Yrogirg wrote:
>
> >Does it make things more clear to you?
>
> yes, that's exactly what I needed.
>
> >PS: I still hate Expression(?) since with a little more thought one can
> almost always do better (and in the end faster).
>
> I have the same feeling, but I guess that might require mathematical
> knowledge and level of abstraction I don't posses by now.
> To make things clear I want to solve Hamiltonian equations. And I want
> to do it using formal solution involving exponent of an operator.
> http://en.wikipedia.org/wiki/Symplectic_integrator formula (3) is that
> formal solution. And I need approximation of this exponent to be done
> symbolically since employing numerical differentiation operator
> promises to be a pain.
>
Do you realize that the 'exp' in this formula is just a shorthand
notation for solution of appropriate differential equation?
If you want to do some symbolic manipulation on it than you
may define an operator and work in Expression(Integer).
However, you should use some different name than 'exp', say:
my_exp := oprator 'my_exp
my_exp(PoissonBracket(f, Hamiltonian))
Note: I have used here _value_ of PoissonBracket(f, Hamiltonian)
as argument to 'my_exp'. In FriCAS it is not possible to
perform symbolic manipulations on _functions_, one have to
use expressions. Note that if f is sufficiently general
then this expression will behave like a function.
OTOH there are only few simplifications possible for 'my_exp'
and you would have to code them (for example as rewrite rules),
so you may prefer using series instead of expressions.
BTW: If your functions are simple enough FriCAS may be able
to find closed form solution and effectively compute 'my_exp'.
However, most interstiong equations do not have closed form
solution and even if they have it is possible that FriCAS
will miss it. So I would not count on being able to
compute 'my_exp' explicitely. But if you have explicit f
and explicit Hamiltonian it makes sense to try.
> >After that my next question will be: How can that be made algorithmic in
> general? You probably want an answer in finite time. Or would you
> accept
> approximate values for this "exp"?
>
> Yes approximate will do, ultimately all these symbolic stuff is
> required to produce a numeric functions. So its not to be some real
> exponent, but rather a finite term series (at least ~32 terms in the
> series). Well, lets call it MyExp to avoid confusion.
>
Then you may wish to look at FriCAS power series machinery. There
is a function that given a (nonsingular) differential equation will
produce power series solution. Of course, in practice you get
only finitely many terms, but you may ask for more and the number
of terms is only limited by computer memory and time.
However to use solver for power series you need a single
parameter for expansion (time is good candidate). You can
keep other parameters symbolic.
--
Waldek Hebisch
[email protected]
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