vode {deSolve} | R Documentation |
Solves the initial value problem for stiff or nonstiff systems of ordinary differential equations (ODE) in the form:
dy/dt = f(t,y)
The R function vode
provides an interface to the Fortran ODE
solver of the same name, written by Peter N. Brown, Alan C. Hindmarsh
and George D. Byrne.
The system of ODE's is written as an R function or be defined in compiled code that has been dynamically loaded.
In contrast to lsoda
, the user has to specify whether or
not the problem is stiff and choose the appropriate solution method.
vode
is very similar to lsode
, but uses a
variable-coefficient method rather than the fixed-step-interpolate
methods in lsode
. In addition, in vode it is possible
to choose whether or not a copy of the Jacobian is saved for reuse in
the corrector iteration algorithm; In lsode
, a copy is not
kept.
vode(y, times, func, parms, rtol = 1e-6, atol = 1e-8, jacfunc = NULL, jactype = "fullint", mf = NULL, verbose = FALSE, tcrit = NULL, hmin = 0, hmax = NULL, hini = 0, ynames = TRUE, maxord = NULL, 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 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 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:
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 .
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
vode() 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 dydot(i)/dy(j), or
a string giving the name of a function or subroutine in
‘dllname’ that computes the jacobian (see Details below 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 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; overruled if mf is not NULL .
|
mf |
the "method flag" passed to function vode - overrules
jactype - provides more options than jactype - see
details.
|
verbose |
if TRUE: full output to the screen, e.g. will output the settings of vectors istate and rstate - see details. |
tcrit |
if not NULL , then vode cannot integrate
past tcrit . The Fortran routine dvode 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, 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 multi-D models.
|
maxord |
the maximum order to be allowed. NULL uses the default,
i.e. order 12 if implicit Adams method (meth = 1), order 5 if BDF
method (meth = 2). 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 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 func , present in the shared library.
|
... |
additional arguments passed to func and
jacfunc allowing this to be a generic function.
|
Before using the integrator vode
, the user has to decide
whether or not the problem is stiff.
If the problem is nonstiff, use method flag mf
= 10, which
selects a nonstiff (Adams) method, no Jacobian used.
If the problem is stiff, there are four standard choices which can be
specified with jactype
or mf
.
The options for jactype are
mf
= 22,
jacfunc
, corresponds to mf
= 21,
jacfunc
; the size of the bands specified by
bandup
and banddown
, corresponds to mf
= 24,
bandup
and banddown
,
corresponds to mf
= 25.
More options are available when specifying mf directly.
The legal values of mf
are 10, 11, 12, 13, 14, 15, 20, 21, 22,
23, 24, 25, -11, -12, -14, -15, -21, -22, -24, -25.
mf
is a signed two-digit integer, mf = JSV*(10*METH +
MITER)
, where
MITER = 1 means chord iteration with a user-supplied full (NEQ by NEQ) Jacobian.
MITER = 2 means chord iteration with an internally generated
(difference quotient) full Jacobian (using NEQ extra calls to
func
per df/dy value).
MITER = 3 means chord iteration with an internally generated
diagonal Jacobian approximation (using 1 extra call to func
per df/dy evaluation).
MITER = 4 means chord iteration with a user-supplied banded Jacobian.
MITER = 5 means chord iteration with an internally generated
banded Jacobian (using ML+MU+1 extra calls to func
per
df/dy evaluation).
If MITER = 1 or 4, the user must supply a subroutine jacfunc
.
The example for integrator lsode
demonstrates how to
specify both a banded and full jacobian.
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, vode will return an
error code. See lsoda
for details.
Models may be defined in compiled C or Fortran code, as well as
in an R-function. See package vignette "compiledCode"
for details.
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:
atol
or
atol(i)
= 0).
For more information, see the comments in the original code dvode.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
, plus an
additional column (the first) for the time value. There will be one
row for each element in times
unless the Fortran routine `vode'
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
vode returned. Normal is istate[1] = 2
. If verbose =
TRUE
, the settings of istate and rstate will be written to the
screen.
Karline Soetaert <k.soetaert@nioo.knaw.nl>
P. N. Brown, G. D. Byrne, and A. C. Hindmarsh, 1989. VODE: A Variable
Coefficient ODE Solver, SIAM J. Sci. Stat. Comput., 10, pp. 1038-1051.
Also, LLNL Report UCRL-98412, June 1988.
G. D. Byrne and A. C. Hindmarsh, 1975. A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations. ACM Trans. Math. Software, 1, pp. 71-96.
A. C. Hindmarsh and G. D. Byrne, 1977. EPISODE: An Effective Package for the Integration of Systems of Ordinary Differential Equations. LLNL Report UCID-30112, Rev. 1.
G. D. Byrne and A. C. Hindmarsh, 1976. EPISODEB: An Experimental Package for the Integration of Systems of Ordinary Differential Equations with Banded Jacobians. LLNL Report UCID-30132, April 1976.
A. C. Hindmarsh, 1983. ODEPACK, a Systematized Collection of ODE Solvers. in Scientific Computing, R. S. Stepleman et al., eds., North-Holland, Amsterdam, pp. 55-64.
K. R. Jackson and R. Sacks-Davis, 1980. An Alternative Implementation of Variable Step-Size Multistep Formulas for Stiff ODEs. ACM Trans. Math. Software, 6, pp. 295-318.
Netlib: http://www.netlib.org
rk
, rk4
and euler
for
Runge-Kutta integrators.
lsoda
, lsode
,
lsodes
, lsodar
,
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.
## The famous Lorenz equations: chaos in the earth's atmosphere ## Lorenz 1963. J. Atmos. Sci. 20, 130-141. chaos <- function(t, state, parameters) { with(as.list(c(state)),{ dx <- -8/3*x+y*z dy <- -10*(y-z) dz <- -x*y+28*y-z list(c(dx, dy, dz)) }) } state <- c(x = 1, y = 1, z = 1) times <- seq(0, 100, 0.01) out <- as.data.frame(vode(state, times, chaos, 0)) plot(out$x, out$y, type = "l", main = "Lorenz butterfly")