Solver for Ordinary Differential Equations (ODE) With Sparse Jacobian
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, 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 R-function 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 dll-functions whose names are
specified by |
ipar |
only when ‘dllname’ is specified: a vector with
integer values passed to the dll-functions 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 two-columned 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 j-th
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 non-zero 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 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 ‘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.), 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.
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, 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.