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Solve fully implicit differential equations — variable order method

```
[t,y] =
ode15i(odefun,tspan,y0,yp0)
```

```
[t,y] =
ode15i(odefun,tspan,y0,yp0,options)
```

```
[t,y,te,ye,ie]
= ode15i(odefun,tspan,y0,yp0,options)
```

`sol = ode15i(___)`

`[`

also
uses the integration settings defined by `t`

,`y`

] =
ode15i(`odefun`

,`tspan`

,`y0`

,`yp0`

,`options`

)`options`

,
which is an argument created using the `odeset`

function.
For example, use the `AbsTol`

and `RelTol`

options
to specify absolute and relative error tolerances, or the `Jacobian`

option
to provide the Jacobian matrix.

`[`

additionally
finds where functions of `t`

,`y`

,`te`

,`ye`

,`ie`

]
= ode15i(`odefun`

,`tspan`

,`y0`

,`yp0`

,`options`

)`(t,y,y')`

, called event
functions, are zero. In the output, `te`

is the time
of the event, `ye`

is the solution at the time of
the event, and `ie`

is the index of the triggered
event.

For each event function, specify whether the integration is
to terminate at a zero and whether the direction of the zero crossing
matters. Do this by setting the `'Events'`

property
to a function, such as `myEventFcn`

or `@myEventFcn`

,
and creating a corresponding function: [`value`

,`isterminal`

,`direction`

]
= `myEventFcn`

(`t`

,`y`

,`yp`

).
For more information, see ODE Event Location.

returns
a structure that you can use with `sol`

= ode15i(___)`deval`

to evaluate
the solution at any point on the interval `[t0 tf]`

.
You can use any of the input argument combinations in previous syntaxes.

Providing the Jacobian matrix to

`ode15i`

is critical for reliability and efficiency. Alternatively, if the system is large and sparse, then providing the Jacobian sparsity pattern also assists the solver. In either case, use`odeset`

to pass in the matrices using the`Jacobian`

or`JPattern`

options.

`ode15i`

is a variable-step, variable-order
(VSVO) solver based on the backward differentiation formulas (BDFs)
of orders 1 to 5. `ode15i`

is designed to be used
with fully implicit differential equations and index-1 differential
algebraic equations (DAEs). The helper function `decic`

computes
consistent initial conditions that are suitable to be used with `ode15i`

[1].

[1] Lawrence F. Shampine, "Solving
0 = F(t, y(t), y′(t)) in MATLAB," *Journal
of Numerical Mathematics*, Vol.10, No.4, 2002, pp. 291-310.

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