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Initial commit.
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45 {\operator David Ashley}{\creatim\yr2000\mo10\dy18\hr4\min48}{\revtim\yr2000\mo11\dy13\hr3\min50}{\printim\yr2000\mo11\dy13\hr3\min7}{\version9}{\edmins6}{\nofpages55}{\nofwords25439}{\nofchars145003}{\*\company Visteon Automotive Systems}{\vern57443}}
46 \widowctrl\ftnbj\aenddoc\hyphcaps0\formshade \fet0\sectd \linex0\endnhere {\footer \pard\plain \s16\widctlpar\brdrt\brdrs\brdrw15\brsp20 \tqc\tx4320\tqr\tx8640 \f4\fs20 RAP_C_IMP.RTF\tab Page {\field{\*\fldinst {\cs17 PAGE }}{\fldrslt {\cs17\lang1024 1}
47 }}{\cs17 Of }{\field{\*\fldinst {\cs17 NUMPAGES \\* MERGEFORMAT }}{\fldrslt {\cs17\lang1024 55}}}{\cs17 \tab Dave Ashley (DTASHLEY@AOL.COM)
48 \par $Revision: 1.1 $\tab \tab $Date: 2001/09/25 21:44:55 $}
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51 {\pntxtb (}{\pntxta )}}{\*\pnseclvl9\pnlcrm\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}\pard\plain \qc\widctlpar \f4\fs20 {\fs72
52 \par
53 \par User\rquote s Manual To Accompany \ldblquote RAP\rdblquote Rational Approximation Software
54 \par }\pard \widctlpar {\fs72
55 \par }
56 \par
57 \par
58 \par
59 \par
60 \par
61 \par
62 \par
63 \par
64 \par
65 \par
66 \par
67 \par
68 \par
69 \par
70 \par
71 \par \pard \qc\widctlpar David T. Ashley, (DTASHLEY@AOL.COM)
72 \par Joseph P. DeVoe (JDEVOE@VISTEON.COM)
73 \par Cory Pratt (CORY_PRATT@3COM.COM)
74 \par Karl Perttunen (KPERTTUN@VISTEON.COM)
75 \par Anatoly Zhigljvasky (ZHIGLJAVSKYAA@CARDIFF.AC.UK)
76 \par David Eppstein (EPPSTEIN@ICS.UCI.EDU)
77 \par David M. Einstein (DEINST@WORLD.STD.COM)
78 \par 10/18/2000
79 \par \pard \widctlpar
80 \par \pard \qc\widctlpar \page {\b\fs28 TABLE OF CONTENTS
81 \par }\pard \widctlpar
82 \par \pard\plain \s20\sb120\sa120\widctlpar\tqr\tldot\tx8640 \b\caps\f4\fs20 {\field\fldedit{\*\fldinst TOC \\o "1-5" }{\fldrslt {\lang1024 1. Introduction And Overview\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721621 }{\field{\*\fldinst {
83 \lang1024 PAGEREF _Toc498721621 }}{\fldrslt {\lang1024 5}}}}}{\lang1024
84 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 1.1 Scope Of This Document\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721622 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721622 }}{\fldrslt {\lang1024 5}}}}}{
85 \lang1024
86 \par 1.2 Overview Of RAP Functionality\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721623 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721623 }}{\fldrslt {\lang1024 5}}}}}{\lang1024
87 \par 1.3 RAP Absolute Maximums\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721624 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721624 }}{\fldrslt {\lang1024 5}}}}}{\lang1024
88 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 1.3.1 Maximum Data Sizes\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721625 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721625 }}{\fldrslt {\lang1024 6}}}}}{\lang1024
89
90 \par 1.3.2 Maximum Length Of Standard Input Stream\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721626 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721626 }}{\fldrslt {\lang1024 6}}}}}{\lang1024
91 \par 1.3.3 Maximum Number Of Farey Neighbors Generated\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721627 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721627 }}{\fldrslt {\lang1024 6}}}}}{\lang1024
92 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 1.4 RAP Authors And Algorithm Authors\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721628 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721628 }}{\fldrslt {\lang1024 6
93 }}}}}{\lang1024
94 \par 1.5 Statement Of Reliability And Intended Use\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721629 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721629 }}{\fldrslt {\lang1024 7}}}}}{\lang1024
95 \par 1.6 Embedded Version Control Information\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721630 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721630 }}{\fldrslt {\lang1024 7}}}}}{\lang1024
96 \par 1.7 Bug Reports\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721631 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721631 }}{\fldrslt {\lang1024 7}}}}}{\lang1024
97 \par }\pard\plain \s20\sb120\sa120\widctlpar\tqr\tldot\tx8640 \b\caps\f4\fs20 {\lang1024 2. Portability Of RAP Source Code\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721632 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721632 }}{\fldrslt {
98 \lang1024 7}}}}}{\lang1024
99 \par 3. Invoking RAP And Specifying Commands\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721633 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721633 }}{\fldrslt {\lang1024 8}}}}}{\lang1024
100 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 3.1 RAP Invocation Modes\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721634 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721634 }}{\fldrslt {\lang1024 8}}}}}{
101 \lang1024
102 \par 3.2 Token Concatenation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721635 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721635 }}{\fldrslt {\lang1024 10}}}}}{\lang1024
103 \par 3.3 Case Sensitivity Of RAP\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721636 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721636 }}{\fldrslt {\lang1024 10}}}}}{\lang1024
104 \par 3.4 Comments In Batch Mode\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721637 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721637 }}{\fldrslt {\lang1024 10}}}}}{\lang1024
105 \par 3.5 Saving RAP Output To A File\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721638 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721638 }}{\fldrslt {\lang1024 10}}}}}{\lang1024
106 \par 3.6 Error Behavior Of RAP\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721639 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721639 }}{\fldrslt {\lang1024 10}}}}}{\lang1024
107 \par 3.7 Specification Of Integers To RAP\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721640 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721640 }}{\fldrslt {\lang1024 11}}}}}{\lang1024
108 \par 3.8 Specification Of Rational Numbers To RAP\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721641 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721641 }}{\fldrslt {\lang1024 11}}}}}{\lang1024
109 \par }\pard\plain \s20\sb120\sa120\widctlpar\tqr\tldot\tx8640 \b\caps\f4\fs20 {\lang1024 4. Detailed Descriptions Of Commands\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721642 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721642 }}{\fldrslt {
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111 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.1 Integer Sum [+]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721643 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721643 }}{\fldrslt {\lang1024 12}}}}}{\lang1024
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113 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.1.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721644 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721644 }}{\fldrslt {\lang1024 12}}}}}
114 {\lang1024
115 \par 4.1.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721645 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721645 }}{\fldrslt {\lang1024 12}}}}}{\lang1024
116 \par 4.1.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721646 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721646 }}{\fldrslt {\lang1024 12}}}}}{\lang1024
117 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.2 Integer Difference [-]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721647 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721647 }}{\fldrslt {\lang1024 13}}}}}{
118 \lang1024
119 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.2.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721648 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721648 }}{\fldrslt {\lang1024 13}}}}}
120 {\lang1024
121 \par 4.2.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721649 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721649 }}{\fldrslt {\lang1024 13}}}}}{\lang1024
122 \par 4.2.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721650 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721650 }}{\fldrslt {\lang1024 13}}}}}{\lang1024
123 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.3 Integer Product [*]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721651 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721651 }}{\fldrslt {\lang1024 14}}}}}{
124 \lang1024
125 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.3.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721652 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721652 }}{\fldrslt {\lang1024 14}}}}}
126 {\lang1024
127 \par 4.3.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721653 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721653 }}{\fldrslt {\lang1024 14}}}}}{\lang1024
128 \par 4.3.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721654 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721654 }}{\fldrslt {\lang1024 14}}}}}{\lang1024
129 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.4 Integer Quotient [/]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721655 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721655 }}{\fldrslt {\lang1024 15}}}}}{
130 \lang1024
131 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.4.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721656 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721656 }}{\fldrslt {\lang1024 15}}}}}
132 {\lang1024
133 \par 4.4.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721657 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721657 }}{\fldrslt {\lang1024 15}}}}}{\lang1024
134 \par 4.4.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721658 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721658 }}{\fldrslt {\lang1024 15}}}}}{\lang1024
135 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.5 Integer Remainder [%]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721659 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721659 }}{\fldrslt {\lang1024 16}}}}}{
136 \lang1024
137 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.5.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721660 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721660 }}{\fldrslt {\lang1024 16}}}}}
138 {\lang1024
139 \par 4.5.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721661 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721661 }}{\fldrslt {\lang1024 16}}}}}{\lang1024
140 \par 4.5.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721662 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721662 }}{\fldrslt {\lang1024 16}}}}}{\lang1024
141 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.6 Integer Raised To An Integer Power [**]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721663 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721663 }}{\fldrslt {
142 \lang1024 17}}}}}{\lang1024
143 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.6.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721664 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721664 }}{\fldrslt {\lang1024 17}}}}}
144 {\lang1024
145 \par 4.6.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721665 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721665 }}{\fldrslt {\lang1024 17}}}}}{\lang1024
146 \par 4.6.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721666 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721666 }}{\fldrslt {\lang1024 17}}}}}{\lang1024
147 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.7 Greatest Common Divisor [GCD]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721667 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721667 }}{\fldrslt {\lang1024 18}}}
148 }}{\lang1024
149 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.7.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721668 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721668 }}{\fldrslt {\lang1024 18}}}}}
150 {\lang1024
151 \par 4.7.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721669 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721669 }}{\fldrslt {\lang1024 18}}}}}{\lang1024
152 \par 4.7.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721670 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721670 }}{\fldrslt {\lang1024 18}}}}}{\lang1024
153 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.8 Decimal Approximation [DAP]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721671 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721671 }}{\fldrslt {\lang1024 19}}}}}
154 {\lang1024
155 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.8.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721672 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721672 }}{\fldrslt {\lang1024 19}}}}}
156 {\lang1024
157 \par 4.8.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721673 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721673 }}{\fldrslt {\lang1024 19}}}}}{\lang1024
158 \par 4.8.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721674 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721674 }}{\fldrslt {\lang1024 19}}}}}{\lang1024
159 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.9 Rational Number Sum [+]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721675 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721675 }}{\fldrslt {\lang1024 22}}}}}{
160 \lang1024
161 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.9.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721676 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721676 }}{\fldrslt {\lang1024 22}}}}}
162 {\lang1024
163 \par 4.9.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721677 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721677 }}{\fldrslt {\lang1024 22}}}}}{\lang1024
164 \par 4.9.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721678 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721678 }}{\fldrslt {\lang1024 22}}}}}{\lang1024
165 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.10 Rational Number Difference [-]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721679 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721679 }}{\fldrslt {\lang1024 23}
166 }}}}{\lang1024
167 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.10.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721680 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721680 }}{\fldrslt {\lang1024 23}}}
168 }}{\lang1024
169 \par 4.10.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721681 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721681 }}{\fldrslt {\lang1024 23}}}}}{\lang1024
170 \par 4.10.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721682 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721682 }}{\fldrslt {\lang1024 23}}}}}{\lang1024
171 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.11 Rational Number Product [*]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721683 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721683 }}{\fldrslt {\lang1024 24}}}
172 }}{\lang1024
173 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.11.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721684 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721684 }}{\fldrslt {\lang1024 24}}}
174 }}{\lang1024
175 \par 4.11.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721685 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721685 }}{\fldrslt {\lang1024 24}}}}}{\lang1024
176 \par 4.11.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721686 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721686 }}{\fldrslt {\lang1024 24}}}}}{\lang1024
177 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.12 Rational Number Quotient [/]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721687 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721687 }}{\fldrslt {\lang1024 25}}}
178 }}{\lang1024
179 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.12.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721688 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721688 }}{\fldrslt {\lang1024 25}}}
180 }}{\lang1024
181 \par 4.12.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721689 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721689 }}{\fldrslt {\lang1024 25}}}}}{\lang1024
182 \par 4.12.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721690 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721690 }}{\fldrslt {\lang1024 25}}}}}{\lang1024
183 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.13 Rational Number Raised To An Integer Power [**]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721691 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721691 }
184 }{\fldrslt {\lang1024 26}}}}}{\lang1024
185 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.13.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721692 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721692 }}{\fldrslt {\lang1024 26}}}
186 }}{\lang1024
187 \par 4.13.2 Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721693 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721693 }}{\fldrslt {\lang1024 26}}}}}{\lang1024
188 \par 4.13.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721694 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721694 }}{\fldrslt {\lang1024 26}}}}}{\lang1024
189 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.14 Continued Fraction Partial Quotients And Convergents Of A Rational Number [CF]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721695 }{\field{\*\fldinst {\lang1024
190 PAGEREF _Toc498721695 }}{\fldrslt {\lang1024 27}}}}}{\lang1024
191 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.14.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721696 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721696 }}{\fldrslt {\lang1024 27}}}
192 }}{\lang1024
193 \par 4.14.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721697 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721697 }}{\fldrslt {\lang1024 27}}}}}{\lang1024
194 \par 4.14.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721698 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721698 }}{\fldrslt {\lang1024 27}}}}}{\lang1024
195 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.15 Rational Number With Smallest Denominator In An Interval [MIND]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721699 }{\field{\*\fldinst {\lang1024 PAGEREF
196 _Toc498721699 }}{\fldrslt {\lang1024 30}}}}}{\lang1024
197 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.15.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721700 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721700 }}{\fldrslt {\lang1024 30}}}
198 }}{\lang1024
199 \par 4.15.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721701 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721701 }}{\fldrslt {\lang1024 30}}}}}{\lang1024
200 \par 4.15.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721702 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721702 }}{\fldrslt {\lang1024 30}}}}}{\lang1024
201 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.16 Enclosing Rational Numbers In The Farey Series Of Order N [FN]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721703 }{\field{\*\fldinst {\lang1024 PAGEREF
202 _Toc498721703 }}{\fldrslt {\lang1024 32}}}}}{\lang1024
203 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.16.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721704 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721704 }}{\fldrslt {\lang1024 32}}}
204 }}{\lang1024
205 \par 4.16.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721705 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721705 }}{\fldrslt {\lang1024 32}}}}}{\lang1024
206 \par 4.16.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721706 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721706 }}{\fldrslt {\lang1024 32}}}}}{\lang1024
207 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.17 Upper Bound On Distance Between Farey Terms In An Interval [FNDMAX]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721707 }{\field{\*\fldinst {\lang1024 PAGEREF
208 _Toc498721707 }}{\fldrslt {\lang1024 41}}}}}{\lang1024
209 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.17.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721708 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721708 }}{\fldrslt {\lang1024 41}}}
210 }}{\lang1024
211 \par 4.17.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721709 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721709 }}{\fldrslt {\lang1024 41}}}}}{\lang1024
212 \par 4.17.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721710 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721710 }}{\fldrslt {\lang1024 41}}}}}{\lang1024
213 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.18 Enclosing Rational Numbers In A Rectangular Area Of The Integer Lattice [FAB]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721711 }{\field{\*\fldinst {\lang1024
214 PAGEREF _Toc498721711 }}{\fldrslt {\lang1024 44}}}}}{\lang1024
215 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.18.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721712 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721712 }}{\fldrslt {\lang1024 44}}}
216 }}{\lang1024
217 \par 4.18.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721713 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721713 }}{\fldrslt {\lang1024 44}}}}}{\lang1024
218 \par 4.18.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721714 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721714 }}{\fldrslt {\lang1024 44}}}}}{\lang1024
219 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 4.19 Upper Bound On Distance Between Members Of F}{\lang1024\sub A,B}{\lang1024 In An Interval [FABDMAX]\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721715 }
220 {\field{\*\fldinst {\lang1024 PAGEREF _Toc498721715 }}{\fldrslt {\lang1024 50}}}}}{\lang1024
221 \par }\pard\plain \s22\li400\widctlpar\tqr\tldot\tx8640 \i\f4\fs20 {\lang1024 4.19.1 Command Line Invocation Forms\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721716 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721716 }}{\fldrslt {\lang1024 50}}}
222 }}{\lang1024
223 \par 4.19.2 Detailed Algorithm Description\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721717 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721717 }}{\fldrslt {\lang1024 50}}}}}{\lang1024
224 \par 4.19.3 Example Invocation\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721718 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721718 }}{\fldrslt {\lang1024 50}}}}}{\lang1024
225 \par }\pard\plain \s20\sb120\sa120\widctlpar\tqr\tldot\tx8640 \b\caps\f4\fs20 {\lang1024 5. Digits Of Useful Constants\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721719 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721719 }}{\fldrslt {\lang1024 54
226 }}}}}{\lang1024
227 \par }\pard\plain \s21\li200\widctlpar\tqr\tldot\tx8640 \scaps\f4\fs20 {\lang1024 5.1 Digits Of }{\lang1024 {\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}}{\lang1024 \tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721720 }
228 {\field{\*\fldinst {\lang1024 PAGEREF _Toc498721720 }}{\fldrslt {\lang1024 54}}}}}{\lang1024
229 \par 5.2 Digits Of e\tab }{\field{\*\fldinst {\lang1024 GOTOBUTTON _Toc498721721 }{\field{\*\fldinst {\lang1024 PAGEREF _Toc498721721 }}{\fldrslt {\lang1024 55}}}}}{\lang1024
230 \par }\pard\plain \widctlpar \f4\fs20 }}\pard\plain \widctlpar \f4\fs20
231 \par
232 \par \page
233 \par {\*\bkmkstart _Toc498721621}{\pntext\pard\plain\b\f5\fs28\kerning28 1.\tab}\pard\plain \s1\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl1\pndec\pnprev1\pnstart1\pnsp144 {\pntxta .}}\b\f5\fs28\kerning28 Introduction And Overview{\*\bkmkend _Toc498721621}
234 \par \pard\plain \widctlpar \f4\fs20
235 \par {\*\bkmkstart _Toc498721622}{\pntext\pard\plain\b\i\f5 1.1\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Scope Of This Document{\*\bkmkend _Toc498721622}
236 \par \pard\plain \widctlpar \f4\fs20 This manual accompanies the rational approximation software program submitted to CALGO (the ACM collected algorithms), in support of a paper entitled \ldblquote {\i On Best Rational Approximations Using Large Integers}
237 \rdblquote submitted to TOMS (ACM Transactions On Mathematical Software) in late 2000. The software program is named RAP (mnemonic for {\i r}ational {\i ap}
238 proximation), and this acronym will be used throughout this document to refer to the software program. This document explains the operation and limitations of RAP.
239 \par
240 \par {\*\bkmkstart _Toc498721623}{\pntext\pard\plain\b\i\f5 1.2\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Overview Of RAP Functionality{\*\bkmkend _Toc498721623}
241 \par \pard\plain \widctlpar \f4\fs20
242 RAP implements a set of useful algorithms on very large integers and rational numbers with large integer components. RAP implements operations on long integers by representing all integers as character strings, and performing multiplication, division, an
243 d other primitive operations using the same \ldblquote long-hand\rdblquote techniques taught in elementary school.
244 \par
245 \par RAP is able to implement the following operations on integers and rational numbers. Each distinct operation includes the command
246 code for the operation used to specify it to RAP, in square brackets. Each operation is explained in detail later in the manual.
247 \par
248 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}\pard \fi-360\li1080\widctlpar{\*\pn \pnlvlblt\pnf1\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}{\b Integer Sum [+].}
249 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Integer Difference [-].}
250 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Integer Product [*].}
251 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Integer Quotient [/].}
252 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Integer Remainder [%].}
253 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Integer Raised To An Integer Power [**].}
254 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Greatest Common Divisor [GCD].}
255 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Decimal Approximation [DAP].}
256 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Rational Number Sum [+].}
257 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Rational Number Difference [-].}
258 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Rational Number Product [*].}
259 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Rational Number Quotient [/].}
260 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Rational Number Raised To An Integer Power [**].}
261 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Continued Fraction Partial Quotients And Convergents Of A Rational Number [CF].}
262 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Rational Number With The Smallest Denominator In An Interval [MIND].}
263 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Enclosing Rational Numbers In The Farey Series Of Order N [FN].}
264 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Upper Bound On Distance Between Farey Terms In An Interval [FNDMAX].}
265 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Enclosing Rational Numbers In A Rectangular Area Of The Integer Lattice [FAB].}
266 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Upper Bound On Distance Between Members Of F}{\b\sub A,B}{\b In An Interval [FABDMAX].}
267 \par \pard \widctlpar
268 \par {\*\bkmkstart _Toc498721624}{\pntext\pard\plain\b\i\f5 1.3\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 RAP Absolute Maximums{\*\bkmkend _Toc498721624}
269 \par \pard\plain \widctlpar \f4\fs20
270 \par {\*\bkmkstart _Toc498721625}{\pntext\pard\plain\f5 1.3.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Maximum Data Sizes{\*\bkmkend _Toc498721625}
271 \par \pard\plain \widctlpar \f4\fs20 To avoid confusion in the discussion which follows, integers which are in binary format and are primitive data types in the \lquote C\rquote programming language are called {\i native}
272 integers; and integers which are maintained by RAP as long strings of decimal digits are called {\i synthetic} integers.
273 \par
274 \par RAP is designed to freely operate within the Farey series of up to order 2{\super 1536}. (2{\super 1536} is about 2.4 {{\field{\*\fldinst SYMBOL 180 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}} 10{\super 462}
275 .) Because RAP manipulates synthetic integers as decimal character strings, the limit of 2{\super 1536}
276 is enforced by prohibiting synthetic integers as input which exceed 463 digits (meaning that RAP will actually accept synthetic integers up to and including 10{\super 463}-1, a number somewhat larger than 2{\super 1536}
277 .). Additionally, no internal intermediate result or output result in RAP may exceed 4,000 decimal digits.
278 \par
279 \par The limits of 2{\super 1536} and 4,000 were chosen out of convenience rather than necessity. The absolute maximums
280 are specified as compile-time preprocessor constants, and allow RAP to operate more efficiently (and allow the code to be more compact) than if internal data structures were allowed to grow arbitrarily. In RAP, {\i every}
281 synthetic integer used for applying the algorithms presented in the TOMS paper has 4,000 digits when it is created\emdash
282 it is never necessary to dynamically resize the storage allocated for such a synthetic integer. Additionally, the limit of 4,000 keeps RAP comfortably clear of platform-specific data size limits for native integers used to index into strings\emdash
283 the ANSI \lquote C\rquote standard guarantees that an unsigned native integer variable can attain at least 65,535 on any platform. RAP can be recompiled with larger upper limits, but this is neither recommended nor supported.
284 \par
285 \par If the data size limits of RAP are exceeded, RAP will announce the error and abort gracefully. There is no known circumstance where RAP will give incorrect results due to an undetected synthetic integer size overflow.
286 \par
287 \par {\*\bkmkstart _Toc498721626}{\pntext\pard\plain\f5 1.3.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Maximum Length Of Standard Input Stream{\*\bkmkend _Toc498721626}
288 \par \pard\plain \widctlpar \f4\fs20
289 If RAP is used in batch mode (described later), the standard input stream must not exceed 31,999 characters before RAP encounters the end-of-file (this is because RAP buffers the input stream before processing it). If this maximum is violated, RAP will a
290 nnounce the error and gracefully abort.
291 \par
292 \par {\*\bkmkstart _Toc498721627}{\pntext\pard\plain\f5 1.3.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Maximum Number Of Farey Neighbors Generated{\*\bkmkend _Toc498721627}
293 \par \pard\plain \widctlpar \f4\fs20 RAP will generate no more than 10,000 Farey neighbors on either side of a rational number. This limit is enforced to avoid platform-specific issues with native integers if RAP is ported.
294 \par
295 \par {\*\bkmkstart _Toc498721628}{\pntext\pard\plain\b\i\f5 1.4\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 RAP Authors And Algorithm Authors{\*\bkmkend _Toc498721628}
296 \par \pard\plain \widctlpar \f4\fs20
297 The RAP software was developed exclusively by Dave Ashley. However, the algorithms that RAP applies were developed by all of the contributors listed below. Note that David Eppstein and David M. Einstein are not authors on the TOMS paper\emdash via the {
298 \f11 sci.math} newsgroup, they contributed approaches for finding a rational number with the smallest denominator within an interval. All of the contributors listed below should be listed as authors for the algorithms in CALGO.
299 \par
300 \par
301 \par \trowd \trgaph108\trleft-108\trkeep\trhdr\trbrdrt\brdrs\brdrw15\brdrcf15 \trbrdrl\brdrs\brdrw15\brdrcf15 \trbrdrb\brdrs\brdrw15\brdrcf15 \trbrdrr\brdrs\brdrw15\brdrcf15 \trbrdrh\brdrs\brdrw15\brdrcf15 \trbrdrv\brdrs\brdrw15\brdrcf15 \clbrdrt
302 \brdrs\brdrw15\brdrcf1 \clbrdrl\brdrs\brdrw15\brdrcf1 \clbrdrr\brdrs\brdrw15\brdrcf1 \clcfpat1\clcbpat8\clshdng6000 \cellx2250\clbrdrt\brdrs\brdrw15\brdrcf1 \clbrdrl\brdrs\brdrw15\brdrcf1 \clbrdrr\brdrs\brdrw15\brdrcf1 \clcfpat1\clcbpat8\clshdng6000
303 \cellx4950\clbrdrt\brdrs\brdrw15\brdrcf1 \clbrdrl\brdrs\brdrw15\brdrcf1 \clbrdrr\brdrs\brdrw15\brdrcf1 \clcfpat1\clcbpat8\clshdng6000 \cellx8748 \pard \qc\keep\keepn\widctlpar\intbl {\b\cf8 Algorithm\cell E-mail\cell Home\cell }\pard \widctlpar\intbl {
304 \b\cf8 \row }\trowd \trgaph108\trleft-108\trkeep\trhdr\trbrdrt\brdrs\brdrw15\brdrcf15 \trbrdrl\brdrs\brdrw15\brdrcf15 \trbrdrb\brdrs\brdrw15\brdrcf15 \trbrdrr\brdrs\brdrw15\brdrcf15 \trbrdrh\brdrs\brdrw15\brdrcf15 \trbrdrv\brdrs\brdrw15\brdrcf15 \clbrdrl
305 \brdrs\brdrw15\brdrcf1 \clbrdrb\brdrs\brdrw15\brdrcf1 \clbrdrr\brdrs\brdrw15\brdrcf1 \clcfpat1\clcbpat8\clshdng6000 \cellx2250\clbrdrl\brdrs\brdrw15\brdrcf1 \clbrdrb\brdrs\brdrw15\brdrcf1 \clbrdrr\brdrs\brdrw15\brdrcf1 \clcfpat1\clcbpat8\clshdng6000
306 \cellx4950\clbrdrl\brdrs\brdrw15\brdrcf1 \clbrdrb\brdrs\brdrw15\brdrcf1 \clbrdrr\brdrs\brdrw15\brdrcf1 \clcfpat1\clcbpat8\clshdng6000 \cellx8748 \pard \qc\keep\keepn\widctlpar\intbl {\b\cf8 Author\cell Address\cell Page\cell }\pard \widctlpar\intbl {
307 \b\cf8 \row }\trowd \trgaph108\trleft-108\trkeep\trbrdrt\brdrs\brdrw15\brdrcf15 \trbrdrl\brdrs\brdrw15\brdrcf15 \trbrdrb\brdrs\brdrw15\brdrcf15 \trbrdrr\brdrs\brdrw15\brdrcf15 \trbrdrh\brdrs\brdrw15\brdrcf15 \trbrdrv\brdrs\brdrw15\brdrcf15 \clbrdrl
308 \brdrs\brdrw15\brdrcf15 \clbrdrb\brdrs\brdrw15\brdrcf15 \clbrdrr\brdrs\brdrw15\brdrcf15 \cellx2250\clbrdrl\brdrs\brdrw15\brdrcf15 \clbrdrb\brdrs\brdrw15\brdrcf15 \clbrdrr\brdrs\brdrw15\brdrcf15 \cellx4950\clbrdrl\brdrs\brdrw15\brdrcf15 \clbrdrb
309 \brdrs\brdrw15\brdrcf15 \clbrdrr\brdrs\brdrw15\brdrcf15 \cellx8748 \pard \keep\keepn\widctlpar\intbl David T. Ashley\cell dtashley@aol.com\cell N/A\cell \pard \widctlpar\intbl \row \trowd \trgaph108\trleft-108\trkeep\trbrdrt\brdrs\brdrw15\brdrcf15
310 \trbrdrl\brdrs\brdrw15\brdrcf15 \trbrdrb\brdrs\brdrw15\brdrcf15 \trbrdrr\brdrs\brdrw15\brdrcf15 \trbrdrh\brdrs\brdrw15\brdrcf15 \trbrdrv\brdrs\brdrw15\brdrcf15 \clbrdrt\brdrs\brdrw15\brdrcf15 \clbrdrl\brdrs\brdrw15\brdrcf15 \clbrdrb
311 \brdrs\brdrw15\brdrcf15 \clbrdrr\brdrs\brdrw15\brdrcf15 \cellx2250\clbrdrt\brdrs\brdrw15\brdrcf15 \clbrdrl\brdrs\brdrw15\brdrcf15 \clbrdrb\brdrs\brdrw15\brdrcf15 \clbrdrr\brdrs\brdrw15\brdrcf15 \cellx4950\clbrdrt\brdrs\brdrw15\brdrcf15 \clbrdrl
312 \brdrs\brdrw15\brdrcf15 \clbrdrb\brdrs\brdrw15\brdrcf15 \clbrdrr\brdrs\brdrw15\brdrcf15 \cellx8748 \pard \keep\keepn\widctlpar\intbl Joseph P. DeVoe\cell jdevoe@visteon.com\cell N/A\cell \pard \widctlpar\intbl \row \pard \keep\keepn\widctlpar\intbl
313 Cory Pratt\cell cory_pratt@3com.com\cell N/A\cell \pard \widctlpar\intbl \row \pard \keep\keepn\widctlpar\intbl Karl Perttunen\cell kperttun@visteon.com\cell N/A\cell \pard \widctlpar\intbl \row \pard \keep\keepn\widctlpar\intbl Anatoly Zhigljavsky\cell
314 zhigljavskyaa@cardiff.ac.uk\cell http://www.cf.ac.uk/maths/zhigljavskyaa/\cell \pard \widctlpar\intbl \row \pard \keep\keepn\widctlpar\intbl David Eppstein\cell eppstein@ics.uci.edu\cell http://www.ics.uci.edu/~eppstein/\cell \pard \widctlpar\intbl \row
315 \trowd \trgaph108\trleft-108\trkeep\trbrdrt\brdrs\brdrw15\brdrcf15 \trbrdrl\brdrs\brdrw15\brdrcf15 \trbrdrb\brdrs\brdrw15\brdrcf15 \trbrdrr\brdrs\brdrw15\brdrcf15 \trbrdrh\brdrs\brdrw15\brdrcf15 \trbrdrv\brdrs\brdrw15\brdrcf15 \clbrdrt
316 \brdrs\brdrw15\brdrcf15 \clbrdrl\brdrs\brdrw15\brdrcf15 \clbrdrb\brdrs\brdrw15\brdrcf15 \clbrdrr\brdrs\brdrw15\brdrcf15 \cellx2250\clbrdrt\brdrs\brdrw15\brdrcf15 \clbrdrl\brdrs\brdrw15\brdrcf15 \clbrdrb\brdrs\brdrw15\brdrcf15 \clbrdrr
317 \brdrs\brdrw15\brdrcf15 \cellx4950\clbrdrt\brdrs\brdrw15\brdrcf15 \clbrdrl\brdrs\brdrw15\brdrcf15 \clbrdrb\brdrs\brdrw15\brdrcf15 \clbrdrr\brdrs\brdrw15\brdrcf15 \cellx8748 \pard \keep\keepn\widctlpar\intbl David M. Einstein\cell deinst@world.std.com
318 \cell N/A\cell \pard \widctlpar\intbl \row \pard \widctlpar
319 \par {\*\bkmkstart _Toc498721629}{\pntext\pard\plain\b\i\f5 1.5\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Statement Of Reliability And Intended Use{\*\bkmkend _Toc498721629}
320 \par \pard\plain \widctlpar \f4\fs20
321 RAP is a research-grade tool, and is intended only to demonstrate for research purposes the algorithms presented in the TOMS paper. RAP has not been subjected to standard software engineering practices which help to minimize the probability of defects, s
322 uch as unit-testing or peer review. RAP is research-grade, intended only to illustrate concepts, and absolutely unsuitable for any application where injury, loss of life, or financial losses could occur because of improper operation of the software. RAP
323 is provided \ldblquote as-is\rdblquote with no warranty of any kind.
324 \par
325 \par {\*\bkmkstart _Toc498721630}{\pntext\pard\plain\b\i\f5 1.6\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Embedded Version Control Information{\*\bkmkend _Toc498721630}
326 \par \pard\plain \widctlpar \f4\fs20
327 For convenience, both this document (in the footer) and the RAP software (information emitted on every invocation) contain embedded version control information, which is inserted automatically by a version control tool. Versions of this document which ha
328 ve different version control information in the footer are necessarily different. Executable copies of RAP which emit different version control information when the software is invoked are necessarily different.
329 \par
330 \par This document and the RAP program are version-controlled separately. This means that this document generally won\rquote
331 t have the same revision number as the RAP executable, and there is no way to make the association between document version and RAP executable version. However, the most current versions of each (i.e. the versions which \ldblquote belong\rdblquote
332 together) are distributed through CALGO.
333 \par
334 \par {\*\bkmkstart _Toc498721631}{\pntext\pard\plain\cs17\b\i\f5 1.7\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 {\cs17 Bug Reports{\*\bkmkend _Toc498721631}
335 \par }\pard\plain \widctlpar \f4\fs20
336 The probability of finding a bug in the operation of the RAP program is very low. Because RAP performs a small number of repetitive operations on large data (adding digits and processing carries, for example), RAP does not fit the profile of software whi
337 ch is good at concealing defects. Defects which would become apparent with large data would generally also become apparent with small data, and the program was exercised with much small data.
338 \par
339 \par Please submit all bug reports to Dave Ashley (and the other contributors if you have difficulty in reaching Dave). If the bug can be reproduced, it will be fixed and RAP will be re-released to CALGO.
340 \par
341 \par {\*\bkmkstart _Toc498721632}{\pntext\pard\plain\b\f5\fs28\kerning28 2.\tab}\pard\plain \s1\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl1\pndec\pnprev1\pnstart1\pnsp144 {\pntxta .}}\b\f5\fs28\kerning28 Portability Of RAP Source Code{\*\bkmkend _Toc498721632}
342
343 \par \pard\plain \widctlpar \f4\fs20
344 RAP is provided to CALGO with a binary executable for Win32-capable platforms (this should include Windows 95, Windows 98, Windows ME, Windows NT, and Windows 2000). Use of RAP on any of these platforms should not require recompilation because the binary
345 executable is already available. This section contains portability notes in the event RAP must be compiled for other platforms.
346 \par
347 \par RAP is written in ANSI \lquote C\rquote
348 , in a single software module. Because only one software module is used, no header files are included. Enumerated below are requirements on any target system and issues which may affect portability favorably or unfavorably.
349 \par
350 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}\pard \fi-360\li1080\widctlpar{\*\pn \pnlvlblt\pnf1\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}
351 RAP was compiled for Win32 as a Win32 console application under Microsoft Visual C++ Version 6.0. The file RAP_C.C contains a function named {\i main_c()}, which is the main function of the program,
352 called from a C++ wrapper in another file. For recompilation on another platform, only the file RAP_C.C is required, and the function {\i main_c()} must be renamed to {\i main()}.
353 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}The only known possible portability issue with RAP is the use of the preprocessor symbol \ldblquote __LINE__\rdblquote
354 , which is used with a home-made assertion mechanism to emit the line number where an assertion fails. If \ldblquote __LINE__\rdblquote won\rquote t compile, it would be necessary t
355 o either find the name of a similar symbol for the target system, or else replace all occurrences of __LINE__ with a constant integer, such as 0.
356 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}The C++ record oriented comment delimiter \ldblquote //\rdblquote is not used, although many \lquote C\rquote compilers allow its use. Only \ldblquote /*\rdblquote and \ldblquote */\rdblquote are used.
357 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}Nested comments are not used, because not all compiler support them.{\cs19\super \chftn {\footnote \pard\plain \s18\widctlpar \f4\fs20 {\cs19\super \chftn } A nested comment is multiple occurrences of \ldblquote /*
358 \rdblquote before a matching \ldblquote */\rdblquote . For example, \ldblquote /* /* */ */\rdblquote is a nested comment. A compiler that tolerates nested comments is sometimes helpful because it allows large blocks of code containing comme
359 nts to be removed from the compilation stream by enclosing the block in \ldblquote /*\rdblquote and \ldblquote */\rdblquote .}}
360 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}RAP assumes the existence of the standard dynamic memory allocation functions {\i malloc()}, {\i realloc()}, and {\i free()}. {\i malloc()}, {\i realloc()}, and {\i free()} must work correctly on the target system.
361
362 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}For command-line parameters, the target system must support {\i argv} and {\i argc} in the standard way.
363 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}Both signed and unsigned integers in the target system must be at least 16 bits, and must be able to attain a positive value of at least 32,100.
364 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}On input in batch mode, the {\i getchar()} family of functions is used to read from the standard input stream. The target system must support these.
365 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}On output, the {\i printf()} family of functions is used. The target system must support these.
366 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}The character set of the target machine must have the digits \lquote 0\rquote through \lquote 9\rquote as contiguous and increasing (RAP performs some calculations making this assumption\emdash
367 this is a nearly universal feature of most character sets).
368 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}RAP performs no console manipulation and does not write to {\i stderr} (this should increase its portability).
369 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}RAP should port to Unix systems with no changes whatsoever, except the possible issue with __LINE__ noted above.
370 \par \pard \widctlpar
371 \par {\*\bkmkstart _Toc498721633}{\pntext\pard\plain\b\f5\fs28\kerning28 3.\tab}\pard\plain \s1\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl1\pndec\pnprev1\pnstart1\pnsp144 {\pntxta .}}\b\f5\fs28\kerning28 Invoking RAP And Specifying Commands
372 {\*\bkmkend _Toc498721633}
373 \par \pard\plain \widctlpar \f4\fs20
374 \par {\*\bkmkstart _Toc498721634}{\pntext\pard\plain\b\i\f5 3.1\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 RAP Invocation Modes{\*\bkmkend _Toc498721634}
375 \par \pard\plain \widctlpar \f4\fs20 RAP operates in two different modes, depending on the command line parameters when RAP is invoked.
376 \par
377 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}\pard \fi-360\li1080\widctlpar{\*\pn \pnlvlblt\pnf1\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}{\b Interactive Mode.}
378 If all of the information that RAP requires to perform an individual command is present on the command line, RAP will perform the command and exit. This allows RAP to be used as an interactive calculator to answer impromptu queries.
379 \par \pard \widctlpar
380 \par \pard \li1080\widctlpar For example, invoking RAP with the command line:
381 \par
382 \par {\b rap fn 3.14159265359 255}
383 \par
384 \par will cause RAP to provide the two best rational approximations to {{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}} with a maximum denominator of 255.{\cs19\super \chftn {\footnote \pard\plain \s18\widctlpar \f4\fs20 {\cs19\super
385 \chftn } This statement should be qualified somewhat. RAP will not operate on irrational numbers\emdash every number which RAP can accept as input is rational. When using RAP to find rational numbers which are near an
386 irrational, a rational approximation of the irrational must be used (in this case, 3.1415926539 for {{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}}). RAP will provide the terms of F{\sub 255}
387 which enclose 3.1415926536, which are not required to be the rational numbers which enclose {{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}}
388 . RAP will provide an incorrect result if and only if there is a term of the Farey series of interest in between the irrational number one is trying to approximate and the rational approximation supplied to RAP. The probability of one getting
389 \ldblquote unlucky\rdblquote in this way increases with a)increasing order of the Farey series, because the terms are, on average, closer together, or b)decreasing precision of the rational number specified to RAP. It is tedious to enter large nu
390 mbers of digits for irrational numbers: commercial symbolic manipulation software such as {\i Mathematica}
391 may provide the ability to calculate arbitrarily many partial quotients of irrational numbers, and so may be more convenient in this regard than RAP.}}
392 \par
393 \par On a Win32 system, invoking RAP in interactive mode will require opening a DOS shell and then invoking the RAP program.
394 \par
395 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}\pard \fi-360\li1080\widctlpar{\*\pn \pnlvlblt\pnf1\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}{\b Batch Mode.} If RAP is invoked with the single parameter \ldblquote BATCH\rdblquote
396 , it will take input from the standard input stream until end-of-file on the stream, treating each token in the same way it would a command-line parameter. The batch mode allows numbers to be specified
397 which are too long for the operating system to tolerate on the command-line.
398 \par \pard \widctlpar
399 \par \pard \li1080\widctlpar For example, preparing an input file named RAP_IN.TXT with the following contents and invoking RAP with the command line \ldblquote {\f3 rap batch <RAP_IN.TXT}\rdblquote
400 will give the same results as the command-line invocation described above.{\cs19\super \chftn {\footnote \pard\plain \s18\widctlpar \f4\fs20 {\cs19\super \chftn } On a command line, with many operating systems, \ldblquote <\rdblquote
401 will cause the standard input for the program to be taken from the specified file.}}
402 \par
403 \par \pard \li1440\widctlpar {\f11 fn
404 \par 3.14159265359
405 \par 255
406 \par }\pard \li1080\widctlpar
407 \par It is also noteworthy that some computing platforms can be \ldblquote tricked\rdblquote into accepting input from the keyboard which is longer than allowed on a command line. For example, on a Win32 platform, the invocation:
408 \par
409 \par \pard \li1440\widctlpar {\f11 rap batch | more
410 \par }\pard \li1080\widctlpar
411 \par will allow input to be entered from the keyboard (stdin) which is much longer than allowed on the command line, but the input must be terminated with CONTROL-Z. The \ldblquote {\f11 | more}\rdblquote
412 pipelining command causes all output to go to a file before it is displayed, and eliminates display problems due to intermingling of input and output.
413 \par \pard \widctlpar
414 \par Invoking RAP with no parameters will result in a help message. Supplying RAP with unexpected input will result in an error message.
415 \par
416 \par {\*\bkmkstart _Toc498721635}{\pntext\pard\plain\b\i\f5 3.2\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Token Concatenation{\*\bkmkend _Toc498721635}
417 \par \pard\plain \widctlpar \f4\fs20 When preparing an input file for use in the batch mode of RAP, it may occur that a number to be supplied to RAP exceeds a convenient length for a line of text. For this reason, RAP treats the backslash (\ldblquote \\
418 \rdblquote ) as a token concatenation operator. RAP first parses its input, identifying tokens (groups of letters, digits, and symbols separated by spaces, tabs, and newline characters
419 ), then tokens ending in the backslash character are concatenated with the following token. Most readers of this document will have substantial experience with programming languages and scripting languages, so this concept doesn\rquote
420 t need further explanation.
421 \par
422 \par For example, the input file RAP_IN.TXT specified in the example above could be prepared as shown below with no change in RAP\rquote s behavior.
423 \par
424 \par \pard \li720\widctlpar {\f11 f\\
425 \par n
426 \par 3.14\\ 1592\\ 65\\
427 \par 359
428 \par 255
429 \par }\pard \widctlpar
430 \par Token concatenation is also applied in interactive mode, although this is probably of no practical value.
431 \par
432 \par {\*\bkmkstart _Toc498721636}{\pntext\pard\plain\b\i\f5 3.3\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Case Sensitivity Of RAP{\*\bkmkend _Toc498721636}
433 \par \pard\plain \widctlpar \f4\fs20 RAP is case-insensitive in all input.
434 \par
435 \par However, an operating system may be case-sensitive. In the example above involving batch mode, the command line \ldblquote {\f3 rap batch <rap_in.txt}\ldblquote might not work correctly, as an operating system may treat {\f11 RAP_IN.TXT}
436 as a different file than {\f11 rap_in.txt}. It is the operating system, not RAP, which arranges for redirection of the standard input and output.
437 \par
438 \par {\*\bkmkstart _Toc498721637}{\pntext\pard\plain\b\i\f5 3.4\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Comments In Batch Mode{\*\bkmkend _Toc498721637}
439 \par \pard\plain \widctlpar \f4\fs20 RAP has no provision for comments in the input file when used in batch mode.
440 \par
441 \par {\*\bkmkstart _Toc498721638}{\pntext\pard\plain\b\i\f5 3.5\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Saving RAP Output To A File{\*\bkmkend _Toc498721638}
442 \par \pard\plain \widctlpar \f4\fs20 All RAP output is written to the standard output stream, which is the console by default. To redirect RAP output to a file, include \ldblquote {\f11 >filename}\rdblquote
443 on the command line (Win32 DOS boxes and Unix). Output may be concatenated to a file using \ldblquote {\f11 >>filename}\rdblquote (Win32 DOS boxes and Unix). Output may also be displayed a screen at a time using \ldblquote {\f11 | more}\rdblquote .
444
445 \par
446 \par {\*\bkmkstart _Toc498721639}{\pntext\pard\plain\b\i\f5 3.6\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Error Behavior Of RAP{\*\bkmkend _Toc498721639}
447 \par \pard\plain \widctlpar \f4\fs20 RAP will respond to errors in five different ways.
448 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}\pard \fi-360\li1080\widctlpar{\*\pn \pnlvlblt\pnf1\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}{\b Input parsing errors.} Some feature of the input command or data violates RAP\rquote
449 s parsing rules (illegal characters, ill-formed numbers, etc.). The diagnostic messages provided in these cases should be adequate to locate the problem.
450 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Command Template Matching Errors.} RAP contains an internal \ldblquote template\rdblquote
451 table which is a list of all commands and the types of arguments that each will accept. If RAP can parse all input parameters, but can\rquote t match the number and type of parameters to the requested command, RAP will issue a nebulous catchall
452 \ldblquote template matching\rdblquote error message. For example, \ldblquote {\f11 rap gcd 2 2/3}\rdblquote will fail because the \ldblquote gcd\rdblquote command won\rquote t accept a non-integer argument. Similarly, \ldblquote {\f11 rap 2 3 4}
453 \rdblquote will fail because the \ldblquote gcd\rdblquote command can only accept two arguments.
454 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Context-Sensitive Error Messages.}
455 There are a small number of situations where RAP will issue specific error messages. In these cases, the information provided in the error message should be adequate to understand the nature of the error.
456 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b NAN.} Any internal calculation in RAP which overflows or produces an undefined result will cause the value of NAN ({\i N}ot {\i A} {\i N}
457 umber) to be assigned to the result. Any operation with NAN as an input will produce NAN as an output (NANs propagate). For example, exponentiating 1000000 to the 1000{\super th} power (\ldblquote rap ** 1000000 1000\rdblquote ) will produce NAN.
458 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}{\b Assertion Failures.} The source code of RAP is liberally decorated with calls to the function {\i asAssert()}, which behaves similarly to the standard C-language {\i assert()}
459 macro. If an assertion fails, it represents a serious internal software error. Only a line number will be displayed. The source code must be consulted to determine the nature of the error. (An assertion failure is a bug\emdash
460 please report any such failures as bugs.)
461 \par \pard \widctlpar
462 \par {\*\bkmkstart _Toc498721640}{\pntext\pard\plain\b\i\f5 3.7\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Specification Of Integers To RAP{\*\bkmkend _Toc498721640}
463 \par \pard\plain \widctlpar \f4\fs20 An integer is specified as a series of digits 0-9, with an optional leading unary \ldblquote -\ldblquote sign, and optional commas.
464 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}\pard \fi-360\li1080\widctlpar{\*\pn \pnlvlblt\pnf1\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}Commas in integers are simply discarded by RAP. Integers are not parsed
465 to determine if the commas are placed correctly within the integer. Example: \ldblquote 1234\rdblquote , \ldblquote 12,34\rdblquote , \ldblquote 1,234\rdblquote , and \ldblquote 1,,2,,3,,4\rdblquote
466 are treated equivalently by RAP; each represents the integer 1,234.
467 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}The integer {\i zero} must be specified as a single instance of the digit \ldblquote 0\rdblquote . Multiple instances are illegal. A unary \ldblquote -\ldblquote is illegal.
468 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}Leading 0\rquote s on non-zero integers are not allowed.
469 \par \pard \widctlpar
470 \par {\*\bkmkstart _Toc498721641}{\pntext\pard\plain\b\i\f5 3.8\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Specification Of Rational Numbers To RAP{\*\bkmkend _Toc498721641}
471 \par \pard\plain \widctlpar \f4\fs20 A rational number may be specified to RAP in two different ways.
472 \par
473 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}\pard \fi-360\li1080\widctlpar{\*\pn \pnlvlblt\pnf1\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}Two integers separated by a forward slash (\ldblquote /\rdblquote
474 ). Rational numbers may never contain whitespace. Only the numerator may contain an optional \lquote -\lquote -sign.
475 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}A floating-point constant with or without a positive or negative exponent. For example, \_3.121934204e-3 is legal and will be treated internally by RAP as the rational number \_3,121,934,204/1,000,000,000,000.{
476 \cs19\super \chftn {\footnote \pard\plain \s18\widctlpar \f4\fs20 {\cs19\super \chftn } This isn\rquote t completely true. RAP will remove any g.c.d. from numerator and denominator before using a rational number internally.}} RAP will also accept \_
477 0.003121934204 or \_3121934204/1000000000000 and treat it equivalently.
478 \par \pard \widctlpar
479 \par RAP will not accept other intuitively plausible specifications of rational numbers. For example, RAP\rquote s parser will not accept 1e30/5.
480 \par
481 \par \page
482 \par {\*\bkmkstart _Toc498721642}{\pntext\pard\plain\b\f5\fs28\kerning28 4.\tab}\pard\plain \s1\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl1\pndec\pnprev1\pnstart1\pnsp144 {\pntxta .}}\b\f5\fs28\kerning28 Detailed Descriptions Of Commands{\*\bkmkend _Toc498721642}
483
484 \par \pard\plain \widctlpar \f4\fs20
485 Commands are polymorphic in that RAP will differentiate between integers and rational numbers and may use slightly different algorithms or format results differently depending on whether input arguments are integers or rational numbers. Note that every i
486 nteger is a rational number, so RAP will always accept integers where rational numbers are specified. RAP will also accept rational numbers where integers are required, if the rational number reduces to an integer.
487 \par
488 \par {\*\bkmkstart _Toc498721643}{\pntext\pard\plain\b\i\f5 4.1\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Integer Sum [+]{\*\bkmkend _Toc498721643}
489 \par \pard\plain \widctlpar \f4\fs20
490 \par {\*\bkmkstart _Toc498721644}{\pntext\pard\plain\f5 4.1.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721644}
491 \par \pard\plain \widctlpar \f4\fs20 {\b rap + I1 I2
492 \par }\pard \li720\widctlpar Adds integer I1 to integer I2 to produce an integer result.
493 \par \pard \widctlpar
494 \par {\*\bkmkstart _Toc498721645}{\pntext\pard\plain\f5 4.1.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721645}
495 \par \pard\plain \widctlpar \f4\fs20 Adds integers to produce an integer result. The algorithm applied is addition of ASCII digits.
496 \par
497 \par {\*\bkmkstart _Toc498721646}{\pntext\pard\plain\f5 4.1.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721646}
498 \par \pard\plain \widctlpar \f4\fs20 Output from the invocation of RAP to add two large integers is reproduced below.
499 \par
500 \par {\f11\fs16 C:\\>rap + 3,142,991,002 7,934,333
501 \par ------------------------------------------------------------------------------
502 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
503 \par ------------------------------------------------------------------------------
504 \par arg1: 3,142,991,002 ( 10 digits)
505 \par ------------------------------------------------------------------------------
506 \par arg2: 7,934,333 ( 7 digits)
507 \par ------------------------------------------------------------------------------
508 \par arg1 + arg2: 3,150,925,335 ( 10 digits)
509 \par ------------------------------------------------------------------------------
510 \par RAP execution ends.
511 \par ------------------------------------------------------------------------------}
512 \par
513 \par \page
514 \par {\*\bkmkstart _Toc498721647}{\pntext\pard\plain\b\i\f5 4.2\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Integer Difference [-]{\*\bkmkend _Toc498721647}
515 \par \pard\plain \widctlpar \f4\fs20
516 \par {\*\bkmkstart _Toc498721648}{\pntext\pard\plain\f5 4.2.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721648}
517 \par \pard\plain \widctlpar \f4\fs20 {\b rap - I1 I2
518 \par }\pard \li720\widctlpar Subtracts integer I2 from integer I1 to produce an integer result.
519 \par
520 \par {\*\bkmkstart _Toc498721649}{\pntext\pard\plain\f5 4.2.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721649}
521 \par \pard\plain \widctlpar \f4\fs20 Subtracts integers to produce an integer result. The algorithm applied is subtraction of ASCII digits.
522 \par
523 \par {\*\bkmkstart _Toc498721650}{\pntext\pard\plain\f5 4.2.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721650}
524 \par \pard\plain \widctlpar \f4\fs20 Output from the invocation of RAP to subtract two large integers is reproduced below.
525 \par
526 \par {\f11\fs16 C:\\>rap - -343926469248723687426946 284622838352848
527 \par ------------------------------------------------------------------------------
528 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
529 \par ------------------------------------------------------------------------------
530 \par arg1: - 343,926,469,248,723,687,426,946 ( 24 digits)
531 \par ------------------------------------------------------------------------------
532 \par arg2: 284,622,838,352,848 ( 15 digits)
533 \par ------------------------------------------------------------------------------
534 \par arg1 - arg2: - 343,926,469,533,346,525,779,794 ( 24 digits)
535 \par ------------------------------------------------------------------------------
536 \par RAP execution ends.
537 \par ------------------------------------------------------------------------------}
538 \par
539 \par \page
540 \par {\*\bkmkstart _Toc498721651}{\pntext\pard\plain\b\i\f5 4.3\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Integer Product [*]{\*\bkmkend _Toc498721651}
541 \par \pard\plain \widctlpar \f4\fs20
542 \par {\*\bkmkstart _Toc498721652}{\pntext\pard\plain\f5 4.3.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721652}
543 \par \pard\plain \widctlpar \f4\fs20 {\b rap * I1 I2
544 \par }\pard \li720\widctlpar Multiplies integer I1 by integer I2 to produce an integer result.
545 \par {\*\bkmkstart _Toc498721653}{\pntext\pard\plain\f5 4.3.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721653}
546 \par \pard\plain \widctlpar \f4\fs20 Multiplies integers to produce an integer result. The algorithm used is essentially long-hand multiplication of ASCII digits (multiplication by a single digit, shifting, addition).
547 \par
548 \par {\*\bkmkstart _Toc498721654}{\pntext\pard\plain\f5 4.3.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721654}
549 \par \pard\plain \widctlpar \f4\fs20 Output from the invocation of RAP to multiply two large integers is reproduced below.
550 \par
551 \par {\f11\fs16 C:\\>rap * -294328649236462394616946 826482348525
552 \par ------------------------------------------------------------------------------
553 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
554 \par ------------------------------------------------------------------------------
555 \par arg1: - 294,328,649,236,462,394,616,946 ( 24 digits)
556 \par ------------------------------------------------------------------------------
557 \par arg2: 826,482,348,525 ( 12 digits)
558 \par ------------------------------------------------------------------------------
559 \par arg1 * arg2: - 243,257,433, ( 36 digits)
560 \par 259,142,387,965,858,847,843,104,650
561 \par ------------------------------------------------------------------------------
562 \par RAP execution ends.
563 \par ------------------------------------------------------------------------------}
564 \par
565 \par
566 \par \page
567 \par {\*\bkmkstart _Toc498721655}{\pntext\pard\plain\b\i\f5 4.4\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Integer Quotient [/]{\*\bkmkend _Toc498721655}
568 \par \pard\plain \widctlpar \f4\fs20
569 \par {\*\bkmkstart _Toc498721656}{\pntext\pard\plain\f5 4.4.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721656}
570 \par \pard\plain \widctlpar \f4\fs20 {\b rap / I1 I2
571 \par }\pard \li720\widctlpar Divides integer I1 by non-zero integer I2 to produce an integer result.
572 \par \pard \widctlpar
573 \par {\*\bkmkstart _Toc498721657}{\pntext\pard\plain\f5 4.4.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721657}
574 \par \pard\plain \widctlpar \f4\fs20 Divides two integers to produce an integer quotient. The result produced is specified by the equation below. The algorithm is long-hand division of ASCII digits (shifting, trial subtraction).
575 \par
576 \par \pard \qc\widctlpar {\pard\plain \qc\widctlpar \f4\fs20 {\object\objemb\objw1460\objh680{\*\objclass Equation.3}{\*\objdata 01050000020000000b0000004571756174696f6e2e3300000000000000000000100000
577 d0cf11e0a1b11ae1000000000000000000000000000000003e000300feff0900060000000000000000000000010000000100000000000000001000000200000001000000feffffff0000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
578 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
579 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
580 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
581 fffffffffffffffffdffffff05000000feffffff0400000006000000fefffffffeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
582 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
583 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
584 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
585 ffffffffffffffffffffffffffffffff52006f006f007400200045006e00740072007900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000016000500ffffffffffffffff0200000002ce020000000000c000000000000046000000000000000000000000a006
586 1cdf674dc00103000000800400000000000001004f006c00650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a000201ffffffffffffffffffffffff00000000000000000000000000000000000000000000000000000000
587 0000000000000000000000001400000000000000030050004900430000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a0002010100000004000000ffffffff0000000000000000000000000000000000000000000000000000
588 00000000000000000000010000004c0000000000000003004d004500540041000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000c000201ffffffffffffffffffffffff000000000000000000000000000000000000000000000000
589 000000000000000000000000030000007002000000000000feffffff02000000feffffff0400000005000000060000000700000008000000090000000a0000000b0000000c000000feffffff0e000000fefffffffeffffff11000000feffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
590 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
591 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
592 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
593 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff010000020000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004c00000000000000000008000f0a0000af040000b4050000a80200000000
594 0000000000000000000000000000e8030000e80300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000008000f0aaf0400000100090000033301000003001500000000000500
595 00000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c02400420091200000026060f001a00ffffffff000010000000c0ffffffb8ffffffe0080000f80300000b00000026060f000c004d617468547970650000e00009000000
596 fa02000010000000000000002200040000002d0100000500000014022002760605000000130220021e0810000000fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d01010008000000320a1a03360801000000fa0008000000320ada03360801000000fb0008000000320aaa0136080100
597 0000fa0008000000320a1a03d40501000000ea0008000000320ada03d40501000000eb0008000000320aaa01d40501000000ea0008000000320a80027e04010000003d0015000000fb0280fe0000000000009001000000000402001054696d6573204e657720526f6d616e0000a9040000002d01020004000000f0010100
598 08000000320aac03500701000000320008000000320a8e014a070100000031000a000000320a80023a0006000000526573756c7415000000fb0280fe0000000000009001010000000402001054696d6573204e657720526f6d616e0000a9040000002d01010004000000f001020008000000320aac039c06010000004900
599 08000000320a8e01c0060100000049000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02000000000102022253797374656d0000040000002d01020004000000f00101000300000000001500000000000000000000000000000000000100feff030a0000ffffffff02ce02000000
600 0000c000000000000046170000004d6963726f736f6674204571756174696f6e20332e30000c0000004453204571756174696f6e000b0000004571756174696f6e2e3300f439b271000000000000000000000000000000000000000000000000000000000000000000000000000000000300000000000000000000000000
601 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010043006f006d0070004f0062006a00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000120002000300000005000000ffff
602 ffff0000000000000000000000000000000000000000000000000000000000000000000000000d000000660000000000000003004f0062006a0049006e0066006f0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000012000201ffffffff06000000
603 ffffffff0000000000000000000000000000000000000000000000000000000000000000000000000f00000004000000000000004500710075006100740069006f006e0020004e00610074006900760065000000000000000000000000000000000000000000000000000000000000000000000020000200ffffffffffff
604 ffffffffffff0000000000000000000000000000000000000000000000000000000000000000000000001000000074000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffff
605 ffffffffffffffff0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001c000000020006e55800000000000000409e4900a08e49000000000003010103010a010281520002816500028173000281750002816c000281740002863d00029804eb029804
606 ef0306000001030e0000011283490002883100000112834900028832000000000296f0f80296fbf80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
607 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
608 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
609 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000105000000000000}{\result {\dn28 {\pict\wmetafile8\picw2575\pich1199\picwgoal1460\pichgoal680
610 010009000003330100000300150000000000050000000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c02400420091200000026060f001a00ffffffff000010000000c0ffffffb8ffffffe0080000f80300000b00000026060f
611 000c004d617468547970650000e00009000000fa02000010000000000000002200040000002d0100000500000014022002760605000000130220021e0810000000fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d01010008000000320a1a03360801000000fa0008000000320ada033608
612 01000000fb0008000000320aaa01360801000000fa0008000000320a1a03d40501000000ea0008000000320ada03d40501000000eb0008000000320aaa01d40501000000ea0008000000320a80027e04010000003d0015000000fb0280fe0000000000009001000000000402001054696d6573204e657720526f6d616e0000
613 a9040000002d01020004000000f001010008000000320aac03500701000000320008000000320a8e014a070100000031000a000000320a80023a0006000000526573756c7415000000fb0280fe0000000000009001010000000402001054696d6573204e657720526f6d616e0000a9040000002d01010004000000f0010200
614 08000000320aac039c0601000000490008000000320a8e01c0060100000049000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02000000000102022253797374656d0000040000002d01020004000000f0010100030000000000150000000001003634310000000000270000000000
615 0100000000000180000000000100ffffff0000000000ffffff000000000000000000000000000000000000000000000000000000000018000000000000000000000000000000ffffff00ffffff00bd077d000200db00ffffff0000000000ffffff00ffffff00ffffff00ffffff}}}}}
616 \par \pard \widctlpar
617 \par {\*\bkmkstart _Toc498721658}{\pntext\pard\plain\f5 4.4.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721658}
618 \par \pard\plain \widctlpar \f4\fs20 Output from the invocation of RAP to divide two large integers is reproduced below. Note that the remainder is also supplied.
619 \par
620 \par {\f11\fs16 C:\\>rap / 9274639462975692736497259623964932 287463864289
621 \par ------------------------------------------------------------------------------
622 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
623 \par ------------------------------------------------------------------------------
624 \par dividend: 9,274,639, ( 34 digits)
625 \par 462,975,692,736,497,259,623,964,932
626 \par ------------------------------------------------------------------------------
627 \par divisor: 287,463,864,289 ( 12 digits)
628 \par ------------------------------------------------------------------------------
629 \par dividend % divisor: 275,260,681,237 ( 12 digits)
630 \par ------------------------------------------------------------------------------
631 \par dividend / divisor: 32,263,670,725,762,915,010,255 ( 23 digits)
632 \par ------------------------------------------------------------------------------
633 \par RAP execution ends.
634 \par ------------------------------------------------------------------------------}
635 \par
636 \par \page
637 \par {\*\bkmkstart _Toc498721659}{\pntext\pard\plain\b\i\f5 4.5\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Integer Remainder [%]{\*\bkmkend _Toc498721659}
638 \par \pard\plain \widctlpar \f4\fs20
639 \par {\*\bkmkstart _Toc498721660}{\pntext\pard\plain\f5 4.5.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721660}
640 \par \pard\plain \widctlpar \f4\fs20 {\b rap % I1 I2
641 \par }\pard \li720\widctlpar Divides integer I1 by non-zero integer I2 to produce an integer remainder.
642 \par \pard \widctlpar
643 \par {\*\bkmkstart _Toc498721661}{\pntext\pard\plain\f5 4.5.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721661}
644 \par \pard\plain \widctlpar \f4\fs20 RAP will divide two integers to produce an integer remainder (the modulo remainder function). The result produced is specified by the equation below. The algorithm employed is long-hand division of ASCII digits.
645 \par \pard \qc\widctlpar {\pard\plain \qc\widctlpar \f4\fs20 {\object\objemb\objw2180\objh680{\*\objclass Equation.3}{\*\objdata 01050000020000000b0000004571756174696f6e2e3300000000000000000000100000
646 d0cf11e0a1b11ae1000000000000000000000000000000003e000300feff0900060000000000000000000000010000000100000000000000001000000200000001000000feffffff0000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
647 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
648 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
649 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
650 fffffffffffffffffdffffff05000000feffffff0400000006000000fefffffffeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
651 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
652 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
653 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
654 ffffffffffffffffffffffffffffffff52006f006f007400200045006e00740072007900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000016000500ffffffffffffffff0200000002ce020000000000c000000000000046000000000000000000000000a006
655 1cdf674dc00103000000000500000000000001004f006c00650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a000201ffffffffffffffffffffffff00000000000000000000000000000000000000000000000000000000
656 0000000000000000000000001400000000000000030050004900430000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a0002010100000004000000ffffffff0000000000000000000000000000000000000000000000000000
657 00000000000000000000010000004c0000000000000003004d004500540041000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000c000201ffffffffffffffffffffffff000000000000000000000000000000000000000000000000
658 00000000000000000000000003000000c002000000000000feffffff02000000feffffff0400000005000000060000000700000008000000090000000a0000000b0000000c0000000d000000feffffff0f000000fefffffffeffffff1200000013000000feffffffffffffffffffffffffffffffffffffffffffffffffff
659 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
660 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
661 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
662 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff010000020000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004c0000000000000000000800050f0000af04000084080000a80200000000
663 0000000000000000000000000000e8030000e8030000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000800050faf0400000100090000035b01000003001500000000000500
664 00000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c024004a00d1200000026060f001a00ffffffff000010000000c0ffffffb8ffffff600d0000f80300000b00000026060f000c004d617468547970650000e00009000000
665 fa02000010000000000000002200040000002d0100000500000014022002f60a05000000130220029e0c10000000fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d01010008000000320a1a03b60c01000000fa0008000000320ada03b60c01000000fb0008000000320aaa01b60c0100
666 0000fa0008000000320a1a03540a01000000ea0008000000320ada03540a01000000eb0008000000320aaa01540a01000000ea0008000000320a80023607010000002d0008000000320a80027e04010000003d0015000000fb0280fe0000000000009001000000000402001054696d6573204e657720526f6d616e000081
667 040000002d01020004000000f001010008000000320aac03d00b01000000320008000000320a8e01ca0b01000000310008000000320a8002160901000000320008000000320a80024c060100000031000a000000320a80023a0006000000526573756c7415000000fb0280fe000000000000900101000000040200105469
668 6d6573204e657720526f6d616e000081040000002d01010004000000f001020008000000320aac031c0b01000000490008000000320a8e01400b01000000490008000000320a8002620801000000490008000000320a8002c2050100000049000a00000026060f000a00ffffffff01000000000010000000fb0210000700
669 00000000bc02000000000102022253797374656d0000040000002d01020004000000f001010003000000000000000100feff030a0000ffffffff02ce020000000000c000000000000046170000004d6963726f736f6674204571756174696f6e20332e30000c0000004453204571756174696f6e000b0000004571756174
670 696f6e2e3300f439b2710000000000000000000000000000000000000000000000000000000000000000000000000000010043006f006d0070004f0062006a00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000120002000300000005000000ffff
671 ffff0000000000000000000000000000000000000000000000000000000000000000000000000e000000660000000000000003004f0062006a0049006e0066006f0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000012000201ffffffff06000000
672 ffffffff0000000000000000000000000000000000000000000000000000000000000000000000001000000004000000000000004500710075006100740069006f006e0020004e00610074006900760065000000000000000000000000000000000000000000000000000000000000000000000020000200ffffffffffff
673 ffffffffffff000000000000000000000000000000000000000000000000000000000000000000000000110000008c000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffff
674 ffffffffffffffff000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001c0000000200
675 06e5700000000000000078814900a08349000000000003010103010a010281520002816500028173000281750002816c000281740002863d001283490002883100028612221283490002883200029804eb029808ef0306000001030e0000011283490002883100000112834900028832000000000296f0f80296fbf80002
676 9804ef000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
677 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
678 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000105000000000000}{\result {\dn28 {\pict\wmetafile8\picw3845\pich1199\picwgoal2180\pichgoal680
679 0100090000035b0100000300150000000000050000000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c024004a00d1200000026060f001a00ffffffff000010000000c0ffffffb8ffffff600d0000f80300000b00000026060f
680 000c004d617468547970650000e00009000000fa02000010000000000000002200040000002d0100000500000014022002f60a05000000130220029e0c10000000fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d01010008000000320a1a03b60c01000000fa0008000000320ada03b60c
681 01000000fb0008000000320aaa01b60c01000000fa0008000000320a1a03540a01000000ea0008000000320ada03540a01000000eb0008000000320aaa01540a01000000ea0008000000320a80023607010000002d0008000000320a80027e04010000003d0015000000fb0280fe0000000000009001000000000402001054
682 696d6573204e657720526f6d616e000081040000002d01020004000000f001010008000000320aac03d00b01000000320008000000320a8e01ca0b01000000310008000000320a8002160901000000320008000000320a80024c060100000031000a000000320a80023a0006000000526573756c7415000000fb0280fe0000
683 000000009001010000000402001054696d6573204e657720526f6d616e000081040000002d01010004000000f001020008000000320aac031c0b01000000490008000000320a8e01400b01000000490008000000320a8002620801000000490008000000320a8002c2050100000049000a00000026060f000a00ffffffff01
684 000000000010000000fb021000070000000000bc02000000000102022253797374656d0000040000002d01020004000000f001010003000000000000000000000000000000000000000000000000ffffff000000000000000000ffffff00ffffff000000000000000000ffffff00ffffff0000000000ffffff00ffffff0000
685 000000650002000000000000000000000000000f000000000000000000000000000000000000000000000000000000000000000000000000000000000000}}}}}
686 \par \pard \widctlpar
687 \par {\*\bkmkstart _Toc498721662}{\pntext\pard\plain\f5 4.5.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721662}
688 \par \pard\plain \widctlpar \f4\fs20 Output from the invocation of RAP to divide two large integers and thereby produce the integer remainder is reproduced below. Note that the quotient is also supplied.
689 \par
690 \par {\f11\fs16 C:\\>rap % 987243769234692364923 23984236496
691 \par ------------------------------------------------------------------------------
692 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
693 \par ------------------------------------------------------------------------------
694 \par dividend: 987,243,769,234,692,364,923 ( 21 digits)
695 \par ------------------------------------------------------------------------------
696 \par divisor: 23,984,236,496 ( 11 digits)
697 \par ------------------------------------------------------------------------------
698 \par dividend / divisor: 41,162,192,901 ( 11 digits)
699 \par ------------------------------------------------------------------------------
700 \par dividend % divisor: 3,136,050,027 ( 10 digits)
701 \par ------------------------------------------------------------------------------
702 \par RAP execution ends.
703 \par ------------------------------------------------------------------------------
704 \par }
705 \par \page
706 \par {\*\bkmkstart _Ref497419035}{\*\bkmkstart _Toc498721663}{\pntext\pard\plain\b\i\f5 4.6\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Integer Raised To An Integer Power [**]{\*\bkmkend _Ref497419035}
707 {\*\bkmkend _Toc498721663}
708 \par \pard\plain \widctlpar \f4\fs20
709 \par {\*\bkmkstart _Toc498721664}{\pntext\pard\plain\f5 4.6.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721664}
710 \par \pard\plain \widctlpar \f4\fs20 {\b rap ** I1 I2
711 \par }\pard \li720\widctlpar Raises integer I1 to the power of non-negative integer I2.
712 \par \pard \widctlpar
713 \par {\*\bkmkstart _Toc498721665}{\pntext\pard\plain\f5 4.6.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721665}
714 \par \pard\plain \widctlpar \f4\fs20 RAP will exponentiate a non-negative integer to a positive integral value, according to the equation below.
715 \par
716 \par \pard \qc\widctlpar {\pard\plain \qc\widctlpar \f4\fs20 {\object\objemb\objw1359\objh360{\*\objclass Equation.3}{\*\objdata 01050000020000000b0000004571756174696f6e2e3300000000000000000000100000
717 d0cf11e0a1b11ae1000000000000000000000000000000003e000300feff0900060000000000000000000000010000000100000000000000001000000200000001000000feffffff0000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
718 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
719 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
720 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
721 fffffffffffffffffdffffff05000000feffffff0400000006000000fefffffffeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
722 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
723 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
724 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
725 ffffffffffffffffffffffffffffffff52006f006f007400200045006e00740072007900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000016000500ffffffffffffffff0200000002ce020000000000c000000000000046000000000000000000000000a006
726 1cdf674dc00103000000800400000000000001004f006c00650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a000201ffffffffffffffffffffffff00000000000000000000000000000000000000000000000000000000
727 0000000000000000000000001400000000000000030050004900430000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a0002010100000004000000ffffffff0000000000000000000000000000000000000000000000000000
728 00000000000000000000010000004c0000000000000003004d004500540041000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000c000201ffffffffffffffffffffffff000000000000000000000000000000000000000000000000
729 000000000000000000000000030000005402000000000000feffffff02000000feffffff0400000005000000060000000700000008000000090000000a0000000b0000000c000000feffffff0e000000fefffffffeffffff11000000feffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
730 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
731 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
732 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
733 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff010000020000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004c00000000000000000008005e0900007b0200004f050000680100000000
734 0000000000000000000000000000e8030000e80300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000008005e097b0200000100090000032601000002001500000000000500
735 00000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c02400280081200000026060f001a00ffffffff000010000000c0ffffffb7ffffff40080000f70100000b00000026060f000c004d617468547970650000500015000000
736 fb0220ff0000000000009001000000000402001054696d6573204e657720526f6d616e0000d3040000002d01000008000000320af400850701000000320015000000fb0280fe0000000000009001000000000402001054696d6573204e657720526f6d616e0000d3040000002d01010004000000f001000008000000320a
737 a0014c060100000031000a000000320aa0013a0006000000526573756c7415000000fb0220ff0000000000009001010000000402001054696d6573204e657720526f6d616e0000d3040000002d01000004000000f001010008000000320af4000f0701000000490015000000fb0280fe0000000000009001010000000402
738 001054696d6573204e657720526f6d616e0000d3040000002d01010004000000f001000008000000320aa001c20501000000490010000000fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d01000004000000f001010008000000320aa0017e04010000003d000a00000026060f000a00
739 ffffffff01000000000010000000fb021000070000000000bc02000000000102022253797374656d0000040000002d01010004000000f001000003000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100feff030a0000ffffffff02ce02000000
740 0000c000000000000046170000004d6963726f736f6674204571756174696f6e20332e30000c0000004453204571756174696f6e000b0000004571756174696f6e2e3300f439b271000000000000000000000000000000000000000000000000000000000000000000000000000000000300000000000000000000000000
741 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010043006f006d0070004f0062006a00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000120002000300000005000000ffff
742 ffff0000000000000000000000000000000000000000000000000000000000000000000000000d000000660000000000000003004f0062006a0049006e0066006f0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000012000201ffffffff06000000
743 ffffffff0000000000000000000000000000000000000000000000000000000000000000000000000f00000004000000000000004500710075006100740069006f006e0020004e00610074006900760065000000000000000000000000000000000000000000000000000000000000000000000020000200ffffffffffff
744 ffffffffffff0000000000000000000000000000000000000000000000000000000000000000000000001000000060000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffff
745 ffffffffffffffff0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001c000000020006e54400000000000000989c4900c09c49000000000003010103010a010281520002816500028173000281750002816c000281740002863d0012834900028831
746 00030f00000b1101128349000288320000000a029804ef00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
747 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
748 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
749 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000105000000000000}{\result {\dn10 {\pict\wmetafile8\picw2398\pich635\picwgoal1359\pichgoal360
750 010009000003260100000200150000000000050000000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c02400280081200000026060f001a00ffffffff000010000000c0ffffffb7ffffff40080000f70100000b00000026060f
751 000c004d617468547970650000500015000000fb0220ff0000000000009001000000000402001054696d6573204e657720526f6d616e0000d3040000002d01000008000000320af400850701000000320015000000fb0280fe0000000000009001000000000402001054696d6573204e657720526f6d616e0000d304000000
752 2d01010004000000f001000008000000320aa0014c060100000031000a000000320aa0013a0006000000526573756c7415000000fb0220ff0000000000009001010000000402001054696d6573204e657720526f6d616e0000d3040000002d01000004000000f001010008000000320af4000f0701000000490015000000fb
753 0280fe0000000000009001010000000402001054696d6573204e657720526f6d616e0000d3040000002d01010004000000f001000008000000320aa001c20501000000490010000000fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d01000004000000f001010008000000320aa0017e04
754 010000003d000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02000000000102022253797374656d0000040000002d01010004000000f00100000300000000000000000000000000000000000000000000006200000000000000000000343937000000000000000000000000000000
755 00000222020000000000000000000276020000000000ffffff0000000000ffffff00ffffff00ffffff00000000000000000000000000321325000000000040958c00000000000000000000000000000000}}}}}
756 \par \pard \widctlpar
757 \par The algorithm applied is to examine the bit pattern of the exponent and to repeatedly square the argument and to selectively multiply in the repeated square. For example, I1{\super 5} can be rewritten as (I1{\super 4})(I1{\super 1}
758 ), and calculated using the following steps:
759 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}\pard \fi-360\li1800\widctlpar{\*\pn \pnlvlblt\pnf1\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}multiplier := I1
760 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}result := 1
761 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}result := result * multiplier
762 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}multiplier := multiplier * multiplier
763 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}multiplier := multiplier * multiplier
764 \par {\pntext\pard\plain\f1\fs20 \'b7\tab}result := result * multiplier
765 \par \pard \widctlpar
766 \par {\*\bkmkstart _Toc498721666}{\pntext\pard\plain\f5 4.6.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721666}
767 \par \pard\plain \widctlpar \f4\fs20
768 \par {\f11\fs16 C:\\>rap ** 117 54
769 \par ------------------------------------------------------------------------------
770 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
771 \par ------------------------------------------------------------------------------
772 \par arg: 117 ( 3 digits)
773 \par ------------------------------------------------------------------------------
774 \par exponent: 54 ( 2 digits)
775 \par ------------------------------------------------------------------------------
776 \par arg ** exponent: 4,808, ( 112 digits)
777 \par 797,999,524,921,631,067,632,458,892,
778 \par 309,152,880,430,748,856,242,309,594,
779 \par 970,691,360,198,993,667,007,419,994,
780 \par 123,210,723,254,173,781,195,143,529
781 \par ------------------------------------------------------------------------------
782 \par RAP execution ends.
783 \par ------------------------------------------------------------------------------
784 \par }
785 \par \page
786 \par {\*\bkmkstart _Toc498721667}{\pntext\pard\plain\b\i\f5 4.7\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Greatest Common Divisor [GCD]{\*\bkmkend _Toc498721667}
787 \par \pard\plain \widctlpar \f4\fs20
788 \par {\*\bkmkstart _Toc498721668}{\pntext\pard\plain\f5 4.7.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721668}
789 \par \pard\plain \widctlpar \f4\fs20 {\b rap gcd I1 I2
790 \par }\pard \li720\widctlpar Calculates the g.c.d. of positive integers I1 and I2 using Euclid\rquote s algorithm.
791 \par \pard \widctlpar
792 \par {\*\bkmkstart _Toc498721669}{\pntext\pard\plain\f5 4.7.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721669}
793 \par \pard\plain \widctlpar \f4\fs20 Calculates the greatest common divisor of two positive integers using Euclid\rquote s algorithm. Euclid\rquote s algorithm and its proof are not discussed explicitly in the TOMS paper. Olds\rquote
794 CF book explains the relationship between Euclid\rquote s algorithm and the apparatus of continued fractions, and supplies a proof. A search of the web would also reveal many pages that discuss Euclid\rquote s g.c.d. algorithm in great detail.
795 \par
796 \par {\*\bkmkstart _Toc498721670}{\pntext\pard\plain\f5 4.7.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721670}
797 \par \pard\plain \widctlpar \f4\fs20 Output from the invocation of RAP to calculate the g.c.d. of two integers is reproduced below.
798 \par
799 \par {\f11\fs16 C:\\ >rap gcd 13433 34048
800 \par ------------------------------------------------------------------------------
801 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
802 \par ------------------------------------------------------------------------------
803 \par arg1: 13,433 ( 5 digits)
804 \par ------------------------------------------------------------------------------
805 \par arg2: 34,048 ( 5 digits)
806 \par ------------------------------------------------------------------------------
807 \par gcd(arg1, arg2): 133 ( 3 digits)
808 \par ------------------------------------------------------------------------------
809 \par RAP execution ends.
810 \par ------------------------------------------------------------------------------
811 \par }
812 \par \page
813 \par {\*\bkmkstart _Ref497594880}{\*\bkmkstart _Toc498721671}{\pntext\pard\plain\b\i\f5 4.8\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Decimal Approximation [DAP]{\*\bkmkend _Ref497594880}
814 {\*\bkmkend _Toc498721671}
815 \par \pard\plain \widctlpar \f4\fs20
816 \par {\*\bkmkstart _Toc498721672}{\pntext\pard\plain\f5 4.8.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721672}
817 \par \pard\plain \widctlpar \f4\fs20 {\b rap dap R1
818 \par }\pard \li720\widctlpar Performs long division on a rational number R1 to obtain the more familiar decimal approximation. By default, four lines ({{\field{\*\fldinst SYMBOL 187 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}}
819 100) of significant figures beyond the decimal point are displayed.
820 \par \pard \widctlpar {\b rap dap R1 D}
821 \par \pard \li720\widctlpar Displays R1 as a rational approximation with D as the denominator.{\b
822 \par }\pard \widctlpar
823 \par {\*\bkmkstart _Toc498721673}{\pntext\pard\plain\f5 4.8.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721673}
824 \par \pard\plain \widctlpar \f4\fs20 The DAP command provides crude functionality to display a more familiar and intuitive decimal approximation of a rational number.
825 \par
826 \par Let h/k be the rational number to be displayed in a more familiar form, and let N/D be an approximation to h/k such that:
827 \par
828 \par \pard \qc\widctlpar {\pard\plain \qc\widctlpar \f4\fs20 {\object\objemb\objw1520\objh620{\*\objclass Equation.3}{\*\objdata 01050000020000000b0000004571756174696f6e2e3300000000000000000000100000
829 d0cf11e0a1b11ae1000000000000000000000000000000003e000300feff0900060000000000000000000000010000000100000000000000001000000200000001000000feffffff0000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
830 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
831 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
832 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
833 fffffffffffffffffdffffff05000000feffffff0400000006000000fefffffffeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
834 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
835 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
836 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
837 ffffffffffffffffffffffffffffffff52006f006f007400200045006e00740072007900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000016000500ffffffffffffffff0200000002ce020000000000c000000000000046000000000000000000000000c0a7
838 23df674dc00103000000800400000000000001004f006c00650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a000201ffffffffffffffffffffffff00000000000000000000000000000000000000000000000000000000
839 0000000000000000000000001400000000000000030050004900430000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a0002010100000004000000ffffffff0000000000000000000000000000000000000000000000000000
840 00000000000000000000010000004c0000000000000003004d004500540041000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000c000201ffffffffffffffffffffffff000000000000000000000000000000000000000000000000
841 000000000000000000000000030000007402000000000000feffffff02000000feffffff0400000005000000060000000700000008000000090000000a0000000b0000000c000000feffffff0e000000fefffffffeffffff11000000feffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
842 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
843 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
844 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
845 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff010000020000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004c0000000000000000000800790a000045040000f00500006c0200000000
846 0000000000000000000000000000e8030000e8030000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000800790a450400000100090000033501000003001500000000000500
847 00000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c02e00380091200000026060f001a00ffffffff000010000000c0ffffffb8ffffff40090000980300000b00000026060f000c004d617468547970650000c00009000000
848 fa02000010000000000000002200040000002d010000050000001402000240000500000013020002a60105000000140200025003050000001302000244040500000014020002ee0505000000130200023a0915000000fb0280fe0000000000009001010000000402001054696d6573204e657720526f6d616e0000870400
849 00002d01010008000000320a8c03100701000000440008000000320a6e011a06010000004e0008000000320a8c036a03010000006b0008000000320a6e016d0301000000680008000000320a8c036f0001000000440008000000320a6e016c00010000004e0015000000fb0280fe00000000000090010000000004020010
850 54696d6573204e657720526f6d616e000087040000002d01020004000000f001010008000000320a6e018a0801000000310010000000fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d01010004000000f001020008000000320a6e018e07010000002b0008000000320a6002b0040100
851 00003c0008000000320a6002120201000000a3000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02000000000102022253797374656d0000040000002d01020004000000f001010003000000000000000000000000000000000000000100feff030a0000ffffffff02ce02000000
852 0000c000000000000046170000004d6963726f736f6674204571756174696f6e20332e30000c0000004453204571756174696f6e000b0000004571756174696f6e2e3300f439b271000000000000000000000000000000000000000000000000000000000000000000000000000000000300000000000000000000000000
853 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010043006f006d0070004f0062006a00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000120002000300000005000000ffff
854 ffff0000000000000000000000000000000000000000000000000000000000000000000000000d000000660000000000000003004f0062006a0049006e0066006f0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000012000201ffffffff06000000
855 ffffffff0000000000000000000000000000000000000000000000000000000000000000000000000f00000004000000000000004500710075006100740069006f006e0020004e00610074006900760065000000000000000000000000000000000000000000000000000000000000000000000020000200ffffffffffff
856 ffffffffffff0000000000000000000000000000000000000000000000000000000000000000000000001000000068000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffff
857 ffffffffffffffff0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001c0000000200b2d84c00000000000000246e4900207a49000000000003010103010a01030e00000112834e00000112834400000002866422030e00000112836800000112836b
858 00000002863c00030e00000112834e0002862b0002883100000112834400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
859 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
860 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
861 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000105000000000000}{\result {\dn24 {\pict\wmetafile8\picw2681\pich1093\picwgoal1520\pichgoal620
862 010009000003350100000300150000000000050000000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c02e00380091200000026060f001a00ffffffff000010000000c0ffffffb8ffffff40090000980300000b00000026060f
863 000c004d617468547970650000c00009000000fa02000010000000000000002200040000002d010000050000001402000240000500000013020002a60105000000140200025003050000001302000244040500000014020002ee0505000000130200023a0915000000fb0280fe000000000000900101000000040200105469
864 6d6573204e657720526f6d616e000087040000002d01010008000000320a8c03100701000000440008000000320a6e011a06010000004e0008000000320a8c036a03010000006b0008000000320a6e016d0301000000680008000000320a8c036f0001000000440008000000320a6e016c00010000004e0015000000fb0280
865 fe0000000000009001000000000402001054696d6573204e657720526f6d616e000087040000002d01020004000000f001010008000000320a6e018a0801000000310010000000fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d01010004000000f001020008000000320a6e018e070100
866 00002b0008000000320a6002b004010000003c0008000000320a6002120201000000a3000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02000000000102022253797374656d0000040000002d01020004000000f001010003000000000000000000000000000000000000a0000000
867 000000400000c903c900c903c900ffffff0000000000000000000101000000000000000000000000000088450000370d5f000000000038343800000000000000000000000000000000}}}}}
868 \par \pard \widctlpar
869 \par The choice of N given below will meet this inequality.
870 \par
871 \par \pard \qc\widctlpar {\pard\plain \qc\widctlpar \f4\fs20 {\object\objemb\objw1080\objh680{\*\objclass Equation.3}{\*\objdata 01050000020000000b0000004571756174696f6e2e3300000000000000000000100000
872 d0cf11e0a1b11ae1000000000000000000000000000000003e000300feff0900060000000000000000000000010000000100000000000000001000000200000001000000feffffff0000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
873 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
874 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
875 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
876 fffffffffffffffffdffffff05000000feffffff0400000006000000fefffffffeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
877 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
878 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
879 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
880 ffffffffffffffffffffffffffffffff52006f006f007400200045006e00740072007900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000016000500ffffffffffffffff0200000002ce020000000000c000000000000046000000000000000000000000c0a7
881 23df674dc00103000000400400000000000001004f006c00650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a000201ffffffffffffffffffffffff00000000000000000000000000000000000000000000000000000000
882 0000000000000000000000001400000000000000030050004900430000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a0002010100000004000000ffffffff0000000000000000000000000000000000000000000000000000
883 00000000000000000000010000004c0000000000000003004d004500540041000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000c000201ffffffffffffffffffffffff000000000000000000000000000000000000000000000000
884 000000000000000000000000030000001802000000000000feffffff02000000feffffff0400000005000000060000000700000008000000090000000a0000000b000000feffffff0d000000fefffffffeffffff10000000feffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
885 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
886 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
887 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
888 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff010000020000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004c000000000000000000080071070000af04000038040000a80200000000
889 0000000000000000000000000000e8030000e80300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000008007107af0400000100090000030401000003001500000000000500
890 00000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c024004c0061200000026060f001a00ffffffff000010000000c0ffffffb8ffffff80060000f80300000b00000026060f000c004d617468547970650000e00009000000
891 fa02000010000000000000002200040000002d0100000500000014022002b8030500000013022002c60510000000fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d01010008000000320a1a03de0501000000fa0008000000320ada03de0501000000fb0008000000320aaa01de050100
892 0000fa0008000000320a1a03160301000000ea0008000000320ada03160301000000eb0008000000320aaa01160301000000ea0008000000320a8002d801010000003d0015000000fb0280fe0000000000009001010000000402001054696d6573204e657720526f6d616e0000b0040000002d01020004000000f0010100
893 08000000320aac035f04010000006b0008000000320a8e01de0302000000446808000000320a80024c00010000004e000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02000000000102022253797374656d0000040000002d01010004000000f001020003000000000000000000
894 788e4900000000000000000000000000000000000000000000000000000000000000000000000000000000000100feff030a0000ffffffff02ce020000000000c000000000000046170000004d6963726f736f6674204571756174696f6e20332e30000c0000004453204571756174696f6e000b0000004571756174696f
895 6e2e3300f439b2710000000000000000000000000000000000000000000000000000000000000000000000000000000003000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001c0000000200b2d83800000000000000
896 5c9d49008c6f49000000000003010103010a0112834e0002863d000306000001030e0000011283440012836800000112010043006f006d0070004f0062006a00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000120002000300000005000000ffff
897 ffff0000000000000000000000000000000000000000000000000000000000000000000000000c000000660000000000000003004f0062006a0049006e0066006f0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000012000201ffffffff06000000
898 ffffffff0000000000000000000000000000000000000000000000000000000000000000000000000e00000004000000000000004500710075006100740069006f006e0020004e00610074006900760065000000000000000000000000000000000000000000000000000000000000000000000020000200ffffffffffff
899 ffffffffffff0000000000000000000000000000000000000000000000000000000000000000000000000f00000054000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffff
900 ffffffffffffffff000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000836b000000000296f0f80296fbf80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
901 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
902 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
903 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
904 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000105000000000000}{\result {\dn28 {\pict\wmetafile8\picw1905\pich1199\picwgoal1080\pichgoal680
905 010009000003040100000300150000000000050000000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c024004c0061200000026060f001a00ffffffff000010000000c0ffffffb8ffffff80060000f80300000b00000026060f
906 000c004d617468547970650000e00009000000fa02000010000000000000002200040000002d0100000500000014022002b8030500000013022002c60510000000fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d01010008000000320a1a03de0501000000fa0008000000320ada03de05
907 01000000fb0008000000320aaa01de0501000000fa0008000000320a1a03160301000000ea0008000000320ada03160301000000eb0008000000320aaa01160301000000ea0008000000320a8002d801010000003d0015000000fb0280fe0000000000009001010000000402001054696d6573204e657720526f6d616e0000
908 b0040000002d01020004000000f001010008000000320aac035f04010000006b0008000000320a8e01de0302000000446808000000320a80024c00010000004e000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02000000000102022253797374656d0000040000002d0101000400
909 0000f001020003000000000000000000788e490000000000000000000002010000ffffff0000400000060f000000ffff00ffffff0000000000000000000210000000000000000000000000000071000000000000006d000000d002d000010200000000000001010000000000}}}}}
910 \par \pard \widctlpar
911 \par DAP operates by accepting h, k, and D as i
912 nput parameters, and choosing N as specified above. Note that if D is chosen to be a power of ten, the digits of N will give the decimal form of a rational number, where the last digit will be truncated rather than rounded (note the form of the inequalit
913 y above).
914 \par
915 \par DAP is described as crude because it is up to the user of RAP to put the decimal point in the right place, and because it will not truncate trailing zeros or round.
916 \par
917 \par The power of ten specified effectively sets the \ldblquote number of decimal places\rdblquote that
918 the decimal approximation is displayed to. A value of D=1e2 will display two decimal places, a value of 1e100 will display 100 decimal places, etc. It would be possible to specify D as other than a power of ten, and in some cases this may be give useful
919 information. For example, in microcontroller work, if one is performing h/2{\super q} scaling, choosing D=2{\super q} would give the required value of h.
920 \par
921 \par
922 \par {\*\bkmkstart _Toc498721674}{\pntext\pard\plain\f5 4.8.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721674}
923 \par \pard\plain \widctlpar \f4\fs20 Output from the DAP command when applied to the rational number 2,043,926/7 is reproduced below. Note that the default choice of D is 10{\super 108}
924 (the right length so that the imaginary decimal point occurs between lines, which allows easier interpretation). From that output below, it can be seen that 2,043,926/7 {{\field{\*\fldinst SYMBOL 187 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}}
925 291,989.428571428571\'85 .
926 \par
927 \par {\f11\fs16 C:\\>rap dap 2043926/7}
928 \par {\f11\fs16 ------------------------------------------------------------------------------
929 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
930 \par ------------------------------------------------------------------------------
931 \par arg_h: 2,043,926 ( 7 digits)
932 \par ------------------------------------------------------------------------------
933 \par arg_k: 7 ( 1 digit)
934 \par ------------------------------------------------------------------------------
935 \par N: 291,989, ( 114 digits)
936 \par 428,571,428,571,428,571,428,571,428,
937 \par 571,428,571,428,571,428,571,428,571,
938 \par 428,571,428,571,428,571,428,571,428,
939 \par 571,428,571,428,571,428,571,428,571
940 \par ------------------------------------------------------------------------------
941 \par D: 1, ( 109 digits)
942 \par 000,000,000,000,000,000,000,000,000,
943 \par 000,000,000,000,000,000,000,000,000,
944 \par 000,000,000,000,000,000,000,000,000,
945 \par 000,000,000,000,000,000,000,000,000
946 \par ------------------------------------------------------------------------------
947 \par RAP execution ends.
948 \par ------------------------------------------------------------------------------
949 \par }
950 \par The value of D used can be changed by invoking DAP with a third parameter, as shown in the output below. In the output below, 216 decimal places are displayed.
951 \par
952 \par {\f11\fs16 C:\\>rap dap 2043926/7 1e216
953 \par ------------------------------------------------------------------------------
954 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
955 \par ------------------------------------------------------------------------------
956 \par arg_h: 2,043,926 ( 7 digits)
957 \par ------------------------------------------------------------------------------
958 \par arg_k: 7 ( 1 digit)
959 \par ------------------------------------------------------------------------------
960 \par N: 291,989, ( 222 digits)
961 \par 428,571,428,571,428,571,428,571,428,
962 \par 571,428,571,428,571,428,571,428,571,
963 \par 428,571,428,571,428,571,428,571,428,
964 \par 571,428,571,428,571,428,571,428,571,
965 \par 428,571,428,571,428,571,428,571,428,
966 \par 571,428,571,428,571,428,571,428,571,
967 \par 428,571,428,571,428,571,428,571,428,
968 \par 571,428,571,428,571,428,571,428,571
969 \par ------------------------------------------------------------------------------
970 \par D: 1, ( 217 digits)
971 \par 000,000,000,000,000,000,000,000,000,
972 \par 000,000,000,000,000,000,000,000,000,
973 \par 000,000,000,000,000,000,000,000,000,
974 \par 000,000,000,000,000,000,000,000,000,
975 \par 000,000,000,000,000,000,000,000,000,
976 \par 000,000,000,000,000,000,000,000,000,
977 \par 000,000,000,000,000,000,000,000,000,
978 \par 000,000,000,000,000,000,000,000,000
979 \par ------------------------------------------------------------------------------
980 \par RAP execution ends.
981 \par ------------------------------------------------------------------------------
982 \par }
983 \par Finally, DAP can also be used to with denominators not an integral power of 10. In the example invocation below, DAP is invoked with a denominator of 2{\super 16}=65536. The output below shows that (for microcontro
984 ller work), if a rational approximation were performed by multiplying by an integer and shifting right a number of bits, using a multiplier of 28,036 and then shifting right by 16 bits would be a good approximation of 3/7.
985 \par
986 \par {\f11\fs16 ------------------------------------------------------------------------------
987 \par RAP execution ends.
988 \par ------------------------------------------------------------------------------
989 \par
990 \par C:\\>rap dap 3/7 65536
991 \par ------------------------------------------------------------------------------
992 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
993 \par ------------------------------------------------------------------------------
994 \par arg_h: 3 ( 1 digit)
995 \par ------------------------------------------------------------------------------
996 \par arg_k: 7 ( 1 digit)
997 \par ------------------------------------------------------------------------------
998 \par N: 28,086 ( 5 digits)
999 \par ------------------------------------------------------------------------------
1000 \par D: 65,536 ( 5 digits)
1001 \par ------------------------------------------------------------------------------
1002 \par RAP execution ends.
1003 \par ------------------------------------------------------------------------------
1004 \par }
1005 \par \page
1006 \par {\*\bkmkstart _Toc498721675}{\pntext\pard\plain\b\i\f5 4.9\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Rational Number Sum [+]{\*\bkmkend _Toc498721675}
1007 \par \pard\plain \widctlpar \f4\fs20
1008 \par {\*\bkmkstart _Toc498721676}{\pntext\pard\plain\f5 4.9.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721676}
1009 \par \pard\plain \widctlpar \f4\fs20 {\b rap + R1 R2
1010 \par }\pard \li720\widctlpar Forms the sum of R1 and R2, presenting the sum in lowest terms.
1011 \par \pard \widctlpar
1012 \par {\*\bkmkstart _Toc498721677}{\pntext\pard\plain\f5 4.9.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721677}
1013 \par \pard\plain \widctlpar \f4\fs20 Forms the sum of two arbitrary rational numbers using standard algebraic techniques. The sum is presented as a rational number in lowest terms.
1014 \par
1015 \par {\*\bkmkstart _Toc498721678}{\pntext\pard\plain\f5 4.9.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721678}
1016 \par \pard\plain \widctlpar \f4\fs20 Output from the invocation of RAP to form the sum of 7/91 and 3.14 is reproduced below. Note that the inputs and outputs are presented in reduced form.
1017 \par
1018 \par {\f11\fs16 C:\\ >rap + 7/91 3.14
1019 \par ------------------------------------------------------------------------------
1020 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
1021 \par ------------------------------------------------------------------------------
1022 \par arg1_h: 1 ( 1 digit)
1023 \par ------------------------------------------------------------------------------
1024 \par arg1_k: 13 ( 2 digits)
1025 \par ------------------------------------------------------------------------------
1026 \par arg2_h: 157 ( 3 digits)
1027 \par ------------------------------------------------------------------------------
1028 \par arg2_k: 50 ( 2 digits)
1029 \par ------------------------------------------------------------------------------
1030 \par result_h: 2,091 ( 4 digits)
1031 \par ------------------------------------------------------------------------------
1032 \par result_k: 650 ( 3 digits)
1033 \par ------------------------------------------------------------------------------
1034 \par RAP execution ends.
1035 \par ------------------------------------------------------------------------------
1036 \par }
1037 \par \page
1038 \par {\*\bkmkstart _Toc498721679}{\pntext\pard\plain\b\i\f5 4.10\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Rational Number Difference [-]{\*\bkmkend _Toc498721679}
1039 \par \pard\plain \widctlpar \f4\fs20
1040 \par {\*\bkmkstart _Toc498721680}{\pntext\pard\plain\f5 4.10.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721680}
1041 \par \pard\plain \widctlpar \f4\fs20 {\b rap - R1 R2
1042 \par }\pard \li720\widctlpar Subtracts R2 from R1, presenting the difference in lowest terms.
1043 \par \pard \widctlpar
1044 \par {\*\bkmkstart _Toc498721681}{\pntext\pard\plain\f5 4.10.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721681}
1045 \par \pard\plain \widctlpar \f4\fs20 Forms the difference of two arbitrary rational numbers, using standard algebraic techniques. The sum is presented as a rational number in lowest terms.
1046 \par
1047 \par {\*\bkmkstart _Toc498721682}{\pntext\pard\plain\f5 4.10.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721682}
1048 \par \pard\plain \widctlpar \f4\fs20 Output from the invocation of RAP to subtract two rational numbers is reproduced below.
1049 \par
1050 \par {\f11\fs16 C:\\>rap - 92493234924/233472634 872434/23643
1051 \par ------------------------------------------------------------------------------
1052 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
1053 \par ------------------------------------------------------------------------------
1054 \par arg1_h: 46,246,617,462 ( 11 digits)
1055 \par ------------------------------------------------------------------------------
1056 \par arg1_k: 116,736,317 ( 9 digits)
1057 \par ------------------------------------------------------------------------------
1058 \par arg2_h: 872,434 ( 6 digits)
1059 \par ------------------------------------------------------------------------------
1060 \par arg2_k: 23,643 ( 5 digits)
1061 \par ------------------------------------------------------------------------------
1062 \par result_h: 991,564,044,668,488 ( 15 digits)
1063 \par ------------------------------------------------------------------------------
1064 \par result_k: 2,759,996,742,831 ( 13 digits)
1065 \par ------------------------------------------------------------------------------
1066 \par RAP execution ends.
1067 \par ------------------------------------------------------------------------------
1068 \par }
1069 \par \page
1070 \par {\*\bkmkstart _Toc498721683}{\pntext\pard\plain\b\i\f5 4.11\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Rational Number Product [*]{\*\bkmkend _Toc498721683}
1071 \par \pard\plain \widctlpar \f4\fs20
1072 \par {\*\bkmkstart _Toc498721684}{\pntext\pard\plain\f5 4.11.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721684}
1073 \par \pard\plain \widctlpar \f4\fs20 {\b rap * R1 R2
1074 \par }\pard \li720\widctlpar Forms the product of R1 and R2, presenting the product in lowest terms.
1075 \par \pard \widctlpar
1076 \par {\*\bkmkstart _Toc498721685}{\pntext\pard\plain\f5 4.11.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721685}
1077 \par \pard\plain \widctlpar \f4\fs20 Forms the product of two arbitrary rational numbers, using standard algebraic techniques. The product is presented as a rational number in lowest terms.
1078 \par
1079 \par {\*\bkmkstart _Toc498721686}{\pntext\pard\plain\f5 4.11.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721686}
1080 \par \pard\plain \widctlpar \f4\fs20 Output from the invocation of RAP to multiply two rational numbers is reproduced below.
1081 \par
1082 \par {\f11\fs16 C:\\>rap * 0.14 7/3
1083 \par ------------------------------------------------------------------------------
1084 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
1085 \par ------------------------------------------------------------------------------
1086 \par arg1_h: 7 ( 1 digit)
1087 \par ------------------------------------------------------------------------------
1088 \par arg1_k: 50 ( 2 digits)
1089 \par ------------------------------------------------------------------------------
1090 \par arg2_h: 7 ( 1 digit)
1091 \par ------------------------------------------------------------------------------
1092 \par arg2_k: 3 ( 1 digit)
1093 \par ------------------------------------------------------------------------------
1094 \par result_h: 49 ( 2 digits)
1095 \par ------------------------------------------------------------------------------
1096 \par result_k: 150 ( 3 digits)
1097 \par ------------------------------------------------------------------------------
1098 \par RAP execution ends.
1099 \par ------------------------------------------------------------------------------
1100 \par }
1101 \par \page
1102 \par {\*\bkmkstart _Toc498721687}{\pntext\pard\plain\b\i\f5 4.12\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Rational Number Quotient [/]{\*\bkmkend _Toc498721687}
1103 \par \pard\plain \widctlpar \f4\fs20
1104 \par {\*\bkmkstart _Toc498721688}{\pntext\pard\plain\f5 4.12.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721688}
1105 \par \pard\plain \widctlpar \f4\fs20 {\b rap / R1 R2
1106 \par }\pard \li720\widctlpar Divides R1 by R2, presenting the quotient in lowest terms.
1107 \par \pard \widctlpar
1108 \par {\*\bkmkstart _Toc498721689}{\pntext\pard\plain\f5 4.12.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721689}
1109 \par \pard\plain \widctlpar \f4\fs20 Forms the quotient of two arbitrary rational numbers, which will also be a rational number. The quotient is presented as a rational number in lowest terms.
1110 \par
1111 \par The quotient is rephrased in integer terms:
1112 \par
1113 \par \pard \qc\widctlpar {\pard\plain \qc\widctlpar \f4\fs20 {\object\objemb\objw859\objh1240{\*\objclass Equation.3}{\*\objdata 01050000020000000b0000004571756174696f6e2e3300000000000000000000100000
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1146 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000105000000000000}{\result {\dn56 {\pict\wmetafile8\picw1516\pich2187\picwgoal859\pichgoal1240
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1151 fb021000070000000000bc02000000000102022253797374656d0000040000002d01020004000000f0010300030000000000000000ffffff00ffffff0030393700018000003439370036353300000001006f6334003439370001460100ffffff00ffffff00000000000000000000000000d002d000ffffff0000000000ffff
1152 ff000000000000000000ffffff00ffffff}}}}}
1153 \par \pard \widctlpar
1154 \par {\*\bkmkstart _Toc498721690}{\pntext\pard\plain\f5 4.12.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721690}
1155 \par \pard\plain \widctlpar \f4\fs20 Output from the invocation of RAP to form the rational quotient of two rational numbers is reproduced below.
1156 \par
1157 \par {\f11\fs16 C:\\>rap / 3.14 -157/2
1158 \par ------------------------------------------------------------------------------
1159 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
1160 \par ------------------------------------------------------------------------------
1161 \par arg1_h: 157 ( 3 digits)
1162 \par ------------------------------------------------------------------------------
1163 \par arg1_k: 50 ( 2 digits)
1164 \par ------------------------------------------------------------------------------
1165 \par arg2_h: - 157 ( 3 digits)
1166 \par ------------------------------------------------------------------------------
1167 \par arg2_k: 2 ( 1 digit)
1168 \par ------------------------------------------------------------------------------
1169 \par result_h: - 1 ( 1 digit)
1170 \par ------------------------------------------------------------------------------
1171 \par result_k: 25 ( 2 digits)
1172 \par ------------------------------------------------------------------------------
1173 \par RAP execution ends.
1174 \par ------------------------------------------------------------------------------
1175 \par }
1176 \par \page
1177 \par {\*\bkmkstart _Toc498721691}{\pntext\pard\plain\b\i\f5 4.13\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Rational Number Raised To An Integer Power [**]{\*\bkmkend _Toc498721691}
1178 \par \pard\plain \widctlpar \f4\fs20
1179 \par {\*\bkmkstart _Toc498721692}{\pntext\pard\plain\f5 4.13.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721692}
1180 \par \pard\plain \widctlpar \f4\fs20 {\b rap ** R1 I1
1181 \par }\pard \li720\widctlpar Raises R1 to the positive power of I1. The result is presented in lowest terms.
1182 \par \pard \widctlpar
1183 \par {\*\bkmkstart _Toc498721693}{\pntext\pard\plain\f5 4.13.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Algorithm Description{\*\bkmkend _Toc498721693}
1184 \par \pard\plain \widctlpar \f4\fs20 Raises a non-negative rational number to a positive integral power. The result is presented in lowest terms. First, the rational number is reduced so that the numerator and denominator are coprime. Then, the nu
1185 merator and denominator (each integers) are each raised to an integer power using the algorithm described in section {\field{\*\fldinst REF _Ref497419035 \\n }{\fldrslt 4.6}}.
1186 \par
1187 \par {\*\bkmkstart _Toc498721694}{\pntext\pard\plain\f5 4.13.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721694}
1188 \par \pard\plain \widctlpar \f4\fs20 Output from the invocation of RAP to raise a rational number to an integer power is reproduced below.
1189 \par
1190 \par {\f11\fs16 C:\\>rap ** 3.14 21
1191 \par ------------------------------------------------------------------------------
1192 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
1193 \par ------------------------------------------------------------------------------
1194 \par arg_h: 157 ( 3 digits)
1195 \par ------------------------------------------------------------------------------
1196 \par arg_k: 50 ( 2 digits)
1197 \par ------------------------------------------------------------------------------
1198 \par exponent: 21 ( 2 digits)
1199 \par ------------------------------------------------------------------------------
1200 \par arg_h ** exponent: 12,998,483,899,984,396,303, ( 47 digits)
1201 \par 172,397,297,010,470,279,141,762,157
1202 \par ------------------------------------------------------------------------------
1203 \par arg_k ** exponent: 476,837,158, ( 36 digits)
1204 \par 203,125,000,000,000,000,000,000,000
1205 \par ------------------------------------------------------------------------------
1206 \par RAP execution ends.
1207 \par ------------------------------------------------------------------------------
1208 \par }
1209 \par \page
1210 \par {\*\bkmkstart _Toc498721695}{\pntext\pard\plain\b\i\f5 4.14\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Continued Fraction Partial Quotients And Convergents Of A Rational Number [CF]
1211 {\*\bkmkend _Toc498721695}
1212 \par \pard\plain \widctlpar \f4\fs20
1213 \par {\*\bkmkstart _Toc498721696}{\pntext\pard\plain\f5 4.14.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721696}
1214 \par \pard\plain \widctlpar \f4\fs20 {\b rap cf R1
1215 \par }\pard \li720\widctlpar Forms the partial quotients and convergents of non-negative rational number R1.
1216 \par \pard \widctlpar
1217 \par {\*\bkmkstart _Toc498721697}{\pntext\pard\plain\f5 4.14.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721697}
1218 \par \pard\plain \widctlpar \f4\fs20 Forms the continued fraction partial quotients of an arbitrary non-negative rational number. The partial quotients are then used to calculate the convergents.
1219 \par
1220 \par {\*\bkmkstart _Toc498721698}{\pntext\pard\plain\f5 4.14.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721698}
1221 \par \pard\plain \widctlpar \f4\fs20 The output below shows the invocation of RAP to form the partial quotients and convergents of a 12-digit rational approximation to {{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}} (obtained using the
1222 \ldblquote {{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}}\rdblquote -key on a pocket calculator).{\cs19\super \chftn {\footnote \pard\plain \s18\widctlpar \f4\fs20 {\cs19\super \chftn }
1223 The approximation used in this example is relatively crude (12 digits). Many web sites list the value of {{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}}
1224 to thousands of decimal places, and when using RAP with Farey series of large order it is recommended to use a much more precise value of {{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}}
1225 . RAP has been tested to accommodate about 460 digits of {{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}}, resulting in about 900 partial quotients. This precision should be adequate for practical rational approximations.}}
1226 \par
1227 \par {\f11\fs16 C:\\>rap cf 3.14159265359
1228 \par ------------------------------------------------------------------------------
1229 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
1230 \par ------------------------------------------------------------------------------
1231 \par ************************ Inputs To CF Calculation ************************
1232 \par ------------------------------------------------------------------------------
1233 \par h_in: 314,159,265,359 ( 12 digits)
1234 \par ------------------------------------------------------------------------------
1235 \par k_in: 100,000,000,000 ( 12 digits)
1236 \par ------------------------------------------------------------------------------
1237 \par ************************** CF Partial Quotients **************************
1238 \par ------------------------------------------------------------------------------
1239 \par a(0): 3 ( 1 digit)
1240 \par ------------------------------------------------------------------------------
1241 \par a(1): 7 ( 1 digit)
1242 \par ------------------------------------------------------------------------------
1243 \par a(2): 15 ( 2 digits)
1244 \par ------------------------------------------------------------------------------
1245 \par a(3): 1 ( 1 digit)
1246 \par ------------------------------------------------------------------------------
1247 \par a(4): 292 ( 3 digits)
1248 \par ------------------------------------------------------------------------------
1249 \par a(5): 1 ( 1 digit)
1250 \par ------------------------------------------------------------------------------
1251 \par a(6): 1 ( 1 digit)
1252 \par ------------------------------------------------------------------------------
1253 \par a(7): 1 ( 1 digit)
1254 \par ------------------------------------------------------------------------------
1255 \par a(8): 2 ( 1 digit)
1256 \par ------------------------------------------------------------------------------
1257 \par a(9): 1 ( 1 digit)
1258 \par ------------------------------------------------------------------------------
1259 \par a(10): 4 ( 1 digit)
1260 \par ------------------------------------------------------------------------------
1261 \par a(11): 1 ( 1 digit)
1262 \par ------------------------------------------------------------------------------
1263 \par a(12): 1 ( 1 digit)
1264 \par ------------------------------------------------------------------------------
1265 \par a(13): 1 ( 1 digit)
1266 \par ------------------------------------------------------------------------------
1267 \par a(14): 1 ( 1 digit)
1268 \par ------------------------------------------------------------------------------
1269 \par a(15): 1 ( 1 digit)
1270 \par ------------------------------------------------------------------------------
1271 \par a(16): 3 ( 1 digit)
1272 \par ------------------------------------------------------------------------------
1273 \par a(17): 1 ( 1 digit)
1274 \par ------------------------------------------------------------------------------
1275 \par a(18): 68 ( 2 digits)
1276 \par ------------------------------------------------------------------------------
1277 \par a(19): 2 ( 1 digit)
1278 \par ------------------------------------------------------------------------------
1279 \par a(20): 4 ( 1 digit)
1280 \par ------------------------------------------------------------------------------
1281 \par a(21): 2 ( 1 digit)
1282 \par ------------------------------------------------------------------------------
1283 \par ***************************** CF Convergents *****************************
1284 \par ------------------------------------------------------------------------------
1285 \par p(0): 3 ( 1 digit)
1286 \par q(0): 1 ( 1 digit)
1287 \par ------------------------------------------------------------------------------
1288 \par p(1): 22 ( 2 digits)
1289 \par q(1): 7 ( 1 digit)
1290 \par ------------------------------------------------------------------------------
1291 \par p(2): 333 ( 3 digits)
1292 \par q(2): 106 ( 3 digits)
1293 \par ------------------------------------------------------------------------------
1294 \par p(3): 355 ( 3 digits)
1295 \par q(3): 113 ( 3 digits)
1296 \par ------------------------------------------------------------------------------
1297 \par p(4): 103,993 ( 6 digits)
1298 \par q(4): 33,102 ( 5 digits)
1299 \par ------------------------------------------------------------------------------
1300 \par p(5): 104,348 ( 6 digits)
1301 \par q(5): 33,215 ( 5 digits)
1302 \par ------------------------------------------------------------------------------
1303 \par p(6): 208,341 ( 6 digits)
1304 \par q(6): 66,317 ( 5 digits)
1305 \par ------------------------------------------------------------------------------
1306 \par p(7): 312,689 ( 6 digits)
1307 \par q(7): 99,532 ( 5 digits)
1308 \par ------------------------------------------------------------------------------
1309 \par p(8): 833,719 ( 6 digits)
1310 \par q(8): 265,381 ( 6 digits)
1311 \par ------------------------------------------------------------------------------
1312 \par p(9): 1,146,408 ( 7 digits)
1313 \par q(9): 364,913 ( 6 digits)
1314 \par ------------------------------------------------------------------------------
1315 \par p(10): 5,419,351 ( 7 digits)
1316 \par q(10): 1,725,033 ( 7 digits)
1317 \par ------------------------------------------------------------------------------
1318 \par p(11): 6,565,759 ( 7 digits)
1319 \par q(11): 2,089,946 ( 7 digits)
1320 \par ------------------------------------------------------------------------------
1321 \par p(12): 11,985,110 ( 8 digits)
1322 \par q(12): 3,814,979 ( 7 digits)
1323 \par ------------------------------------------------------------------------------
1324 \par p(13): 18,550,869 ( 8 digits)
1325 \par q(13): 5,904,925 ( 7 digits)
1326 \par ------------------------------------------------------------------------------
1327 \par p(14): 30,535,979 ( 8 digits)
1328 \par q(14): 9,719,904 ( 7 digits)
1329 \par ------------------------------------------------------------------------------
1330 \par p(15): 49,086,848 ( 8 digits)
1331 \par q(15): 15,624,829 ( 8 digits)
1332 \par ------------------------------------------------------------------------------
1333 \par p(16): 177,796,523 ( 9 digits)
1334 \par q(16): 56,594,391 ( 8 digits)
1335 \par ------------------------------------------------------------------------------
1336 \par p(17): 226,883,371 ( 9 digits)
1337 \par q(17): 72,219,220 ( 8 digits)
1338 \par ------------------------------------------------------------------------------
1339 \par p(18): 15,605,865,751 ( 11 digits)
1340 \par q(18): 4,967,501,351 ( 10 digits)
1341 \par ------------------------------------------------------------------------------
1342 \par p(19): 31,438,614,873 ( 11 digits)
1343 \par q(19): 10,007,221,922 ( 11 digits)
1344 \par ------------------------------------------------------------------------------
1345 \par p(20): 141,360,325,243 ( 12 digits)
1346 \par q(20): 44,996,389,039 ( 11 digits)
1347 \par ------------------------------------------------------------------------------
1348 \par p(21): 314,159,265,359 ( 12 digits)
1349 \par q(21): 100,000,000,000 ( 12 digits)
1350 \par ------------------------------------------------------------------------------
1351 \par RAP execution ends.
1352 \par ------------------------------------------------------------------------------
1353 \par }
1354 \par \page
1355 \par {\*\bkmkstart _Toc498721699}{\pntext\pard\plain\b\i\f5 4.15\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Rational Number With Smallest Denominator In An Interval [MIND]{\*\bkmkend _Toc498721699}
1356
1357 \par \pard\plain \widctlpar \f4\fs20
1358 \par {\*\bkmkstart _Toc498721700}{\pntext\pard\plain\f5 4.15.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721700}
1359 \par \pard\plain \widctlpar \f4\fs20 {\b rap mind R1 R2
1360 \par }\pard \li720\widctlpar Locates the rational number in the interval [R1, R2] with the smallest denominator.
1361 \par \pard \widctlpar
1362 \par {\*\bkmkstart _Toc498721701}{\pntext\pard\plain\f5 4.15.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721701}
1363 \par \pard\plain \widctlpar \f4\fs20
1364 Locates the rational number with the smallest denominator in an interval of the Farey series. This information is useful to provide an upper bound on the distance between Farey terms in the interval. The algorithm used is to find the best approximation
1365 with the smallest denominator in the interval to the midpoint of the interval, (R1+R2)/2. This algorithm is described fully in the paper.
1366 \par
1367 \par {\*\bkmkstart _Toc498721702}{\pntext\pard\plain\f5 4.15.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721702}
1368 \par \pard\plain \widctlpar \f4\fs20 The invocation below shows RAP used to find the rational number with the smallest denominator in the interval [0.385, 0.386]. This rational number is 22/57.
1369 \par
1370 \par {\f11\fs16 C:\\>rap mind 0.385 0.386
1371 \par ------------------------------------------------------------------------------
1372 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
1373 \par ------------------------------------------------------------------------------
1374 \par l_h: 77 ( 2 digits)
1375 \par ------------------------------------------------------------------------------
1376 \par l_k: 200 ( 3 digits)
1377 \par ------------------------------------------------------------------------------
1378 \par r_h: 193 ( 3 digits)
1379 \par ------------------------------------------------------------------------------
1380 \par r_k: 500 ( 3 digits)
1381 \par ------------------------------------------------------------------------------
1382 \par midpoint_h: 771 ( 3 digits)
1383 \par ------------------------------------------------------------------------------
1384 \par midpoint_k: 2,000 ( 4 digits)
1385 \par ------------------------------------------------------------------------------
1386 \par ***************** CF Representation Of Interval Midpoint *****************
1387 \par ************************ Inputs To CF Calculation ************************
1388 \par ------------------------------------------------------------------------------
1389 \par h_in: 771 ( 3 digits)
1390 \par ------------------------------------------------------------------------------
1391 \par k_in: 2,000 ( 4 digits)
1392 \par ------------------------------------------------------------------------------
1393 \par ************************** CF Partial Quotients **************************
1394 \par ------------------------------------------------------------------------------
1395 \par a(0): 0 ( 1 digit)
1396 \par ------------------------------------------------------------------------------
1397 \par a(1): 2 ( 1 digit)
1398 \par ------------------------------------------------------------------------------
1399 \par a(2): 1 ( 1 digit)
1400 \par ------------------------------------------------------------------------------
1401 \par a(3): 1 ( 1 digit)
1402 \par ------------------------------------------------------------------------------
1403 \par a(4): 2 ( 1 digit)
1404 \par ------------------------------------------------------------------------------
1405 \par a(5): 6 ( 1 digit)
1406 \par ------------------------------------------------------------------------------
1407 \par a(6): 3 ( 1 digit)
1408 \par ------------------------------------------------------------------------------
1409 \par a(7): 3 ( 1 digit)
1410 \par ------------------------------------------------------------------------------
1411 \par a(8): 2 ( 1 digit)
1412 \par ------------------------------------------------------------------------------
1413 \par ***************************** CF Convergents *****************************
1414 \par ------------------------------------------------------------------------------
1415 \par p(0): 0 ( 1 digit)
1416 \par q(0): 1 ( 1 digit)
1417 \par ------------------------------------------------------------------------------
1418 \par p(1): 1 ( 1 digit)
1419 \par q(1): 2 ( 1 digit)
1420 \par ------------------------------------------------------------------------------
1421 \par p(2): 1 ( 1 digit)
1422 \par q(2): 3 ( 1 digit)
1423 \par ------------------------------------------------------------------------------
1424 \par p(3): 2 ( 1 digit)
1425 \par q(3): 5 ( 1 digit)
1426 \par ------------------------------------------------------------------------------
1427 \par p(4): 5 ( 1 digit)
1428 \par q(4): 13 ( 2 digits)
1429 \par ------------------------------------------------------------------------------
1430 \par p(5): 32 ( 2 digits)
1431 \par q(5): 83 ( 2 digits)
1432 \par ------------------------------------------------------------------------------
1433 \par p(6): 101 ( 3 digits)
1434 \par q(6): 262 ( 3 digits)
1435 \par ------------------------------------------------------------------------------
1436 \par p(7): 335 ( 3 digits)
1437 \par q(7): 869 ( 3 digits)
1438 \par ------------------------------------------------------------------------------
1439 \par p(8): 771 ( 3 digits)
1440 \par q(8): 2,000 ( 4 digits)
1441 \par ------------------------------------------------------------------------------
1442 \par ******** A Rational Number With Smallest Denominator In Interval ********
1443 \par ------------------------------------------------------------------------------
1444 \par result_h: 22 ( 2 digits)
1445 \par ------------------------------------------------------------------------------
1446 \par result_k: 57 ( 2 digits)
1447 \par ------------------------------------------------------------------------------
1448 \par RAP execution ends.
1449 \par ------------------------------------------------------------------------------
1450 \par }
1451 \par \page
1452 \par {\*\bkmkstart _Toc498721703}{\pntext\pard\plain\b\i\f5 4.16\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Enclosing Rational Numbers In The Farey Series Of Order N [FN]{\*\bkmkend _Toc498721703}
1453
1454 \par \pard\plain \widctlpar \f4\fs20
1455 \par {\*\bkmkstart _Toc498721704}{\pntext\pard\plain\f5 4.16.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721704}
1456 \par \pard\plain \widctlpar \f4\fs20 {\b rap fn R1 ORDER
1457 \par }\pard \li720\widctlpar
1458 Locates the immediate left and right Farey neighbors of non-negative rational number R1, in the Farey series of order ORDER. Any decimal approximations are presented with the default denominator (1E108, to give four lines of digits after the decimal poin
1459 t).
1460 \par \pard \widctlpar {\b
1461 \par rap fn R1 ORDER NNEIGHBORS D
1462 \par }\pard \li720\widctlpar Same as form immediately above, except allows specification of the number of Farey neighbors on both the left and right to generate, and the denominator to use in any decimal approximations presented (see the {\i DAP}
1463 command, section {\field{\*\fldinst REF _Ref497594880 \\n }{\fldrslt 4.8}}). A value of NNEIGHBORS greater than 10,000 is treated as 10,000.
1464 \par \pard \widctlpar
1465 \par {\*\bkmkstart _Toc498721705}{\pntext\pard\plain\f5 4.16.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721705}
1466 \par \pard\plain \widctlpar \f4\fs20 Applies the continued fraction algorithms described in the TOMS paper to obtain Farey neighbors to an arbitrary non-negative rational number R1.
1467 There are two cases to consider: either the supplied rational number R1 is in the Farey series of order ORDER, or it is not. The algorithm will announce clearly which case applies. In either case, the algorithm applied is nearly identical.
1468 \par
1469 \par {\b\ul NOTE:}
1470 The implementation of this algorithm in the RAP program is subtly different than specified in the TOMS paper. The TOMS paper outlines an algorithm where the continued fraction decomposition of R1 is carried out only until the necessary partial quotient
1471 s and convergents are obtained. However, the RAP implementation will form {\i all}
1472 partial quotients and convergents regardless of the value of ORDER. This has no effect on the Farey neighbors obtained, but it means that specifying R1 very precisely may noticeably slow the program, regardless of the value of ORDER. The results will a
1473 lways be as expected, but the software may be more sluggish for more precisely specified R1.
1474 \par
1475 \par {\*\bkmkstart _Toc498721706}{\pntext\pard\plain\f5 4.16.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721706}
1476 \par \pard\plain \widctlpar \f4\fs20 The example invocation below is intended to generate the 10 best approximations to {{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}}{\cs19\super \chftn {\footnote \pard\plain \s18\widctlpar \f4\fs20 {
1477 \cs19\super \chftn } {{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}} to 1,000 decimal places is supplied in section {\field{\*\fldinst REF _Ref497727390 \\n }{\fldrslt 5.1}}.}} in the Farey series of order 65,
1478 535. The output below includes narrative explanations in a different font and with shading.
1479 \par
1480 \par {\f11\fs16 C:\\>rap fn 3.1415926535897932384626433832795028841971693993751058209749445923078 164062862089 65535 5 1e108
1481 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 In the command-line invocation above, the \ldblquote 1e108\rdblquote tells RAP to use this large power-of-ten denominator as the denomi
1482 nator when presenting the decimal approximations of numbers. There are 27 digits per line, so 1e108 is a convenient value to position the decimal point between lines and to give 108 decimal places of precision.
1483 \par \pard \widctlpar {\f11\fs16 ------------------------------------------------------------------------------
1484 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
1485 \par ------------------------------------------------------------------------------
1486 \par **************** Rational Number h_in/k_in To Approximate ****************
1487 \par ------------------------------------------------------------------------------
1488 \par h_in: 31,415,926,535,897,932,384,626,433, ( 80 digits)
1489 \par 832,795,028,841,971,693,993,751,058,
1490 \par 209,749,445,923,078,164,062,862,089
1491 \par ------------------------------------------------------------------------------
1492 \par k_in: 10,000,000,000,000,000,000,000,000, ( 80 digits)
1493 \par 000,000,000,000,000,000,000,000,000,
1494 \par 000,000,000,000,000,000,000,000,000
1495 \par ------------------------------------------------------------------------------
1496 \par *********************** Other Solution Parameters ***********************
1497 \par ------------------------------------------------------------------------------
1498 \par Order: 65,535 ( 5 digits)
1499 \par ------------------------------------------------------------------------------
1500 \par NNEIGHBORS: 5 ( 1 digit)
1501 \par ------------------------------------------------------------------------------
1502 \par DAP Denominator: 1, ( 109 digits)
1503 \par 000,000,000,000,000,000,000,000,000,
1504 \par 000,000,000,000,000,000,000,000,000,
1505 \par 000,000,000,000,000,000,000,000,000,
1506 \par 000,000,000,000,000,000,000,000,000
1507 \par ------------------------------------------------------------------------------
1508 \par *************** Continued Fraction Expansion Of h_in/k_in ***************
1509 \par ------------------------------------------------------------------------------
1510 \par ************************ Inputs To CF Calculation ************************
1511 \par ------------------------------------------------------------------------------
1512 \par h_in: 31,415,926,535,897,932,384,626,433, ( 80 digits)
1513 \par 832,795,028,841,971,693,993,751,058,
1514 \par 209,749,445,923,078,164,062,862,089
1515 \par ------------------------------------------------------------------------------
1516 \par k_in: 10,000,000,000,000,000,000,000,000, ( 80 digits)
1517 \par 000,000,000,000,000,000,000,000,000,
1518 \par 000,000,000,000,000,000,000,000,000
1519 \par ------------------------------------------------------------------------------
1520 \par ************************** CF Partial Quotients **************************
1521 \par ------------------------------------------------------------------------------
1522 \par a(0): 3 ( 1 digit)
1523 \par ------------------------------------------------------------------------------
1524 \par a(1): 7 ( 1 digit)
1525 \par ------------------------------------------------------------------------------
1526 \par a(2): 15 ( 2 digits)
1527 \par ------------------------------------------------------------------------------
1528 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 RAP has generated 148 partial quotients. Most of them are deleted for space. They can be easily reproduced by running RAP with the command-line reproduced above.
1529 \par \pard \widctlpar {\f11\fs16 ------------------------------------------------------------------------------
1530 \par a(145): 1 ( 1 digit)
1531 \par ------------------------------------------------------------------------------
1532 \par a(146): 2 ( 1 digit)
1533 \par ------------------------------------------------------------------------------
1534 \par a(147): 16 ( 2 digits)
1535 \par ------------------------------------------------------------------------------
1536 \par ***************************** CF Convergents *****************************
1537 \par ------------------------------------------------------------------------------
1538 \par p(0): 3 ( 1 digit)
1539 \par q(0): 1 ( 1 digit)
1540 \par ------------------------------------------------------------------------------
1541 \par p(1): 22 ( 2 digits)
1542 \par q(1): 7 ( 1 digit)
1543 \par ------------------------------------------------------------------------------
1544 \par p(2): 333 ( 3 digits)
1545 \par q(2): 106 ( 3 digits)
1546 \par ------------------------------------------------------------------------------
1547 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 RAP has generated 148 convergents. Most of them are deleted for space. They can be easily reproduced by running RAP with the command-line reproduced above.
1548 \par \pard \widctlpar {\f11\fs16 ------------------------------------------------------------------------------
1549 \par p(145): 725,955,471,712,149,333,468,082, ( 78 digits)
1550 \par 037,080,816,297,117,171,943,844,090,
1551 \par 600,740,370,938,512,424,999,369,433
1552 \par q(145): 231,078,803,575,194,322,803,693, ( 78 digits)
1553 \par 901,261,807,173,022,671,765,576,571,
1554 \par 012,554,067,457,355,871,085,907,056
1555 \par ------------------------------------------------------------------------------
1556 \par p(146): 1,918,123,191,511,611,440,697,396, ( 79 digits)
1557 \par 987,232,138,284,053,407,628,119,185,
1558 \par 475,563,067,186,535,358,691,468,291
1559 \par q(146): 610,557,574,776,550,354,824,769, ( 78 digits)
1560 \par 131,171,137,051,686,083,014,651,464,
1561 \par 311,715,370,783,915,258,057,130,809
1562 \par ------------------------------------------------------------------------------
1563 \par p(147): 31,415,926,535,897,932,384,626,433, ( 80 digits)
1564 \par 832,795,028,841,971,693,993,751,058,
1565 \par 209,749,445,923,078,164,062,862,089
1566 \par q(147): 10,000,000,000,000,000,000,000,000, ( 80 digits)
1567 \par 000,000,000,000,000,000,000,000,000,
1568 \par 000,000,000,000,000,000,000,000,000
1569 \par ------------------------------------------------------------------------------
1570 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500
1571 The first rational number below is the highest-order convergent with a denominator which is not larger than 65,535. The second rational number is an intermediate fraction chosen according to a formula in the TOMS paper. It is proved in the paper that th
1572 ese two numbers are the two Farey neighbors to the rational number of interest (if the rational number of interest is not in the Fa
1573 rey series). The two numbers are not necessarily in ascending order (it depends on whether the convergent is even or odd). If the rational number of interest is already in the Farey series of interest, the convergent will be this rational number, and th
1574 e intermediate fraction will be its left or right Farey neighbor.
1575 \par \pard \widctlpar {\f11\fs16 ***************** Highest-Order Convergent With q(i)<=N *****************
1576 \par ------------------------------------------------------------------------------
1577 \par p(5): 104,348 ( 6 digits)
1578 \par q(5): 33,215 ( 5 digits)
1579 \par ------------------------------------------------------------------------------
1580 \par ******************* Accompanying Intermediate Fraction *******************
1581 \par ------------------------------------------------------------------------------
1582 \par intermediate_h: 103,993 ( 6 digits)
1583 \par intermediate_k: 33,102 ( 5 digits)
1584 \par ------------------------------------------------------------------------------
1585 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 The line below indicates that the rational number of interest (3.14\'85) is not in the Farey series of interest (F{\sub 65,535}
1586 ).. In the neighbors presented just after this in the output, the neighbors are subscripted so that any number with a negative subscript is less than the rational number of interest, any number with a positive subscript is larger than the rational number
1587 of interest, and the subscript \ldblquote 0\rdblquote is reserved for the ratio
1588 nal number of interest if it appears in the Farey series of interest. In this case, the rational number of interest is not in the Farey series of interest, so there will not be a number presented with subscript \ldblquote 0\rdblquote .
1589 \par \pard \widctlpar {\f11\fs16 ******************************************************************************
1590 \par ************** h_in/k_in IS NOT In Farey Series Of Interest **************
1591 \par ******************************************************************************
1592 \par ------------------------------------------------------------------------------
1593 \par ************************ Farey Neighbor Index -5 ************************
1594 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 The subscript (or index) of \ldblquote -5\rdblquote here indicates that this is the fifth Farey neighbor to the left of the rational number of interest.
1595 \par \pard \widctlpar {\f11\fs16 ------------------------------------------------------------------------------
1596 \par h(-5): 205,501 ( 6 digits)
1597 \par ------------------------------------------------------------------------------
1598 \par k(-5): 65,413 ( 5 digits)
1599 \par ------------------------------------------------------------------------------
1600 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 The rational number below is the rational number 205,501/65,413 expressed as a decimal approximation. From the output below, one knows that 205,501/65,413 is 3.141592649\'85.
1601 \par \pard \widctlpar {\f11\fs16 DAP_N(-5): 3, ( 109 digits)
1602 \par 141,592,649,779,095,898,368,825,768,
1603 \par 578,111,384,587,161,573,387,552,932,
1604 \par 903,245,532,233,653,860,853,347,193,
1605 \par 982,847,446,226,285,294,971,947,472
1606 \par ------------------------------------------------------------------------------
1607 \par DAP_D(-5): 1, ( 109 digits)
1608 \par 000,000,000,000,000,000,000,000,000,
1609 \par 000,000,000,000,000,000,000,000,000,
1610 \par 000,000,000,000,000,000,000,000,000,
1611 \par 000,000,000,000,000,000,000,000,000
1612 \par ------------------------------------------------------------------------------
1613 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500
1614 The rational number below is the error (approximation minus actual) of the approximation, expressed as a lowest-terms rational number. This rational number might not be useful or intuitive for Farey series of large orders.
1615 \par \pard \widctlpar {\f11\fs16 error_h(-5): - 2,492,691,451,075,568,916,304, ( 76 digits)
1616 \par 621,221,639,894,419,213,237,970,674,
1617 \par 340,506,166,311,945,843,997,827,757
1618 \par ------------------------------------------------------------------------------
1619 \par error_k(-5): 654, ( 84 digits)
1620 \par 130,000,000,000,000,000,000,000,000,
1621 \par 000,000,000,000,000,000,000,000,000,
1622 \par 000,000,000,000,000,000,000,000,000
1623 \par ------------------------------------------------------------------------------
1624 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 The rational number below is the error (approximation minus actual) of the approxi
1625 mation, expressed as a decimal approximation. Because the imaginary decimal point is positioned to the left of the fourth line up, it is easy to see that the approximation error is about 3.81 {{\field{\*\fldinst SYMBOL 180 \\f "Symbol" \\s 10}{\fldrslt
1626 \f1\fs20}}} 10{\super -9}.
1627 \par \pard \widctlpar {\f11\fs16 ERROR_DAP_N(-5): - 3,810,697,340,093,817,614, ( 100 digits)
1628 \par 701,391,499,610,007,825,987,552,888,
1629 \par 071,699,060,074,162,545,432,861,706,
1630 \par 017,152,553,773,714,705,028,052,527
1631 \par ------------------------------------------------------------------------------
1632 \par ERROR_DAP_D(-5): 1, ( 109 digits)
1633 \par 000,000,000,000,000,000,000,000,000,
1634 \par 000,000,000,000,000,000,000,000,000,
1635 \par 000,000,000,000,000,000,000,000,000,
1636 \par 000,000,000,000,000,000,000,000,000
1637 \par ------------------------------------------------------------------------------
1638 \par ************************ Farey Neighbor Index -4 ************************
1639 \par ------------------------------------------------------------------------------
1640 \par h(-4): 102,928 ( 6 digits)
1641 \par ------------------------------------------------------------------------------
1642 \par k(-4): 32,763 ( 5 digits)
1643 \par ------------------------------------------------------------------------------
1644 \par DAP_N(-4): 3, ( 109 digits)
1645 \par 141,592,650,245,703,995,360,620,211,
1646 \par 824,314,012,758,294,417,483,136,464,
1647 \par 914,690,351,921,374,721,484,601,532,
1648 \par 216,219,515,917,345,786,405,396,331
1649 \par ------------------------------------------------------------------------------
1650 \par DAP_D(-4): 1, ( 109 digits)
1651 \par 000,000,000,000,000,000,000,000,000,
1652 \par 000,000,000,000,000,000,000,000,000,
1653 \par 000,000,000,000,000,000,000,000,000,
1654 \par 000,000,000,000,000,000,000,000,000
1655 \par ------------------------------------------------------------------------------
1656 \par error_h(-4): - 1,095,623,958,717,515,851,663, ( 76 digits)
1657 \par 863,529,949,518,610,317,265,920,126,
1658 \par 021,096,777,809,889,191,550,621,907
1659 \par ------------------------------------------------------------------------------
1660 \par error_k(-4): 327, ( 84 digits)
1661 \par 630,000,000,000,000,000,000,000,000,
1662 \par 000,000,000,000,000,000,000,000,000,
1663 \par 000,000,000,000,000,000,000,000,000
1664 \par ------------------------------------------------------------------------------
1665 \par ERROR_DAP_N(-4): - 3,344,089,243,102,023,171, ( 100 digits)
1666 \par 455,188,871,438,874,981,891,969,356,
1667 \par 060,254,240,386,441,684,801,607,367,
1668 \par 783,780,484,082,654,213,594,603,668
1669 \par ------------------------------------------------------------------------------
1670 \par ERROR_DAP_D(-4): 1, ( 109 digits)
1671 \par 000,000,000,000,000,000,000,000,000,
1672 \par 000,000,000,000,000,000,000,000,000,
1673 \par 000,000,000,000,000,000,000,000,000,
1674 \par 000,000,000,000,000,000,000,000,000
1675 \par ------------------------------------------------------------------------------
1676 \par ************************ Farey Neighbor Index -3 ************************
1677 \par ------------------------------------------------------------------------------
1678 \par h(-3): 103,283 ( 6 digits)
1679 \par ------------------------------------------------------------------------------
1680 \par k(-3): 32,876 ( 5 digits)
1681 \par ------------------------------------------------------------------------------
1682 \par DAP_N(-3): 3, ( 109 digits)
1683 \par 141,592,651,174,108,772,356,734,395,
1684 \par 911,911,424,747,536,196,617,593,381,
1685 \par 189,925,781,725,270,714,198,807,640,
1686 \par 832,218,031,390,680,131,402,847,061
1687 \par ------------------------------------------------------------------------------
1688 \par DAP_D(-3): 1, ( 109 digits)
1689 \par 000,000,000,000,000,000,000,000,000,
1690 \par 000,000,000,000,000,000,000,000,000,
1691 \par 000,000,000,000,000,000,000,000,000,
1692 \par 000,000,000,000,000,000,000,000,000
1693 \par ------------------------------------------------------------------------------
1694 \par error_h(-3): - 198,545,106,269,244,659,671, ( 75 digits)
1695 \par 742,342,052,165,352,934,639,947,425,
1696 \par 930,696,041,779,430,432,663,509,491
1697 \par ------------------------------------------------------------------------------
1698 \par error_k(-3): 82, ( 83 digits)
1699 \par 190,000,000,000,000,000,000,000,000,
1700 \par 000,000,000,000,000,000,000,000,000,
1701 \par 000,000,000,000,000,000,000,000,000
1702 \par ------------------------------------------------------------------------------
1703 \par ERROR_DAP_N(-3): - 2,415,684,466,105,908,987, ( 100 digits)
1704 \par 367,591,459,449,633,202,757,512,439,
1705 \par 785,018,810,582,545,692,087,401,259,
1706 \par 167,781,968,609,319,868,597,152,938
1707 \par ------------------------------------------------------------------------------
1708 \par ERROR_DAP_D(-3): 1, ( 109 digits)
1709 \par 000,000,000,000,000,000,000,000,000,
1710 \par 000,000,000,000,000,000,000,000,000,
1711 \par 000,000,000,000,000,000,000,000,000,
1712 \par 000,000,000,000,000,000,000,000,000
1713 \par ------------------------------------------------------------------------------
1714 \par ************************ Farey Neighbor Index -2 ************************
1715 \par ------------------------------------------------------------------------------
1716 \par h(-2): 103,638 ( 6 digits)
1717 \par ------------------------------------------------------------------------------
1718 \par k(-2): 32,989 ( 5 digits)
1719 \par ------------------------------------------------------------------------------
1720 \par DAP_N(-2): 3, ( 109 digits)
1721 \par 141,592,652,096,153,263,208,948,437,
1722 \par 357,907,181,181,605,989,875,413,016,
1723 \par 460,032,131,922,762,132,832,156,173,
1724 \par 269,877,838,067,234,532,723,028,888
1725 \par ------------------------------------------------------------------------------
1726 \par DAP_D(-2): 1, ( 109 digits)
1727 \par 000,000,000,000,000,000,000,000,000,
1728 \par 000,000,000,000,000,000,000,000,000,
1729 \par 000,000,000,000,000,000,000,000,000,
1730 \par 000,000,000,000,000,000,000,000,000
1731 \par ------------------------------------------------------------------------------
1732 \par error_h(-2): - 492,736,891,436,441,425,710, ( 75 digits)
1733 \par 075,206,467,804,213,159,853,659,281,
1734 \par 424,471,556,425,554,269,757,454,021
1735 \par ------------------------------------------------------------------------------
1736 \par error_k(-2): 329, ( 84 digits)
1737 \par 890,000,000,000,000,000,000,000,000,
1738 \par 000,000,000,000,000,000,000,000,000,
1739 \par 000,000,000,000,000,000,000,000,000
1740 \par ------------------------------------------------------------------------------
1741 \par ERROR_DAP_N(-2): - 1,493,639,975,253,694,945, ( 100 digits)
1742 \par 921,595,703,015,563,409,499,692,804,
1743 \par 514,912,460,385,054,273,454,052,726,
1744 \par 730,122,161,932,765,467,276,971,111
1745 \par ------------------------------------------------------------------------------
1746 \par ERROR_DAP_D(-2): 1, ( 109 digits)
1747 \par 000,000,000,000,000,000,000,000,000,
1748 \par 000,000,000,000,000,000,000,000,000,
1749 \par 000,000,000,000,000,000,000,000,000,
1750 \par 000,000,000,000,000,000,000,000,000
1751 \par ------------------------------------------------------------------------------
1752 \par ************************ Farey Neighbor Index -1 ************************
1753 \par ------------------------------------------------------------------------------
1754 \par h(-1): 103,993 ( 6 digits)
1755 \par ------------------------------------------------------------------------------
1756 \par k(-1): 33,102 ( 5 digits)
1757 \par ------------------------------------------------------------------------------
1758 \par DAP_N(-1): 3, ( 109 digits)
1759 \par 141,592,653,011,902,604,072,261,494,
1760 \par 773,729,684,007,008,639,961,331,641,
1761 \par 592,653,011,902,604,072,261,494,773,
1762 \par 729,684,007,008,639,961,331,641,592
1763 \par ------------------------------------------------------------------------------
1764 \par DAP_D(-1): 1, ( 109 digits)
1765 \par 000,000,000,000,000,000,000,000,000,
1766 \par 000,000,000,000,000,000,000,000,000,
1767 \par 000,000,000,000,000,000,000,000,000,
1768 \par 000,000,000,000,000,000,000,000,000
1769 \par ------------------------------------------------------------------------------
1770 \par error_h(-1): - 95,646,678,897,952,106,366, ( 74 digits)
1771 \par 590,522,363,473,507,290,573,764,429,
1772 \par 563,079,472,866,693,404,430,435,039
1773 \par ------------------------------------------------------------------------------
1774 \par error_k(-1): 165, ( 84 digits)
1775 \par 510,000,000,000,000,000,000,000,000,
1776 \par 000,000,000,000,000,000,000,000,000,
1777 \par 000,000,000,000,000,000,000,000,000
1778 \par ------------------------------------------------------------------------------
1779 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 This is the last approximation on the left of the number of interest (note the subscript of -1). The error is about 5.78 {{\field{\*\fldinst SYMBOL 180 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}} 10{
1780 \super -10}.
1781 \par \pard \widctlpar {\f11\fs16 ERROR_DAP_N(-1): - 577,890,634,390,381,888, ( 99 digits)
1782 \par 505,773,200,190,160,759,413,774,179,
1783 \par 382,291,580,405,212,334,024,714,126,
1784 \par 270,315,992,991,360,038,668,358,407
1785 \par ------------------------------------------------------------------------------
1786 \par ERROR_DAP_D(-1): 1, ( 109 digits)
1787 \par 000,000,000,000,000,000,000,000,000,
1788 \par 000,000,000,000,000,000,000,000,000,
1789 \par 000,000,000,000,000,000,000,000,000,
1790 \par 000,000,000,000,000,000,000,000,000
1791 \par ------------------------------------------------------------------------------
1792 \par ************************* Farey Neighbor Index 1 *************************
1793 \par ------------------------------------------------------------------------------
1794 \par h(1): 104,348 ( 6 digits)
1795 \par ------------------------------------------------------------------------------
1796 \par k(1): 33,215 ( 5 digits)
1797 \par ------------------------------------------------------------------------------
1798 \par DAP_N(1): 3, ( 109 digits)
1799 \par 141,592,653,921,421,044,708,715,941,
1800 \par 592,653,921,421,044,708,715,941,592,
1801 \par 653,921,421,044,708,715,941,592,653,
1802 \par 921,421,044,708,715,941,592,653,921
1803 \par ------------------------------------------------------------------------------
1804 \par DAP_D(1): 1, ( 109 digits)
1805 \par 000,000,000,000,000,000,000,000,000,
1806 \par 000,000,000,000,000,000,000,000,000,
1807 \par 000,000,000,000,000,000,000,000,000,
1808 \par 000,000,000,000,000,000,000,000,000
1809 \par ------------------------------------------------------------------------------
1810 \par error_h(1): 22,030,035,168,926,600,048, ( 74 digits)
1811 \par 742,623,402,782,036,799,511,720,312,
1812 \par 634,430,732,991,756,130,407,142,773
1813 \par ------------------------------------------------------------------------------
1814 \par error_k(1): 66, ( 83 digits)
1815 \par 430,000,000,000,000,000,000,000,000,
1816 \par 000,000,000,000,000,000,000,000,000,
1817 \par 000,000,000,000,000,000,000,000,000
1818 \par ------------------------------------------------------------------------------
1819 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 This is the first approximation on the right of the number of interest (note the subscript of 1). The error is about 3.32 {{\field{\*\fldinst SYMBOL 180 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}} 10{
1820 \super -10}.
1821 \par \pard \widctlpar {\f11\fs16 ERROR_DAP_N(1): 331,627,806,246,072,558, ( 99 digits)
1822 \par 313,151,037,223,875,309,340,835,771,
1823 \par 678,976,828,736,892,309,655,383,753,
1824 \par 921,421,044,708,715,941,592,653,921
1825 \par ------------------------------------------------------------------------------
1826 \par ERROR_DAP_D(1): 1, ( 109 digits)
1827 \par 000,000,000,000,000,000,000,000,000,
1828 \par 000,000,000,000,000,000,000,000,000,
1829 \par 000,000,000,000,000,000,000,000,000,
1830 \par 000,000,000,000,000,000,000,000,000
1831 \par ------------------------------------------------------------------------------
1832 \par ************************* Farey Neighbor Index 2 *************************
1833 \par ------------------------------------------------------------------------------
1834 \par h(2): 104,703 ( 6 digits)
1835 \par ------------------------------------------------------------------------------
1836 \par k(2): 33,328 ( 5 digits)
1837 \par ------------------------------------------------------------------------------
1838 \par DAP_N(2): 3, ( 109 digits)
1839 \par 141,592,654,824,771,963,514,162,265,
1840 \par 962,554,008,641,382,621,219,395,103,
1841 \par 216,514,642,342,774,843,975,036,005,
1842 \par 760,921,747,479,596,735,477,676,428
1843 \par ------------------------------------------------------------------------------
1844 \par DAP_D(2): 1, ( 109 digits)
1845 \par 000,000,000,000,000,000,000,000,000,
1846 \par 000,000,000,000,000,000,000,000,000,
1847 \par 000,000,000,000,000,000,000,000,000,
1848 \par 000,000,000,000,000,000,000,000,000
1849 \par ------------------------------------------------------------------------------
1850 \par error_h(2): 25,724,606,842,823,138,326, ( 74 digits)
1851 \par 287,954,922,172,961,411,016,545,749,
1852 \par 091,904,142,228,184,257,058,268,613
1853 \par ------------------------------------------------------------------------------
1854 \par error_k(2): 20, ( 83 digits)
1855 \par 830,000,000,000,000,000,000,000,000,
1856 \par 000,000,000,000,000,000,000,000,000,
1857 \par 000,000,000,000,000,000,000,000,000
1858 \par ------------------------------------------------------------------------------
1859 \par ERROR_DAP_N(2): 1,234,978,725,051,518,882, ( 100 digits)
1860 \par 683,051,124,444,213,221,844,289,282,
1861 \par 241,570,050,034,958,437,688,827,105,
1862 \par 760,921,747,479,596,735,477,676,428
1863 \par ------------------------------------------------------------------------------
1864 \par ERROR_DAP_D(2): 1, ( 109 digits)
1865 \par 000,000,000,000,000,000,000,000,000,
1866 \par 000,000,000,000,000,000,000,000,000,
1867 \par 000,000,000,000,000,000,000,000,000,
1868 \par 000,000,000,000,000,000,000,000,000
1869 \par ------------------------------------------------------------------------------
1870 \par ************************* Farey Neighbor Index 3 *************************
1871 \par ------------------------------------------------------------------------------
1872 \par h(3): 105,058 ( 6 digits)
1873 \par ------------------------------------------------------------------------------
1874 \par k(3): 33,441 ( 5 digits)
1875 \par ------------------------------------------------------------------------------
1876 \par DAP_N(3): 3, ( 109 digits)
1877 \par 141,592,655,722,017,882,240,363,625,
1878 \par 489,668,371,161,149,487,156,484,554,
1879 \par 887,712,688,017,702,819,891,749,648,
1880 \par 634,909,243,144,642,803,743,907,179
1881 \par ------------------------------------------------------------------------------
1882 \par DAP_D(3): 1, ( 109 digits)
1883 \par 000,000,000,000,000,000,000,000,000,
1884 \par 000,000,000,000,000,000,000,000,000,
1885 \par 000,000,000,000,000,000,000,000,000,
1886 \par 000,000,000,000,000,000,000,000,000
1887 \par ------------------------------------------------------------------------------
1888 \par error_h(3): 713,037,243,125,707,426,197, ( 75 digits)
1889 \par 501,440,495,624,581,154,970,862,407,
1890 \par 768,778,886,343,115,573,828,881,751
1891 \par ------------------------------------------------------------------------------
1892 \par error_k(3): 334, ( 84 digits)
1893 \par 410,000,000,000,000,000,000,000,000,
1894 \par 000,000,000,000,000,000,000,000,000,
1895 \par 000,000,000,000,000,000,000,000,000
1896 \par ------------------------------------------------------------------------------
1897 \par ERROR_DAP_N(3): 2,132,224,643,777,720,242, ( 100 digits)
1898 \par 210,165,486,963,980,087,781,378,733,
1899 \par 912,768,095,709,886,413,605,540,748,
1900 \par 634,909,243,144,642,803,743,907,179
1901 \par ------------------------------------------------------------------------------
1902 \par ERROR_DAP_D(3): 1, ( 109 digits)
1903 \par 000,000,000,000,000,000,000,000,000,
1904 \par 000,000,000,000,000,000,000,000,000,
1905 \par 000,000,000,000,000,000,000,000,000,
1906 \par 000,000,000,000,000,000,000,000,000
1907 \par ------------------------------------------------------------------------------
1908 \par ************************* Farey Neighbor Index 4 *************************
1909 \par ------------------------------------------------------------------------------
1910 \par h(4): 105,413 ( 6 digits)
1911 \par ------------------------------------------------------------------------------
1912 \par k(4): 33,554 ( 5 digits)
1913 \par ------------------------------------------------------------------------------
1914 \par DAP_N(4): 3, ( 109 digits)
1915 \par 141,592,656,613,220,480,419,622,101,
1916 \par 686,833,164,451,332,180,962,031,352,
1917 \par 446,802,169,637,003,039,876,020,742,
1918 \par 683,435,655,957,560,946,533,945,282
1919 \par ------------------------------------------------------------------------------
1920 \par DAP_D(4): 1, ( 109 digits)
1921 \par 000,000,000,000,000,000,000,000,000,
1922 \par 000,000,000,000,000,000,000,000,000,
1923 \par 000,000,000,000,000,000,000,000,000,
1924 \par 000,000,000,000,000,000,000,000,000
1925 \par ------------------------------------------------------------------------------
1926 \par error_h(4): 507,240,388,383,122,319,587, ( 75 digits)
1927 \par 197,801,118,240,889,866,838,496,415,
1928 \par 033,545,748,517,641,517,362,732,847
1929 \par ------------------------------------------------------------------------------
1930 \par error_k(4): 167, ( 84 digits)
1931 \par 770,000,000,000,000,000,000,000,000,
1932 \par 000,000,000,000,000,000,000,000,000,
1933 \par 000,000,000,000,000,000,000,000,000
1934 \par ------------------------------------------------------------------------------
1935 \par ERROR_DAP_N(4): 3,023,427,241,956,978,718, ( 100 digits)
1936 \par 407,330,280,254,162,781,586,925,531,
1937 \par 471,857,577,329,186,633,589,811,842,
1938 \par 683,435,655,957,560,946,533,945,282
1939 \par ------------------------------------------------------------------------------
1940 \par ERROR_DAP_D(4): 1, ( 109 digits)
1941 \par 000,000,000,000,000,000,000,000,000,
1942 \par 000,000,000,000,000,000,000,000,000,
1943 \par 000,000,000,000,000,000,000,000,000,
1944 \par 000,000,000,000,000,000,000,000,000
1945 \par ------------------------------------------------------------------------------
1946 \par ************************* Farey Neighbor Index 5 *************************
1947 \par ------------------------------------------------------------------------------
1948 \par h(5): 105,768 ( 6 digits)
1949 \par ------------------------------------------------------------------------------
1950 \par k(5): 33,667 ( 5 digits)
1951 \par ------------------------------------------------------------------------------
1952 \par DAP_N(5): 3, ( 109 digits)
1953 \par 141,592,657,498,440,609,498,915,852,
1954 \par 318,293,878,278,432,886,803,100,959,
1955 \par 396,441,619,389,907,030,623,459,173,
1956 \par 671,547,806,457,361,808,298,927,733
1957 \par ------------------------------------------------------------------------------
1958 \par DAP_D(5): 1, ( 109 digits)
1959 \par 000,000,000,000,000,000,000,000,000,
1960 \par 000,000,000,000,000,000,000,000,000,
1961 \par 000,000,000,000,000,000,000,000,000,
1962 \par 000,000,000,000,000,000,000,000,000
1963 \par ------------------------------------------------------------------------------
1964 \par error_h(5): 1,315,924,310,406,781,852,151, ( 76 digits)
1965 \par 289,763,977,338,978,312,383,123,252,
1966 \par 365,404,107,727,450,495,622,049,637
1967 \par ------------------------------------------------------------------------------
1968 \par error_k(5): 336, ( 84 digits)
1969 \par 670,000,000,000,000,000,000,000,000,
1970 \par 000,000,000,000,000,000,000,000,000,
1971 \par 000,000,000,000,000,000,000,000,000
1972 \par ------------------------------------------------------------------------------
1973 \par ERROR_DAP_N(5): 3,908,647,371,036,272,469, ( 100 digits)
1974 \par 038,790,994,081,263,487,427,995,138,
1975 \par 421,497,027,082,090,624,337,250,273,
1976 \par 671,547,806,457,361,808,298,927,733
1977 \par ------------------------------------------------------------------------------
1978 \par ERROR_DAP_D(5): 1, ( 109 digits)
1979 \par 000,000,000,000,000,000,000,000,000,
1980 \par 000,000,000,000,000,000,000,000,000,
1981 \par 000,000,000,000,000,000,000,000,000,
1982 \par 000,000,000,000,000,000,000,000,000
1983 \par ------------------------------------------------------------------------------
1984 \par RAP execution ends.
1985 \par ------------------------------------------------------------------------------
1986 \par }
1987 \par {\b\i\ul Practical Note:} For Farey series of large order, the terms become quite dense. If an irrational number to be approximated isn\rquote
1988 t specified precisely enough, it is easy to accidentally generate Farey neighbors which enclose the rational approximation specified, but do not enclose the irrational number.
1989 \par
1990 \par For example, using 3.141592654 as a rational approximation to {{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}} to obtain the best approximations to {{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}} in F{\sub
1991 65,535} yields 104,348/33,215 and 104,703/33,328 as the two best approximations to {{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}}. However, it is easy to show that
1992 \par
1993 \par \pard \qc\widctlpar {\pard\plain \qc\widctlpar \f4\fs20 {\object\objemb\objw3861\objh660{\*\objclass Equation.3}{\*\objdata 01050000020000000b0000004571756174696f6e2e3300000000000000000000100000
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2034 \par \pard \widctlpar
2035 \par In other words, the rational numbers in F{\sub 65,535} generated enclose the rational approximation of {{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}} used, but they do not enclose {{\field{\*\fldinst SYMBOL 112 \\f "Symbol"
2036 \\s 10}{\fldrslt\f1\fs20}}}.
2037 \par
2038 \par There are three approaches to mitigate this type of problem.
2039 \par
2040 \par The first approach is analytic and practical. It can be shown that the distance between two consecutive terms of the Farey series of any order is 1/ab, where a and b are the denominators of the neighboring terms. It follows immediately that in a Farey s
2041 eries of order N, the minimum distance between terms is 1/(N{\super 2}-N). Although it won\rquote t be done here, this informa
2042 tion can be used to derive the required precision of the rational approximation to the irrational number of interest used to apply the CF algorithms to obtain Farey neighbors, so that at most one Farey term appears between the irrational and the rational
2043 approximation used to apply the algorithms. In other words, one could derive a rule of thumb for how many decimal places are appropriate as a function of the order of the Farey series.
2044 \par
2045 \par The second (analytic and usually impractical) approach would be to cal
2046 culate the continued fraction partial quotients required symbolically rather than numerically. Unfortunately, doing this by hand is generally not practical (although it can be done for many algebraic numbers relatively easily). (Commercial software such
2047 as {\i Mathematica} will do this automatically, even for transcendentals, but as of this writing {\i Mathematica}
2048 costs about $1,500.) Every case where the wrong Farey neighbors are located corresponds to an error in the partial quotients and convergents used\emdash or to
2049 put it another way, the continued fraction representation of the rational approximation used differs from the true continued fraction representation of the irrational relatively [too] early in the partial quotients and convergents. (This gets into techn
2050 ical detail and won\rquote t be discussed here.)
2051 \par
2052 \par The third approach i
2053 s practical. If the irrational number can be bounded by rational numbers, and if the CF algorithms give the same result at both of the rational bounds, then the rational approximation of the irrational is specified precisely enough. The case of 3.141592
2054 654 is atypical because it is rounded rather than truncated (it came from a pocket calculator). The actual inequality at this number of decimal places which confines {{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}} is:
2055 \par \pard \qc\widctlpar {\pard\plain \qc\widctlpar \f4\fs20 {\object\objemb\objw3221\objh279{\*\objclass Equation.3}{\*\objdata 01050000020000000b0000004571756174696f6e2e3300000000000000000000100000
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2057 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2058 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2059 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2060 fffffffffffffffffdffffff05000000feffffff0400000006000000fefffffffeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2061 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2062 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2063 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2064 ffffffffffffffffffffffffffffffff52006f006f007400200045006e00740072007900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000016000500ffffffffffffffff0200000002ce020000000000c000000000000046000000000000000000000000a070
2065 34df674dc00103000000800400000000000001004f006c00650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a000201ffffffffffffffffffffffff00000000000000000000000000000000000000000000000000000000
2066 0000000000000000000000001400000000000000030050004900430000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a0002010100000004000000ffffffff0000000000000000000000000000000000000000000000000000
2067 00000000000000000000010000004c0000000000000003004d004500540041000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000c000201ffffffffffffffffffffffff000000000000000000000000000000000000000000000000
2068 000000000000000000000000030000001402000000000000feffffff02000000feffffff0400000005000000060000000700000008000000090000000a0000000b000000feffffff0d000000fefffffffeffffff1000000011000000feffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2069 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2070 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2071 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2072 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff010000020000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004c00000000000000000008002f160000ed010000940c0000170100000000
2073 0000000000000000000000000000e8030000e80300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000008002f16ed0100000100090000030501000002001500000000000500
2074 00000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c02c00120141200000026060f001a00ffffffff000010000000c0ffffffc6ffffffe0130000860100000b00000026060f000c004d617468547970650000300015000000
2075 fb0280fe0000000000009001000000000402001054696d6573204e657720526f6d616e0000bf040000002d0100000c000000320a60012a0d090000003134313539323635340008000000320a6001ca0c010000002e0008000000320a60010a0c0100000033000c000000320a60014e010900000031343135393236353300
2076 08000000320a6001ee00010000002e0008000000320a60012e0001000000330010000000fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d01010004000000f001000008000000320a6001de0a010000003c0008000000320a60015c08010000003c0010000000fb0280fe000000000000
2077 9001010000020002001053796d626f6c0002040000002d01000004000000f001010008000000320a600176090100000070000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02000000000102022253797374656d0000040000002d01010004000000f00100000300000000001100
2078 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100feff030a0000ffffffff02ce020000000000c000000000000046170000004d6963726f736f6674204571756174696f6e20332e30000c0000004453204571756174696f6e000b0000004571756174696f
2079 6e2e3300f439b2710000000000000000000000000000000000000000000000000000000000000000000000000000000003000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001c0000000200d5d17000000000000000
2080 d4bd4900a08f49000000000003010103010a010288330002822e00028831000288340002883100028835000288390002010043006f006d0070004f0062006a00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000120002000300000005000000ffff
2081 ffff0000000000000000000000000000000000000000000000000000000000000000000000000c000000660000000000000003004f0062006a0049006e0066006f0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000012000201ffffffff06000000
2082 ffffffff0000000000000000000000000000000000000000000000000000000000000000000000000e00000004000000000000004500710075006100740069006f006e0020004e00610074006900760065000000000000000000000000000000000000000000000000000000000000000000000020000200ffffffffffff
2083 ffffffffffff0000000000000000000000000000000000000000000000000000000000000000000000000f0000008c000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffff
2084 ffffffffffffffff00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000088320002883600028835000288330002863c000284c00302863c000288330002822e000288310002883400028831000288350002883900028832000288360002883500028834
2085 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
2086 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
2087 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
2088 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000105000000000000}{\result {\dn6 {\pict\wmetafile8\picw5679\pich493\picwgoal3220\pichgoal279
2089 010009000003050100000200150000000000050000000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c02c00120141200000026060f001a00ffffffff000010000000c0ffffffc6ffffffe0130000860100000b00000026060f
2090 000c004d617468547970650000300015000000fb0280fe0000000000009001000000000402001054696d6573204e657720526f6d616e0000bf040000002d0100000c000000320a60012a0d090000003134313539323635340008000000320a6001ca0c010000002e0008000000320a60010a0c0100000033000c000000320a
2091 60014e01090000003134313539323635330008000000320a6001ee00010000002e0008000000320a60012e0001000000330010000000fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d01010004000000f001000008000000320a6001de0a010000003c0008000000320a60015c08010000
2092 003c0010000000fb0280fe0000000000009001010000020002001053796d626f6c0002040000002d01000004000000f001010008000000320a600176090100000070000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02000000000102022253797374656d0000040000002d010100
2093 04000000f001000003000000000011000000000000035202000000000000000100796d62000e46000000000000350d5f0002601100383438000008000000320a0002fa0700000000}}}}}
2094 \par \pard \widctlpar
2095 \par If the CF algorithms presented give the same results for both 3.141592653 and 3.141592654, then the correct Farey neighbors of {{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}} are identified.
2096 \par
2097 \par Using 3.141592653 as a rational approximation gives 103,638/32,989 and 103,993/33,102 as the best approximations in F{\sub 65535}. Using 3.141592654 as a rational approximation gives 104,348/33,215 and 104,703/33,328 as the best approximations in F{\sub
2098 65,535}. The results are not identical, so neither pair of neighbors can be trusted to be the correct pair.
2099 \par
2100 \par
2101 \par \page
2102 \par {\*\bkmkstart _Toc498721707}{\pntext\pard\plain\b\i\f5 4.17\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Upper Bound On Distance Between Farey Terms In An Interval [FNDMAX]{\*\bkmkend _Toc498721707}
2103
2104 \par \pard\plain \widctlpar \f4\fs20
2105 \par {\*\bkmkstart _Toc498721708}{\pntext\pard\plain\f5 4.17.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721708}
2106 \par \pard\plain \widctlpar \f4\fs20 {\b rap fndmax LL UL N
2107 \par }\pard \li720\widctlpar Calculates an upper bound on the maximum distance between terms of F{\sub N} in the interval [LL,UL] using the techniques developed in the TOMS paper. LL and UL must both be in F{\sub N}, and LL and UL must be non-negative.
2108 \par \pard \widctlpar
2109 \par {\*\bkmkstart _Toc498721709}{\pntext\pard\plain\f5 4.17.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721709}
2110 \par \pard\plain \widctlpar \f4\fs20 Applies the continued fraction algorithms described in the TOMS paper to find the rational number with the the smallest denominator in the interval [LL, UL], and to place an upper bound on the distance between terms in F{
2111 \sub N}.
2112 \par
2113 \par The results in the TOMS paper state the distance in terms of an upper bound on the distance between r{\sub A} and r{\sub I}, i.e.
2114 \par
2115 \par \pard \qc\widctlpar {\pard\plain \qc\widctlpar \f4\fs20 {\object\objemb\objw3900\objh700{\*\objclass Equation.3}{\*\objdata 01050000020000000b0000004571756174696f6e2e3300000000000000000000120000
2116 d0cf11e0a1b11ae1000000000000000000000000000000003e000300feff0900060000000000000000000000010000000100000000000000001000000200000001000000feffffff0000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2117 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2118 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2119 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2120 fffffffffffffffffdffffff06000000feffffff040000000500000007000000fefffffffeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2121 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2122 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2123 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2124 ffffffffffffffffffffffffffffffff52006f006f007400200045006e00740072007900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000016000500ffffffffffffffff0200000002ce020000000000c000000000000046000000000000000000000000a070
2125 34df674dc00103000000400600000000000001004f006c00650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a000201ffffffffffffffffffffffff00000000000000000000000000000000000000000000000000000000
2126 0000000000000000000000001400000000000000030050004900430000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a0002010100000004000000ffffffff0000000000000000000000000000000000000000000000000000
2127 00000000000000000000010000004c0000000000000003004d004500540041000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000c000201ffffffffffffffffffffffff000000000000000000000000000000000000000000000000
2128 00000000000000000000000003000000a803000000000000feffffff02000000feffffff0400000005000000060000000700000008000000090000000a0000000b0000000c0000000d0000000e0000000f0000001000000011000000feffffff13000000fefffffffeffffff160000001700000018000000feffffffffff
2129 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2130 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2131 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2132 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff010000020400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004c0000000000000000000800df1a0000d20400003c0f0000bc0200000000
2133 0000000000000000000000000000e8030000e8030000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000800df1ad2040000010009000003d001000003001500000000000500
2134 00000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c02600460181200000026060f001a00ffffffff000010000000c0ffffffb6ffffff20180000160400000b00000026060f000c004d617468547970650000e00009000000
2135 fa02000010000000000000002200040000002d0100000500000014021c0276000500000013021c0276010500000014024a004800050000001302ee0348000500000014024a009c01050000001302ee039c010500000014021c02ee050500000013021c021e1815000000fb0280fe00000000000090010100000000020010
2136 54696d6573204e657720526f6d616e00008d040000002d01010009000000320a8a019600010000006800c00009000000320aa8039600010000006b00aa0009000000320aa0022903010000007200950009000000320aa803c806010000007100c00009000000320aa803d00c010000007100c00009000000320aa8033b10
2137 010000006b00aa0009000000320aa803ab14010000007100c00015000000fb0220ff0000000000009001010000000002001054696d6573204e657720526f6d616e00008d040000002d01020004000000f001010009000000320a0003b4030100000049004a000c000000320a0804b507030000004d494e00ba004a009400
2138 0c000000320a0804bd0d030000004d494e00ba004a0094000c000000320a08041b11030000004d415800ba00880088000c000000320a08049815030000004d494e00ba004a00940010000000fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d01010004000000f001020009000000320a
2139 a0020802010000002d00d30009000000320aa002ae04010000003c00d30009000000320aa8038a13010000002d00d30015000000fb0280fe0000000000009001000000000002001054696d6573204e657720526f6d616e00008d040000002d01020004000000f001010009000000320a8a01ac0e010000003100c0000900
2140 0000320aa8030206010000003200c0000d000000320aa803b109040000006d6178282b01aa00c000800009000000320aa803ab0f010000002c00600009000000320aa803861701000000290080000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02000000000102022253797374
2141 656d0000040000002d01010004000000f00102000300000000000000000000000000000000000000000000000000000000000100feff030a0000ffffffff02ce020000000000c000000000000046170000004d6963726f736f6674204571756174696f6e20332e30000c0000004453204571756174696f6e000b00000045
2142 71756174696f6e2e3300f439b2710000000000000000000000000000000000000000000000000000000000000000000000000000000003000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001c0000000200bbd4bc00
2143 0000f4894a0054804a0000000000587a4a0002010102010a010304000001030e000001128368000112836b0000000296610296620002862d128372030f01000b011283490011000a02863c030e0000010288310001028832128371030f01000b0112834d12834912834e0011000a02980812826d12826112827802822812
2144 8371030f01000b0112834d12834912834e0011000a02822c12836b030f01000b0112834d1283411283580011000a02862d128371030f0100010043006f006d0070004f0062006a00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000120002000300
2145 000005000000ffffffff00000000000000000000000000000000000000000000000000000000000000000000000012000000660000000000000003004f0062006a0049006e0066006f0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000012000201
2146 ffffffff06000000ffffffff0000000000000000000000000000000000000000000000000000000000000000000000001400000004000000000000004500710075006100740069006f006e0020004e0061007400690076006500000000000000000000000000000000000000000000000000000000000000000000002000
2147 0200ffffffffffffffffffffffff00000000000000000000000000000000000000000000000000000000000000000000000015000000d80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
2148 00000000ffffffffffffffffffffffff0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000b0112834d12834912834e0011000a0282290000000001000000000000000000000000000000000000000000000000000000000000000000000000000000
2149 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
2150 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
2151 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
2152 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000105000000000000}{\result {\dn28 {\pict\wmetafile8\picw6879\pich1234\picwgoal3900\pichgoal700
2153 010009000003d00100000300150000000000050000000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c02600460181200000026060f001a00ffffffff000010000000c0ffffffb6ffffff20180000160400000b00000026060f
2154 000c004d617468547970650000e00009000000fa02000010000000000000002200040000002d0100000500000014021c0276000500000013021c0276010500000014024a004800050000001302ee0348000500000014024a009c01050000001302ee039c010500000014021c02ee050500000013021c021e1815000000fb02
2155 80fe0000000000009001010000000002001054696d6573204e657720526f6d616e00008d040000002d01010009000000320a8a019600010000006800c00009000000320aa8039600010000006b00aa0009000000320aa0022903010000007200950009000000320aa803c806010000007100c00009000000320aa803d00c01
2156 0000007100c00009000000320aa8033b10010000006b00aa0009000000320aa803ab14010000007100c00015000000fb0220ff0000000000009001010000000002001054696d6573204e657720526f6d616e00008d040000002d01020004000000f001010009000000320a0003b4030100000049004a000c000000320a0804
2157 b507030000004d494e00ba004a0094000c000000320a0804bd0d030000004d494e00ba004a0094000c000000320a08041b11030000004d415800ba00880088000c000000320a08049815030000004d494e00ba004a00940010000000fb0280fe0000000000009001000000020002001053796d626f6c0002040000002d0101
2158 0004000000f001020009000000320aa0020802010000002d00d30009000000320aa002ae04010000003c00d30009000000320aa8038a13010000002d00d30015000000fb0280fe0000000000009001000000000002001054696d6573204e657720526f6d616e00008d040000002d01020004000000f001010009000000320a
2159 8a01ac0e010000003100c00009000000320aa8030206010000003200c0000d000000320aa803b109040000006d6178282b01aa00c000800009000000320aa803ab0f010000002c00600009000000320aa803861701000000290080000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc
2160 02000000000102022253797374656d0000040000002d01010004000000f001020003000000000000303638005f546f0034393800313838}}}}}
2161 \par \pard \widctlpar
2162 \par However, this command calculates the distance as an upper bound on the distance between consecutive Farey terms, which is twice the result above. The formula used by this command to obtain a result is:
2163 \par
2164 \par \pard \qc\widctlpar {\pard\plain \qc\widctlpar \f4\fs20 {\object\objemb\objw3700\objh680{\*\objclass Equation.3}{\*\objdata 01050000020000000b0000004571756174696f6e2e3300000000000000000000100000
2165 d0cf11e0a1b11ae1000000000000000000000000000000003e000300feff0900060000000000000000000000010000000100000000000000001000000200000001000000feffffff0000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2166 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2167 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2168 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2169 fffffffffffffffffdffffff05000000feffffff0400000006000000fefffffffeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2170 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2171 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2172 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2173 ffffffffffffffffffffffffffffffff52006f006f007400200045006e00740072007900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000016000500ffffffffffffffff0200000002ce020000000000c0000000000000460000000000000000000000006098
2174 3ddf674dc00103000000800500000000000001004f006c00650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a000201ffffffffffffffffffffffff00000000000000000000000000000000000000000000000000000000
2175 0000000000000000000000001400000000000000030050004900430000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a0002010100000004000000ffffffff0000000000000000000000000000000000000000000000000000
2176 00000000000000000000010000004c0000000000000003004d004500540041000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000c000201ffffffffffffffffffffffff000000000000000000000000000000000000000000000000
2177 000000000000000000000000030000002003000000000000feffffff02000000feffffff0400000005000000060000000700000008000000090000000a0000000b0000000c0000000d0000000e0000000f000000feffffff11000000fefffffffeffffff1400000015000000feffffffffffffffffffffffffffffffffff
2178 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2179 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2180 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2181 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff010000020400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004c00000000000000000008007e190000af040000740e0000a80200000000
2182 0000000000000000000000000000e8030000e80300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000008007e19af0400000100090000038c01000003001500000000000500
2183 00000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c02400420171200000026060f001a00ffffffff000010000000c0ffffffb6ffffffe0160000f60300000b00000026060f000c004d617468547970650000e00009000000
2184 fa02000010000000000000002200040000002d010000050000001402fc016005050000001302fc01ca1615000000fb0280fe0000000000009001010000000002001054696d6573204e657720526f6d616e000087040000002d01010010000000320a8002340006000000726573756c749500aa009500c0006b006b000900
2185 0000320a88037405010000007100c00009000000320a88037c0b010000007100c00009000000320a8803e70e010000006b00aa0009000000320a88035713010000007100c00015000000fb0220ff0000000000009001010000000002001054696d6573204e657720526f6d616e000087040000002d01020004000000f001
2186 01000c000000320ae8036106030000004d494e00ba004a0094000c000000320ae803690c030000004d494e00ba004a0094000c000000320ae803c70f030000004d415800ba00880088000c000000320ae8034414030000004d494e00ba004a00940010000000fb0280fe0000000000009001000000020002001053796d62
2187 6f6c0002040000002d01010004000000f001020009000000320a80021a04010000003d00d30009000000320a88033612010000002d00d30015000000fb0280fe0000000000009001000000000002001054696d6573204e657720526f6d616e000087040000002d01020004000000f001010009000000320a6a01bb0d0100
2188 00003100c0000d000000320a88035d08040000006d6178282b01aa00c000800009000000320a8803570e010000002c00600009000000320a8803321601000000290080000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02000000000102022253797374656d0000040000002d01
2189 010004000000f00102000300000000000000000000000000000000000000000000000000000000000000000000000000010043006f006d0070004f0062006a00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000120002000300000005000000ffff
2190 ffff00000000000000000000000000000000000000000000000000000000000000000000000010000000660000000000000003004f0062006a0049006e0066006f0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000012000201ffffffff06000000
2191 ffffffff0000000000000000000000000000000000000000000000000000000000000000000000001200000004000000000000004500710075006100740069006f006e0020004e00610074006900760065000000000000000000000000000000000000000000000000000000000000000000000020000200ffffffffffff
2192 ffffffffffff00000000000000000000000000000000000000000000000000000000000000000000000013000000b8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffff
2193 ffffffffffffffff0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100feff030a0000ffffffff02ce020000000000c000000000000046170000004d6963726f736f6674204571756174696f6e20332e30000c0000004453204571756174696f6e
2194 000b0000004571756174696f6e2e3300f439b2710000000000000000000000000000000000000000000000000000000000000000000000000000000003000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001c000000
2195 0200bbd49c000000448b4a00cc894a0000000000a4874a0002010102010a0112837212836512837312837512836c12837402863d030e0000010288310001128371030f01000b0112834d12834912834e0011000a02980812826d128261128278028228128371030f01000b0112834d12834912834e0011000a02822c1283
2196 6b030f01000b0112834d1283411283580011000a02862d128371030f01000b0112834d12834912834e0011000a028229000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
2197 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000105000000000000}{\result {\dn28 {\pict\wmetafile8\picw6526\pich1199\picwgoal3700\pichgoal680
2198 0100090000038c0100000300150000000000050000000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c02400420171200000026060f001a00ffffffff000010000000c0ffffffb6ffffffe0160000f60300000b00000026060f
2199 000c004d617468547970650000e00009000000fa02000010000000000000002200040000002d010000050000001402fc016005050000001302fc01ca1615000000fb0280fe0000000000009001010000000002001054696d6573204e657720526f6d616e000087040000002d01010010000000320a80023400060000007265
2200 73756c749500aa009500c0006b006b0009000000320a88037405010000007100c00009000000320a88037c0b010000007100c00009000000320a8803e70e010000006b00aa0009000000320a88035713010000007100c00015000000fb0220ff0000000000009001010000000002001054696d6573204e657720526f6d616e
2201 000087040000002d01020004000000f00101000c000000320ae8036106030000004d494e00ba004a0094000c000000320ae803690c030000004d494e00ba004a0094000c000000320ae803c70f030000004d415800ba00880088000c000000320ae8034414030000004d494e00ba004a00940010000000fb0280fe00000000
2202 00009001000000020002001053796d626f6c0002040000002d01010004000000f001020009000000320a80021a04010000003d00d30009000000320a88033612010000002d00d30015000000fb0280fe0000000000009001000000000002001054696d6573204e657720526f6d616e000087040000002d01020004000000f0
2203 01010009000000320a6a01bb0d010000003100c0000d000000320a88035d08040000006d6178282b01aa00c000800009000000320a8803570e010000002c00600009000000320a8803321601000000290080000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc020000000001020222
2204 53797374656d0000040000002d01010004000000f00102000300000000000034393800353131006f633400383731}}}}}
2205 \par \pard \widctlpar
2206 \par {\*\bkmkstart _Toc498721710}{\pntext\pard\plain\f5 4.17.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721710}
2207 \par \pard\plain \widctlpar \f4\fs20 The example invocation below is the same example as at the end of the TOMS paper. Note that the upper bound obtained is twice the value in the TOMS paper, for the reasons discussed above.
2208 \par
2209 \par {\f11\fs16 c:\\>rap fndmax 0.385 0.386 500
2210 \par ------------------------------------------------------------------------------
2211 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
2212 \par ------------------------------------------------------------------------------
2213 \par l_h: 77 ( 2 digits)
2214 \par ------------------------------------------------------------------------------
2215 \par l_k: 200 ( 3 digits)
2216 \par ------------------------------------------------------------------------------
2217 \par r_h: 193 ( 3 digits)
2218 \par ------------------------------------------------------------------------------
2219 \par r_k: 500 ( 3 digits)
2220 \par ------------------------------------------------------------------------------
2221 \par k_max: 500 ( 3 digits)
2222 \par ------------------------------------------------------------------------------
2223 \par dap_D: 1, ( 109 digits)
2224 \par 000,000,000,000,000,000,000,000,000,
2225 \par 000,000,000,000,000,000,000,000,000,
2226 \par 000,000,000,000,000,000,000,000,000,
2227 \par 000,000,000,000,000,000,000,000,000
2228 \par ------------------------------------------------------------------------------
2229 \par midpoint_h: 771 ( 3 digits)
2230 \par ------------------------------------------------------------------------------
2231 \par midpoint_k: 2,000 ( 4 digits)
2232 \par ------------------------------------------------------------------------------
2233 \par ***************** CF Representation Of Interval Midpoint *****************
2234 \par ************************ Inputs To CF Calculation ************************
2235 \par ------------------------------------------------------------------------------
2236 \par h_in: 771 ( 3 digits)
2237 \par ------------------------------------------------------------------------------
2238 \par k_in: 2,000 ( 4 digits)
2239 \par ------------------------------------------------------------------------------
2240 \par ************************** CF Partial Quotients **************************
2241 \par ------------------------------------------------------------------------------
2242 \par a(0): 0 ( 1 digit)
2243 \par ------------------------------------------------------------------------------
2244 \par a(1): 2 ( 1 digit)
2245 \par ------------------------------------------------------------------------------
2246 \par a(2): 1 ( 1 digit)
2247 \par ------------------------------------------------------------------------------
2248 \par a(3): 1 ( 1 digit)
2249 \par ------------------------------------------------------------------------------
2250 \par a(4): 2 ( 1 digit)
2251 \par ------------------------------------------------------------------------------
2252 \par a(5): 6 ( 1 digit)
2253 \par ------------------------------------------------------------------------------
2254 \par a(6): 3 ( 1 digit)
2255 \par ------------------------------------------------------------------------------
2256 \par a(7): 3 ( 1 digit)
2257 \par ------------------------------------------------------------------------------
2258 \par a(8): 2 ( 1 digit)
2259 \par ------------------------------------------------------------------------------
2260 \par ***************************** CF Convergents *****************************
2261 \par ------------------------------------------------------------------------------
2262 \par p(0): 0 ( 1 digit)
2263 \par q(0): 1 ( 1 digit)
2264 \par ------------------------------------------------------------------------------
2265 \par p(1): 1 ( 1 digit)
2266 \par q(1): 2 ( 1 digit)
2267 \par ------------------------------------------------------------------------------
2268 \par p(2): 1 ( 1 digit)
2269 \par q(2): 3 ( 1 digit)
2270 \par ------------------------------------------------------------------------------
2271 \par p(3): 2 ( 1 digit)
2272 \par q(3): 5 ( 1 digit)
2273 \par ------------------------------------------------------------------------------
2274 \par p(4): 5 ( 1 digit)
2275 \par q(4): 13 ( 2 digits)
2276 \par ------------------------------------------------------------------------------
2277 \par p(5): 32 ( 2 digits)
2278 \par q(5): 83 ( 2 digits)
2279 \par ------------------------------------------------------------------------------
2280 \par p(6): 101 ( 3 digits)
2281 \par q(6): 262 ( 3 digits)
2282 \par ------------------------------------------------------------------------------
2283 \par p(7): 335 ( 3 digits)
2284 \par q(7): 869 ( 3 digits)
2285 \par ------------------------------------------------------------------------------
2286 \par p(8): 771 ( 3 digits)
2287 \par q(8): 2,000 ( 4 digits)
2288 \par ------------------------------------------------------------------------------
2289 \par ******** A Rational Number With Smallest Denominator In Interval ********
2290 \par ------------------------------------------------------------------------------
2291 \par result_h: 22 ( 2 digits)
2292 \par ------------------------------------------------------------------------------
2293 \par result_k: 57 ( 2 digits)
2294 \par ------------------------------------------------------------------------------
2295 \par ***** Upper Bound On Distance Between Farey Terms As Rational Number *****
2296 \par ------------------------------------------------------------------------------
2297 \par error_ub_h: 1 ( 1 digit)
2298 \par ------------------------------------------------------------------------------
2299 \par error_up_k: 25,251 ( 5 digits)
2300 \par ------------------------------------------------------------------------------
2301 \par ** Upper Bound On Distance Between Farey Terms As Decimal Approximation **
2302 \par ------------------------------------------------------------------------------
2303 \par dap_h: 39,602,391,984,475,862,342,085, ( 104 digits)
2304 \par 461,961,902,498,910,934,220,426,913,
2305 \par 785,592,649,796,047,681,279,949,308,
2306 \par 938,259,870,896,202,130,608,688,764
2307 \par ------------------------------------------------------------------------------
2308 \par dap_k: 1, ( 109 digits)
2309 \par 000,000,000,000,000,000,000,000,000,
2310 \par 000,000,000,000,000,000,000,000,000,
2311 \par 000,000,000,000,000,000,000,000,000,
2312 \par 000,000,000,000,000,000,000,000,000
2313 \par ------------------------------------------------------------------------------
2314 \par RAP execution ends.
2315 \par ------------------------------------------------------------------------------}
2316 \par \page
2317 \par {\*\bkmkstart _Toc498721711}{\pntext\pard\plain\b\i\f5 4.18\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Enclosing Rational Numbers In A Rectangular Area Of The Integer Lattice [FAB]
2318 {\*\bkmkend _Toc498721711}
2319 \par \pard\plain \widctlpar \f4\fs20
2320 \par {\*\bkmkstart _Toc498721712}{\pntext\pard\plain\f5 4.18.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721712}
2321 \par \pard\plain \widctlpar \f4\fs20 {\b rap fab R1 HMAX KMAX
2322 \par }\pard \li720\widctlpar Finds the two rational numbers in {\pard\plain \li720\widctlpar \f4\fs20 {\object\objemb\objw1120\objh380{\*\objclass Equation.3}{\*\objdata 01050000020000000b0000004571756174696f6e2e3300000000000000000000100000
2323 d0cf11e0a1b11ae1000000000000000000000000000000003e000300feff0900060000000000000000000000010000000100000000000000001000000200000001000000feffffff0000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2324 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2325 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2326 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2327 fffffffffffffffffdffffff05000000feffffff0400000006000000fefffffffeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2328 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2329 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2330 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2331 ffffffffffffffffffffffffffffffff52006f006f007400200045006e00740072007900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000016000500ffffffffffffffff0200000002ce020000000000c0000000000000460000000000000000000000006098
2332 3ddf674dc00103000000800400000000000001004f006c00650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a000201ffffffffffffffffffffffff00000000000000000000000000000000000000000000000000000000
2333 0000000000000000000000001400000000000000030050004900430000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a0002010100000004000000ffffffff0000000000000000000000000000000000000000000000000000
2334 00000000000000000000010000004c0000000000000003004d004500540041000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000c000201ffffffffffffffffffffffff000000000000000000000000000000000000000000000000
2335 000000000000000000000000030000007802000000000000feffffff02000000feffffff0400000005000000060000000700000008000000090000000a0000000b0000000c000000feffffff0e000000fefffffffeffffff11000000feffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2336 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2337 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2338 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
2339 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff010000020400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004c0000000000000000000800b70700009e020000600400007c0100000000
2340 0000000000000000000000000000e8030000e8030000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000800b7079e0200000100090000033701000003001500000000000500
2341 00000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c02600200071200000026060f001a00ffffffff000010000000c0ffffffc0ffffffc0060000200200000b00000026060f000c004d617468547970650000800009000000
2342 fa02000009000000000000002200040000002d0100000500000014021c0117040500000013021c01b8040500000014021c01c4040500000013021c017e050500000014021c0172050500000013021c01fa050500000014021c011b060500000013021c01a30615000000fb0280fe00000000000090010100000000020010
2343 54696d6573204e657720526f6d616e00008d040000002d01010009000000320a60014c00010000004600ea0015000000fb0220ff0000000000009001010000000002001054696d6573204e657720526f6d616e00008d040000002d01020004000000f00101000d000000320adf012b01040000004b4d41589400ba008800
2344 880009000000320adf010204010000004800a10009000000320adf01b104010000004d00ba0009000000320adf016d05010000004100880009000000320adf010406010000005800880015000000fb0220ff0000000000009001000000000002001054696d6573204e657720526f6d616e00008d040000002d0101000400
2345 0000f001020009000000320adf01b903010000002c0038000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02000000000102022253797374656d0000040000002d01020004000000f0010100030000000000000000000000000000000100feff030a0000ffffffff02ce02000000
2346 0000c000000000000046170000004d6963726f736f6674204571756174696f6e20332e30000c0000004453204571756174696f6e000b0000004571756174696f6e2e3300f439b271000000000000000000000000000000000000000000000000000000000000000000000000000000000300000000000000000000000000
2347 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010043006f006d0070004f0062006a00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000120002000300000005000000ffff
2348 ffff0000000000000000000000000000000000000000000000000000000000000000000000000d000000660000000000000003004f0062006a0049006e0066006f0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000012000201ffffffff06000000
2349 ffffffff0000000000000000000000000000000000000000000000000000000000000000000000000f00000004000000000000004500710075006100740069006f006e0020004e00610074006900760065000000000000000000000000000000000000000000000000000000000000000000000020000200ffffffffffff
2350 ffffffffffff0000000000000000000000000000000000000000000000000000000000000000000000001000000058000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffff
2351 ffffffffffffffff0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001c0000000200b2d53c000000f4894a0054804a0000000000587a4a0002010102010a01128346030f01000b0112834b12834d12834112835802822c32834806110032834d0611
2352 003283410611003283580611000011000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
2353 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
2354 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
2355 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000105000000000000}{\result {\dn16 {\pict\wmetafile8\picw1975\pich670\picwgoal1120\pichgoal380
2356 010009000003370100000300150000000000050000000902000000000400000002010100050000000102ffffff00040000002e01180005000000310201000000050000000b0200000000050000000c02600200071200000026060f001a00ffffffff000010000000c0ffffffc0ffffffc0060000200200000b00000026060f
2357 000c004d617468547970650000800009000000fa02000009000000000000002200040000002d0100000500000014021c0117040500000013021c01b8040500000014021c01c4040500000013021c017e050500000014021c0172050500000013021c01fa050500000014021c011b060500000013021c01a30615000000fb02
2358 80fe0000000000009001010000000002001054696d6573204e657720526f6d616e00008d040000002d01010009000000320a60014c00010000004600ea0015000000fb0220ff0000000000009001010000000002001054696d6573204e657720526f6d616e00008d040000002d01020004000000f00101000d000000320adf
2359 012b01040000004b4d41589400ba008800880009000000320adf010204010000004800a10009000000320adf01b104010000004d00ba0009000000320adf016d05010000004100880009000000320adf010406010000005800880015000000fb0220ff0000000000009001000000000002001054696d6573204e657720526f
2360 6d616e00008d040000002d01010004000000f001020009000000320adf01b903010000002c0038000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02000000000102022253797374656d0000040000002d01020004000000f00101000300000000000000005f546f00343938003837
2361 3000c903c900c903c900380d5f006f6334}}}}} which enclose the non-negative rational number R1.
2362 \par \pard \widctlpar {\b rap fab R1 HMAX KMAX NNEIGHBORS D
2363 \par }\pard \li720\widctlpar Same as above but provides NNEIGHBORS terms in F{\sub KMAX,HMAX} on both the left and right, and will emit decimal approximations with denominator D.
2364 \par {\*\bkmkstart _Toc498721713}{\pntext\pard\plain\f5 4.18.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721713}
2365 \par \pard\plain \widctlpar \f4\fs20 Applies the continued fraction algorithms described in the TOMS paper to obtain neighbors to an arbitrary non-negative rational number R1 in F{\sub KMAX,HMAX}
2366 . There are two cases to consider: either the supplied rational number R1 is in F{\sub KMAX,HMAX}, or it is not. The algorithm will announce clearly which case applies. In either case, the algorithm applied is nearly identical.
2367 \par
2368 \par {\*\bkmkstart _Toc498721714}{\pntext\pard\plain\f5 4.18.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721714}
2369 \par \pard\plain \widctlpar \f4\fs20 The invocation below shows RAP used to find the neighbors to 0.31830989 (approximately 1/{{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}}) in the rectangular area of the integer lattice bounded by h{
2370 {\field{\*\fldinst SYMBOL 163 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}}193 and k{{\field{\*\fldinst SYMBOL 163 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}}500. Although it isn\rquote t shown below, RAP handles all of the boundary cases at the \ldblquote
2371 corner\rdblquote near h/k=193/500. The output below includes narrative explanations in a different font and with shading.
2372 \par
2373 \par {\f11\fs16 c:\\>rap fab 0.31830989 193 500 2 1e54
2374 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 In the command-line invocation above, \ldblquote 2\rdblquote tells RAP to display 2 neighbors on each side of the rational number to be approximated, and \ldblquote 1e54\rdblquote
2375 is the DAP denominator, which indicates that two lines of digits after the decimal point are desired (see the DAP command).
2376 \par \pard \widctlpar {\f11\fs16 ------------------------------------------------------------------------------
2377 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
2378 \par ------------------------------------------------------------------------------
2379 \par **************** Rational Number h_in/k_in To Approximate ****************
2380 \par ------------------------------------------------------------------------------
2381 \par h_in: 31,830,989 ( 8 digits)
2382 \par ------------------------------------------------------------------------------
2383 \par k_in: 100,000,000 ( 9 digits)
2384 \par ------------------------------------------------------------------------------
2385 \par *********************** Other Solution Parameters ***********************
2386 \par ------------------------------------------------------------------------------
2387 \par hmax: 193 ( 3 digits)
2388 \par ------------------------------------------------------------------------------
2389 \par kmax: 500 ( 3 digits)
2390 \par ------------------------------------------------------------------------------
2391 \par NNEIGHBORS: 2 ( 1 digit)
2392 \par ------------------------------------------------------------------------------
2393 \par DAP Denominator: 1, ( 55 digits)
2394 \par 000,000,000,000,000,000,000,000,000,
2395 \par 000,000,000,000,000,000,000,000,000
2396 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 Below are the continued fraction partial quotients and convergents of the rational number to be approximated. These are necessary to economically locate the immediate Farey neighbors along the
2397 \ldblquote right wall\rdblquote of the rectangular area of the integer lattice, if this proves necessary.
2398 \par \pard \widctlpar {\f11\fs16 ------------------------------------------------------------------------------
2399 \par *************** Continued Fraction Expansion Of h_in/k_in ***************
2400 \par ------------------------------------------------------------------------------
2401 \par ************************ Inputs To CF Calculation ************************
2402 \par ------------------------------------------------------------------------------
2403 \par h_in: 31,830,989 ( 8 digits)
2404 \par ------------------------------------------------------------------------------
2405 \par k_in: 100,000,000 ( 9 digits)
2406 \par ------------------------------------------------------------------------------
2407 \par ************************** CF Partial Quotients **************************
2408 \par ------------------------------------------------------------------------------
2409 \par a(0): 0 ( 1 digit)
2410 \par ------------------------------------------------------------------------------
2411 \par a(1): 3 ( 1 digit)
2412 \par ------------------------------------------------------------------------------
2413 \par a(2): 7 ( 1 digit)
2414 \par ------------------------------------------------------------------------------
2415 \par a(3): 15 ( 2 digits)
2416 \par ------------------------------------------------------------------------------
2417 \par a(4): 1 ( 1 digit)
2418 \par ------------------------------------------------------------------------------
2419 \par a(5): 256 ( 3 digits)
2420 \par ------------------------------------------------------------------------------
2421 \par a(6): 3 ( 1 digit)
2422 \par ------------------------------------------------------------------------------
2423 \par a(7): 5 ( 1 digit)
2424 \par ------------------------------------------------------------------------------
2425 \par a(8): 5 ( 1 digit)
2426 \par ------------------------------------------------------------------------------
2427 \par a(9): 13 ( 2 digits)
2428 \par ------------------------------------------------------------------------------
2429 \par ***************************** CF Convergents *****************************
2430 \par ------------------------------------------------------------------------------
2431 \par p(0): 0 ( 1 digit)
2432 \par q(0): 1 ( 1 digit)
2433 \par ------------------------------------------------------------------------------
2434 \par p(1): 1 ( 1 digit)
2435 \par q(1): 3 ( 1 digit)
2436 \par ------------------------------------------------------------------------------
2437 \par p(2): 7 ( 1 digit)
2438 \par q(2): 22 ( 2 digits)
2439 \par ------------------------------------------------------------------------------
2440 \par p(3): 106 ( 3 digits)
2441 \par q(3): 333 ( 3 digits)
2442 \par ------------------------------------------------------------------------------
2443 \par p(4): 113 ( 3 digits)
2444 \par q(4): 355 ( 3 digits)
2445 \par ------------------------------------------------------------------------------
2446 \par p(5): 29,034 ( 5 digits)
2447 \par q(5): 91,213 ( 5 digits)
2448 \par ------------------------------------------------------------------------------
2449 \par p(6): 87,215 ( 5 digits)
2450 \par q(6): 273,994 ( 6 digits)
2451 \par ------------------------------------------------------------------------------
2452 \par p(7): 465,109 ( 6 digits)
2453 \par q(7): 1,461,183 ( 7 digits)
2454 \par ------------------------------------------------------------------------------
2455 \par p(8): 2,412,760 ( 7 digits)
2456 \par q(8): 7,579,909 ( 7 digits)
2457 \par ------------------------------------------------------------------------------
2458 \par p(9): 31,830,989 ( 8 digits)
2459 \par q(9): 100,000,000 ( 9 digits)
2460 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500
2461 Below are the continued fraction partial quotients and convergents of the reciprocal of the rational number to be approximated. These are necessary to economically locate the immediate Farey neighbors along the \ldblquote top wall\rdblquote
2462 of the rectangular area of the integer lattice, if this proves necessary.
2463 \par \pard \widctlpar {\f11\fs16 ------------------------------------------------------------------------------
2464 \par *************** Continued Fraction Expansion Of k_in/h_in ***************
2465 \par ------------------------------------------------------------------------------
2466 \par ************************ Inputs To CF Calculation ************************
2467 \par ------------------------------------------------------------------------------
2468 \par h_in: 100,000,000 ( 9 digits)
2469 \par ------------------------------------------------------------------------------
2470 \par k_in: 31,830,989 ( 8 digits)
2471 \par ------------------------------------------------------------------------------
2472 \par ************************** CF Partial Quotients **************************
2473 \par ------------------------------------------------------------------------------
2474 \par a(0): 3 ( 1 digit)
2475 \par ------------------------------------------------------------------------------
2476 \par a(1): 7 ( 1 digit)
2477 \par ------------------------------------------------------------------------------
2478 \par a(2): 15 ( 2 digits)
2479 \par ------------------------------------------------------------------------------
2480 \par a(3): 1 ( 1 digit)
2481 \par ------------------------------------------------------------------------------
2482 \par a(4): 256 ( 3 digits)
2483 \par ------------------------------------------------------------------------------
2484 \par a(5): 3 ( 1 digit)
2485 \par ------------------------------------------------------------------------------
2486 \par a(6): 5 ( 1 digit)
2487 \par ------------------------------------------------------------------------------
2488 \par a(7): 5 ( 1 digit)
2489 \par ------------------------------------------------------------------------------
2490 \par a(8): 13 ( 2 digits)
2491 \par ------------------------------------------------------------------------------
2492 \par ***************************** CF Convergents *****************************
2493 \par ------------------------------------------------------------------------------
2494 \par p(0): 3 ( 1 digit)
2495 \par q(0): 1 ( 1 digit)
2496 \par ------------------------------------------------------------------------------
2497 \par p(1): 22 ( 2 digits)
2498 \par q(1): 7 ( 1 digit)
2499 \par ------------------------------------------------------------------------------
2500 \par p(2): 333 ( 3 digits)
2501 \par q(2): 106 ( 3 digits)
2502 \par ------------------------------------------------------------------------------
2503 \par p(3): 355 ( 3 digits)
2504 \par q(3): 113 ( 3 digits)
2505 \par ------------------------------------------------------------------------------
2506 \par p(4): 91,213 ( 5 digits)
2507 \par q(4): 29,034 ( 5 digits)
2508 \par ------------------------------------------------------------------------------
2509 \par p(5): 273,994 ( 6 digits)
2510 \par q(5): 87,215 ( 5 digits)
2511 \par ------------------------------------------------------------------------------
2512 \par p(6): 1,461,183 ( 7 digits)
2513 \par q(6): 465,109 ( 6 digits)
2514 \par ------------------------------------------------------------------------------
2515 \par p(7): 7,579,909 ( 7 digits)
2516 \par q(7): 2,412,760 ( 7 digits)
2517 \par ------------------------------------------------------------------------------
2518 \par p(8): 100,000,000 ( 9 digits)
2519 \par q(8): 31,830,989 ( 8 digits)
2520 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 Below are the continued fraction partial quotients and convergents of the corner point [i.e. (k{\sub MAX}, h{\sub MAX}
2521 )]. These are necessary to economically locate the immediate Farey neighbor just to the left on the number line, along the \ldblquote right wall\rdblquote of the rectangular region of the integer lattice.
2522 \par \pard \widctlpar {\f11\fs16 ------------------------------------------------------------------------------
2523 \par ************** Continued Fraction Expansion Of Corner Point **************
2524 \par ------------------------------------------------------------------------------
2525 \par ************************ Inputs To CF Calculation ************************
2526 \par ------------------------------------------------------------------------------
2527 \par h_in: 193 ( 3 digits)
2528 \par ------------------------------------------------------------------------------
2529 \par k_in: 500 ( 3 digits)
2530 \par ------------------------------------------------------------------------------
2531 \par ************************** CF Partial Quotients **************************
2532 \par ------------------------------------------------------------------------------
2533 \par a(0): 0 ( 1 digit)
2534 \par ------------------------------------------------------------------------------
2535 \par a(1): 2 ( 1 digit)
2536 \par ------------------------------------------------------------------------------
2537 \par a(2): 1 ( 1 digit)
2538 \par ------------------------------------------------------------------------------
2539 \par a(3): 1 ( 1 digit)
2540 \par ------------------------------------------------------------------------------
2541 \par a(4): 2 ( 1 digit)
2542 \par ------------------------------------------------------------------------------
2543 \par a(5): 3 ( 1 digit)
2544 \par ------------------------------------------------------------------------------
2545 \par a(6): 1 ( 1 digit)
2546 \par ------------------------------------------------------------------------------
2547 \par a(7): 8 ( 1 digit)
2548 \par ------------------------------------------------------------------------------
2549 \par ***************************** CF Convergents *****************************
2550 \par ------------------------------------------------------------------------------
2551 \par p(0): 0 ( 1 digit)
2552 \par q(0): 1 ( 1 digit)
2553 \par ------------------------------------------------------------------------------
2554 \par p(1): 1 ( 1 digit)
2555 \par q(1): 2 ( 1 digit)
2556 \par ------------------------------------------------------------------------------
2557 \par p(2): 1 ( 1 digit)
2558 \par q(2): 3 ( 1 digit)
2559 \par ------------------------------------------------------------------------------
2560 \par p(3): 2 ( 1 digit)
2561 \par q(3): 5 ( 1 digit)
2562 \par ------------------------------------------------------------------------------
2563 \par p(4): 5 ( 1 digit)
2564 \par q(4): 13 ( 2 digits)
2565 \par ------------------------------------------------------------------------------
2566 \par p(5): 17 ( 2 digits)
2567 \par q(5): 44 ( 2 digits)
2568 \par ------------------------------------------------------------------------------
2569 \par p(6): 22 ( 2 digits)
2570 \par q(6): 57 ( 2 digits)
2571 \par ------------------------------------------------------------------------------
2572 \par p(7): 193 ( 3 digits)
2573 \par q(7): 500 ( 3 digits)
2574 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 Below are the continued fraction partial quotients and convergents of the reciprocal of the corner point [i.e. (k{\sub MAX}, h{\sub MAX}
2575 )]. These are necessary to economically locate the immediate Farey neighbor just to the right on the number line, along the \ldblquote top wall\rdblquote of the rectangular region of the integer lattice.
2576 \par \pard \widctlpar {\f11\fs16 ------------------------------------------------------------------------------
2577 \par ******** Continued Fraction Expansion Of Corner Point Reciprocal ********
2578 \par ------------------------------------------------------------------------------
2579 \par ************************ Inputs To CF Calculation ************************
2580 \par ------------------------------------------------------------------------------
2581 \par h_in: 500 ( 3 digits)
2582 \par ------------------------------------------------------------------------------
2583 \par k_in: 193 ( 3 digits)
2584 \par ------------------------------------------------------------------------------
2585 \par ************************** CF Partial Quotients **************************
2586 \par ------------------------------------------------------------------------------
2587 \par a(0): 2 ( 1 digit)
2588 \par ------------------------------------------------------------------------------
2589 \par a(1): 1 ( 1 digit)
2590 \par ------------------------------------------------------------------------------
2591 \par a(2): 1 ( 1 digit)
2592 \par ------------------------------------------------------------------------------
2593 \par a(3): 2 ( 1 digit)
2594 \par ------------------------------------------------------------------------------
2595 \par a(4): 3 ( 1 digit)
2596 \par ------------------------------------------------------------------------------
2597 \par a(5): 1 ( 1 digit)
2598 \par ------------------------------------------------------------------------------
2599 \par a(6): 8 ( 1 digit)
2600 \par ------------------------------------------------------------------------------
2601 \par ***************************** CF Convergents *****************************
2602 \par ------------------------------------------------------------------------------
2603 \par p(0): 2 ( 1 digit)
2604 \par q(0): 1 ( 1 digit)
2605 \par ------------------------------------------------------------------------------
2606 \par p(1): 3 ( 1 digit)
2607 \par q(1): 1 ( 1 digit)
2608 \par ------------------------------------------------------------------------------
2609 \par p(2): 5 ( 1 digit)
2610 \par q(2): 2 ( 1 digit)
2611 \par ------------------------------------------------------------------------------
2612 \par p(3): 13 ( 2 digits)
2613 \par q(3): 5 ( 1 digit)
2614 \par ------------------------------------------------------------------------------
2615 \par p(4): 44 ( 2 digits)
2616 \par q(4): 17 ( 2 digits)
2617 \par ------------------------------------------------------------------------------
2618 \par p(5): 57 ( 2 digits)
2619 \par q(5): 22 ( 2 digits)
2620 \par ------------------------------------------------------------------------------
2621 \par p(6): 500 ( 3 digits)
2622 \par q(6): 193 ( 3 digits)
2623 \par ------------------------------------------------------------------------------
2624 \par ************************* Corner Point Neighbors *************************
2625 \par ------------------------------------------------------------------------------
2626 \par corner_pred_h: 22 ( 2 digits)
2627 \par corner_pred_k: 57 ( 2 digits)
2628 \par ------------------------------------------------------------------------------
2629 \par corner_succ_h: 171 ( 3 digits)
2630 \par corner_succ_k: 443 ( 3 digits)
2631 \par ------------------------------------------------------------------------------
2632 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 The lines below advise that the rational number to approximated is not in the rectangular area of the integer lattice (i.e. can\rquote
2633 t be represented subject to the constraints). (This is the most typical case, as we seldom seek neighbors to a number we can represent exactly.) This means that none of the neighbors will be subscripted \ldblquote 0\rdblquote .
2634 \par \pard \widctlpar {\f11\fs16 ******************************************************************************
2635 \par ******** h_in/k_in IS NOT In Rectangular Farey Series Of Interest ********
2636 \par ******************************************************************************
2637 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 The statement below advises that the rational number to be approximated is less than h{\sub MAX}/k{\sub MAX}
2638 \par (i.e. to the left of the corner point) in the rectangular area of the integer lattice formed by the constraints. All boundary cases are covered when the neighbors span the corner.
2639 \par \pard \widctlpar {\f11\fs16 ------------------------------------------------------------------------------
2640 \par *********************** 0 <= h_in/k_in < hmax/kmax ***********************
2641 \par ------------------------------------------------------------------------------
2642 \par ***************** Highest-Order Convergent With q(i)<=N *****************
2643 \par ------------------------------------------------------------------------------
2644 \par p(4): 113 ( 3 digits)
2645 \par q(4): 355 ( 3 digits)
2646 \par ------------------------------------------------------------------------------
2647 \par ******************* Accompanying Intermediate Fraction *******************
2648 \par ------------------------------------------------------------------------------
2649 \par intermediate_h: 106 ( 3 digits)
2650 \par intermediate_k: 333 ( 3 digits)
2651 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 Below is a typical neighbor. The subscript of \ldblquote -2\rdblquote indicates it is the second neighbor to the left on the number line from the number to be approximated, subject to the constrai
2652 nts.
2653 \par \pard \widctlpar {\f11\fs16 ------------------------------------------------------------------------------
2654 \par ****************** Rectangular Farey Neighbor Index -2 ******************
2655 \par ------------------------------------------------------------------------------
2656 \par h(-2): 120 ( 3 digits)
2657 \par ------------------------------------------------------------------------------
2658 \par k(-2): 377 ( 3 digits)
2659 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 Below, the number 120/377 is rephrased as a rational number with a larger denominator which is a power of ten. This says that this number is 0.318302\'85
2660 \par \pard \widctlpar {\f11\fs16
2661 \par ------------------------------------------------------------------------------
2662 \par DAP_N(-2): 318,302,387,267,904,509,283,819,628, ( 54 digits)
2663 \par 647,214,854,111,405,835,543,766,578
2664 \par ------------------------------------------------------------------------------
2665 \par DAP_D(-2): 1, ( 55 digits)
2666 \par 000,000,000,000,000,000,000,000,000,
2667 \par 000,000,000,000,000,000,000,000,000
2668 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 The error (the difference between the approximation and the number to be approximated) is supplied as a rational number.
2669 \par \pard \widctlpar {\f11\fs16 ------------------------------------------------------------------------------
2670 \par error_h(-2): - 282,853 ( 6 digits)
2671 \par ------------------------------------------------------------------------------
2672 \par error_k(-2): 37,700,000,000 ( 11 digits)
2673 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 The error just above is restated with a power-of-ten denominator. This says that the error is about -0.0000075\'85
2674 \par \pard \widctlpar {\f11\fs16 ------------------------------------------------------------------------------
2675 \par ERROR_DAP_N(-2): - 7,502,732,095,490,716,180,371, ( 49 digits)
2676 \par 352,785,145,888,594,164,456,233,421
2677 \par ------------------------------------------------------------------------------
2678 \par ERROR_DAP_D(-2): 1, ( 55 digits)
2679 \par 000,000,000,000,000,000,000,000,000,
2680 \par 000,000,000,000,000,000,000,000,000
2681 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 The same information described above is repeated for each neighbor.
2682 \par \pard \widctlpar {\f11\fs16 ------------------------------------------------------------------------------
2683 \par ****************** Rectangular Farey Neighbor Index -1 ******************
2684 \par ------------------------------------------------------------------------------
2685 \par h(-1): 113 ( 3 digits)
2686 \par ------------------------------------------------------------------------------
2687 \par k(-1): 355 ( 3 digits)
2688 \par ------------------------------------------------------------------------------
2689 \par DAP_N(-1): 318,309,859,154,929,577,464,788,732, ( 54 digits)
2690 \par 394,366,197,183,098,591,549,295,774
2691 \par ------------------------------------------------------------------------------
2692 \par DAP_D(-1): 1, ( 55 digits)
2693 \par 000,000,000,000,000,000,000,000,000,
2694 \par 000,000,000,000,000,000,000,000,000
2695 \par ------------------------------------------------------------------------------
2696 \par error_h(-1): - 219 ( 3 digits)
2697 \par ------------------------------------------------------------------------------
2698 \par error_k(-1): 7,100,000,000 ( 10 digits)
2699 \par ------------------------------------------------------------------------------
2700 \par ERROR_DAP_N(-1): - 30,845,070,422,535,211,267, ( 47 digits)
2701 \par 605,633,802,816,901,408,450,704,225
2702 \par ------------------------------------------------------------------------------
2703 \par ERROR_DAP_D(-1): 1, ( 55 digits)
2704 \par 000,000,000,000,000,000,000,000,000,
2705 \par 000,000,000,000,000,000,000,000,000
2706 \par ------------------------------------------------------------------------------
2707 \par ******************* Rectangular Farey Neighbor Index 1 *******************
2708 \par ------------------------------------------------------------------------------
2709 \par h(1): 106 ( 3 digits)
2710 \par ------------------------------------------------------------------------------
2711 \par k(1): 333 ( 3 digits)
2712 \par ------------------------------------------------------------------------------
2713 \par DAP_N(1): 318,318,318,318,318,318,318,318,318, ( 54 digits)
2714 \par 318,318,318,318,318,318,318,318,318
2715 \par ------------------------------------------------------------------------------
2716 \par DAP_D(1): 1, ( 55 digits)
2717 \par 000,000,000,000,000,000,000,000,000,
2718 \par 000,000,000,000,000,000,000,000,000
2719 \par ------------------------------------------------------------------------------
2720 \par error_h(1): 280,663 ( 6 digits)
2721 \par ------------------------------------------------------------------------------
2722 \par error_k(1): 33,300,000,000 ( 11 digits)
2723 \par ------------------------------------------------------------------------------
2724 \par ERROR_DAP_N(1): 8,428,318,318,318,318,318,318, ( 49 digits)
2725 \par 318,318,318,318,318,318,318,318,318
2726 \par ------------------------------------------------------------------------------
2727 \par ERROR_DAP_D(1): 1, ( 55 digits)
2728 \par 000,000,000,000,000,000,000,000,000,
2729 \par 000,000,000,000,000,000,000,000,000
2730 \par ------------------------------------------------------------------------------
2731 \par ******************* Rectangular Farey Neighbor Index 2 *******************
2732 \par ------------------------------------------------------------------------------
2733 \par h(2): 99 ( 2 digits)
2734 \par ------------------------------------------------------------------------------
2735 \par k(2): 311 ( 3 digits)
2736 \par ------------------------------------------------------------------------------
2737 \par DAP_N(2): 318,327,974,276,527,331,189,710,610, ( 54 digits)
2738 \par 932,475,884,244,372,990,353,697,749
2739 \par ------------------------------------------------------------------------------
2740 \par DAP_D(2): 1, ( 55 digits)
2741 \par 000,000,000,000,000,000,000,000,000,
2742 \par 000,000,000,000,000,000,000,000,000
2743 \par ------------------------------------------------------------------------------
2744 \par error_h(2): 562,421 ( 6 digits)
2745 \par ------------------------------------------------------------------------------
2746 \par error_k(2): 31,100,000,000 ( 11 digits)
2747 \par ------------------------------------------------------------------------------
2748 \par ERROR_DAP_N(2): 18,084,276,527,331,189,710,610, ( 50 digits)
2749 \par 932,475,884,244,372,990,353,697,749
2750 \par ------------------------------------------------------------------------------
2751 \par ERROR_DAP_D(2): 1, ( 55 digits)
2752 \par 000,000,000,000,000,000,000,000,000,
2753 \par 000,000,000,000,000,000,000,000,000
2754 \par ------------------------------------------------------------------------------
2755 \par RAP execution ends.
2756 \par ------------------------------------------------------------------------------}
2757 \par \page
2758 \par {\*\bkmkstart _Toc498721715}{\pntext\pard\plain\b\i\f5 4.19\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Upper Bound On Distance Between Members Of F{\sub A,B} In An Interval [FABDMAX]
2759 {\*\bkmkend _Toc498721715}
2760 \par \pard\plain \widctlpar \f4\fs20
2761 \par {\*\bkmkstart _Toc498721716}{\pntext\pard\plain\f5 4.19.1\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Command Line Invocation Forms{\*\bkmkend _Toc498721716}
2762 \par \pard\plain \widctlpar \f4\fs20 {\b rap fab LL UL HMAX KMAX
2763 \par }\pard \li720\widctlpar Calculates an upper bound on the maximum distance between terms of F{\sub KMAX,HMAX} in the interval [LL,UL] using the techniques developed in the TOMS paper. LL and UL must be non-negative, and must both be in F{\sub KMAX,HMAX}
2764 . HMAX and KMAX must both be positive.
2765 \par
2766 \par {\*\bkmkstart _Toc498721717}{\pntext\pard\plain\f5 4.19.2\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Detailed Algorithm Description{\*\bkmkend _Toc498721717}
2767 \par \pard\plain \widctlpar \f4\fs20 The TOMS paper develops several ways of bounding the maximum distance between terms in F{\sub KMAX,HMAX}
2768 , and which is applicable depends on whether HMAX is the dominant constraint, or KMAX is the dominant constraint. The RAP program output clearly explains how it calculates an upper bound on the distance between terms in F{\sub KMAX,HMAX}.
2769 \par
2770 \par {\*\bkmkstart _Toc498721718}{\pntext\pard\plain\f5 4.19.3\tab}\pard\plain \s3\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl3\pndec\pnprev1\pnstart1\pnsp144 {\pntxtb .}}\f5 Example Invocation{\*\bkmkend _Toc498721718}
2771 \par \pard\plain \widctlpar \f4\fs20 The example invocation below demonstrates error bounds over the interval [0.385, 2.160] with h{\sub MAX} {{\field{\*\fldinst SYMBOL 163 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}} 193 and k{\sub MAX} {{\field{\*\fldinst SYMBOL
2772 163 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}} 500. This is the same example used in the TOMS paper. Explanatory remarks are included in a different font and with shading.
2773 \par
2774 \par {\f11\fs16 c:\\>rap fabdmax 0.385 2.160 193 500
2775 \par ------------------------------------------------------------------------------
2776 \par RAP ($Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $) execution begins.
2777 \par ------------------------------------------------------------------------------
2778 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 The section below just echoes the command-line or batch inputs.
2779 \par \pard \widctlpar {\f11\fs16 ******************************************************************************
2780 \par ******************************************************************************
2781 \par **************************** Input Parameters ****************************
2782 \par ******************************************************************************
2783 \par ******************************************************************************
2784 \par ------------------------------------------------------------------------------
2785 \par l_h: 77 ( 2 digits)
2786 \par ------------------------------------------------------------------------------
2787 \par l_k: 200 ( 3 digits)
2788 \par ------------------------------------------------------------------------------
2789 \par r_h: 54 ( 2 digits)
2790 \par ------------------------------------------------------------------------------
2791 \par r_k: 25 ( 2 digits)
2792 \par ------------------------------------------------------------------------------
2793 \par h_max: 193 ( 3 digits)
2794 \par ------------------------------------------------------------------------------
2795 \par k_max: 500 ( 3 digits)
2796 \par ------------------------------------------------------------------------------
2797 \par dap_D: 1, ( 109 digits)
2798 \par 000,000,000,000,000,000,000,000,000,
2799 \par 000,000,000,000,000,000,000,000,000,
2800 \par 000,000,000,000,000,000,000,000,000,
2801 \par 000,000,000,000,000,000,000,000,000
2802 \par ------------------------------------------------------------------------------
2803 \par ******************************************************************************
2804 \par ******************************************************************************
2805 \par ************************* Case Selection Results *************************
2806 \par ******************************************************************************
2807 \par ******************************************************************************
2808 \par ------------------------------------------------------------------------------
2809 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500
2810 The program decides which cases should be evaluated to give an error bound. Even though the highest-numbered case will always give the larger bounds, unnecessary cases which apply are evaluated in case the user is interested in different upper bounds ove
2811 r different portions of the interval.
2812 \par \pard \widctlpar {\f11\fs16 Case I applies and will be evaluated.
2813 \par Case II applies and will be evaluated.
2814 \par Case III applies and will be evaluated.
2815 \par ------------------------------------------------------------------------------
2816 \par ******************************************************************************
2817 \par ******************************************************************************
2818 \par ************************ Start Of Case I Results ************************
2819 \par ******************************************************************************
2820 \par ******************************************************************************
2821 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 These are the Case I results. Note that the right endpoint of the interval is truncated at HMAX/KMAX.
2822 \par \pard \widctlpar {\f11\fs16 ------------------------------------------------------------------------------
2823 \par ***************** Possibly Truncated Interval For Case I *****************
2824 \par ------------------------------------------------------------------------------
2825 \par l_h: 77 ( 2 digits)
2826 \par ------------------------------------------------------------------------------
2827 \par l_k: 200 ( 3 digits)
2828 \par ------------------------------------------------------------------------------
2829 \par r_h: 193 ( 3 digits)
2830 \par ------------------------------------------------------------------------------
2831 \par r_k: 500 ( 3 digits)
2832 \par ------------------------------------------------------------------------------
2833 \par midpoint_h: 771 ( 3 digits)
2834 \par ------------------------------------------------------------------------------
2835 \par midpoint_k: 2,000 ( 4 digits)
2836 \par ------------------------------------------------------------------------------
2837 \par ***************** CF Representation Of Interval Midpoint *****************
2838 \par ************************ Inputs To CF Calculation ************************
2839 \par ------------------------------------------------------------------------------
2840 \par h_in: 771 ( 3 digits)
2841 \par ------------------------------------------------------------------------------
2842 \par k_in: 2,000 ( 4 digits)
2843 \par ------------------------------------------------------------------------------
2844 \par ************************** CF Partial Quotients **************************
2845 \par ------------------------------------------------------------------------------
2846 \par a(0): 0 ( 1 digit)
2847 \par ------------------------------------------------------------------------------
2848 \par a(1): 2 ( 1 digit)
2849 \par ------------------------------------------------------------------------------
2850 \par a(2): 1 ( 1 digit)
2851 \par ------------------------------------------------------------------------------
2852 \par a(3): 1 ( 1 digit)
2853 \par ------------------------------------------------------------------------------
2854 \par a(4): 2 ( 1 digit)
2855 \par ------------------------------------------------------------------------------
2856 \par a(5): 6 ( 1 digit)
2857 \par ------------------------------------------------------------------------------
2858 \par a(6): 3 ( 1 digit)
2859 \par ------------------------------------------------------------------------------
2860 \par a(7): 3 ( 1 digit)
2861 \par ------------------------------------------------------------------------------
2862 \par a(8): 2 ( 1 digit)
2863 \par ------------------------------------------------------------------------------
2864 \par ***************************** CF Convergents *****************************
2865 \par ------------------------------------------------------------------------------
2866 \par p(0): 0 ( 1 digit)
2867 \par q(0): 1 ( 1 digit)
2868 \par ------------------------------------------------------------------------------
2869 \par p(1): 1 ( 1 digit)
2870 \par q(1): 2 ( 1 digit)
2871 \par ------------------------------------------------------------------------------
2872 \par p(2): 1 ( 1 digit)
2873 \par q(2): 3 ( 1 digit)
2874 \par ------------------------------------------------------------------------------
2875 \par p(3): 2 ( 1 digit)
2876 \par q(3): 5 ( 1 digit)
2877 \par ------------------------------------------------------------------------------
2878 \par p(4): 5 ( 1 digit)
2879 \par q(4): 13 ( 2 digits)
2880 \par ------------------------------------------------------------------------------
2881 \par p(5): 32 ( 2 digits)
2882 \par q(5): 83 ( 2 digits)
2883 \par ------------------------------------------------------------------------------
2884 \par p(6): 101 ( 3 digits)
2885 \par q(6): 262 ( 3 digits)
2886 \par ------------------------------------------------------------------------------
2887 \par p(7): 335 ( 3 digits)
2888 \par q(7): 869 ( 3 digits)
2889 \par ------------------------------------------------------------------------------
2890 \par p(8): 771 ( 3 digits)
2891 \par q(8): 2,000 ( 4 digits)
2892 \par ------------------------------------------------------------------------------
2893 \par ******** A Rational Number With Smallest Denominator In Interval ********
2894 \par ------------------------------------------------------------------------------
2895 \par result_h: 22 ( 2 digits)
2896 \par ------------------------------------------------------------------------------
2897 \par result_k: 57 ( 2 digits)
2898 \par ------------------------------------------------------------------------------
2899 \par ***** Upper Bound On Distance Between Farey Terms As Rational Number *****
2900 \par ------------------------------------------------------------------------------
2901 \par error_ub_h: 1 ( 1 digit)
2902 \par ------------------------------------------------------------------------------
2903 \par error_up_k: 25,251 ( 5 digits)
2904 \par ------------------------------------------------------------------------------
2905 \par ** Upper Bound On Distance Between Farey Terms As Decimal Approximation **
2906 \par ------------------------------------------------------------------------------
2907 \par dap_h: 39,602,391,984,475,862,342,085, ( 104 digits)
2908 \par 461,961,902,498,910,934,220,426,913,
2909 \par 785,592,649,796,047,681,279,949,308,
2910 \par 938,259,870,896,202,130,608,688,764
2911 \par ------------------------------------------------------------------------------
2912 \par dap_k: 1, ( 109 digits)
2913 \par 000,000,000,000,000,000,000,000,000,
2914 \par 000,000,000,000,000,000,000,000,000,
2915 \par 000,000,000,000,000,000,000,000,000,
2916 \par 000,000,000,000,000,000,000,000,000
2917 \par ------------------------------------------------------------------------------
2918 \par ******************************************************************************
2919 \par ******************************************************************************
2920 \par ************************ Start Of Case II Results ************************
2921 \par ******************************************************************************
2922 \par ******************************************************************************
2923 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 These are the Case II results.
2924 \par \pard \widctlpar {\f11\fs16 ------------------------------------------------------------------------------
2925 \par ************************** Case II Right Limit **************************
2926 \par ------------------------------------------------------------------------------
2927 \par case_2_right_h: 1 ( 1 digit)
2928 \par ------------------------------------------------------------------------------
2929 \par case_2_right_k: 1 ( 1 digit)
2930 \par ------------------------------------------------------------------------------
2931 \par ***** Upper Bound On Distance Between Farey Terms As Rational Number *****
2932 \par ------------------------------------------------------------------------------
2933 \par error_ub_h: 1 ( 1 digit)
2934 \par ------------------------------------------------------------------------------
2935 \par error_up_k: 194 ( 3 digits)
2936 \par ------------------------------------------------------------------------------
2937 \par ** Upper Bound On Distance Between Farey Terms As Decimal Approximation **
2938 \par ------------------------------------------------------------------------------
2939 \par dap_h: 5,154,639,175,257,731,958,762,886, ( 106 digits)
2940 \par 597,938,144,329,896,907,216,494,845,
2941 \par 360,824,742,268,041,237,113,402,061,
2942 \par 855,670,103,092,783,505,154,639,175
2943 \par ------------------------------------------------------------------------------
2944 \par dap_k: 1, ( 109 digits)
2945 \par 000,000,000,000,000,000,000,000,000,
2946 \par 000,000,000,000,000,000,000,000,000,
2947 \par 000,000,000,000,000,000,000,000,000,
2948 \par 000,000,000,000,000,000,000,000,000
2949 \par ------------------------------------------------------------------------------
2950 \par ******************************************************************************
2951 \par ******************************************************************************
2952 \par *********************** Start Of Case III Results ***********************
2953 \par ******************************************************************************
2954 \par ******************************************************************************
2955 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 These are the Case III results.
2956 \par \pard \widctlpar {\f11\fs16 ------------------------------------------------------------------------------
2957 \par ***** Upper Bound On Distance Between Farey Terms As Rational Number *****
2958 \par ------------------------------------------------------------------------------
2959 \par error_ub_h: 1 ( 1 digit)
2960 \par ------------------------------------------------------------------------------
2961 \par error_up_k: 89 ( 2 digits)
2962 \par ------------------------------------------------------------------------------
2963 \par ** Upper Bound On Distance Between Farey Terms As Decimal Approximation **
2964 \par ------------------------------------------------------------------------------
2965 \par dap_h: 11,235,955,056,179,775,280,898,876, ( 107 digits)
2966 \par 404,494,382,022,471,910,112,359,550,
2967 \par 561,797,752,808,988,764,044,943,820,
2968 \par 224,719,101,123,595,505,617,977,528
2969 \par ------------------------------------------------------------------------------
2970 \par dap_k: 1, ( 109 digits)
2971 \par 000,000,000,000,000,000,000,000,000,
2972 \par 000,000,000,000,000,000,000,000,000,
2973 \par 000,000,000,000,000,000,000,000,000,
2974 \par 000,000,000,000,000,000,000,000,000
2975 \par ------------------------------------------------------------------------------
2976 \par ******************************************************************************
2977 \par ******************************************************************************
2978 \par ********************** Start Of Cumulative Results **********************
2979 \par ******************************************************************************
2980 \par ******************************************************************************
2981 \par ------------------------------------------------------------------------------
2982 \par }\pard \widctlpar\box\brdrs\brdrw15\brsp20 \shading500 These are the cumulative results. The largest error upper bound anywhere over the interval is supplied.
2983 \par \pard \widctlpar {\f11\fs16 ******************************************************************************
2984 \par ******************************************************************************
2985 \par ************************ Largest Error (Case III) ************************
2986 \par ******************************************************************************
2987 \par ******************************************************************************
2988 \par ------------------------------------------------------------------------------
2989 \par ***** Upper Bound On Distance Between Farey Terms As Rational Number *****
2990 \par ------------------------------------------------------------------------------
2991 \par error_ub_h: 1 ( 1 digit)
2992 \par ------------------------------------------------------------------------------
2993 \par error_up_k: 89 ( 2 digits)
2994 \par ------------------------------------------------------------------------------
2995 \par ** Upper Bound On Distance Between Farey Terms As Decimal Approximation **
2996 \par ------------------------------------------------------------------------------
2997 \par dap_h: 11,235,955,056,179,775,280,898,876, ( 107 digits)
2998 \par 404,494,382,022,471,910,112,359,550,
2999 \par 561,797,752,808,988,764,044,943,820,
3000 \par 224,719,101,123,595,505,617,977,528
3001 \par ------------------------------------------------------------------------------
3002 \par dap_k: 1, ( 109 digits)
3003 \par 000,000,000,000,000,000,000,000,000,
3004 \par 000,000,000,000,000,000,000,000,000,
3005 \par 000,000,000,000,000,000,000,000,000,
3006 \par 000,000,000,000,000,000,000,000,000
3007 \par ------------------------------------------------------------------------------
3008 \par RAP execution ends.
3009 \par ------------------------------------------------------------------------------}
3010 \par \page
3011 \par {\*\bkmkstart _Toc498721719}{\pntext\pard\plain\b\f5\fs28\kerning28 5.\tab}\pard\plain \s1\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl1\pndec\pnprev1\pnstart1\pnsp144 {\pntxta .}}\b\f5\fs28\kerning28 Digits Of Useful Constants{\*\bkmkend _Toc498721719}
3012 \par \pard\plain \widctlpar \f4\fs20 This section contains base-10 rational approximations of useful constants to many decimal places. These can be manually entered or pasted into files used as RAP input.
3013 \par
3014 \par {\*\bkmkstart _Ref497727390}{\*\bkmkstart _Toc498721720}{\pntext\pard\plain\b\i\f5 5.1\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Digits Of {{\field{\*\fldinst SYMBOL 112 \\f "Symbol"
3015 \\s 12}{\fldrslt\f1\fs24}}}{\*\bkmkend _Ref497727390}{\*\bkmkend _Toc498721720}
3016 \par \pard\plain \widctlpar \f4\fs20 The first 1,000 digits of {{\field{\*\fldinst SYMBOL 112 \\f "Symbol" \\s 10}{\fldrslt\f1\fs20}}} were obtained from the web site {\i http://www.ex.ac.uk/cimt/general/pi10000.htm}
3017 . The numbers in bold and in brackets show how many digits have been presented up to that point. This information has not been verified for accuracy.
3018 \par
3019 \par \pard\plain \s29\widctlpar\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631 \f11\fs20 {\fs16 \tab 3.1415926535 8979323846 2643383279 5028841971 6939937510
3020 \par \tab 5820974944 5923078164 0628620899 8628034825 3421170679
3021 \par \tab 8214808651 3282306647 0938446095 5058223172 5359408128
3022 \par \tab 4811174502 8410270193 8521105559 6446229489 5493038196
3023 \par \tab 4428810975 6659334461 2847564823 3786783165 2712019091
3024 \par \tab 4564856692 3460348610 4543266482 1339360726 0249141273\tab
3025 \par \tab 7245870066 0631558817 4881520920 9628292540 9171536436\tab
3026 \par \tab 7892590360 0113305305 4882046652 1384146951 9415116094\tab
3027 \par \tab 3305727036 5759591953 0921861173 8193261179 3105118548\tab
3028 \par \tab 0744623799 6274956735 1885752724 8912279381 8301194912\tab
3029 \par \tab }{\b\fs16 [}{\b\i\fs16 500]}{\fs16
3030 \par \tab 9833673362 4406566430 8602139494 6395224737 1907021798\tab
3031 \par \tab 6094370277 0539217176 2931767523 8467481846 7669405132\tab
3032 \par \tab 0005681271 4526356082 7785771342 7577896091 7363717872\tab
3033 \par \tab 1468440901 2249534301 4654958537 1050792279 6892589235\tab
3034 \par \tab 4201995611 2129021960 8640344181 5981362977 4771309960\tab
3035 \par \tab 5187072113 4999999837 2978049951 0597317328 1609631859\tab
3036 \par \tab 5024459455 3469083026 4252230825 3344685035 2619311881\tab
3037 \par \tab 7101000313 7838752886 5875332083 8142061717 7669147303\tab
3038 \par \tab 5982534904 2875546873 1159562863 8823537875 9375195778\tab
3039 \par \tab 1857780532 1712268066 1300192787 6611195909 2164201989\tab
3040 \par \tab }{\b\fs16 [}{\b\i\fs16 1000]}{\fs16
3041 \par }\pard\plain \widctlpar \f4\fs20
3042 \par \page
3043 \par {\*\bkmkstart _Toc498721721}{\pntext\pard\plain\b\i\f5 5.2\tab}\pard\plain \s2\sb240\sa60\keepn\widctlpar{\*\pn \pnlvl2\pndec\pnprev1\pnstart1\pnsp144 }\b\i\f5 Digits Of e{\*\bkmkend _Toc498721721}
3044 \par \pard\plain \widctlpar \f4\fs20 The first 1,000 digits of {\i e} were obtained from the web site {\i http://fermi.udw.ac.za/physics/e.html}. This information has not been verified for accuracy.
3045 \par
3046 \par \pard\plain \s29\widctlpar\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631 \f11\fs20 {\fs16 2.7182818284590452353602874713526624977572470936999595749669676277240766303535
3047 \par 475945713821785251664274274663919320030599218174135966290435729003342952605956
3048 \par 307381323286279434907632338298807531952510190115738341879307021540891499348841
3049 \par 675092447614606680822648001684774118537423454424371075390777449920695517027618
3050 \par 386062613313845830007520449338265602976067371132007093287091274437470472306969
3051 \par 772093101416928368190255151086574637721112523897844250569536967707854499699679
3052 \par 468644549059879316368892300987931277361782154249992295763514822082698951936680
3053 \par 331825288693984964651058209392398294887933203625094431173012381970684161403970
3054 \par 198376793206832823764648042953118023287825098194558153017567173613320698112509
3055 \par 961818815930416903515988885193458072738667385894228792284998920868058257492796
3056 \par 104841984443634632449684875602336248270419786232090021609902353043699418491463
3057 \par 140934317381436405462531520961836908887070167683964243781405927145635490613031
3058 \par 07208510383750510115747704171898610687396965521267154688957035035
3059 \par }\pard\plain \widctlpar \f4\fs20
3060 \par
3061 \par }
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