daspk {deSolve} | R Documentation |
Solves either:
y' = f(t,y,...)
or
F(t,y,y') = 0
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.
daspk(y, times, func = NULL, parms, dy = NULL, res = NULL, nalg = 0, rtol = 1e-6, atol = 1e-8, jacfunc = NULL, jacres = NULL, jactype = "fullint", 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, ...)
y |
the initial (state) values for the DE system. If y
has a name attribute, the names will be used to label the output
matrix.
|
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 |
cannot be used if the model is a DAE system. If an ODE
system, func should be an R-function that computes the
values of the derivatives in the ODE system (the model
definition) at time t.
func must be defined as: yprime = func(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 , unless
ynames is FALSE. 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 .
Note that it is not possible to define func as a compiled
function in a dynamically loaded shared library. Use res
instead.
|
parms |
vector or list of parameters used in func ,
jacfunc , or res
|
dy |
the initial derivatives of the state variables of the DE system. Ignored if an ODE. |
res |
if a DAE system: either an R-function that computes the
residual function F(t,y,y') of the DAE system (the model
defininition) at time t , or a character string giving the
name of a compiled function in a dynamically loaded shared library.
If res is a user-supplied R-function, it must be called as:
F = func(t, y, dy, parms, ...) .
Here t is the current time point in the integration, y
is the current estimate of the variables in the ODE system,
dy are the corresponding rates of change. If the initial
y or dy have a names attribute, the names will be
available inside func , unless ynames is FALSE .
parms is a vector of parameters.
The return value of res should be a list, whose first element
is a vector containing the residuals of the DAE system,
i.e. delta = F(t,y,y'), and whose next elements contain output
variables that are required at each point in times .
If res is a string, then dllname must give the name of
the shared library (without extension) which must be loaded before
daspk() is called (see package vignette for more
information).
|
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 estini = 1.
|
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 NULL , an R function that computes the
jacobian of the system of differential equations. Only used in case
the system is an ODE (y' = f(t,y)), specified by func . 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.
|
jacres |
jacres and not jacfunc should be used if
the system is specified by the residual function F(t,y,y'),
i.e. jacres is used in conjunction with res .
If jacres is an R-function, the calling sequence for
jacres is identical to that of res , but with extra
parameter cj . Thus it should be called as: jacres =
func(t, y, dy, parms, cj, ...) . Here t is the current time
point in the integration, y is the current estimate of the
variables in the ODE system, y' are the corresponding rates of
change and cj is a scalar, which is normally proportional to
the inverse of the stepsize. If the initial y or dy have a names
attribute, the names will be available inside jacres, unless
ynames is FALSE . parms is a vector of
parameters (which may have a names attribute).
If the jacobian is a full matrix, jacres should return the
matrix dG/dy + cj*dG/dyprime, where the ith row is the sum of the
derivatives of G_i with respect to y_j and the scaled
derivatives of G_i with respect to dy_j.
Ff the jacobian is banded, jacres should return only the
nonzero bands of the jacobian, rotated rowwise. See details for the
calling sequence when jacres is a string.
|
jactype |
the structure of the jacobian, one of
"fullint" , "fullusr" , "bandusr" or
"bandint" - either full or banded and estimated internally or
by the user.
|
estini |
only if a DAE system, and if initial values of y
and dy are not consistent (i.e. F(t,y,dy) is not = 0), setting
estini = 1 or 2, will solve for them. If estini = 1: dy
and the algebraic variables are estimated from y ; in this
case, the number of algebraic equations must be given (nalg ).
If estini = 2: y will be estimated from dy .
|
verbose |
if TRUE: full output to the screen, e.g. will output the settings of vector istate and rstate - see details. |
tcrit |
the Fortran routine daspk 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 |
if FALSE: names of state variables are not passed to
function func ; this may speed up the simulation especially
for large models.
|
maxord |
the maximum order to be allowed. Reduce maxord
to save storage space ( <= 5)
|
bandup |
number of non-zero bands above the diagonal, in case
the jacobian is banded (and jactype one of
"bandint","bandusr")
|
banddown |
number of non-zero bands below the diagonal, in case
the jacobian is banded (and jactype one of
"bandint","bandusr")
|
maxsteps |
maximal number of steps during one call to the
solver; will be recalculated to be at least 500 and a multiple of
500; the solver will give a warning if more than 500 steps are
taken, but it will continue till maxsteps steps.
|
dllname |
a string giving the name of the shared library
(without extension) that contains all the compiled function or
subroutine definitions referred to in func /or res and
jacfunc . See package vignette.
|
initfunc |
if not NULL, the name of the initialisation function (which initialises values of parameters), as provided in ‘dllname’. See package vignette. |
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 calculed in the dll - you have
to perform this check in the code - See package vignette.
|
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.
|
... |
additional arguments passed to func ,
jacfunc , res and jacres , allowing this to be a
generic function.
|
The daspk solver uses the backward differentiation formulas of orders
one through five (specified with maxord
) to solve either:
y' = f(t,y,...)
for y = Y, or
F(t,y,y') = 0
for y = Y and y' = YPRIME.
ODEs are specified in func
, DAEs are specified in res
.
If a DAE system, Values for Y and YPRIME at the initial time
must be given as input. Ideally,these values should be consistent,
that is, if T, Y, YPRIME are the given initial values, they should
satisfy F(T,Y,YPRIME) = 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:
daspk
, the default,
jacfunc
or jacres
,
jacfunc
or jacres
; the size of the bands
specified by bandup
and banddown
,
daspk
; the size of the bands specified by bandup
and
banddown
.
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.
res and jacres may be defined in compiled C or Fortran code, as
well as in an R-function. See package vignette for details. Examples
in Fortran are in the ‘dynload’ subdirectory of the
deSolve
package directory.
The output will have the attributes istate, and rstate, two vectors with several useful elements.
If verbose
= TRUE, the settings of istate and rstate will be
written to the screen.
The following elements of istate are meaningful:
For more information about possible options, see the comments in the original code daspk.f.
A matrix 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.
The output will have the attributes istate
, and rstate
,
two vectors with several useful elements. See details. The first
element of istate returns the conditions under which the last call to
lsoda returned. Normal is istate[1] = 2
. If verbose
=
TRUE, the settings of istate and rstate will be written to the screen.
In this version, the krylov method is not (yet) supported.
Karline Soetaert <k.soetaert@nioo.knaw.nl>
L. R. Petzold, A Description of DASSL: A Differential/Algebraic System Solver, in Scientific Computing, R. S. Stepleman et al. (Eds.), North-Holland, Amsterdam, 1983, pp. 65-68.
K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic 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. 40-91.
P. N. Brown, A. C. Hindmarsh, and L. R. Petzold, Using Krylov Methods in the Solution of Large-Scale Differential-Algebraic Systems, SIAM J. Sci. Comp., 15 (1994), pp. 1467-1488.
P. N. Brown, A. C. Hindmarsh, and L. R. Petzold, Consistent Initial Condition Calculation for Differential-Algebraic Systems, LLNL Report UCRL-JC-122175, August 1995; submitted to SIAM J. Sci. Comp.
Netlib: http://www.netlib.org
lsoda
, lsodar
,
lsode
, lsodes
, vode
## ==================================================== ## 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 1s-t order rate, r ## ==================================================== ## Dissociation constant K <- 1 ## parameters pars <- c( ka = 1e6, # forward rate r = 1, prod = 0.1) ## ==================================== ## Chemical problem formulation 1: ODE ## ==================================== 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 ## dB-dA = -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)) }) } 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 <- as.data.frame(daspk(y = y, times = times, func = Fun_ODE, parms = pars, atol = 1e-10, rtol = 1e-10)) ## Initial rate of change dy <- c(dA = 0, dB = 0, dD = 0) ## DAE model solved with daspk DAE <- as.data.frame(daspk(y = y, dy = dy, times = times, res = Res_DAE, parms = pars, atol = 1e-10, rtol = 1e-10)) ## ================ ## plotting output ## ================ opa <- par(mfrow = c(2,2)) for (i in 2:5) { plot(ODE$time,ODE[,i],xlab = "time", ylab = "conc",main = names(ODE)[i],type = "l") points(DAE$time,DAE[,i],col = "red") } legend("bottomright",lty = c(1,NA),pch = c(NA,1), col = c("black","red"),legend = c("ODE","DAE")) # difference between both implementations: max(abs(ODE-DAE)) ## ====================================== ## 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(nc = 3, nr = 3, 0) DAE2 <- as.data.frame(daspk(y = y, dy = dy, times = times, res = Res_DAE, jacres = jacres_DAE, jactype = "fullusr", parms = pars, atol = 1e-10, rtol = 1e-10)) max(abs(DAE-DAE2)) ## See \dynload subdirectory for a FORTRAN implementation of this model