Quotient - Maple Help

OreTools

 Quotient
 compute the right or left quotient of two Ore polynomials
 Remainder
 compute the right or left remainder of two Ore polynomials

 Calling Sequence Quotient['right'](Poly1, Poly2, A, 'R') Quotient(Poly1, Poly2, A, 'R') Quotient['left'](Poly1, Poly2, A, 'R') Remainder['right'](Poly1, Poly2, A, 'Q') Remainder(Poly1, Poly2, A, 'Q') Remainder['left'](Poly1, Poly2, A, 'Q')

Parameters

 Poly1 - Ore polynomial; to define an Ore polynomial, use the OrePoly structure. Poly2 - nonzero Ore polynomial; to define an Ore polynomial, use the OrePoly structure. A - Ore algebra; to define an Ore algebra, use the SetOreRing function. Q, R - (optional) unevaluated names.

Description

 • The Quotient['right'](Poly1, Poly2, A) or Quotient(Poly1, Poly2, A) calling sequence returns the right quotient Q of Poly1 and Poly2 such that:

$\mathrm{Poly1}=Q\mathrm{Poly2}+R$

 where the degree of the right remainder R is less than that of Poly2.
 If the fourth argument 'R' is specified, it is assigned the right remainder defined above.
 • The Quotient['left'](Poly1, Poly2, A) calling sequence returns the right quotient Q of Poly1 and Poly2 such that:

$\mathrm{Poly1}=\mathrm{Poly2}Q+R$

 where the degree of the left remainder R is less than that of Poly2.
 If the fourth argument 'R' is specified, it is assigned the left remainder defined above.
 • The Remainder['right'](Poly1, Poly2, A) or Remainder(Poly1, Poly2, A) calling sequence returns the right remainder R of Poly1 and Poly2 such that:

$\mathrm{Poly1}=Q\mathrm{Poly2}+R$

 where the degree of R is less than that of Poly2 and Q is the right quotient.
 If the fourth argument 'Q' is specified, it is assigned the right quotient defined above.
 • The Remainder['left'](Poly1, Poly2, A) calling sequence returns the left remainder R of Poly1 and Poly2 such that:

$\mathrm{Poly1}=\mathrm{Poly2}Q+R$

 where the degree of R is less than that of Poly2 and Q is the left quotient.

Examples

 > $\mathrm{with}\left(\mathrm{OreTools}\right):$
 > $A≔\mathrm{SetOreRing}\left(x,'\mathrm{differential}'\right)$
 ${A}{≔}{\mathrm{UnivariateOreRing}}{}\left({x}{,}{\mathrm{differential}}\right)$ (1)
 > $\mathrm{Poly1}≔\mathrm{OrePoly}\left(\frac{1}{x},0,\frac{x}{x+1},\frac{{x}^{2}}{x+2}\right)$
 ${\mathrm{Poly1}}{≔}{\mathrm{OrePoly}}{}\left(\frac{{1}}{{x}}{,}{0}{,}\frac{{x}}{{x}{+}{1}}{,}\frac{{{x}}^{{2}}}{{x}{+}{2}}\right)$ (2)
 > $\mathrm{Poly2}≔\mathrm{OrePoly}\left(x,\frac{x+1}{x},x\right)$
 ${\mathrm{Poly2}}{≔}{\mathrm{OrePoly}}{}\left({x}{,}\frac{{x}{+}{1}}{{x}}{,}{x}\right)$ (3)
 > $\mathrm{R1}≔\mathrm{Remainder}\left['\mathrm{right}'\right]\left(\mathrm{Poly1},\mathrm{Poly2},A\right)$
 ${\mathrm{R1}}{≔}{\mathrm{OrePoly}}{}\left(\frac{{{x}}^{{2}}{+}{4}{}{x}{+}{2}}{{x}{}\left({x}{+}{1}\right){}\left({x}{+}{2}\right)}{,}{-}\frac{{{x}}^{{4}}{-}{{x}}^{{2}}{-}{2}{}{x}{-}{1}}{\left({x}{+}{2}\right){}{{x}}^{{2}}}\right)$ (4)
 > $\mathrm{R2}≔\mathrm{Remainder}\left(\mathrm{Poly1},\mathrm{Poly2},A\right)$
 ${\mathrm{R2}}{≔}{\mathrm{OrePoly}}{}\left(\frac{{{x}}^{{2}}{+}{4}{}{x}{+}{2}}{{x}{}\left({x}{+}{1}\right){}\left({x}{+}{2}\right)}{,}{-}\frac{{{x}}^{{4}}{-}{{x}}^{{2}}{-}{2}{}{x}{-}{1}}{\left({x}{+}{2}\right){}{{x}}^{{2}}}\right)$ (5)
 > $\mathrm{Minus}\left(\mathrm{R1},\mathrm{R2}\right)$
 ${\mathrm{OrePoly}}{}\left({0}\right)$ (6)
 > $\mathrm{R3}≔\mathrm{Remainder}\left['\mathrm{left}'\right]\left(\mathrm{Poly1},\mathrm{Poly2},A\right)$
 ${\mathrm{R3}}{≔}{\mathrm{OrePoly}}{}\left(\frac{{5}{}{{x}}^{{9}}{+}{40}{}{{x}}^{{8}}{+}{151}{}{{x}}^{{7}}{+}{287}{}{{x}}^{{6}}{+}{302}{}{{x}}^{{5}}{+}{193}{}{{x}}^{{4}}{+}{58}{}{{x}}^{{3}}{-}{30}{}{{x}}^{{2}}{-}{32}{}{x}{-}{8}}{{{x}}^{{3}}{}{\left({x}{+}{1}\right)}^{{3}}{}{\left({x}{+}{2}\right)}^{{4}}}{,}{-}\frac{{{x}}^{{8}}{+}{6}{}{{x}}^{{7}}{+}{13}{}{{x}}^{{6}}{+}{22}{}{{x}}^{{5}}{+}{23}{}{{x}}^{{4}}{+}{12}{}{{x}}^{{3}}{+}{{x}}^{{2}}{-}{8}{}{x}{-}{4}}{{\left({x}{+}{1}\right)}^{{2}}{}{\left({x}{+}{2}\right)}^{{3}}{}{{x}}^{{2}}}\right)$ (7)
 > $R≔\mathrm{Remainder}\left['\mathrm{right}'\right]\left(\mathrm{Poly1},\mathrm{Poly2},A,'Q'\right)$
 ${R}{≔}{\mathrm{OrePoly}}{}\left(\frac{{{x}}^{{2}}{+}{4}{}{x}{+}{2}}{{x}{}\left({x}{+}{1}\right){}\left({x}{+}{2}\right)}{,}{-}\frac{{{x}}^{{4}}{-}{{x}}^{{2}}{-}{2}{}{x}{-}{1}}{\left({x}{+}{2}\right){}{{x}}^{{2}}}\right)$ (8)
 > $Q$
 ${\mathrm{OrePoly}}{}\left({-}\frac{{{x}}^{{2}}{+}{x}{+}{1}}{{x}{}\left({x}{+}{1}\right){}\left({x}{+}{2}\right)}{,}\frac{{x}}{{x}{+}{2}}\right)$ (9)
 > $\mathrm{Minus}\left(\mathrm{Poly1},\mathrm{Add}\left(\mathrm{Multiply}\left(Q,\mathrm{Poly2},A\right),R\right)\right)$
 ${\mathrm{OrePoly}}{}\left({0}\right)$ (10)
 > $R≔\mathrm{Remainder}\left['\mathrm{left}'\right]\left(\mathrm{Poly1},\mathrm{Poly2},A,'Q'\right)$
 ${R}{≔}{\mathrm{OrePoly}}{}\left(\frac{{5}{}{{x}}^{{9}}{+}{40}{}{{x}}^{{8}}{+}{151}{}{{x}}^{{7}}{+}{287}{}{{x}}^{{6}}{+}{302}{}{{x}}^{{5}}{+}{193}{}{{x}}^{{4}}{+}{58}{}{{x}}^{{3}}{-}{30}{}{{x}}^{{2}}{-}{32}{}{x}{-}{8}}{{{x}}^{{3}}{}{\left({x}{+}{1}\right)}^{{3}}{}{\left({x}{+}{2}\right)}^{{4}}}{,}{-}\frac{{{x}}^{{8}}{+}{6}{}{{x}}^{{7}}{+}{13}{}{{x}}^{{6}}{+}{22}{}{{x}}^{{5}}{+}{23}{}{{x}}^{{4}}{+}{12}{}{{x}}^{{3}}{+}{{x}}^{{2}}{-}{8}{}{x}{-}{4}}{{\left({x}{+}{1}\right)}^{{2}}{}{\left({x}{+}{2}\right)}^{{3}}{}{{x}}^{{2}}}\right)$ (11)
 > $Q$
 ${\mathrm{OrePoly}}{}\left({-}\frac{{4}{}{{x}}^{{2}}{+}{5}{}{x}{+}{2}}{\left({x}{+}{1}\right){}{\left({x}{+}{2}\right)}^{{2}}{}{x}}{,}\frac{{x}}{{x}{+}{2}}\right)$ (12)
 > $\mathrm{Minus}\left(\mathrm{Poly1},\mathrm{Add}\left(\mathrm{Multiply}\left(\mathrm{Poly2},Q,A\right),R\right)\right)$
 ${\mathrm{OrePoly}}{}\left({0}\right)$ (13)
 > $B≔\mathrm{SetOreRing}\left(x,'\mathrm{shift}'\right)$
 ${B}{≔}{\mathrm{UnivariateOreRing}}{}\left({x}{,}{\mathrm{shift}}\right)$ (14)
 > $\mathrm{Poly1}≔\mathrm{OrePoly}\left(\frac{1}{x},0,\frac{x}{x+1},\frac{{x}^{2}}{x+2}\right)$
 ${\mathrm{Poly1}}{≔}{\mathrm{OrePoly}}{}\left(\frac{{1}}{{x}}{,}{0}{,}\frac{{x}}{{x}{+}{1}}{,}\frac{{{x}}^{{2}}}{{x}{+}{2}}\right)$ (15)
 > $\mathrm{Poly2}≔\mathrm{OrePoly}\left(x,\frac{x+1}{x},x\right)$
 ${\mathrm{Poly2}}{≔}{\mathrm{OrePoly}}{}\left({x}{,}\frac{{x}{+}{1}}{{x}}{,}{x}\right)$ (16)
 > $Q≔\mathrm{Quotient}\left(\mathrm{Poly1},\mathrm{Poly2},B\right)$
 ${Q}{≔}{\mathrm{OrePoly}}{}\left(\frac{{1}}{{\left({x}{+}{1}\right)}^{{2}}}{,}\frac{{{x}}^{{2}}}{\left({x}{+}{2}\right){}\left({x}{+}{1}\right)}\right)$ (17)
 > $Q≔\mathrm{Quotient}\left['\mathrm{right}'\right]\left(\mathrm{Poly1},\mathrm{Poly2},B,'R'\right)$
 ${Q}{≔}{\mathrm{OrePoly}}{}\left(\frac{{1}}{{\left({x}{+}{1}\right)}^{{2}}}{,}\frac{{{x}}^{{2}}}{\left({x}{+}{2}\right){}\left({x}{+}{1}\right)}\right)$ (18)
 > $R$
 ${\mathrm{OrePoly}}{}\left(\frac{{2}{}{x}{+}{1}}{{x}{}{\left({x}{+}{1}\right)}^{{2}}}{,}{-}\frac{{{x}}^{{4}}{+}{{x}}^{{3}}{+}{x}{+}{2}}{{x}{}\left({x}{+}{1}\right){}\left({x}{+}{2}\right)}\right)$ (19)
 > $\mathrm{Minus}\left(\mathrm{Poly1},\mathrm{Add}\left(\mathrm{Multiply}\left(Q,\mathrm{Poly2},B\right),R\right)\right)$
 ${\mathrm{OrePoly}}{}\left({0}\right)$ (20)
 > $Q≔\mathrm{Quotient}\left['\mathrm{left}'\right]\left(\mathrm{Poly1},\mathrm{Poly2},B\right)$
 ${Q}{≔}{\mathrm{OrePoly}}{}\left(\frac{{1}}{\left({x}{-}{1}\right){}{\left({x}{-}{2}\right)}^{{2}}}{,}\frac{{x}{-}{2}}{{x}}\right)$ (21)
 > $Q≔\mathrm{Quotient}\left['\mathrm{left}'\right]\left(\mathrm{Poly1},\mathrm{Poly2},B,'R'\right)$
 ${Q}{≔}{\mathrm{OrePoly}}{}\left(\frac{{1}}{\left({x}{-}{1}\right){}{\left({x}{-}{2}\right)}^{{2}}}{,}\frac{{x}{-}{2}}{{x}}\right)$ (22)
 > $R$
 ${\mathrm{OrePoly}}{}\left(\frac{{{x}}^{{3}}{-}{6}{}{{x}}^{{2}}{+}{8}{}{x}{-}{4}}{{x}{}\left({x}{-}{1}\right){}{\left({x}{-}{2}\right)}^{{2}}}{,}{-}\frac{{{x}}^{{5}}{-}{4}{}{{x}}^{{4}}{+}{5}{}{{x}}^{{3}}{-}{2}{}{{x}}^{{2}}{+}{x}{+}{1}}{{{x}}^{{2}}{}{\left({x}{-}{1}\right)}^{{2}}}\right)$ (23)
 > $\mathrm{Minus}\left(\mathrm{Poly1},\mathrm{Add}\left(\mathrm{Multiply}\left(\mathrm{Poly2},Q,B\right),R\right)\right)$
 ${\mathrm{OrePoly}}{}\left({0}\right)$ (24)