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%$Header$ |
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|
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\chapter{\cdcmzerolongtitle{}} |
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|
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\label{cdcm0} |
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|
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|
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`''---From A Chinese Restaurant |
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Fortune Cookie, 01/26/01 |
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|
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\beginchapterquote{``Simplicity of character is the natural result |
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of profound thought.''} |
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{From A Chinese Restaurant fortune cookie, 01/26/01} |
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|
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\section{Introduction} |
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%Section Tag: INT0 |
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\label{cdcm0:sint0} |
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|
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\emph{The Iju Tool Set} includes a number of DOS console-mode utilities. |
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Although these utilities present a humble interface (text-only input and output), |
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some are very powerful. Because the internal architecture of the |
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\index{Microsoft \emph{Windows}@Microsoft Windows}\emph{Windows} family of |
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operating systems has changed over the years so that DOS console-mode |
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applications now use the Win32 memory model, these applications can now |
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allocate at least hundreds of megabytes of memory and are generally unfettered |
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in their operation. Despite the humble interface, console-mode applications |
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can now be very powerful indeed. |
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|
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Many of these utilities are very similar to---and sometimes use the |
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same software components as---Tcl/Tk extensions that perform the same |
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functions. |
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|
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The first part of this chapter describes command-line option formats which |
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are common across more than one utility. The second part of the chapter describes |
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each of the utilities in detail. |
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
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\subsection{Notational Conventions} |
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%Subsection tag: NCO0 |
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\label{ctin0:sccl0:snco0} |
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|
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We use the angle brackets (`\texttt{<}' and `\texttt{>}') to denote a |
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parameter which must occur and be completed with the information |
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indicated. For example, we would use \texttt{<winfname>} to indicate |
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that the name of a file must be provided in a format acceptable to |
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Windows. |
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|
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We use the square brackets (`\texttt{[}' and `\texttt{]}') to denote |
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an optional parameter. Square brackets may be nested arbitrarily |
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deep---for example, `\texttt{[lastname [firstname]]}' would |
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indicate that both first name and last name may be omitted, but |
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the first name may not occur alone without the last name. |
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|
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In all cases, the parameter types described in Section |
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\ctinzeroxrefhyphen{}\ref{ctin0:sccl0} |
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are used. |
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|
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In some cases it may be necessary to distinguish between the |
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\emph{name} of a parameter and the \emph{type} of the parameter. |
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For example, two parameters may both be of type \texttt{uint32} |
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but may serve different functions---one may be, for example, |
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a dividend and the other a divisor. In these cases we use |
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the underscore character (`\texttt{\_}') to separate |
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the name and the type. For example, in such a case we may |
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use a description such as \texttt{divisor\_uint32}. |
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|
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\subsection{The \texttt{-help} Option} |
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%Subsection tag: hel0 |
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\label{cdcm0:sccl0:shel0} |
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|
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\index{help@\texttt{-help}} |
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\index{-help@\texttt{-help}} |
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The command-line option \texttt{-help} (which must be used alone with no |
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other parameters) will cause a full set of help information to be written |
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to the standard output stream. |
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|
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Any command line which is not recognized (i.e. that contains no parameters or is |
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otherwise invalid) will cause a message to be displayed to the effect that |
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the utility may be re-run with the \texttt{-help} parameter to obtain help |
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information. |
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|
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\subsection{The \texttt{-verbose} And \texttt{-debug} Options} |
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%Subsection tag: ver0 |
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\label{cdcm0:sccl0:sver0} |
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|
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\index{verbose@\texttt{-verbose}} |
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\index{-verbose@\texttt{-verbose}} |
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\index{debug@\texttt{-debug}} |
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\index{-debug@\texttt{-debug}} |
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The command-line option \texttt{-verbose} will cause intermediate results and other |
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useful information to be written to the standard output as the utility runs. |
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Because it is very often |
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used, the \texttt{-verbose} option |
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can always be abbreviated \texttt{-v}. |
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For utilities which perform arithmetic, these intermediate results may be useful |
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in verifying hand calculations or for other purposes. |
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|
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The \texttt{-debug} option will cause debugging information to be written to |
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the standard output. Because the \texttt{-debug} option is very rarely used, |
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it does not have an abbreviated form. The debugging information consists of |
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information that may be useful in diagnosing the internal problem if a utility |
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ever is found to contain a software defect. |
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|
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The \texttt{-verbose} and \texttt{-debug} options are potentially orthogonal. |
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The information output enabled by the \texttt{-verbose} option may be different |
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than the information output enabled by the \texttt{-debug} option. The options |
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may be used together. |
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|
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\subsection{The \texttt{-noformat} Option} |
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%Subsection tag: nfo0 |
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\label{cdcm0:sccl0:snfo0} |
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|
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\index{noformat@\texttt{-noformat}} |
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\index{-noformat@\texttt{-noformat}} |
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The command-line option \texttt{-noformat} will cause formatting of the output to be |
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suppressed. Because it is very often used, this option can always be abbreviated |
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\texttt{-nf}. |
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|
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Normally, each command will format the output to be very aesthetic and |
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human-readable. This involves adding commas and comments and splitting |
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long integers across multiple lines. However, when piping data from one |
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utility to another or spawning these utilities from a script or other program, |
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it may be desirable to suppress all but the output data. |
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|
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The \texttt{-verbose} and \texttt{-debug} options override (i.e. cancel) |
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the \texttt{-noformat} option. The \texttt{-noformat} option will not be |
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honored if it appears on the same command line as the |
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\texttt{-verbose} option or the \texttt{-debug} option. |
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Invocation Of The Utilities} |
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%Section tag: INV0 |
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|
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The most common way to invoke any of these command-line utilities is |
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to open a console box (in Windows), and then to run these utilities |
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from the command line or from a batch file. |
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|
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These utilities could also be run from a Tcl/Tk script, or spawned from |
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a Windows application. |
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|
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These utilities do not write to the standard error stream (only to the |
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standard output), so output can always be redirected. |
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|
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These utilities will generate a return code of 0 if no errors are detected, or |
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a return code of 4 if any errors are detected. |
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{File CRC And Hash Function Utilities} |
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%Section tag: CHF0 |
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\label{cdcm0:schf0} |
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{The \emph{crc32} Utility} |
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%Subection tag: CRC0 |
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\label{cdcm0:schf0:scrc0} |
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|
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\index{crc32@\emph{crc32}} |
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\begin{dosutilcommandname}{crc32}% |
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calculates the CRC-32 and file size (in bytes) of a file. |
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The CRC-32 is written to the standard output in hexadecimal |
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notation, and the file size is written to the standard output |
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in decimal notation with commas. |
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\end{dosutilcommandname} |
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|
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\begin{dosutilcommandsynopsis} |
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\dosutilcommandsynopsisline{crc32}{winfname [$<$options$>$]} |
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\dosutilcommandsynopsisline{crc32}{-help} |
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\end{dosutilcommandsynopsis} |
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|
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\begin{dosutilcommanddescription} |
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The \emph{crc32} utility calculates the CRC-32 and length of a file. |
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This utility relies on C-language code obtained from a web page |
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maintained by \index{Ellingson, Richard A.}Richard A. Ellingson |
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\cite{bibref:w:ellingsoncrc32pages}. |
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The utility was checked against the values calculated by |
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\index{WinZip@\emph{WinZip}}\emph{WinZip} |
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\cite{bibref:s:winzip}. |
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\end{dosutilcommanddescription} |
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|
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\begin{dosutilcommandsampleinvocations} |
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\begin{scriptsize} |
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\begin{verbatim} |
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C:\>crc32 t1.bin |
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0x9567CF44 (116,785,153) |
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|
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C:\>crc32 t1.bin -noformat |
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9567CF44 |
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116785153 |
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\end{verbatim} |
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\end{scriptsize} |
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\end{dosutilcommandsampleinvocations} |
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|
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\begin{dosutilcommandseealso} |
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See also the \emph{crc32} Tcl extension, |
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Section \cxtnzeroxrefhyphen{}\ref{cxtn0:scrc0:scrc0}. |
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\end{dosutilcommandseealso} |
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Arbitrary-Length Integer Utilities} |
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%Section tag: ALI0 |
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\label{cdcm0:sali0} |
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{The \emph{intadd} Utility} |
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%Subection tag: ADD0 |
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\label{cdcm0:sali0:sadd0} |
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|
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\index{intadd@\emph{intadd}} |
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\begin{dosutilcommandname}{intadd}% |
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adds two integers. (Mnemonic: \emph{int}eger |
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\emph{add}ition.) |
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\end{dosutilcommandname} |
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|
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\begin{dosutilcommandsynopsis} |
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\dosutilcommandsynopsisline{intadd}{sint sint [$<$options$>$]} |
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\end{dosutilcommandsynopsis} |
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|
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\begin{dosutilcommanddescription} |
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The \emph{intadd} utility calculates the sum of two integers. |
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\end{dosutilcommanddescription} |
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|
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\begin{dosutilcommandsampleinvocations} |
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\begin{scriptsize} |
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\begin{verbatim} |
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C:\>intadd 1e20 4912471263493214 |
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------------------------------------------------------------------------------ |
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arg1: 100,000,000,000,000,000,000 ( 21 digits) |
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------------------------------------------------------------------------------ |
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arg2: 4,912,471,263,493,214 ( 16 digits) |
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------------------------------------------------------------------------------ |
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arg1 + arg2: 100,004,912,471,263,493,214 ( 21 digits) |
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------------------------------------------------------------------------------ |
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\end{verbatim} |
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\end{scriptsize} |
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\end{dosutilcommandsampleinvocations} |
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|
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\begin{dosutilcommandseealso} |
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See also the \emph{arbint intadd} Tcl extension, |
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Section \cxtnzeroxrefhyphen{}\ref{cxtn0:sarb0:sadd0}. |
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\end{dosutilcommandseealso} |
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{The \emph{intsub} Utility} |
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%Subection tag: SUB0 |
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\label{cdcm0:sali0:ssub0} |
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|
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\index{intsub@\emph{intsub}} |
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\begin{dosutilcommandname}{intsub}% |
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subtracts two integers. (Mnemonic: \emph{int}eger |
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\emph{sub}traction.) |
273 |
\end{dosutilcommandname} |
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|
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\begin{dosutilcommandsynopsis} |
276 |
\dosutilcommandsynopsisline{intsub}{sint sint [$<$options$>$]} |
277 |
\end{dosutilcommandsynopsis} |
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|
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\begin{dosutilcommanddescription} |
280 |
The \emph{intsub} utility calculates the difference of two integers. |
281 |
\end{dosutilcommanddescription} |
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|
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\begin{dosutilcommandsampleinvocations} |
284 |
\begin{scriptsize} |
285 |
\begin{verbatim} |
286 |
C:\>intsub 1e110 99 |
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------------------------------------------------------------------------------ |
288 |
arg1: 100, ( 111 digits) |
289 |
000,000,000,000,000,000,000,000,000, |
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000,000,000,000,000,000,000,000,000, |
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000,000,000,000,000,000,000,000,000, |
292 |
000,000,000,000,000,000,000,000,000 |
293 |
------------------------------------------------------------------------------ |
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arg2: 99 ( 2 digits) |
295 |
------------------------------------------------------------------------------ |
296 |
arg1 - arg2: 99, ( 110 digits) |
297 |
999,999,999,999,999,999,999,999,999, |
298 |
999,999,999,999,999,999,999,999,999, |
299 |
999,999,999,999,999,999,999,999,999, |
300 |
999,999,999,999,999,999,999,999,901 |
301 |
------------------------------------------------------------------------------ |
302 |
|
303 |
C:\>intsub 1e110 99 -nf |
304 |
10000000000000000000000000000000000000000000000000000000000000000000000000000000 |
305 |
0000000000000000000000000000000 |
306 |
99 |
307 |
99999999999999999999999999999999999999999999999999999999999999999999999999999999 |
308 |
999999999999999999999999999901 |
309 |
\end{verbatim} |
310 |
\end{scriptsize} |
311 |
\end{dosutilcommandsampleinvocations} |
312 |
|
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\begin{dosutilcommandseealso} |
314 |
See also the \emph{arbint intsub} Tcl extension, |
315 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:sarb0:ssub0}. |
316 |
\end{dosutilcommandseealso} |
317 |
|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{The \emph{intmul} Utility} |
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%Subsection tag: MUL0 |
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\label{cdcm0:sali0:smul0} |
325 |
|
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\index{intmul@\emph{intmul}} |
327 |
\begin{dosutilcommandname}{intmul}% |
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calculates the product of |
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two integers. (Mnemonic: \emph{int}eger |
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\emph{mul}tiplication.) |
331 |
\end{dosutilcommandname} |
332 |
|
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\begin{dosutilcommandsynopsis} |
334 |
\dosutilcommandsynopsisline{intmul}{sint sint [$<$options$>$]} |
335 |
\end{dosutilcommandsynopsis} |
336 |
|
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\begin{dosutilcommanddescription} |
338 |
The \emph{intmul} utility calculates product of two |
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integers. |
340 |
\end{dosutilcommanddescription} |
341 |
|
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|
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\begin{dosutilcommandsampleinvocations} |
344 |
\begin{scriptsize} |
345 |
\begin{verbatim} |
346 |
C:\>intmul -916439216486321439672164132964 4321846.2864e28 |
347 |
------------------------------------------------------------------------------ |
348 |
arg1: - 916, ( 30 digits) |
349 |
439,216,486,321,439,672,164,132,964 |
350 |
------------------------------------------------------------------------------ |
351 |
arg2: 43,218,462, ( 35 digits) |
352 |
864,000,000,000,000,000,000,000,000 |
353 |
------------------------------------------------------------------------------ |
354 |
arg1 * arg2: - 39,607,094,244, ( 65 digits) |
355 |
827,339,704,438,441,915,017,392,248, |
356 |
896,000,000,000,000,000,000,000,000 |
357 |
------------------------------------------------------------------------------ |
358 |
|
359 |
C:\>intmul -916439216486321439672164132964 4321846.2864e28 -nf |
360 |
-916439216486321439672164132964 |
361 |
43218462864000000000000000000000000 |
362 |
-39607094244827339704438441915017392248896000000000000000000000000 |
363 |
\end{verbatim} |
364 |
\end{scriptsize} |
365 |
\end{dosutilcommandsampleinvocations} |
366 |
|
367 |
\begin{dosutilcommandseealso} |
368 |
See also the \emph{arbint intmul} Tcl extension, |
369 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:sarb0:smul0}. |
370 |
\end{dosutilcommandseealso} |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
373 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
375 |
\subsection{The \emph{intdiv} Utility} |
376 |
%Subsection tag: DIV0 |
377 |
\label{cdcm0:sali0:sdiv0} |
378 |
|
379 |
\index{intdiv@\emph{intdiv}} |
380 |
\begin{dosutilcommandname}{intdiv}% |
381 |
calculates the integer quotient (i.e. without remainder) of |
382 |
two integers. |
383 |
(Mnemonic: \emph{int}eger |
384 |
\emph{div}ision.) |
385 |
\end{dosutilcommandname} |
386 |
|
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\begin{dosutilcommandsynopsis} |
388 |
\dosutilcommandsynopsisline{intdiv}{sint sint [$<$options$>$]} |
389 |
\end{dosutilcommandsynopsis} |
390 |
|
391 |
\begin{dosutilcommanddescription} |
392 |
The \emph{intdiv} utility calculates the integer quotient |
393 |
of two integers. The mapping that is made is |
394 |
more precisely described in |
395 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:sarb0:sdiv0} in the |
396 |
description of the \emph{arbint intdiv} Tcl extension. |
397 |
\end{dosutilcommanddescription} |
398 |
|
399 |
\begin{dosutilcommandsampleinvocations} |
400 |
\begin{scriptsize} |
401 |
\begin{verbatim} |
402 |
C:\>intdiv 296491826436721 91843682163 |
403 |
------------------------------------------------------------------------------ |
404 |
arg1: 296,491,826,436,721 ( 15 digits) |
405 |
------------------------------------------------------------------------------ |
406 |
arg2: 91,843,682,163 ( 11 digits) |
407 |
------------------------------------------------------------------------------ |
408 |
arg1 / arg2: 3,228 ( 4 digits) |
409 |
------------------------------------------------------------------------------ |
410 |
|
411 |
C:\>intdiv 296491826436721 91843682163 -nf |
412 |
296491826436721 |
413 |
91843682163 |
414 |
3228 |
415 |
\end{verbatim} |
416 |
\end{scriptsize} |
417 |
\end{dosutilcommandsampleinvocations} |
418 |
|
419 |
\begin{dosutilcommandseealso} |
420 |
See also the \emph{arbint intdiv} Tcl extension, |
421 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:sarb0:sdiv0}. |
422 |
\end{dosutilcommandseealso} |
423 |
|
424 |
|
425 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
426 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
427 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
428 |
\subsection{The \emph{intmod} Utility} |
429 |
%Subsection tag: MOD0 |
430 |
\label{cdcm0:sali0:smod0} |
431 |
|
432 |
\index{intmod@\emph{intmod}} |
433 |
\begin{dosutilcommandname}{intmod}% |
434 |
calculates the integer modulo (i.e. remainder of division) of |
435 |
two integers. |
436 |
(Mnemonic: \emph{int}eger |
437 |
\emph{mod}ulo.) |
438 |
\end{dosutilcommandname} |
439 |
|
440 |
\begin{dosutilcommandsynopsis} |
441 |
\dosutilcommandsynopsisline{intmod}{sint sint [$<$options$>$]} |
442 |
\end{dosutilcommandsynopsis} |
443 |
|
444 |
\begin{dosutilcommanddescription} |
445 |
The \emph{intmod} utility calculates the integer modulo |
446 |
of two integers. The mapping that is made is |
447 |
more precisely described in |
448 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:sarb0:smod0} in the |
449 |
description of the \emph{arbint intmod} Tcl extension. |
450 |
\end{dosutilcommanddescription} |
451 |
|
452 |
\begin{dosutilcommandsampleinvocations} |
453 |
\begin{scriptsize} |
454 |
\begin{verbatim} |
455 |
C:\>intmod 2836549213654 15432154 |
456 |
------------------------------------------------------------------------------ |
457 |
arg1: 2,836,549,213,654 ( 13 digits) |
458 |
------------------------------------------------------------------------------ |
459 |
arg2: 15,432,154 ( 8 digits) |
460 |
------------------------------------------------------------------------------ |
461 |
arg1 % arg2: 11,283,376 ( 8 digits) |
462 |
------------------------------------------------------------------------------ |
463 |
|
464 |
C:\>intmod 2836549213654 0 |
465 |
------------------------------------------------------------------------------ |
466 |
arg1: 2,836,549,213,654 ( 13 digits) |
467 |
------------------------------------------------------------------------------ |
468 |
arg2: 0 ( 1 digit ) |
469 |
------------------------------------------------------------------------------ |
470 |
arg1 % arg2: GMP_INTS_EF_INTOVF_POS |
471 |
------------------------------------------------------------------------------ |
472 |
|
473 |
C:\>intmod 2836549213654 0 -nf |
474 |
2836549213654 |
475 |
0 |
476 |
GMP_INTS_EF_INTOVF_POS |
477 |
\end{verbatim} |
478 |
\end{scriptsize} |
479 |
\end{dosutilcommandsampleinvocations} |
480 |
|
481 |
\begin{dosutilcommandseealso} |
482 |
See also the \emph{arbint intmod} Tcl extension, |
483 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:sarb0:smod0}. |
484 |
\end{dosutilcommandseealso} |
485 |
|
486 |
|
487 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
488 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
489 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
490 |
\subsection{The \emph{intgcd} Utility} |
491 |
%Subsection tag: GCD0 |
492 |
\label{cdcm0:sali0:sgcd0} |
493 |
|
494 |
\index{intgcd@\emph{intgcd}} |
495 |
\begin{dosutilcommandname}{intgcd}% |
496 |
calculates the g.c.d. of two arbitrary integers using |
497 |
Euclid's algorithm. (Mnemonic: \emph{int}eger |
498 |
\emph{g}reatest \emph{c}ommon \emph{d}ivisor.) |
499 |
\end{dosutilcommandname} |
500 |
|
501 |
\begin{dosutilcommandsynopsis} |
502 |
\dosutilcommandsynopsisline{intgcd}{sint\_1 sint\_2 [$<$options$>$]} |
503 |
\end{dosutilcommandsynopsis} |
504 |
|
505 |
\begin{dosutilcommanddescription} |
506 |
The \emph{intgcd} utility calculates the g.c.d. of two |
507 |
integers. |
508 |
|
509 |
If either or both integers are 0, the result will be 1. |
510 |
If either or both integers are negative, the absolute |
511 |
value of the negative argument(s) will be used in the |
512 |
calculation of the g.c.d. |
513 |
|
514 |
The algorithm employed is the |
515 |
\emph{Modern Euclidian Algorithm} as |
516 |
described in \cite{bibref:b:knuthclassic2ndedvol2}, p. 337. |
517 |
Although faster algorithms for computer |
518 |
implementation do exist, this is the simplest |
519 |
and most direct algorithm. |
520 |
\end{dosutilcommanddescription} |
521 |
|
522 |
\begin{dosutilcommandsampleinvocations} |
523 |
\begin{scriptsize} |
524 |
\begin{verbatim} |
525 |
C:\>intgcd 8326413249250 12935482154386850 |
526 |
------------------------------------------------------------------------------ |
527 |
arg1: 8,326,413,249,250 ( 13 digits) |
528 |
------------------------------------------------------------------------------ |
529 |
arg2: 12,935,482,154,386,850 ( 17 digits) |
530 |
------------------------------------------------------------------------------ |
531 |
gcd(arg1, arg2): 50 ( 2 digits) |
532 |
------------------------------------------------------------------------------ |
533 |
|
534 |
C:\>intgcd 8326413249250 12935482154386850 -nf |
535 |
8326413249250 |
536 |
12935482154386850 |
537 |
50 |
538 |
\end{verbatim} |
539 |
\end{scriptsize} |
540 |
\end{dosutilcommandsampleinvocations} |
541 |
|
542 |
\begin{dosutilcommandseealso} |
543 |
See also the \emph{arbint intgcd} Tcl extension, |
544 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:sarb0:sgcd0}. |
545 |
\end{dosutilcommandseealso} |
546 |
|
547 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
548 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
549 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
550 |
\subsection{The \emph{intlcm} Utility} |
551 |
%Subsection tag: LCM0 |
552 |
\label{cdcm0:sali0:slcm0} |
553 |
|
554 |
\index{intlcm@\emph{intlcm}} |
555 |
\begin{dosutilcommandname}{intlcm}% |
556 |
calculates the l.c.m. (least common multiple) |
557 |
of two integers. (Mnemonic: \emph{int}eger |
558 |
\emph{l}east \emph{c}ommon \emph{m}ultiple.) |
559 |
\end{dosutilcommandname} |
560 |
|
561 |
\begin{dosutilcommandsynopsis} |
562 |
\dosutilcommandsynopsisline{intlcm}{sint\_1 sint\_2 [$<$options$>$]} |
563 |
\end{dosutilcommandsynopsis} |
564 |
|
565 |
\begin{dosutilcommanddescription} |
566 |
The \emph{intlcm} utility calculates the l.c.m. of two |
567 |
integers. |
568 |
|
569 |
The mapping from arguments to output and the details |
570 |
of how the l.c.m. is calculated are described |
571 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:sarb0:slcm0}. |
572 |
\end{dosutilcommanddescription} |
573 |
|
574 |
\begin{dosutilcommandsampleinvocations} |
575 |
\begin{scriptsize} |
576 |
\begin{verbatim} |
577 |
C:\>intlcm 12 69 |
578 |
------------------------------------------------------------------------------ |
579 |
arg1: 12 ( 2 digits) |
580 |
------------------------------------------------------------------------------ |
581 |
arg2: 69 ( 2 digits) |
582 |
------------------------------------------------------------------------------ |
583 |
lcm(arg1, arg2): 276 ( 3 digits) |
584 |
------------------------------------------------------------------------------ |
585 |
\end{verbatim} |
586 |
\end{scriptsize} |
587 |
\end{dosutilcommandsampleinvocations} |
588 |
|
589 |
\begin{dosutilcommandseealso} |
590 |
See also the \emph{arbint intlcm} Tcl extension, |
591 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:sarb0:slcm0}. |
592 |
\end{dosutilcommandseealso} |
593 |
|
594 |
|
595 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
596 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
597 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
598 |
\subsection{The \emph{intexp} Utility} |
599 |
%Subsection tag: IXP0 |
600 |
\label{cdcm0:sali0:sixp0} |
601 |
|
602 |
\index{intexp@\emph{intexp}} |
603 |
\begin{dosutilcommandname}{intexp}% |
604 |
exponentiates a signed integer to a non-negative integer |
605 |
power. (Mnemonic: \emph{int}eger |
606 |
\emph{exp}onent.) |
607 |
\end{dosutilcommandname} |
608 |
|
609 |
\begin{dosutilcommandsynopsis} |
610 |
\dosutilcommandsynopsisline{intexp}{sint\_base uint32\_exponent [$<$options$>$]} |
611 |
\end{dosutilcommandsynopsis} |
612 |
|
613 |
\begin{dosutilcommanddescription} |
614 |
The \emph{intexp} utility exponentiates an integer to a non-negative |
615 |
integer power. For convenience and simplicity, $0^0$ produces 1, |
616 |
which is inconsistent with the mathematical definition; but all other |
617 |
exponentiations are as expected. The mapping is more precisely described |
618 |
with the \emph{arbint intexp} Tcl extension in |
619 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:sarb0:sixp0}. |
620 |
\end{dosutilcommanddescription} |
621 |
|
622 |
\begin{dosutilcommandsampleinvocations} |
623 |
\begin{scriptsize} |
624 |
\begin{verbatim} |
625 |
C:\>intexp -22 41 |
626 |
------------------------------------------------------------------------------ |
627 |
base: - 22 ( 2 digits) |
628 |
------------------------------------------------------------------------------ |
629 |
exponent: 41 ( 2 digits) |
630 |
------------------------------------------------------------------------------ |
631 |
base ** exponent: - 10, ( 56 digits) |
632 |
947,877,107,572,929,152,919,737,180, |
633 |
202,022,857,988,400,441,953,615,872 |
634 |
------------------------------------------------------------------------------ |
635 |
|
636 |
C:\>intexp -22 41 -nf |
637 |
-22 |
638 |
41 |
639 |
-10947877107572929152919737180202022857988400441953615872 |
640 |
\end{verbatim} |
641 |
\end{scriptsize} |
642 |
\end{dosutilcommandsampleinvocations} |
643 |
|
644 |
\begin{dosutilcommandseealso} |
645 |
See also the \emph{arbint intexp} Tcl extension, |
646 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:sarb0:sixp0}. |
647 |
\end{dosutilcommandseealso} |
648 |
|
649 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
650 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
651 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
652 |
\subsection{The \emph{intfac} Utility} |
653 |
%Subsection tag: IFC0 |
654 |
\label{cdcm0:sali0:sifc0} |
655 |
|
656 |
\index{intfac@\emph{intfac}} |
657 |
\begin{dosutilcommandname}{intfac}% |
658 |
calculates the \index{factorial function}factorial of a non-negative |
659 |
integer. (Mnemonic: \emph{int}eger |
660 |
\emph{fac}torial.) |
661 |
\end{dosutilcommandname} |
662 |
|
663 |
\begin{dosutilcommandsynopsis} |
664 |
\dosutilcommandsynopsisline{intfac}{uint32 [$<$options$>$]} |
665 |
\end{dosutilcommandsynopsis} |
666 |
|
667 |
\begin{dosutilcommanddescription} |
668 |
The \emph{intfac} utility calculates the factorial of |
669 |
a non-negative integer. 0! is defined to be 1. |
670 |
|
671 |
This utility is useful for calculating the factorial of |
672 |
integers where the result would be quite large (larger than |
673 |
could be displayed on a pocket calculator, for example). |
674 |
\end{dosutilcommanddescription} |
675 |
|
676 |
\begin{dosutilcommandremarks} |
677 |
At the present time (07/2001, Version 1.04), the \emph{intfac} utility |
678 |
is constrained by the division operations required to convert from the |
679 |
binary internal integer format to a base-10 ASCII representation. |
680 |
The \emph{intfac} utility will not calculate factorials of over about |
681 |
1,000 decimal digits without a fair delay. In the future, algorithmic |
682 |
improvements may raise this limit. |
683 |
\end{dosutilcommandremarks} |
684 |
|
685 |
\begin{dosutilcommandsampleinvocations} |
686 |
\begin{scriptsize} |
687 |
\begin{verbatim} |
688 |
C:\>intfac 47 |
689 |
------------------------------------------------------------------------------ |
690 |
arg: 47 ( 2 digits) |
691 |
------------------------------------------------------------------------------ |
692 |
arg!: 258,623, ( 60 digits) |
693 |
241,511,168,180,642,964,355,153,611, |
694 |
979,969,197,632,389,120,000,000,000 |
695 |
------------------------------------------------------------------------------ |
696 |
|
697 |
C:\>intfac 47 -noformat |
698 |
47 |
699 |
258623241511168180642964355153611979969197632389120000000000 |
700 |
\end{verbatim} |
701 |
\end{scriptsize} |
702 |
\end{dosutilcommandsampleinvocations} |
703 |
|
704 |
\begin{dosutilcommandseealso} |
705 |
See also the \emph{arbint intfac} Tcl extension, |
706 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:sarb0:sfac0}. |
707 |
\end{dosutilcommandseealso} |
708 |
|
709 |
|
710 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
711 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
712 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
713 |
\section{Rational Number Arithmetic Utilities} |
714 |
%Section tag: RNU0 |
715 |
\label{cdcm0:srnu0} |
716 |
|
717 |
|
718 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
719 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
720 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
721 |
\subsection{The \emph{rnadd} Utility} |
722 |
%Subsection tag: RNA0 |
723 |
\label{cdcm0:srnu0:srna0} |
724 |
|
725 |
\index{rnadd@\emph{rnadd}} |
726 |
\begin{dosutilcommandname}{rnadd}% |
727 |
adds two rational numbers to produce a normalized rational result. |
728 |
(Mnemonic: \emph{r}ational \emph{n}umber |
729 |
\emph{add}ition.) |
730 |
\end{dosutilcommandname} |
731 |
|
732 |
\begin{dosutilcommandsynopsis} |
733 |
\dosutilcommandsynopsisline{rnadd}{srn srn [$<$options$>$]} |
734 |
\end{dosutilcommandsynopsis} |
735 |
|
736 |
\begin{dosutilcommanddescription} |
737 |
The \emph{rnadd} utility calculates the sum of two arbitrary |
738 |
rational numbers. |
739 |
\end{dosutilcommanddescription} |
740 |
|
741 |
\begin{dosutilcommandsampleinvocations} |
742 |
\begin{scriptsize} |
743 |
\begin{verbatim} |
744 |
C:\>rnadd 3.29 42/67 |
745 |
------------------------------------------------------------------------------ |
746 |
arg1_num: 329 ( 3 digits) |
747 |
------------------------------------------------------------------------------ |
748 |
arg1_den: 100 ( 3 digits) |
749 |
------------------------------------------------------------------------------ |
750 |
arg2_num: 42 ( 2 digits) |
751 |
------------------------------------------------------------------------------ |
752 |
arg2_den: 67 ( 2 digits) |
753 |
------------------------------------------------------------------------------ |
754 |
result_num: 26,243 ( 5 digits) |
755 |
------------------------------------------------------------------------------ |
756 |
result_den: 6,700 ( 4 digits) |
757 |
------------------------------------------------------------------------------ |
758 |
\end{verbatim} |
759 |
\end{scriptsize} |
760 |
\end{dosutilcommandsampleinvocations} |
761 |
|
762 |
\begin{dosutilcommandseealso} |
763 |
See also the \emph{arbint rnadd} Tcl extension, |
764 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:srne0:srad0}. |
765 |
\end{dosutilcommandseealso} |
766 |
|
767 |
|
768 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
769 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
770 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
771 |
\subsection{The \emph{rnsub} Utility} |
772 |
%Subsection tag: RSB0 |
773 |
\label{cdcm0:srnu0:srsb0} |
774 |
|
775 |
\index{rnsub@\emph{rnsub}} |
776 |
\begin{dosutilcommandname}{rnsub}% |
777 |
subtracts two rational numbers to produce a normalized rational result. |
778 |
(Mnemonic: \emph{r}ational \emph{n}umber |
779 |
\emph{sub}traction.) |
780 |
\end{dosutilcommandname} |
781 |
|
782 |
\begin{dosutilcommandsynopsis} |
783 |
\dosutilcommandsynopsisline{rnsub}{srn srn [$<$options$>$]} |
784 |
\end{dosutilcommandsynopsis} |
785 |
|
786 |
\begin{dosutilcommanddescription} |
787 |
The \emph{rnsub} utility calculates the difference of two arbitrary |
788 |
rational numbers. The second argument is subtracted from the first, |
789 |
i.e. $arg_2 - arg_1$ is calculated. |
790 |
\end{dosutilcommanddescription} |
791 |
|
792 |
\begin{dosutilcommandsampleinvocations} |
793 |
\begin{scriptsize} |
794 |
\begin{verbatim} |
795 |
C:\>rnsub 3.29 42/67 |
796 |
------------------------------------------------------------------------------ |
797 |
arg1_num: 329 ( 3 digits) |
798 |
------------------------------------------------------------------------------ |
799 |
arg1_den: 100 ( 3 digits) |
800 |
------------------------------------------------------------------------------ |
801 |
arg2_num: 42 ( 2 digits) |
802 |
------------------------------------------------------------------------------ |
803 |
arg2_den: 67 ( 2 digits) |
804 |
------------------------------------------------------------------------------ |
805 |
result_num: 17,843 ( 5 digits) |
806 |
------------------------------------------------------------------------------ |
807 |
result_den: 6,700 ( 4 digits) |
808 |
------------------------------------------------------------------------------ |
809 |
\end{verbatim} |
810 |
\end{scriptsize} |
811 |
\end{dosutilcommandsampleinvocations} |
812 |
|
813 |
\begin{dosutilcommandseealso} |
814 |
See also the \emph{arbint rnsub} Tcl extension, |
815 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:srne0:srsb0}. |
816 |
\end{dosutilcommandseealso} |
817 |
|
818 |
|
819 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
820 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
821 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
822 |
\subsection{The \emph{rnmul} Utility} |
823 |
%Subsection tag: RMU0 |
824 |
\label{cdcm0:srnu0:srmu0} |
825 |
|
826 |
\index{rnmul@\emph{rnmul}} |
827 |
\begin{dosutilcommandname}{rnmul}% |
828 |
calculates the product of two rational numbers. |
829 |
(Mnemonic: \emph{r}ational |
830 |
\emph{n}umber \emph{mul}tiplication.) |
831 |
\end{dosutilcommandname} |
832 |
|
833 |
\begin{dosutilcommandsynopsis} |
834 |
\dosutilcommandsynopsisline{rnmul}{srn srn [$<$options$>$]} |
835 |
\end{dosutilcommandsynopsis} |
836 |
|
837 |
\begin{dosutilcommanddescription} |
838 |
The \emph{rnmul} utility calculates the product |
839 |
of two rational numbers. |
840 |
\end{dosutilcommanddescription} |
841 |
|
842 |
\begin{dosutilcommandsampleinvocations} |
843 |
\begin{scriptsize} |
844 |
\begin{verbatim} |
845 |
E:\>rnmul 3.14 64/207 |
846 |
------------------------------------------------------------------------------ |
847 |
arg1_num: 314 ( 3 digits) |
848 |
------------------------------------------------------------------------------ |
849 |
arg1_den: 100 ( 3 digits) |
850 |
------------------------------------------------------------------------------ |
851 |
arg2_num: 64 ( 2 digits) |
852 |
------------------------------------------------------------------------------ |
853 |
arg2_den: 207 ( 3 digits) |
854 |
------------------------------------------------------------------------------ |
855 |
result_num: 5,024 ( 4 digits) |
856 |
------------------------------------------------------------------------------ |
857 |
result_den: 5,175 ( 4 digits) |
858 |
------------------------------------------------------------------------------ |
859 |
\end{verbatim} |
860 |
\end{scriptsize} |
861 |
\end{dosutilcommandsampleinvocations} |
862 |
|
863 |
\begin{dosutilcommandseealso} |
864 |
See also the \emph{arbint rnmul} Tcl extension, |
865 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:srne0:srmu0}. |
866 |
\end{dosutilcommandseealso} |
867 |
|
868 |
|
869 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
870 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
871 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
872 |
\subsection{The \emph{rndiv} Utility} |
873 |
%Subsection tag: RDV0 |
874 |
\label{cdcm0:srnu0:srdv0} |
875 |
|
876 |
\index{rndiv@\emph{rndiv}} |
877 |
\begin{dosutilcommandname}{rndiv}% |
878 |
calculates the quotient of two rational numbers. |
879 |
(Mnemonic: \emph{r}ational |
880 |
\emph{n}umber \emph{div}ision.) |
881 |
\end{dosutilcommandname} |
882 |
|
883 |
\begin{dosutilcommandsynopsis} |
884 |
\dosutilcommandsynopsisline{rndiv}{srn srn [$<$options$>$]} |
885 |
\end{dosutilcommandsynopsis} |
886 |
|
887 |
\begin{dosutilcommanddescription} |
888 |
The \emph{rnmul} utility calculates the quotient |
889 |
of two rational numbers. The first argument is divided |
890 |
by the second; i.e. the quotient $arg_1 / arg_2$ is calculated. |
891 |
\end{dosutilcommanddescription} |
892 |
|
893 |
\begin{dosutilcommandsampleinvocations} |
894 |
\begin{scriptsize} |
895 |
\begin{verbatim} |
896 |
E:\>rndiv 3.14 34/291 |
897 |
------------------------------------------------------------------------------ |
898 |
arg1_num: 314 ( 3 digits) |
899 |
------------------------------------------------------------------------------ |
900 |
arg1_den: 100 ( 3 digits) |
901 |
------------------------------------------------------------------------------ |
902 |
arg2_num: 34 ( 2 digits) |
903 |
------------------------------------------------------------------------------ |
904 |
arg2_den: 291 ( 3 digits) |
905 |
------------------------------------------------------------------------------ |
906 |
result_num: 45,687 ( 5 digits) |
907 |
------------------------------------------------------------------------------ |
908 |
result_den: 1,700 ( 4 digits) |
909 |
------------------------------------------------------------------------------ |
910 |
\end{verbatim} |
911 |
\end{scriptsize} |
912 |
\end{dosutilcommandsampleinvocations} |
913 |
|
914 |
\begin{dosutilcommandseealso} |
915 |
See also the \emph{arbint rndiv} Tcl extension, |
916 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:srne0:srdv0}. |
917 |
\end{dosutilcommandseealso} |
918 |
|
919 |
|
920 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
921 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
922 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
923 |
\subsection{The \emph{rndeq} Utility} |
924 |
%Subsection tag: RNQ0 |
925 |
\label{cdcm0:srnu0:srnq0} |
926 |
|
927 |
\index{rndeq@\emph{rndeq}} |
928 |
\begin{dosutilcommandname}{rndeq}% |
929 |
calculates the decimal equivalent of a rational number. |
930 |
(Mnemonic: \emph{r}ational \emph{n}umber |
931 |
\emph{d}ecimal \emph{eq}uivalent.) |
932 |
\end{dosutilcommandname} |
933 |
|
934 |
\begin{dosutilcommandsynopsis} |
935 |
\dosutilcommandsynopsisline{rndeq}{srn [$<$options$>$]} |
936 |
\dosutilcommandsynopsisline{rndeq}{srn uint [$<$options$>$]} |
937 |
\end{dosutilcommandsynopsis} |
938 |
|
939 |
\begin{dosutilcommanddescription} |
940 |
The \emph{rndeq} utility calculates the decimal equivalent |
941 |
of a rational number using a large integer |
942 |
algorithm. |
943 |
|
944 |
Let $a/b$ be a rational number that we wish to express |
945 |
as a decimal number, let $N$ be an integral power |
946 |
of 10, and let $x$ be an integer chosen so that $x/N$ |
947 |
is an approximation to $a/b$. |
948 |
|
949 |
If we choose |
950 |
|
951 |
\begin{equation} |
952 |
\label{eq:cdcm0:sali0:srnq0:01} |
953 |
x = sgn \left( {\frac{a}{b}} \right) |
954 |
\left\lfloor {\left| \frac{a}{b} \right| N} \right\rfloor , |
955 |
\end{equation} |
956 |
|
957 |
it can be shown that |
958 |
|
959 |
\begin{equation} |
960 |
\label{eq:cdcm0:sali0:srnq0:02} |
961 |
\left| {\frac{a}{b}} \right| - \frac{1}{N} |
962 |
< |
963 |
\frac{\left\lfloor {\left| {\frac{a}{b}} \right| N} \right\rfloor}{N} |
964 |
\leq |
965 |
\left|{\frac{a}{b}}\right|. |
966 |
\end{equation} |
967 |
|
968 |
In other words, the result if $x$ is chosen as specified in |
969 |
(\ref{eq:cdcm0:sali0:srnq0:01}), $x/N$ will be at or up to one numerator |
970 |
count less in magnitude than $a/b$. Stated differently, |
971 |
$x/N$ is ``truncated down'' in magnitude. |
972 |
|
973 |
The \emph{rndeq} utility performs the mapping implied by |
974 |
(\ref{eq:cdcm0:sali0:srnq0:01}) and |
975 |
(\ref{eq:cdcm0:sali0:srnq0:02}), by default using |
976 |
$N=10^{108}$. The value of $N=10^{108}$ was chosen |
977 |
because it produces an attractive display format at |
978 |
27 digits per line, and also places the imaginary decimal |
979 |
point between lines so it is easier to interpret the output of |
980 |
this utility. An example is interpreted in the sample invocations |
981 |
below. |
982 |
|
983 |
This utility will accept an optional second parameter, which |
984 |
is a value of $N$ to override the default of $10^{108}$. This |
985 |
optional second parameter can be used to provide more decimal places, |
986 |
or to provide a different type of mapping.\footnote{Please be aware that integers |
987 |
may be specified using scientific notation. For example, to effectively double |
988 |
the number of decimal places displayed by \texttt{rndeq} in the |
989 |
invocation example below, one might enter the command line |
990 |
\texttt{rndeq 5345/319 1e216}. We would regret it if this fact were unknown |
991 |
and some desperate user were to enter the digit ``1'' followed by |
992 |
216 zeros!} |
993 |
|
994 |
It is not required that the second parameter, if supplied, be a power of |
995 |
10---the mappings described by the equations above will be carried |
996 |
out with \emph{any} supplied denominator. |
997 |
However, it is not intuitively obvious how this utility would be |
998 |
useful with a denominator that is not a power of 10. |
999 |
\end{dosutilcommanddescription} |
1000 |
|
1001 |
\begin{dosutilcommandsampleinvocations} |
1002 |
\begin{scriptsize} |
1003 |
\begin{verbatim} |
1004 |
C:\>rndeq 5345/319 |
1005 |
------------------------------------------------------------------------------ |
1006 |
arg_num: 5,345 ( 4 digits) |
1007 |
------------------------------------------------------------------------------ |
1008 |
arg_den: 319 ( 3 digits) |
1009 |
------------------------------------------------------------------------------ |
1010 |
dap_num: 16, ( 110 digits) |
1011 |
755,485,893,416,927,899,686,520,376, |
1012 |
175,548,589,341,692,789,968,652,037, |
1013 |
617,554,858,934,169,278,996,865,203, |
1014 |
761,755,485,893,416,927,899,686,520 |
1015 |
------------------------------------------------------------------------------ |
1016 |
dap_den: 1, ( 109 digits) |
1017 |
000,000,000,000,000,000,000,000,000, |
1018 |
000,000,000,000,000,000,000,000,000, |
1019 |
000,000,000,000,000,000,000,000,000, |
1020 |
000,000,000,000,000,000,000,000,000 |
1021 |
------------------------------------------------------------------------------ |
1022 |
\end{verbatim} |
1023 |
\end{scriptsize} |
1024 |
|
1025 |
It can be seen from the sample invocation above that |
1026 |
5345/319$\approx$16.755---note how the imaginary decimal point is placed |
1027 |
between lines of \texttt{dap\_num}. It can also be seen that the |
1028 |
decimal representation repeats, i.e. |
1029 |
|
1030 |
\begin{equation} |
1031 |
\frac{5345}{319} = 16.\overline{7554858934169278996865203761}. |
1032 |
\end{equation} |
1033 |
\end{dosutilcommandsampleinvocations} |
1034 |
|
1035 |
|
1036 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1037 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1038 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1039 |
\subsection{The \emph{rnred} Utility} |
1040 |
%Subsection tag: RNR0 |
1041 |
\label{cdcm0:srnu0:srnr0} |
1042 |
|
1043 |
\index{rnred@\emph{rnred}} |
1044 |
\begin{dosutilcommandname}{rnred}% |
1045 |
reduces a rational number to lowest terms and normalizes it. |
1046 |
(Mnemonic: \emph{r}ational |
1047 |
\emph{n}umber \emph{red}uce and normalize.) |
1048 |
\end{dosutilcommandname} |
1049 |
|
1050 |
\begin{dosutilcommandsynopsis} |
1051 |
\dosutilcommandsynopsisline{rnred}{srn [$<$options$>$]} |
1052 |
\end{dosutilcommandsynopsis} |
1053 |
|
1054 |
\begin{dosutilcommanddescription} |
1055 |
The \emph{rnred} reduces a rational number to its lowest terms |
1056 |
and normalizes it. This process is the same process |
1057 |
described in the description of the \emph{arbint rnred} Tcl |
1058 |
extension, Section \cxtnzeroxrefhyphen{}\ref{cxtn0:srne0:srnr0}. |
1059 |
\end{dosutilcommanddescription} |
1060 |
|
1061 |
\begin{dosutilcommandsampleinvocations} |
1062 |
\begin{scriptsize} |
1063 |
\begin{verbatim} |
1064 |
C:\>rnred 422.414 |
1065 |
------------------------------------------------------------------------------ |
1066 |
numerator: 211,207 ( 6 digits) |
1067 |
------------------------------------------------------------------------------ |
1068 |
denominator: 500 ( 3 digits) |
1069 |
------------------------------------------------------------------------------ |
1070 |
|
1071 |
C:\>rnred 422.414 -nf |
1072 |
211207 |
1073 |
500 |
1074 |
\end{verbatim} |
1075 |
\end{scriptsize} |
1076 |
\end{dosutilcommandsampleinvocations} |
1077 |
|
1078 |
\begin{dosutilcommandseealso} |
1079 |
See also the \emph{arbint rnred} Tcl extension, |
1080 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:srne0:srnr0}. |
1081 |
\end{dosutilcommandseealso} |
1082 |
|
1083 |
|
1084 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1085 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1086 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1087 |
\section[Number Theory, CF, And Approximation Utilities] |
1088 |
{Number Theory, Continued Fraction, And Best Rational Approximation |
1089 |
Utilities} |
1090 |
%Section tag: nth0 |
1091 |
\label{cdcm0:snth0} |
1092 |
|
1093 |
|
1094 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1095 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1096 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1097 |
\subsection{The \emph{cfratnum} Utility} |
1098 |
%Subsection tag: CFR0 |
1099 |
\label{cdcm0:snth0:scfr0} |
1100 |
|
1101 |
\index{cfratnum@\emph{cfratnum}} |
1102 |
\begin{dosutilcommandname}{cfratnum}% |
1103 |
calculates the continued fraction partial quotients and convergents |
1104 |
of a non-negative rational number. (Mnemonic: \emph{c}ontinued |
1105 |
\emph{f}raction partial quotients and convergents of |
1106 |
a \emph{rat}ional \emph{num}ber.) |
1107 |
\end{dosutilcommandname} |
1108 |
|
1109 |
\begin{dosutilcommandsynopsis} |
1110 |
\dosutilcommandsynopsisline{cfratnum}{urn [$<$options$>$]} |
1111 |
\end{dosutilcommandsynopsis} |
1112 |
|
1113 |
\begin{dosutilcommanddescription} |
1114 |
The \emph{cfratnum} utility calculates the continued fraction |
1115 |
partial quotients and convergents of a non-negative rational number. |
1116 |
This information can be useful in a variety of contexts, including |
1117 |
finding best rational approximations. |
1118 |
|
1119 |
Additionally, this DOS utility provides a decimal approximation |
1120 |
of each convergent, using a denominator of |
1121 |
$10^{108}$ (see the description of the \emph{rndeq} utility, Section |
1122 |
\ref{cdcm0:srnu0:srnq0}). |
1123 |
\end{dosutilcommanddescription} |
1124 |
|
1125 |
\begin{dosutilcommandsampleinvocations} |
1126 |
\begin{scriptsize} |
1127 |
\begin{verbatim} |
1128 |
C:\>cfratnum 3.14 |
1129 |
****************************************************************************** |
1130 |
******************* Continued Fraction Representation ******************* |
1131 |
****************************************************************************** |
1132 |
Input Numerator: 314 ( 3 digits) |
1133 |
------------------------------------------------------------------------------ |
1134 |
Input Denominator: 100 ( 3 digits) |
1135 |
------------------------------------------------------------------------------ |
1136 |
dividend(0): 314 ( 3 digits) |
1137 |
------------------------------------------------------------------------------ |
1138 |
divisor(0): 100 ( 3 digits) |
1139 |
------------------------------------------------------------------------------ |
1140 |
a(0): 3 ( 1 digit ) |
1141 |
------------------------------------------------------------------------------ |
1142 |
p(0): 3 ( 1 digit ) |
1143 |
------------------------------------------------------------------------------ |
1144 |
q(0): 1 ( 1 digit ) |
1145 |
------------------------------------------------------------------------------ |
1146 |
dap_h(0): 3, ( 109 digits) |
1147 |
000,000,000,000,000,000,000,000,000, |
1148 |
000,000,000,000,000,000,000,000,000, |
1149 |
000,000,000,000,000,000,000,000,000, |
1150 |
000,000,000,000,000,000,000,000,000 |
1151 |
------------------------------------------------------------------------------ |
1152 |
dap_k(0): 1, ( 109 digits) |
1153 |
000,000,000,000,000,000,000,000,000, |
1154 |
000,000,000,000,000,000,000,000,000, |
1155 |
000,000,000,000,000,000,000,000,000, |
1156 |
000,000,000,000,000,000,000,000,000 |
1157 |
------------------------------------------------------------------------------ |
1158 |
dividend(1): 100 ( 3 digits) |
1159 |
------------------------------------------------------------------------------ |
1160 |
divisor(1): 14 ( 2 digits) |
1161 |
------------------------------------------------------------------------------ |
1162 |
a(1): 7 ( 1 digit ) |
1163 |
------------------------------------------------------------------------------ |
1164 |
p(1): 22 ( 2 digits) |
1165 |
------------------------------------------------------------------------------ |
1166 |
q(1): 7 ( 1 digit ) |
1167 |
------------------------------------------------------------------------------ |
1168 |
dap_h(1): 3, ( 109 digits) |
1169 |
142,857,142,857,142,857,142,857,142, |
1170 |
857,142,857,142,857,142,857,142,857, |
1171 |
142,857,142,857,142,857,142,857,142, |
1172 |
857,142,857,142,857,142,857,142,857 |
1173 |
------------------------------------------------------------------------------ |
1174 |
dap_k(1): 1, ( 109 digits) |
1175 |
000,000,000,000,000,000,000,000,000, |
1176 |
000,000,000,000,000,000,000,000,000, |
1177 |
000,000,000,000,000,000,000,000,000, |
1178 |
000,000,000,000,000,000,000,000,000 |
1179 |
------------------------------------------------------------------------------ |
1180 |
dividend(2): 14 ( 2 digits) |
1181 |
------------------------------------------------------------------------------ |
1182 |
divisor(2): 2 ( 1 digit ) |
1183 |
------------------------------------------------------------------------------ |
1184 |
a(2): 7 ( 1 digit ) |
1185 |
------------------------------------------------------------------------------ |
1186 |
p(2): 157 ( 3 digits) |
1187 |
------------------------------------------------------------------------------ |
1188 |
q(2): 50 ( 2 digits) |
1189 |
------------------------------------------------------------------------------ |
1190 |
dap_h(2): 3, ( 109 digits) |
1191 |
140,000,000,000,000,000,000,000,000, |
1192 |
000,000,000,000,000,000,000,000,000, |
1193 |
000,000,000,000,000,000,000,000,000, |
1194 |
000,000,000,000,000,000,000,000,000 |
1195 |
------------------------------------------------------------------------------ |
1196 |
dap_k(2): 1, ( 109 digits) |
1197 |
000,000,000,000,000,000,000,000,000, |
1198 |
000,000,000,000,000,000,000,000,000, |
1199 |
000,000,000,000,000,000,000,000,000, |
1200 |
000,000,000,000,000,000,000,000,000 |
1201 |
------------------------------------------------------------------------------ |
1202 |
\end{verbatim} |
1203 |
\end{scriptsize} |
1204 |
|
1205 |
Note in the sample invocation above that each convergent |
1206 |
also includes a decimal approximation. For example, |
1207 |
in the invocation above, it can be seen that |
1208 |
22/7 $\approx$ 3.142857142857 \ldots{}. |
1209 |
|
1210 |
Note also that at each round of calculation of partial |
1211 |
quotients and convergents, the remainder is not shown because |
1212 |
it becomes the divisor of the next round, and so it would be |
1213 |
redundant to show it. Note also that the final non-zero remainder |
1214 |
(which is the g.c.d. of the numerator and denominator) |
1215 |
appears as the final divisor. In the sample |
1216 |
invocation above, it can be seen that the g.c.d. of |
1217 |
314 and 100 is the integer labeled as |
1218 |
\texttt{divisor(2)}, which has the value of 2. |
1219 |
\end{dosutilcommandsampleinvocations} |
1220 |
|
1221 |
\begin{dosutilcommandseealso} |
1222 |
See also the \emph{arbint cfratnum} Tcl extension, |
1223 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:snth0:scfr0}. |
1224 |
\end{dosutilcommandseealso} |
1225 |
|
1226 |
|
1227 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1228 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1229 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1230 |
\subsection{The \emph{cfbrapab} Utility} |
1231 |
%Subsection tag: BRA0 |
1232 |
\label{cdcm0:snth0:sbra0} |
1233 |
|
1234 |
\index{cfbrapab@\emph{cfbrapab}} |
1235 |
\begin{dosutilcommandname}{cfbrapab}% |
1236 |
calculates the best rational approximation to a |
1237 |
rational |
1238 |
number in the Farey series of order |
1239 |
$k_{MAX}$, or in a rectangular region of the integer lattice |
1240 |
defined by $k \leq k_{MAX}$ and $h \leq h_{MAX}$. |
1241 |
(Mnemonic: \emph{c}ontinued |
1242 |
\emph{f}raction \emph{b}est \emph{r}ational \emph{ap}proximation |
1243 |
in $F_{a, \overline{b}}$.) |
1244 |
\end{dosutilcommandname} |
1245 |
|
1246 |
\begin{dosutilcommanddescription} |
1247 |
Except for display format, |
1248 |
this utility behaves the same way and |
1249 |
accepts the same options as the \emph{arbint cfbrapab} |
1250 |
Tcl extension, described in Section \cxtnzeroxrefhyphen{}\ref{cxtn0:snth0:sbra0}. |
1251 |
Please see the documentation of this Tcl extension, which also applies |
1252 |
to this DOS command-line utility. |
1253 |
\end{dosutilcommanddescription} |
1254 |
|
1255 |
|
1256 |
\begin{dosutilcommandsampleinvocations} |
1257 |
\begin{scriptsize} |
1258 |
\begin{verbatim} |
1259 |
C:\swprojs\cfbrapab\Release>cfbrapab 1.6093 255 255 |
1260 |
------------------------------------------------------------------------------ |
1261 |
MAJOR MODE: Finding closest rational number(s) under the constraints. |
1262 |
------------------------------------------------------------------------------ |
1263 |
RI_IN Numerator: 16,093 ( 5 digits) |
1264 |
------------------------------------------------------------------------------ |
1265 |
RI_IN Denominator: 10,000 ( 5 digits) |
1266 |
------------------------------------------------------------------------------ |
1267 |
K_MAX: 255 ( 3 digits) |
1268 |
------------------------------------------------------------------------------ |
1269 |
H_MAX: 255 ( 3 digits) |
1270 |
------------------------------------------------------------------------------ |
1271 |
approx_num(-1): 243 ( 3 digits) |
1272 |
------------------------------------------------------------------------------ |
1273 |
approx_den(-1): 151 ( 3 digits) |
1274 |
------------------------------------------------------------------------------ |
1275 |
dap_num(-1): 1, ( 109 digits) |
1276 |
609,271,523,178,807,947,019,867,549, |
1277 |
668,874,172,185,430,463,576,158,940, |
1278 |
397,350,993,377,483,443,708,609,271, |
1279 |
523,178,807,947,019,867,549,668,874 |
1280 |
------------------------------------------------------------------------------ |
1281 |
dap_den(-1): 1, ( 109 digits) |
1282 |
000,000,000,000,000,000,000,000,000, |
1283 |
000,000,000,000,000,000,000,000,000, |
1284 |
000,000,000,000,000,000,000,000,000, |
1285 |
000,000,000,000,000,000,000,000,000 |
1286 |
------------------------------------------------------------------------------ |
1287 |
error_num(-1): - 43 ( 2 digits) |
1288 |
------------------------------------------------------------------------------ |
1289 |
error_den(-1): 1,510,000 ( 7 digits) |
1290 |
------------------------------------------------------------------------------ |
1291 |
\end{verbatim} |
1292 |
\end{scriptsize} |
1293 |
|
1294 |
Note in the sample invocation above that the |
1295 |
decimal equivalent of the approximation is also provided. |
1296 |
Note also that the different parameters of this utility produce |
1297 |
different output formats which are not documented here, for |
1298 |
brevity. For example, using the \texttt{-pred} or \texttt{-succ} |
1299 |
options does not produce decimal approximation information, because |
1300 |
these options are not normally used for approximation---such |
1301 |
behaviors are inconsequential and not documented. |
1302 |
\end{dosutilcommandsampleinvocations} |
1303 |
|
1304 |
|
1305 |
\begin{dosutilcommandseealso} |
1306 |
This extension uses the continued fraction |
1307 |
algorithms presented in Chapter \ccfrzeroxrefhyphen{}\ref{ccfr0}. |
1308 |
See also the \emph{arbint cfbrapab} Tcl extension, |
1309 |
Section \cxtnzeroxrefhyphen{}\ref{cxtn0:snth0:sbra0}. |
1310 |
\end{dosutilcommandseealso} |
1311 |
|
1312 |
|
1313 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1314 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1315 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1316 |
\section{Authors And Acknowledgements} |
1317 |
%Section tag: ACK0 |
1318 |
|
1319 |
|
1320 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1321 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1322 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1323 |
\section{Exercises} |
1324 |
%Section tag: EXE0 |
1325 |
|
1326 |
|
1327 |
|
1328 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1329 |
|
1330 |
\noindent\begin{figure}[!b] |
1331 |
\noindent\rule[-0.25in]{\textwidth}{1pt} |
1332 |
\begin{tiny} |
1333 |
\begin{verbatim} |
1334 |
$HeadURL$ |
1335 |
$Revision$ |
1336 |
$Date$ |
1337 |
$Author$ |
1338 |
\end{verbatim} |
1339 |
\end{tiny} |
1340 |
\noindent\rule[0.25in]{\textwidth}{1pt} |
1341 |
\end{figure} |
1342 |
|
1343 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1344 |
% |
1345 |
%End of file C_DCM0.TEX |