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# impl

differential algebraic equation

### Syntax

`y = impl([type,] y0, ydot0, t0, t [,atol [,rtol]], res, adda [,jac])`

### Arguments

y0, ydot0

real vectors or matrices (initial conditions)

t0

a real scalar (initial time)

t

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

externals (function or character string or list)

type

string `'adams'` or `'stiff'`

atol, rtol

real scalars or real vectors of the same size as `y`

jac

external (function or character string or list).

### Description

Solution of the linear implicit differential equation

A(t,y) dy/dt=g(t,y), y(t0)=y0

`t0` is the initial instant, `y0` is the vector of initial conditions. Vector `ydot0` of the time derivative of `y` at `t0` must also be given. The input `res` is an external i.e. a function with specified syntax, or the name a Fortran subroutine or a C function (character string) with specified syntax or a list.

If `res` is a function, its syntax must be as follows:

```r = res(t,y,ydot)
```

where `t` is a real scalar (time) and `y` and `ydot` are real vector (state and derivative of the state). This function must return `r=g(t,y)-A(t,y)*ydot`.

If `res` is a character string, it refers to the name of a Fortran subroutine or a C function. See `SCI/modules/differential_equations/sci_gateway/fortran/Ex-impl.f` for an example to do that.

`res` can also be a list: see the help of ode.

The input `adda` is also an external.

If `adda` is a function, its syntax must be as follows:

```r = adda(t,y,p)
```

and it must return `r=A(t,y)+p` where `p` is a matrix to be added to `A(t,y)`.

If `adda` is a character string, it refers to the name of a Fortran subroutine or a C function. See `SCI/modules/differential_equations/sci_gateway/fortran/Ex-impl.f` for an example to do that.

`adda` can also be a list: see the help of ode.

The input `jac` is also an external.

If `jac` is a function, its syntax must be as follows:

```j = jac(t,y,ydot)
```

and it must return the Jacobian of `r=g(t,y)-A(t,y)*ydot` with respect to `y`.

If `jac` is a character string, it refers to the name of a Fortran subroutine or a C function. See `SCI/modules/differential_equations/sci_gateway/fortran/Ex-impl.f` for an example to do that.

`jac` can also be a list: see the help of ode.

### Examples

```y = impl([1;0;0], [-0.04;0.04;0], 0, 0.4, 'resid', 'aplusp');
// Using hot restart
//[x1,w,iw] = impl([1;0;0], [-0.04;0.04;0], 0, 0.2, 'resid', 'aplusp');
// hot start from previous call
//x1 = impl([1;0;0],[-0.04;0.04;0], 0.2, 0.4, 'resid', 'aplusp', w, iw);
//max(abs(x1-x))```

• dassl — differential algebraic equation
• ode — ordinary differential equation solver
• external — Scilab Object, external function or routine