R: Solver for Ordinary Differential Equations (ODE), Switching...

lsoda {deSolve} | R Documentation

## 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 `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 `.C`, `.Fortran`, `.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 `optim`, `nls`, `nlm` or `nlme`

`lsoda` differs from the other integrators (except `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

`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` | 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 `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 `lsoda()` 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.
`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. When `rootfunc` is provided, then `lsodar` will be called.
`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 `lsoda` cannot integrate past `tcrit`. The FORTRAN routine `lsoda` 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 large 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 <= 5. 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

All the hard work is done by the FORTRAN subroutine `lsoda`, 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 12 November 2003 version of lsoda, from Netlib.

`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 Jacobian can be specified by `jactype` which can take the following values:

"fullint"

  a full Jacobian, calculated internally by lsoda, the default,

"fullusr"

  a full Jacobian, specified by user function `jacfunc`,

"bandusr"

  a banded Jacobian, specified by user function `jacfunc` the size of the bands specified by `bandup` and `banddown`,

"bandint"

  banded Jacobian, calculated by lsoda; the size of the bands specified by `bandup` and `banddown`.

If `jactype` = "fullusr" or "bandusr" then the user must supply a subroutine `jacfunc`.

The following description of error control is adapted from the documentation of the lsoda source code (input arguments `rtol` and `atol`, above):

The input parameters `rtol`, and `atol` determine the error control performed by the solver. The solver will control the vector e of estimated local errors in y, according to an inequality of the form max-norm of ( e/ewt ) <= 1, where ewt is a vector of positive error weights. The values of `rtol` and `atol` should all be non-negative. The form of ewt is:

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

### Note

The ‘demo’ directory contains some examples of using `gnls` to estimate parameters in a dynamic model.

### Author(s)

R. Woodrow Setzer <setzer.woodrow@epa.gov>

### References

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

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

Netlib: https://www.netlib.org

### See Also

- `rk`, `rkMethod`, `rk4` and `euler` for Runge-Kutta integrators.

- `lsode`, which can also find a root

- `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:
    ##   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

[Package deSolve version 1.32 Index]