The MinimalEquation constructs minimal equation (y1-consequence) for the given system corresponding to the matrix M and Ore algebra A and a given function with the index n of its undetermined functions. The minimal equation is a scalar equation whose solution is the component of some solution of the given system corresponding to the given function. The output is OrePoly L, such that $Ly$ is the y1-consequence, i.e. the minimal equation such that its solution is the n-th element of some solution of the given system.

The ReducedSystem reduces the given system corresponding to the matrix M and Ore algebra A with respect to a given subset ns of its undetermined functions. The reduced system is a system which contains the given subset of the undetermined functions of the given system among its undetermined functions and components of its closed form solution corresponding to the given subset are components of some solution of the given system. It addresses the following problem: given a subset of the components of solutions to be found and an appropriate class of functions, find all solutions whose specified components are in the given class (more precisely we are interested in computing those components only). For example, given a differential system, find all the rational functions that are first and second components of a solution of the system. The output is $\left[R\,s\right]$, where R is the matrix of the reduced system and s is the set of pairs, the first element of each is the index out of ns and the second one is the index of the same undetermined function in the reduced system.

$\mathrm{with}\left(\mathrm{OreTools}\right)\:$

$\mathrm{with}\left(\mathrm{OreTools}\left[\mathrm{Consequences}\right]\right)\:$

$A≔\mathrm{SetOreRing}\left(x\,'\mathrm{differential}'\right)$

$A≔\mathrm{UnivariateOreRing}\left(x\,\mathrm{differential}\right)$

$M≔\mathrm{Matrix}\left(\left[\left[0\,0\,1\right]\,\left[0\,1\,0\right]\,\left[1\,0\,0\right]\right]\right)$

$M≔\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right]$

$\mathrm{ReducedSystem}\left(M\,\left\{1\,3\right\}\,A\right)$

$\left[\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\,\left\{\left[1\,1\right]\,\left[3\,2\right]\right\}\right]$

$\mathrm{MinimalEquation}\left(M\,1\,A\right)$

$\mathrm{OrePoly}\left(\mathrm{-1}\,0\,1\right)$

$\mathrm{MinimalEquation}\left(M\,2\,A\right)$

$\mathrm{OrePoly}\left(\mathrm{-1}\,1\right)$

$A≔\mathrm{SetOreRing}\left(n\,'\mathrm{shift}'\right)$

$A≔\mathrm{UnivariateOreRing}\left(n\,\mathrm{shift}\right)$

$M≔\mathrm{Matrix}\left(\left[\left[1\,-\frac{1}{n{\left(n+1\right)}^2}\,-n\,1\right]\,\left[1\,\frac{n^2+2n-1}{n\left(n+1\right)}\,n\,-1\right]\,\left[1\,0\,1\,0\right]\,\left[n+1\,0\,0\,\frac{n+1}{n}\right]\right]\right)$

$M≔\left[\begin{array}{cccc}1& -\frac{1}{n{\left(n+1\right)}^2}& -n& 1\\ 1& \frac{n^2+2n-1}{n\left(n+1\right)}& n& \mathrm{-1}\\ 1& 0& 1& 0\\ n+1& 0& 0& \frac{n+1}{n}\end{array}\right]$

$\mathrm{ReducedSystem}\left(M\,\left\{1\,2\right\}\,A\right)$

$\left[\left[\begin{array}{ccc}0& 1& 0\\ -\frac{n^4+6n^3+13n^2+14n+4}{n\left(n^2+3n+2\right)\left(n+2\right)}& \frac{2n^4+11n^3+21n^2+17n+4}{n\left(n^2+3n+2\right)\left(n+2\right)}& \frac{3n^2+6n+2}{n^2\left(n^4+5n^3+9n^2+7n+2\right)}\\ 2& \mathrm{-1}& \frac{n^3+3n^2+n-2}{n\left(n^2+2n+1\right)}\end{array}\right]\,\left\{\left[1\,1\right]\,\left[2\,3\right]\right\}\right]$