lsode {deSolve}R Documentation

General Solver for Ordinary Differential Equations (ODE)

Description

Solves the initial value problem for stiff or nonstiff systems of ordinary differential equations (ODE) in the form:

dy/dt = f(t,y)

.

The R function lsode 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.

In contrast to lsoda, the user has to specify whether or not the problem is stiff and choose the appropriate solution method.

lsode is very similar to vode, but uses a fixed-step-interpolate method rather than the variable-coefficient method in vode. In addition, in vode it is possible to choose whether or not a copy of the Jacobian is saved for reuse in the corrector iteration algorithm; In lsode, a copy is not kept.

Usage

lsode(y, times, func, parms, rtol = 1e-6, atol = 1e-6,  
  jacfunc = NULL, jactype = "fullint", mf = NULL,
  verbose = FALSE, tcrit = NULL, hmin = 0, hmax = NULL, hini = 0,
  ynames = TRUE, maxord = NULL, bandup = NULL, banddown = NULL,
  maxsteps = 5000, dllname = NULL, initfunc = dllname,
  initpar = parms, rpar = NULL, ipar = NULL, nout = 0,
  outnames = NULL, ...)

Arguments

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 time sequence for which output is wanted; the first value of times must be the initial time; if only one step is to be taken; set times = NULL.
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. The derivatives should be specified in the same order as the state variables y.
If func is a string, then dllname must give the name of the shared library (without extension) which must be loaded before lsode() is called. See package vignette "compiledCode" 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; overruled if mfis not NULL.
mf the "method flag" passed to function lsode - overrules jactype - provides more options than jactype - see details.
verbose if TRUE: full output to the screen, e.g. will output the settings of vectors istate and rstate - see details.
tcrit if not NULL, then lsode cannot integrate past tcrit. The Fortran routine lsode 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 multi-D models.
maxord the maximum order to be allowed. NULL uses the default, i.e. order 12 if implicit Adams method (meth = 1), order 5 if BDF method (meth = 2). 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 func and jacfunc. See package vignette "compiledCode".
initfunc if not NULL, the name of the initialisation function (which initialises values of parameters), as provided in ‘dllname’. See package vignette "compiledCode".
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 "compiledCode".
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.

Details

The work is done by the Fortran subroutine lsode, 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 lsode, from Netlib.

Before using the integrator lsode, the user has to decide whether or not the problem is stiff.

If the problem is nonstiff, use method flag mf = 10, which selects a nonstiff (Adams) method, no Jacobian used.
If the problem is stiff, there are four standard choices which can be specified with jactype or mf.

The options for jactype are

jactype = "fullint"
a full jacobian, calculated internally by lsode, corresponds to mf = 22,
jactype = "fullusr"
a full jacobian, specified by user function jacfunc, corresponds to mf = 21,
jactype = "bandusr"
a banded jacobian, specified by user function jacfunc; the size of the bands specified by bandup and banddown, corresponds to mf = 24,
jactype = "bandint"
a banded jacobian, calculated by lsode; the size of the bands specified by bandup and banddown, corresponds to mf = 25.

More options are available when specifying mf directly.
The legal values of mf are 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25.
mf is a positive two-digit integer, mf = (10*METH + MITER), where

METH
indicates the basic linear multistep method: METH = 1 means the implicit Adams method. METH = 2 means the method based on backward differentiation formulas (BDF-s).
MITER
indicates the corrector iteration method: MITER = 0 means functional iteration (no Jacobian matrix is involved). MITER = 1 means chord iteration with a user-supplied full (NEQ by NEQ) Jacobian. MITER = 2 means chord iteration with an internally generated (difference quotient) full Jacobian (using NEQ extra calls to func per df/dy value). MITER = 3 means chord iteration with an internally generated diagonal Jacobian approximation (using 1 extra call to func per df/dy evaluation). MITER = 4 means chord iteration with a user-supplied banded Jacobian. MITER = 5 means chord iteration with an internally generated banded Jacobian (using ML+MU+1 extra calls to func per df/dy evaluation).

If MITER = 1 or 4, the user must supply a subroutine jacfunc.

Inspection of the example below shows how to specify both a banded and full jacobian.

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 "compiledCode" for details.

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:

For more information, see the comments in the original code lsode.f

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 an additional column (the first) for the time value. There will be one row for each element in times unless the Fortran routine `lsode' 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 lsode returned. Normal is istate[1] = 2. If verbose = TRUE, the settings of istate and rstate will be written to the screen

Author(s)

Karline Soetaert <k.soetaert@nioo.knaw.nl>

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.

See Also

diagnostics to print diagnostic messages.

Examples

## ========================================
## Example 1:
##   Various ways to solve the same model.
## ========================================

## the model, 5 state variables
f1 <- function  (t, y, parms)
{
  ydot <- vector(len = 5)

  ydot[1] <-  0.1*y[1] -0.2*y[2]
  ydot[2] <- -0.3*y[1] +0.1*y[2] -0.2*y[3]
  ydot[3] <-           -0.3*y[2] +0.1*y[3] -0.2*y[4]
  ydot[4] <-                     -0.3*y[3] +0.1*y[4] -0.2*y[5]
  ydot[5] <-                               -0.3*y[4] +0.1*y[5]

  return(list(ydot))
}

## the jacobian, written as a full matrix
fulljac <- function  (t, y, parms)
{
   jac <- matrix(nrow = 5, ncol = 5, byrow = TRUE,
                data = c(0.1, -0.2,  0  ,  0  ,  0  ,
                        -0.3,  0.1, -0.2,  0  ,  0  ,
                         0  , -0.3,  0.1, -0.2,  0  ,
                         0  ,  0  , -0.3,  0.1, -0.2,
                         0  ,  0  ,  0  , -0.3,  0.1)    )
   return(jac)
}

## the jacobian, written in banded form
bandjac <- function  (t, y, parms)
{
   jac <- matrix(nrow = 3, ncol = 5, byrow = TRUE,
                 data = c( 0  , -0.2, -0.2, -0.2, -0.2,
                           0.1,  0.1,  0.1,  0.1,  0.1,
                          -0.3, -0.3, -0.3, -0.3,    0)    )
   return(jac)
}

## initial conditions and output times
yini  <- 1:5
times <- 1:20

## default: stiff method, internally generated, full jacobian
out   <- lsode(yini, times, f1, parms = 0, jactype = "fullint")

## stiff method, user-generated full jacobian
out2  <- lsode(yini, times, f1, parms = 0, jactype = "fullusr",
              jacfunc = fulljac)

## stiff method, internally-generated banded jacobian
## one nonzero band above (up) and below(down) the diagonal
out3  <- lsode(yini, times, f1, parms = 0, jactype = "bandint",
                              bandup = 1, banddown = 1)

## stiff method, user-generated banded jacobian
out4  <- lsode(yini, times, f1, parms = 0, jactype = "bandusr",
              jacfunc = bandjac, bandup = 1, banddown = 1)

## non-stiff method
out5  <- lsode(yini, times, f1, parms = 0, mf = 10)

## =====================================
## Example 2:
##   diffusion on a 2-D grid
##   partially specified jacobian
## =====================================

diffusion2D <- function(t, Y, par)
{
   y <- matrix(nr = n, nc = n, data = Y)
   dY   <- r*y     # production

   ## diffusion in X-direction; boundaries = 0-concentration
   Flux <- -Dx * rbind(y[1,],(y[2:n,]-y[1:(n-1),]),-y[n,])/dx
   dY   <- dY - (Flux[2:(n+1),]-Flux[1:n,])/dx

   ## diffusion in Y-direction
   Flux <- -Dy * cbind(y[,1],(y[,2:n]-y[,1:(n-1)]),-y[,n])/dy
   dY    <- dY - (Flux[,2:(n+1)]-Flux[,1:n])/dy

   return(list(as.vector(dY)))
}

## parameters
dy    <- dx <- 1   # grid size
Dy    <- Dx <- 1   # diffusion coeff, X- and Y-direction
r     <- 0.025     # production rate
times <- c(0, 1)

n  <- 50
y  <- matrix(nr = n, nc = n, 0.)

pa <- par(ask = FALSE)

## initial condition
for (i in 1:n) {
  for (j in 1:n) {
    dst <- (i-n/2)^2+(j-n/2)^2
    y[i,j] <- max(0.,1.-1./(n*n)*(dst-n)^2)
  }
}
filled.contour(y, color.palette = terrain.colors)

## jacfunc need not be estimated exactly
## a crude approximation, with a smaller bandwidth will do.
## Here the half-bandwidth 1 is used, whereas the true
## half-bandwidths are equal to n.
## This corresponds to ignoring the y-direction coupling in the ODEs.

print(system.time(
  for (i in 1:20) {
    out  <-  lsode(func = diffusion2D, y = as.vector(y), times = times,
              parms = NULL, jactype = "bandint", bandup = 1, banddown = 1)

    filled.contour(matrix(nr = n, nc = n, out[2,-1]), zlim = c(0,1),
                  color.palette = terrain.colors, main = i)

    y <- out[2,-1]
  }
))
par(ask = pa)

[Package deSolve version 1.3 Index]