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, ...)
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 .
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 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 Details
below 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 given by 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 output the settings of vectors *istate* and *rstate* - 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 |
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 during one call to 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. |
initfunc |
if not NULL, the name of the initialisation function (which initialises values of parameters), as provided in ‘dllname’. See package vignette. |
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. |
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 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.
Models may be defined in compiled C or Fortran code, as well as in an R-function. See package vignette 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:
rstate contains the following:
For more information, see the comments in the original code lsodes.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 `lsodes' 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 lsoda 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>
ode
, lsoda
, lsode
, lsodar
, vode
, daspk
, rk
.
# Various ways to solve the same model. ############################### ## The example from lsodes 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)