LinearOperators[MinimalAnnihilator]  construct the minimal annihilator

Calling Sequence


MinimalAnnihilator(L, expr, x, case)


Parameters


L



completely factored Ore operator

expr



Maple expression

x



name of the independent variable

case



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





Description


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Given a factored Ore operator L that is an annihilator for the expression expr, the LinearOperators[MinimalAnnihilator] function returns the minimal annihilator in nonfactored form for expr.

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A completely factored Ore operator is an operator that can be factored into a product of factors of degree at most one.

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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 .

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An Ore operator is a structure that consists of the keyword OrePoly with a sequence of coefficients starting with the one of degree zero. The coefficients must be rational functions in x. For example, in the differential case with the differential operator D, OrePoly(2/x, x, x+1, 1) represents the operator .

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

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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 ringoperation 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.


Note: The operator L must annihilate expr, that is, satisfy L(expr)=0.



Examples


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References



Abramov, S. A., and Zima, E. V. "Minimal Completely Factorable Annihilators." In Proceedings of ISSAC '97, pp. 290297. Edited by Wolfgang Kuchlin. New York: ACM Press, 1997.



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