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LinearOperators

  

FactoredMinimalAnnihilator

  

construct the minimal annihilator in the completely factored form

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

FactoredMinimalAnnihilator(expr, x, case)

Parameters

expr

-

Maple expression

x

-

name of the independent variable

case

-

parameter indicating the case of the equation ('differential' or 'shift')

Description

• 

Given a d'Alembertian term expr, the LinearOperators[FactoredMinimalAnnihilator] function returns a factored Ore operator that is the minimal annihilator in the completely factored form for expr. That is, applying this operator to expr yields zero.

• 

A completely factored Ore operator is an operator that can be factored into a product of linear factors.

• 

A completely factored Ore operator is represented by a structure that consists of the keyword FactoredOrePoly and a sequence of lists. Each list consists of two elements and describes a first degree factor. The first element provides the zero degree coefficient and the second element provides the first degree coefficient. For example, in the differential case with a differential operator D, FactoredOrePoly([-1, x], [x, 0], [4, x^2], [0, 1]) describes the operator 1+xDxDx2+4D.

• 

There are routines in the package that convert between Ore operators and the corresponding Maple expressions. See LinearOperators[converters].

• 

The expression expr must be a d'Alembertian term. The main property of a d'Alembertian term is that it is annihilated by a linear operator that can be written as a composition of operators of the first degree. The set of d'Alembertian terms has a ring structure. The package recognizes some basic d'Alembertian terms and their ring-operation closure terms. The result of the substitution of a rational term for the independent variable in the d'Alembertian term is also a d'Alembertian term.

Examples

exprxlnxx+1

expr:=xlnxx+1

(1)

LLinearOperators[FactoredMinimalAnnihilator]expr,x,'differential'

L:=FactoredOrePoly1x1x,1,1x,1

(2)

LinearOperators[Apply]L,expr,x,'differential'

0

(3)

exprΓnn2

expr:=Γnn2

(4)

LLinearOperators[FactoredMinimalAnnihilator]expr,n,'shift'

L:=FactoredOrePolyn2+2n+1n,1

(5)

simplifyLinearOperators[Apply]L,expr,n,'shift'

0

(6)

References

  

Abramov, S.A., and  Zima, E.V. "Minimal Completely Factorable Annihilators." Proc. ISSAC'97. 1997.

See Also

LinearOperators

LinearOperators[Apply]

LinearOperators[converters]

 


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