R: Solver for Ordinary Differential Equations (ODE) for COMPLEX...

zvode {deSolve} | R Documentation

## Solver for Ordinary Differential Equations (ODE) for COMPLEX variables

### Description

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

dy/dt = f(t,y)

where dy and y are complex variables.

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

### Usage

    zvode(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, ...)

### 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. y has to be complex
`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`. They should be complex numbers. If `func` is a string, then `dllname` must give the name of the shared library (without extension) which must be loaded before `zvode()` 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). Its elements should be complex numbers. 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 `zvode` - 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 `zvode` 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"`
`...` | additional arguments passed to `func` and `jacfunc` allowing this to be a generic function.

### Details

see `vode`, the double precision version, for details.

### 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 ‘zvode’ 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.

The following text is adapted from the zvode.f source code:

When using `zvode` for a stiff system, it should only be used for the case in which the function f is analytic, that is, when each f(i) is an analytic function of each y(j). Analyticity means that the partial derivative df(i)/dy(j) is a unique complex number, and this fact is critical in the way `zvode` solves the dense or banded linear systems that arise in the stiff case. For a complex stiff ODE system in which f is not analytic, `zvode` is likely to have convergence failures, and for this problem one should instead use `ode` on the equivalent real system (in the real and imaginary parts of y).

### 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.

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.

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.

Netlib: http://www.netlib.org

### See Also

`vode` for the double precision version

### Examples

    ## =======================================================================
    ## Example 1 - very simple example
    ## df/dt = 1i*f, where 1i is the imaginary unit
    ## The initial value is f(0) = 1 = 1+0i
    ## =======================================================================

    ZODE <- function(Time, f, Pars) {
      df <-  1i*f
      return(list(df))
    }

    pars    <- NULL
    yini    <- c(f = 1+0i)
    times   <- seq(0, 2*pi, length = 100)
    out     <- zvode(func = ZODE, y = yini, parms = pars, times = times,
      atol = 1e-10, rtol = 1e-10)

    # The analytical solution to this ODE is the exp-function:
    # f(t) = exp(1i*t)
    #      = cos(t)+1i*sin(t)  (due to Euler's equation)

    analytical.solution  <- exp(1i * times)

    ## compare numerical and analytical solution
    tail(cbind(out[,2], analytical.solution))

    ## =======================================================================
    ## Example 2 - example in "zvode.f",
    ## df/dt = 1i*f        (same as above ODE)
    ## dg/dt = -1i*g*g*f   (an additional ODE depending on f)
    ##
    ## Initial values are
    ## g(0) = 1/2.1 and
    ## z(0) = 1
    ## =======================================================================

    ZODE2<-function(Time,State,Pars) {
      with(as.list(State), {
        df <- 1i * f
        dg <- -1i * g*g * f
        return(list(c(df, dg)))
      })
    }

    yini    <- c(f = 1 + 0i, g = 1/2.1 + 0i)
    times   <- seq(0, 2*pi, length = 100)
    out     <- zvode(func = ZODE2, y = yini, parms = NULL, times = times,
      atol = 1e-10, rtol = 1e-10)

    ## The analytical solution is
    ## f(t) = exp(1i*t)   (same as above)
    ## g(t) = 1/(f(t) + 1.1)

    analytical <- cbind(f = exp(1i * times), g = 1/(exp(1i * times) + 1.1))

    ## compare numerical solution and the two analytical ones:
    tail(cbind(out[,2], analytical[,1]))

[Package deSolve version 1.28 Index]