R: Solver for Ordinary Differential Equations (ODE) With Sparse...

lsodes {deSolve} | R Documentation

## Solver for Ordinary Differential Equations (ODE) With Sparse Jacobian

### Description

Solves the initial value problem for stiff systems 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.

The R function `lsodes` provides an interface to the FORTRAN ODE solver of the same name, written by Alan C. Hindmarsh and Andrew H. Sherman.

The system of ODE's is written as an R function or be defined in compiled code that has been dynamically loaded.

### Usage

    lsodes(y, times, func, parms, rtol = 1e-6, atol = 1e-6,
      jacvec = NULL, sparsetype = "sparseint", nnz = NULL,
      inz = NULL,  rootfunc = NULL,
      verbose = FALSE, nroot = 0, tcrit = NULL, hmin = 0,
      hmax = NULL, hini = 0, ynames = TRUE, maxord = NULL,
      maxsteps = 5000, lrw = NULL, liw = NULL, dllname = NULL,
      initfunc = dllname, initpar = parms, rpar = NULL,
      ipar = NULL, nout = 0, outnames = NULL, forcings=NULL,
      initforc = NULL, fcontrol=NULL, events=NULL, lags = NULL,
      ...)

### Arguments

`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: `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 must 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 `lsodes()` 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.
`jacvec` | if not `NULL`, an R function that computes a column of 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 column of the Jacobian (see vignette `"compiledCode"` for more about this option). The R calling sequence for `jacvec` is identical to that of `func`, but with extra parameter `j`, denoting the column number. Thus, `jacvec` should be called as: `jacvec = func(t, y, j, parms)` and `jacvec` should return a vector containing column `j` of the Jacobian, i.e. its i-th value is dydot(i)/dy(j). If this function is absent, `lsodes` will generate the Jacobian by differences.
`sparsetype` | the sparsity structure of the Jacobian, one of "sparseint" or "sparseusr", "sparsejan", ..., The sparsity can be estimated internally by lsodes (first option) or given by the user (last two). See details.
`nnz` | the number of nonzero elements in the sparse Jacobian (if this is unknown, use an estimate).
`inz` | if `sparsetype` equal to "sparseusr", a two-columned matrix with the (row, column) indices to the nonzero elements in the sparse Jacobian. If `sparsetype` = "sparsejan", a vector with the elements ian followed by he elements jan as used in the lsodes code. See details. In all other cases, ignored.
`rootfunc` | if not `NULL`, an R function that computes the function whose root has to be estimated or a string giving the name of a function or subroutine in ‘dllname’ that computes the root function. The R calling sequence for `rootfunc` is identical to that of `func`. `rootfunc` should return a vector with the function values whose root is sought.
`verbose` | if `TRUE`: full output to the screen, e.g. will print the `diagnostiscs` of the integration - see details.
`nroot` | only used if ‘dllname’ is specified: the number of constraint functions whose roots are desired during the integration; if `rootfunc` is an R-function, the solver estimates the number of roots.
`tcrit` | if not `NULL`, then `lsodes` cannot integrate past `tcrit`. The FORTRAN routine `lsodes` 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 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.
`maxsteps` | maximal number of steps per output interval taken by the solver.
`lrw` | the length of the real work array rwork; due to the sparsicity, this cannot be readily predicted. If `NULL`, a guess will be made, and if not sufficient, `lsodes` will return with a message indicating the size of rwork actually required. Therefore, some experimentation may be necessary to estimate the value of `lrw`. For instance, if you get the error:
    DLSODES- RWORK length is insufficient to proceed.
      Length needed is .ge. LENRW (=I1), exceeds LRW (=I2)
      In above message,  I1 =     27627   I2 =     25932
set `lrw` equal to 27627 or a higher value
`liw` | the length of the integer work array iwork; due to the sparsicity, this cannot be readily predicted. If `NULL`, a guess will be made, and if not sufficient, `lsodes` will return with a message indicating the size of iwork actually required. Therefore, some experimentation may be necessary to estimate the value of `liw`.
`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 calculated 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. 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`.
`events` | A matrix or data frame that specifies events, i.e. when the value of a state variable is suddenly changed. See events for more information.
`lags` | A list that specifies timelags, i.e. the number of steps that has to be kept. To be used for delay differential equations. See timelags, dede for more information.
`...` | additional arguments passed to `func` and `jacfunc` allowing this to be a generic function.

### Details

The work is done by the FORTRAN subroutine `lsodes`, whose documentation should be consulted for details (it is included as comments in the source file ‘src/opkdmain.f’). The implementation is based on the November, 2003 version of lsodes, from Netlib.

`lsodes` is applied for stiff problems, where the Jacobian has a sparse structure.

There are several choices depending on whether `jacvec` is specified and depending on the setting of `sparsetype`.

If function `jacvec` is present, then it should return the j-th column of the Jacobian matrix.

There are also several choices for the sparsity specification, selected by argument `sparsetype`.

- `sparsetype` = `"sparseint"`. 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`). In this case, `inz` need not be specified.

- `sparsetype` = `"sparseusr"`. The sparsity is determined by the user. In this case, `inz` should be a `matrix`, containing indices (row, column) to the nonzero elements in the Jacobian matrix. The number of nonzeros `nnz` will be set equal to the number of rows in `inz`.

- `sparsetype` = `"sparsejan"`. The sparsity is also determined by the user. In this case, `inz` should be a `vector`, containting the `ian` and `jan` elements of the sparse storage format, as used in the sparse solver. Elements of `ian` should be the first `n+1` elements of this vector, and contain the starting locations in `jan` of columns 1.. n. `jan` contains the row indices of the nonzero locations of the Jacobian, reading in columnwise order. The number of nonzeros `nnz` will be set equal to the length of `inz` - (n+1).

- `sparsetype` = `"1D"`, `"2D"`, `"3D"`. The sparsity is estimated by the solver, based on numerical differences. Assumes finite differences in a 1D, 2D or 3D regular grid - used by functions `ode.1D`, `ode.2D`, `ode.3D`. Similar are `"2Dmap"`, and `"3Dmap"`, which also include a mapping variable (passed in nnz).

The input parameters `rtol`, and `atol` determine the error control performed by the solver. See `lsoda` for details.

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 ‘doc/examples/dynload’ subdirectory of the `deSolve` package directory.

`lsodes` can find the root of at least one of a set of constraint functions `rootfunc` of the independent and dependent variables. It then returns the solution at the root if that occurs sooner than the specified stop condition, and otherwise returns the solution according the specified stop condition.

Caution: Because of numerical errors in the function `rootfun` due to roundoff and integration error, `lsodes` may return false roots, or return the same root at two or more nearly equal values of `time`.

### 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 ‘lsodes’ 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>

### References

Alan C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (Eds.), North-Holland, Amsterdam, 1983, pp. 55-64.

S. C. Eisenstat, M. C. Gursky, M. H. Schultz, and A. H. Sherman, Yale Sparse Matrix Package: I. The Symmetric Codes, Int. J. Num. Meth. Eng., 18 (1982), pp. 1145-1151.

S. C. Eisenstat, M. C. Gursky, M. H. Schultz, and A. H. Sherman, Yale Sparse Matrix Package: II. The Nonsymmetric Codes, Research Report No. 114, Dept. of Computer Sciences, Yale University, 1977.

### See Also

- `rk`,

- `rk4` and `euler` for Runge-Kutta integrators.

- `lsoda`, `lsode`, `lsodar`, `vode`, `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.

### Examples

    ## Various ways to solve the same model.

    ## =======================================================================
    ## The example from lsodes source code
    ## A chemical model
    ## =======================================================================

    n  <- 12
    y  <- rep(1, n)
    dy <- rep(0, n)

    times <- c(0, 0.1*(10^(0:4)))

    rtol <- 1.0e-4
    atol <- 1.0e-6

    parms <- c(rk1  = 0.1,   rk2 = 10.0, rk3 = 50.0,  rk4 = 2.5,  rk5 = 0.1,
               rk6  = 10.0,  rk7 = 50.0, rk8 = 2.5,   rk9 = 50.0, rk10 = 5.0,
               rk11 = 50.0, rk12 = 50.0,rk13 = 50.0, rk14 = 30.0,
               rk15 = 100.0,rk16 = 2.5, rk17 = 100.0,rk18 = 2.5,
               rk19 = 50.0, rk20 = 50.0)

    #
    chemistry <- function (time, Y, pars) {
      with (as.list(pars), {
        dy[1] <- -rk1 *Y[1]
        dy[2] <-  rk1 *Y[1]        + rk11*rk14*Y[4]  + rk19*rk14*Y[5]  -
                  rk3 *Y[2]*Y[3]   - rk15*Y[2]*Y[12] - rk2*Y[2]
        dy[3] <-  rk2 *Y[2]        - rk5 *Y[3]       - rk3*Y[2]*Y[3]   -
                  rk7*Y[10]*Y[3]   + rk11*rk14*Y[4]   + rk12*rk14*Y[6]
        dy[4] <-  rk3 *Y[2]*Y[3]   - rk11*rk14*Y[4]  - rk4*Y[4]
        dy[5] <-  rk15*Y[2]*Y[12]  - rk19*rk14*Y[5]  - rk16*Y[5]
        dy[6] <-  rk7 *Y[10]*Y[3]  - rk12*rk14*Y[6]  - rk8*Y[6]
        dy[7] <-  rk17*Y[10]*Y[12] - rk20*rk14*Y[7]  - rk18*Y[7]
        dy[8] <-  rk9 *Y[10]       - rk13*rk14*Y[8]  - rk10*Y[8]
        dy[9] <-  rk4 *Y[4]        + rk16*Y[5]       + rk8*Y[6]         +
                  rk18*Y[7]
        dy[10] <- rk5 *Y[3]        + rk12*rk14*Y[6]  + rk20*rk14*Y[7]   +
                  rk13*rk14*Y[8]   - rk7 *Y[10]*Y[3] - rk17*Y[10]*Y[12] -
                  rk6 *Y[10]       - rk9*Y[10]
        dy[11] <- rk10*Y[8]
        dy[12] <- rk6 *Y[10]       + rk19*rk14*Y[5]  + rk20*rk14*Y[7]   -
                  rk15*Y[2]*Y[12]  - rk17*Y[10]*Y[12]
        return(list(dy))
      })
    }

    ## =======================================================================
    ## application 1. lsodes estimates the structure of the Jacobian
    ##                and calculates the Jacobian by differences
    ## =======================================================================
    out <- lsodes(func = chemistry, y = y, parms = parms, times = times,
                  atol = atol, rtol = rtol, verbose = TRUE)

    ## =======================================================================
    ## application 2. the structure of the Jacobian is input
    ##                lsodes calculates the Jacobian by differences
    ##                this is not so efficient...
    ## =======================================================================

    ## elements of Jacobian that are not zero
    nonzero <-  matrix(nc = 2, byrow = TRUE, data = c(
      1, 1,   2, 1,    # influence of sp1 on rate of change of others
      2, 2,   3, 2,   4, 2,   5, 2,  12, 2,
      2, 3,   3, 3,   4, 3,   6, 3,  10, 3,
      2, 4,   3, 4,   4, 4,   9, 4,  # d (dyi)/dy4
      2, 5,   5, 5,   9, 5,  12, 5,
      3, 6,   6, 6,   9, 6,  10, 6,
      7, 7,   9, 7,  10, 7,  12, 7,
      8, 8,  10, 8,  11, 8,
      3,10,   6,10,   7,10,   8,10,  10,10,  12,10,
      2,12,   5,12,   7,12,  10,12,  12,12)
    )

    ## when run, the default length of rwork is too small
    ## lsodes will tell the length actually needed
    # out2 <- lsodes(func = chemistry, y = y, parms = parms, times = times,
    #              inz = nonzero, atol = atol,rtol = rtol)  #gives warning
    out2 <- lsodes(func = chemistry, y = y, parms = parms, times = times,
                sparsetype = "sparseusr", inz = nonzero,
                 atol = atol, rtol = rtol, verbose = TRUE, lrw = 353)

    ## =======================================================================
    ## application 3. lsodes estimates the structure of the Jacobian
    ##                the Jacobian (vector) function is input
    ## =======================================================================
    chemjac <- function (time, Y, j, pars) {
      with (as.list(pars), {
        PDJ <- rep(0,n)

        if (j == 1){
           PDJ[1] <- -rk1
           PDJ[2] <- rk1
        } else if (j == 2) {
           PDJ[2] <- -rk3*Y[3] - rk15*Y[12] - rk2
           PDJ[3] <- rk2 - rk3*Y[3]
           PDJ[4] <- rk3*Y[3]
           PDJ[5] <- rk15*Y[12]
           PDJ[12] <- -rk15*Y[12]
        } else if (j == 3) {
           PDJ[2] <- -rk3*Y[2]
           PDJ[3] <- -rk5 - rk3*Y[2] - rk7*Y[10]
           PDJ[4] <- rk3*Y[2]
           PDJ[6] <- rk7*Y[10]
           PDJ[10] <- rk5 - rk7*Y[10]
        } else if (j == 4) {
           PDJ[2] <- rk11*rk14
           PDJ[3] <- rk11*rk14
           PDJ[4] <- -rk11*rk14 - rk4
           PDJ[9] <- rk4
        } else if (j == 5) {
           PDJ[2] <- rk19*rk14
           PDJ[5] <- -rk19*rk14 - rk16
           PDJ[9] <- rk16
           PDJ[12] <- rk19*rk14
        } else if (j == 6) {
           PDJ[3] <- rk12*rk14
           PDJ[6] <- -rk12*rk14 - rk8
           PDJ[9] <- rk8
           PDJ[10] <- rk12*rk14
        } else if (j == 7) {
           PDJ[7] <- -rk20*rk14 - rk18
           PDJ[9] <- rk18
           PDJ[10] <- rk20*rk14
           PDJ[12] <- rk20*rk14
        } else if (j == 8) {
           PDJ[8] <- -rk13*rk14 - rk10
           PDJ[10] <- rk13*rk14
           PDJ[11] <- rk10
        } else if (j == 10) {
           PDJ[3] <- -rk7*Y[3]
           PDJ[6] <- rk7*Y[3]
           PDJ[7] <- rk17*Y[12]
           PDJ[8] <- rk9
           PDJ[10] <- -rk7*Y[3] - rk17*Y[12] - rk6 - rk9
           PDJ[12] <- rk6 - rk17*Y[12]
        } else if (j == 12) {
           PDJ[2] <- -rk15*Y[2]
           PDJ[5] <- rk15*Y[2]
           PDJ[7] <- rk17*Y[10]
           PDJ[10] <- -rk17*Y[10]
           PDJ[12] <- -rk15*Y[2] - rk17*Y[10]
        }
        return(PDJ)
      })
    }

    out3 <- lsodes(func = chemistry, y = y, parms = parms, times = times,
                  jacvec = chemjac, atol = atol, rtol = rtol)

    ## =======================================================================
    ## application 4. The structure of the Jacobian (nonzero elements) AND
    ##                the Jacobian (vector) function is input
    ## =======================================================================
    out4 <- lsodes(func = chemistry, y = y, parms = parms, times = times,
                   lrw = 351, sparsetype = "sparseusr", inz = nonzero,
                   jacvec = chemjac, atol = atol, rtol = rtol,
                   verbose = TRUE)

    # The sparsejan variant
    # note: errors in inz may cause R to break, so this is not without danger...
    # out5 <- lsodes(func = chemistry, y = y, parms = parms, times = times,
    #               jacvec = chemjac, atol = atol, rtol = rtol, sparsetype = "sparsejan",
    #               inz = c(1,3,8,13,17,21,25,29,32,32,38,38,43,                   # ian
    #               1,2, 2,3,4,5,12, 2,3,4,6,10, 2,3,4,9, 2,5,9,12, 3,6,9,10,      # jan
    #               7,9,10,12, 8,10,11, 3,6,7,8,10,12, 2,5,7,10,12), lrw = 343)

[Package deSolve version 1.21 Index]