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 = 1e6, atol = 1e6,
jacvec = NULL, sparsetype = "sparseint", nnz = NULL,
inz = NULL, rootfunc = NULL,
verbose = FALSE, nroot = 0, 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, events=NULL, lags = NULL,
...)
y 
the initial (state) values for the ODE system. If 
times 
time sequence for which output is wanted; the first
value of 
func 
either an Rfunction that computes the values of the
derivatives in the ODE system (the model definition) at time
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 
jacvec 
if not The R
calling sequence for 
sparsetype 
the sparsity structure of the Jacobian, one of "sparseint" or "sparseusr", "sparsejan", ..., The sparsity can be estimated internally by lsodes (first option) or given by the user (last two). See details. 
nnz 
the number of nonzero elements in the sparse Jacobian (if this is unknown, use an estimate). 
inz 
if 
rootfunc 
if not 
verbose 
if 
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 
maxord 
the maximum order to be allowed. 
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 For instance, if you get the error: DLSODES RWORK length is insufficient to proceed. Length needed is .ge. LENRW (=I1), exceeds LRW (=I2) In above message, I1 = 27627 I2 = 25932 set 
liw 
the length of the integer work array iwork; due to the
sparsicity, this cannot be readily predicted. If 
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 dllfunctions whose names are
specified by 
ipar 
only when ‘dllname’ is specified: a vector with
integer values passed to the dllfunctions 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 twocolumned 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 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 several choices depending on whether jacvec
is specified and depending on the setting of sparsetype
.
If function jacvec
is present, then it should return the jth
column of the Jacobian matrix.
There are also several choices for the sparsity specification, selected by
argument sparsetype
.
sparsetype
= "sparseint"
. The sparsity is estimated
by the solver, based on numerical differences.
In this case, it is advisable to provide an estimate of the number
of nonzero elements in the Jacobian (nnz
).
This value can be approximate; upon return the number of nonzero
elements actually required will be known (1st element of attribute
dims
).
In this case, inz
need not be specified.
sparsetype
= "sparseusr"
. The sparsity is determined by
the user. In this case, inz
should be a matrix
, containing indices
(row, column) to the nonzero elements in the Jacobian matrix.
The number of nonzeros nnz
will be set equal to the number of rows
in inz
.
sparsetype
= "sparsejan"
. The sparsity is also determined by
the user.
In this case, inz
should be a vector
, containting the ian
and
jan
elements of the sparse storage format, as used in the sparse solver.
Elements of ian
should be the first n+1
elements of this vector, and
contain the starting locations in jan
of columns 1.. n.
jan
contains the row indices of the nonzero locations of
the Jacobian, reading in columnwise order.
The number of nonzeros nnz
will be set equal to the length of inz
 (n+1).
sparsetype
= "1D"
, "2D"
, "3D"
.
The sparsity is estimated by the solver, based on numerical differences.
Assumes finite differences in a 1D, 2D or 3D regular grid  used by
functions ode.1D
, ode.2D
, ode.3D
.
Similar are "2Dmap"
, and "3Dmap"
, which also include a
mapping variable (passed in nnz).
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 Rfunction. 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 ‘doc/examples/dynload’ subdirectory
of the deSolve
package directory.
lsodes
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, lsodes
may
return false roots, or return the same root at two or more
nearly equal values of time
.
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 ‘lsodes’
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 <karline.soetaert@nioz.nl>
Alan C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (Eds.), NorthHolland, Amsterdam, 1983, pp. 5564.
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. 11451151.
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
,
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 1D models,
ode.2D
for integrating 2D models,
ode.3D
for integrating 3D 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 < 1.0e4
atol < 1.0e6
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, 8,10, 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 = 353)
## =======================================================================
## 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
## =======================================================================
out4 < lsodes(func = chemistry, y = y, parms = parms, times = times,
lrw = 351, sparsetype = "sparseusr", inz = nonzero,
jacvec = chemjac, atol = atol, rtol = rtol,
verbose = TRUE)
# The sparsejan variant
# note: errors in inz may cause R to break, so this is not without danger...
# out5 < lsodes(func = chemistry, y = y, parms = parms, times = times,
# jacvec = chemjac, atol = atol, rtol = rtol, sparsetype = "sparsejan",
# inz = c(1,3,8,13,17,21,25,29,32,32,38,38,43, # ian
# 1,2, 2,3,4,5,12, 2,3,4,6,10, 2,3,4,9, 2,5,9,12, 3,6,9,10, # jan
# 7,9,10,12, 8,10,11, 3,6,7,8,10,12, 2,5,7,10,12), lrw = 343)