--- pubs/books/ucbka/trunk/c_cfs0/c_cfs0.tex	2016/10/06 03:15:02	4
+++ pubs/books/ucbka/trunk/c_cfs0/c_cfs0.tex	2017/07/03 01:59:16	140
@@ -1,129 +1,129 @@
-%$Header: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_cfs0/c_cfs0.tex,v 1.2 2001/07/01 21:10:59 dtashley Exp $
-
-\chapter[Solutions: \ccfrzeroxrefcomma{}Chapter \ref{ccfr0}]
-        {Solutions: \ccfrzeroxrefcomma{}Chapter \ref{ccfr0}, \ccfrzerolongtitle{}}
-
-\label{ccfs0}
-
-\vworkexercisechapterheader{}
-\begin{vworkexercisesolution}{\ref{exe:cfr0:sexe0:a01}}
-Placeholder.
-
-\end{vworkexercisesolution}
-\vworkexerciseseparator
-\begin{vworkexercisesolution}{\ref{exe:cfr0:sexe0:b01}}
-Note that $1.609344 > 255/255$ (i.e. beyond the ``corner point''
-in the sense suggested by Fig. \cfryzeroxrefhyphen{}\ref{fig:cfry0:ili0:00}), 
-so it is necessary to find
-the Farey neighbors of $1.609344^{-1}$ in $F_{255}$
-(Algorithm \ccfrzeroxrefhyphen{}\ref{alg:ccfr0:scba0:cfenclosingneighborsfab}),
-and then invert and re-order the results.
-
-Table \ref{tbl:ccfs0:exe:cfr0:sexe0:b01:01} 
-shows the application of 
-Algorithm \ccfrzeroxrefhyphen{}\ref{alg:ccfr0:scrn0:akgenalg}
-to form the continued fraction partial quotients and
-convergents of $1.609344^{-1}$ = 1,000,000/1,609,344.
-
-\begin{table}
-\caption{Continued Fraction Partial Quotients And Convergents Of $1.609344^{-1}$, 
-         The Reciprocal Of The Conversion Factor From Miles
-         To Kilometers (Solution To Exercise \ccfrzeroxrefhyphen{}\ref{exe:cfr0:sexe0:b01})}
-\label{tbl:ccfs0:exe:cfr0:sexe0:b01:01}
-\begin{center}
-\begin{tabular}{|c|c|c|c|c|c|c|}
-\hline
-\small{Index}     &      \small{Dividend}  &       \small{Divisor} &  $a_k$ &   \small{Remainder} & $p_k$ & $q_k$ \\
-\small{(k)}       &                        &                       &        &                     &       &       \\
-\hline
-\hline
-       \small{-1} & \small{N/A}            & \small{1,000,000}     &     \small{N/A} &   \small{1,609,344}   &      \small{1} &       \small{0}  \\
-\hline
-       \small{0}  & \small{1,000,000}      & \small{1,609,344}     &     \small{0}   &   \small{1,000,000}   &      \small{0} &       \small{1}  \\
-\hline
-       \small{1}  & \small{1,609,344}      & \small{1,000,000}     &     \small{1}   &     \small{609,344}   &      \small{1} &       \small{1}  \\
-\hline
-       \small{2}  & \small{1,000,000}      &   \small{609,344}     &     \small{1}   &     \small{390,656}   &      \small{1} &       \small{2}  \\
-\hline
-       \small{3}  &   \small{609,344}      &   \small{390,656}     &     \small{1}   &     \small{218,688}   &      \small{2} &       \small{3}  \\
-\hline
-       \small{4}  &   \small{390,656}      &   \small{218,688}     &     \small{1}   &     \small{171,968}   &      \small{3} &       \small{5}  \\
-\hline
-       \small{5}  &   \small{218,688}      &   \small{171,968}     &     \small{1}   &      \small{46,720}   &      \small{5} &       \small{8}  \\
-\hline
-       \small{6}  &   \small{171,968}      &    \small{46,720}     &     \small{3}   &      \small{31,808}   &     \small{18} &      \small{29}  \\
-\hline
-       \small{7}  &    \small{46,720}      &    \small{31,808}     &     \small{1}   &      \small{14,912}   &     \small{23} &      \small{37}  \\
-\hline
-       \small{8}  &    \small{31,808}      &    \small{14,912}     &     \small{2}   &       \small{1,984}   &     \small{64} &     \small{103}  \\
-\hline
-       \small{9}  &    \small{14,912}      &     \small{1,984}     &     \small{7}   &       \small{1,024}   &    \small{471} &     \small{758}  \\
-\hline
-      \small{10}  &     \small{1,984}      &     \small{1,024}     &     \small{1}   &         \small{960}   &    \small{535} &     \small{861}  \\
-\hline
-      \small{11}  &     \small{1,024}      &       \small{960}     &     \small{1}   &          \small{64}   &  \small{1,006} &   \small{1,619} \\
-\hline
-      \small{12}  &       \small{960}      &        \small{64}     &    \small{15}   &           \small{0}   & \small{15,625} &  \small{25,146} \\
-\hline
-\end{tabular}
-\end{center}
-\end{table}
-
-From the table, the highest-ordered convergent with a denominator not greater
-than 255 is 64/103.  Applying 
-Theorem \ccfrzeroxrefhyphen{}\ref{thm:ccfr0:scba0:convergentbetterappthanlesserdenominator}
-yields 151/243 as the other neighbor to 
-$1.609344^{-1}$ in $F_{255}$.  Inverting and ordering these 
-fractions yields
-
-\begin{equation}
-\frac{243}{151} < 1.609344 < \frac{103}{64}.
-\end{equation}
-
-Either rational approximation is quite good, but 103/64 is closer
-to 1.609344 than 243/151, and so we choose 103/64 as the best
-rational approximation under the constraints.
-\end{vworkexercisesolution}
-\vworkexerciseseparator
-\begin{vworkexercisesolution}{\ref{exe:cfr0:sexe0:b02}}
-To find the best rational approximation to 1.609344 with
-a numerator and denominator no larger than 65,535, first consider the
-possibility that 1.609344 is already a rational number that can
-be expressed under these constraints.  Examining the final
-convergent\footnote{For a rational number, the final convergent
-is the rational number in lowest terms.  This is established
-by Theorem \ccfrzeroxrefhyphen{}\ref{thm:ccfr0:scnv0:evenslessthanoddsgreaterthan}.}
-of Table \ref{tbl:ccfs0:exe:cfr0:sexe0:b01:01} verifies
-that 1.609344 = 25,146/15,625.  Thus, the best rational 
-approximation under
-the constraints is 25,146/15,625, which is precisely equal to the
-number to be approximated.
-\end{vworkexercisesolution}
-\vworkexercisechapterfooter
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\vfill
-\noindent\begin{figure}[!b]
-\noindent\rule[-0.25in]{\textwidth}{1pt}
-\begin{tiny}
-\begin{verbatim}
-$RCSfile: c_cfs0.tex,v $
-$Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_cfs0/c_cfs0.tex,v $
-$Revision: 1.2 $
-$Author: dtashley $
-$Date: 2001/07/01 21:10:59 $
-\end{verbatim}
-\end{tiny}
-\noindent\rule[0.25in]{\textwidth}{1pt}
-\end{figure}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% $Log: c_cfs0.tex,v $
-% Revision 1.2  2001/07/01 21:10:59  dtashley
-% Safety check-in after major re-org.
-%
-% Revision 1.1.1.1  2001/07/01 00:14:34  dtashley
-% Initial import.
-%
-%
-%End of file C_CFS0.TEX
+%$Header$
+
+\chapter[Solutions: \ccfrzeroxrefcomma{}Chapter \ref{ccfr0}]
+        {Solutions: \ccfrzeroxrefcomma{}Chapter \ref{ccfr0}, \ccfrzerolongtitle{}}
+
+\label{ccfs0}
+
+\vworkexercisechapterheader{}
+\begin{vworkexercisesolution}{\ref{exe:cfr0:sexe0:a01}}
+Placeholder.
+
+\end{vworkexercisesolution}
+\vworkexerciseseparator
+\begin{vworkexercisesolution}{\ref{exe:cfr0:sexe0:b01}}
+Note that $1.609344 > 255/255$ (i.e. beyond the ``corner point''
+in the sense suggested by Fig. \cfryzeroxrefhyphen{}\ref{fig:cfry0:ili0:00}), 
+so it is necessary to find
+the Farey neighbors of $1.609344^{-1}$ in $F_{255}$
+(Algorithm \ccfrzeroxrefhyphen{}\ref{alg:ccfr0:scba0:cfenclosingneighborsfab}),
+and then invert and re-order the results.
+
+Table \ref{tbl:ccfs0:exe:cfr0:sexe0:b01:01} 
+shows the application of 
+Algorithm \ccfrzeroxrefhyphen{}\ref{alg:ccfr0:scrn0:akgenalg}
+to form the continued fraction partial quotients and
+convergents of $1.609344^{-1}$ = 1,000,000/1,609,344.
+
+\begin{table}
+\caption{Continued Fraction Partial Quotients And Convergents Of $1.609344^{-1}$, 
+         The Reciprocal Of The Conversion Factor From Miles
+         To Kilometers (Solution To Exercise \ccfrzeroxrefhyphen{}\ref{exe:cfr0:sexe0:b01})}
+\label{tbl:ccfs0:exe:cfr0:sexe0:b01:01}
+\begin{center}
+\begin{tabular}{|c|c|c|c|c|c|c|}
+\hline
+\small{Index}     &      \small{Dividend}  &       \small{Divisor} &  $a_k$ &   \small{Remainder} & $p_k$ & $q_k$ \\
+\small{(k)}       &                        &                       &        &                     &       &       \\
+\hline
+\hline
+       \small{-1} & \small{N/A}            & \small{1,000,000}     &     \small{N/A} &   \small{1,609,344}   &      \small{1} &       \small{0}  \\
+\hline
+       \small{0}  & \small{1,000,000}      & \small{1,609,344}     &     \small{0}   &   \small{1,000,000}   &      \small{0} &       \small{1}  \\
+\hline
+       \small{1}  & \small{1,609,344}      & \small{1,000,000}     &     \small{1}   &     \small{609,344}   &      \small{1} &       \small{1}  \\
+\hline
+       \small{2}  & \small{1,000,000}      &   \small{609,344}     &     \small{1}   &     \small{390,656}   &      \small{1} &       \small{2}  \\
+\hline
+       \small{3}  &   \small{609,344}      &   \small{390,656}     &     \small{1}   &     \small{218,688}   &      \small{2} &       \small{3}  \\
+\hline
+       \small{4}  &   \small{390,656}      &   \small{218,688}     &     \small{1}   &     \small{171,968}   &      \small{3} &       \small{5}  \\
+\hline
+       \small{5}  &   \small{218,688}      &   \small{171,968}     &     \small{1}   &      \small{46,720}   &      \small{5} &       \small{8}  \\
+\hline
+       \small{6}  &   \small{171,968}      &    \small{46,720}     &     \small{3}   &      \small{31,808}   &     \small{18} &      \small{29}  \\
+\hline
+       \small{7}  &    \small{46,720}      &    \small{31,808}     &     \small{1}   &      \small{14,912}   &     \small{23} &      \small{37}  \\
+\hline
+       \small{8}  &    \small{31,808}      &    \small{14,912}     &     \small{2}   &       \small{1,984}   &     \small{64} &     \small{103}  \\
+\hline
+       \small{9}  &    \small{14,912}      &     \small{1,984}     &     \small{7}   &       \small{1,024}   &    \small{471} &     \small{758}  \\
+\hline
+      \small{10}  &     \small{1,984}      &     \small{1,024}     &     \small{1}   &         \small{960}   &    \small{535} &     \small{861}  \\
+\hline
+      \small{11}  &     \small{1,024}      &       \small{960}     &     \small{1}   &          \small{64}   &  \small{1,006} &   \small{1,619} \\
+\hline
+      \small{12}  &       \small{960}      &        \small{64}     &    \small{15}   &           \small{0}   & \small{15,625} &  \small{25,146} \\
+\hline
+\end{tabular}
+\end{center}
+\end{table}
+
+From the table, the highest-ordered convergent with a denominator not greater
+than 255 is 64/103.  Applying 
+Theorem \ccfrzeroxrefhyphen{}\ref{thm:ccfr0:scba0:convergentbetterappthanlesserdenominator}
+yields 151/243 as the other neighbor to 
+$1.609344^{-1}$ in $F_{255}$.  Inverting and ordering these 
+fractions yields
+
+\begin{equation}
+\frac{243}{151} < 1.609344 < \frac{103}{64}.
+\end{equation}
+
+Either rational approximation is quite good, but 103/64 is closer
+to 1.609344 than 243/151, and so we choose 103/64 as the best
+rational approximation under the constraints.
+\end{vworkexercisesolution}
+\vworkexerciseseparator
+\begin{vworkexercisesolution}{\ref{exe:cfr0:sexe0:b02}}
+To find the best rational approximation to 1.609344 with
+a numerator and denominator no larger than 65,535, first consider the
+possibility that 1.609344 is already a rational number that can
+be expressed under these constraints.  Examining the final
+convergent\footnote{For a rational number, the final convergent
+is the rational number in lowest terms.  This is established
+by Theorem \ccfrzeroxrefhyphen{}\ref{thm:ccfr0:scnv0:evenslessthanoddsgreaterthan}.}
+of Table \ref{tbl:ccfs0:exe:cfr0:sexe0:b01:01} verifies
+that 1.609344 = 25,146/15,625.  Thus, the best rational 
+approximation under
+the constraints is 25,146/15,625, which is precisely equal to the
+number to be approximated.
+\end{vworkexercisesolution}
+\vworkexercisechapterfooter
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\vfill
+\noindent\begin{figure}[!b]
+\noindent\rule[-0.25in]{\textwidth}{1pt}
+\begin{tiny}
+\begin{verbatim}
+$RCSfile: c_cfs0.tex,v $
+$Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_cfs0/c_cfs0.tex,v $
+$Revision: 1.2 $
+$Author: dtashley $
+$Date: 2001/07/01 21:10:59 $
+\end{verbatim}
+\end{tiny}
+\noindent\rule[0.25in]{\textwidth}{1pt}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% $Log: c_cfs0.tex,v $
+% Revision 1.2  2001/07/01 21:10:59  dtashley
+% Safety check-in after major re-org.
+%
+% Revision 1.1.1.1  2001/07/01 00:14:34  dtashley
+% Initial import.
+%
+%
+%End of file C_CFS0.TEX
