# Iterative steady-state solver for ordinary differential equations (ODE) and a full or banded Jacobian. Estimates the steady-state condition for a system of ordinary differential equations (ODE) written in the form: [REMOVE_ME]$$ dy/dt = f(t,y) [REMOVE_ME_2]$$ i.e. finds the values of `y` for which f(t,y) = 0. Uses a newton-raphson method, implemented in Fortran 77. The system of ODE's is written as an function or defined in compiled code that has been dynamically loaded. ## Description Estimates the steady-state condition for a system of ordinary differential equations (ODE) written in the form: $$ dy/dt = f(t,y) $$ i.e. finds the values of `y` for which f(t,y) = 0. Uses a newton-raphson method, implemented in Fortran 77. The system of ODE's is written as an function or defined in compiled code that has been dynamically loaded. ```r stode(y, time = 0, func, parms = NULL, rtol = 1e-6, atol = 1e-8, ctol = 1e-8, jacfunc = NULL, jactype = "fullint", verbose = FALSE, bandup = 1, banddown = 1, positive = FALSE, maxiter = 100, ynames = TRUE, dllname = NULL, initfunc = dllname, initpar = parms, rpar = NULL, ipar = NULL, nout = 0, outnames = NULL, forcings = NULL, initforc = NULL, fcontrol = NULL, times = time, ...) ``` ## Arguments - `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, times`: time for which steady-state is wanted; the default is `times`=0. (note- since version 1.7, 'times' has been added as an alias to 'time'). - `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 `stode()` is called. see Details for more information. - `parms`: other parameters passed to `func` and `jacfunc`. - `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. - `jacfunc`: if not `NULL`, either a user-supplied function that estimates the Jacobian of the system of differential equations dydot(i)/dy(j), or a character string giving the name of a compiled function in a dynamically loaded shared library as provided in `dllname`. In some circumstances, supplying `jacfunc` can speed up the computations. The 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 and Fortran store matrices). If the Jacobian is banded, `jacfunc` should return a matrix containing only the nonzero bands of the jacobian, (dydot/dy), rotated row-wise. - `jactype`: the structure of the Jacobian, one of "fullint", "fullusr", "bandusr", or "bandint" - either full or banded and estimated internally or by the user. - `verbose`: if `TRUE`: full output to the screen, e.g. will output the steady-state settings. - `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. - `positive`: either a logical or a vector with indices of the state variables that have to be non-negative; if `TRUE`, all state variables `y` 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` and `jacfunc`. - `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` 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 calculated 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. - `forcings`: only used if dllname is specified: a vector with the forcing function values, or 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. This feature is here for compatibility with models defined in compiled code from package deSolve; see deSolve's 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 deSolve's package vignette `"compiledCode"`. - `fcontrol`: A list of control parameters for the forcing functions. See deSolve's package vignette `"compiledCode"`. - `...`: additional arguments passed to `func` and `jacfunc` allowing this to be a generic function. ## Returns 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). ## Author(s) Karline Soetaert ## Examples ```r ## ======================================================================= ## Example 1. A simple sediment biogeochemical model ## ======================================================================= model<-function(t, y, pars) { with (as.list(c(y, pars)),{ Min = r*OM oxicmin = Min*(O2/(O2+ks)) anoxicmin = Min*(1-O2/(O2+ks))* SO4/(SO4+ks2) dOM = Flux - oxicmin - anoxicmin dO2 = -oxicmin -2*rox*HS*(O2/(O2+ks)) + D*(BO2-O2) dSO4 = -0.5*anoxicmin +rox*HS*(O2/(O2+ks)) + D*(BSO4-SO4) dHS = 0.5*anoxicmin -rox*HS*(O2/(O2+ks)) + D*(BHS-HS) list(c(dOM, dO2, dSO4, dHS), SumS = SO4+HS) }) } # parameter values pars <- c(D = 1, Flux = 100, r = 0.1, rox = 1, ks = 1, ks2 = 1, BO2 = 100, BSO4 = 10000, BHS = 0) # initial conditions y<-c(OM = 1, O2 = 1, SO4 = 1, HS = 1) # direct iteration - enforces positivitiy.. ST <- stode(y = y, func = model, parms = pars, pos = TRUE) ST ## ======================================================================= ## Example 2. 1000 simultaneous equations ## ======================================================================= model <- function (time, OC, parms, decay, ing) { # model describing organic Carbon (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 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 print( system.time( ss <- stode(runif(N), func = model, parms = NULL, positive = TRUE, decay = 5, ing = 20))) plot(ss$y[1:N], dist, ylim = rev(range(dist)), type = "l", lwd = 2, xlab = "Nonlocal exchange", ylab = "sediment depth", main = "stode, full jacobian") ## ======================================================================= ## Example 3. Solving a system of linear equations ## ======================================================================= # this example is included to demonstrate how to use the "jactype" option. # (and that stode is quite efficient). A <- matrix(nrow = 500, ncol = 500, runif(500*500)) B <- 1:500 # this is how one would solve this in R print(system.time(X1 <- solve(A, B))) # to use stode: # 1. create a function that receives the current estimate of x # and that returns the difference A%*%x-b, as a list: fun <- function (t, x, p) # t and p are dummies here... list(A%*%x-B) # 2. jfun returns the Jacobian: here this equals "A" jfun <- function (t, x, p) # all input parameters are dummies A # 3. solve with jactype="fullusr" (a full Jacobian, specified by user) print (system.time( X <- stode(y = 1:500, func = fun, jactype = "fullusr", jacfunc = jfun) )) # the results are the same (within precision) sum((X1-X$y)^2) ``` ## References 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 LINPACK code: Dongarra, J.J., J.R. Bunch, C.B. Moler and G.W. Stewart, 1979. LINPACK user's guide, SIAM, Philadelphia. ## Details The work is done by a Fortran 77 routine that implements the Newton-Raphson method. It uses code from LINPACK. The form of the Jacobian can be specified by `jactype` which can take the following values: * jactype = "fullint" : a full jacobian, calculated internally by the solver, the default. * 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 the solver; the size of the bands specified by `bandup` and `banddown`. if `jactype`= "fullusr" or "bandusr" then the user must supply a subroutine `jacfunc`. The input parameters `rtol`, `atol` and `ctol` determine the error control performed by the solver. The solver will control the vector e of estimated local errors in y , according to an inequality of the form max-norm of ( e /ewt ) $<=$ 1, where ewt is a vector of positive error weights. The values of `rtol` and `atol` should all be non-negative. The form of ewt is: $$ \mathbf{rtol} \times \mathrm{abs}(\mathbf{y}) + \mathbf{atol}\bold{rtol} * abs(\bold{y}) + \bold{atol} $$ where multiplication of two vectors is element-by-element. In addition, the solver will stop if between two iterations, the maximal change in the values of y is less than `ctol`. Models may be defined in compiled C or Fortran code, as well as in . If `func` or `jacfunc` are a string, then they are assumed to be compiled code. In this case, `dllname` must give the name of the shared library (without extension) which must be loaded before `stode()` is called. See vignette("rooSolve") for how a model has to be specified in compiled code. Also, vignette("compiledCode") from package `deSolve` contains examples of how to do this. If `func` is a user-supplied R-function , it must be called as: yprime = func(t, y, parms,...). t is the `time` at which the steady-state should be estimated, `y` is the current estimate of the variables in the ode system. 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 contains output variables whose values at steady-state are also required. An example is given below: `model<-function(t,y,pars)` `{` `with (as.list(c(y,pars)),{` `Min = r*OM` `oxicmin = Min*(O2/(O2+ks))` `anoxicmin = Min*(1-O2/(O2+ks))* SO4/(SO4+ks2` `dOM = Flux - oxicmin - anoxicmin` `dO2 = -oxicmin -2*rox*HS*(O2/(O2+ks)) + D*(BO2-O2)` `dSO4 = -0.5*anoxicmin +rox*HS*(O2/(O2+ks)) + D*(BSO4-SO4)` `dHS = 0.5*anoxicmin -rox*HS*(O2/(O2+ks)) + D*(BHS-HS)` `list(c(dOM,dO2,dSO4,dHS),SumS=SO4+HS)` `})` `}` This model can be solved as follows: `pars <- c(D=1,Flux=100,r=0.1,rox =1,` `ks=1,ks2=1,BO2=100,BSO4=10000,BHS = 0)` `y<-c(OM=1,O2=1,SO4=1,HS=1)` `ST <- stode(y=y,func=model,parms=pars,pos=TRUE))` ## See Also `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. `stodes`, iterative steady-state solver for ODEs with arbitrary sparse Jacobian. `runsteady`, steady-state solver by dynamically running to steady-state ## Note The implementation of `stode` and substantial parts of the help file is similar to the implementation of the integration routines (e.g. `lsode`) from package deSolve.

stode function