\begin{patch}{OperatorXmpPagePatch1} \begin{paste}{OperatorXmpPageFull1}{OperatorXmpPageEmpty1} \pastebutton{OperatorXmpPageFull1}{\hidepaste} \tab{5}\spadcommand{R := SQMATRIX(2, INT)\bound{r }} \indentrel{3}\begin{verbatim} (1) SquareMatrix(2,Integer) Type: Domain \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty1} \begin{paste}{OperatorXmpPageEmpty1}{OperatorXmpPagePatch1} \pastebutton{OperatorXmpPageEmpty1}{\showpaste} \tab{5}\spadcommand{R := SQMATRIX(2, INT)\bound{r }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch2} \begin{paste}{OperatorXmpPageFull2}{OperatorXmpPageEmpty2} \pastebutton{OperatorXmpPageFull2}{\hidepaste} \tab{5}\spadcommand{t := operator("tilde") :: OP(R)\free{r }\bound{t }} \indentrel{3}\begin{verbatim} (2) tilde Type: Operator SquareMatrix(2,Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty2} \begin{paste}{OperatorXmpPageEmpty2}{OperatorXmpPagePatch2} \pastebutton{OperatorXmpPageEmpty2}{\showpaste} \tab{5}\spadcommand{t := operator("tilde") :: OP(R)\free{r }\bound{t }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch3} \begin{paste}{OperatorXmpPageFull3}{OperatorXmpPageEmpty3} \pastebutton{OperatorXmpPageFull3}{\hidepaste} \tab{5}\spadcommand{)set expose add constructor Operator\free{t }\bound{expose }} \indentrel{3}\begin{verbatim} \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty3} \begin{paste}{OperatorXmpPageEmpty3}{OperatorXmpPagePatch3} \pastebutton{OperatorXmpPageEmpty3}{\showpaste} \tab{5}\spadcommand{)set expose add constructor Operator\free{t }\bound{expose }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch4} \begin{paste}{OperatorXmpPageFull4}{OperatorXmpPageEmpty4} \pastebutton{OperatorXmpPageFull4}{\hidepaste} \tab{5}\spadcommand{evaluate(t, m +-> transpose m)\free{expose }\free{t }\bound{evt }} \indentrel{3}\begin{verbatim} (3) tilde Type: Operator SquareMatrix(2,Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty4} \begin{paste}{OperatorXmpPageEmpty4}{OperatorXmpPagePatch4} \pastebutton{OperatorXmpPageEmpty4}{\showpaste} \tab{5}\spadcommand{evaluate(t, m +-> transpose m)\free{expose }\free{t }\bound{evt }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch5} \begin{paste}{OperatorXmpPageFull5}{OperatorXmpPageEmpty5} \pastebutton{OperatorXmpPageFull5}{\hidepaste} \tab{5}\spadcommand{s : R := matrix [[0, 1], [1, 0]]\bound{s }} \indentrel{3}\begin{verbatim} Ú0 1¿ (4) ³ ³ À1 0Ù Type: SquareMatrix(2,Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty5} \begin{paste}{OperatorXmpPageEmpty5}{OperatorXmpPagePatch5} \pastebutton{OperatorXmpPageEmpty5}{\showpaste} \tab{5}\spadcommand{s : R := matrix [[0, 1], [1, 0]]\bound{s }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch6} \begin{paste}{OperatorXmpPageFull6}{OperatorXmpPageEmpty6} \pastebutton{OperatorXmpPageFull6}{\hidepaste} \tab{5}\spadcommand{rho := t * s\free{evt s }\bound{rho }} \indentrel{3}\begin{verbatim} Ú0 1¿ (5) tilde³ ³ À1 0Ù Type: Operator SquareMatrix(2,Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty6} \begin{paste}{OperatorXmpPageEmpty6}{OperatorXmpPagePatch6} \pastebutton{OperatorXmpPageEmpty6}{\showpaste} \tab{5}\spadcommand{rho := t * s\free{evt s }\bound{rho }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch7} \begin{paste}{OperatorXmpPageFull7}{OperatorXmpPageEmpty7} \pastebutton{OperatorXmpPageFull7}{\hidepaste} \tab{5}\spadcommand{z := rho**4 - 1\free{rho }\bound{z }} \indentrel{3}\begin{verbatim} (6) Ú0 1¿ Ú0 1¿ Ú0 1¿ Ú0 1¿ - 1 + tilde³ ³tilde³ ³tilde³ ³tilde³ ³ À1 0Ù À1 0Ù À1 0Ù À1 0Ù Type: Operator SquareMatrix(2,Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty7} \begin{paste}{OperatorXmpPageEmpty7}{OperatorXmpPagePatch7} \pastebutton{OperatorXmpPageEmpty7}{\showpaste} \tab{5}\spadcommand{z := rho**4 - 1\free{rho }\bound{z }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch8} \begin{paste}{OperatorXmpPageFull8}{OperatorXmpPageEmpty8} \pastebutton{OperatorXmpPageFull8}{\hidepaste} \tab{5}\spadcommand{m:R := matrix [[1, 2], [3, 4]]\bound{m }} \indentrel{3}\begin{verbatim} Ú1 2¿ (7) ³ ³ À3 4Ù Type: SquareMatrix(2,Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty8} \begin{paste}{OperatorXmpPageEmpty8}{OperatorXmpPagePatch8} \pastebutton{OperatorXmpPageEmpty8}{\showpaste} \tab{5}\spadcommand{m:R := matrix [[1, 2], [3, 4]]\bound{m }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch9} \begin{paste}{OperatorXmpPageFull9}{OperatorXmpPageEmpty9} \pastebutton{OperatorXmpPageFull9}{\hidepaste} \tab{5}\spadcommand{z m\free{z m }} \indentrel{3}\begin{verbatim} Ú0 0¿ (8) ³ ³ À0 0Ù Type: SquareMatrix(2,Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty9} \begin{paste}{OperatorXmpPageEmpty9}{OperatorXmpPagePatch9} \pastebutton{OperatorXmpPageEmpty9}{\showpaste} \tab{5}\spadcommand{z m\free{z m }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch10} \begin{paste}{OperatorXmpPageFull10}{OperatorXmpPageEmpty10} \pastebutton{OperatorXmpPageFull10}{\hidepaste} \tab{5}\spadcommand{rho m\free{rho m }} \indentrel{3}\begin{verbatim} Ú3 1¿ (9) ³ ³ À4 2Ù Type: SquareMatrix(2,Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty10} \begin{paste}{OperatorXmpPageEmpty10}{OperatorXmpPagePatch10} \pastebutton{OperatorXmpPageEmpty10}{\showpaste} \tab{5}\spadcommand{rho m\free{rho m }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch11} \begin{paste}{OperatorXmpPageFull11}{OperatorXmpPageEmpty11} \pastebutton{OperatorXmpPageFull11}{\hidepaste} \tab{5}\spadcommand{rho rho m\free{rho m }} \indentrel{3}\begin{verbatim} Ú4 3¿ (10) ³ ³ À2 1Ù Type: SquareMatrix(2,Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty11} \begin{paste}{OperatorXmpPageEmpty11}{OperatorXmpPagePatch11} \pastebutton{OperatorXmpPageEmpty11}{\showpaste} \tab{5}\spadcommand{rho rho m\free{rho m }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch12} \begin{paste}{OperatorXmpPageFull12}{OperatorXmpPageEmpty12} \pastebutton{OperatorXmpPageFull12}{\hidepaste} \tab{5}\spadcommand{(rho**3) m\free{rho m }} \indentrel{3}\begin{verbatim} Ú2 4¿ (11) ³ ³ À1 3Ù Type: SquareMatrix(2,Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty12} \begin{paste}{OperatorXmpPageEmpty12}{OperatorXmpPagePatch12} \pastebutton{OperatorXmpPageEmpty12}{\showpaste} \tab{5}\spadcommand{(rho**3) m\free{rho m }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch13} \begin{paste}{OperatorXmpPageFull13}{OperatorXmpPageEmpty13} \pastebutton{OperatorXmpPageFull13}{\hidepaste} \tab{5}\spadcommand{b := t * s - s * t\free{s evt }\bound{b }} \indentrel{3}\begin{verbatim} Ú0 1¿ Ú0 1¿ (12) - ³ ³tilde + tilde³ ³ À1 0Ù À1 0Ù Type: Operator SquareMatrix(2,Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty13} \begin{paste}{OperatorXmpPageEmpty13}{OperatorXmpPagePatch13} \pastebutton{OperatorXmpPageEmpty13}{\showpaste} \tab{5}\spadcommand{b := t * s - s * t\free{s evt }\bound{b }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch14} \begin{paste}{OperatorXmpPageFull14}{OperatorXmpPageEmpty14} \pastebutton{OperatorXmpPageFull14}{\hidepaste} \tab{5}\spadcommand{b m\free{b m }} \indentrel{3}\begin{verbatim} Ú1 - 3¿ (13) ³ ³ À3 - 1Ù Type: SquareMatrix(2,Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty14} \begin{paste}{OperatorXmpPageEmpty14}{OperatorXmpPagePatch14} \pastebutton{OperatorXmpPageEmpty14}{\showpaste} \tab{5}\spadcommand{b m\free{b m }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch15} \begin{paste}{OperatorXmpPageFull15}{OperatorXmpPageEmpty15} \pastebutton{OperatorXmpPageFull15}{\hidepaste} \tab{5}\spadcommand{L n == n = 0 => 1 n = 1 => x (2*n-1)/n * x * L(n-1) - (n-1)/n * L(n-2) \bound{l }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty15} \begin{paste}{OperatorXmpPageEmpty15}{OperatorXmpPagePatch15} \pastebutton{OperatorXmpPageEmpty15}{\showpaste} \tab{5}\spadcommand{L n == n = 0 => 1 n = 1 => x (2*n-1)/n * x * L(n-1) - (n-1)/n * L(n-2) \bound{l }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch16} \begin{paste}{OperatorXmpPageFull16}{OperatorXmpPageEmpty16} \pastebutton{OperatorXmpPageFull16}{\hidepaste} \tab{5}\spadcommand{dx := operator("D") :: OP(POLY FRAC INT)\bound{dx }} \indentrel{3}\begin{verbatim} (15) D Type: Operator Polynomial Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty16} \begin{paste}{OperatorXmpPageEmpty16}{OperatorXmpPagePatch16} \pastebutton{OperatorXmpPageEmpty16}{\showpaste} \tab{5}\spadcommand{dx := operator("D") :: OP(POLY FRAC INT)\bound{dx }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch17} \begin{paste}{OperatorXmpPageFull17}{OperatorXmpPageEmpty17} \pastebutton{OperatorXmpPageFull17}{\hidepaste} \tab{5}\spadcommand{evaluate(dx, p +-> D(p, 'x))\free{dx }\bound{edx }} \indentrel{3}\begin{verbatim} (16) D Type: Operator Polynomial Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty17} \begin{paste}{OperatorXmpPageEmpty17}{OperatorXmpPagePatch17} \pastebutton{OperatorXmpPageEmpty17}{\showpaste} \tab{5}\spadcommand{evaluate(dx, p +-> D(p, 'x))\free{dx }\bound{edx }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch18} \begin{paste}{OperatorXmpPageFull18}{OperatorXmpPageEmpty18} \pastebutton{OperatorXmpPageFull18}{\hidepaste} \tab{5}\spadcommand{E n == (1 - x**2) * dx**2 - 2 * x * dx + n*(n+1)\free{edx }\bound{E }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty18} \begin{paste}{OperatorXmpPageEmpty18}{OperatorXmpPagePatch18} \pastebutton{OperatorXmpPageEmpty18}{\showpaste} \tab{5}\spadcommand{E n == (1 - x**2) * dx**2 - 2 * x * dx + n*(n+1)\free{edx }\bound{E }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch19} \begin{paste}{OperatorXmpPageFull19}{OperatorXmpPageEmpty19} \pastebutton{OperatorXmpPageFull19}{\hidepaste} \tab{5}\spadcommand{L 15\free{L }} \indentrel{3}\begin{verbatim} (18) 9694845 15 35102025 13 50702925 11 ÄÄÄÄÄÄÄ x - ÄÄÄÄÄÄÄÄ x + ÄÄÄÄÄÄÄÄ x 2048 2048 2048 + 37182145 9 14549535 7 2909907 5 255255 3 - ÄÄÄÄÄÄÄÄ x + ÄÄÄÄÄÄÄÄ x - ÄÄÄÄÄÄÄ x + ÄÄÄÄÄÄ x 2048 2048 2048 2048 + 6435 - ÄÄÄÄ x 2048 Type: Polynomial Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty19} \begin{paste}{OperatorXmpPageEmpty19}{OperatorXmpPagePatch19} \pastebutton{OperatorXmpPageEmpty19}{\showpaste} \tab{5}\spadcommand{L 15\free{L }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch20} \begin{paste}{OperatorXmpPageFull20}{OperatorXmpPageEmpty20} \pastebutton{OperatorXmpPageFull20}{\hidepaste} \tab{5}\spadcommand{E 15\free{E }} \indentrel{3}\begin{verbatim} 2 2 (19) 240 - 2x D - (x - 1)D Type: Operator Polynomial Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty20} \begin{paste}{OperatorXmpPageEmpty20}{OperatorXmpPagePatch20} \pastebutton{OperatorXmpPageEmpty20}{\showpaste} \tab{5}\spadcommand{E 15\free{E }} \end{paste}\end{patch} \begin{patch}{OperatorXmpPagePatch21} \begin{paste}{OperatorXmpPageFull21}{OperatorXmpPageEmpty21} \pastebutton{OperatorXmpPageFull21}{\hidepaste} \tab{5}\spadcommand{(E 15)(L 15)\free{L E }} \indentrel{3}\begin{verbatim} (20) 0 Type: Polynomial Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{OperatorXmpPageEmpty21} \begin{paste}{OperatorXmpPageEmpty21}{OperatorXmpPagePatch21} \pastebutton{OperatorXmpPageEmpty21}{\showpaste} \tab{5}\spadcommand{(E 15)(L 15)\free{L E }} \end{paste}\end{patch}