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Solver for Ordinary Differential Equations (ODE) for COMPLEX variables

zvode.Rd

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 \(\partial\dot{y}_i/\partial y_j\), 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 \(\dot{dy}/dy\), 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.

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.

Author

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

Details

see vode, the double precision version, for details.

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).

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))
#>                                    analytical.solution
#>  [95,] 0.9500711-3.120334e-01i 0.9500711-3.120334e-01i
#>  [96,] 0.9679487-2.511480e-01i 0.9679487-2.511480e-01i
#>  [97,] 0.9819287-1.892512e-01i 0.9819287-1.892512e-01i
#>  [98,] 0.9919548-1.265924e-01i 0.9919548-1.265925e-01i
#>  [99,] 0.9979867-6.342392e-02i 0.9979867-6.342392e-02i
#> [100,] 1.0000000+3.256413e-09i 1.0000000-2.449294e-16i


## =======================================================================
## 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]))
#>                           [,1]                    [,2]
#>  [95,] 0.9500711-3.120334e-01i 0.9500711-3.120334e-01i
#>  [96,] 0.9679487-2.511480e-01i 0.9679487-2.511480e-01i
#>  [97,] 0.9819287-1.892512e-01i 0.9819287-1.892512e-01i
#>  [98,] 0.9919548-1.265925e-01i 0.9919548-1.265925e-01i
#>  [99,] 0.9979867-6.342392e-02i 0.9979867-6.342392e-02i
#> [100,] 1.0000000+1.984294e-09i 1.0000000-2.449294e-16i