lsodar: General Solver for Ordinary Differential Equations (ODE), Switching Automatically Between Stiff and Non-stiff Methods and With Root Finding

Description

Solving initial value problems for stiff or non-stiff systems of first-order ordinary differential equations (ODEs) and including root-finding.

The Rfunction lsodar provides an interface to the Fortran ODE solver of the same name, written by Alan C. Hindmarsh and Linda R. Petzold. The system of ODE's is written as an Rfunction or be defined in compiled code that has been dynamically loaded. - see description of lsoda for details.

lsodar differs from lsode in two respects.

It switches automatically between stiff and nonstiff methods (similar as lsoda).
It finds the root of at least one of a set of constraint functions g(i) of the independent and dependent variables.

Usage

lsodar(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, ...)

Arguments

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 fu

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 Rfunction, 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

jactype

the structure of the jacobian, one of "fullint", "fullusr", "bandusr" or "bandint" - either full or banded and estimated internally or by user.

rootfunc

if not NULL, an Rfunction that computes the function whose root has to be estimated or a string giving the name of a function or subroutine in dllname that computes the root function. The Rcalling sequence for

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.

nroot

only used if dllname is specified: the number of constraint functions whose roots are desired during the integration; if rootfunc is an R-function, the solver estimates the number of roots.

tcrit

if not NULL, then lsodar cannot integrate past tcrit. The Fortran routine lsodar overshoots its targets (times points in the vector times), and interpolates values for the desire

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 specif

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.

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

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 checke

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.

Value

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 `lsodar' returns with an unrecoverable error or has found a root, in which case the last row will contain the function value at the root. 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. If a root has been found, the output will also have the attribute iroot, an integer indicating which root has been found.

Details

The work is done by the Fortran subroutine lsodar, 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 lsodar, from Netlib. lsodar switches automatically between stiff and nonstiff methods (similar as lsoda). 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. It finds the root of at least one of a set of constraint functions g(i) 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. The form of the jacobian can be specified by jactype which can take the following values: [object Object],[object Object],[object Object],[object Object] If jactype = "fullusr" or "bandusr" then the user must supply a subroutine jacfunc.

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. Examples in Fortran are in the dynload subdirectory of the deSolve package directory. The output will have the attributes istate and rstate, and if a root was found iroot, three 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:

el 1 : returns the conditions under which the last call to lsodar returned. 2 if lsodar was successful, 3 if lsodar was succesful and one or more roots were found - seeiroot. -1 if excess work done, -2 means excess accuracy requested. (Tolerances too small), -3 means illegal input detected. (See printed message.), -4 means repeated error test failures. (Check all input), -5 means repeated convergence failures. (Perhaps bad Jacobian supplied or wrong choice of MF or tolerances.), -6 means error weight became zero during problem. (Solution component i vanished, and atol or atol(i) = 0.)
el 12 : The number of steps taken for the problem so far.
el 13 : The number of function evaluations for the problem so far.,
el 14 : The number of Jacobian evaluations and LU decompositions so far.,
el 15 : The method order last used (successfully).,
el 16 : The order to be attempted on the next step.,
el 17 : if el 1 = -4,-5: the largest component in the error vector,
el 18 : The length of rwork actually required.,
el 19 : The length of IUSER actually required.,
el 20 : The method indicator for the last succesful step, 1 = adams (nonstiff), 2 = bdf (stiff),
el 21 : The current method indicator to be attempted on th next step, 1 = adams (nonstiff), 2 = bdf (stiff),

rstate contains the following:

1: The step size in t last used (successfully).
2: The step size to be attempted on the next step.
3: A tolerance scale factor, greater than 1.0, computed when a request for too much accuracy was detected.
4: the value of t at the time of the last method switch, if any.

iroot is a vector, its length equal to the number of constraint functions it will have a value of 1 for the constraint function whose root that has been found and 0 otherwise.

References

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. Linda R. Petzold, Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations, Siam J. Sci. Stat. Comput. 4 (1983), pp. 136-148.

Kathie L. Hiebert and Lawrence F. Shampine, Implicitly Defined Output Points for Solutions of ODEs, Sandia Report SAND80-0180, February 1980. Netlib: http://www.netlib.org

Examples

Run this code

## ===================================================
## Example 1:
##   from lsodar source code
## ===================================================

Fun <- function (t, y, parms)
{
  ydot <- vector(len = 3)
  ydot[1] <- -.04*y[1] + 1.e4*y[2]*y[3]
  ydot[3] <- 3.e7*y[2]*y[2]
  ydot[2] <- -ydot[1]-ydot[3]
  return(list(ydot,ytot = sum(y)))
}

rootFun <- function (t, y, parms)
{
  yroot <- vector(len = 2)
  yroot[1] <- y[1] - 1.e-4
  yroot[2] <- y[3] - 1.e-2
  return(yroot)
}

y     <- c(1, 0, 0)
times <- c(0, 0.4*10^(0:8))
Out   <- NULL
ny    <- length(y)

out   <- lsodar(y = y, times = times, fun = Fun, rootfun = rootFun,
                rtol = 1e-4, atol = c(1e-6, 1e-10, 1e-6), parms = NULL)
print(paste("root is found for eqn", which(attributes(out)$iroot == 1)))
print(out[nrow(out),])
  
## ===================================================
## Example 2:
##   using lsodar to estimate steady-state conditions
## ===================================================

## Bacteria (Bac) are growing on a substrate (Sub)
model <- function(t, state, pars)
{
  with (as.list(c(state, pars)), {
    ##       substrate uptake             death  respiration
    dBact =  gmax*eff*Sub/(Sub+ks)*Bact - dB*Bact - rB*Bact
    dSub  = -gmax    *Sub/(Sub+ks)*Bact + dB*Bact            + input

    return(list(c(dBact,dSub)))
  })
}

## root is the condition where sum of |rates of change|
## is very small

rootfun <- function (t, state, pars)
{
  dstate <- unlist(model(t, state, pars)) # rate of change vector
  return(sum(abs(dstate)) - 1e-10)
}

pars <- list(Bini = 0.1, Sini = 100, gmax = 0.5, eff = 0.5,
             ks = 0.5, rB = 0.01, dB = 0.01, input = 0.1)

tout    <- c(0, 1e10)
state   <- c(Bact = pars$Bini, Sub = pars$Sini)
out     <- lsodar(state, tout, model, pars, rootfun = rootfun)
print(out)

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