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, forcings=NULL, initforc = NULL, fcontrol=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: 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 , 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 .
The derivatives should be specified in the same order as the specification
of the state variables y .
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 defined as:
res <- function(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 derivatives. If the initial
y or dy have a names attribute, the names will be
available inside res , 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 "compiledCode"
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 derivatives
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.
If 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
print the diagnostiscs of the integration - 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 |
logical, 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 per output interval taken by 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 res and
jacres . 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 res and jacres .
|
ipar |
only when ‘dllname’ is specified: a vector with
integer values passed to the dll-functions whose names are specified
by res and jacres .
|
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 res , 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
"compiledCode" .
|
outnames |
only used if ‘dllname’ is specified and
nout > 0: the names of output variables calculated in the
compiled function res , 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 .
|
... |
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 "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 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.
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.
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
rk
, rk4
and euler
for
Runge-Kutta integrators.
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 1-D models,
ode.2D
for integrating 2-D models,
ode.3D
for integrating 3-D models,
diagnostics
to print diagnostic messages.
## ======================================================================= ## 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 ## 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 ## 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)) par(mfrow = opa) ## ======================================================================= ## 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 ## ======================================================================= ## 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 = 2*3/pars["K"]) ## Initial rate of change dy <- c(dA = 0, dB = 0, dD = 0) # production increases with time prod <- matrix(nc=2,data=c(seq(0,100,by=10),0.1*(1+runif(11)*1))) ODE_dll <- as.data.frame(daspk(y=y,dy=dy,times=times,res="chemres", dllname="deSolve", initfunc="initparms", initforc="initforcs", parms=pars, forcings=prod, atol=1e-10,rtol=1e-10,nout=2, outnames=c("CONC","Prod"))) opa <- par(mfrow = c(1,2)) plot(ODE_dll$time,ODE_dll$Prod,xlab = "time", ylab = "/day",main = "production rate",type = "l") plot(ODE_dll$time,ODE_dll$D,xlab = "time", ylab = "conc",main = "D",type = "l") par(mfrow = opa)