LinearFunctionalSystems - Maple Programming Help

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LinearFunctionalSystems

 Properties
 return the basic properties of the system

 Calling Sequence Properties(sys, vars) Properties(A, b, x, case) Properties(A, x, case)

Parameters

 sys - list of equations; linear functional system vars - list of function variables such as [y1(x), y2(x), ...]; variables to solve for A - Matrix with rational elements b - Vector with rational elements x - independent variable case - name indicating the case of the system; one of 'differential', 'difference', or 'qdifference'

Description

 • The Properties function returns the basic properties of the specified linear functional system of equations with polynomial coefficients. A test for the correctness of the input is also performed.
 • The result returned by this function is a table that contains the different properties of the given system. The table has the following indices (where applicable):

 'method'    - either 'quasimodular'  if the quasimodular version of the EG-algorithm is invoked or 'ordinary' if the ordinary version is invoked 'input'     - either 'lists' for list input type or 'matrices' for matrix input type 'case'      - 'differential', 'difference', or 'qdifference' 'order'     - order of the system 'var'       - independent variable 'q_par'     - name of the parameter q in the q-difference case extracted from the given system, or it's value in case the parameter is a number 'system'    - system in list form 'functions' - functions to be solved for 'fnames'    - names of the functions to be solved for 'shifts'    - shifts of function variables in the difference case 'qshifts'   - q-shifts of function variables in the q-difference case 'mat'       - matrix of the matrix input form 'vec'       - vector of the matrix input form 'matdenom'  - list of denominators of the rows of the matrix for the matrix input form

 • The error conditions associated with Properties are the following:

 1. At least two arguments must be specified. 2. For the matrix input type: (a) the matrix must be square (b) the number of arguments is at most four (c) if the number of arguments is three, the third argument must specify the case of the system (d) the second argument must be a name for the 3-argument case, and a vector for the 4-argument case (e) vector and matrix sizes must be the same (f) if the number of arguments is four, the third argument must be a name specifying the independent variable, the fourth argument must specify the case of the system 3. For the list input type: (a) the number of arguments is two (b) the first argument must be a list of equations, the second argument must be a list of functions to be solved for (c) the independent variable must be specified only once (d) the system must be linear (e) the number of equations must be equal to the number of functions to be solved for 4. The functions to be solved for must appear in equations. 5. There exist unknown function names. 6. There exist extra independent variables in the system. 7. The type of the system is not supported.

 • This function is part of the LinearFunctionalSystems package, and so it can be used in the form Properties(..) only after executing the command with(LinearFunctionalSystems). However, it can always be accessed through the long form of the command by using the form LinearFunctionalSystems[Properties](..).

Examples

 > $\mathrm{with}\left(\mathrm{LinearFunctionalSystems}\right):$
 > $\mathrm{sys}≔\left[\mathrm{diff}\left(\mathrm{y1}\left(x\right),x\right)\left(-1+{x}^{2}\right)+\left(6+2x\right)\mathrm{y1}\left(x\right)+6\mathrm{y3}\left(x\right)-6\mathrm{y4}\left(x\right)=0,\mathrm{diff}\left(\mathrm{y2}\left(x\right),x\right)\cdot 2x{\left(1+{x}^{2}\right)}^{2}-\left(4+x\right)\mathrm{y2}\left(x\right){\left(1+{x}^{2}\right)}^{2}+4\mathrm{y3}\left(x\right){\left(1+{x}^{2}\right)}^{2}=-4{x}^{2}-\left(4+x\right)\left(1+{x}^{2}\right),\mathrm{diff}\left(\mathrm{y3}\left(x\right),x\right)\left(1+{x}^{2}\right)\cdot 2x-\left(4+3x\right)\left(1+{x}^{2}\right)\mathrm{y2}\left(x\right)+4\left(1+{x}^{2}\right)\mathrm{y3}\left(x\right)=-\left(4+3x\right),\mathrm{diff}\left(\mathrm{y4}\left(x\right),x\right)\left(1+{x}^{2}\right)\left({x}^{2}-1\right)\cdot 2x-\left(1+{x}^{2}\right)\cdot 2x\left(-7+{x}^{2}-2x\right)\mathrm{y1}\left(x\right)-\mathrm{y2}\left(x\right)\left(4+3x\right)\left(1+{x}^{2}\right)\left({x}^{2}-1\right)+\mathrm{y3}\left(x\right)\left(6x+2{x}^{2}-2\right)\left(1+{x}^{2}\right)\cdot 2-\mathrm{y4}\left(x\right)\cdot 6\left(1+{x}^{2}\right)\cdot 2x=-\left(4+3x\right)\left({x}^{2}-1\right)\right]$
 ${\mathrm{sys}}{≔}\left[\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y1}}{}\left({x}\right)\right){}\left({{x}}^{{2}}{-}{1}\right){+}\left({6}{+}{2}{}{x}\right){}{\mathrm{y1}}{}\left({x}\right){+}{6}{}{\mathrm{y3}}{}\left({x}\right){-}{6}{}{\mathrm{y4}}{}\left({x}\right){=}{0}{,}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y2}}{}\left({x}\right)\right){}{x}{}{\left({{x}}^{{2}}{+}{1}\right)}^{{2}}{-}\left({4}{+}{x}\right){}{\mathrm{y2}}{}\left({x}\right){}{\left({{x}}^{{2}}{+}{1}\right)}^{{2}}{+}{4}{}{\mathrm{y3}}{}\left({x}\right){}{\left({{x}}^{{2}}{+}{1}\right)}^{{2}}{=}{-}{4}{}{{x}}^{{2}}{-}\left({4}{+}{x}\right){}\left({{x}}^{{2}}{+}{1}\right){,}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y3}}{}\left({x}\right)\right){}\left({{x}}^{{2}}{+}{1}\right){}{x}{-}\left({4}{+}{3}{}{x}\right){}\left({{x}}^{{2}}{+}{1}\right){}{\mathrm{y2}}{}\left({x}\right){+}{4}{}\left({{x}}^{{2}}{+}{1}\right){}{\mathrm{y3}}{}\left({x}\right){=}{-}{4}{-}{3}{}{x}{,}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y4}}{}\left({x}\right)\right){}\left({{x}}^{{2}}{+}{1}\right){}\left({{x}}^{{2}}{-}{1}\right){}{x}{-}{2}{}\left({{x}}^{{2}}{+}{1}\right){}{x}{}\left({{x}}^{{2}}{-}{2}{}{x}{-}{7}\right){}{\mathrm{y1}}{}\left({x}\right){-}{\mathrm{y2}}{}\left({x}\right){}\left({4}{+}{3}{}{x}\right){}\left({{x}}^{{2}}{+}{1}\right){}\left({{x}}^{{2}}{-}{1}\right){+}{2}{}{\mathrm{y3}}{}\left({x}\right){}\left({2}{}{{x}}^{{2}}{+}{6}{}{x}{-}{2}\right){}\left({{x}}^{{2}}{+}{1}\right){-}{12}{}{\mathrm{y4}}{}\left({x}\right){}\left({{x}}^{{2}}{+}{1}\right){}{x}{=}{-}\left({4}{+}{3}{}{x}\right){}\left({{x}}^{{2}}{-}{1}\right)\right]$ (1)
 > $\mathrm{vars}≔\left[\mathrm{y1}\left(x\right),\mathrm{y2}\left(x\right),\mathrm{y3}\left(x\right),\mathrm{y4}\left(x\right)\right]:$
 > $\mathrm{props}≔\mathrm{Properties}\left(\mathrm{sys},\mathrm{vars}\right)$
 ${\mathrm{props}}{≔}{table}{}\left(\left[{\mathrm{q_par}}{=}{\mathrm{__q}}{,}{\mathrm{case}}{=}{\mathrm{differential}}{,}{\mathrm{f_names}}{=}\left[{\mathrm{y1}}{,}{\mathrm{y2}}{,}{\mathrm{y3}}{,}{\mathrm{y4}}\right]{,}{\mathrm{var}}{=}{x}{,}{\mathrm{input}}{=}{\mathrm{lists}}{,}{\mathrm{system0}}{=}\left[\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y1}}{}\left({x}\right)\right){}{{x}}^{{2}}{+}{2}{}{\mathrm{y1}}{}\left({x}\right){}{x}{-}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y1}}{}\left({x}\right){+}{6}{}{\mathrm{y1}}{}\left({x}\right){+}{6}{}{\mathrm{y3}}{}\left({x}\right){-}{6}{}{\mathrm{y4}}{}\left({x}\right){,}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y2}}{}\left({x}\right)\right){}{{x}}^{{5}}{-}{\mathrm{y2}}{}\left({x}\right){}{{x}}^{{5}}{+}{4}{}{\mathrm{y3}}{}\left({x}\right){}{{x}}^{{4}}{-}{4}{}{\mathrm{y2}}{}\left({x}\right){}{{x}}^{{4}}{+}{4}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y2}}{}\left({x}\right)\right){}{{x}}^{{3}}{-}{2}{}{\mathrm{y2}}{}\left({x}\right){}{{x}}^{{3}}{+}{8}{}{\mathrm{y3}}{}\left({x}\right){}{{x}}^{{2}}{-}{8}{}{\mathrm{y2}}{}\left({x}\right){}{{x}}^{{2}}{+}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y2}}{}\left({x}\right)\right){}{x}{-}{\mathrm{y2}}{}\left({x}\right){}{x}{+}{4}{}{\mathrm{y3}}{}\left({x}\right){-}{4}{}{\mathrm{y2}}{}\left({x}\right){,}{-}{3}{}{\mathrm{y2}}{}\left({x}\right){}{{x}}^{{3}}{+}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y3}}{}\left({x}\right)\right){}{{x}}^{{3}}{+}{4}{}{\mathrm{y3}}{}\left({x}\right){}{{x}}^{{2}}{-}{4}{}{\mathrm{y2}}{}\left({x}\right){}{{x}}^{{2}}{-}{3}{}{\mathrm{y2}}{}\left({x}\right){}{x}{+}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y3}}{}\left({x}\right)\right){}{x}{+}{4}{}{\mathrm{y3}}{}\left({x}\right){-}{4}{}{\mathrm{y2}}{}\left({x}\right){,}{-}{2}{}{\mathrm{y1}}{}\left({x}\right){}{{x}}^{{5}}{-}{3}{}{\mathrm{y2}}{}\left({x}\right){}{{x}}^{{5}}{+}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y4}}{}\left({x}\right)\right){}{{x}}^{{5}}{+}{4}{}{\mathrm{y1}}{}\left({x}\right){}{{x}}^{{4}}{+}{4}{}{\mathrm{y3}}{}\left({x}\right){}{{x}}^{{4}}{-}{4}{}{\mathrm{y2}}{}\left({x}\right){}{{x}}^{{4}}{+}{12}{}{\mathrm{y1}}{}\left({x}\right){}{{x}}^{{3}}{+}{12}{}{\mathrm{y3}}{}\left({x}\right){}{{x}}^{{3}}{-}{12}{}{\mathrm{y4}}{}\left({x}\right){}{{x}}^{{3}}{+}{4}{}{\mathrm{y1}}{}\left({x}\right){}{{x}}^{{2}}{+}{14}{}{\mathrm{y1}}{}\left({x}\right){}{x}{+}{12}{}{\mathrm{y3}}{}\left({x}\right){}{x}{-}{12}{}{\mathrm{y4}}{}\left({x}\right){}{x}{+}{3}{}{\mathrm{y2}}{}\left({x}\right){}{x}{-}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y4}}{}\left({x}\right)\right){}{x}{-}{4}{}{\mathrm{y3}}{}\left({x}\right){+}{4}{}{\mathrm{y2}}{}\left({x}\right)\right]{,}{\mathrm{system}}{=}\left[\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y1}}{}\left({x}\right)\right){}{{x}}^{{2}}{+}{2}{}{\mathrm{y1}}{}\left({x}\right){}{x}{-}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y1}}{}\left({x}\right){+}{6}{}{\mathrm{y1}}{}\left({x}\right){+}{6}{}{\mathrm{y3}}{}\left({x}\right){-}{6}{}{\mathrm{y4}}{}\left({x}\right){,}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y2}}{}\left({x}\right)\right){}{{x}}^{{5}}{-}{\mathrm{y2}}{}\left({x}\right){}{{x}}^{{5}}{+}{4}{}{\mathrm{y3}}{}\left({x}\right){}{{x}}^{{4}}{-}{4}{}{\mathrm{y2}}{}\left({x}\right){}{{x}}^{{4}}{+}{4}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y2}}{}\left({x}\right)\right){}{{x}}^{{3}}{-}{2}{}{\mathrm{y2}}{}\left({x}\right){}{{x}}^{{3}}{+}{8}{}{\mathrm{y3}}{}\left({x}\right){}{{x}}^{{2}}{-}{8}{}{\mathrm{y2}}{}\left({x}\right){}{{x}}^{{2}}{+}{{x}}^{{3}}{+}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y2}}{}\left({x}\right)\right){}{x}{-}{\mathrm{y2}}{}\left({x}\right){}{x}{+}{8}{}{{x}}^{{2}}{+}{4}{}{\mathrm{y3}}{}\left({x}\right){-}{4}{}{\mathrm{y2}}{}\left({x}\right){+}{x}{+}{4}{,}{-}{3}{}{\mathrm{y2}}{}\left({x}\right){}{{x}}^{{3}}{+}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y3}}{}\left({x}\right)\right){}{{x}}^{{3}}{+}{4}{}{\mathrm{y3}}{}\left({x}\right){}{{x}}^{{2}}{-}{4}{}{\mathrm{y2}}{}\left({x}\right){}{{x}}^{{2}}{-}{3}{}{\mathrm{y2}}{}\left({x}\right){}{x}{+}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y3}}{}\left({x}\right)\right){}{x}{+}{4}{}{\mathrm{y3}}{}\left({x}\right){-}{4}{}{\mathrm{y2}}{}\left({x}\right){+}{3}{}{x}{+}{4}{,}{-}{2}{}{\mathrm{y1}}{}\left({x}\right){}{{x}}^{{5}}{-}{3}{}{\mathrm{y2}}{}\left({x}\right){}{{x}}^{{5}}{+}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y4}}{}\left({x}\right)\right){}{{x}}^{{5}}{+}{4}{}{\mathrm{y1}}{}\left({x}\right){}{{x}}^{{4}}{+}{4}{}{\mathrm{y3}}{}\left({x}\right){}{{x}}^{{4}}{-}{4}{}{\mathrm{y2}}{}\left({x}\right){}{{x}}^{{4}}{+}{12}{}{\mathrm{y1}}{}\left({x}\right){}{{x}}^{{3}}{+}{12}{}{\mathrm{y3}}{}\left({x}\right){}{{x}}^{{3}}{-}{12}{}{\mathrm{y4}}{}\left({x}\right){}{{x}}^{{3}}{+}{4}{}{\mathrm{y1}}{}\left({x}\right){}{{x}}^{{2}}{+}{3}{}{{x}}^{{3}}{+}{14}{}{\mathrm{y1}}{}\left({x}\right){}{x}{+}{12}{}{\mathrm{y3}}{}\left({x}\right){}{x}{-}{12}{}{\mathrm{y4}}{}\left({x}\right){}{x}{+}{3}{}{\mathrm{y2}}{}\left({x}\right){}{x}{-}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y4}}{}\left({x}\right)\right){}{x}{+}{4}{}{{x}}^{{2}}{-}{4}{}{\mathrm{y3}}{}\left({x}\right){+}{4}{}{\mathrm{y2}}{}\left({x}\right){-}{3}{}{x}{-}{4}\right]{,}{\mathrm{functions}}{=}\left[{\mathrm{y1}}{}\left({x}\right){,}{\mathrm{y2}}{}\left({x}\right){,}{\mathrm{y3}}{}\left({x}\right){,}{\mathrm{y4}}{}\left({x}\right)\right]{,}{\mathrm{method}}{=}{\mathrm{ordinary}}{,}{\mathrm{order}}{=}{1}\right]\right)$ (2)
 > $\mathrm{indices}\left(\mathrm{props}\right)$
 $\left[{\mathrm{q_par}}\right]{,}\left[{\mathrm{case}}\right]{,}\left[{\mathrm{f_names}}\right]{,}\left[{\mathrm{var}}\right]{,}\left[{\mathrm{input}}\right]{,}\left[{\mathrm{system0}}\right]{,}\left[{\mathrm{system}}\right]{,}\left[{\mathrm{functions}}\right]{,}\left[{\mathrm{method}}\right]{,}\left[{\mathrm{order}}\right]$ (3)