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The Satellite
package contains Maple procedures
to determine satellite unknowns in linear differential systems.
These procedures implement partial algorithms, so they
cannot be applied to all differential systems and thus
they solve the problem in some cases. In other cases
they do not give any answer (nor positive, nor negative).
We consider a differential system S of the form
y'=Ay or Ary(r)+Ar-1y(r-1)+ … +A1y'+A0y=0
where A, A_0, …, A_r are n x
n matrices over K=Q(x),
y=(y1,…,yn)T is a vector of unknowns.
We assume that some unknowns (entries of the vector y) are selected.
Denote the set of selected unknowns by s.
Definition. The unselected unknown yj is called satellite unknown for the set of selected unknowns s in S if minimal subfield of a Picard–Vessio field over K for S, that contains all selected components of all solutions to S, also contains the yj component of any solution.
The Satellite
package exports two procedures:
Testing
;Determination
.
Testing(A, s, v)
A
— square matrix of the normal differential system y'=Ays
— set of positive integers — indices of selected unknownsv
— positive integer - index of the testing unknown
Testing
procedure determines whether the unknown of index
v
(yv
)
of differential system y'=Ay is a satellite
for the set of selected unknowns s.
Testing
is an implementation of partial algorithm
and so it returns “true” if it is able to determine that
yv
is a satellite;
otherwise it returns “FAIL”. It also
returns “FAIL” in a case when
yv
is a satellite for s
but the partial algorithm cannot determine this.
Testing
procedure cannot determine
if y<sub>v
/<sub> is not a satellite.
Determination(M, s)
M
— square matrix of the normal differential system y'=Ay or a list of high-order differential system matricess
— set of positive integers — indices of selected unknowns
Determination
procedure is based on Testing
procedure
and intended to determine the set of satellite unknowns for a given
system and a fixed set of selected unknowns.
It builds a partition of the set of unselected unknown indices that contains three parts:
Testing
algorithm cannot determine if they are satellite;sat.mpl — the Maple code of the package
sat_sample.pdf — the pdf copy of Maple session with examples