LinearOperators
FactoredAnnihilator
construct completely factorable annihilator
Calling Sequence
Parameters
Description
Examples
References
FactoredAnnihilator(expr, x, case)
expr

an algebraic expression
x
the name of the independent variable
case
a parameter indicating the case of the equation ('differential' or 'shift')
Given an algebraic expression expr, the LinearOperators[FactoredAnnihilator] function returns a completely factored Ore operator that is an annihilator for expr. That is, applying this operator to expr yields zero. If such an operator does not exist, the function returns $\mathrm{1}$.
A completely factored Ore operator is an operator that can be factored into a product of factors of degree at most one.
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 $\left(1+\mathrm{xD}\right)\left(x\right)\left({x}^{2}\mathrm{D}+4\right)\left(\mathrm{D}\right)$.
There are routines in the package that convert between Ore operators and the corresponding Maple expressions. See LinearOperators[converters].
To get the completely factorable annihilator, 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.
$\mathrm{expr}\u2254\mathrm{exp}\left(\frac{x}{x1}\right)+\mathrm{sqrt}\left({x}^{2}+1\right)$
${\mathrm{expr}}{\u2254}{{\ⅇ}}^{\frac{{x}}{{x}{}{1}}}{+}\sqrt{{{x}}^{{2}}{+}{1}}$
$L\u2254\mathrm{LinearOperators}\left[\mathrm{FactoredAnnihilator}\right]\left(\mathrm{expr}\,x\,'\mathrm{differential}'\right)$
${L}{\u2254}{\mathrm{FactoredOrePoly}}{}\left(\left[\frac{{{x}}^{{4}}{+}{6}{}{{x}}^{{2}}{}{2}{}{x}{+}{3}}{\left({{x}}^{{3}}{}{{x}}^{{2}}{+}{x}{+}{1}\right){}\left({{x}}^{{2}}{+}{1}\right){}\left({x}{}{1}\right)}{\,}{1}\right]{\,}\left[\frac{{1}}{{\left({x}{}{1}\right)}^{{2}}}{\,}{1}\right]\right)$
$\mathrm{LinearOperators}\left[\mathrm{Apply}\right]\left(L\,\mathrm{expr}\,x\,'\mathrm{differential}'\right)$
${0}$
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.
See Also
LinearOperators[Apply]
LinearOperators[converters]
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