Solver for Ordinary Differential Equations (ODE) With Sparse Jacobian

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 $\partial\dot{y}_i/\partial y_j$, 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 $\partial\dot{y}_i/\partial y_j$. 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.

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

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.

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.

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)
#> 
#> -------------------- 
#> Time settings 
#> --------------------
#>  
#>   Normal computation of output values of y(t) at t = TOUT 
#> 
#> -------------------- 
#> Integration settings 
#> --------------------
#>  
#>   Model function an R-function:  
#>   Jacobian not specified 
#> 
#> 
#> -------------------- 
#> Integration method 
#> --------------------
#>  
#>   The Jacobian will be generated internally,
#>       its structure (indices to nonzero elements) will be obtained from NEQ+1 calls to func 
#> 
#> --------------------
#> lsodes return code
#> --------------------
#> 
#>   return code (idid) =  2 
#>   Integration was successful.
#> 
#> --------------------
#> INTEGER values
#> --------------------
#> 
#>   1 The return code : 2 
#>   2 The number of steps taken for the problem so far: 245 
#>   3 The number of function evaluations for the problem so far: 383 
#>   5 The method order last used (successfully): 1 
#>   6 The order of the method to be attempted on the next step: 1 
#>   7 If return flag =-4,-5: the largest component in error vector 0 
#>   8 The length of the real work array actually required: 353 
#>   9 The length of the integer work array actually required: 30 
#>  14 The number of Jacobian evaluations and LU decompositions so far: 5 
#>  17 The number of nonzero elements in the sparse Jacobian: 44 
#>  
#> --------------------
#> RSTATE values
#> --------------------
#> 
#>   1 The step size in t last used (successfully): 900 
#>   2 The step size to be attempted on the next step: 900 
#>   3 The current value of the independent variable which the solver has reached: 1782.292 
#>   4 Tolerance scale factor > 1.0 computed when requesting too much accuracy: 0 
#>  

## =======================================================================
## 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)
#> 
#> -------------------- 
#> Time settings 
#> --------------------
#>  
#>   Normal computation of output values of y(t) at t = TOUT 
#> 
#> -------------------- 
#> Integration settings 
#> --------------------
#>  
#>   Model function an R-function:  
#>   Jacobian not specified 
#> 
#> 
#> -------------------- 
#> Integration method 
#> --------------------
#>  
#>   The user has supplied indices to nonzero elements of Jacobian,
#>         the Jacobian will be estimated internally, by differences 
#> 
#> --------------------
#> lsodes return code
#> --------------------
#> 
#>   return code (idid) =  2 
#>   Integration was successful.
#> 
#> --------------------
#> INTEGER values
#> --------------------
#> 
#>   1 The return code : 2 
#>   2 The number of steps taken for the problem so far: 245 
#>   3 The number of function evaluations for the problem so far: 383 
#>   5 The method order last used (successfully): 1 
#>   6 The order of the method to be attempted on the next step: 1 
#>   7 If return flag =-4,-5: the largest component in error vector 0 
#>   8 The length of the real work array actually required: 353 
#>   9 The length of the integer work array actually required: 85 
#>  14 The number of Jacobian evaluations and LU decompositions so far: 5 
#>  17 The number of nonzero elements in the sparse Jacobian: 44 
#>  
#> --------------------
#> RSTATE values
#> --------------------
#> 
#>   1 The step size in t last used (successfully): 900 
#>   2 The step size to be attempted on the next step: 900 
#>   3 The current value of the independent variable which the solver has reached: 1782.292 
#>   4 Tolerance scale factor > 1.0 computed when requesting too much accuracy: 0 
#>  
                            
## =======================================================================
## 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)
#> 
#> -------------------- 
#> Time settings 
#> --------------------
#>  
#>   Normal computation of output values of y(t) at t = TOUT 
#> 
#> -------------------- 
#> Integration settings 
#> --------------------
#>  
#>   Model function an R-function:  
#>   Jacobian specified as an R-function:  
#> 
#> 
#> -------------------- 
#> Integration method 
#> --------------------
#>  
#>   The user has supplied indices to nonzero elements of Jacobian,
#>       and a Jacobian function 
#> 
#> --------------------
#> lsodes return code
#> --------------------
#> 
#>   return code (idid) =  2 
#>   Integration was successful.
#> 
#> --------------------
#> INTEGER values
#> --------------------
#> 
#>   1 The return code : 2 
#>   2 The number of steps taken for the problem so far: 245 
#>   3 The number of function evaluations for the problem so far: 343 
#>   5 The method order last used (successfully): 1 
#>   6 The order of the method to be attempted on the next step: 1 
#>   7 If return flag =-4,-5: the largest component in error vector 0 
#>   8 The length of the real work array actually required: 343 
#>   9 The length of the integer work array actually required: 85 
#>  14 The number of Jacobian evaluations and LU decompositions so far: 5 
#>  17 The number of nonzero elements in the sparse Jacobian: 44 
#>  
#> --------------------
#> RSTATE values
#> --------------------
#> 
#>   1 The step size in t last used (successfully): 900 
#>   2 The step size to be attempted on the next step: 900 
#>   3 The current value of the independent variable which the solver has reached: 1788.582 
#>   4 Tolerance scale factor > 1.0 computed when requesting too much accuracy: 0 
#>  
               
# 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)