@@ -1,77 +1,77 @@
-\documentstyle[12pt]{article}
-\begin{document}
-\begin{center} {\Large INEQ} \end{center}
-
-\begin{center} Herbert Melenk \\ Konrad-Zuse-Zentrum fuer
-Informationstechnik \\
-Heilbronner Str. 10, D10711 Berlin -- Wilmersdorf\\ Germany \\
-melenk@zib-berlin.de \end{center}
-
-This package supports the operator {\bf ineq\_solve} that
-tries to solves single inequalities and sets of coupled inequalities.
-The following types of systems are supported
-\footnote{For linear optimization problems please use the operator
-{\bf simplex} of the {\bf linalg} package}:
-\begin{itemize}
-\item only numeric coefficients (no parametric system),
-\item a linear system of mixed equations and $<=$ -- $>=$
-     inequalities, applying the method of Fourier and Motzkin
-     \footnote{described by G.B. Dantzig in {\em Linear Programming
-      and Extensions.}},
-\item a univariate inequality with $<=$, $>=$, $>$ or $<$ operator
-     and polynomial or rational left--hand and right--hand sides,
-     or a system of such inequalities with only one variable.
-\end{itemize}
-
-Syntax:
-\begin{center}
-{\tt INEQ\_SOLVE($<$expr$>$ [,$<$vl$>$])}
-\end{center}
-where $<$expr$>$ is an inequality or a list of coupled inequalities
-and equations, and the optional argument $<$vl$>$ is a single
-variable (kernel) or a list of variables (kernels). If not
-specified, they are extracted automatically from $<$expr$>$.
-For multivariate input an explicit variable list specifies the
-elimination sequence: the last member is the most specific one.
-
-An error message occurs if the input cannot be processed by the
-currently implemented algorithms.
-
-The result is a list. It is empty if the system has no feasible solution.
-Otherwise the result presents the admissible ranges as set
-of equations where each variable is equated to
-one expression or to an interval.
-The most specific variable is the first one in the result list and
-each form contains only preceding variables (resolved form).
-The interval limits can be formal {\bf max} or {\bf min} expressions.
-Algebraic numbers are encoded as rounded number approximations.
-
-\noindent
-{\bf Examples}:
-
-\begin{verbatim}
-ineq_solve({(2*x^2+x-1)/(x-1) >=  (x+1/2)^2, x>0});
-
-{x=(0 .. 0.326583),x=(1 .. 2.56777)}
-
- reg:=
- {a + b - c>=0, a - b + c>=0, - a + b + c>=0, 0>=0, 2>=0,
-  2*c - 2>=0, a - b + c>=0, a + b - c>=0, - a + b + c - 2>=0,
-  2>=0, 0>=0, 2*b - 2>=0, k + 1>=0, - a - b - c + k>=0,
-   - a - b - c + k + 2>=0, - 2*b + k>=0,
-   - 2*c + k>=0, a + b + c - k>=0,
-  2*b + 2*c - k - 2>=0, a + b + c - k>=0}$
-
-ineq_solve (reg,{k,a,b,c});
-
-{c=(1 .. infinity),
-
- b=(1 .. infinity),
-
- a=(max( - b + c,b - c) .. b + c - 2),
-
- k=a + b + c}
-\end{verbatim}
-
-\end{document}
-
+\documentstyle[12pt]{article}
+\begin{document}
+\begin{center} {\Large INEQ} \end{center}
+
+\begin{center} Herbert Melenk \\ Konrad-Zuse-Zentrum fuer
+Informationstechnik \\
+Heilbronner Str. 10, D10711 Berlin -- Wilmersdorf\\ Germany \\
+melenk@zib-berlin.de \end{center}
+
+This package supports the operator {\bf ineq\_solve} that
+tries to solves single inequalities and sets of coupled inequalities.
+The following types of systems are supported
+\footnote{For linear optimization problems please use the operator
+{\bf simplex} of the {\bf linalg} package}:
+\begin{itemize}
+\item only numeric coefficients (no parametric system),
+\item a linear system of mixed equations and $<=$ -- $>=$
+     inequalities, applying the method of Fourier and Motzkin
+     \footnote{described by G.B. Dantzig in {\em Linear Programming
+      and Extensions.}},
+\item a univariate inequality with $<=$, $>=$, $>$ or $<$ operator
+     and polynomial or rational left--hand and right--hand sides,
+     or a system of such inequalities with only one variable.
+\end{itemize}
+
+Syntax:
+\begin{center}
+{\tt INEQ\_SOLVE($<$expr$>$ [,$<$vl$>$])}
+\end{center}
+where $<$expr$>$ is an inequality or a list of coupled inequalities
+and equations, and the optional argument $<$vl$>$ is a single
+variable (kernel) or a list of variables (kernels). If not
+specified, they are extracted automatically from $<$expr$>$.
+For multivariate input an explicit variable list specifies the
+elimination sequence: the last member is the most specific one.
+
+An error message occurs if the input cannot be processed by the
+currently implemented algorithms.
+
+The result is a list. It is empty if the system has no feasible solution.
+Otherwise the result presents the admissible ranges as set
+of equations where each variable is equated to
+one expression or to an interval.
+The most specific variable is the first one in the result list and
+each form contains only preceding variables (resolved form).
+The interval limits can be formal {\bf max} or {\bf min} expressions.
+Algebraic numbers are encoded as rounded number approximations.
+
+\noindent
+{\bf Examples}:
+
+\begin{verbatim}
+ineq_solve({(2*x^2+x-1)/(x-1) >=  (x+1/2)^2, x>0});
+
+{x=(0 .. 0.326583),x=(1 .. 2.56777)}
+
+ reg:=
+ {a + b - c>=0, a - b + c>=0, - a + b + c>=0, 0>=0, 2>=0,
+  2*c - 2>=0, a - b + c>=0, a + b - c>=0, - a + b + c - 2>=0,
+  2>=0, 0>=0, 2*b - 2>=0, k + 1>=0, - a - b - c + k>=0,
+   - a - b - c + k + 2>=0, - 2*b + k>=0,
+   - 2*c + k>=0, a + b + c - k>=0,
+  2*b + 2*c - k - 2>=0, a + b + c - k>=0}$
+
+ineq_solve (reg,{k,a,b,c});
+
+{c=(1 .. infinity),
+
+ b=(1 .. infinity),
+
+ a=(max( - b + c,b - c) .. b + c - 2),
+
+ k=a + b + c}
+\end{verbatim}
+
+\end{document}
+