daspk {deSolve}  R Documentation 
Solver for Differential Algebraic Equations (DAE)
Description
Solves either:
a system of ordinary differential equations (ODE) of the form
y' = f(t, y, ...)
or
a system of differential algebraic equations (DAE) of the form
F(t,y,y') = 0
or
a system of linearly implicit DAES in the form
M y' = f(t, y)
using a combination of backward differentiation formula (BDF) and a direct linear system solution method (dense or banded).
The R function daspk
provides an interface to the FORTRAN DAE
solver of the same name, written by Linda R. Petzold, Peter N. Brown,
Alan C. Hindmarsh and Clement W. Ulrich.
The system of DE'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.
Usage
daspk(y, times, func = NULL, parms, nind = c(length(y), 0, 0),
dy = NULL, res = NULL, nalg = 0,
rtol = 1e6, atol = 1e6, jacfunc = NULL,
jacres = NULL, jactype = "fullint", mass = NULL, estini = NULL,
verbose = FALSE, tcrit = NULL, hmin = 0, hmax = NULL,
hini = 0, ynames = TRUE, maxord = 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 DE system. If 
times 
time sequence for which output is wanted; the first
value of 
func 
to be used if the model is an ODE, or a DAE written in linearly
implicit form (M y' = f(t, y)).
The return value of Note that it is not possible to define 
parms 
vector or list of parameters used in 
nind 
if a DAE system: a threevalued vector with the number of variables of index 1, 2, 3 respectively. The equations must be defined such that the index 1 variables precede the index 2 variables which in turn precede the index 3 variables. The sum of the variables of different index should equal N, the total number of variables. Note that this has been added for consistency with radau. If used, then the variables are weighed differently than in the original daspk code, i.e. index 2 variables are scaled with 1/h, index 3 variables are scaled with 1/h^2. In some cases this allows daspk to solve index 2 or index 3 problems. 
dy 
the initial derivatives of the state variables of the DE system. Ignored if an ODE. 
res 
if a DAE system: either an Rfunction that computes the
residual function If Here The return value of If 
nalg 
if a DAE system: the number of algebraic equations (equations not involving derivatives). Algebraic equations should always be the last, i.e. preceeded by the differential equations. Only used if 
rtol 
relative error tolerance, either a scalar or a vector, one value for each y, 
atol 
absolute error tolerance, either a scalar or a vector, one value for each y. 
jacfunc 
if not If the Jacobian is a full matrix, If the Jacobian is banded, 
jacres 
If If the Jacobian is a full matrix, If the Jacobian is banded, 
jactype 
the structure of the Jacobian, one of

mass 
the mass matrix.
If not If 
estini 
only if a DAE system, and if initial values of 
verbose 
if TRUE: full output to the screen, e.g. will
print the 
tcrit 
the FORTRAN routine 
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 
hmax 
an optional maximum value of the integration stepsize. If
not specified, 
hini 
initial step size to be attempted; if 0, the initial step size is determined by the solver 
ynames 
logical, if 
maxord 
the maximum order to be allowed. Reduce 
bandup 
number of nonzero bands above the diagonal, in case
the Jacobian is banded (and 
banddown 
number of nonzero bands below the diagonal, in case
the Jacobian is banded (and 
maxsteps 
maximal number of steps per output interval taken by the
solver; will be recalculated to be at least 500 and a multiple of
500; if 
dllname 
a string giving the name of the shared library
(without extension) that contains all the compiled function or
subroutine definitions referred to in 
initfunc 
if not 
initpar 
only when ‘dllname’ is specified and an
initialisation function 
rpar 
only when ‘dllname’ is specified: a vector with
double precision values passed to the dllfunctions whose names are
specified by 
ipar 
only when ‘dllname’ is specified: a vector with
integer values passed to the dllfunctions whose names are specified
by 
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 
outnames 
only used if ‘dllname’ is specified and

forcings 
only used if ‘dllname’ is specified: a list with
the forcing function data sets, each present as a twocolumned matrix,
with (time,value); interpolation outside the interval
[min( See forcings or package vignette 
initforc 
if not 
fcontrol 
A list of control parameters for the forcing functions.
See forcings or vignette 
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 
Details
The daspk solver uses the backward differentiation formulas of orders
one through five (specified with maxord
) to solve either:
an ODE system of the form
y' = f(t,y,...)
or
a DAE system of the form
y' = M f(t,y,...)
or
a DAE system of the form
F(t,y,y') = 0
. The index of the DAE should be preferable <= 1.
ODEs are specified using argument func
,
DAEs are specified using argument res
.
If a DAE system, Values for y and y' (argument dy
)
at the initial time must be given as input. Ideally, these values should be consistent,
that is, if t, y, y' are the given initial values, they should
satisfy F(t,y,y') = 0.
However, if consistent values are not
known, in many cases daspk can solve for them: when estini
= 1,
y' and algebraic variables (their number specified with nalg
)
will be estimated, when estini
= 2, y will be estimated.
The form of the Jacobian can be specified by
jactype
. This is one of:
 jactype = "fullint":
a full Jacobian, calculated internally by
daspk
, the default, jactype = "fullusr":
a full Jacobian, specified by user function
jacfunc
orjacres
, jactype = "bandusr":
a banded Jacobian, specified by user function
jacfunc
orjacres
; the size of the bands specified bybandup
andbanddown
, jactype = "bandint":
a banded Jacobian, calculated by
daspk
; the size of the bands specified bybandup
andbanddown
.
If jactype
= "fullusr" or "bandusr" then the user must supply a
subroutine jacfunc
.
If jactype = "fullusr" or "bandusr" then the user must supply a
subroutine jacfunc
or jacres
.
The input parameters rtol
, and atol
determine the
error control performed by the solver. If the request for
precision exceeds the capabilities of the machine, daspk
will return
an error code. See lsoda
for details.
When the index of the variables is specified (argument nind
),
and higher index variables
are present, then the equations are scaled such that equations corresponding
to index 2 variables are multiplied with 1/h, for index 3 they are multiplied
with 1/h^2, where h is the time step. This is not in the standard DASPK code,
but has been added for consistency with solver radau. Because of this,
daspk can solve certain index 2 or index 3 problems.
res and jacres may be defined in compiled C or FORTRAN code, as
well as in an Rfunction. See package vignette "compiledCode"
for details. Examples
in FORTRAN are in the ‘dynload’ subdirectory of the
deSolve
package directory.
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 Rfunction. 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
or
res
, plus an additional column (the first) for the time value.
There will be one row for each element in times
unless the
FORTRAN routine ‘daspk’ returns with an unrecoverable error. If
y
has a names attribute, it will be used to label the columns
of the output value.
Note
In this version, the Krylov method is not (yet) supported.
From deSolve
version 1.10.4 and above, the following changes were made
the argument list to
daspk
now also includesnind
, the index of each variable. This is used to scale the variables, such thatdaspk
in R can also solve certain index 2 or index 3 problems, which the original Fortran version may not be able to solve.the default of
atol
was changed from 1e8 to 1e6, to be consistent with the other solvers.the multiple warnings from daspk when the number of steps exceed 500 were toggled off unless
verbose
isTRUE
Author(s)
Karline Soetaert <karline.soetaert@nioz.nl>
References
L. R. Petzold, A Description of DASSL: A Differential/Algebraic System Solver, in Scientific Computing, R. S. Stepleman et al. (Eds.), NorthHolland, Amsterdam, 1983, pp. 6568.
K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of InitialValue Problems in DifferentialAlgebraic Equations, Elsevier, New York, 1989.
P. N. Brown and A. C. Hindmarsh, Reduced Storage Matrix Methods in Stiff ODE Systems, J. Applied Mathematics and Computation, 31 (1989), pp. 4091. doi:10.1016/00963003(89)901100
P. N. Brown, A. C. Hindmarsh, and L. R. Petzold, Using Krylov Methods in the Solution of LargeScale DifferentialAlgebraic Systems, SIAM J. Sci. Comp., 15 (1994), pp. 14671488. doi:10.1137/0915088
P. N. Brown, A. C. Hindmarsh, and L. R. Petzold, Consistent Initial Condition Calculation for DifferentialAlgebraic Systems, LLNL Report UCRLJC122175, August 1995; submitted to SIAM J. Sci. Comp.
Netlib: https://netlib.org
See Also

radau
for integrating DAEs up to index 3, 
rk
, 
lsoda
,lsode
,lsodes
,lsodar
,vode
, 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 1D models, 
ode.2D
for integrating 2D models, 
ode.3D
for integrating 3D models,
diagnostics
to print diagnostic messages.
Examples
## =======================================================================
## Coupled chemical reactions including an equilibrium
## modeled as (1) an ODE and (2) as a DAE
##
## The model describes three chemical species A,B,D:
## subjected to equilibrium reaction D < > A + B
## D is produced at a constant rate, prod
## B is consumed at 1st order rate, r
## Chemical problem formulation 1: ODE
## =======================================================================
## Dissociation constant
K < 1
## parameters
pars < c(
ka = 1e6, # forward rate
r = 1,
prod = 0.1)
Fun_ODE < function (t, y, pars)
{
with (as.list(c(y, pars)), {
ra < ka*D # forward rate
rb < ka/K *A*B # backward rate
## rates of changes
dD < ra + rb + prod
dA < ra  rb
dB < ra  rb  r*B
return(list(dy = c(dA, dB, dD),
CONC = A+B+D))
})
}
## =======================================================================
## Chemical problem formulation 2: DAE
## 1. get rid of the fast reactions ra and rb by taking
## linear combinations : dD+dA = prod (res1) and
## dBdA = r*B (res2)
## 2. In addition, the equilibrium condition (eq) reads:
## as ra = rb : ka*D = ka/K*A*B = > K*D = A*B
## =======================================================================
Res_DAE < function (t, y, yprime, pars)
{
with (as.list(c(y, yprime, pars)), {
## residuals of lumped rates of changes
res1 < dD  dA + prod
res2 < dB + dA  r*B
## and the equilibrium equation
eq < K*D  A*B
return(list(c(res1, res2, eq),
CONC = A+B+D))
})
}
## =======================================================================
## Chemical problem formulation 3: Mass * Func
## Based on the DAE formulation
## =======================================================================
Mass_FUN < function (t, y, pars) {
with (as.list(c(y, pars)), {
## as above, but without the
f1 < prod
f2 <  r*B
## and the equilibrium equation
f3 < K*D  A*B
return(list(c(f1, f2, f3),
CONC = A+B+D))
})
}
Mass < matrix(nrow = 3, ncol = 3, byrow = TRUE,
data=c(1, 0, 1, # dA + 0 + dB
1, 1, 0, # dA + dB +0
0, 0, 0)) # algebraic
times < seq(0, 100, by = 2)
## Initial conc; D is in equilibrium with A,B
y < c(A = 2, B = 3, D = 2*3/K)
## ODE model solved with daspk
ODE < daspk(y = y, times = times, func = Fun_ODE,
parms = pars, atol = 1e10, rtol = 1e10)
## Initial rate of change
dy < c(dA = 0, dB = 0, dD = 0)
## DAE model solved with daspk
DAE < daspk(y = y, dy = dy, times = times,
res = Res_DAE, parms = pars, atol = 1e10, rtol = 1e10)
MASS< daspk(y=y, times=times, func = Mass_FUN, parms = pars, mass = Mass)
## ================
## plotting output
## ================
plot(ODE, DAE, xlab = "time", ylab = "conc", type = c("l", "p"),
pch = c(NA, 1))
legend("bottomright", lty = c(1, NA), pch = c(NA, 1),
col = c("black", "red"), legend = c("ODE", "DAE"))
# difference between both implementations:
max(abs(ODEDAE))
## =======================================================================
## same DAE model, now with the Jacobian
## =======================================================================
jacres_DAE < function (t, y, yprime, pars, cj)
{
with (as.list(c(y, yprime, pars)), {
## res1 = dD  dA + prod
PD[1,1] < 1*cj # d(res1)/d(A)cj*d(res1)/d(dA)
PD[1,2] < 0 # d(res1)/d(B)cj*d(res1)/d(dB)
PD[1,3] < 1*cj # d(res1)/d(D)cj*d(res1)/d(dD)
## res2 = dB + dA  r*B
PD[2,1] < 1*cj
PD[2,2] < r 1*cj
PD[2,3] < 0
## eq = K*D  A*B
PD[3,1] < B
PD[3,2] < A
PD[3,3] < K
return(PD)
})
}
PD < matrix(ncol = 3, nrow = 3, 0)
DAE2 < daspk(y = y, dy = dy, times = times,
res = Res_DAE, jacres = jacres_DAE, jactype = "fullusr",
parms = pars, atol = 1e10, rtol = 1e10)
max(abs(DAEDAE2))
## See \dynload subdirectory for a FORTRAN implementation of this model
## =======================================================================
## The chemical model as a DLL, with production a forcing function
## =======================================================================
times < seq(0, 100, by = 2)
pars < c(K = 1, ka = 1e6, r = 1)
## Initial conc; D is in equilibrium with A,B
y < c(A = 2, B = 3, D = as.double(2*3/pars["K"]))
## Initial rate of change
dy < c(dA = 0, dB = 0, dD = 0)
# production increases with time
prod < matrix(ncol = 2,
data = c(seq(0, 100, by = 10), 0.1*(1+runif(11)*1)))
ODE_dll < daspk(y = y, dy = dy, times = times, res = "chemres",
dllname = "deSolve", initfunc = "initparms",
initforc = "initforcs", parms = pars, forcings = prod,
atol = 1e10, rtol = 1e10, nout = 2,
outnames = c("CONC","Prod"))
plot(ODE_dll, which = c("Prod", "D"), xlab = "time",
ylab = c("/day", "conc"), main = c("production rate","D"))