Gcrd

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\vspace*{1mm}\noindent{\bf Gcrd}(u,v:LODO(C,v)).
The arguments are from a non-commutative ring of ordinary differential operators.  The meaning of the parameters is as follows.

$C$: Coefficient type.

$v$: Differential variable in $\fracsm{d}{dv}$.

\vspace*{1mm}\noindent{\bf Specification.}
A generator for their sum right-ideal is returned.

\vspace*{1mm}\noindent{\bf Examples.} The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is given in a separate window.

\4{\tt d1:=D(x)+1/2+1/x;  d2:=D(x)-1/2;}

\4{\tt Gcrd(d1,d2|LODO(RATF Q,x)|);}

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The differential operators, $D=\fracsm{d}{dx}$:
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$\displaystyle
D+\fracsm{x+2}{2x},\2 D-\fracsm{1}{2}$

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The Lclm:

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$D^2+\fracsm{x+2}{x^2+x}D-\fracsm{x^2+3x+4}{4x^2+4x}$}}

 


\vspace*{1mm}\noindent{\bf Gcrd}(u,v:LDO(C,vs).
The arguments are from a non-commutative ring of partial differential operators.

$C$ determines the coefficient type.

$v=\{v_1,v_2,\ldots\}$ determines the differential operators
$\fracsm{\partial}{\partial v_1},\fracsm{\partial}{\partial v_2},\ldots$.


\vspace*{1mm}\noindent{\bf Specification.}
The generators for their intersection left-ideal are returned.

\vspace*{1mm}\noindent{\bf Gcrd}(u,v:MODULE).

\vspace*{1mm}\noindent{\bf Examples.}

 

\vspace*{1mm}\noindent{\bf Gcrd}$(u,v:LODE(C,y,x)$.
The arguments are linear homogeneous ordinary differential polynomials. The meaning of the parameters is as follows.

$C:$ Coefficient type.

$y:$ Dependent variable.

$x:$ Independent variable.


\vspace*{1mm}\noindent{\bf Specification.}
The greatest common right divisor of $u$ and $v$ is returned.

\vspace*{1mm}\noindent{\bf Examples.} The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is given in a separate window.


\4{\tt depend y,x;}

\4{\tt deq1:=df(y,x)-y/x;  deq2:=df(y,x)-2*y/x;}

\4{\tt Gcrd(deq1,deq2|LODE(RATF Q,y,x)|);}

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The Lclm:

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$y''-\fracsm{2}{x}y'+\fracsm{2}{x^2}y=0$}}

 


\vspace*{1mm}\noindent{\bf Gcrd}(u,v:LDFMOD(C,z,x,O)).
The arguments $u$ and $v$ have the type $LDFMOD$ they represents differential polynomials in the dependent variables $z$ and independent variables $x$. The meaning of its parameters is as follows.

$C$ determines the type of the coefficients.

$z=\{z_1,z_2,\ldots \}$ denotes the dependent variables in decreasing order.

$x=\{x_1,x_2,\ldots \}$ denotes the independent variables in decreasing order.

$O:LEX|GRLEX$ determines the applied term ordering.

\vspace*{1mm}\noindent{\bf Specification.}
The greatest common right divisor of $u$ and $v$ is returned.


\vspace*{1mm}\noindent{\bf Examples.} The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window.


{\tt depend z,x,y;}

\4{\tt
l1:={df(z,y)+z/y,df(z,x)+z/x};  l2:={df(z,y)+z/(x+y),df(z,x)+z/(x+y)};}


\5{\tt Gcrd(l1,l2|LDFMOD(RATF Q,{z},{y,x},GRLEX)|); }

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The Lclm:
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$\displaystyle
z_y-\fracsm{x^2}{y^2}z_x-\fracsm{x-y}{y^2},\2
z_{xx}+\fracsm{4x+2y}{x^2+xy}z_x+\fracsm{2}{x^2+xy}z$

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Term Order: $GRLEX,z>y>x$  }}

 

\vspace*{1mm}\noindent{\bf Related Functions: $Lclm(u,v)$}

 

 


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{\bf References}

D.~Grigoriev, F.~Schwarz, {\em Factoring and Solving Linear Partial Differential Equations}, Computing {\bf 73}, 179-197 (2004).

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