R: Solver for Ordinary Differential Equations (ODE)

vode {deSolve}

R Documentation

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 vode provides an interface to the FORTRAN ODE solver of the same name, written by Peter N. Brown, Alan C. Hindmarsh and George D. Byrne.

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.

vode is very similar to lsode, but uses a variable-coefficient method rather than the fixed-step-interpolate methods in lsode. 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

vode(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, forcings=NULL,
  initforc = NULL, fcontrol=NULL, events=NULL, lags = 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: `func <- function(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 must 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 `vode()` 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 `\partial\dot{y}_i/\partial y_j`, or a string giving the name of a function or subroutine in ‘dllname’ that computes the Jacobian (see vignette `"compiledCode"` 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 `\partial\dot{y}/\partial y`, 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 `mf` is not `NULL`.
`mf`	the "method flag" passed to function vode - overrules `jactype` - provides more options than `jactype` - see details.
`verbose`	if TRUE: full output to the screen, e.g. will print the `diagnostiscs` of the integration - see details.
`tcrit`	if not `NULL`, then `vode` cannot integrate past `tcrit`. The FORTRAN routine `dvode` 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`	logical; 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 calculated 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. These names will be used to label the output matrix.
`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(`times`), max(`times`)] is done by taking the value at the closest data extreme. See forcings or package vignette `"compiledCode"`.
`initforc`	if not `NULL`, the name of the forcing function initialisation function, as provided in ‘dllname’. It MUST be present if `forcings` has been given a value. See forcings or package vignette `"compiledCode"`.
`fcontrol`	A list of control parameters for the forcing functions. forcings or package vignette `"compiledCode"`
`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 `func` and `jacfunc` allowing this to be a generic function.

Details

Before using the integrator vode, 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

jac = "fullint":: a full Jacobian, calculated internally by vode, corresponds to mf = 22,
jac = "fullusr":: a full Jacobian, specified by user function jacfunc, corresponds to mf = 21,
jac = "bandusr":: a banded Jacobian, specified by user function jacfunc; the size of the bands specified by bandup and banddown, corresponds to mf = 24,
jac = "bandint":: a banded Jacobian, calculated by vode; 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, -11, -12, -14, -15, -21, -22, -24, -25.

mf is a signed two-digit integer, mf = JSV*(10*METH + MITER), where

JSV = SIGN(mf)

indicates the Jacobian-saving strategy: JSV = 1 means a copy of the Jacobian is saved for reuse in the corrector iteration algorithm. JSV = -1 means a copy of the Jacobian is not saved.

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.

The example for integrator lsode demonstrates how to specify both a banded and full Jacobian.

The input parameters rtol, and atol determine the error control performed by the solver. If the request for precision exceeds the capabilities of the machine, vode will return an error code. See lsoda for details.

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.

Value

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 ‘vode’ returns with an unrecoverable error. If y has a names attribute, it will be used to label the columns of the output value.

Note

From version 1.10.4, the default of atol was changed from 1e-8 to 1e-6, to be consistent with the other solvers.

Author(s)

Karline Soetaert <karline.soetaert@nioz.nl>

References

P. N. Brown, G. D. Byrne, and A. C. Hindmarsh, 1989. VODE: A Variable Coefficient ODE Solver, SIAM J. Sci. Stat. Comput., 10, pp. 1038-1051.
Also, LLNL Report UCRL-98412, June 1988. doi:10.1137/0910062

G. D. Byrne and A. C. Hindmarsh, 1975. A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations. ACM Trans. Math. Software, 1, pp. 71-96. doi:10.1145/355626.355636

A. C. Hindmarsh and G. D. Byrne, 1977. EPISODE: An Effective Package for the Integration of Systems of Ordinary Differential Equations. LLNL Report UCID-30112, Rev. 1.

G. D. Byrne and A. C. Hindmarsh, 1976. EPISODEB: An Experimental Package for the Integration of Systems of Ordinary Differential Equations with Banded Jacobians. LLNL Report UCID-30132, April 1976.

A. C. Hindmarsh, 1983. ODEPACK, a Systematized Collection of ODE Solvers. in Scientific Computing, R. S. Stepleman et al., eds., North-Holland, Amsterdam, pp. 55-64.

K. R. Jackson and R. Sacks-Davis, 1980. An Alternative Implementation of Variable Step-Size Multistep Formulas for Stiff ODEs. ACM Trans. Math. Software, 6, pp. 295-318. doi:10.1145/355900.355903

Netlib: https://netlib.org

Examples

## =======================================================================
## ex. 1
## The famous Lorenz equations: chaos in the earth's atmosphere
## Lorenz 1963. J. Atmos. Sci. 20, 130-141.
## =======================================================================

chaos <- function(t, state, parameters) {
  with(as.list(c(state)), {

    dx     <- -8/3 * x + y * z
    dy     <- -10 * (y - z)
    dz     <- -x * y + 28 * y - z

    list(c(dx, dy, dz))
  })
}

state <- c(x = 1, y = 1, z = 1)
times <- seq(0, 100, 0.01)

out   <- vode(state, times, chaos, 0)

plot(out, type = "l")   # all versus time
plot(out[,"x"], out[,"y"], type = "l", main = "Lorenz butterfly",
  xlab = "x", ylab = "y")


## =======================================================================
## ex. 2
## SCOC model, in FORTRAN  - to see the FORTRAN code:
## browseURL(paste(system.file(package="deSolve"),
##                             "/doc/examples/dynload/scoc.f",sep=""))
## example from Soetaert and Herman, 2009, chapter 3. (simplified)
## =======================================================================

## Forcing function data
Flux <- matrix(ncol = 2, byrow = TRUE, data = c(
  1,  0.654, 11, 0.167,  21, 0.060, 41, 0.070, 73, 0.277, 83, 0.186,
  93, 0.140,103, 0.255, 113, 0.231,123, 0.309,133, 1.127,143, 1.923,
  153,1.091,163, 1.001, 173, 1.691,183, 1.404,194, 1.226,204, 0.767,
  214,0.893,224, 0.737, 234, 0.772,244, 0.726,254, 0.624,264, 0.439,
  274,0.168,284, 0.280, 294, 0.202,304, 0.193,315, 0.286,325, 0.599,
  335,1.889,345, 0.996, 355, 0.681,365, 1.135))

parms <- c(k = 0.01)

meanDepo <- mean(approx(Flux[,1], Flux[,2], xout = seq(1, 365, by = 1))$y)

Yini <- c(y = as.double(meanDepo/parms))

times <- 1:365
out <- vode(Yini, times, func = "scocder",
    parms = parms, dllname = "deSolve",
    initforc = "scocforc", forcings = Flux,
    initfunc = "scocpar", nout = 2,
    outnames = c("Mineralisation", "Depo"))

matplot(out[,1], out[,c("Depo", "Mineralisation")], 
        type = "l", col = c("red", "blue"), xlab = "time", ylab = "Depo")

## Constant interpolation of forcing function - left side of interval
fcontrol <- list(method = "constant")

out2 <- vode(Yini, times, func = "scocder",
    parms = parms, dllname = "deSolve",
    initforc = "scocforc",  forcings = Flux, fcontrol = fcontrol,
    initfunc = "scocpar", nout = 2,
    outnames = c("Mineralisation", "Depo"))
matplot(out2[,1], out2[,c("Depo", "Mineralisation")], 
        type = "l", col = c("red", "blue"), xlab = "time", ylab = "Depo")

## Constant interpolation of forcing function - middle of interval
fcontrol <- list(method = "constant", f = 0.5)

out3 <- vode(Yini, times, func = "scocder",
    parms = parms, dllname = "deSolve",
    initforc = "scocforc",  forcings = Flux, fcontrol = fcontrol,
    initfunc = "scocpar", nout = 2,
    outnames = c("Mineralisation", "Depo"))

matplot(out3[,1], out3[,c("Depo", "Mineralisation")], 
        type = "l", col = c("red", "blue"), xlab = "time", ylab = "Depo")

plot(out, out2, out3)

[Package deSolve version 1.40 Index]