Solver for Ordinary Differential Equations (ODE), Switching Automatically Between Stiff and Non-stiff Methods
Solving initial value problems for stiff or non-stiff systems of first-order ordinary differential equations (ODEs).
The R function lsoda provides an interface to the FORTRAN ODE
solver of the same name, written by Linda R. Petzold and Alan
C. Hindmarsh.
The system of ODE's is written as an R function (which may, of
course, use .C, .Fortran,
.Call, etc., to call foreign code) or be defined in
compiled code that has been dynamically loaded. A vector of
parameters is passed to the ODEs, so the solver may be used as part of
a modeling package for ODEs, or for parameter estimation using any
appropriate modeling tool for non-linear models in R such as
optim, nls, nlm or
nlme
lsoda differs from the other integrators (except lsodar)
in that it switches automatically between stiff and nonstiff methods.
This means that the user does not have to determine whether the
problem is stiff or not, and the solver will automatically choose the
appropriate method. It always starts with the nonstiff method.
lsoda(y, times, func, parms, rtol = 1e-6, atol = 1e-6, jacfunc = NULL, jactype = "fullint", rootfunc = NULL, verbose = FALSE, nroot = 0, tcrit = NULL, hmin = 0, hmax = NULL, hini = 0, ynames = TRUE, maxordn = 12, maxords = 5, bandup = NULL, banddown = NULL, maxsteps = 5000, 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 |
times at which explicit estimates for |
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 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 |
jacfunc |
if not In some circumstances, supplying
If the Jacobian is a full matrix, If the Jacobian is banded, |
jactype |
the structure of the Jacobian, one of |
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 |
maxordn |
the maximum order to be allowed in case the method is
non-stiff. Should be <= 12. Reduce |
maxords |
the maximum order to be allowed in case the method is stiff. Should be <= 5. 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 |
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 |
All the hard work is done by the FORTRAN subroutine lsoda,
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 12 November 2003 version of lsoda, from Netlib.
lsoda switches automatically between stiff and nonstiff
methods. This means that the user does not have to determine whether
the problem is stiff or not, and the solver will automatically choose
the appropriate method. It always starts with the nonstiff method.
The form of the Jacobian can be specified by jactype which can
take the following values:
a full Jacobian, calculated internally by lsoda, the default,
a full Jacobian, specified by user function jacfunc,
a banded Jacobian, specified by user function jacfunc
the size of the bands specified by bandup and banddown,
banded Jacobian, calculated by lsoda; the size of the bands
specified by bandup and banddown.
If jactype = "fullusr" or "bandusr" then the user must supply a
subroutine jacfunc.
The following description of error control is adapted from the
documentation of the lsoda source code
(input arguments rtol and atol, above):
The input parameters rtol, and atol determine the error
control performed by the solver. The solver will control the vector
e of estimated local errors in y, according to an
inequality of the form max-norm of ( e/ewt )
<= 1, where ewt is a vector of positive error weights. The
values of rtol and atol should all be non-negative. The
form of ewt is:
\bold{rtol} * abs(\bold{y}) + \bold{atol}
where multiplication of two vectors is element-by-element.
If the request for precision exceeds the capabilities of the machine,
the FORTRAN subroutine lsoda will return an error code; under some
circumstances, the R function lsoda will attempt a reasonable
reduction of precision in order to get an answer. It will write a
warning if it does so.
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.
The ‘demo’ directory contains some examples of using
gnls to estimate parameters in a
dynamic model.
R. Woodrow Setzer <setzer.woodrow@epa.gov>
Hindmarsh, Alan C. (1983) ODEPACK, A Systematized Collection of ODE Solvers; in p.55–64 of Stepleman, R.W. et al.[ed.] (1983) Scientific Computing, North-Holland, Amsterdam.
Petzold, Linda R. (1983) Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations. Siam J. Sci. Stat. Comput. 4, 136–148.
Netlib: http://www.netlib.org
lsode, which can also find a root
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:
## A simple resource limited Lotka-Volterra-Model
##
## Note:
## 1. parameter and state variable names made
## accessible via "with" function
## 2. function sigimp accessible through lexical scoping
## (see also ode and rk examples)
## =======================================================================
SPCmod <- function(t, x, parms) {
with(as.list(c(parms, x)), {
import <- sigimp(t)
dS <- import - b*S*P + g*C #substrate
dP <- c*S*P - d*C*P #producer
dC <- e*P*C - f*C #consumer
res <- c(dS, dP, dC)
list(res)
})
}
## Parameters
parms <- c(b = 0.0, c = 0.1, d = 0.1, e = 0.1, f = 0.1, g = 0.0)
## vector of timesteps
times <- seq(0, 100, length = 101)
## external signal with rectangle impulse
signal <- as.data.frame(list(times = times,
import = rep(0,length(times))))
signal$import[signal$times >= 10 & signal$times <= 11] <- 0.2
sigimp <- approxfun(signal$times, signal$import, rule = 2)
## Start values for steady state
y <- xstart <- c(S = 1, P = 1, C = 1)
## Solving
out <- lsoda(xstart, times, SPCmod, parms)
## Plotting
mf <- par("mfrow")
plot(out, main = c("substrate", "producer", "consumer"))
plot(out[,"P"], out[,"C"], type = "l", xlab = "producer", ylab = "consumer")
par(mfrow = mf)
## =======================================================================
## Example 2:
## from lsoda source code
## =======================================================================
## names makes this easier to read, but may slow down execution.
parms <- c(k1 = 0.04, k2 = 1e4, k3 = 3e7)
my.atol <- c(1e-6, 1e-10, 1e-6)
times <- c(0,4 * 10^(-1:10))
lsexamp <- function(t, y, p) {
yd1 <- -p["k1"] * y[1] + p["k2"] * y[2]*y[3]
yd3 <- p["k3"] * y[2]^2
list(c(yd1, -yd1-yd3, yd3), c(massbalance = sum(y)))
}
exampjac <- function(t, y, p) {
matrix(c(-p["k1"], p["k1"], 0,
p["k2"]*y[3],
- p["k2"]*y[3] - 2*p["k3"]*y[2],
2*p["k3"]*y[2],
p["k2"]*y[2], -p["k2"]*y[2], 0
), 3, 3)
}
## measure speed (here and below)
system.time(
out <- lsoda(c(1, 0, 0), times, lsexamp, parms, rtol = 1e-4,
atol = my.atol, hmax = Inf)
)
out
## This is what the authors of lsoda got for the example:
## the output of this program (on a cdc-7600 in single precision)
## is as follows..
##
## at t = 4.0000e-01 y = 9.851712e-01 3.386380e-05 1.479493e-02
## at t = 4.0000e+00 y = 9.055333e-01 2.240655e-05 9.444430e-02
## at t = 4.0000e+01 y = 7.158403e-01 9.186334e-06 2.841505e-01
## at t = 4.0000e+02 y = 4.505250e-01 3.222964e-06 5.494717e-01
## at t = 4.0000e+03 y = 1.831975e-01 8.941774e-07 8.168016e-01
## at t = 4.0000e+04 y = 3.898730e-02 1.621940e-07 9.610125e-01
## at t = 4.0000e+05 y = 4.936363e-03 1.984221e-08 9.950636e-01
## at t = 4.0000e+06 y = 5.161831e-04 2.065786e-09 9.994838e-01
## at t = 4.0000e+07 y = 5.179817e-05 2.072032e-10 9.999482e-01
## at t = 4.0000e+08 y = 5.283401e-06 2.113371e-11 9.999947e-01
## at t = 4.0000e+09 y = 4.659031e-07 1.863613e-12 9.999995e-01
## at t = 4.0000e+10 y = 1.404280e-08 5.617126e-14 1.000000e+00
## Using the analytic Jacobian speeds up execution a little :
system.time(
outJ <- lsoda(c(1, 0, 0), times, lsexamp, parms, rtol = 1e-4,
atol = my.atol, jacfunc = exampjac, jactype = "fullusr", hmax = Inf)
)
all.equal(as.data.frame(out), as.data.frame(outJ)) # TRUE
diagnostics(out)
diagnostics(outJ) # shows what lsoda did internallyPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.