Martin R wrote:
>
> Am Samstag, 13. August 2016 22:08:16 UTC+2 schrieb Waldek Hebisch:
>
> Well, Sage uses Maxima as its default integrator. There are whole
> > classes of functions that FriCAS can integrate and Maxima can not
> > (the opposite happens, but is rare). Also, it is not hard
> > to find examples where Maxima gives nonelemetary answer when
> > elementary integral exists. FriCAS answers are irredundant:
> > nonelementary parts are necessary to express the answer.
>
>
> integration is one (and so far the only) part of sage which actually uses
> FriCAS (optionally).
>
>
> > FriCAS has solver for differential linear ODE-s of higher
> > order and for systems. IIUC Sage (via Maxima) is limited to
> > order 2.
> >
>
> Great, I added an example from one of the input files. (I know nothing
> hardly anything about ODE's.)
>
> I belive that FriCAS limit command is stronger than Maxima
> > and Sympy. The difference here is probably smaller than in
> > case of integrator, but still there is reason to call
> > FriCAS limit.
> >
>
> OK, I'll check!
>
>
> > I wonder if Sage has symbolic Jordan decomposition? FriCAS
> > has (under name generalizedEigenvectors).
>
>
> I don't know what you mean here. Sage has Jordan decomposition over
> algebraic numbers.
>
> I checked generalizedEigenvectors matrix [[1, x], [0, 1]] but this gives a
> wrong result:
>
> (5) -> m := matrix([[1,x],[0,1]])
>
> +1 x+
> (5) | |
> +0 1+
> Type:
> Matrix(Polynomial(Integer))
> (6) -> generalizedEigenvectors m
>
> +0+ +1+
> (6) [[eigval= 1,geneigvec= [| |,| |]]]
> +1+ +0+
> Type: List(Record(eigval:
> Union(Fraction(Polynomial(Integer)),SuchThat(Symbol,Polynomial(Integer))),geneigvec:
>
> List(Matrix(Fraction(Polynomial(Integer))))))
Why do you think it is wrong? The first vector is an eigenvector,
the second is in kernel of (A - 1)^2 and linearly independent of the
first. This gives Jordan form:
[ 1 lambda ]
[ ]
[ 0 1 ]
Matrices with different lambda are equivalent, so this is
Jordan form of input matrix. More generally, generalized
eigenvectors corresponding to lambda gives you basis of
subspace where (A - lambda) is nilpotent. Since in
your case m - 1 is nilpotent the result is rather trivial.
--
Waldek Hebisch
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