ode_root ordinary differential equation solver with roots finding

## Syntax

    [y, rd, w, iw] = ode("root", y0, t0, t [,rtol [,atol]], f [,jac], ng, g [,w,iw])

## Arguments

y0

  a real vector or matrix (initial conditions).

t0

  a real scalar (initial time).

t

  a real vector (times at which the solution is computed).

f

  an external i.e. function or character string or list.

rtol, atol

  a real constants or real vectors of the same size as `y`.

jac

  an external i.e. function or character string or list.

ng

  an integer.

g

  an external i.e. function or character string or list.

y

  a real vector or matrix. The solution.

rd

  a real vector.

w, iw

  vectors of real numbers. See ode() optional output

## Description

With this syntax (first argument equal to `"root"`) `ode` computes the solution of the differential equation `dy/dt=f(t,y)` until the state `y(t)` crosses the surface `g(t,y)=0`.

`g` should give the equation of the surface. It is an external i.e. a function with specified syntax, or the name of a Fortran subroutine or a C function (character string) with specified syntax or a list.

If `g` is a function the syntax should be as follows:

    z = g(t,y)

where `t` is a real scalar (time) and `y` a real vector (state). It returns a vector of size `ng` which corresponds to the `ng` constraints. If `g` is a character string it refers to the name of a Fortran subroutine or a C function, with the following calling sequence: `g(n,t,y,ng,gout)` where `ng` is the number of constraints and `gout` is the value of `g` (output of the program). If `g` is a list the same conventions as for `f` apply (see ode help).

Output `rd` is a `1 x k` vector. The first entry contains the stopping time. Other entries indicate which components of `g` have changed sign. `k` larger than 2 indicates that more than one surface (`(k-1)` surfaces) have been simultaneously traversed.

Other arguments and other options are the same as for `ode`, see the ode help.

## Examples

    // Integration of the differential equation
    // dy/dt=y , y(0)=1, and finds the minimum time t such that y(t)=2
    deff("ydot = f(t,y)", "ydot=y")
    deff("z = g(t,y)", "z=y-2")
    y0 = 1; ng = 1;
    [y,rd] = ode("root", y0, 0, 2, f, ng, g)

    deff("z = g(t,y)", "z = y-[2;2;33]")
    [y,rd] = ode("root", 1, 0, 2, f, 3, g)

## See also

- ode
- ode_optional_output
- ode_discrete
- dasrt
- daskr