LinearOperators
FactoredGCRD
return the greatest common right divisor in the completely factored form
Calling Sequence
Parameters
Description
Examples
References
FactoredGCRD(U, V, x, case)
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')
Given a completely factored Ore operator U and a nonfactored 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].
$a\u2254\mathrm{FactoredOrePoly}\left(\left[1\,x\right]\,\left[3\,x\right]\right)$
${a}{\u2254}{\mathrm{FactoredOrePoly}}{}\left(\left[{\mathrm{1}}{\,}{x}\right]{\,}\left[{3}{\,}{x}\right]\right)$
$b\u2254\mathrm{OrePoly}\left(0\,0\,2{x}^{3}\+4{x}^{2}\,{x}^{4}\right)$
${b}{\u2254}{\mathrm{OrePoly}}{}\left({0}{\,}{0}{\,}{2}{}{{x}}^{{3}}{+}{4}{}{{x}}^{{2}}{\,}{{x}}^{{4}}\right)$
${\mathrm{LinearOperators}}_{\mathrm{FactoredGCRD}}\left(a\,b\,x\,\'\mathrm{differential}\'\right)$
${\mathrm{FactoredOrePoly}}{}\left(\left[{}\frac{{1}}{{x}}{\,}{1}\right]\right)$
Abramov, S.A., and Zima, E.V. "Minimal Completely Factorable Annihilators." Proc. ISSAC'97. 1997.
See Also
LinearOperators[converters]
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