\name{lsoda}
\alias{lsoda}
\title{
  Solver for Ordinary Differential Equations (ODE),
  Switching Automatically Between Stiff and Non-stiff Methods
}
\description{
  
  Solving initial value problems for stiff or non-stiff systems of
  first-order ordinary differential equations (ODEs).
  
  The \R function \code{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 \code{\link{.C}}, \code{\link{.Fortran}},
  \code{\link{.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
  \code{\link{optim}}, \code{\link{nls}}, \code{\link{nlm}} or
  \code{\link[nlme]{nlme}}
    
  \code{lsoda} differs from the other integrators (except \code{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.
}
\usage{
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,...)
}
\arguments{
  \item{y }{the initial (state) values for the ODE system. If \code{y}
    has a name attribute, the names will be used to label the output
    matrix.
  }
  \item{times }{times at which explicit estimates for \code{y} are
    desired.  The first value in \code{times} must be the initial time.
  }
  \item{func }{either an \R-function that computes the values of the
    derivatives in the ODE system (the \emph{model definition}) at time
    t, or a character string giving the name of a compiled function in a
    dynamically loaded shared library.

    If \code{func} is an \R-function, it must be defined as:
     \code{func <- function(t, y, parms,...)}.  \code{t} is the current
    time point in the integration, \code{y} is the current estimate of
    the variables in the ODE system.  If the initial values \code{y} has
    a \code{names} attribute, the names will be available inside \code{func}.
    \code{parms} is a vector or list of parameters; ... (optional) are
    any other arguments passed to the function.

    The return value of \code{func} should be a list, whose first
    element is a vector containing the derivatives of \code{y} with
    respect to \code{time}, and whose next elements are global values
    that are required at each point in \code{times}.  The derivatives
    must be specified in the \bold{same order} as the state variables \code{y}.

    If \code{func} is a string, then \code{dllname} must give the name
    of the shared library (without extension) which must be loaded
    before \code{lsoda()} is called. See package vignette
    \code{"compiledCode"} for more
    details.
  }
  \item{parms }{vector or list of parameters used in \code{func} or
    \code{jacfunc}.
  }
  \item{rtol }{relative error tolerance, either a scalar or an array as
    long as \code{y}. See details.
  }
  \item{atol }{absolute error tolerance, either a scalar or an array as
    long as \code{y}. See details.
  }
  \item{jacfunc }{if not \code{NULL}, an \R function, that computes the
    Jacobian of the system of differential equations 
    \eqn{\partial\dot{y}_i/\partial y_j}{dydot(i)/dy(j)}, or
    a string giving the name of a function or subroutine in
    \file{dllname} that computes the Jacobian (see vignette
    \code{"compiledCode"} for more about this option).
    
    In some circumstances, supplying
    \code{jacfunc} can speed up the computations, if the system is
    stiff.  The \R calling sequence for \code{jacfunc} is identical to
    that of \code{func}.

    If the Jacobian is a full matrix, \code{jacfunc} should return a
    matrix \eqn{\partial\dot{y}/\partial y}{dydot/dy}, where the ith row contains the derivative of
    \eqn{dy_i/dt} with respect to \eqn{y_j}, or a vector containing the
    matrix elements by columns (the way \R and FORTRAN store matrices).
    
    If the Jacobian is banded, \code{jacfunc} should return a matrix
    containing only the nonzero bands of the Jacobian, rotated
    row-wise. See first example of \link{lsode}.
  }
  \item{jactype }{the structure of the Jacobian, one of \code{"fullint"},
    \code{"fullusr"}, \code{"bandusr"} or \code{"bandint"} - either
    full or banded and estimated internally or by user.
  }
  \item{rootfunc }{if not \code{NULL}, an \R function that computes the
    function whose root has to be estimated or a string giving the name
    of a function or subroutine in \file{dllname} that computes the root
    function.  The \R calling sequence for \code{rootfunc} is identical
    to that of \code{func}.  \code{rootfunc} should return a vector with
    the function values whose root is sought.  When \code{rootfunc} is
    provided, then \code{lsodar} will be called.
  }
  \item{verbose }{if \code{TRUE}: full output to the screen, e.g. will
    print the \code{diagnostiscs} of the integration - see details.
  }
  \item{nroot }{only used if \file{dllname}  is specified: the number of
    constraint functions whose roots are desired during the integration;
    if \code{rootfunc} is an R-function, the solver estimates the number
    of roots.
  }
  \item{tcrit }{if not \code{NULL}, then \code{lsoda} cannot integrate
    past \code{tcrit}. The FORTRAN routine \code{lsoda} overshoots its
    targets (times points in the vector \code{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 \code{tcrit}.
  }
  \item{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 \code{hmin} if you don't know why!
  }
  \item{hmax }{an optional maximum value of the integration stepsize. If
    not specified, \code{hmax} is set to the largest difference in
    \code{times}, to avoid that the simulation possibly ignores
    short-term events. If 0, no maximal size is specified.
  }
  \item{hini }{initial step size to be attempted; if 0, the initial step
    size is determined by the solver.
  }
  \item{ynames }{logical, if \code{FALSE}: names of state variables are not
    passed to function \code{func}; this may speed up the simulation especially
    for large models.
  }
  \item{maxordn }{the maximum order to be allowed in case the method is
    non-stiff. Should be <= 12. Reduce \code{maxord} to save storage space.
  }
  \item{maxords }{the maximum order to be allowed in case the method is
    stiff. Should be <= 5. Reduce maxord to save storage space.
  }
  \item{bandup }{number of non-zero bands above the diagonal, in case
    the Jacobian is banded.
  }
  \item{banddown }{number of non-zero bands below the diagonal, in case
    the Jacobian is banded.
  }
  \item{maxsteps }{maximal number of steps per output interval taken by the
    solver.
  }
  \item{dllname }{a string giving the name of the shared library
    (without extension) that contains all the compiled function or
    subroutine definitions refered to in \code{func} and
    \code{jacfunc}. See package vignette \code{"compiledCode"}.
  }
  \item{initfunc }{if not \code{NULL}, the name of the initialisation function
    (which initialises values of parameters), as provided in
    \file{dllname}. See package vignette \code{"compiledCode"}.
  }
  \item{initpar }{only when \file{dllname} is specified and an
    initialisation function \code{initfunc} is in the dll: the
    parameters passed to the initialiser, to initialise the common
    blocks (FORTRAN) or global variables (C, C++).
  }
  \item{rpar }{only when \file{dllname} is specified: a vector with
    double precision values passed to the dll-functions whose names are
    specified by \code{func} and \code{jacfunc}.
  }
  \item{ipar }{only when \file{dllname} is specified: a vector with
    integer values passed to the dll-functions whose names are specified
    by \code{func} and \code{jacfunc}.
  }
  \item{nout }{only used if \code{dllname} is specified and the model is
    defined in compiled code: the number of output variables calculated
    in the compiled function \code{func}, present in the shared
    library. Note: it is not automatically checked whether this is
    indeed the number of output variables calculated in the dll - you have
    to perform this check in the code. See package vignette
    \code{"compiledCode"}.
  }
  \item{outnames }{only used if \file{dllname} is specified and
    \code{nout} > 0: the names of output variables calculated in the
    compiled function \code{func}, present in the shared library.
    These names will be used to label the output matrix.
  }
  \item{forcings }{only used if \file{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(\code{times}), max(\code{times})] is done by taking the value at
    the closest data extreme.

    See \link{forcings} or package vignette \code{"compiledCode"}.
  }
  \item{initforc }{if not \code{NULL}, the name of the forcing function
    initialisation function, as provided in
    \file{dllname}. It MUST be present if \code{forcings} has been given a
    value.
    See \link{forcings} or package vignette \code{"compiledCode"}.
  }
  \item{fcontrol }{A list of control parameters for the forcing functions.
    See \link{forcings} or vignette \code{compiledCode}.
  }
  \item{events }{A matrix or data frame that specifies events, i.e. when the value of a 
   state variable is suddenly changed. See \link{events} for more information.
  }
  \item{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 \link{timelags}, \link{dede} for more information.
  }
  \item{... }{additional arguments passed to \code{func} and
    \code{jacfunc} allowing this to be a generic function.
  }
}
\value{
  A matrix of class \code{deSolve} with up to as many rows as elements
  in \code{times} and as many columns as elements in \code{y} plus the number of "global"
  values returned in the next elements of the return from \code{func},
  plus and additional column for the time value.  There will be a row
  for each element in \code{times} unless the FORTRAN routine `lsoda'
  returns with an unrecoverable error. If \code{y} has a names
  attribute, it will be used to label the columns of the output value.
    
}
\author{R. Woodrow Setzer <setzer.woodrow@epa.gov>}
\examples{
## =======================================================================
## 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 internally

}
\references{

  Hindmarsh, Alan C. (1983) ODEPACK, A Systematized Collection of ODE
  Solvers; in p.55--64 of Stepleman, R.W. et al.[ed.] (1983)
  \emph{Scientific Computing}, North-Holland, Amsterdam.

  Petzold, Linda R. (1983) Automatic Selection of Methods for Solving
  Stiff and Nonstiff Systems of Ordinary Differential Equations.
  \emph{Siam J. Sci. Stat. Comput.} \bold{4}, 136--148.
  \doi{10.1137/0904010}

  Netlib: \url{https://netlib.org}
}
\details{
  All the hard work is done by the FORTRAN subroutine \code{lsoda},
  whose documentation should be consulted for details (it is included as
  comments in the source file \file{src/opkdmain.f}). The implementation
  is based on the 12 November 2003 version of lsoda, from Netlib.

  \code{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 \bold{Jacobian} can be specified by \code{jactype} which can
  take the following values:
  \describe{
    \item{"fullint"}{a full Jacobian, calculated internally by lsoda, the default,}
    \item{"fullusr"}{a full Jacobian, specified by user function \code{jacfunc},}
    \item{"bandusr"}{a banded Jacobian, specified by user function \code{jacfunc}
      the size of the bands specified by \code{bandup} and \code{banddown},}
    \item{"bandint"}{banded Jacobian, calculated by lsoda; the size of the bands
      specified by \code{bandup} and \code{banddown}.}
  }
  If \code{jactype} = "fullusr" or "bandusr" then the user must supply a
  subroutine \code{jacfunc}.
    
  The following description of \bold{error control} is adapted from the
  documentation of the lsoda source code
  (input arguments \code{rtol} and \code{atol}, above):
    
  The input parameters \code{rtol}, and \code{atol} determine the error
  control performed by the solver.  The solver will control the vector
  \bold{e} of estimated local errors in \bold{y}, according to an
  inequality of the form max-norm of ( \bold{e}/\bold{ewt} ) \eqn{\leq}{
  <= } 1, where \bold{ewt} is a vector of positive error weights.  The
  values of \code{rtol} and \code{atol} should all be non-negative.  The
  form of \bold{ewt} is:
  
  \deqn{\mathbf{rtol} \times \mathrm{abs}(\mathbf{y}) +
  \mathbf{atol}}{\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 \code{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 \code{\link{diagnostics}}. If \code{verbose} = \code{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.

  \bold{Models} may be defined in compiled C or FORTRAN code, as well as
  in an R-function. See package vignette \code{"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 \link{forcings}.

  Examples in both C and FORTRAN are in the \file{dynload} subdirectory
  of the \code{deSolve} package directory.
  

    }
\seealso{
  \itemize{
    \item \code{\link{rk}}, \code{\link{rkMethod}}, \code{\link{rk4}} and \code{\link{euler}} for
      Runge-Kutta integrators.
    \item  \code{\link{lsode}}, which can also find a root
    \item  \code{\link{lsodes}}, \code{\link{lsodar}}, \code{\link{vode}},
      \code{\link{daspk}} for other solvers of the Livermore family,
    \item \code{\link{ode}} for a general interface to most of the ODE solvers,
    \item \code{\link{ode.band}} for solving models with a banded
      Jacobian,
    \item \code{\link{ode.1D}} for integrating 1-D models,
    \item \code{\link{ode.2D}} for integrating 2-D models,
    \item \code{\link{ode.3D}} for integrating 3-D models,
  }

  \code{\link{diagnostics}} to print diagnostic messages.
}
\note{
  The \file{demo} directory contains some examples of using
  \code{\link[nlme]{gnls}} to estimate parameters in a
  dynamic model.
}
\keyword{math}

