the initial (state) values for the DAE or ODE 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
either an R-function that computes the values of the
derivatives in the DAE or 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
should 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
gamd() is called. See deSolve package vignette "compiledCode"
for more details.
parms
vector or list of parameters used in func or
jacfunc.
nind
if a DAE system: a three-valued 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.
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 dydot(i)/dy(j), or
a string giving the name of a function or subroutine in
‘dllname’ that computes the Jacobian (see vignette
"compiledCode" from package deSolve, 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 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.
jactype
the structure of the Jacobian, one of
"fullint", "fullusr", "bandusr" or
"bandint" - either full or banded and estimated internally or
by user.
mass
the mass matrix.
If not NULL, the problem is a linearly
implicit DAE and defined as M dy/dt = f(t,y).
If the mass-matrix M is full, it should be of dimension
n*n where n is the number of y-values;
if banded the number of rows should be less than n,
and the mass-matrix is stored diagonal-wise with element (i, j)
stored in mass(i - j + mumas + 1, j).
If mass = NULL then the model is an ODE (default)
massup
number of non-zero bands above the diagonal of the mass
matrix, in case it is banded.
massdown
number of non-zero bands below the diagonal of the mass
matrix, in case it is banded.
verbose
if TRUE: full output to the screen, e.g. will
print the diagnostiscs of the integration - see details.
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 set equal to 1e-6. Usually 1e-3 to 1e-5 is good for stiff equations
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.
minord
the minimum order to be allowed, >= 3 and <= 9.
NULL uses the default, 3.
maxord
the maximum order to be allowed, >= minord
and <= 9. NULL uses the default, 9.
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 taken by the solver,
for the entire integration. This is different from the settings
of this argument in the solvers from package deSolve!
maxnewtit
A four-valued vector, with the maximal number of
splitting-Newton iterations for the solution of the iplicit system
in each step for order 3, 5, 7 and 9 respectively.
The default is c(10,18,26,36).
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 vignette "compiledCode"
from package deSolve.
initfunc
if not NULL, the name of the initialisation function
(which initialises values of parameters), as provided in
‘dllname’. See vignette "compiledCode"
from package deSolve.
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 vignette "compiledCode"
from package deSolve.
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.
...
additional arguments passed to func and
jacfunc allowing this to be a generic function.
Details
The work is done by the FORTRAN 90 subroutine gamd, whose
documentation should be consulted for details. The implementation
is based on the Fortran 90 version from 2007/24/05.
There are four standard choices for the jacobian which can be specified with
jactype.
The options for jactype are
jactype = "fullint"
a full Jacobian, calculated internally by
the solver.
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"
a banded Jacobian, calculated by gamd;
the size of the bands specified by bandup and
banddown.
Inspection of the example below shows how to specify both a banded and
full Jacobian.
The input parameters rtol, and atol determine the
error control performed by the solver, which roughly keeps the
local error of y(i) below rtol(i)*abs(y(i))+atol(i).
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") from the deSolve package 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" from package
deSolve for details.
Information about linking forcing functions to compiled code is in
forcings (from package deSolve).
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 ‘gamd’
returns with an unrecoverable error. If y has a names
attribute, it will be used to label the columns of the output value.
Author(s)
Karline Soetaert <karline.soetaert@nioz.nl>
Francesca Mazzia
References
L.Brugnano, D.Trigiante,
Solving Differential Problems by Multistep Initial and Boundary Value Methods,
Gordon & Breach, Amsterdam, 1998.
F.Iavernaro, F.Mazzia,
Block-Boundary Value Methods for the solution of Ordinary Differential Equation.
Siam J. Sci. Comput. 21 (1) (1999) 323–339.
F.Iavernaro, F.Mazzia,
Solving Ordinary Differential Equations by Generalized Adams Methods:
properties and implementation techniques, proceedings of NUMDIFF8,
Appl. Num. Math. 28 (2-4) (1998) 107-126.
See Also
bimd another DAE solver from package deTestSet,
mebdfi another DAE solver from package deTestSet,
daspk another DAE solver from package deSolve,
ode for a general interface to most of the ODE solvers
from package deSolve,
ode.1D for integrating 1-D models,
ode.2D for integrating 2-D models,
ode.3D for integrating 3-D models,
mebdfi for integrating DAE models,
dopri853 for the Dormand-Prince Runge-Kutta method of order 8(53)
diagnostics to print diagnostic messages.
Examples
## =======================================================================
## Example 1:
## Various ways to solve the same model.
## =======================================================================
## the model, 5 state variables
f1 <- function (t, y, parms)
{
ydot <- vector(len = 5)
ydot[1] <- 0.1*y[1] -0.2*y[2]
ydot[2] <- -0.3*y[1] +0.1*y[2] -0.2*y[3]
ydot[3] <- -0.3*y[2] +0.1*y[3] -0.2*y[4]
ydot[4] <- -0.3*y[3] +0.1*y[4] -0.2*y[5]
ydot[5] <- -0.3*y[4] +0.1*y[5]
return(list(ydot))
}
## the Jacobian, written as a full matrix
fulljac <- function (t, y, parms)
{
jac <- matrix(nrow = 5, ncol = 5, byrow = TRUE,
data = c(0.1, -0.2, 0 , 0 , 0 ,
-0.3, 0.1, -0.2, 0 , 0 ,
0 , -0.3, 0.1, -0.2, 0 ,
0 , 0 , -0.3, 0.1, -0.2,
0 , 0 , 0 , -0.3, 0.1) )
return(jac)
}
## the Jacobian, written in banded form
bandjac <- function (t, y, parms)
{
jac <- matrix(nrow = 3, ncol = 5, byrow = TRUE,
data = c( 0 , -0.2, -0.2, -0.2, -0.2,
0.1, 0.1, 0.1, 0.1, 0.1,
-0.3, -0.3, -0.3, -0.3, 0) )
return(jac)
}
## initial conditions and output times
yini <- 1:5
times <- 1:20
## default: stiff method, internally generated, full Jacobian
out <- gamd(yini, times, f1, parms = 0, jactype = "fullint")
plot(out)
## stiff method, user-generated full Jacobian
out2 <- gamd(yini, times, f1, parms = 0, jactype = "fullusr",
jacfunc = fulljac)
## stiff method, internally-generated banded Jacobian
## one nonzero band above (up) and below(down) the diagonal
out3 <- gamd(yini, times, f1, parms = 0, jactype = "bandint",
bandup = 1, banddown = 1)
## stiff method, user-generated banded Jacobian
out4 <- gamd(yini, times, f1, parms = 0, jactype = "bandusr",
jacfunc = bandjac, bandup = 1, banddown = 1)
## =======================================================================
## Example 2:
## stiff problem from chemical kinetics
## =======================================================================
Chemistry <- function (t, y, p) {
dy1 <- -.04*y[1] + 1.e4*y[2]*y[3]
dy2 <- .04*y[1] - 1.e4*y[2]*y[3] - 3.e7*y[2]^2
dy3 <- 3.e7*y[2]^2
list(c(dy1,dy2,dy3))
}
times <- 10^(seq(0, 10, by = 0.1))
yini <- c(y1 = 1.0, y2 = 0, y3 = 0)
out <- gamd(func = Chemistry, times = times, y = yini, parms = NULL)
plot(out, log = "x", type = "l", lwd = 2)
## =============================================================================
## Example 3: DAE
## Car axis problem, index 3 DAE, 8 differential, 2 algebraic equations
## from
## F. Mazzia and C. Magherini. Test Set for Initial Value Problem Solvers,
## release 2.4. Department
## of Mathematics, University of Bari and INdAM, Research Unit of Bari,
## February 2008.
## Available at http://www.dm.uniba.it/~testset.
## =============================================================================
## Problem is written as M*y = f(t,y,p).
## caraxisfun implements the right-hand side:
caraxisfun <- function(t, y, parms) {
with(as.list(y), {
yb <- r * sin(w * t)
xb <- sqrt(L * L - yb * yb)
Ll <- sqrt(xl^2 + yl^2)
Lr <- sqrt((xr - xb)^2 + (yr - yb)^2)
dxl <- ul; dyl <- vl; dxr <- ur; dyr <- vr
dul <- (L0-Ll) * xl/Ll + 2 * lam2 * (xl-xr) + lam1*xb
dvl <- (L0-Ll) * yl/Ll + 2 * lam2 * (yl-yr) + lam1*yb - k * g
dur <- (L0-Lr) * (xr-xb)/Lr - 2 * lam2 * (xl-xr)
dvr <- (L0-Lr) * (yr-yb)/Lr - 2 * lam2 * (yl-yr) - k * g
c1 <- xb * xl + yb * yl
c2 <- (xl - xr)^2 + (yl - yr)^2 - L * L
list(c(dxl, dyl, dxr, dyr, dul, dvl, dur, dvr, c1, c2))
})
}
eps <- 0.01; M <- 10; k <- M * eps^2/2;
L <- 1; L0 <- 0.5; r <- 0.1; w <- 10; g <- 1
yini <- c(xl = 0, yl = L0, xr = L, yr = L0,
ul = -L0/L, vl = 0,
ur = -L0/L, vr = 0,
lam1 = 0, lam2 = 0)
# the mass matrix
Mass <- diag(nrow = 10, 1)
Mass[5,5] <- Mass[6,6] <- Mass[7,7] <- Mass[8,8] <- M * eps * eps/2
Mass[9,9] <- Mass[10,10] <- 0
Mass
# index of the variables: 4 of index 1, 4 of index 2, 2 of index 3
index <- c(4, 4, 2)
times <- seq(0, 3, by = 0.01)
out <- gamd(y = yini, mass = Mass, times = times, func = caraxisfun,
parms = NULL, nind = index)
plot(out, which = 1:4, type = "l", lwd = 2)