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(odefun,[t0 tfinal],y0,yp0,...)
The following table lists the input arguments for ode15i
.
 A function handle that evaluates the left side of the differential equations, which are of the form f(t,y,y′) = 0. 
 A vector specifying the interval of integration, 
 Vectors of initial conditions for y and y′ respectively. 
 Optional integration argument created using the 
The following table lists the output arguments for ode15i
.
 Column vector of time points 
 Solution array. Each row in 
[T,Y] = ode15i(odefun,tspan,y0,yp0)
with tspan
= [t0 tf]
integrates the system of differential equations f(t,y,y′)
= 0 from time t0
to tf
with
initial conditions y0
and yp0
. odefun
is
a function handle. Function ode15i
solves ODEs
and DAEs of index 1. The initial conditions must be consistent, meaning
that f(t0,y0,yp0)
= 0. You can use the function decic
to compute
consistent initial conditions close to guessed values. Function odefun(t,y,yp)
,
for a scalar t
and column vectors y
and yp
,
must return a column vector corresponding to f(t,y,y′).
Each row in the solution array Y
corresponds to
a time returned in the column vector T
. To obtain
solutions at specific times t0,t1,...,tf
(all increasing
or all decreasing), use tspan = [t0,t1,...,tf]
.
Parameterizing Functions explains how to provide additional
parameters to the function odefun
, if necessary.
[T,Y] = ode15i(odefun,tspan,y0,yp0,options)
solves as above with default integration parameters replaced by property
values specified in options
, an argument created
with the odeset
function. Commonly used options
include a scalar relative error tolerance RelTol
(1e3
by
default) and a vector of absolute error tolerances AbsTol
(all
components 1e6
by default). See odeset
for
details.
[T,Y,TE,YE,IE] = ode15i(odefun,tspan,y0,yp0,options...)
with the 'Events'
property in options
set
to a function events
, solves as above while also
finding where functions of (t,y,y′),
called event functions, are zero. The function events
is
of the form [value,isterminal,direction] = events(t,y,yp)
and includes the necessary event
functions. Code the function events
so that the i
th
element of each output vector corresponds to the i
th
event. For the i
th event function in events
:
value(i)
is the value of the function.
isterminal(i) = 1
if the integration
is to terminate at a zero of this event function and 0
otherwise.
direction(i) = 0
if all zeros are
to be computed (the default), +1
if only the zeros
where the event function increases, and 1
if only
the zeros where the event function decreases.
Output TE
is a column vector of times at
which events occur. Rows of YE
are the corresponding
solutions, and indices in vector IE
specify which
event occurred. See Integrator Options in the MATLAB^{®} Mathematics
documentation for more information.
sol = ode15i(odefun,[t0 tfinal],y0,yp0,...)
returns
a structure that can be used with deval
to evaluate
the solution at any point between t0
and tfinal
.
The structure sol
always includes these fields:
 Steps chosen by the solver. If you specify the 
 Each column 
If you specify the Events
option and events
are detected, sol
also includes these fields:
 Points at which events, if any, occurred. 
 Solutions that correspond to events in 
 Indices into the vector returned by the function specified
in the 
ode15i
accepts the following parameters
in options
. For more information, see odeset
and Changing ODE Integration Properties in
the MATLAB Mathematics documentation.
Error control 

Solver output 

Event location 

Step size 

Jacobian matrix 

If you specify an output function as the value of the OutputFcn
property,
the solver calls it with the computed solution after each time step.
Four output functions are provided: odeplot
, odephas2
, odephas3
, odeprint
.
When you call the solver with no output arguments, it calls the default odeplot
to
plot the solution as it is computed. odephas2
and odephas3
produce
two and threedimensional phase plane plots, respectively. odeprint
displays
the solution components on the screen. By default, the ODE solver
passes all components of the solution to the output function. You
can pass only specific components by providing a vector of indices
as the value of the OutputSel
property. For example,
if you call the solver with no output arguments and set the value
of OutputSel
to [1,3]
, the solver
plots solution components 1 and 3 as they are computed.
The Jacobian matrices ∂f/∂y and ∂f/∂y′ are critical to reliability and efficiency. You can provide these matrices as one of the following:
Function of the form [dfdy,dfdyp] = FJAC(t,y,yp)
that computes the Jacobian matrices.
If FJAC
returns an empty matrix []
for
either dfdy
or dfdyp
, then ode15i
approximates
that matrix by finite differences.
Cell array of two constant matrices {dfdy,dfdyp}
,
either of which could be empty.
Use odeset
to set the Jacobian
option
to the function or cell array. If you do not set the Jacobian
option, ode15i
approximates
both Jacobian matrices by finite differences.
For ode15i
, Vectorized
is
a twoelement cell array. Set the first element to 'on'
if odefun(t,[y1,y2,...],yp)
returns [odefun(t,y1,yp),odefun(t,y2,yp),...]
.
Set the second element to 'on'
if odefun(t,y,[yp1,yp2,...])
returns [odefun(t,y,yp1),odefun(t,y,yp2),...]
.
The default value of Vectorized
is {'off','off'}
.
For ode15i
, JPattern
is
also a twoelement sparse matrix cell array. If ∂f/∂y or
∂f/∂y′
is a sparse matrix, set JPattern
to the sparsity
patterns, {SPDY,SPDYP}
. A sparsity pattern of ∂f/∂y is
a sparse matrix SPDY
with SPDY(i,j) =
1
if component i
of f(t,y,yp)
depends
on component j
of y
, and 0
otherwise.
Use SPDY = []
to
indicate that ∂f/∂y is
a full matrix. Similarly for ∂f/∂y′
and SPDYP
. The default value of JPattern
is {[],[]}
.
This example uses a helper function, decic
, to hold fixed the initial value for y(t0) and compute a consistent initial value for y'(t0) for the Weissinger implicit ODE. The Weissinger function evaluates the residual of the implicit ODE.
t0 = 1; y0 = sqrt(3/2); yp0 = 0; [y0,yp0] = decic(@weissinger,t0,y0,1,yp0,0);
Use ode15i
to solve the ODE, and then plot the numerical solution, y
, against the analytical solution, ytrue
.
[t,y] = ode15i(@weissinger,[1 10],y0,yp0);
ytrue = sqrt(t.^2 + 0.5);
plot(t,y,t,ytrue,'o')
The files, ihb1dae.m
and iburgersode.m
,
are examples of implicit ODEs.
decic
 deval
 ode113
 ode15s
 ode23
 ode23s
 ode23t
 ode23tb
 ode45
 odeget
 odeset