# radau: Implicit Runge-Kutta RADAU IIA In deSolve: Solvers for Initial Value Problems of Differential Equations ('ODE', 'DAE', 'DDE')

## Description

Solves the initial value problem for stiff or nonstiff systems of ordinary differential equations (ODE) in the form:

dy/dt = f(t,y)

or linearly implicit differential algebraic equations in the form:

M dy/dt = f(t,y)

.

The R function `radau` provides an interface to the Fortran solver RADAU5, written by Ernst Hairer and G. Wanner, which implements the 3-stage RADAU IIA method. It implements the implicit Runge-Kutta method of order 5 with step size control and continuous output. The system of ODEs or DAEs is written as an R function or can be defined in compiled code that has been dynamically loaded.

## Usage

 ```1 2 3 4 5 6 7 8 9``` ```radau(y, times, func, parms, nind = c(length(y), 0, 0), rtol = 1e-6, atol = 1e-6, jacfunc = NULL, jactype = "fullint", mass = NULL, massup = NULL, massdown = NULL, rootfunc = NULL, verbose = FALSE, nroot = 0, hmax = NULL, hini = 0, ynames = TRUE, bandup = NULL, banddown = NULL, maxsteps = 5000, 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 the right-hand side of the equation M dy/dt = f(t,y) if a DAE. (if `mass` is supplied then the problem is assumed a DAE). `func` can also be 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 `radau()` is called. See deSolve package vignette `"compiledCode"` for more details. `parms ` vector or list of parameters used in `func` or `jacfunc`. `nind ` if a DAE system: a three-valued vector with the number of variables of index 1, 2, 3 respectively. The equations must be defined such that the index 1 variables precede the index 2 variables which in turn precede the index 3 variables. The sum of the variables of different index should equal N, the total number of variables. This has implications on the scaling of the variables, i.e. index 2 variables are scaled by 1/h, index 3 variables are scaled by 1/h^2. `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. `jacfunc ` if not `NULL`, an R function that computes 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 Jacobian (see vignette `"compiledCode"` from package deSolve, for more about this option). In some circumstances, supplying `jacfunc` can speed up the computations, if the system is stiff. The R 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 R and FORTRAN store matrices). If the Jacobian is banded, `jacfunc` should return a matrix containing only the nonzero bands of the Jacobian, rotated row-wise. See example. `jactype ` the structure of the Jacobian, one of `"fullint"`, `"fullusr"`, `"bandusr"` or `"bandint"` - either full or banded and estimated internally or by user. `mass ` the mass matrix. If not `NULL`, the problem is a linearly implicit DAE and defined as M dy/dt = f(t,y). If the mass-matrix M is full, it should be of dimension n*n where n is the number of y-values; if banded the number of rows should be less than n, and the mass-matrix is stored diagonal-wise with element (i, j) stored in `mass(i - j + mumas + 1, j)`. If `mass = NULL` then the model is an ODE (default) `massup ` number of non-zero bands above the diagonal of the `mass` matrix, in case it is banded. `massdown ` number of non-zero bands below the diagonal of the `mass` matrix, in case it is banded. `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. `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 set equal to 1e-6. Usually 1e-3 to 1e-5 is good for stiff equations `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. `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. `maxsteps ` average maximal number of steps per output interval taken by the solver. This argument is defined such as to ensure compatibility with the Livermore-solvers. RADAU only accepts the maximal number of steps for the entire integration, and this is calculated as `length(times) * maxsteps`. `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 vignette `"compiledCode"` from package `deSolve`. `initfunc ` if not `NULL`, the name of the initialisation function (which initialises values of parameters), as provided in ‘dllname’. See vignette `"compiledCode"` from package `deSolve`. `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 calculed in the DLL - you have to perform this check in the code - See vignette `"compiledCode"` from package `deSolve`. `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 `RADAU5`, whose documentation should be consulted for details. The implementation is based on the Fortran 77 version from January 18, 2002.

There are four standard choices for the Jacobian which can be specified with `jactype`.

The options for jactype are

jactype = "fullint"

a full Jacobian, calculated internally by the solver.

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 radau; the size of the bands specified by `bandup` and `banddown`.

Inspection of the example below shows how to specify both a banded and full Jacobian.

The input parameters `rtol`, and `atol` determine the error control performed by the solver, which roughly keeps the local error of y(i) below rtol(i)*abs(y(i))+atol(i).

The diagnostics of the integration can be printed to screen by calling `diagnostics`. If `verbose` = `TRUE`, the diagnostics will be written to the screen at the end of the integration.

See vignette("deSolve") from the `deSolve` package 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"` from package `deSolve` for details.

Information about linking forcing functions to compiled code is in forcings (from package `deSolve`).

`radau` 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, `radau` 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 returns with an unrecoverable error. If `y` has a names attribute, it will be used to label the columns of the output value.

Karline Soetaert

## References

E. Hairer and G. Wanner, 1996. Solving Ordinary Differential Equations II. Stiff and Differential-algebraic problems. Springer series in computational mathematics 14, Springer-Verlag, second edition.

• `ode` for a general interface to most of the ODE solvers ,

• `ode.1D` for integrating 1-D models,

• `ode.2D` for integrating 2-D models,

• `ode.3D` for integrating 3-D models,

• `daspk` for integrating DAE models up to index 1

`diagnostics` to print diagnostic messages.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136``` ```## ======================================================================= ## Example 1: ODE ## Various ways to solve the same model. ## ======================================================================= ## the model, 5 state variables f1 <- function (t, y, parms) { ydot <- vector(len = 5) ydot[1] <- 0.1*y[1] -0.2*y[2] ydot[2] <- -0.3*y[1] +0.1*y[2] -0.2*y[3] ydot[3] <- -0.3*y[2] +0.1*y[3] -0.2*y[4] ydot[4] <- -0.3*y[3] +0.1*y[4] -0.2*y[5] ydot[5] <- -0.3*y[4] +0.1*y[5] return(list(ydot)) } ## the Jacobian, written as a full matrix fulljac <- function (t, y, parms) { jac <- matrix(nrow = 5, ncol = 5, byrow = TRUE, data = c(0.1, -0.2, 0 , 0 , 0 , -0.3, 0.1, -0.2, 0 , 0 , 0 , -0.3, 0.1, -0.2, 0 , 0 , 0 , -0.3, 0.1, -0.2, 0 , 0 , 0 , -0.3, 0.1)) return(jac) } ## the Jacobian, written in banded form bandjac <- function (t, y, parms) { jac <- matrix(nrow = 3, ncol = 5, byrow = TRUE, data = c( 0 , -0.2, -0.2, -0.2, -0.2, 0.1, 0.1, 0.1, 0.1, 0.1, -0.3, -0.3, -0.3, -0.3, 0)) return(jac) } ## initial conditions and output times yini <- 1:5 times <- 1:20 ## default: stiff method, internally generated, full Jacobian out <- radau(yini, times, f1, parms = 0) plot(out) ## stiff method, user-generated full Jacobian out2 <- radau(yini, times, f1, parms = 0, jactype = "fullusr", jacfunc = fulljac) ## stiff method, internally-generated banded Jacobian ## one nonzero band above (up) and below(down) the diagonal out3 <- radau(yini, times, f1, parms = 0, jactype = "bandint", bandup = 1, banddown = 1) ## stiff method, user-generated banded Jacobian out4 <- radau(yini, times, f1, parms = 0, jactype = "bandusr", jacfunc = bandjac, bandup = 1, banddown = 1) ## ======================================================================= ## Example 2: ODE ## stiff problem from chemical kinetics ## ======================================================================= Chemistry <- function (t, y, p) { dy1 <- -.04*y[1] + 1.e4*y[2]*y[3] dy2 <- .04*y[1] - 1.e4*y[2]*y[3] - 3.e7*y[2]^2 dy3 <- 3.e7*y[2]^2 list(c(dy1, dy2, dy3)) } times <- 10^(seq(0, 10, by = 0.1)) yini <- c(y1 = 1.0, y2 = 0, y3 = 0) out <- radau(func = Chemistry, times = times, y = yini, parms = NULL) plot(out, log = "x", type = "l", lwd = 2) ## ============================================================================= ## Example 3: DAE ## Car axis problem, index 3 DAE, 8 differential, 2 algebraic equations ## from ## F. Mazzia and C. Magherini. Test Set for Initial Value Problem Solvers, ## release 2.4. Department ## of Mathematics, University of Bari and INdAM, Research Unit of Bari, ## February 2008. ## Available at http://www.dm.uniba.it/~testset. ## ============================================================================= ## Problem is written as M*y' = f(t,y,p). ## caraxisfun implements the right-hand side: caraxisfun <- function(t, y, parms) { with(as.list(y), { yb <- r * sin(w * t) xb <- sqrt(L * L - yb * yb) Ll <- sqrt(xl^2 + yl^2) Lr <- sqrt((xr - xb)^2 + (yr - yb)^2) dxl <- ul; dyl <- vl; dxr <- ur; dyr <- vr dul <- (L0-Ll) * xl/Ll + 2 * lam2 * (xl-xr) + lam1*xb dvl <- (L0-Ll) * yl/Ll + 2 * lam2 * (yl-yr) + lam1*yb - k * g dur <- (L0-Lr) * (xr-xb)/Lr - 2 * lam2 * (xl-xr) dvr <- (L0-Lr) * (yr-yb)/Lr - 2 * lam2 * (yl-yr) - k * g c1 <- xb * xl + yb * yl c2 <- (xl - xr)^2 + (yl - yr)^2 - L * L list(c(dxl, dyl, dxr, dyr, dul, dvl, dur, dvr, c1, c2)) }) } eps <- 0.01; M <- 10; k <- M * eps^2/2; L <- 1; L0 <- 0.5; r <- 0.1; w <- 10; g <- 1 yini <- c(xl = 0, yl = L0, xr = L, yr = L0, ul = -L0/L, vl = 0, ur = -L0/L, vr = 0, lam1 = 0, lam2 = 0) # the mass matrix Mass <- diag(nrow = 10, 1) Mass[5,5] <- Mass[6,6] <- Mass[7,7] <- Mass[8,8] <- M * eps * eps/2 Mass[9,9] <- Mass[10,10] <- 0 Mass # index of the variables: 4 of index 1, 4 of index 2, 2 of index 3 index <- c(4, 4, 2) times <- seq(0, 3, by = 0.01) out <- radau(y = yini, mass = Mass, times = times, func = caraxisfun, parms = NULL, nind = index) plot(out, which = 1:4, type = "l", lwd = 2) ```

### Example output

```      [,1] [,2] [,3] [,4]  [,5]  [,6]  [,7]  [,8] [,9] [,10]
[1,]    1    0    0    0 0e+00 0e+00 0e+00 0e+00    0     0
[2,]    0    1    0    0 0e+00 0e+00 0e+00 0e+00    0     0
[3,]    0    0    1    0 0e+00 0e+00 0e+00 0e+00    0     0
[4,]    0    0    0    1 0e+00 0e+00 0e+00 0e+00    0     0
[5,]    0    0    0    0 5e-04 0e+00 0e+00 0e+00    0     0
[6,]    0    0    0    0 0e+00 5e-04 0e+00 0e+00    0     0
[7,]    0    0    0    0 0e+00 0e+00 5e-04 0e+00    0     0
[8,]    0    0    0    0 0e+00 0e+00 0e+00 5e-04    0     0
[9,]    0    0    0    0 0e+00 0e+00 0e+00 0e+00    0     0
[10,]    0    0    0    0 0e+00 0e+00 0e+00 0e+00    0     0
```

deSolve documentation built on May 10, 2018, 3 p.m.