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LinearOperators

 FactoredGCRD
 return the greatest common right divisor in the completely factored form

 Calling Sequence FactoredGCRD(U, V, x, case)

Parameters

 U - a completely factored Ore operator V - an Ore operator x - the name of the independent variable case - a parameter indicating the case of the equation ('differential' or 'shift')

Description

 • Given a completely factored Ore operator U and a non-factored Ore operator V, the LinearOperators[FactoredGCRD] function returns the greatest common right divisor (GCRD) in the completely factored form.
 • 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 $\left(-1+\mathrm{xD}\right)\left(x\right)\left({x}^{2}\mathrm{D}+4\right)\left(\mathrm{D}\right)$.
 • 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 $\frac{2}{x}+x\mathrm{D}+\left(x+1\right){\mathrm{D}}^{2}+{\mathrm{D}}^{3}$.
 • There are routines in the package that convert between Ore operators and the corresponding Maple expressions. See LinearOperators[converters].

Examples

 > $a≔\mathrm{FactoredOrePoly}\left(\left[-1,x\right],\left[3,x\right]\right)$
 ${a}{≔}{\mathrm{FactoredOrePoly}}{}\left(\left[{-1}{,}{x}\right]{,}\left[{3}{,}{x}\right]\right)$ (1)
 > $b≔\mathrm{OrePoly}\left(0,0,2{x}^{3}+4{x}^{2},{x}^{4}\right)$
 ${b}{≔}{\mathrm{OrePoly}}{}\left({0}{,}{0}{,}{2}{}{{x}}^{{3}}{+}{4}{}{{x}}^{{2}}{,}{{x}}^{{4}}\right)$ (2)
 > $\mathrm{LinearOperators}\left[\mathrm{FactoredGCRD}\right]\left(a,b,x,'\mathrm{differential}'\right)$
 ${\mathrm{FactoredOrePoly}}{}\left(\left[{-}\frac{{1}}{{x}}{,}{1}\right]\right)$ (3)

References

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