lsoda {deSolve} | R Documentation |
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", verbose=FALSE, 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, ...)
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 |
times at which explicit estimates for y are desired. The first value in times must be the initial time. |
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 lsoda() 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. |
jacfunc |
if not NULL , an R function that computes
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 jacobian (see Details
below for more about this option). In some circumstances, supplying
jacfunc can speed up
the computations, if the system is stiff. The R calling sequence for
jacfunc is identical to that of func .
If the jacobian is a full matrix, jacfunc should return a matrix dydot/dy, where the ith
row contains the derivative of dy_i/dt with respect to y_j,
or a vector containing the matrix elements by columns (the way R and Fortran store matrices).
If the jacobian is banded, jacfunc should return a matrix containing only the
nonzero bands of the jacobian, rotated row-wise. See first example of lsode. |
jactype |
the structure of the jacobian, one of "fullint", "fullusr", "bandusr" or "bandint" - either full or banded and estimated internally or by user |
verbose |
a logical value that, when TRUE, triggers more verbose output from the ODE solver. Will output the settings of vectors *istate* and *rstate* - see details |
tcrit |
if not NULL , then lsoda cannot integrate past tcrit . The Fortran routine lsoda 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 large models |
maxordn |
the maximum order to be allowed in case the method is non-stiff. Should be <=12. Reduce maxord to save storage space |
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 during one call to 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 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 |
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:
jacfunc
jacfunc
; the size of the bands specified by bandup
and banddown
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 that documentation (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:
rtol * abs(y) + 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.
Models may be defined in compiled C or Fortran code, as well as in R. See package vignette for details.
Examples in both C and Fortran are in the ‘dynload’ subdirectory of
the deSolve
package directory.
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:
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 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 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
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
ode
, lsode
, lsodes
,
lsodar
, vode
, daspk
, rk
.
######################################### ## Example 1: Lotka-volterra model ######################################### ## A simple resource limited Lotka-Volterra-Model ## Note: ## 1. parameter and state variable names made ## accessible via "with" statement ## 2. function sigimp accessible through lexical scoping ## (see also ode and rk examples) lvmodel <-function(t, x, parms) { with(as.list(c(parms,x)), { import <- sigimp(t) dS <- import - b*S*P + g*K #substrate dP <- c*S*P - d*K*P #producer dK <- e*P*K - f*K #consumer res<-c(dS, dP, dK) 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$import, rule=2) ## Start values for steady state y<-xstart <- c(S=1, P=1, K=1) ## Solving out2 <- as.data.frame(lsoda(xstart, times, lvmodel, parms)) mf <- par(mfrow=c(2,2)) plot (out2$time, out2$S, type="l", ylab="substrate") plot (out2$time, out2$P, type="l", ylab="producer") plot (out2$time, out2$K, type="l", ylab="consumer") plot (out2$P, out2$K, 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) { 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 ) } ## measure speed (here and below) system.time( out <- lsoda(c(1,0,0),times,lsexamp, parms, rtol=1e-4, atol= my.atol) ) 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) ) all.equal(out, outJ) # TRUE