PROFESSOR: We can get to documentation for the MATLAB
ODE Suite by entering this command at the MATLAB
prompt-- doc ode45.
This will bring us to an extensive documentation
for MATLAB ode45 that includes among other things
this chart that compares MATLAB ODE solvers.
There are seven of them and this compares
their various attributes.
As we've said before, MATLAB ode45 is the workhorse.
It's a nonstiff solver with medium accuracy
that is the first one you should try,
and we use it most of the time.
I have a soft heart in my heart for MATLAB ode23.
It's a nonstiff solver with low accuracy,
but its accuracy that's appropriate
for graphics work because the step size it chooses
is appropriate for most graphics work.
MATLAB ode113 we haven't talked about it yet,
there could be a comma between the 1 and the 13
here because this is a variable order
method where the order varies all the way from 1 to 13.
It's a multi-step method that saves history.
If, you know about these things, it's an Adams-Moulton method.
I associate this with worked done at Jet Propulsion
Laboratory years ago for computing orbits of planets
and satellites, which of course are very smooth
and go on for years.
It can have very high accuracy requirements.
Then there are the stiff solvers.
There are four of them-- 15s, 23s, and the twins,
the trapezoid rules.
15s is the primary stiff solver, low to medium accuracy.
If you find ode45 is slow, taking
lots of steps-- indication that the problem is stiff-- try 15s.
23s can be as a low order method, low accuracy,
and used at crude error tolerances.
We haven't talked about mass matrices.
This is where there's a matrix in front
of the derivative term, and this can be used with constant mass
matrices.
And then the two routines with T's in their name
are based on the trapezoidal rule,
and they're for use with problems
without any numerical damping.
You can see the documentation for more details
on the trapezoid methods.
That's the MATLAB ODE Suite seven solvers,
three for nonstiff problems and four for stiff problems.
You may well get through with never using anything
but ode45 may well serve all your needs.
There's a second chart in the documentation that
summarizes all the options that are available through the ODE
Set function.
We've briefly mentioned the tolerances, RelTol and AbsTol--
the output function-- these are available in all seven
of the solvers.
There are various other functions--
various other options-- available for more
specialized work, including event handling,
providing a Jacobian to the stiff solvers
so they don't have to work so hard
taking numerical differences, options
associated with the mass matrices,
providing a limit on the step size.
These are all our options that can
be specified through ode Set for more specialized work
with the ODE solvers.