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Solver for Ordinary Differential Equations (ODE), Switching Automatically Between Stiff and Non-stiff Methods and With Root Finding

lsodar.Rd

Solving initial value problems for stiff or non-stiff systems of first-order ordinary differential equations (ODEs) and including root-finding.

The R function lsodar provides an interface to the FORTRAN ODE solver of the same name, written by Alan C. Hindmarsh and Linda R. Petzold.

The system of ODE's is written as an R function or be defined in compiled code that has been dynamically loaded. - see description of lsoda for details.

lsodar differs from lsode in two respects.

  • It switches automatically between stiff and nonstiff methods (similar as lsoda).

  • It finds the root of at least one of a set of constraint functions g(i) of the independent and dependent variables.

Two uses of lsodar are:

  • To stop the simulation when a certain condition is met

  • To trigger events, i.e. sudden changes in one of the state variables when a certain condition is met.

when a particular condition is met.

Usage

lsodar(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 lsodar() 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.

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

a logical value that, when TRUE, 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 lsodar cannot integrate past tcrit. The FORTRAN routine lsodar 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.

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

If a root has been found, the output will have the attribute iroot, an integer indicating which root has been found.

Author

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.

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

Kathie L. Hiebert and Lawrence F. Shampine, Implicitly Defined Output Points for Solutions of ODEs, Sandia Report SAND80-0180, February 1980.

Netlib: https://netlib.org

Details

The work is done by the FORTRAN subroutine lsodar, 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 lsodar, from Netlib.

lsodar switches automatically between stiff and nonstiff methods (similar as lsoda). 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.

lsodar 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, lsodar may return false roots, or return the same root at two or more nearly equal values of time.

The form of the Jacobian can be specified by jactype which can take the following values:

jactype = "fullint":

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

jactype = "fullusr":

a full Jacobian, specified by user function jacfunc,

jactype = "bandusr":

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

jactype = "bandint":

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

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

The input parameters rtol, and atol determine the error control performed by the solver. See lsoda for details.

The output will have the attribute iroot, if a root was found iroot is a vector, its length equal to the number of constraint functions it will have a value of 1 for the constraint function whose root that has been found and 0 otherwise.

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.

See also

  • roots for more examples on roots and events

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

  • lsoda, lsode, lsodes, 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:
##   from lsodar source code
## =======================================================================

Fun <- function (t, y, parms) {
  ydot <- vector(len = 3)
  ydot[1] <- -.04*y[1] + 1.e4*y[2]*y[3]
  ydot[3] <- 3.e7*y[2]*y[2]
  ydot[2] <- -ydot[1] - ydot[3]
  return(list(ydot, ytot = sum(y)))
}

rootFun <- function (t, y, parms) {
  yroot <- vector(len = 2)
  yroot[1] <- y[1] - 1.e-4
  yroot[2] <- y[3] - 1.e-2
  return(yroot)
}

y     <- c(1, 0, 0)
times <- c(0, 0.4*10^(0:8))

out   <- lsodar(y = y, times = times, fun = Fun, rootfun = rootFun,
                rtol = 1e-4, atol = c(1e-6, 1e-10, 1e-6), parms = NULL)
print(paste("root is found for eqn", which(attributes(out)$iroot == 1)))
#> [1] "root is found for eqn 2"
print(out[nrow(out),])
#>         time            1            2            3         ytot 
#> 2.639981e-01 9.899653e-01 3.470563e-05 1.000000e-02 1.000000e+00 

diagnostics(out)
#> 
#> --------------------
#> lsodar return code
#> --------------------
#> 
#>   return code (idid) =  3 
#>   Integration was successful and a root was found before reaching the end.
#> 
#> --------------------
#> INTEGER values
#> --------------------
#> 
#>   1 The return code : 3 
#>   2 The number of steps taken for the problem so far: 49 
#>   3 The number of function evaluations for the problem so far: 104 
#>   5 The method order last used (successfully): 2 
#>   6 The order of the method to be attempted on the next step: 3 
#>   7 If return flag =-4,-5: the largest component in error vector 0 
#>   8 The length of the real work array actually required: 74 
#>   9 The length of the integer work array actually required: 23 
#>  14 The number of Jacobian evaluations and LU decompositions so far: 4 
#>  15 The method indicator for the last succesful step,
#>            1=adams (nonstiff), 2= bdf (stiff): 2 
#>  16 The current method indicator to be attempted on the next step,
#>            1=adams (nonstiff), 2= bdf (stiff): 2 
#>  
#> --------------------
#> RSTATE values
#> --------------------
#> 
#>   1 The step size in t last used (successfully): 0.05896259 
#>   2 The step size to be attempted on the next step: 0.1220619 
#>   3 The current value of the independent variable which the solver has reached: 0.2721339 
#>   4 Tolerance scale factor > 1.0 computed when requesting too much accuracy: 0 
#>   5 The value of t at the time of the last method switch, if any: 0.006009229 
#>  
#> --------------------
#> ROOT
#> --------------------
#> 
#>  root found for root equation: 2 
#> 
#>  at time : 0.264 
  
## =======================================================================
## Example 2:
##   using lsodar to estimate steady-state conditions
## =======================================================================

## Bacteria (Bac) are growing on a substrate (Sub)
model <- function(t, state, pars) {
  with (as.list(c(state, pars)), {
    ##        substrate uptake             death     respiration
    dBact <-  gmax*eff*Sub/(Sub+ks)*Bact - dB*Bact - rB*Bact
    dSub  <- -gmax    *Sub/(Sub+ks)*Bact + dB*Bact            + input

    return(list(c(dBact,dSub)))
  })
}

## root is the condition where sum of |rates of change|
## is very small

rootfun <- function (t, state, pars) {
  dstate <- unlist(model(t, state, pars)) # rate of change vector
  return(sum(abs(dstate)) - 1e-10)
}

pars <- list(Bini = 0.1, Sini = 100, gmax = 0.5, eff = 0.5,
             ks = 0.5, rB = 0.01, dB = 0.01, input = 0.1)

tout    <- c(0, 1e10)
state   <- c(Bact = pars$Bini, Sub = pars$Sini)
out     <- lsodar(state, tout, model, pars, rootfun = rootfun)
print(out)
#>       time     Bact          Sub
#> 1    0.000 0.100000 100.00000000
#> 2 2583.657 3.333333   0.04347826


## =======================================================================
## Example 3:
##   using lsodar to trigger an event
## =======================================================================

## a state variable is decaying at a first-order rate. 
## when it reaches the value 0.1, a random amount is added.

derivfun <- function (t,y,parms)
  list (-0.05 * y)

rootfun <- function (t,y,parms)
  return(y - 0.1) 

eventfun <- function(t,y,parms)
  return(y + runif(1))  

yini <- 0.8
times <- 0:200

out <- lsodar(func=derivfun, y = yini, times=times, 
  rootfunc = rootfun, events = list(func=eventfun, root = TRUE))

plot(out, type = "l", lwd = 2, main = "lsodar with event")