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 eigenring
 compute the endomorphisms of the solution space
 endomorphism_charpoly
 give the characteristic polynomial of an endomorphism

 Calling Sequence eigenring(L, domain) endomorphism_charpoly(L, r, domain)

Parameters

 L - differential operator r - differential operator in the output of eigenring domain - list containing two names

Description

 • The input L is a differential operator. Denote V(L) as the solution space of L. eigenring computes a basis (a vector space) of the set of all operators r for which r(V(L)) is a subset of V(L). So r is an endomorphism of the solution space V(L). The characteristic polynomial of this map can be computed by the command endomorphism_charpoly(L,r).
 • For endomorphisms r, the product of L and r is divisible on the right by L. If the optional third argument is the equation verify=true then eigenring checks if the output satisfies this condition. This should not be necessary though.
 • The argument domain describes the differential algebra. If this argument is the list $\left[\mathrm{Dt},t\right]$ then the differential operators are notated with the symbols $\mathrm{Dt}$ and $t$. They are viewed as elements of the differential algebra C(t)[Dt] where $C$ is the field of constants.
 • If the argument domain is omitted then the differential specified by the environment variable _Envdiffopdomain will be used. If this environment variable is not set, then the argument domain may not be omitted.
 • These functions are part of the DEtools package, and so they can be used in the form eigenring(..) and endomorphism_charpoly(..) only after executing the command with(DEtools).  However, they can always be accessed through the long form of the command by using DEtools[eigenring](..) or DEtools[endomorphism_charpoly](..).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$

Take the differential ring C(x)[Dx]:

 > $A≔\left[\mathrm{Dx},x\right]$
 ${A}{≔}\left[{\mathrm{Dx}}{,}{x}\right]$ (1)
 > $L≔{\mathrm{Dx}}^{4}+2x{\mathrm{Dx}}^{2}+2\mathrm{Dx}+{x}^{2}-4$
 ${L}{≔}{{\mathrm{Dx}}}^{{4}}{+}{2}{}{x}{}{{\mathrm{Dx}}}^{{2}}{+}{{x}}^{{2}}{+}{2}{}{\mathrm{Dx}}{-}{4}$ (2)

Compute a basis v for the endomorphisms$r:V\left(L\right)\to V\left(L\right)$. Compute an eigenvalue $e$ of $r$. Then compute the greatest common right divisor $G$. Then the solution space $V\left(G\right)$ is the kernel of$r-e:V\left(L\right)\to V\left(L\right)$.

 > $v≔\mathrm{eigenring}\left(L,A\right)$
 ${v}{≔}\left[{1}{,}{{\mathrm{Dx}}}^{{2}}{+}{x}\right]$ (3)
 > $\mathbf{for}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}r\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}v\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{print}\left('r'=r\right);\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}F≔\mathrm{factor}\left(\mathrm{endomorphism_charpoly}\left(L,r,A\right)\right);\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{for}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}e\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\left\{\mathrm{solve}\left(F,x\right)\right\}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{print}\left(G=\mathrm{GCRD}\left(r-e,L,A\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end do}$
 ${r}{=}{1}$
 ${G}{=}{{\mathrm{Dx}}}^{{4}}{+}{2}{}{x}{}{{\mathrm{Dx}}}^{{2}}{+}{2}{}{\mathrm{Dx}}{+}{{x}}^{{2}}{-}{4}$
 ${r}{=}{{\mathrm{Dx}}}^{{2}}{+}{x}$
 ${G}{=}{{\mathrm{Dx}}}^{{2}}{+}{x}{+}{2}$
 ${G}{=}{{\mathrm{Dx}}}^{{2}}{+}{x}{-}{2}$ (4)

References

 For a description of the method used, see:
 van der Put, M., and Singer, M. F. Galois Theory of Linear Differential Equations, Vol. 328. Springer: 2003. An electronic version of this book is available at http://www4.ncsu.edu/~singer/ms_papers.html.
 van Hoeij, M. "Rational Solutions of the Mixed Differential Equation and its Application to Factorization of Differential Operators." ISSAC '96 Proceedings. (1996): 219-225.