stodes {rootSolve} | R Documentation |
Estimates the steady-state condition for a system of ordinary differential equations (ODE) in the form:
dy/dt = f(t,y)
and where the jacobian matrix df/dy has an arbitrary sparse structure.
Uses a newton-raphson method, implemented in Fortran.
The system of ODE's is written as an R function or defined in compiled code that has been dynamically loaded.
stodes(y, time=0, func, parms=NULL, rtol=1e-6, atol=1e-8, ctol=1e-8, sparsetype="sparseint",verbose=FALSE, nnz=NULL, inz=NULL, lrw=NULL, ngp=NULL, positive=FALSE, maxiter=100, ynames=TRUE, dllname=NULL, initfunc=dllname, initpar=parms, rpar=NULL, ipar=NULL, nout=0, outnames = NULL, ...)
y |
the initial guess of (state) values for the ode system, a vector.
If y has a name attribute, the names will be used to label the
output matrix.
|
time |
time for which steady-state is wanted; the default is
time =0.
|
func |
either a user-supplied function that computes the values of the
derivatives in the ode system (the model definition) at time
time , or a character string giving the name of a
compiled function in a dynamically loaded shared library.
If func is a user-supplied function, it must be called as:
yprime = func(t, y, parms) . t is the time point
at which the steady-state is wanted, 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 of parameters (which may have a names attribute).
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 (possibly with a
names attribute) are global values that are required as
output.
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 stodes() is called. see Details for more information.
|
parms |
other parameters passed to func .
|
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. |
ctol |
if between two iterations, the maximal change in y is less than this amount, steady-state is assumed to be reached. |
sparsetype |
the sparsity pattern, to date only "sparseint", sparse jacobian, estimated internally by stodes. |
verbose |
if TRUE: full output to the screen, e.g. will output the steady-state settings. |
nnz |
the number of nonzero elements in the sparse Jacobian (if
this is unknown, use an estimate); If NULL, a guess will be made, and
if not sufficient, stodes will return with a message indicating
the size actually required.
If a solution is found, the minimal value of nnz actually required
is returned by the solver (1st element of attribute dims ).
|
inz |
(row,column) indices to the nonzero elements in the sparse
Jacobian. If this is NULL, the sparsity will be determined by stodes .
|
lrw |
the length of the work array of solver; due to the sparsicity,
this cannot be readily predicted. If NULL, a guess will be made, and
if not sufficient, stodes will return with a message indicating
the size actually required. Therefore, some experimentation may be
necessary to estimate the value of lrw
If a solution is found, the minimal value of lrw actually required
is returned by the solver (3rd element of attribute dims ).
|
ngp |
number of groups of independent state variables. Due to the
sparsicity, this cannot be readily predicted. If NULL, a guess will be
made, and if not sufficient, stodes will return with a message
indicating the size actually required. Therefore, some experimentation
may be necessary to estimate the value of ngp
If a solution is found, the minimal value of ngp actually required
is returned by the solver (2nd element of attribute dims .
|
positive |
either a logical or a vector with indices of the state variables that have to be non-negative; if TRUE, the state variables are forced to be non-negative numbers. |
maxiter |
maximal number of iterations during one call to 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.
|
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 .
|
initfunc |
if not NULL, the name of the initialisation function (which initialises values of parameters), as provided in ‘dllname’. See details. |
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 .
|
ipar |
only when ‘dllname’ is specified: a vector with integer
values passed to the dll-functions whose names are specified by func .
|
nout |
only used if ‘dllname’ is specified: the number of output
variables calculated in the compiled function func , present in the
shared library.
|
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 allowing this to be
a generic function.
|
The work is done by a Fortran 77 routine that implements the Newton-Raphson method.
stodes
is to be used for problems, where the Jacobian has a sparse
structure.
There are two choices for the sparsity specification, depending on whether
inz
is present.
inz
is present, the sparsity is determined by
the user. inz
should contain indices (row, column) to the
nonzero elements in the Jacobian matrix.
In this case, nnz
will be set equal to the number of rows
in inz
.
inz
is NOT present, the sparsity is estimated
by the solver, based on numerical differences.
In this case, it is advisable to provide an estimate of the number
of non-zero elements in the Jacobian (nnz
).
This value can be approximate; upon return the number of nonzero
elements actually required will be known (1st element of attribute
dims
).
Either way, the Jacobian itself is always generated by the solver (i.e. there is no provision to provide an analytic Jacobian).
This is done by perturbing simulataneously a combination of state variables that do not affect each other.
This significantly reduces computing time. The number of groups with
independent state variables can be given by ngp
The input parameters rtol
, atol
and ctol
determine
the error control performed by the solver. See help for stode
for details.
Models may be defined in compiled C or Fortran code, as well as in R. See package vignette for details on how to write models in compiled code.
A list containing
y |
a vector with the state variable values from the last iteration
during estimation of steady-state condition of the system of equations.
If y has a names attribute, it will be used to label the output
values.
|
... |
the number of "global" values returned. |
The output will have the attribute steady
, which returns TRUE
,
if steady-state has been reached and the attribute precis
with an
estimate of the precision attained during each iteration, the mean absolute
rate of change (sum(abs(dy))/n).
Karline Soetaert <k.soetaert@nioo.knaw.nl>
For a description of the Newton-Raphson method, e.g.
Press, WH, Teukolsky, SA, Vetterling, WT, Flannery, BP, 1996. Numerical Recipes in FORTRAN. The Art of Scientific computing. 2nd edition. Cambridge University Press.
The algorithm uses linear algebra routines from the Yale sparse matrix package:
Eisenstat, S.C., Gursky, M.C., Schultz, M.H., Sherman, A.H., 1982. Yale Sparse Matrix Package. i. The symmetric codes. Int. J. Num. meth. Eng. 18, 1145-1151.
steady
, for a general interface to most of the steady-state
solvers
steady.band
, to find the steady-state of ODE models with a
banded Jacobian
steady.1D
, steady.2D
,
steady.3D
, steady-state solvers for 1-D, 2-D and 3-D
partial differential equations.
stode
, iterative steady-state solver for ODEs with full
or banded Jacobian.
runsteady
, steady-state solver by dynamically running to
steady-state
## ======================================================================= ## 1000 simultaneous equations ## ======================================================================= model <- function (time,OC,parms,decay,ing) { # model describing C in a sediment, # Upper boundary = imposed flux, lower boundary = zero-gradient Flux <- v * c(OC[1] ,OC) + # advection -Kz*diff(c(OC[1],OC,OC[N]))/dx # diffusion; Flux[1]<- flux # imposed flux # Rate of change= Flux gradient and first-order consumption dOC <- -diff(Flux)/dx - decay*OC # Fraction of OC in first 5 layers is translocated to mean depth # (layer N/2) dOC[1:5] <- dOC[1:5] - ing*OC[1:5] dOC[N/2] <- dOC[N/2] + ing*sum(OC[1:5]) list(dOC) } v <- 0.1 # cm/yr flux <- 10 dx <- 0.01 N <- 1000 dist <- seq(dx/2,by=dx,len=N) Kz <- 1 #bioturbation (diffusion), cm2/yr ss <- stodes(runif(N),func=model,parms=NULL, positive=TRUE, decay=5,ing=20,verbose=TRUE) plot(ss$y[1:N],dist,ylim=rev(range(dist)),type="l",lwd=2, xlab="Nonlocal exchange",ylab="sediment depth", main="stodes, sparse jacobian")