@@ -1,362 +1,362 @@
-\section{Matrix Normal Forms}
-
-\begin{Operator}{Smithex}
-The operator \name{smithex} computes the Smith normal form S of a
-\nameref{matrix} A (say). It returns \{S,P,$P^-1$\} where $P*S*P^-1 = A$.
-
-\begin{Syntax}
-\name{smithex}\(\meta{matrix},\meta{variable}\)
-
-\meta{matrix} :- a rectangular \nameref{matrix} of univariate polynomials in 
-                 \meta{variable}.
-\meta{variable} :- the variable.
-\end{Syntax}
-
-\begin{Examples}
- a := mat((x,x+1),(0,3*x^2)); &
-\begin{multilineoutput}{6cm}
-      [x  x + 1]
-      [        ]
- a := [      2 ]
-      [0  3*x  ]
-\end{multilineoutput}\\
-
- smithex(a,x); &
-\begin{multilineoutput}{6cm}
-   [1  0 ]    [1    0]    [x   x + 1]
-\{  [     ],   [      ],   [         ]  \}
-   [    3]    [   2  ]    [         ]
-   [0  x ]    [3*x  1]    [-3    -3 ]
-\end{multilineoutput}\\
-\end{Examples}
-\end{Operator}
-
-
-\begin{Operator}{Smithex\_int}
-The operator \name{smithex\_int} performs the same task as \name{smithex}
-but on matrices containing only integer entries. Namely, 
-\name{smithex\_int} returns \{S,P,$P^-1$\} where S is the smith normal
-form of the input \nameref{matrix} (A say), and $P*S*P^-1 = A$.
-
-\begin{Syntax}
-\name{smithex\_int}\(\meta{matrix}\)
-
-\meta{matrix} :- a rectangular \nameref{matrix} of integer entries.
-\end{Syntax}
-
-\begin{Examples}
- a := mat((9,-36,30),(-36,192,-180),(30,-180,180)); &
-\begin{multilineoutput}{6cm}
-     [ 9   -36    30 ]
-     [               ]
-a := [-36  192   -180]
-     [               ]
-     [30   -180  180 ]
-\end{multilineoutput}\\
-
- smithex_int(a); &
-\begin{multilineoutput}{6cm}
-  [3  0   0 ]    [-17  -5   -4 ]    [1   -24  30 ]
-  [         ]    [             ]    [            ]
-\{ [0  12  0 ],   [64   19   15 ],   [-1  25   -30] \}
-  [         ]    [             ]    [            ] 
-  [0  0   60]    [-50  -15  -12]    [0   -1    1 ] 
-\end{multilineoutput}\\
-\end{Examples}
-\end{Operator}
-
-
-\begin{Operator}{Frobenius}
-The operator \name{frobenius} computes the \name{frobenius} normal form F of a 
-\nameref{matrix} (A say). It returns \{F,P,$P^-1$\} where $P*F*P^-1 = A$.
-
-\begin{Syntax}
-\name{frobenius}\(\meta{matrix}\)
-
-\meta{matrix} :- a square \nameref{matrix}.
-\end{Syntax}
-
-Field Extensions:
-
-By default, calculations are performed in the rational numbers. To 
-extend this field the \nameref{arnum} package can be used. The package must
-first be loaded by load\_package arnum;. The field can now be extended
-by using the defpoly command. For example, defpoly sqrt2**2-2; will
-extend the field to include the square root of 2 (now defined by sqrt2).
-
-Modular Arithmetic: 
-
-\name{Frobenius} can also be calculated in a modular base. To do this 
-first type on  modular;. Then setmod p; (where p is a prime) will set 
-the modular base of calculation to p. By further typing on balanced\_mod
-the answer will appear using a symmetric modular representation. See 
-\nameref{ratjordan} for an example.
-
-\begin{Examples}
- a := mat((x,x^2),(3,5*x)); &
-\begin{multilineoutput}{6cm}
-     [    2 ]
-     [x  x  ]
-a := [      ]
-     [3  5*x]
-\end{multilineoutput}\\
- frobenius(a);&
-\begin{multilineoutput}{6cm}
-   [         2]    [1  x]    [       - x ]
-\{  [0   - 2*x ],   [    ],   [1     -----]  \}
-   [          ]    [0  3]    [        3  ]
-   [1    6*x  ]              [           ]
-                             [        1  ]
-                             [0      --- ]
-                             [        3  ]
-
-\end{multilineoutput}\\
- load\_package arnum;\\
- defpoly sqrt2**2-2;\\
- a := mat((sqrt2,5),(7*sqrt2,sqrt2));&
-\begin{multilineoutput}{6cm}
-     [ sqrt2     5  ]
-a := [              ]
-     [7*sqrt2  sqrt2]
-\end{multilineoutput}\\
-
- frobenius(a); &
-\begin{multilineoutput}{6cm}
-  [0  35*sqrt2 - 2]    [1   sqrt2 ]    [           1  ]
-\{ [               ],   [          ],   [1       - --- ]  \}
-  [1    2*sqrt2   ]    [1  7*sqrt2]    [           7  ]
-                                       [              ]
-                                       [     1        ]
-                                       [0   ----*sqrt2]
-                                       [     14       ]
-
-\end{multilineoutput}\\
-\end{Examples}
-\end{Operator}
-
-
-
-\begin{Operator}{Ratjordan}
-The operator \name{ratjordan} computes the rational Jordan normal form R
-of a \nameref{matrix} (A say). It returns \{R,P,$P^-1$\} where $P*R*P^-1 = A$.
-
-\begin{Syntax}
-\name{ratjordan}\(\meta{matrix}\)
-
-\meta{matrix} :- a square \nameref{matrix}.
-\end{Syntax}
-
-Field Extensions:
-
-By default, calculations are performed in the rational numbers. To 
-extend this field the \name{arnum} package can be used. The package must
-first be loaded by  load\_package arnum;. The field can now be extended
-by using the defpoly command. For example, defpoly sqrt2**2-2; will
-extend the field to include the square root of 2 (now defined by sqrt2).
-See \nameref{frobenius} for an example.
-
-Modular Arithmetic:
-
-\name{ratjordan} can also be calculated in a modular base. To do this 
-first type on  modular;. Then setmod p; (where p is a prime) will set 
-the modular base of calculation to p. By further typing on balanced\_mod
-the answer will appear using a symmetric modular representation.
-
-\begin{Examples}
- a := mat((5,4*x),(2,x^2));&
-\begin{multilineoutput}{6cm}
-     [5  4*x]
-     [      ]
-a := [    2 ]
-     [2  x  ]
-\end{multilineoutput}\\
-
- ratjordan(a); &
-\begin{multilineoutput}{6cm}
-  [0  x*( - 5*x + 8)]   [1  5]    [        -5 ]  
-\{ [                 ],  [    ],   [1     -----] \}
-  [        2        ]   [0  2]    [        2  ]
-  [1      x  + 5    ]             [           ]
-                                  [        1  ]
-                                  [0     -----]
-                                  [        2  ]
-\end{multilineoutput}\\
- on modular; \\
- setmod 23; \\
- a := mat((12,34),(56,78)); &
-\begin{multilineoutput}{6cm}
-     [12  11]
-a := [      ]
-     [10  9 ]
-\end{multilineoutput}\\
-
- ratjordan(a); &
-\begin{multilineoutput}{6cm}
-  [15  0]   [16  8]   [1  21]
-\{ [     ],  [     ],  [     ]  \}
-  [0   6]   [19  4]   [1  4 ]
-\end{multilineoutput}\\
-
- on balanced\_mod;\\
- ratjordan(a);&
-\begin{multilineoutput}{6cm}
-  [- 8  0]   [ - 7  8]   [1  - 2]
-\{ [      ],  [       ],  [      ]  \}
-  [ 0   6]   [ - 4  4]   [1   4 ]
-\end{multilineoutput}\\
-
-\end{Examples}
-\end{Operator}
-
-
-
-\begin{Operator}{Jordansymbolic}
-The operator \name{jordansymbolic} computes the Jordan normal form J
-of a \nameref{matrix} (A say). It returns \{J,L,P,$P^-1$\} where
-$P*J*P^-1 = A$. L = \{ll,mm\} where mm is a name and ll is a list of
-irreducible factors of p(mm).
-
-\begin{Syntax}
-\name{jordansymbolic}\(\meta{matrix}\)
-
-\meta{matrix} :- a square \nameref{matrix}.
-\end{Syntax}
-
-Field Extensions:
-
-By default, calculations are performed in the rational numbers. To 
-extend this field the \nameref{arnum} package can be used. The package must
-first be loaded by  load\_package arnum;. The field can now be extended
-by using the defpoly command. For example, defpoly sqrt2**2-2; will
-extend the field to include the square root of 2 (now defined by sqrt2).
-See \nameref{frobenius} for an example.
-
-Modular Arithmetic:
-
-\name{jordansymbolic} can also be calculated in a modular base. To do this 
-first type on  modular;. Then setmod p; (where p is a prime) will set 
-the modular base of calculation to p. By further typing on balanced\_mod
-the answer will appear using a symmetric modular representation. See 
-\nameref{ratjordan} for an example.
-
-% Extras:
-
-% If using \name{xr}, the X interface for REDUCE, then the appearance of the
-% output can be improved by switching on looking\_good. This
-% converts any lambdas to a greek font.
-
-\begin{Examples}
-
- a := mat((1,y),(2,5*y)); &
-\begin{multilineoutput}{6cm}
-     [1   y ]
-a := [      ]
-     [2  5*y]
-\end{multilineoutput}\\
-
- jordansymbolic(a); &
-\begin{multilineoutput}{6cm}
-\{
-
- [lambda11     0    ]
- [                  ]
- [   0      lambda12]
-
- ,
-
-         2
- {{lambda  - 5*lambda*y - lambda + 3*y},lambda},
-
-
- [lambda11 - 5*y  lambda12 - 5*y]
- [                              ]
- [      2               2       ]
-
- ,
-
-
- [ 2*lambda11 - 5*y - 1    5*lambda11*y - lambda11 - y + 1 ]
- [----------------------  ---------------------------------]
- [       2                              2                  ]
- [   25*y  - 2*y + 1             2*(25*y  - 2*y + 1)       ]
- [                                                         ]
- [ 2*lambda12 - 5*y - 1    5*lambda12*y - lambda12 - y + 1 ]
- [----------------------  ---------------------------------]
- [       2                              2                  ]
- [   25*y  - 2*y + 1             2*(25*y  - 2*y + 1)       ]
-
- \}
-
-\end{multilineoutput}\\
-\end{Examples}
-\end{Operator}
-
-
-
-\begin{Operator}{Jordan}
-The operator \name{jordan} computes the Jordan normal form J
-of a \nameref{matrix} (A say). It returns \{J,P,$P^-1$\} where $P*J*P^-1 = A$.
-
-\begin{Syntax}
-\name{jordan}\(\meta{matrix}\)
-
-\meta{matrix} :- a square \nameref{matrix}.
-\end{Syntax}
-
-Field Extensions:
-By default, calculations are performed in the rational numbers. To 
-extend this field the \name{arnum} package can be used. The package must
-first be loaded by  load\_package arnum;. The field can now be extended
-by using the defpoly command. For example, defpoly sqrt2**2-2; will
-extend the field to include the square root of 2 (now defined by sqrt2).
-See \nameref{frobenius} for an example.
-
-Modular Arithmetic:
-\name{Jordan} can also be calculated in a modular base. To do this 
-first type on  modular;. Then setmod p; (where p is a prime) will set 
-the modular base of calculation to p. By further typing on balanced\_mod
-the answer will appear using a symmetric modular representation. See
-\nameref{ratjordan} for an example.
-
-\begin{Examples}
-
- a := mat((1,x),(0,x)); &
-\begin{multilineoutput}{6cm}
-     [1  x]
-a := [    ]
-     [0  x]
-\end{multilineoutput}\\
-
- jordan(a);&
-\begin{multilineoutput}{6cm}
-\{
-
- [1  0]
- [    ]
- [0  x]
-
- ,
-
-
- [   1           x       ]
- [-------  --------------]
- [ x - 1     2           ]
- [          x  - 2*x + 1 ]
- [                       ]
- [               1       ]
- [   0        -------    ]
- [             x - 1     ]
-
- ,
-
-
- [x - 1   - x ]
- [            ]
- [  0    x - 1]
-
- \}
-\end{multilineoutput}\\
-\end{Examples}
-
-\end{Operator}
-
+\section{Matrix Normal Forms}
+
+\begin{Operator}{Smithex}
+The operator \name{smithex} computes the Smith normal form S of a
+\nameref{matrix} A (say). It returns \{S,P,$P^-1$\} where $P*S*P^-1 = A$.
+
+\begin{Syntax}
+\name{smithex}\(\meta{matrix},\meta{variable}\)
+
+\meta{matrix} :- a rectangular \nameref{matrix} of univariate polynomials in 
+                 \meta{variable}.
+\meta{variable} :- the variable.
+\end{Syntax}
+
+\begin{Examples}
+ a := mat((x,x+1),(0,3*x^2)); &
+\begin{multilineoutput}{6cm}
+      [x  x + 1]
+      [        ]
+ a := [      2 ]
+      [0  3*x  ]
+\end{multilineoutput}\\
+
+ smithex(a,x); &
+\begin{multilineoutput}{6cm}
+   [1  0 ]    [1    0]    [x   x + 1]
+\{  [     ],   [      ],   [         ]  \}
+   [    3]    [   2  ]    [         ]
+   [0  x ]    [3*x  1]    [-3    -3 ]
+\end{multilineoutput}\\
+\end{Examples}
+\end{Operator}
+
+
+\begin{Operator}{Smithex\_int}
+The operator \name{smithex\_int} performs the same task as \name{smithex}
+but on matrices containing only integer entries. Namely, 
+\name{smithex\_int} returns \{S,P,$P^-1$\} where S is the smith normal
+form of the input \nameref{matrix} (A say), and $P*S*P^-1 = A$.
+
+\begin{Syntax}
+\name{smithex\_int}\(\meta{matrix}\)
+
+\meta{matrix} :- a rectangular \nameref{matrix} of integer entries.
+\end{Syntax}
+
+\begin{Examples}
+ a := mat((9,-36,30),(-36,192,-180),(30,-180,180)); &
+\begin{multilineoutput}{6cm}
+     [ 9   -36    30 ]
+     [               ]
+a := [-36  192   -180]
+     [               ]
+     [30   -180  180 ]
+\end{multilineoutput}\\
+
+ smithex_int(a); &
+\begin{multilineoutput}{6cm}
+  [3  0   0 ]    [-17  -5   -4 ]    [1   -24  30 ]
+  [         ]    [             ]    [            ]
+\{ [0  12  0 ],   [64   19   15 ],   [-1  25   -30] \}
+  [         ]    [             ]    [            ] 
+  [0  0   60]    [-50  -15  -12]    [0   -1    1 ] 
+\end{multilineoutput}\\
+\end{Examples}
+\end{Operator}
+
+
+\begin{Operator}{Frobenius}
+The operator \name{frobenius} computes the \name{frobenius} normal form F of a 
+\nameref{matrix} (A say). It returns \{F,P,$P^-1$\} where $P*F*P^-1 = A$.
+
+\begin{Syntax}
+\name{frobenius}\(\meta{matrix}\)
+
+\meta{matrix} :- a square \nameref{matrix}.
+\end{Syntax}
+
+Field Extensions:
+
+By default, calculations are performed in the rational numbers. To 
+extend this field the \nameref{arnum} package can be used. The package must
+first be loaded by load\_package arnum;. The field can now be extended
+by using the defpoly command. For example, defpoly sqrt2**2-2; will
+extend the field to include the square root of 2 (now defined by sqrt2).
+
+Modular Arithmetic: 
+
+\name{Frobenius} can also be calculated in a modular base. To do this 
+first type on  modular;. Then setmod p; (where p is a prime) will set 
+the modular base of calculation to p. By further typing on balanced\_mod
+the answer will appear using a symmetric modular representation. See 
+\nameref{ratjordan} for an example.
+
+\begin{Examples}
+ a := mat((x,x^2),(3,5*x)); &
+\begin{multilineoutput}{6cm}
+     [    2 ]
+     [x  x  ]
+a := [      ]
+     [3  5*x]
+\end{multilineoutput}\\
+ frobenius(a);&
+\begin{multilineoutput}{6cm}
+   [         2]    [1  x]    [       - x ]
+\{  [0   - 2*x ],   [    ],   [1     -----]  \}
+   [          ]    [0  3]    [        3  ]
+   [1    6*x  ]              [           ]
+                             [        1  ]
+                             [0      --- ]
+                             [        3  ]
+
+\end{multilineoutput}\\
+ load\_package arnum;\\
+ defpoly sqrt2**2-2;\\
+ a := mat((sqrt2,5),(7*sqrt2,sqrt2));&
+\begin{multilineoutput}{6cm}
+     [ sqrt2     5  ]
+a := [              ]
+     [7*sqrt2  sqrt2]
+\end{multilineoutput}\\
+
+ frobenius(a); &
+\begin{multilineoutput}{6cm}
+  [0  35*sqrt2 - 2]    [1   sqrt2 ]    [           1  ]
+\{ [               ],   [          ],   [1       - --- ]  \}
+  [1    2*sqrt2   ]    [1  7*sqrt2]    [           7  ]
+                                       [              ]
+                                       [     1        ]
+                                       [0   ----*sqrt2]
+                                       [     14       ]
+
+\end{multilineoutput}\\
+\end{Examples}
+\end{Operator}
+
+
+
+\begin{Operator}{Ratjordan}
+The operator \name{ratjordan} computes the rational Jordan normal form R
+of a \nameref{matrix} (A say). It returns \{R,P,$P^-1$\} where $P*R*P^-1 = A$.
+
+\begin{Syntax}
+\name{ratjordan}\(\meta{matrix}\)
+
+\meta{matrix} :- a square \nameref{matrix}.
+\end{Syntax}
+
+Field Extensions:
+
+By default, calculations are performed in the rational numbers. To 
+extend this field the \name{arnum} package can be used. The package must
+first be loaded by  load\_package arnum;. The field can now be extended
+by using the defpoly command. For example, defpoly sqrt2**2-2; will
+extend the field to include the square root of 2 (now defined by sqrt2).
+See \nameref{frobenius} for an example.
+
+Modular Arithmetic:
+
+\name{ratjordan} can also be calculated in a modular base. To do this 
+first type on  modular;. Then setmod p; (where p is a prime) will set 
+the modular base of calculation to p. By further typing on balanced\_mod
+the answer will appear using a symmetric modular representation.
+
+\begin{Examples}
+ a := mat((5,4*x),(2,x^2));&
+\begin{multilineoutput}{6cm}
+     [5  4*x]
+     [      ]
+a := [    2 ]
+     [2  x  ]
+\end{multilineoutput}\\
+
+ ratjordan(a); &
+\begin{multilineoutput}{6cm}
+  [0  x*( - 5*x + 8)]   [1  5]    [        -5 ]  
+\{ [                 ],  [    ],   [1     -----] \}
+  [        2        ]   [0  2]    [        2  ]
+  [1      x  + 5    ]             [           ]
+                                  [        1  ]
+                                  [0     -----]
+                                  [        2  ]
+\end{multilineoutput}\\
+ on modular; \\
+ setmod 23; \\
+ a := mat((12,34),(56,78)); &
+\begin{multilineoutput}{6cm}
+     [12  11]
+a := [      ]
+     [10  9 ]
+\end{multilineoutput}\\
+
+ ratjordan(a); &
+\begin{multilineoutput}{6cm}
+  [15  0]   [16  8]   [1  21]
+\{ [     ],  [     ],  [     ]  \}
+  [0   6]   [19  4]   [1  4 ]
+\end{multilineoutput}\\
+
+ on balanced\_mod;\\
+ ratjordan(a);&
+\begin{multilineoutput}{6cm}
+  [- 8  0]   [ - 7  8]   [1  - 2]
+\{ [      ],  [       ],  [      ]  \}
+  [ 0   6]   [ - 4  4]   [1   4 ]
+\end{multilineoutput}\\
+
+\end{Examples}
+\end{Operator}
+
+
+
+\begin{Operator}{Jordansymbolic}
+The operator \name{jordansymbolic} computes the Jordan normal form J
+of a \nameref{matrix} (A say). It returns \{J,L,P,$P^-1$\} where
+$P*J*P^-1 = A$. L = \{ll,mm\} where mm is a name and ll is a list of
+irreducible factors of p(mm).
+
+\begin{Syntax}
+\name{jordansymbolic}\(\meta{matrix}\)
+
+\meta{matrix} :- a square \nameref{matrix}.
+\end{Syntax}
+
+Field Extensions:
+
+By default, calculations are performed in the rational numbers. To 
+extend this field the \nameref{arnum} package can be used. The package must
+first be loaded by  load\_package arnum;. The field can now be extended
+by using the defpoly command. For example, defpoly sqrt2**2-2; will
+extend the field to include the square root of 2 (now defined by sqrt2).
+See \nameref{frobenius} for an example.
+
+Modular Arithmetic:
+
+\name{jordansymbolic} can also be calculated in a modular base. To do this 
+first type on  modular;. Then setmod p; (where p is a prime) will set 
+the modular base of calculation to p. By further typing on balanced\_mod
+the answer will appear using a symmetric modular representation. See 
+\nameref{ratjordan} for an example.
+
+% Extras:
+
+% If using \name{xr}, the X interface for REDUCE, then the appearance of the
+% output can be improved by switching on looking\_good. This
+% converts any lambdas to a greek font.
+
+\begin{Examples}
+
+ a := mat((1,y),(2,5*y)); &
+\begin{multilineoutput}{6cm}
+     [1   y ]
+a := [      ]
+     [2  5*y]
+\end{multilineoutput}\\
+
+ jordansymbolic(a); &
+\begin{multilineoutput}{6cm}
+\{
+
+ [lambda11     0    ]
+ [                  ]
+ [   0      lambda12]
+
+ ,
+
+         2
+ {{lambda  - 5*lambda*y - lambda + 3*y},lambda},
+
+
+ [lambda11 - 5*y  lambda12 - 5*y]
+ [                              ]
+ [      2               2       ]
+
+ ,
+
+
+ [ 2*lambda11 - 5*y - 1    5*lambda11*y - lambda11 - y + 1 ]
+ [----------------------  ---------------------------------]
+ [       2                              2                  ]
+ [   25*y  - 2*y + 1             2*(25*y  - 2*y + 1)       ]
+ [                                                         ]
+ [ 2*lambda12 - 5*y - 1    5*lambda12*y - lambda12 - y + 1 ]
+ [----------------------  ---------------------------------]
+ [       2                              2                  ]
+ [   25*y  - 2*y + 1             2*(25*y  - 2*y + 1)       ]
+
+ \}
+
+\end{multilineoutput}\\
+\end{Examples}
+\end{Operator}
+
+
+
+\begin{Operator}{Jordan}
+The operator \name{jordan} computes the Jordan normal form J
+of a \nameref{matrix} (A say). It returns \{J,P,$P^-1$\} where $P*J*P^-1 = A$.
+
+\begin{Syntax}
+\name{jordan}\(\meta{matrix}\)
+
+\meta{matrix} :- a square \nameref{matrix}.
+\end{Syntax}
+
+Field Extensions:
+By default, calculations are performed in the rational numbers. To 
+extend this field the \name{arnum} package can be used. The package must
+first be loaded by  load\_package arnum;. The field can now be extended
+by using the defpoly command. For example, defpoly sqrt2**2-2; will
+extend the field to include the square root of 2 (now defined by sqrt2).
+See \nameref{frobenius} for an example.
+
+Modular Arithmetic:
+\name{Jordan} can also be calculated in a modular base. To do this 
+first type on  modular;. Then setmod p; (where p is a prime) will set 
+the modular base of calculation to p. By further typing on balanced\_mod
+the answer will appear using a symmetric modular representation. See
+\nameref{ratjordan} for an example.
+
+\begin{Examples}
+
+ a := mat((1,x),(0,x)); &
+\begin{multilineoutput}{6cm}
+     [1  x]
+a := [    ]
+     [0  x]
+\end{multilineoutput}\\
+
+ jordan(a);&
+\begin{multilineoutput}{6cm}
+\{
+
+ [1  0]
+ [    ]
+ [0  x]
+
+ ,
+
+
+ [   1           x       ]
+ [-------  --------------]
+ [ x - 1     2           ]
+ [          x  - 2*x + 1 ]
+ [                       ]
+ [               1       ]
+ [   0        -------    ]
+ [             x - 1     ]
+
+ ,
+
+
+ [x - 1   - x ]
+ [            ]
+ [  0    x - 1]
+
+ \}
+\end{multilineoutput}\\
+\end{Examples}
+
+\end{Operator}
+