,12, 10,12, 12,12) ) ## when run, the default length of rwork is too small ## lsodes will tell the length actually needed # out2 <- lsodes(func = chemistry, y = y, parms = parms, times = times, # inz = nonzero, atol = atol,rtol = rtol) #gives warning out2 <- lsodes(func = chemistry, y = y, parms = parms, times = times, sparsetype = "sparseusr", inz = nonzero, atol = atol, rtol = rtol, verbose = TRUE, lrw = 351) ## ======================================================================= ## application 3. lsodes estimates the structure of the Jacobian ## the Jacobian (vector) function is input ## ======================================================================= chemjac <- function (time, Y, j, pars) { with (as.list(pars), { PDJ <- rep(0,n) if (j == 1){ PDJ[1] <- -rk1 PDJ[2] <- rk1 } else if (j == 2) { PDJ[2] <- -rk3*Y[3] - rk15*Y[12] - rk2 PDJ[3] <- rk2 - rk3*Y[3] PDJ[4] <- rk3*Y[3] PDJ[5] <- rk15*Y[12] PDJ[12] <- -rk15*Y[12] } else if (j == 3) { PDJ[2] <- -rk3*Y[2] PDJ[3] <- -rk5 - rk3*Y[2] - rk7*Y[10] PDJ[4] <- rk3*Y[2] PDJ[6] <- rk7*Y[10] PDJ[10] <- rk5 - rk7*Y[10] } else if (j == 4) { PDJ[2] <- rk11*rk14 PDJ[3] <- rk11*rk14 PDJ[4] <- -rk11*rk14 - rk4 PDJ[9] <- rk4 } else if (j == 5) { PDJ[2] <- rk19*rk14 PDJ[5] <- -rk19*rk14 - rk16 PDJ[9] <- rk16 PDJ[12] <- rk19*rk14 } else if (j == 6) { PDJ[3] <- rk12*rk14 PDJ[6] <- -rk12*rk14 - rk8 PDJ[9] <- rk8 PDJ[10] <- rk12*rk14 } else if (j == 7) { PDJ[7] <- -rk20*rk14 - rk18 PDJ[9] <- rk18 PDJ[10] <- rk20*rk14 PDJ[12] <- rk20*rk14 } else if (j == 8) { PDJ[8] <- -rk13*rk14 - rk10 PDJ[10] <- rk13*rk14 PDJ[11] <- rk10 } else if (j == 10) { PDJ[3] <- -rk7*Y[3] PDJ[6] <- rk7*Y[3] PDJ[7] <- rk17*Y[12] PDJ[8] <- rk9 PDJ[10] <- -rk7*Y[3] - rk17*Y[12] - rk6 - rk9 PDJ[12] <- rk6 - rk17*Y[12] } else if (j == 12) { PDJ[2] <- -rk15*Y[2] PDJ[5] <- rk15*Y[2] PDJ[7] <- rk17*Y[10] PDJ[10] <- -rk17*Y[10] PDJ[12] <- -rk15*Y[2] - rk17*Y[10] } return(PDJ) }) } out3 <- lsodes(func = chemistry, y = y, parms = parms, times = times, jacvec = chemjac, atol = atol, rtol = rtol) ## ======================================================================= ## application 4. The structure of the Jacobian (nonzero elements) AND ## the Jacobian (vector) function is input ## not very efficient... ## ======================================================================= out4 <- lsodes(func = chemistry, y = y, parms = parms, times = times, lrw = 351, sparsetype = "sparseusr", inz = nonzero, jacvec = chemjac, atol = atol, rtol = rtol, verbose = TRUE)