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Table of Contents

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
*A _{r}y*

where A, A_0, …, A_r are *n* `x`

*n* matrices over *K*=** Q**(

**Definition.**
The unselected unknown *y _{j}* is called

The `Satellite`

package exports two procedures:

`Testing`

;`Determination`

.

`Testing(A, s, v)`

`A`

— square matrix of the normal differential system*y*'=*Ay*`s`

— set of positive integers — indices of selected unknowns`v`

— positive integer - index of the testing unknown

`Testing`

procedure determines whether the unknown of index
`v`

(*y*_{v})
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
*y*_{v} is a satellite;
otherwise it returns “**FAIL**”. It also
returns “**FAIL**” in a case when
*y*_{v} 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 matrices`s`

— 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:

- indices of unselected unknowns that are satellite for selected unknowns
*s*; - indices of unselected unknowns for which
`Testing`

algorithm cannot determine if they are satellite; - indices of unselected unknowns that are not satellite for sure.

sat.mpl — the Maple code of the package

sat_sample.pdf — the pdf copy of Maple session with examples

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