R: Solver for Ordinary Differential Equations (ODE)

lsode {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 `lsode` provides an interface to the FORTRAN ODE solver of the same name, written by Alan C. Hindmarsh and Andrew H. Sherman.

It combines parts of the code `lsodar` and can thus find the root of at least one of a set of constraint functions g(i) of the independent and dependent variables. This can be used to stop the simulation or to trigger events, i.e. a sudden change in one of the state variables.

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, rootfunc = NULL,
      verbose = FALSE, nroot = 0, 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 `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 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 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 `mf`is not `NULL`.
`mf` | the "method flag" passed to function lsode - overrules `jactype` - provides more options than `jactype` - see details.
`rootfunc` | if not `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 ‘dllname’ that computes the root function. The R calling sequence for `rootfunc` is identical to that of `func`. `rootfunc` should return a vector with the function values whose root is sought.
`verbose` | if TRUE: full output to the screen, e.g. will print the `diagnostiscs` of the integration - 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 `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` | 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. See forcings or 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

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.

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.

`lsode` can find the root of at least one of a set of constraint functions `rootfunc` 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.

Caution: Because of numerical errors in the function `rootfun` due to roundoff and integration error, `lsode` may return false roots, or return the same root at two or more nearly equal values of `time`.

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

### Author(s)

Karline Soetaert <karline.soetaert@nioz.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

- `rk`,

- `rk4` and `euler` for Runge-Kutta integrators.

- `lsoda`, `lsodes`, `lsodar`, `vode`, `daspk` for other solvers of the Livermore family,

- `ode` for a general interface to most of the ODE solvers,

- `ode.band` for solving models with a banded Jacobian,

- `ode.1D` for integrating 1-D models,

- `ode.2D` for integrating 2-D models,

- `ode.3D` for integrating 3-D models,

`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(nrow = n, ncol = 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(nrow = n, ncol = 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(nrow = n, ncol = n, out[2,-1]), zlim = c(0,1),
                      color.palette = terrain.colors, main = i)

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

[Package deSolve version 1.32 Index]