R: Solving & Simulation of a ODE/solved system (and solving...

rxSolve {RxODE} | R Documentation

## Solving & Simulation of a ODE/solved system (and solving options) equation

### Description

This uses RxODE family of objects, file, or model specification to solve a ODE system. There are many options for a solved RxODE model, the first are the required `object`, and `events` with the some-times optional `params` and `inits`.

### Usage

    rxSolve(
      object,
      params = NULL,
      events = NULL,
      inits = NULL,
      scale = NULL,
      method = c("liblsoda", "lsoda", "dop853", "indLin"),
      transitAbs = NULL,
      atol = 1e-08,
      rtol = 1e-06,
      maxsteps = 70000L,
      hmin = 0,
      hmax = NA_real_,
      hmaxSd = 0,
      hini = 0,
      maxordn = 12L,
      maxords = 5L,
      ...,
      cores,
      covsInterpolation = c("locf", "linear", "nocb", "midpoint"),
      addCov = FALSE,
      matrix = FALSE,
      sigma = NULL,
      sigmaDf = NULL,
      sigmaLower = -Inf,
      sigmaUpper = Inf,
      nCoresRV = 1L,
      sigmaIsChol = FALSE,
      sigmaSeparation = c("auto", "lkj", "separation"),
      sigmaXform = c("identity", "variance", "log", "nlmixrSqrt", "nlmixrLog",
        "nlmixrIdentity"),
      nDisplayProgress = 10000L,
      amountUnits = NA_character_,
      timeUnits = "hours",
      stiff,
      theta = NULL,
      thetaLower = -Inf,
      thetaUpper = Inf,
      eta = NULL,
      addDosing = FALSE,
      stateTrim = Inf,
      updateObject = FALSE,
      omega = NULL,
      omegaDf = NULL,
      omegaIsChol = FALSE,
      omegaSeparation = c("auto", "lkj", "separation"),
      omegaXform = c("variance", "identity", "log", "nlmixrSqrt", "nlmixrLog",
        "nlmixrIdentity"),
      omegaLower = -Inf,
      omegaUpper = Inf,
      nSub = 1L,
      thetaMat = NULL,
      thetaDf = NULL,
      thetaIsChol = FALSE,
      nStud = 1L,
      dfSub = 0,
      dfObs = 0,
      returnType = c("rxSolve", "matrix", "data.frame", "data.frame.TBS", "data.table",
        "tbl", "tibble"),
      seed = NULL,
      nsim = NULL,
      minSS = 10L,
      maxSS = 1000L,
      infSSstep = 12,
      strictSS = TRUE,
      istateReset = TRUE,
      subsetNonmem = TRUE,
      maxAtolRtolFactor = 0.1,
      from = NULL,
      to = NULL,
      by = NULL,
      length.out = NULL,
      iCov = NULL,
      keep = NULL,
      indLinPhiTol = 1e-07,
      indLinPhiM = 0L,
      indLinMatExpType = c("expokit", "Al-Mohy", "arma"),
      indLinMatExpOrder = 6L,
      drop = NULL,
      idFactor = TRUE,
      mxhnil = 0,
      hmxi = 0,
      warnIdSort = TRUE,
      warnDrop = TRUE,
      ssAtol = 1e-08,
      ssRtol = 1e-06,
      safeZero = TRUE,
      sumType = c("pairwise", "fsum", "kahan", "neumaier", "c"),
      prodType = c("long double", "double", "logify"),
      sensType = c("advan", "autodiff", "forward", "central"),
      linDiff = c(tlag = 1.5e-05, f = 1.5e-05, rate = 1.5e-05, dur = 1.5e-05, tlag2 =
        1.5e-05, f2 = 1.5e-05, rate2 = 1.5e-05, dur2 = 1.5e-05),
      linDiffCentral = c(tlag = TRUE, f = TRUE, rate = TRUE, dur = TRUE, tlag2 = TRUE, f2 =
        TRUE, rate2 = TRUE, dur2 = TRUE),
      resample = NULL,
      resampleID = TRUE,
      maxwhile = 1e+05
    )

    ## Default S3 method:
    rxSolve(
      object,
      params = NULL,
      events = NULL,
      inits = NULL,
      ...,
      theta = NULL,
      eta = NULL
    )

    ## S3 method for class 'rxSolve'
    update(object, ...)

    ## S3 method for class 'RxODE'
    predict(object, ...)

    ## S3 method for class 'rxSolve'
    predict(object, ...)

    ## S3 method for class 'rxEt'
    predict(object, ...)

    ## S3 method for class 'rxParams'
    predict(object, ...)

    ## S3 method for class 'RxODE'
    simulate(object, nsim = 1L, seed = NULL, ...)

    ## S3 method for class 'rxSolve'
    simulate(object, nsim = 1L, seed = NULL, ...)

    ## S3 method for class 'rxParams'
    simulate(object, nsim = 1L, seed = NULL, ...)

    ## S3 method for class 'rxSolve'
    solve(a, b, ...)

    ## S3 method for class 'RxODE'
    solve(a, b, ...)

    ## S3 method for class 'rxParams'
    solve(a, b, ...)

    ## S3 method for class 'rxEt'
    solve(a, b, ...)

    rxControl(..., params = NULL, events = NULL, inits = NULL)

### Arguments

`object` | is a either a RxODE family of objects, or a file-name with a RxODE model specification, or a string with a RxODE model specification.
`params` | a numeric named vector with values for every parameter in the ODE system; the names must correspond to the parameter identifiers used in the ODE specification;
`events` | an `eventTable` object describing the input (e.g., doses) to the dynamic system and observation sampling time points (see `eventTable()`);
`inits` | a vector of initial values of the state variables (e.g., amounts in each compartment), and the order in this vector must be the same as the state variables (e.g., PK/PD compartments);
`scale` | a numeric named vector with scaling for ode parameters of the system. The names must correspond to the parameter identifiers in the ODE specification. Each of the ODE variables will be divided by the scaling factor. For example `scale=c(center=2)` will divide the center ODE variable by 2.
`method` | The method for solving ODEs. Currently this supports: `"liblsoda"` thread safe lsoda. This supports parallel thread-based solving, and ignores user Jacobian specification. `"lsoda"` – LSODA solver. Does not support parallel thread-based solving, but allows user Jacobian specification. `"dop853"` – DOP853 solver. Does not support parallel thread-based solving nor user Jacobain specification `"indLin"` – Solving through inductive linearization. The RxODE dll must be setup specially to use this solving routine.
`transitAbs` | boolean indicating if this is a transit compartment absorption
`atol` | a numeric absolute tolerance (1e-8 by default) used by the ODE solver to determine if a good solution has been achieved; This is also used in the solved linear model to check if prior doses do not add anything to the solution.
`rtol` | a numeric relative tolerance (`1e-6` by default) used by the ODE solver to determine if a good solution has been achieved. This is also used in the solved linear model to check if prior doses do not add anything to the solution.
`maxsteps` | maximum number of (internally defined) steps allowed during one call to the solver. (5000 by default)
`hmin` | The minimum absolute step size allowed. The default value is 0.
`hmax` | The maximum absolute step size allowed. When `hmax=NA` (default), uses the average difference + hmaxSd*sd in times and sampling events. The `hmaxSd` is a user specified parameter and which defaults to zero. When `hmax=NULL` RxODE uses the maximum difference in times in your sampling and events. The value 0 is equivalent to infinite maximum absolute step size.
`hmaxSd` | The number of standard deviations of the time difference to add to hmax. The default is 0
`hini` | The step size to be attempted on the first step. The default value is determined by the solver (when `hini = 0`)
`maxordn` | The maximum order to be allowed for the nonstiff (Adams) method. The default is 12. It can be between 1 and 12.
`maxords` | The maximum order to be allowed for the stiff (BDF) method. The default value is 5. This can be between 1 and 5.
`...` | Other arguments including scaling factors for each compartment. This includes S# = numeric will scale a compartment # by a dividing the compartment amount by the scale factor, like NONMEM.
`cores` | Number of cores used in parallel ODE solving. This is equivalent to calling `setRxThreads()`
`covsInterpolation` | specifies the interpolation method for time-varying covariates. When solving ODEs it often samples times outside the sampling time specified in `events`. When this happens, the time varying covariates are interpolated. Currently this can be: `"linear"` interpolation, which interpolates the covariate by solving the line between the observed covariates and extrapolating the new covariate value. `"constant"` – Last observation carried forward (the default). `"NOCB"` – Next Observation Carried Backward. This is the same method that NONMEM uses. `"midpoint"` Last observation carried forward to midpoint; Next observation carried backward to midpoint.
`addCov` | A boolean indicating if covariates should be added to the output matrix or data frame. By default this is disabled.
`matrix` | A boolean indicating if a matrix should be returned instead of the RxODE's solved object.
`sigma` | Named sigma covariance or Cholesky decomposition of a covariance matrix. The names of the columns indicate parameters that are simulated. These are simulated for every observation in the solved system.
`sigmaDf` | Degrees of freedom of the sigma t-distribution. By default it is equivalent to `Inf`, or a normal distribution.
`sigmaLower` | Lower bounds for simulated unexplained variability (by default -Inf)
`sigmaUpper` | Upper bounds for simulated unexplained variability (by default Inf)
`nCoresRV` | Number of cores used for the simulation of the sigma variables. By default this is 1. To reproduce the results you need to run on the same platform with the same number of cores. This is the reason this is set to be one, regardless of what the number of cores are used in threaded ODE solving.
`sigmaIsChol` | Boolean indicating if the sigma is in the Cholesky decomposition instead of a symmetric covariance
`sigmaSeparation` | separation strategy for sigma; Tells the type of separation strategy when simulating covariance with parameter uncertainty with standard deviations modeled in the `thetaMat` matrix. `"lkj"` simulates the correlation matrix from the `rLKJ1` matrix with the distribution parameter `eta` equal to the degrees of freedom `nu` by `(nu-1)/2` `"separation"` simulates from the identity inverse Wishart covariance matrix with `nu` degrees of freedom. This is then converted to a covariance matrix and augmented with the modeled standard deviations. While computationally more complex than the `"lkj"` prior, it performs better when the covariance matrix size is greater or equal to 10 `"auto"` chooses `"lkj"` when the dimension of the matrix is less than 10 and `"separation"` when greater than equal to 10.
`sigmaXform` | When taking `sigma` values from the `thetaMat` simulations (using the separation strategy for covariance simulation), how should the `thetaMat` values be turned int standard deviation values: `identity` This is when standard deviation values are directly modeled by the `params` and `thetaMat` matrix `variance` This is when the `params` and `thetaMat` simulates the variance that are directly modeled by the `thetaMat` matrix `log` This is when the `params` and `thetaMat` simulates `log(sd)` `nlmixrSqrt` This is when the `params` and `thetaMat` simulates the inverse cholesky decomposed matrix with the `x^2` modeled along the diagonal. This only works with a diagonal matrix. `nlmixrLog` This is when the `params` and `thetaMat` simulates the inverse cholesky decomposed matrix with the `exp(x^2)` along the diagonal. This only works with a diagonal matrix. `nlmixrIdentity` This is when the `params` and `thetaMat` simulates the inverse cholesky decomposed matrix. This only works with a diagonal matrix.
`nDisplayProgress` | An integer indicating the minimum number of c-based solves before a progress bar is shown. By default this is 10,000.
`amountUnits` | This supplies the dose units of a data frame supplied instead of an event table. This is for importing the data as an RxODE event table.
`timeUnits` | This supplies the time units of a data frame supplied instead of an event table. This is for importing the data as an RxODE event table.
`stiff` | a logical (`TRUE` by default) indicating whether the ODE system is stiff or not.
    For stiff ODE systems (`stiff = TRUE`), `RxODE` uses the
    LSODA (Livermore Solver for Ordinary Differential Equations)
    Fortran package, which implements an automatic method switching
    for stiff and non-stiff problems along the integration
    interval, authored by Hindmarsh and Petzold (2003).

    For non-stiff systems (`stiff = FALSE`), `RxODE` uses
    DOP853, an explicit Runge-Kutta method of order 8(5, 3) of
    Dormand and Prince as implemented in C by Hairer and Wanner
    (1993).

    If stiff is not specified, the `method` argument is used instead.
`theta` | A vector of parameters that will be named `THETA\[#\]` and added to parameters
`thetaLower` | Lower bounds for simulated population parameter variability (by default `-Inf`)
`thetaUpper` | Upper bounds for simulated population unexplained variability (by default `Inf`)
`eta` | A vector of parameters that will be named `ETA\[#\]` and added to parameters
`addDosing` | Boolean indicating if the solve should add RxODE EVID and related columns. This will also include dosing information and estimates at the doses. Be default, RxODE only includes estimates at the observations. (default `FALSE`). When `addDosing` is `NULL`, only include `EVID=0` on solve and exclude any model-times or `EVID=2`. If `addDosing` is `NA` the classic `RxODE` EVID events are returned. When `addDosing` is `TRUE` add the event information in NONMEM-style format; If `subsetNonmem=FALSE` RxODE will also include extra event types (`EVID`) for ending infusion and modeled times: `EVID=-1` when the modeled rate infusions are turned off (matches `rate=-1`) `EVID=-2` When the modeled duration infusions are turned off (matches `rate=-2`) `EVID=-10` When the specified `rate` infusions are turned off (matches `rate>0`) `EVID=-20` When the specified `dur` infusions are turned off (matches `dur>0`) `EVID=101,102,103,...` Modeled time where 101 is the first model time, 102 is the second etc.
`stateTrim` | When amounts/concentrations in one of the states are above this value, trim them to be this value. By default Inf. Also trims to -stateTrim for large negative amounts/concentrations. If you want to trim between a range say `c(0, 2000000)` you may specify 2 values with a lower and upper range to make sure all state values are in the reasonable range.
`updateObject` | This is an internally used flag to update the RxODE solved object (when supplying an RxODE solved object) as well as returning a new object. You probably should not modify it's `FALSE` default unless you are willing to have unexpected results.
`omega` | Estimate of Covariance matrix. When omega is a list, assume it is a block matrix and convert it to a full matrix for simulations.
`omegaDf` | The degrees of freedom of a t-distribution for simulation. By default this is `NULL` which is equivalent to `Inf` degrees, or to simulate from a normal distribution instead of a t-distribution.
`omegaIsChol` | Indicates if the `omega` supplied is a Cholesky decomposed matrix instead of the traditional symmetric matrix.
`omegaSeparation` | Omega separation strategy Tells the type of separation strategy when simulating covariance with parameter uncertainty with standard deviations modeled in the `thetaMat` matrix. `"lkj"` simulates the correlation matrix from the `rLKJ1` matrix with the distribution parameter `eta` equal to the degrees of freedom `nu` by `(nu-1)/2` `"separation"` simulates from the identity inverse Wishart covariance matrix with `nu` degrees of freedom. This is then converted to a covariance matrix and augmented with the modeled standard deviations. While computationally more complex than the `"lkj"` prior, it performs better when the covariance matrix size is greater or equal to 10 `"auto"` chooses `"lkj"` when the dimension of the matrix is less than 10 and `"separation"` when greater than equal to 10.
`omegaXform` | When taking `omega` values from the `thetaMat` simulations (using the separation strategy for covariance simulation), how should the `thetaMat` values be turned int standard deviation values: `identity` This is when standard deviation values are directly modeled by the `params` and `thetaMat` matrix `variance` This is when the `params` and `thetaMat` simulates the variance that are directly modeled by the `thetaMat` matrix `log` This is when the `params` and `thetaMat` simulates `log(sd)` `nlmixrSqrt` This is when the `params` and `thetaMat` simulates the inverse cholesky decomposed matrix with the `x^2` modeled along the diagonal. This only works with a diagonal matrix. `nlmixrLog` This is when the `params` and `thetaMat` simulates the inverse cholesky decomposed matrix with the `exp(x^2)` along the diagonal. This only works with a diagonal matrix. `nlmixrIdentity` This is when the `params` and `thetaMat` simulates the inverse cholesky decomposed matrix. This only works with a diagonal matrix.
`omegaLower` | Lower bounds for simulated ETAs (by default -Inf)
`omegaUpper` | Upper bounds for simulated ETAs (by default Inf)
`nSub` | Number between subject variabilities (`ETAs`) simulated for every realization of the parameters.
`thetaMat` | Named theta matrix.
`thetaDf` | The degrees of freedom of a t-distribution for simulation. By default this is `NULL` which is equivalent to `Inf` degrees, or to simulate from a normal distribution instead of a `t`-distribution.
`thetaIsChol` | Indicates if the `theta` supplied is a Cholesky decomposed matrix instead of the traditional symmetric matrix.
`nStud` | Number virtual studies to characterize uncertainty in estimated parameters.
`dfSub` | Degrees of freedom to sample the between subject variability matrix from the inverse Wishart distribution (scaled) or scaled inverse chi squared distribution.
`dfObs` | Degrees of freedom to sample the unexplained variability matrix from the inverse Wishart distribution (scaled) or scaled inverse chi squared distribution.
`returnType` | This tells what type of object is returned. The currently supported types are: `"rxSolve"` (default) will return a reactive data frame that can change easily change different pieces of the solve and update the data frame. This is the currently standard solving method in RxODE, is used for `rxSolve(object, ...)`, `solve(object,...)`, `"data.frame"` – returns a plain, non-reactive data frame; Currently very slightly faster than `returnType="matrix"` `"matrix"` – returns a plain matrix with column names attached to the solved object. This is what is used `object$run` as well as `object$solve` `"data.table"` – returns a `data.table`; The `data.table` is created by reference (ie `setDt()`), which should be fast. `"tbl"` or `"tibble"` returns a tibble format.
`seed` | an object specifying if and how the random number generator should be initialized
`nsim` | represents the number of simulations. For RxODE, if you supply single subject event tables (created with `[eventTable()]`)
`minSS` | Minimum number of iterations for a steady-state dose
`maxSS` | Maximum number of iterations for a steady-state dose
`infSSstep` | Step size for determining if a constant infusion has reached steady state. By default this is large value, 420.
`strictSS` | Boolean indicating if a strict steady-state is required. If a strict steady-state is (`TRUE`) required then at least `minSS` doses are administered and the total number of steady states doses will continue until `maxSS` is reached, or `atol` and `rtol` for every compartment have been reached. However, if ODE solving problems occur after the `minSS` has been reached the whole subject is considered an invalid solve. If `strictSS` is `FALSE` then as long as `minSS` has been reached the last good solve before ODE solving problems occur is considered the steady state, even though either `atol`, `rtol` or `maxSS` have not been achieved.
`istateReset` | When `TRUE`, reset the `ISTATE` variable to 1 for lsoda and liblsoda with doses, like `deSolve`; When `FALSE`, do not reset the `ISTATE` variable with doses.
`subsetNonmem` | subset to NONMEM compatible EVIDs only. By default `TRUE`.
`maxAtolRtolFactor` | The maximum `atol`/`rtol` that FOCEi and other routines may adjust to. By default 0.1
`from` | When there is no observations in the event table, start observations at this value. By default this is zero.
`to` | When there is no observations in the event table, end observations at this value. By default this is 24 + maximum dose time.
`by` | When there are no observations in the event table, this is the amount to increment for the observations between `from` and `to`.
`length.out` | The number of observations to create if there isn't any observations in the event table. By default this is 200.
`iCov` | A data frame of individual non-time varying covariates to combine with the `events` dataset by merge.
`keep` | Columns to keep from either the input dataset or the `iCov` dataset. With the `iCov` dataset, the column is kept once per line. For the input dataset, if any records are added to the data LOCF (Last Observation Carried forward) imputation is performed.
`indLinPhiTol` | the requested accuracy tolerance on exponential matrix.
`indLinPhiM` | the maximum size for the Krylov basis
`indLinMatExpType` | This is them matrix exponential type that is use for RxODE. Currently the following are supported: `Al-Mohy` Uses the exponential matrix method of Al-Mohy Higham (2009) `arma` Use the exponential matrix from RcppArmadillo `expokit` Use the exponential matrix from Roger B. Sidje (1998)
`indLinMatExpOrder` | an integer, the order of approximation to be used, for the `Al-Mohy` and `expokit` values. The best value for this depends on machine precision (and slightly on the matrix). We use `6` as a default.
`drop` | Columns to drop from the output
`idFactor` | This boolean indicates if original ID values should be maintained. This changes the default sequentially ordered ID to a factor with the original ID values in the original dataset. By default this is enabled.
`mxhnil` | maximum number of messages printed (per problem) warning that `T + H = T` on a step (`H` = step size). This must be positive to result in a non-default value. The default value is 0 (or infinite).
`hmxi` | inverse of the maximum absolute value of `H` to are used. hmxi = 0.0 is allowed and corresponds to an infinite `hmax1 (default).` hmin`and`hmxi`may be changed at any time, but will not take effect until the next change of`H`is considered. This option is only considered with`method="liblsoda"'.
`warnIdSort` | Warn if the ID is not present and RxODE assumes the order of the parameters/iCov are the same as the order of the parameters in the input dataset.
`warnDrop` | Warn if column(s) were supposed to be dropped, but were not present.
`ssAtol` | Steady state atol convergence factor. Can be a vector based on each state.
`ssRtol` | Steady state rtol convergence factor. Can be a vector based on each state.
`safeZero` | Use safe zero divide and log routines. By default this is turned on but you may turn it off if you wish.
`sumType` | Sum type to use for `sum()` in RxODE code blocks. `pairwise` uses the pairwise sum (fast, default) `fsum` uses Python's fsum function (most accurate) `kahan` uses Kahan correction `neumaier` uses Neumaier correction `c` uses no correction: default/native summing
`prodType` | Product to use for `prod()` in RxODE blocks `long double` converts to long double, performs the multiplication and then converts back. `double` uses the standard double scale for multiplication.
`sensType` | Sensitivity type for `linCmt()` model: `advan` Use the direct advan solutions `autodiff` Use the autodiff advan solutions `forward` Use forward difference solutions `central` Use central differences
`linDiff` | This gives the linear difference amount for all the types of linear compartment model parameters where sensitivities are not calculated. The named components of this numeric vector are: `"lag"` Central compartment lag `"f"` Central compartment bioavailability `"rate"` Central compartment modeled rate `"dur"` Central compartment modeled duration `"lag2"` Depot compartment lag `"f2"` Depot compartment bioavailability `"rate2"` Depot compartment modeled rate `"dur2"` Depot compartment modeled duration
`linDiffCentral` | This gives the which parameters use central differences for the linear compartment model parameters. The are the same components as `linDiff`
`resample` | A character vector of model variables to resample from the input dataset; This sampling is done with replacement. When `NULL` or `FALSE` no resampling is done. When `TRUE` resampling is done on all covariates in the input dataset
`resampleID` | boolean representing if the resampling should be done on an individual basis `TRUE` (ie. a whole patient is selected) or each covariate is resampled independent of the subject identifier `FALSE`. When `resampleID=TRUE` correlations of parameters are retained, where as when `resampleID=FALSE` ignores patient covariate correaltions. Hence the default is `resampleID=TRUE`.
`maxwhile` | represents the maximum times a while loop is evaluated before exiting. By default this is 100000
`a` | when using `solve()`, this is equivalent to the `object` argument. If you specify `object` later in the argument list it overwrites this parameter.
`b` | when using `solve()`, this is equivalent to the `params` argument. If you specify `params` as a named argument, this overwrites the output

### Details

The rest of the document focus on the different ODE solving methods, followed by the core solving method's options, RxODE event handling options, RxODE's numerical stability options, RxODE's output options, and finally internal RxODE options or compatibility options.

### Value

An “rxSolve” solve object that stores the solved value in a special data.frame or other type as determined by `returnType`. By default this has as many rows as there are sampled time points and as many columns as system variables (as defined by the ODEs and additional assignments in the RxODE model code). It also stores information about the call to allow dynamic updating of the solved object.

The operations for the object are similar to a data-frame, but expand the `$` and `[[""]]` access operators and assignment operators to resolve based on different parameter values, initial conditions, solver parameters, or events (by updating the `time` variable).

You can call the `eventTable()` methods on the solved object to update the event table and resolve the system of equations.

### Author(s)

Matthew Fidler, Melissa Hallow and Wenping Wang

### References

"New Scaling and Squaring Algorithm for the Matrix Exponential", by Awad H. Al-Mohy and Nicholas J. Higham, August 2009

Roger B. Sidje (1998). EXPOKIT: Software package for computing matrix exponentials. ACM - Transactions on Mathematical Software 24(1), 130-156.

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

Petzold, L. R. Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations. Siam J. Sci. Stat. Comput. 4 (1983), pp. 136-148.

Hairer, E., Norsett, S. P., and Wanner, G. Solving ordinary differential equations I, nonstiff problems. 2nd edition, Springer Series in Computational Mathematics, Springer-Verlag (1993).

### See Also

`RxODE()`

[Package RxODE version 1.1.2 Index]