The Euclidean['right'](Poly1, Poly2, A) or Euclidean(Poly1, Poly2, A) calling sequence returns a list [m, S] where m is a positive integer and S is an array with m nonzero Ore polynomials such that:

In addition, Remainder['right'](S[m-1], S[m], A) = 0. S is called the right Euclidean polynomial remainder sequence of Poly1 and Poly2.

If the fourth argument c1 of Euclidean['right'] or Euclidean is specified, it is assigned the first co-sequence of Poly1 and Poly2 so that:

and c1[m+1] Poly1 is a least common left multiple (LCLM) of Poly1 and Poly2.

If the fifth argument c2 of Euclidean['right'] or Euclidean is specified, it is assigned the second co-sequence of Poly1 and Poly2 so that:

and ${\mathrm{c1}}_{m+1}\mathrm{Poly1}=-{\mathrm{c2}}_{m+1}\mathrm{Poly2}$ is an LCLM of Poly1 and Poly2.

The Euclidean['left'](Poly1, Poly2, A) calling sequence returns a list [m, S] where m is a positive integer and S is an array with m nonzero Ore polynomials such that:

In addition, ${\mathrm{Remainder}}_{'\mathrm{left}'}\left(S_{m-1}\,S_m\,A\right)=0$. S is called the left Euclidean polynomial remainder sequence of Poly1 and Poly2.

If the fourth argument c1 of Euclidean['left'] is specified, it is assigned the first co-sequence of Poly1 and Poly2 so that:

and Poly1 c1[m+1] is a least common right multiple (LCRM) of Poly1 and Poly2.

If the fifth argument c2 of Euclidean['left'] is is specified, it is assigned the second co-sequence of Poly1 and Poly2 so that:

and Poly1 c1[m+1]= - Poly2 c2[m+1] is an LCRM of Poly1 and Poly2.

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

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

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

$\mathrm{Ore1}≔\mathrm{OrePoly}\left(1\,3x\,x^2-\left(1+x\right)\,23x+1\right)$

$\mathrm{Ore1}≔\mathrm{OrePoly}\left(1\,3x\,x^2-x-1\,23x+1\right)$

$\mathrm{Ore2}≔\mathrm{OrePoly}\left(x\,x\,x^2+x+1\right)$

$\mathrm{Ore2}≔\mathrm{OrePoly}\left(x\,x\,x^2+x+1\right)$

$U≔\mathrm{Euclidean}\left['\mathrm{right}'\right]\left(\mathrm{Ore1}\,\mathrm{Ore2}\,A\,'\mathrm{c1}'\,'\mathrm{c2}'\right)$

$U≔\left[4\,S\right]$

$m≔U\left[1\right]\;$$S≔U\left[2\right]$

$m≔4$

$S≔S$

$\mathrm{print}\left(S\right)$

$\left[\begin{array}{cccc}\mathrm{OrePoly}\left(1\,3x\,x^2-x-1\,23x+1\right)& \mathrm{OrePoly}\left(x\,x\,x^2+x+1\right)& \mathrm{OrePoly}\left(-\frac{x\left(x^4-x^3-49x^2-7x+20\right)}{{\left(x^2+x+1\right)}^2}\,\frac{2x^5-17x^4+32x^3-14x^2-20x-1}{{\left(x^2+x+1\right)}^2}\right)& \mathrm{OrePoly}\left(\frac{x^{12}+5x^{11}-200x^{10}+598x^9+1451x^8+2135x^7+1781x^6+1343x^5+1850x^4+1420x^3+981x^2-5x-20}{{\left(2x^5-17x^4+32x^3-14x^2-20x-1\right)}^2}\right)\end{array}\right]$

$$\begin{gathered} \mathbf{for}i\mathbf{to}m\mathbf{do} \\ \mathrm{W1}≔\mathrm{Multiply}\left(\mathrm{c1}\left[i\right]\,\mathrm{Ore1}\,A\right); \\ \mathrm{W2}≔\mathrm{Multiply}\left(\mathrm{c2}\left[i\right]\,\mathrm{Ore2}\,A\right); \\ W≔\mathrm{Add}\left(\mathrm{W1}\,\mathrm{W2}\right); \\ C≔\mathrm{Minus}\left(W\,S\left[i\right]\right); \\ \mathrm{print}\left(C\right) \\ \mathbf{end}\mathbf{do}\: \end{gathered}$$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{W3}≔\mathrm{Multiply}\left(\mathrm{c1}\left[5\right]\,\mathrm{Ore1}\,A\right)\:$

$\mathrm{W4}≔\mathrm{Multiply}\left(\mathrm{c2}\left[5\right]\,\mathrm{Ore2}\,A\right)\:$

$\mathrm{Add}\left(\mathrm{W3}\,\mathrm{W4}\right)$

$\mathrm{OrePoly}\left(0\right)$

$U≔\mathrm{Euclidean}\left['\mathrm{left}'\right]\left(\mathrm{Ore1}\,\mathrm{Ore2}\,A\,'\mathrm{c1}'\,'\mathrm{c2}'\right)$

$U≔\left[4\,S\right]$

$m≔U\left[1\right]\;$$S≔U\left[2\right]$

$m≔4$

$S≔S$

$\mathrm{print}\left(S\right)$

$\left[\begin{array}{cccc}\mathrm{OrePoly}\left(1\,3x\,x^2-x-1\,23x+1\right)& \mathrm{OrePoly}\left(x\,x\,x^2+x+1\right)& \mathrm{OrePoly}\left(-\frac{x^7+18x^5+99x^4-12x^3-1059x^2-955x-51}{{\left(x^2+x+1\right)}^3}\,\frac{2x^5-21x^4+20x^3-17x^2-289x-136}{{\left(x^2+x+1\right)}^2}\right)& \mathrm{OrePoly}\left(\frac{x^{12}+x^{11}+44x^{10}+556x^9+1037x^8+1941x^7+11909x^6+36562x^5+72548x^4+88995x^3+76169x^2+39236x+12308}{{\left(2x^5-21x^4+20x^3-17x^2-289x-136\right)}^2}\right)\end{array}\right]$

$$\begin{gathered} \mathbf{for}i\mathbf{to}m\mathbf{do} \\ \mathrm{W1}≔\mathrm{Multiply}\left(\mathrm{Ore1}\,\mathrm{c1}\left[i\right]\,A\right); \\ \mathrm{W2}≔\mathrm{Multiply}\left(\mathrm{Ore2}\,\mathrm{c2}\left[i\right]\,A\right); \\ W≔\mathrm{Add}\left(\mathrm{W1}\,\mathrm{W2}\right); \\ C≔\mathrm{Minus}\left(W\,S\left[i\right]\right); \\ \mathrm{print}\left(C\right) \\ \mathbf{end}\mathbf{do}\: \end{gathered}$$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{W3}≔\mathrm{Multiply}\left(\mathrm{Ore1}\,\mathrm{c1}\left[m+1\right]\,A\right)\:$

$\mathrm{W4}≔\mathrm{Multiply}\left(\mathrm{Ore2}\,\mathrm{c2}\left[m+1\right]\,A\right)\:$

$\mathrm{Add}\left(\mathrm{W3}\,\mathrm{W4}\right)$

$\mathrm{OrePoly}\left(0\right)$