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| /FamilyName (Computer Modern) readonly def |
| /Weight (Bold) readonly def |
| /ItalicAngle 0 def |
| /isFixedPitch false def |
| end readonly def |
| /FontName /CMBX12 def |
| /PaintType 0 def |
| /FontType 1 def |
| /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| /Encoding 256 array |
| 0 1 255 {1 index exch /.notdef put} for |
| dup 40 /parenleft put |
| dup 41 /parenright put |
| dup 45 /hyphen put |
| dup 46 /period put |
| dup 47 /slash put |
| dup 49 /one put |
| dup 50 /two put |
| dup 51 /three put |
| dup 52 /four put |
| dup 53 /five put |
| dup 54 /six put |
| dup 55 /seven put |
| dup 56 /eight put |
| dup 57 /nine put |
| dup 65 /A put |
| dup 66 /B put |
| dup 67 /C put |
| dup 68 /D put |
| dup 69 /E put |
| dup 70 /F put |
| dup 71 /G put |
| dup 72 /H put |
| dup 73 /I put |
| dup 76 /L put |
| dup 77 /M put |
| dup 78 /N put |
| dup 79 /O put |
| dup 80 /P put |
| dup 82 /R put |
| dup 83 /S put |
| dup 84 /T put |
| dup 85 /U put |
| dup 86 /V put |
| dup 97 /a put |
| dup 98 /b put |
| dup 99 /c put |
| dup 100 /d put |
| dup 101 /e put |
| dup 102 /f put |
| dup 103 /g put |
| dup 104 /h put |
| dup 105 /i put |
| dup 106 /j put |
| dup 107 /k put |
| dup 108 /l put |
| dup 109 /m put |
| dup 110 /n put |
| dup 111 /o put |
| dup 112 /p put |
| dup 114 /r put |
| dup 115 /s put |
| dup 116 /t put |
| dup 117 /u put |
| dup 119 /w put |
| dup 120 /x put |
| dup 121 /y put |
| dup 122 /z put |
| readonly def |
| /FontBBox{-53 -251 1139 750}readonly def |
| currentdict end |
| currentfile eexec |
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| cleartomark |
| %%EndFont |
| %%BeginFont: CMMI6 |
| %!PS-AdobeFont-1.1: CMMI6 1.100 |
| %%CreationDate: 1996 Jul 23 07:53:52 |
| % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 11 dict begin |
| /FontInfo 7 dict dup begin |
| /version (1.100) readonly def |
| /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| /FullName (CMMI6) readonly def |
| /FamilyName (Computer Modern) readonly def |
| /Weight (Medium) readonly def |
| /ItalicAngle -14.04 def |
| /isFixedPitch false def |
| end readonly def |
| /FontName /CMMI6 def |
| /PaintType 0 def |
| /FontType 1 def |
| /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| /Encoding 256 array |
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| dup 106 /j put |
| readonly def |
| /FontBBox{11 -250 1241 750}readonly def |
| currentdict end |
| currentfile eexec |
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| cleartomark |
| %%EndFont |
| %%BeginFont: CMTI10 |
| %!PS-AdobeFont-1.1: CMTI10 1.00B |
| %%CreationDate: 1992 Feb 19 19:56:16 |
| % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 11 dict begin |
| /FontInfo 7 dict dup begin |
| /version (1.00B) readonly def |
| /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| /FullName (CMTI10) readonly def |
| /FamilyName (Computer Modern) readonly def |
| /Weight (Medium) readonly def |
| /ItalicAngle -14.04 def |
| /isFixedPitch false def |
| end readonly def |
| /FontName /CMTI10 def |
| /PaintType 0 def |
| /FontType 1 def |
| /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| /Encoding 256 array |
| 0 1 255 {1 index exch /.notdef put} for |
| dup 12 /fi put |
| dup 39 /quoteright put |
| dup 40 /parenleft put |
| dup 41 /parenright put |
| dup 44 /comma put |
| dup 45 /hyphen put |
| dup 46 /period put |
| dup 48 /zero put |
| dup 49 /one put |
| dup 50 /two put |
| dup 51 /three put |
| dup 58 /colon put |
| dup 65 /A put |
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| dup 117 /u put |
| dup 118 /v put |
| dup 119 /w put |
| dup 120 /x put |
| dup 121 /y put |
| dup 122 /z put |
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| readonly def |
| /FontBBox{-163 -250 1146 969}readonly def |
| currentdict end |
| currentfile eexec |
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| cleartomark |
| %%EndFont |
| %%BeginFont: CMMI8 |
| %!PS-AdobeFont-1.1: CMMI8 1.100 |
| %%CreationDate: 1996 Jul 23 07:53:54 |
| % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 11 dict begin |
| /FontInfo 7 dict dup begin |
| /version (1.100) readonly def |
| /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| /FullName (CMMI8) readonly def |
| /FamilyName (Computer Modern) readonly def |
| /Weight (Medium) readonly def |
| /ItalicAngle -14.04 def |
| /isFixedPitch false def |
| end readonly def |
| /FontName /CMMI8 def |
| /PaintType 0 def |
| /FontType 1 def |
| /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| /Encoding 256 array |
| 0 1 255 {1 index exch /.notdef put} for |
| dup 84 /T put |
| dup 100 /d put |
| dup 105 /i put |
| dup 106 /j put |
| dup 107 /k put |
| dup 110 /n put |
| dup 114 /r put |
| dup 115 /s put |
| dup 116 /t put |
| readonly def |
| /FontBBox{-24 -250 1110 750}readonly def |
| currentdict end |
| currentfile eexec |
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| cleartomark |
| %%EndFont |
| %%BeginFont: CMR8 |
| %!PS-AdobeFont-1.1: CMR8 1.0 |
| %%CreationDate: 1991 Aug 20 16:39:40 |
| % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 11 dict begin |
| /FontInfo 7 dict dup begin |
| /version (1.0) readonly def |
| /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| /FullName (CMR8) readonly def |
| /FamilyName (Computer Modern) readonly def |
| /Weight (Medium) readonly def |
| /ItalicAngle 0 def |
| /isFixedPitch false def |
| end readonly def |
| /FontName /CMR8 def |
| /PaintType 0 def |
| /FontType 1 def |
| /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| /Encoding 256 array |
| 0 1 255 {1 index exch /.notdef put} for |
| dup 43 /plus put |
| dup 49 /one put |
| dup 50 /two put |
| dup 51 /three put |
| dup 55 /seven put |
| readonly def |
| /FontBBox{-36 -250 1070 750}readonly def |
| currentdict end |
| currentfile eexec |
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| cleartomark |
| %%EndFont |
| %%BeginFont: CMSY10 |
| %!PS-AdobeFont-1.1: CMSY10 1.0 |
| %%CreationDate: 1991 Aug 15 07:20:57 |
| % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 11 dict begin |
| /FontInfo 7 dict dup begin |
| /version (1.0) readonly def |
| /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| /FullName (CMSY10) readonly def |
| /FamilyName (Computer Modern) readonly def |
| /Weight (Medium) readonly def |
| /ItalicAngle -14.035 def |
| /isFixedPitch false def |
| end readonly def |
| /FontName /CMSY10 def |
| /PaintType 0 def |
| /FontType 1 def |
| /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| /Encoding 256 array |
| 0 1 255 {1 index exch /.notdef put} for |
| dup 0 /minus put |
| dup 2 /multiply put |
| dup 20 /lessequal put |
| dup 21 /greaterequal put |
| dup 50 /element put |
| dup 56 /universal put |
| dup 57 /existential put |
| dup 91 /union put |
| dup 102 /braceleft put |
| dup 103 /braceright put |
| dup 106 /bar put |
| dup 110 /backslash put |
| readonly def |
| /FontBBox{-29 -960 1116 775}readonly def |
| currentdict end |
| currentfile eexec |
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| cleartomark |
| %%EndFont |
| %%BeginFont: CMMI10 |
| %!PS-AdobeFont-1.1: CMMI10 1.100 |
| %%CreationDate: 1996 Jul 23 07:53:57 |
| % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 11 dict begin |
| /FontInfo 7 dict dup begin |
| /version (1.100) readonly def |
| /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| /FullName (CMMI10) readonly def |
| /FamilyName (Computer Modern) readonly def |
| /Weight (Medium) readonly def |
| /ItalicAngle -14.04 def |
| /isFixedPitch false def |
| end readonly def |
| /FontName /CMMI10 def |
| /PaintType 0 def |
| /FontType 1 def |
| /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| /Encoding 256 array |
| 0 1 255 {1 index exch /.notdef put} for |
| dup 58 /period put |
| dup 59 /comma put |
| dup 61 /slash put |
| dup 62 /greater put |
| dup 65 /A put |
| dup 75 /K put |
| dup 76 /L put |
| dup 80 /P put |
| dup 82 /R put |
| dup 83 /S put |
| dup 84 /T put |
| dup 98 /b put |
| dup 99 /c put |
| dup 100 /d put |
| dup 101 /e put |
| dup 103 /g put |
| dup 105 /i put |
| dup 106 /j put |
| dup 107 /k put |
| dup 109 /m put |
| dup 110 /n put |
| dup 111 /o put |
| dup 112 /p put |
| dup 113 /q put |
| dup 114 /r put |
| dup 115 /s put |
| dup 116 /t put |
| dup 118 /v put |
| dup 120 /x put |
| dup 122 /z put |
| readonly def |
| /FontBBox{-32 -250 1048 750}readonly def |
| currentdict end |
| currentfile eexec |
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| cleartomark |
| %%EndFont |
| %%BeginFont: CMTT10 |
| %!PS-AdobeFont-1.1: CMTT10 1.00B |
| %%CreationDate: 1992 Apr 26 10:42:42 |
| % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 11 dict begin |
| /FontInfo 7 dict dup begin |
| /version (1.00B) readonly def |
| /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| /FullName (CMTT10) readonly def |
| /FamilyName (Computer Modern) readonly def |
| /Weight (Medium) readonly def |
| /ItalicAngle 0 def |
| /isFixedPitch true def |
| end readonly def |
| /FontName /CMTT10 def |
| /PaintType 0 def |
| /FontType 1 def |
| /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| /Encoding 256 array |
| 0 1 255 {1 index exch /.notdef put} for |
| dup 35 /numbersign put |
| dup 38 /ampersand put |
| dup 40 /parenleft put |
| dup 41 /parenright put |
| dup 42 /asterisk put |
| dup 44 /comma put |
| dup 45 /hyphen put |
| dup 46 /period put |
| dup 47 /slash put |
| dup 48 /zero put |
| dup 49 /one put |
| dup 50 /two put |
| dup 51 /three put |
| dup 52 /four put |
| dup 53 /five put |
| dup 55 /seven put |
| dup 59 /semicolon put |
| dup 61 /equal put |
| dup 62 /greater put |
| dup 65 /A put |
| dup 66 /B put |
| dup 67 /C put |
| dup 68 /D put |
| dup 69 /E put |
| dup 70 /F put |
| dup 71 /G put |
| dup 72 /H put |
| dup 73 /I put |
| dup 76 /L put |
| dup 77 /M put |
| dup 78 /N put |
| dup 79 /O put |
| dup 80 /P put |
| dup 82 /R put |
| dup 83 /S put |
| dup 84 /T put |
| dup 85 /U put |
| dup 86 /V put |
| dup 87 /W put |
| dup 90 /Z put |
| dup 91 /bracketleft put |
| dup 93 /bracketright put |
| dup 95 /underscore put |
| dup 97 /a put |
| dup 98 /b put |
| dup 99 /c put |
| dup 100 /d put |
| dup 101 /e put |
| dup 102 /f put |
| dup 103 /g put |
| dup 104 /h put |
| dup 105 /i put |
| dup 106 /j put |
| dup 107 /k put |
| dup 108 /l put |
| dup 109 /m put |
| dup 110 /n put |
| dup 111 /o put |
| dup 112 /p put |
| dup 113 /q put |
| dup 114 /r put |
| dup 115 /s put |
| dup 116 /t put |
| dup 117 /u put |
| dup 118 /v put |
| dup 119 /w put |
| dup 120 /x put |
| dup 121 /y put |
| dup 122 /z put |
| dup 123 /braceleft put |
| dup 125 /braceright put |
| readonly def |
| /FontBBox{-4 -235 731 800}readonly def |
| currentdict end |
| currentfile eexec |
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| cleartomark |
| %%EndFont |
| %%BeginFont: CMR10 |
| %!PS-AdobeFont-1.1: CMR10 1.00B |
| %%CreationDate: 1992 Feb 19 19:54:52 |
| % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 11 dict begin |
| /FontInfo 7 dict dup begin |
| /version (1.00B) readonly def |
| /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| /FullName (CMR10) readonly def |
| /FamilyName (Computer Modern) readonly def |
| /Weight (Medium) readonly def |
| /ItalicAngle 0 def |
| /isFixedPitch false def |
| end readonly def |
| /FontName /CMR10 def |
| /PaintType 0 def |
| /FontType 1 def |
| /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| /Encoding 256 array |
| 0 1 255 {1 index exch /.notdef put} for |
| dup 11 /ff put |
| dup 12 /fi put |
| dup 13 /fl put |
| dup 14 /ffi put |
| dup 34 /quotedblright put |
| dup 37 /percent put |
| dup 39 /quoteright put |
| dup 40 /parenleft put |
| dup 41 /parenright put |
| dup 42 /asterisk put |
| dup 43 /plus put |
| dup 44 /comma put |
| dup 45 /hyphen put |
| dup 46 /period put |
| dup 47 /slash put |
| dup 48 /zero put |
| dup 49 /one put |
| dup 50 /two put |
| dup 51 /three put |
| dup 52 /four put |
| dup 53 /five put |
| dup 54 /six put |
| dup 55 /seven put |
| dup 56 /eight put |
| dup 57 /nine put |
| dup 58 /colon put |
| dup 59 /semicolon put |
| dup 61 /equal put |
| dup 63 /question put |
| dup 65 /A put |
| dup 66 /B put |
| dup 67 /C put |
| dup 68 /D put |
| dup 69 /E put |
| dup 70 /F put |
| dup 71 /G put |
| dup 72 /H put |
| dup 73 /I put |
| dup 74 /J put |
| dup 75 /K put |
| dup 76 /L put |
| dup 77 /M put |
| dup 78 /N put |
| dup 79 /O put |
| dup 80 /P put |
| dup 81 /Q put |
| dup 82 /R put |
| dup 83 /S put |
| dup 84 /T put |
| dup 85 /U put |
| dup 86 /V put |
| dup 87 /W put |
| dup 89 /Y put |
| dup 90 /Z put |
| dup 91 /bracketleft put |
| dup 92 /quotedblleft put |
| dup 93 /bracketright put |
| dup 97 /a put |
| dup 98 /b put |
| dup 99 /c put |
| dup 100 /d put |
| dup 101 /e put |
| dup 102 /f put |
| dup 103 /g put |
| dup 104 /h put |
| dup 105 /i put |
| dup 106 /j put |
| dup 107 /k put |
| dup 108 /l put |
| dup 109 /m put |
| dup 110 /n put |
| dup 111 /o put |
| dup 112 /p put |
| dup 113 /q put |
| dup 114 /r put |
| dup 115 /s put |
| dup 116 /t put |
| dup 117 /u put |
| dup 118 /v put |
| dup 119 /w put |
| dup 120 /x put |
| dup 121 /y put |
| dup 122 /z put |
| dup 123 /endash put |
| dup 126 /tilde put |
| dup 127 /dieresis put |
| readonly def |
| /FontBBox{-251 -250 1009 969}readonly def |
| currentdict end |
| currentfile eexec |
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| cleartomark |
| %%EndFont |
| %%BeginFont: CMBX10 |
| %!PS-AdobeFont-1.1: CMBX10 1.00B |
| %%CreationDate: 1992 Feb 19 19:54:06 |
| % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 11 dict begin |
| /FontInfo 7 dict dup begin |
| /version (1.00B) readonly def |
| /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| /FullName (CMBX10) readonly def |
| /FamilyName (Computer Modern) readonly def |
| /Weight (Bold) readonly def |
| /ItalicAngle 0 def |
| /isFixedPitch false def |
| end readonly def |
| /FontName /CMBX10 def |
| /PaintType 0 def |
| /FontType 1 def |
| /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| /Encoding 256 array |
| 0 1 255 {1 index exch /.notdef put} for |
| dup 12 /fi put |
| dup 40 /parenleft put |
| dup 41 /parenright put |
| dup 45 /hyphen put |
| dup 46 /period put |
| dup 48 /zero put |
| dup 49 /one put |
| dup 50 /two put |
| dup 51 /three put |
| dup 52 /four put |
| dup 53 /five put |
| dup 54 /six put |
| dup 55 /seven put |
| dup 56 /eight put |
| dup 57 /nine put |
| dup 65 /A put |
| dup 66 /B put |
| dup 67 /C put |
| dup 68 /D put |
| dup 69 /E put |
| dup 70 /F put |
| dup 71 /G put |
| dup 72 /H put |
| dup 73 /I put |
| dup 76 /L put |
| dup 77 /M put |
| dup 78 /N put |
| dup 79 /O put |
| dup 80 /P put |
| dup 83 /S put |
| dup 84 /T put |
| dup 85 /U put |
| dup 86 /V put |
| dup 97 /a put |
| dup 98 /b put |
| dup 99 /c put |
| dup 100 /d put |
| dup 101 /e put |
| dup 102 /f put |
| dup 103 /g put |
| dup 104 /h put |
| dup 105 /i put |
| dup 106 /j put |
| dup 108 /l put |
| dup 109 /m put |
| dup 110 /n put |
| dup 111 /o put |
| dup 112 /p put |
| dup 113 /q put |
| dup 114 /r put |
| dup 115 /s put |
| dup 116 /t put |
| dup 117 /u put |
| dup 118 /v put |
| dup 120 /x put |
| dup 121 /y put |
| readonly def |
| /FontBBox{-301 -250 1164 946}readonly def |
| currentdict end |
| currentfile eexec |
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| cleartomark |
| %%EndFont |
| %%BeginFont: CMR12 |
| %!PS-AdobeFont-1.1: CMR12 1.0 |
| %%CreationDate: 1991 Aug 20 16:38:05 |
| % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 11 dict begin |
| /FontInfo 7 dict dup begin |
| /version (1.0) readonly def |
| /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| /FullName (CMR12) readonly def |
| /FamilyName (Computer Modern) readonly def |
| /Weight (Medium) readonly def |
| /ItalicAngle 0 def |
| /isFixedPitch false def |
| end readonly def |
| /FontName /CMR12 def |
| /PaintType 0 def |
| /FontType 1 def |
| /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| /Encoding 256 array |
| 0 1 255 {1 index exch /.notdef put} for |
| dup 40 /parenleft put |
| dup 41 /parenright put |
| dup 44 /comma put |
| dup 45 /hyphen put |
| dup 46 /period put |
| dup 48 /zero put |
| dup 50 /two put |
| dup 52 /four put |
| dup 55 /seven put |
| dup 56 /eight put |
| dup 57 /nine put |
| dup 67 /C put |
| dup 69 /E put |
| dup 70 /F put |
| dup 72 /H put |
| dup 73 /I put |
| dup 75 /K put |
| dup 79 /O put |
| dup 82 /R put |
| dup 83 /S put |
| dup 84 /T put |
| dup 90 /Z put |
| dup 97 /a put |
| dup 98 /b put |
| dup 99 /c put |
| dup 100 /d put |
| dup 101 /e put |
| dup 102 /f put |
| dup 104 /h put |
| dup 105 /i put |
| dup 107 /k put |
| dup 108 /l put |
| dup 109 /m put |
| dup 110 /n put |
| dup 111 /o put |
| dup 112 /p put |
| dup 114 /r put |
| dup 115 /s put |
| dup 116 /t put |
| dup 117 /u put |
| dup 118 /v put |
| dup 119 /w put |
| dup 121 /y put |
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| readonly def |
| /FontBBox{-34 -251 988 750}readonly def |
| currentdict end |
| currentfile eexec |
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| %%CreationDate: 1991 Aug 20 16:38:24 |
| % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 11 dict begin |
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| /version (1.0) readonly def |
| /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| /FullName (CMR17) readonly def |
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| end readonly def |
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| currentdict end |
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| %%EndFont |
| TeXDict begin 39139632 55387786 1000 600 600 (cddlibman.dvi) |
| @start /Fa 173[66 82[{}1 90.9091 /MSBM10 rf /Fb 139[51 |
| 51 51 1[51 51 9[51 101[{}6 99.6264 /CMTT12 rf /Fc 255[55{}1 |
| 66.4176 /CMSY8 rf /Fd 133[50 59 3[62 44 44 46 1[62 56 |
| 62 1[31 2[31 3[51 1[50 62 54 13[62 3[84 1[106 67 2[42 |
| 2[70 1[86 81 12[56 56 56 56 56 56 1[56 31 46[{}31 99.6264 |
| /CMBX12 rf /Fe 149[25 106[{}1 49.8132 /CMMI6 rf /Ff 132[46 |
| 37 44 42 60 42 49 30 37 38 42 46 46 51 74 23 42 1[28 |
| 46 42 28 42 46 42 42 46 10[68 68 65 51 66 1[62 70 68 |
| 82 57 1[48 35 68 70 1[62 69 65 1[68 6[28 6[46 46 46 46 |
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| %%Page: 3 3 |
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| b Fi(\000)f Fj(Ax)36 b Fi(\025)e Fp(0)72 b Fo(of)36 b |
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| end |
| %%Page: 4 4 |
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| end |
| %%Page: 5 5 |
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| (is)h(set)f(to)h(b)s(e)f(either)h(the)f(rational)i(t)m(yp)s(e)e |
| Fn(mpq)p 3550 157 V 34 w(t)g Fo(of)g(the)0 270 y(GNU)f(MP)f(Library)f |
| (or)h(the)h(C)f(double)f(arra)m(y)i(of)f(size)h(1.)53 |
| b(This)33 b(abstract)i(t)m(yp)s(e)f(allo)m(ws)i(us)d(to)i(write)f(a)h |
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| (t)m(w)m(o)i(or)e(more)h(di\013eren)m(t)f(arithmetics,)j(see)e(example) |
| g(programs)0 496 y(suc)m(h)d(as)h(simplecdd.c,)g(testlp*.c)h(and)e |
| (testcdd*.c)i(in)e(the)h Fn(src)e Fo(and)h Fn(src-gmp)e |
| Fo(sub)s(directories)i(of)h(the)f(source)0 609 y(distribution.)141 |
| 722 y(There)42 b(is)h(another)g(data)g(t)m(yp)s(e)g(that)g(is)g(used)f |
| (v)m(ery)h(often,)j Fn(dd)p 2419 722 V 34 w(SetFamilyType)p |
| Fo(.)74 b(This)42 b(is)h(to)g(store)g(a)0 835 y(family)35 |
| b(of)h(subsets)e(of)h(a)g(\014nite)g(set.)55 b(Suc)m(h)35 |
| b(a)g(family)h(can)f(represen)m(t)g(the)g(incidence)h(relations)g(b)s |
| (et)m(w)m(een)f(the)0 947 y(set)f(of)g(extreme)h(p)s(oin)m(ts)e(and)g |
| (the)h(set)g(of)g(facets)h(of)f(a)g(p)s(olyhedron.)49 |
| b(Also,)36 b(it)e(can)g(represen)m(t)f(a)h(graph)g(struc-)0 |
| 1060 y(ture)g(b)m(y)g(listing)i(the)e(set)h(of)f(v)m(ertices)i(adjacen) |
| m(t)g(to)f(eac)m(h)h(v)m(ertex)f(\(i.e.)54 b(the)34 b(adjacency)i |
| (list\).)53 b(T)-8 b(o)35 b(implemen)m(t)0 1173 y Fn(dd)p |
| 102 1173 V 34 w(SetFamilyType)p Fo(,)26 b(w)m(e)j(use)f(a)h(separate)h |
| (set)g(library)e(called)i Fn(setoper)p Fo(,)d(that)j(handles)e(the)h |
| (basic)g(set)g(op)s(era-)0 1286 y(tions,)i(This)f(library)g(is)g |
| (brie\015y)f(in)m(tro)s(duced)h(in)g(Section)i(4.6.)0 |
| 1594 y Fn(#define)46 b(dd_FALSE)f(0)0 1707 y(#define)h(dd_TRUE)g(1)0 |
| 1933 y(typedef)g(long)g(dd_rowrange;)0 2046 y(typedef)g(long)g |
| (dd_colrange;)0 2159 y(typedef)g(long)g(dd_bigrange;)0 |
| 2385 y(typedef)g(set_type)f(dd_rowset;)141 b(/*)47 b(set_type)f |
| (defined)f(in)j(setoper.h)d(*/)0 2498 y(typedef)h(set_type)f |
| (dd_colset;)0 2610 y(typedef)h(long)g(*dd_rowindex;)0 |
| 2723 y(typedef)g(int)h(*dd_rowflag;)0 2836 y(typedef)f(long)g |
| (*dd_colindex;)0 2949 y(typedef)g(mytype)g(**dd_Amatrix;)92 |
| b(/*)47 b(mytype)f(is)h(either)f(GMP)h(mpq_t)g(or)g(1-dim)f(double)g |
| (array.)g(*/)0 3062 y(typedef)g(mytype)g(*dd_Arow;)0 |
| 3175 y(typedef)g(set_type)f(*dd_SetVector;)0 3401 y(typedef)h(enum)g({) |
| 95 3514 y(dd_Real,)g(dd_Rational,)e(dd_Integer,)h(dd_Unknown)0 |
| 3627 y(})i(dd_NumberType;)0 3852 y(typedef)f(enum)g({)95 |
| 3965 y(dd_Inequality,)e(dd_Generator,)h(dd_Unspecified)0 |
| 4078 y(})i(dd_RepresentationType;)0 4304 y(typedef)f(enum)g({)95 |
| 4417 y(dd_MaxIndex,)f(dd_MinIndex,)f(dd_MinCutoff,)h(dd_MaxCutoff,)f |
| (dd_MixCutoff,)143 4530 y(dd_LexMin,)h(dd_LexMax,)g(dd_RandomRow)0 |
| 4643 y(})i(dd_RowOrderType;)0 4869 y(typedef)f(enum)g({)95 |
| 4982 y(dd_InProgress,)e(dd_AllFound,)h(dd_RegionEmpty)0 |
| 5094 y(})i(dd_CompStatusType;)0 5320 y(typedef)f(enum)g({)95 |
| 5433 y(dd_DimensionTooLarge,)c(dd_ImproperInputFormat,)1926 |
| 5682 y Fo(5)p eop end |
| %%Page: 6 6 |
| TeXDict begin 6 5 bop 95 44 a Fn(dd_NegativeMatrixSize,)42 |
| b(dd_EmptyVrepresentation,)95 157 y(dd_IFileNotFound,)i |
| (dd_OFileNotOpen,)f(dd_NoLPObjective,)95 270 y(dd_NoRealNumberSupport,) |
| f(dd_NoError)0 383 y(})47 b(dd_ErrorType;)0 609 y(typedef)f(enum)g({)95 |
| 722 y(dd_LPnone=0,)f(dd_LPmax,)g(dd_LPmin)0 835 y(})i |
| (dd_LPObjectiveType;)0 1060 y(typedef)f(enum)g({)95 1173 |
| y(dd_LPSundecided,)e(dd_Optimal,)h(dd_Inconsistent,)e |
| (dd_DualInconsistent,)95 1286 y(dd_StrucInconsistent,)f |
| (dd_StrucDualInconsistent,)95 1399 y(dd_Unbounded,)j(dd_DualUnbounded)0 |
| 1512 y(})i(dd_LPStatusType;)0 1738 y(typedef)f(struct)g(matrixdata)f |
| (*dd_MatrixPtr;)0 1851 y(typedef)h(struct)g(matrixdata)f({)95 |
| 1964 y(dd_rowrange)g(rowsize;)95 2077 y(dd_rowset)h(linset;)191 |
| 2189 y(/*)95 b(a)47 b(subset)f(of)i(rows)e(of)h(linearity)f(\(ie,)g |
| (generators)f(of)382 2302 y(linearity)g(space)h(for)h |
| (V-representation,)c(and)k(equations)382 2415 y(for)g |
| (H-representation.)c(*/)95 2528 y(dd_colrange)i(colsize;)95 |
| 2641 y(dd_RepresentationType)d(representation;)95 2754 |
| y(dd_NumberType)j(numbtype;)95 2867 y(dd_Amatrix)g(matrix;)95 |
| 2980 y(dd_LPObjectiveType)e(objective;)95 3093 y(dd_Arow)j(rowvec;)0 |
| 3206 y(})95 b(dd_MatrixType;)0 3431 y(typedef)46 b(struct)g(setfamily)f |
| (*dd_SetFamilyPtr;)0 3544 y(typedef)h(struct)g(setfamily)f({)95 |
| 3657 y(dd_bigrange)g(famsize;)95 3770 y(dd_bigrange)g(setsize;)95 |
| 3883 y(dd_SetVector)g(set;)0 3996 y(})i(dd_SetFamilyType;)0 |
| 4222 y(typedef)f(struct)g(lpsolution)f(*dd_LPSolutionPtr;)0 |
| 4335 y(typedef)h(struct)g(lpsolution)f({)95 4448 y(dd_DataFileType)f |
| (filename;)95 4561 y(dd_LPObjectiveType)f(objective;)95 |
| 4673 y(dd_LPSolverType)h(solver;)95 4786 y(dd_rowrange)h(m;)95 |
| 4899 y(dd_colrange)g(d;)95 5012 y(dd_NumberType)g(numbtype;)95 |
| 5238 y(dd_LPStatusType)f(LPS;)94 b(/*)48 b(the)f(current)e(solution)h |
| (status)g(*/)95 5351 y(mytype)h(optvalue;)93 b(/*)47 |
| b(optimal)f(value)g(*/)1926 5682 y Fo(6)p eop end |
| %%Page: 7 7 |
| TeXDict begin 7 6 bop 95 44 a Fn(dd_Arow)46 b(sol;)142 |
| b(/*)48 b(primal)e(solution)f(*/)95 157 y(dd_Arow)h(dsol;)94 |
| b(/*)48 b(dual)e(solution)g(*/)95 270 y(dd_colindex)f(nbindex;)93 |
| b(/*)48 b(current)d(basis)i(represented)e(by)i(nonbasic)e(indices)h(*/) |
| 95 383 y(dd_rowrange)f(re;)95 b(/*)47 b(row)g(index)f(as)h(a)h |
| (certificate)d(in)i(the)g(case)f(of)i(inconsistency)c(*/)95 |
| 496 y(dd_colrange)h(se;)95 b(/*)47 b(col)g(index)f(as)h(a)h |
| (certificate)d(in)i(the)g(case)f(of)i(dual)e(inconsistency)e(*/)95 |
| 609 y(long)j(pivots[5];)143 722 y(/*)g(pivots[0]=setup)d(\(to)j(find)g |
| (a)g(basis\),)f(pivots[1]=PhaseI)d(or)k(Criss-Cross,)286 |
| 835 y(pivots[2]=Phase)d(II,)j(pivots[3]=Anticycling,)42 |
| b(pivots[4]=GMP)i(postopt)93 b(*/)95 947 y(long)47 b(total_pivots;)0 |
| 1060 y(})g(dd_LPSolutionType;)0 1455 y Fq(4)135 b(Library)45 |
| b(F)-11 b(unctions)0 1658 y Fo(Here)24 b(w)m(e)g(list)g(some)f(of)h |
| (the)f(most)h(imp)s(ortan)m(t)f(library)g(functions/pro)s(cedures.)37 |
| b(W)-8 b(e)25 b(use)e(the)g(follo)m(wing)i(con)m(v)m(en-)0 |
| 1771 y(tion:)42 b Fn(poly)29 b Fo(is)i(of)g(t)m(yp)s(e)g |
| Fn(dd)p 947 1771 29 4 v 34 w(PolyhedraPtr)p Fo(,)c Fn(matrix)p |
| Fo(,)j Fn(matrix1)e Fo(and)i Fn(matrix2)f Fo(are)i(of)g(t)m(yp)s(e)g |
| Fn(dd)p 3415 1771 V 34 w(MatrixPtr)p Fo(,)0 1884 y Fn(matrixP)p |
| Fo(,)40 b(of)h(t)m(yp)s(e)g Fn(dd)p 832 1884 V 34 w(MatrixPtr*)p |
| Fo(,)h Fn(err)e Fo(is)h(of)h(t)m(yp)s(e)f Fn(dd)p 2125 |
| 1884 V 34 w(ErrorType*)p Fo(,)g Fn(ifile)f Fo(and)h Fn(ofile)e |
| Fo(are)j(of)f(t)m(yp)s(e)0 1996 y Fn(char*)p Fo(,)29 |
| b Fn(A)h Fo(is)g(of)g(t)m(yp)s(e)g Fn(dd)p 872 1996 V |
| 34 w(Amatrix)p Fo(,)e Fn(point)h Fo(and)g Fn(vector)f |
| Fo(are)j(of)f(t)m(yp)s(e)g Fn(dd)p 2611 1996 V 34 w(Arow)p |
| Fo(,)f Fn(d)g Fo(is)h(of)h(t)m(yp)s(e)f Fn(dd)p 3463 |
| 1996 V 34 w(colrange)p Fo(,)0 2109 y Fn(m)37 b Fo(and)h |
| Fn(i)f Fo(are)h(of)g(t)m(yp)s(e)g Fn(dd)p 939 2109 V |
| 34 w(rowrange)p Fo(,)g Fn(x)f Fo(is)h(of)g(t)m(yp)s(e)g |
| Fn(mytype)p Fo(,)g Fn(a)g Fo(is)f(of)h(t)m(yp)s(e)g Fn(signed)47 |
| b(long)f(integer)p Fo(,)38 b Fn(b)g Fo(is)0 2222 y(of)33 |
| b(t)m(yp)s(e)f Fn(double)p Fo(,)g Fn(set)f Fo(is)i(of)f(t)m(yp)s(e)h |
| Fn(set)p 1388 2222 V 34 w(type)p Fo(.)45 b(Also,)34 b |
| Fn(setfam)d Fo(is)i(of)f(t)m(yp)s(e)h Fn(dd)p 2738 2222 |
| V 34 w(SetFamilyPtr)p Fo(,)c Fn(lp)j Fo(is)h(of)f(t)m(yp)s(e)0 |
| 2335 y Fn(dd)p 102 2335 V 34 w(LPPtr)p Fo(,)h Fn(lps)g |
| Fo(is)g(of)h(t)m(yp)s(e)g Fn(dd)p 1116 2335 V 34 w(LPSolutionPtr)p |
| Fo(,)d Fn(solver)g Fo(is)j(of)g(t)m(yp)s(e)f Fn(dd)p |
| 2654 2335 V 34 w(LPSolverType)p Fo(,)f Fn(roworder)f |
| Fo(is)i(of)0 2448 y(t)m(yp)s(e)e Fn(dd)p 307 2448 V 33 |
| w(RowOrderType)p Fo(.)0 2687 y Fd(4.1)112 b(Library)39 |
| b(Initialization)0 2858 y Fn(void)47 b(dd)p 341 2858 |
| V 33 w(set)p 518 2858 V 34 w(global)p 840 2858 V 33 w |
| (constants\(void\))41 b Fo(:)227 2971 y(This)31 b(is)g(to)h(set)g(the)f |
| (global)h(constan)m(ts)h(suc)m(h)e(as)g Fn(dd)p 2032 |
| 2971 V 34 w(zero)p Fo(,)f Fn(dd)p 2409 2971 V 34 w(purezero)f |
| Fo(and)i Fn(dd)p 3130 2971 V 34 w(one)f Fo(for)h(sign)g(recog-)227 |
| 3084 y(nition)h(and)g(basic)g(arithmetic)h(op)s(erations.)46 |
| b(Ev)m(ery)32 b(program)g(to)h(use)f(cddlib)f(m)m(ust)h(call)h(this)f |
| (function)227 3197 y(b)s(efore)37 b(doing)g(an)m(y)h(computation.)62 |
| b(Just)36 b(call)j(this)e(once.)61 b(See)38 b(Section)g(4.3.3)h(for)d |
| (the)i(de\014nitions)e(of)227 3310 y(constan)m(ts.)0 |
| 3487 y Fn(void)47 b(dd)p 341 3487 V 33 w(free)p 566 3487 |
| V 34 w(global)p 888 3487 V 32 w(constants\(void\))42 |
| b Fo(:)227 3600 y(This)e(is)g(to)h(free)f(the)g(global)i(constan)m(ts.) |
| 71 b(This)39 b(should)h(b)s(e)f(called)i(when)f(one)g(do)s(es)g(not)g |
| (use)g(cddlib)227 3713 y(functions)30 b(an)m(ymore.)0 |
| 3951 y Fd(4.2)112 b(Core)38 b(F)-9 b(unctions)0 4123 |
| y Fo(There)33 b(are)i(t)m(w)m(o)g(t)m(yp)s(es)f(of)g(core)g(functions)g |
| (in)f(cddlib.)51 b(The)33 b(\014rst)g(t)m(yp)s(e)h(runs)f(the)h(double) |
| f(description)h(\(DD\))0 4236 y(algorithm)28 b(and)f(do)s(es)g(a)h |
| (represen)m(tation)h(con)m(v)m(ersion)f(of)g(a)g(sp)s(eci\014ed)f(p)s |
| (olyhedron.)38 b(The)27 b(standard)g(header)g(for)0 4349 |
| y(this)32 b(t)m(yp)s(e)g(is)g Fn(dd)p 580 4349 V 34 w(DD*)p |
| Fo(.)45 b(The)32 b(second)g(t)m(yp)s(e)g(solv)m(es)h(one)g(or)f(more)g |
| (linear)h(programs)e(with)h(no)g(sp)s(ecial)h(headers.)0 |
| 4462 y(Both)k(t)m(yp)s(es)f(of)g(computations)h(are)f(non)m(trivial)h |
| (and)f(the)g(users)f(\(esp)s(ecially)i(for)f(the)g(DD)h(algorithm\))h |
| (m)m(ust)0 4575 y(kno)m(w)e(that)i(there)e(is)h(a)g(serous)f(limit)h |
| (in)f(the)h(sizes)g(of)g(problems)e(that)j(can)e(b)s(e)g(practically)i |
| (solv)m(ed.)60 b(Please)0 4687 y(c)m(hec)m(k)32 b(*.ext)g(and)d(*.ine)j |
| (\014les)e(that)h(come)g(with)f(cddlib)g(to)h(get)h(ideas)e(of)h |
| (tractable)h(problems.)141 4800 y(In)k(addition)h(to)g(previously)g |
| (de\014ned)e(ob)5 b(jects,)39 b(the)e(sym)m(b)s(ol)f |
| Fn(roworder)f Fo(is)h(of)h Fn(dd)p 3056 4800 V 34 w(RowOrderType)p |
| Fo(.)56 b(The)0 4913 y(sym)m(b)s(ol)32 b Fn(matrixP)f |
| Fo(is)h(a)h(p)s(oin)m(ter)g(to)g Fp(dd)p 1402 4913 32 |
| 4 v 38 w(MatrixT)m(yp)s(e)p Fo(.)46 b(the)33 b(argumen)m(ts)g |
| Fn(impl)p 2859 4913 29 4 v 33 w(lin)f Fo(and)g Fn(redset)e |
| Fo(are)j(b)s(oth)0 5026 y(a)e(p)s(oin)m(ter)f(to)h Fn(dd)p |
| 602 5026 V 34 w(rowset)e Fo(t)m(yp)s(e,)i(and)e Fn(newpos)g |
| Fo(is)h(a)h(p)s(oin)m(ter)g(to)g Fn(dd)p 2364 5026 V |
| 33 w(rowindex)e Fo(t)m(yp)s(e.)0 5207 y Fn(dd)p 102 5207 |
| V 34 w(PolyhedraPtr)44 b(dd)p 852 5207 V 34 w(DDMatrix2Poly\(matrix,)e |
| (err\))i Fo(:)227 5320 y(Store)32 b(the)g(represen)m(tation)g(giv)m(en) |
| h(b)m(y)e Fn(matrix)f Fo(in)h(a)h(p)s(olyhedra)e(data,)j(and)e |
| (generate)i(the)e(second)h(rep-)227 5433 y(resen)m(tation)46 |
| b(of)e Fn(*poly)p Fo(.)80 b(It)44 b(returns)e(a)j(p)s(oin)m(ter)f(to)g |
| (the)g(data.)82 b Fn(*err)43 b Fo(returns)g Fn(dd)p 3269 |
| 5433 V 34 w(NoError)f Fo(if)h(the)1926 5682 y(7)p eop |
| end |
| %%Page: 8 8 |
| TeXDict begin 8 7 bop 227 44 a Fo(computation)38 b(terminates)g |
| (normally)-8 b(.)60 b(Otherwise,)38 b(it)g(returns)d(a)i(v)-5 |
| b(alue)37 b(according)h(to)g(the)f(error)f(o)s(c-)227 |
| 157 y(cured.)0 345 y Fn(dd)p 102 345 29 4 v 34 w(PolyhedraPtr)44 |
| b(dd)p 852 345 V 34 w(DDMatrix2Poly2\(matrix,)e(roworder,)j(err\))f |
| Fo(:)227 458 y(This)f(is)h(the)f(same)h(function)f(as)h |
| Fn(dd)p 1559 458 V 34 w(DDMatrix2Poly)c Fo(except)k(that)g(the)g |
| (insertion)g(order)f(is)g(sp)s(eci-)227 571 y(\014ed)34 |
| b(b)m(y)h(the)g(user.)53 b(The)34 b(argumen)m(t)h Fn(roworder)e |
| Fo(is)i(of)f Fn(dd)p 2260 571 V 34 w(RowOrderType)e Fo(and)i(tak)m(es)i |
| (one)f(of)g(the)g(v)-5 b(al-)227 683 y(ues:)55 b Fn(dd)p |
| 536 683 V 33 w(MaxIndex)p Fo(,)37 b Fn(dd)p 1111 683 |
| V 34 w(MinIndex)p Fo(,)g Fn(dd)p 1687 683 V 34 w(MinCutoff)p |
| Fo(,)g Fn(dd)p 2311 683 V 34 w(MaxCutoff)p Fo(,)g Fn(dd)p |
| 2935 683 V 33 w(MixCutoff)p Fo(,)g Fn(dd)p 3558 683 V |
| 34 w(LexMin)p Fo(,)227 796 y Fn(dd)p 329 796 V 34 w(LexMax)p |
| Fo(,)d Fn(dd)p 806 796 V 34 w(RandomRow)p Fo(.)51 b(In)33 |
| b(general,)k Fn(dd)p 1905 796 V 34 w(LexMin)c Fo(is)h(the)h(b)s(est)f |
| (c)m(hoice)j(whic)m(h)d(is)g(in)h(fact)g(c)m(hosen)227 |
| 909 y(in)29 b Fn(dd)p 434 909 V 34 w(DDMatrix2Poly)p |
| Fo(.)36 b(If)28 b(y)m(ou)i(kno)m(w)e(that)i(the)f(input)e(is)i(already) |
| h(sorted)e(in)h(the)g(order)f(y)m(ou)h(lik)m(e,)i(use)227 |
| 1022 y Fn(dd)p 329 1022 V 34 w(MinIndex)k Fo(or)i Fn(dd)p |
| 996 1022 V 34 w(MaxIndex)p Fo(.)59 b(If)37 b(the)h(input)e(con)m(tains) |
| i(man)m(y)g(redundan)m(t)e(ro)m(ws)h(\(sa)m(y)h(more)g(than)227 |
| 1135 y(80\045)j(redundan)m(t\),)h(y)m(ou)e(migh)m(t)h(w)m(an)m(t)g(to)g |
| (try)f Fn(dd)p 2009 1135 V 34 w(MaxCutoff)d Fo(whic)m(h)j(migh)m(t)h |
| (result)f(in)g(m)m(uc)m(h)g(faster)227 1248 y(termination,)32 |
| b(see)f([3,)g(16)q(])0 1436 y Fn(boolean)46 b(dd)p 484 |
| 1436 V 34 w(DDInputAppend\(poly,)c(matrix,)k(err\))e |
| Fo(:)227 1549 y(Mo)s(dify)27 b(the)h(input)e(represen)m(tation)i(in)f |
| Fn(*poly)f Fo(b)m(y)h(app)s(ending)f(the)h(matrix)h(of)f |
| Fn(*matrix)p Fo(,)g(and)f(compute)227 1661 y(the)36 b(second)f |
| (represen)m(tation.)56 b(The)34 b(n)m(um)m(b)s(er)g(of)h(columns)g(in)g |
| Fn(*matrix)e Fo(m)m(ust)i(b)s(e)g(equal)g(to)h(the)g(input)227 |
| 1774 y(represen)m(tation.)0 1962 y Fn(boolean)46 b(dd)p |
| 484 1962 V 34 w(LPSolve\(lp,)e(solver,)i(err\))f Fo(:)227 |
| 2075 y(Solv)m(e)28 b Fn(lp)e Fo(b)m(y)h(the)g(algorithm)h |
| Fn(solver)d Fo(and)i(sa)m(v)m(e)h(the)f(solututions)g(in)g |
| Fn(*lp)p Fo(.)39 b(Unlik)m(e)27 b(the)g(earlier)h(v)m(ersions)227 |
| 2188 y(\(dplex,)d(cdd+\),)g(it)f(can)f(deal)h(with)f(equations)h(and)f |
| (totally)i(zero)g(righ)m(t)f(hand)e(sides.)38 b(It)24 |
| b(is)f(recommended)227 2301 y(that)32 b Fn(solver)d Fo(is)i |
| Fn(dd)p 936 2301 V 33 w(DualSimplex)p Fo(,)d(the)j(revised)g(dual)g |
| (simplex)f(metho)s(d)h(that)g(up)s(dates)f(a)h Fj(d)21 |
| b Fi(\002)f Fj(d)31 b Fo(dual)227 2414 y(basis)g(matrix)f(in)g(eac)m(h) |
| i(piv)m(ot)f(\(where)f Fj(d)h Fo(is)f(the)h(column)f(size)h(of)g(lp\).) |
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| (constrain)m(ts)g(\(i.e.)61 b(large)37 b(ro)m(w)227 2790 |
| y(size,)f(can)e(b)s(e)f(a)h(few)g(ten)g(thousands\).)50 |
| b(If)34 b(y)m(our)f(LP)h(has)f(man)m(y)h(v)-5 b(ariables)35 |
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| (arithmetic)h(and)d(v)m(eri\014es)h(whether)f(the)h(output)g(basis)f |
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| (ted)g(in)f(the)h(v)m(ersion)g(093.)42 b(The)30 b(original)i |
| (\(non-crosso)m(v)m(er\))g(v)m(ersion)f(of)227 3505 y(the)g(same)g |
| (function)f(is)g(still)h(a)m(v)-5 b(ailable)33 b(as)e |
| Fn(boolean)46 b(dd)p 2206 3505 V 33 w(LPSolve0)p Fo(.)0 |
| 3692 y Fn(dd)p 102 3692 V 34 w(boolean)g(dd)p 614 3692 |
| V 33 w(Redundant\(matrix,)e(i,)j(point,)f(err\))e Fo(:)227 |
| 3805 y(Chec)m(k)28 b(whether)g Fj(i)p Fo(th)g(data)g(in)g |
| Fn(matrix)e Fo(is)h(redundan)m(t)g(for)h(the)g(represen)m(tation.)41 |
| b(If)27 b(it)h(is)g(nonredundan)m(t,)227 3918 y(it)d(returns)e(a)h |
| (certi\014cate.)41 b(F)-8 b(or)25 b(H-represen)m(tation,)i(it)e(is)f(a) |
| g Fn(point)f Fo(in)h Fj(R)2684 3885 y Fg(d)2748 3918 |
| y Fo(whic)m(h)g(satis\014es)g(all)h(inequalities)227 |
| 4031 y(except)31 b(for)f(the)g Fj(i)p Fo(th)g(inequalit)m(y)-8 |
| b(.)42 b(If)29 b Fj(i)h Fo(is)g(a)g(linearit)m(y)-8 b(,)32 |
| b(it)f(do)s(es)e(nothing)h(and)f(alw)m(a)m(ys)i(returns)e |
| Fn(dd)p 3606 4031 V 34 w(FALSE)p Fo(.)0 4219 y Fn(dd)p |
| 102 4219 V 34 w(rowset)46 b(dd)p 566 4219 V 34 w |
| (RedundantRows\(matrix,)c(err\))i Fo(:)227 4331 y(Returns)24 |
| b(a)h(maximal)h(set)f(of)g(ro)m(w)g(indices)f(suc)m(h)h(that)g(the)g |
| (asso)s(ciated)h(ro)m(ws)f(can)g(b)s(e)f(eliminated)i(without)227 |
| 4444 y(c)m(hanging)32 b(the)e(p)s(olyhedron.)40 b(The)29 |
| b(function)h(w)m(orks)h(for)f(b)s(oth)g(V-)g(and)g(H-represen)m |
| (tations.)0 4632 y Fn(dd)p 102 4632 V 34 w(boolean)46 |
| b(dd)p 614 4632 V 33 w(SRedundant\(matrix,)d(i,)k(point,)g(err\))d |
| Fo(:)227 4745 y(Chec)m(k)32 b(whether)e Fj(i)p Fo(th)i(data)g(in)e |
| Fn(matrix)g Fo(is)h(strongly)h(redundan)m(t)e(for)h(the)g(represen)m |
| (tation.)44 b(If)31 b Fj(i)h Fo(is)f(a)g(lin-)227 4858 |
| y(earit)m(y)-8 b(,)32 b(it)d(do)s(es)f(nothing)h(and)f(alw)m(a)m(ys)j |
| (returns)c Fn(dd)p 2005 4858 V 34 w(FALSE)p Fo(.)h(Here,)h |
| Fj(i)p Fo(th)g(inequalit)m(y)i(in)d(H-represen)m(tation)227 |
| 4971 y(is)h Ff(str)-5 b(ongly)33 b(r)-5 b(e)g(dundant)39 |
| b Fo(if)29 b(it)g(is)g(redundan)m(t)e(and)h(there)h(is)g(no)f(p)s(oin)m |
| (t)h(in)f(the)h(p)s(olyhedron)e(satisfying)i(the)227 |
| 5084 y(inequalit)m(y)c(with)e(equalit)m(y)-8 b(.)41 b(In)22 |
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| (relativ)m(e)h(in)m(terior)e(of)g(the)g(p)s(olyhedron.)41 |
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| %%Page: 9 9 |
| TeXDict begin 9 8 bop 0 44 a Fn(dd)p 102 44 29 4 v 34 |
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| Fj(i)p Fo(th)h(ro)m(w)g(in)g(the)g(input)f(is)h(forced)g(to)h(b)s(e)e |
| (linearit)m(y)j(\(equalit)m(y)g(for)d(H-represen)m(tation\).)227 |
| 270 y(If)f Fj(i)h Fo(is)f(linearit)m(y)i(itself,)g(it)e(do)s(es)g |
| (nothing)h(and)f(alw)m(a)m(ys)h(returns)f Fn(dd)p 2589 |
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| (ro)m(ws)h(that)g(are)g(implicitly)g(linearit)m(y)-8 |
| b(.)70 b(It)40 b(simply)f(calls)h(the)g(library)227 680 |
| y(function)e Fn(dd)p 693 680 V 34 w(ImplicitLinearity)33 |
| b Fo(for)38 b(eac)m(h)h(inequalit)m(y)g(and)f(collects)i(the)e(ro)m(w)g |
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| Fn(dd)p 879 793 V 33 w(TRUE)p Fo(.)0 977 y Fn(dd)p 102 |
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| (MatrixCanonicalize\(matrixP)o(,)c(impl)p 2176 977 V |
| 33 w(lin,)47 b(redset,)f(newpos,)f(err\))g Fo(:)227 1090 |
| y(The)30 b(input)f(is)h(a)h(p)s(oin)m(ter)f Fn(matrixP)e |
| Fo(to)j(a)f(matrix)h(and)e(the)i(function)f(mo)s(di\014es)f(the)h |
| (matrix)h(b)m(y)f(putting)227 1203 y(a)40 b(maximally)h(linear)f(indep) |
| s(enden)m(t)e(linearities)j(\(basis\))g(at)f(the)g(top)g(of)f(the)h |
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| y(matrix)c(will)f(b)s(e)f(returned)g(in)g(the)i(corresp)s(onding)e(ro)m |
| (w)h(sets.)40 b(The)26 b(new)g(p)s(ositions)h(of)g(the)g(original)i(ro) |
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| (linearit)m(y)i(set)f Fn(\(*matrixP\)->linset)22 b Fo(is)28 |
| b(the)g(co)s(dimension)f(of)h(the)g(p)s(oly-)227 1803 |
| y(hedron)i(if)g(it)h(is)f(H-p)s(olyhedron,)g(and)g(is)g(the)h |
| (dimension)e(of)i(linearit)m(y)h(space)f(if)f(it)h(is)f(V-p)s |
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| b(In)27 b(the)h(future,)g(this)g(function)g(is)g(exp)s(ected)227 |
| 2178 y(to)j(b)s(e)f(correctly)i(implemen)m(ted.)0 2362 |
| y Fn(dd)p 102 2362 V 34 w(boolean)46 b(dd)p 614 2362 |
| V 33 w(MatrixCanonicalizeLinearit)o(y\(ma)o(tri)o(xP,)41 |
| b(impl)p 2605 2362 V 34 w(linset,)k(newpos.)94 b(err\))44 |
| b Fo(:)227 2475 y(It)g(do)s(es)g(only)g(the)g(\014rst)f(half)h(of)g |
| Fn(dd)p 1554 2475 V 34 w(boolean)i(dd)p 2066 2475 V 33 |
| w(MatrixCanonicalize)p Fo(,)d(namely)-8 b(,)48 b(it)d(detects)g(all)227 |
| 2588 y(implicit)27 b(linearities)g(and)f(puts)f(a)h(maximally)h(indep)s |
| (enden)m(t)d(linearities)k(at)e(the)g(top)g(of)g(the)g(matrix.)40 |
| b(F)-8 b(or)227 2701 y(example,)32 b(this)e(function)g(can)h(b)s(e)e |
| (used)h(to)h(detect)h(the)e(dimension)g(of)h(an)f(H-p)s(olyhedron.)0 |
| 2885 y Fn(dd)p 102 2885 V 34 w(boolean)46 b(dd)p 614 |
| 2885 V 33 w(MatrixRedundancyRemove\(mat)o(rixP)o(,)c(redset,)k(newpos,) |
| f(err\))g Fo(:)227 2998 y(It)25 b(do)s(es)e(essen)m(tially)j(the)f |
| (second)f(half)g(of)g Fn(dd)p 1757 2998 V 34 w(boolean)46 |
| b(dd)p 2269 2998 V 34 w(MatrixCanonicalize)p Fo(,)20 |
| b(namely)-8 b(,)27 b(it)d(detects)227 3111 y(all)30 b(redundancies.)39 |
| b(This)27 b(function)h(should)g(b)s(e)f(used)h(after)h |
| Fn(dd)p 2421 3111 V 34 w(MatrixCanonicalizeLinear)o(ity)22 |
| b Fo(has)227 3224 y(b)s(een)30 b(called.)0 3408 y Fn(dd)p |
| 102 3408 V 34 w(boolean)46 b(dd)p 614 3408 V 33 w |
| (FindRelativeInterior\(matri)o(x,)c(impl)p 2224 3408 |
| V 33 w(lin,)k(lin)p 2639 3408 V 34 w(basis,)g(lps,)h(err\))d |
| Fo(:)227 3521 y(Computes)39 b(a)h(p)s(oin)m(t)g(in)f(the)h(relativ)m(e) |
| h(in)m(terior)f(of)g(an)g(H-p)s(olyhedron)e(giv)m(en)j(b)m(y)e(matrix,) |
| j(b)m(y)e(solving)227 3634 y(an)32 b(LP)-8 b(.)31 b(The)g(p)s(oin)m(t)h |
| (will)g(b)s(e)f(returned)f(b)m(y)h Fn(lps)p Fo(.)44 b(See)31 |
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| 3747 y(all)47 b(nonempt)m(y)e(faces)h(of)g(an)f(H-p)s(olyhedron)g(and)g |
| (a)h(relativ)m(e)h(in)m(terior)g(p)s(oin)m(t)e(for)g(eac)m(h)i(face.)87 |
| b(The)227 3860 y(former)35 b(returns)f(all)i(implicit)g(linearit)m(y)g |
| (ro)m(ws)g(\(implicit)g(equations\))g(and)f(the)g(latter)h(returns)e(a) |
| i(basis)227 3973 y(of)31 b(the)g(union)e(of)i(linearit)m(y)h(ro)m(ws)f |
| (and)f(implicit)h(linearit)m(y)h(ro)m(ws.)41 b(This)30 |
| b(means)h(that)g(the)f(cardinalit)m(y)i(of)227 4086 y |
| Fn(*lin)p 425 4086 V 34 w(basis)c Fo(is)j(the)f(co)s(dimension)h(of)f |
| (the)h(p)s(olyhedron.)0 4270 y Fn(dd)p 102 4270 V 34 |
| w(boolean)46 b(dd)p 614 4270 V 33 w(ExistsRestrictedFace\(matri)o(x,)c |
| (R,)47 b(S,)g(err\))d Fo(:)227 4383 y(Returns)26 b(the)i(answ)m(er)f |
| (to)g(the)h(F)-8 b(ark)j(as')28 b(t)m(yp)s(e)f(decision)h(problem)e(as) |
| i(to)f(whether)g(there)g(is)g(a)h(p)s(oin)m(t)f(in)f(the)227 |
| 4496 y(p)s(olyhedron)h(giv)m(en)h(b)m(y)g(matrix)g(satisfying)g(all)h |
| (constrain)m(ts)f(in)g Fn(R)f Fo(with)g(equalit)m(y)j(and)d(all)h |
| (constrain)m(ts)h(in)227 4609 y Fn(S)h Fo(with)g(strict)i(inequalit)m |
| (y)-8 b(.)42 b(More)31 b(precisely)-8 b(,)32 b(it)f(is)f(the)h(linear)f |
| (feasibilit)m(y)j(problem:)1111 4795 y Fi(9)p Fo(?)82 |
| b Fj(x)113 b Fo(satisfying)h Fj(b)1979 4809 y Fg(r)2038 |
| 4795 y Fi(\000)19 b Fj(A)2196 4809 y Fg(r)2235 4795 y |
| Fj(x)83 b Fo(=)25 b(0)p Fj(;)41 b Fi(8)p Fj(r)27 b Fi(2)e |
| Fj(R)c Fi([)f Fj(L)1940 4908 y(b)1979 4922 y Fg(s)2036 |
| 4908 y Fi(\000)g Fj(A)2195 4922 y Fg(s)2232 4908 y Fj(x)86 |
| b(>)25 b Fo(0)p Fj(;)41 b Fi(8)p Fj(s)24 b Fi(2)h Fj(S)1940 |
| 5021 y(b)1979 5035 y Fg(t)2029 5021 y Fi(\000)20 b Fj(A)2188 |
| 5035 y Fg(t)2218 5021 y Fj(x)100 b Fi(\025)25 b Fo(0)p |
| Fj(;)41 b Fi(8)p Fj(t)24 b Fi(2)h Fj(T)8 b(;)227 5207 |
| y Fo(where)31 b Fj(L)g Fo(is)g(the)g(set)h(of)f(linearit)m(y)i(ro)m(ws) |
| e(of)g Fn(matrix)p Fo(,)f(and)g Fj(T)44 b Fo(represen)m(ts)31 |
| b(the)g(set)h(of)f(ro)m(ws)g(that)h(are)f(not)227 5320 |
| y(in)e Fj(R)17 b Fi([)f Fj(L)h Fi([)f Fj(S)5 b Fo(.)40 |
| b(Both)29 b Fn(R)g Fo(and)f Fn(S)g Fo(are)h(of)g Fn(dd)p |
| 1683 5320 V 33 w(rowset)e Fo(t)m(yp)s(e.)41 b(The)28 |
| b(set)h Fj(S)k Fo(is)c(supp)s(osed)e(to)i(b)s(e)f(disjoin)m(t)h(from) |
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| %%Page: 10 10 |
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| (linearit)m(y)i(ro)m(ws)e(of)g Fn(matrix)p Fo(,)f(and)g |
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| 1454 y(optimal)c(and)d(the)i(optimal)g(v)-5 b(alue)30 |
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| 1939 y Fo(or)e Fn(dd)p 441 1939 V 33 w(Unspecified)p |
| Fo(,)d(it)j(treats)g(the)g(matrix)g(as)f(if)g(it)h(w)m(ere)g(inequalit) |
| m(y)h(represen)m(tation.)0 2118 y Fn(dd)p 102 2118 V |
| 34 w(SetFamilyPtr)44 b(dd)p 852 2118 V 34 w(Matrix2Adjacency\(matrix,)d |
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| (input)f(ro)m(ws)g(using)h(the)g(LP)f(solv)m(er)i(and)e(without)h |
| (running)e(the)227 2344 y(represen)m(tation)27 b(con)m(v)m(ersion.)41 |
| b(When)26 b(the)g(input)f(is)h(H-represen)m(tation,)j(it)d(giv)m(es)h |
| (the)g(facet)g(graph)e(of)h(the)227 2457 y(p)s(olyhedron.)37 |
| b(F)-8 b(or)24 b(V-represen)m(tation,)j(it)c(giv)m(es)i(the)e(\(v)m |
| (ertex\))i(graph)e(of)g(the)h(p)s(olyhedron.)37 b(It)23 |
| b(is)g(required)227 2570 y(that)28 b(the)f(input)g(matrix)g(is)g(a)h |
| (minimal)f(represen)m(tation.)41 b(Run)26 b(redundancy)g(remo)m(v)-5 |
| b(al)28 b(functions)f(b)s(efore)227 2683 y(calling)32 |
| b(this)e(function,)h(see)g(the)f(sample)h(co)s(de)f(adjacency)-8 |
| b(.c.)0 2862 y Fn(dd)p 102 2862 V 34 w(SetFamilyPtr)44 |
| b(dd)p 852 2862 V 34 w(Matrix2WeakAdjacency\(matr)o(ix,)d(err\))k |
| Fo(:)227 2975 y(Computes)37 b(the)h(w)m(eak)h(adjacency)g(list)f(of)g |
| (input)f(ro)m(ws)h(using)f(the)h(LP)g(solv)m(er)g(and)f(without)h |
| (running)227 3088 y(the)31 b(represen)m(tation)h(con)m(v)m(ersion.)44 |
| b(When)30 b(the)h(input)f(is)h(H-represen)m(tation,)i(it)e(giv)m(es)i |
| (the)e(graph)f(where)227 3201 y(its)g(no)s(des)f(are)h(the)g(facets)h |
| (t)m(w)m(o)f(no)s(des)f(are)h(adjacen)m(t)h(if)f(and)f(only)g(if)h(the) |
| g(asso)s(ciated)h(facets)f(ha)m(v)m(e)h(some)227 3314 |
| y(in)m(tersection.)41 b(F)-8 b(or)26 b(V-represen)m(tation,)j(it)d(giv) |
| m(es)h(the)f(graph)f(where)g(its)h(no)s(des)e(are)i(the)g(v)m(ertices)h |
| (and)e(t)m(w)m(o)227 3427 y(no)s(des)i(are)g(adjacen)m(t)i(if)e(and)g |
| (only)g(if)g(the)h(asso)s(ciated)h(v)m(ertices)g(are)e(on)g(a)h(common) |
| g(facet.)41 b(It)27 b(is)g(required)227 3540 y(that)h(the)f(input)g |
| (matrix)g(is)g(a)h(minimal)f(represen)m(tation.)41 b(Run)26 |
| b(redundancy)g(remo)m(v)-5 b(al)28 b(functions)f(b)s(efore)227 |
| 3653 y(calling)32 b(this)e(function,)h(see)g(the)f(sample)h(co)s(de)f |
| (adjacency)-8 b(.c.)0 3832 y Fn(dd)p 102 3832 V 34 w(MatrixPtr)45 |
| b(dd)p 709 3832 V 34 w(FourierElimination\(matri)o(x,)d(err\))i |
| Fo(:)227 3945 y(Eliminate)32 b(the)f(last)h(v)-5 b(ariable)32 |
| b(from)f(a)g(system)g(of)g(linear)h(inequalities)g(giv)m(en)g(b)m(y)f |
| (matrix)g(b)m(y)g(using)f(the)227 4058 y(F)-8 b(ourier's)21 |
| b(Elimination.)38 b(If)20 b(the)h(input)e(matrix)i(is)f(V-represen)m |
| (tation,)k Fn(*err)19 b Fo(returns)g Fn(dd)p 3275 4058 |
| V 34 w(NotAvailForV)p Fo(.)227 4171 y(This)32 b(function)h(do)s(es)f |
| (not)h(remo)m(v)m(e)i(redundancy)c(and)h(one)h(migh)m(t)h(w)m(an)m(t)f |
| (to)h(call)g(redundancy)d(remo)m(v)-5 b(al)227 4284 y(functions)30 |
| b(afterw)m(ards.)41 b(See)31 b(the)f(sample)h(co)s(de)f(fourier.c.)0 |
| 4463 y Fn(dd)p 102 4463 V 34 w(MatrixPtr)45 b(dd)p 709 |
| 4463 V 34 w(BlockElimination\(matrix,)c(set,)47 b(err\))d |
| Fo(:)227 4576 y(Eliminate)d(a)f(set)g(of)f(v)-5 b(ariables)41 |
| b(from)e(a)h(system)f(of)h(linear)g(inequalities)h(giv)m(en)f(b)m(y)g |
| (matrix)g(b)m(y)f(using)227 4689 y(the)33 b(extreme)g(ra)m(ys)g(of)g |
| (the)g(dual)f(linear)h(system.)48 b(See)33 b(commen)m(ts)g(in)f(the)h |
| (co)s(de)g(cddpro)5 b(j.c)32 b(for)g(details.)227 4802 |
| y(This)27 b(migh)m(t)g(b)s(e)g(a)h(faster)f(w)m(a)m(y)h(to)g(eliminate) |
| h(v)-5 b(ariables)28 b(than)e(the)i(rep)s(eated)f(F)-8 |
| b(ourierElimination)29 b(when)227 4915 y(the)35 b(n)m(um)m(b)s(er)e(of) |
| h(v)-5 b(ariables)35 b(to)g(eliminate)g(is)g(large.)53 |
| b(If)33 b(the)i(input)e(matrix)i(is)f(V-represen)m(tation,)j |
| Fn(*err)227 5028 y Fo(returns)26 b Fn(dd)p 640 5028 V |
| 34 w(NotAvailForV)p Fo(.)d(This)k(function)f(do)s(es)h(not)g(remo)m(v)m |
| (e)h(redundancy)d(and)h(one)i(migh)m(t)f(w)m(an)m(t)h(to)227 |
| 5141 y(call)k(redundancy)d(remo)m(v)-5 b(al)31 b(functions)f(afterw)m |
| (ards.)41 b(See)31 b(the)f(sample)h(co)s(de)f(pro)5 b(jection.c.)0 |
| 5320 y Fn(dd)p 102 5320 V 34 w(rowrange)45 b(dd)p 661 |
| 5320 V 34 w(RayShooting\(matrix,)e(point,)j(vector\))d |
| Fo(:)227 5433 y(Finds)32 b(the)g(index)g(of)g(a)h(halfspace)f(\014rst)g |
| (left)g(b)m(y)g(the)h(ra)m(y)f(starting)h(from)f Fn(point)f |
| Fo(to)m(w)m(ard)i(the)f(direction)1903 5682 y(10)p eop |
| end |
| %%Page: 11 11 |
| TeXDict begin 11 10 bop 227 44 a Fn(vector)p Fo(.)39 |
| b(It)28 b(resolv)m(es)i(tie)f(b)m(y)f(a)h(lexicographic)h(p)s |
| (erturbation.)39 b(Those)28 b(inequalities)i(violated)g(b)m(y)e |
| Fn(point)227 157 y Fo(will)j(b)s(e)f(simply)g(ignored.)0 |
| 401 y Fd(4.3)112 b(Data)38 b(Manipulations)0 572 y Fp(4.3.1)105 |
| b(Num)m(b)s(er)35 b(Assignmen)m(ts)0 744 y Fo(F)-8 b(or)31 |
| b(n)m(um)m(b)s(er)d(assignmen)m(ts,)j(one)f(cannot)h(use)e(suc)m(h)h |
| (expressions)f(as)h Fn(x=\(mytype\)a)p Fo(.)38 b(This)29 |
| b(is)g(b)s(ecause)h(cddlib)0 857 y(uses)37 b(an)g(abstract)h(n)m(um)m |
| (b)s(er)e(t)m(yp)s(e)h(\()p Fn(mytype)p Fo(\))f(so)i(that)f(it)h(can)f |
| (compute)h(with)f(v)-5 b(arious)37 b(n)m(um)m(b)s(er)f(t)m(yp)s(es)h |
| (suc)m(h)0 970 y(as)c(C)f(double)g(and)h(GMP)g(rational.)49 |
| b(User)32 b(can)h(easily)h(add)e(a)h(new)g(n)m(um)m(b)s(er)e(t)m(yp)s |
| (e)i(b)m(y)g(rede\014ning)e(arithmetic)0 1082 y(op)s(erations)g(in)f |
| (cddmp.h)e(and)i(cddmp.c.)0 1295 y Fn(void)47 b(dd)p |
| 341 1295 29 4 v 33 w(init\(x\))d Fo(:)227 1408 y(This)36 |
| b(is)g(to)h(initialize)h(a)f Fn(mytype)d Fo(v)-5 b(ariable)37 |
| b Fn(x)f Fo(and)g(to)h(set)f(it)h(to)g(zero.)59 b(This)35 |
| b(initialization)k(has)d(to)h(b)s(e)227 1521 y(called)32 |
| b(b)s(efore)e(an)m(y)h Fn(mytype)d Fo(v)-5 b(ariable)31 |
| b(to)g(b)s(e)f(used.)0 1708 y Fn(void)47 b(dd)p 341 1708 |
| V 33 w(clear\(x\))d Fo(:)227 1821 y(This)30 b(is)g(to)h(free)g(the)f |
| (space)h(allo)s(cated)i(to)e(a)g Fn(mytype)d Fo(v)-5 |
| b(ariable)31 b Fn(x)p Fo(.)0 2009 y Fn(void)47 b(dd)p |
| 341 2009 V 33 w(set)p 518 2009 V 34 w(si\(x,)f(a\))f |
| Fo(:)227 2122 y(This)30 b(is)g(to)h(set)g(a)g Fn(mytype)e |
| Fo(v)-5 b(ariable)31 b Fn(x)f Fo(to)h(the)g(v)-5 b(alue)30 |
| b(of)h(signed)f(long)h(in)m(teger)h Fn(a)p Fo(.)0 2310 |
| y Fn(void)47 b(dd)p 341 2310 V 33 w(set)p 518 2310 V |
| 34 w(si2\(x,)f(a,)h(b\))e Fo(:)227 2422 y(This)24 b(is)g(to)i(set)f(a)f |
| Fn(mytype)f Fo(v)-5 b(ariable)25 b Fn(x)f Fo(to)i(the)e(v)-5 |
| b(alue)25 b(of)g(the)f(rational)i(expression)e Fn(a/b)p |
| Fo(,)h(where)f Fn(a)g Fo(is)h(signed)227 2535 y(long)31 |
| b(and)f Fn(b)g Fo(is)g(unsigned)g(long)h(in)m(tegers.)0 |
| 2723 y Fn(void)47 b(dd)p 341 2723 V 33 w(set)p 518 2723 |
| V 34 w(d\(x,)g(b\))e Fo(:)227 2836 y(This)33 b(is)g(to)h(set)f(a)h |
| Fn(mytype)d Fo(v)-5 b(ariable)34 b Fn(x)f Fo(to)h(the)f(v)-5 |
| b(alue)34 b(of)f(double)g Fn(b)p Fo(.)49 b(This)32 b(is)h(a)m(v)-5 |
| b(ailable)36 b(only)d(when)f(the)227 2949 y(library)e(is)h(compiled)f |
| (without)h Fn(-DGMPRATIONAL)c Fo(compiler)k(option.)0 |
| 3189 y Fp(4.3.2)105 b(Arithmetic)35 b(Op)s(erations)g(for)g |
| Fn(mytype)e Fp(Num)m(b)s(ers)0 3361 y Fo(Belo)m(w)f Fn(x)p |
| Fo(,)e Fn(y)p Fo(,)h Fn(z)f Fo(are)g(of)h(t)m(yp)s(e)f |
| Fn(mytype)p Fo(.)0 3573 y Fn(void)47 b(dd)p 341 3573 |
| V 33 w(add\(x,)g(y,)g(z\))e Fo(:)227 3686 y(Set)31 b |
| Fn(x)f Fo(to)h(b)s(e)f(the)g(sum)g(of)g Fn(y)g Fo(and)g |
| Fn(z)p Fo(.)0 3874 y Fn(void)47 b(dd)p 341 3874 V 33 |
| w(sub\(x,)g(y,)g(z\))e Fo(:)227 3987 y(Set)31 b Fn(x)f |
| Fo(to)h(b)s(e)f(the)g(substraction)h(of)f Fn(z)g Fo(from)g |
| Fn(y)p Fo(.)0 4174 y Fn(void)47 b(dd)p 341 4174 V 33 |
| w(mul\(x,)g(y,)g(z\))e Fo(:)227 4287 y(Set)31 b Fn(x)f |
| Fo(to)h(b)s(e)f(the)g(m)m(ultiplication)j(of)d Fn(y)g |
| Fo(and)g Fn(z)p Fo(.)0 4475 y Fn(void)47 b(dd)p 341 4475 |
| V 33 w(div\(x,)g(y,)g(z\))e Fo(:)227 4588 y(Set)31 b |
| Fn(x)f Fo(to)h(b)s(e)f(the)g(division)h(of)f Fn(y)g Fo(o)m(v)m(er)i |
| Fn(z)p Fo(.)0 4775 y Fn(void)47 b(dd)p 341 4775 V 33 |
| w(inv\(x,)g(y\))d Fo(:)227 4888 y(Set)31 b Fn(x)f Fo(to)h(b)s(e)f(the)g |
| (reciplo)s(cal)i(of)f Fn(y)p Fo(.)1903 5682 y(11)p eop |
| end |
| %%Page: 12 12 |
| TeXDict begin 12 11 bop 0 44 a Fp(4.3.3)105 b(Prede\014ned)36 |
| b(Constan)m(ts)0 216 y Fo(There)24 b(are)h(sev)m(eral)g |
| Fn(mytype)e Fo(constan)m(ts)i(de\014ned)f(when)f Fn(dd)p |
| 2043 216 29 4 v 34 w(set)p 2221 216 V 33 w(global)p 2542 |
| 216 V 33 w(constants\(void\))d Fo(is)25 b(called.)40 |
| b(Some)0 329 y(constan)m(ts)35 b(dep)s(end)d(on)i(the)g(double)g |
| (constan)m(t)h Fn(dd)p 1784 329 V 34 w(almostzero)c Fo(whic)m(h)i(is)h |
| (normally)g(set)h(to)g(10)3417 296 y Fc(\000)p Fh(7)3546 |
| 329 y Fo(in)e(cdd.h.)0 442 y(This)28 b(v)-5 b(alue)29 |
| b(can)f(b)s(e)g(mo)s(di\014ed)f(dep)s(ending)g(on)i(ho)m(w)f(n)m |
| (umerically)h(delicate)i(y)m(our)d(problems)g(are)h(but)f(an)g(extra)0 |
| 555 y(caution)j(should)f(b)s(e)f(tak)m(en.)0 767 y Fn(mytype)46 |
| b(dd)p 436 767 V 34 w(purezero)d Fo(:)227 880 y(This)30 |
| b(represen)m(ts)g(the)h(mathematical)h(zero)g(0.)0 1068 |
| y Fn(mytype)46 b(dd)p 436 1068 V 34 w(zero)e Fo(:)227 |
| 1181 y(This)36 b(represen)m(ts)g(the)h(largest)h(p)s(ositiv)m(e)f(n)m |
| (um)m(b)s(er)e(whic)m(h)h(should)g(b)s(e)g(considered)g(to)h(b)s(e)f |
| (zero.)60 b(In)35 b(the)227 1293 y(GMPRA)-8 b(TIONAL)34 |
| b(mo)s(de,)h(it)f(is)g(equal)g(to)g Fn(dd)p 1890 1293 |
| V 34 w(purezero)p Fo(.)49 b(In)33 b(the)h(C)f(double)g(mo)s(de,)i(it)f |
| (is)g(set)g(to)g(the)227 1406 y(v)-5 b(alue)31 b(of)g |
| Fn(dd)p 668 1406 V 33 w(almostzero)p Fo(.)0 1594 y Fn(mytype)46 |
| b(dd)p 436 1594 V 34 w(minuszero)d Fo(:)227 1707 y(The)30 |
| b(negativ)m(e)j(of)d Fn(dd)p 978 1707 V 34 w(zero)p Fo(.)0 |
| 1895 y Fn(mytype)46 b(dd)p 436 1895 V 34 w(one)f Fo(:)227 |
| 2007 y(This)30 b(represen)m(ts)g(the)h(mathematical)h(one)f(1.)0 |
| 2248 y Fp(4.3.4)105 b(Sign)36 b(Ev)-6 b(aluation)34 b(and)h(Comparison) |
| h(for)f Fn(mytype)e Fp(Num)m(b)s(ers)0 2419 y Fo(Belo)m(w)f |
| Fn(x)p Fo(,)e Fn(y)p Fo(,)h Fn(z)f Fo(are)g(of)h(t)m(yp)s(e)f |
| Fn(mytype)p Fo(.)0 2632 y Fn(dd)p 102 2632 V 34 w(boolean)46 |
| b(dd)p 614 2632 V 33 w(Positive\(x\))d Fo(:)227 2745 |
| y(Returns)23 b Fn(dd)p 668 2745 V 34 w(TRUE)f Fo(if)h |
| Fn(x)g Fo(is)g(considered)h(p)s(ositiv)m(e,)i(and)c Fn(dd)p |
| 2207 2745 V 34 w(FALSE)g Fo(otherwise.)39 b(In)23 b(the)g(GMPRA)-8 |
| b(TIONAL)227 2858 y(mo)s(de,)23 b(the)d(p)s(ositivit)m(y)i(recognition) |
| g(is)f(exact.)39 b(In)20 b(the)h(C)f(double)g(mo)s(de,)i(this)f(means)f |
| (the)h(v)-5 b(alue)21 b(is)g(strictly)227 2970 y(larger)31 |
| b(than)f Fn(dd)p 799 2970 V 34 w(zero)p Fo(.)227 3121 |
| y Fn(dd)p 329 3121 V 34 w(boolean)46 b(dd)p 841 3121 |
| V 34 w(Negative\(x\))27 b Fo(w)m(orks)j(similarly)-8 |
| b(.)0 3308 y Fn(dd)p 102 3308 V 34 w(boolean)46 b(dd)p |
| 614 3308 V 33 w(Nonpositive\(x\))c Fo(:)227 3421 y(Returns)30 |
| b(the)g(negation)i(of)f Fn(dd)p 1304 3421 V 33 w(Positive\(x\))p |
| Fo(.)38 b Fn(dd)p 2024 3421 V 34 w(Nonnegative\(x\))26 |
| b Fo(w)m(orks)31 b(similarly)-8 b(.)0 3609 y Fn(dd)p |
| 102 3609 V 34 w(boolean)46 b(dd)p 614 3609 V 33 w(EqualToZero\(x\))c |
| Fo(:)227 3722 y(Returns)34 b Fn(dd)p 679 3722 V 34 w(TRUE)g |
| Fo(if)h Fn(x)g Fo(is)g(considered)g(zero,)i(and)d Fn(dd)p |
| 2156 3722 V 34 w(FALSE)g Fo(otherwise.)55 b(In)34 b(the)h(GMPRA)-8 |
| b(TIONAL)227 3835 y(mo)s(de,)28 b(the)f(zero)h(recognition)h(is)f |
| (exact.)41 b(In)26 b(the)i(C)e(double)h(mo)s(de,)h(this)f(means)g(the)h |
| (v)-5 b(alue)27 b(is)h(in)m(b)s(et)m(w)m(een)227 3948 |
| y Fn(dd)p 329 3948 V 34 w(minuszero)g Fo(and)i Fn(dd)p |
| 1096 3948 V 34 w(zero)f Fo(inclusiv)m(e.)0 4135 y Fn(dd)p |
| 102 4135 V 34 w(boolean)46 b(dd)p 614 4135 V 33 w(Larger\(x,)g(y\))f |
| Fo(:)227 4248 y(Returns)31 b Fn(dd)p 676 4248 V 33 w(TRUE)f |
| Fo(if)i Fn(x)f Fo(is)g(strictly)h(larger)g(than)f Fn(y)p |
| Fo(,)g(and)g Fn(dd)p 2349 4248 V 34 w(FALSE)f Fo(otherwise.)43 |
| b(This)31 b(is)g(implemen)m(ted)227 4361 y(as)f(dd)p |
| 445 4361 28 4 v 32 w(P)m(ositiv)m(e\(z\))j(where)d Fn(z)f |
| Fo(is)h(the)g(subtraction)g(of)g Fn(y)g Fo(from)f Fn(x)p |
| Fo(.)40 b Fn(dd)p 2602 4361 29 4 v 34 w(Smaller\(x,)45 |
| b(y\))29 b Fo(w)m(orks)h(similarly)-8 b(.)0 4549 y Fn(dd)p |
| 102 4549 V 34 w(boolean)46 b(dd)p 614 4549 V 33 w(Equal\(x,)g(y\))f |
| Fo(:)227 4662 y(Returns)30 b Fn(dd)p 675 4662 V 33 w(TRUE)g |
| Fo(if)g Fn(x)g Fo(is)g(considered)g(equal)h(to)g Fn(y)p |
| Fo(,)g(and)e Fn(dd)p 2354 4662 V 34 w(FALSE)g Fo(otherwise.)41 |
| b(This)30 b(is)g(implemen)m(ted)227 4774 y(as)h(dd)p |
| 446 4774 28 4 v 32 w(EqualT)-8 b(oZero\(z\))32 b(where)e |
| Fn(z)g Fo(is)g(the)h(subtraction)f(of)h Fn(y)f Fo(from)g |
| Fn(x)p Fo(.)0 5015 y Fp(4.3.5)105 b(P)m(olyhedra)36 b(Data)f |
| (Manipulation)0 5186 y Fn(dd)p 102 5186 29 4 v 34 w(MatrixPtr)45 |
| b(dd)p 709 5186 V 34 w(PolyFile2Matrix)f(\(f,)i(err\))f |
| Fo(:)227 5299 y(Read)28 b(a)f(P)m(olyhedra)h(data)g(from)e(stream)i |
| Fn(f)f Fo(and)f(store)i(it)g(in)f Fn(matrixdata)d Fo(and)j(return)f(a)h |
| (p)s(oin)m(ter)g(to)h(the)227 5412 y(data.)1903 5682 |
| y(12)p eop end |
| %%Page: 13 13 |
| TeXDict begin 13 12 bop 0 44 a Fn(dd)p 102 44 29 4 v |
| 34 w(MatrixPtr)45 b(dd)p 709 44 V 34 w(CopyInequalities\(poly\))39 |
| b Fo(:)227 157 y(Cop)m(y)26 b(the)f(inequalit)m(y)i(represen)m(tation)g |
| (p)s(oin)m(ted)e(b)m(y)g(p)s(oly)g(to)h Fn(matrixdata)d |
| Fo(and)i(return)f Fn(dd)p 3415 157 V 34 w(MatrixPtr)p |
| Fo(.)0 345 y Fn(dd)p 102 345 V 34 w(MatrixPtr)45 b(dd)p |
| 709 345 V 34 w(CopyGenerators\(poly\))40 b Fo(:)227 458 |
| y(Cop)m(y)27 b(the)g(generator)i(represen)m(tation)f(p)s(oin)m(ted)e(b) |
| m(y)h(p)s(oly)g(to)h Fn(matrixdata)c Fo(and)i(return)g |
| Fn(dd)p 3415 458 V 34 w(MatrixPtr)p Fo(.)0 645 y Fn(dd)p |
| 102 645 V 34 w(SetFamilyPtr)44 b(dd)p 852 645 V 34 w |
| (CopyIncidence\(poly\))c Fo(:)227 758 y(Cop)m(y)21 b(the)f(incidence)h |
| (represen)m(tation)g(of)g(the)f(computed)g(represen)m(tation)i(p)s(oin) |
| m(ted)e(b)m(y)g(p)s(oly)g(to)h Fn(setfamily)227 871 y |
| Fo(and)38 b(return)f Fn(dd)p 800 871 V 34 w(SetFamilyPtr)p |
| Fo(.)62 b(The)37 b(computed)i(represen)m(tation)g(is)g |
| Fn(Inequality)c Fo(if)k(the)f(input)g(is)227 984 y Fn(Generator)p |
| Fo(,)28 b(and)i(the)h(vice)g(visa.)0 1172 y Fn(dd)p 102 |
| 1172 V 34 w(SetFamilyPtr)44 b(dd)p 852 1172 V 34 w |
| (CopyAdjacency\(poly\))c Fo(:)227 1285 y(Cop)m(y)21 b(the)f(adjacency)h |
| (represen)m(tation)g(of)g(the)f(computed)g(represen)m(tation)i(p)s(oin) |
| m(ted)e(b)m(y)g(p)s(oly)g(to)h Fn(setfamily)227 1397 |
| y Fo(and)38 b(return)f Fn(dd)p 800 1397 V 34 w(SetFamilyPtr)p |
| Fo(.)62 b(The)37 b(computed)i(represen)m(tation)g(is)g |
| Fn(Inequality)c Fo(if)k(the)f(input)g(is)227 1510 y Fn(Generator)p |
| Fo(,)28 b(and)i(the)h(vice)g(visa.)0 1698 y Fn(dd)p 102 |
| 1698 V 34 w(SetFamilyPtr)44 b(dd)p 852 1698 V 34 w |
| (CopyInputIncidence\(poly\))39 b Fo(:)227 1811 y(Cop)m(y)30 |
| b(the)g(incidence)h(represen)m(tation)g(of)f(the)g(input)f(represen)m |
| (tation)h(p)s(oin)m(ted)g(b)m(y)g(p)s(oly)f(to)i Fn(setfamily)227 |
| 1924 y Fo(and)f(return)f Fn(d)p 736 1924 V 34 w(SetFamilyPtr)p |
| Fo(.)0 2111 y Fn(dd)p 102 2111 V 34 w(SetFamilyPtr)44 |
| b(dd)p 852 2111 V 34 w(CopyInputAdjacency\(poly\))39 |
| b Fo(:)227 2224 y(Cop)m(y)28 b(the)g(adjacency)h(represen)m(tation)f |
| (of)g(the)g(input)f(represen)m(tation)i(p)s(oin)m(ted)f(b)m(y)f(p)s |
| (oly)h(to)g Fn(setfamily)227 2337 y Fo(and)i(return)f |
| Fn(d)p 736 2337 V 34 w(SetFamilyPtr)p Fo(.)0 2525 y Fn(void)47 |
| b(dd)p 341 2525 V 33 w(FreePolyhedra\(poly\))41 b Fo(:)227 |
| 2638 y(F)-8 b(ree)32 b(memory)e(allo)s(cated)i(to)f Fn(poly)p |
| Fo(.)0 2878 y Fp(4.3.6)105 b(LP)36 b(Data)e(Manipulation)0 |
| 3050 y Fn(dd)p 102 3050 V 34 w(LPPtr)46 b(dd)p 518 3050 |
| V 34 w(MakeLPforInteriorFinding)o(\(lp\))39 b Fo(:)227 |
| 3162 y(Set)29 b(up)f(an)g(LP)g(to)i(\014nd)d(an)h(in)m(terior)i(p)s |
| (oin)m(t)e(of)h(the)g(feasible)g(region)g(of)g Fn(lp)f |
| Fo(and)g(return)f(a)i(p)s(oin)m(ter)g(to)g(the)227 3275 |
| y(LP)-8 b(.)36 b(The)f(new)g(LP)g(has)h(one)g(new)f(v)-5 |
| b(ariable)36 b Fj(x)1867 3290 y Fg(d)p Fh(+1)2033 3275 |
| y Fo(and)f(one)h(more)g(constrain)m(t:)52 b(max)15 b |
| Fj(x)3329 3290 y Fg(d)p Fh(+1)3495 3275 y Fo(sub)5 b(ject)36 |
| b(to)227 3388 y Fj(b)20 b Fi(\000)g Fj(Ax)h Fi(\000)f |
| Fj(x)661 3403 y Fg(d)p Fh(+1)816 3388 y Fi(\025)25 b |
| Fo(0)31 b(and)f Fj(x)1217 3403 y Fg(d)p Fh(+1)1373 3388 |
| y Fi(\024)25 b Fj(K)7 b Fo(,)30 b(where)g Fj(K)37 b Fo(is)30 |
| b(a)h(p)s(ositiv)m(e)g(constan)m(t.)0 3576 y Fn(dd)p |
| 102 3576 V 34 w(LPPtr)46 b(dd)p 518 3576 V 34 w(Matrix2LP\(matrix,)d |
| (err\))h Fo(:)227 3689 y(Load)31 b Fn(matrix)d Fo(to)k |
| Fn(lpdata)c Fo(and)i(return)f(a)i(p)s(oin)m(ter)f(to)i(the)e(data.)0 |
| 3876 y Fn(dd)p 102 3876 V 34 w(LPSolutionPtr)44 b(dd)p |
| 900 3876 V 34 w(CopyLPSolution\(lp\))c Fo(:)227 3989 |
| y(Load)32 b(the)g(solutions)g(of)g Fn(lp)f Fo(to)i Fn(lpsolution)c |
| Fo(and)i(return)g(a)h(p)s(oin)m(ter)g(to)g(the)g(data.)46 |
| b(This)31 b(replaces)h(the)227 4102 y(old)f(name)f Fn(dd)p |
| 723 4102 V 34 w(LPSolutionLoad\(lp\))p Fo(.)0 4290 y |
| Fn(void)47 b(dd)p 341 4290 V 33 w(FreeLPData\(lp\))42 |
| b Fo(:)227 4403 y(F)-8 b(ree)32 b(memory)e(allo)s(cated)i(as)f(an)f(LP) |
| g(data)h(p)s(oin)m(ted)g(b)m(y)f Fn(lp)p Fo(.)0 4590 |
| y Fn(void)47 b(dd)p 341 4590 V 33 w(FreeLPSolution\(lps\))41 |
| b Fo(:)227 4703 y(F)-8 b(ree)32 b(memory)e(allo)s(cated)i(as)f(an)f(LP) |
| g(solution)h(data)g(p)s(oin)m(ted)g(b)m(y)f Fn(lps)p |
| Fo(.)0 4943 y Fp(4.3.7)105 b(Matrix)35 b(Manipulation)0 |
| 5115 y Fn(dd)p 102 5115 V 34 w(MatrixPtr)45 b(dd)p 709 |
| 5115 V 34 w(CopyMatrix\(matrix\))40 b Fo(:)227 5228 y(Mak)m(e)32 |
| b(a)f(cop)m(y)g(of)g(matrixdata)g(p)s(oin)m(ted)f(b)m(y)h |
| Fn(matrix)d Fo(and)i(return)f(a)i(p)s(oin)m(ter)g(to)g(the)f(cop)m(y)-8 |
| b(.)1903 5682 y(13)p eop end |
| %%Page: 14 14 |
| TeXDict begin 14 13 bop 0 44 a Fn(dd)p 102 44 29 4 v |
| 34 w(MatrixPtr)45 b(dd)p 709 44 V 34 w(AppendMatrix\(matrix1,)d |
| (matrix2\))h Fo(:)227 157 y(Mak)m(e)32 b(a)e(matrixdata)h(b)m(y)e(cop)m |
| (ying)i Fn(*matrix1)c Fo(and)i(app)s(ending)g(the)h(matrix)g(in)f |
| Fn(*matrix2)f Fo(and)h(return)227 270 y(a)39 b(p)s(oin)m(ter)g(to)h |
| (the)f(data.)67 b(The)38 b(colsize)j(m)m(ust)d(b)s(e)h(equal)g(in)g |
| (the)g(t)m(w)m(o)h(input)e(matrices.)67 b(It)39 b(returns)e(a)227 |
| 383 y Fn(NULL)j Fo(p)s(oin)m(ter)g(if)h(the)g(input)e(matrices)j(are)f |
| (not)g(appropriate.)71 b(Its)41 b Fn(rowsize)e Fo(is)h(set)i(to)f(the)g |
| (sum)e(of)227 496 y(the)k(ro)m(wsizes)h(of)e Fn(matrix1)f |
| Fo(and)h Fn(matrix2)p Fo(.)75 b(The)42 b(new)g(matrixdata)i(inherits)e |
| (ev)m(erything)i(else)f(\(i.e.)227 609 y(n)m(um)m(b)s(ert)m(yp)s(e,)30 |
| b(represen)m(tation,)i(etc\))f(from)f(the)h(\014rst)e(matrix.)0 |
| 794 y Fn(int)47 b(dd)p 293 794 V 34 w(MatrixAppendTo\(&)c(matrix1,)j |
| (matrix2\))d Fo(:)227 907 y(Same)31 b(as)f Fn(dd)p 683 |
| 907 V 34 w(AppendMatrix)d Fo(except)32 b(that)e(the)h(\014rst)f(matrix) |
| g(is)h(mo)s(di\014ed)e(to)i(tak)m(e)h(the)f(result.)0 |
| 1092 y Fn(int)47 b(dd)p 293 1092 V 34 w(MatrixRowRemove\(&)c(matrix,)j |
| (i\))f Fo(:)227 1205 y(Remo)m(v)m(e)32 b(the)f Fj(i)p |
| Fo(th)f(ro)m(w)h(of)g Fn(matrix)p Fo(.)0 1391 y Fn(dd)p |
| 102 1391 V 34 w(MatrixPtr)45 b(dd)p 709 1391 V 34 w |
| (MatrixSubmatrix\(matrix,)c(set\))k Fo(:)227 1504 y(Generate)36 |
| b(the)f(submatrix)f(of)h Fn(matrix)e Fo(b)m(y)i(remo)m(ving)h(the)f(ro) |
| m(ws)f(indexed)h(b)m(y)f Fn(set)g Fo(and)g(return)g(a)h(ma-)227 |
| 1617 y(trixdata)d(p)s(oin)m(ter.)0 1802 y Fn(dd)p 102 |
| 1802 V 34 w(SetFamilyPtr)44 b(dd)p 852 1802 V 34 w |
| (Matrix2Adjacency\(matrix,)d(err\))k Fo(:)227 1915 y(Return)25 |
| b(the)i(adjacency)f(list)h(of)f(the)g(represen)m(tation)h(giv)m(en)g(b) |
| m(y)f Fn(matrix)p Fo(.)38 b(The)25 b(computation)i(is)f(done)g(b)m(y) |
| 227 2028 y(the)h(built-in)g(LP)f(solv)m(er.)41 b(The)26 |
| b(represen)m(tation)i(should)e(b)s(e)g(free)h(of)g(redundancy)e(when)g |
| (this)i(function)g(is)227 2141 y(called.)40 b(See)22 |
| b(the)h(function)f Fn(dd)p 1270 2141 V 33 w(rowset)47 |
| b(dd)p 1734 2141 V 33 w(RedundantRows)19 b Fo(and)j(the)g(example)h |
| (program)f(adjacency)-8 b(.c.)0 2383 y Fd(4.4)112 b(Input/Output)38 |
| b(F)-9 b(unctions)0 2555 y Fn(dd)p 102 2555 V 34 w(MatrixPtr)45 |
| b(dd)p 709 2555 V 34 w(PolyFile2Matrix)f(\(f,)i(err\))f |
| Fo(:)227 2668 y(Read)28 b(a)f(P)m(olyhedra)h(data)g(from)e(stream)i |
| Fn(f)f Fo(and)f(store)i(it)g(in)f Fn(matrixdata)d Fo(and)j(return)f(a)h |
| (p)s(oin)m(ter)g(to)h(the)227 2780 y(data.)0 2966 y Fn(boolean)46 |
| b(dd)p 484 2966 V 34 w(DDFile2File\(ifile,)c(ofile,)47 |
| b(err\))d Fo(:)227 3079 y(Compute)33 b(the)g(represen)m(tation)h(con)m |
| (v)m(ersion)g(for)e(a)i(p)s(olyhedron)d(giv)m(en)j(b)m(y)f(a)g(P)m |
| (olyhedra)g(\014le)g(i\014le,)h(and)227 3192 y(write)d(the)g(other)f |
| (represen)m(tation)i(in)e(a)h(P)m(olyhedra)g(\014le)f(o\014le.)42 |
| b Fn(*err)29 b Fo(returns)g Fn(dd)p 3073 3192 V 34 w(NoError)g |
| Fo(if)h(the)h(com-)227 3305 y(putation)g(terminates)g(normally)-8 |
| b(.)42 b(Otherwise,)30 b(it)h(returns)e(a)i(v)-5 b(alue)31 |
| b(according)g(to)g(the)g(error)f(o)s(ccured.)0 3490 y |
| Fn(void)47 b(dd)p 341 3490 V 33 w(WriteMatrix\(f,)d(matrix\))g |
| Fo(:)227 3603 y(W)-8 b(rite)32 b Fn(matrix)d Fo(to)i(stream)g |
| Fn(f)p Fo(.)0 3788 y Fn(void)47 b(dd)p 341 3788 V 33 |
| w(WriteNumber\(f,)d(x\))h Fo(:)227 3901 y(W)-8 b(rite)27 |
| b Fn(x)e Fo(to)h(stream)f Fn(f)p Fo(.)39 b(If)25 b Fn(x)f |
| Fo(is)i(of)f(GMP)h(mp)s(q)p 1827 3901 28 4 v 31 w(t)g(rational)g |
| Fj(p=q)s Fo(,)h(the)e(output)g(is)g Fj(p=q)s Fo(.)39 |
| b(If)25 b(it)g(is)h(of)f(C)g(double,)227 4014 y(it)31 |
| b(is)g(formated)f(as)h(a)g(double)f(\015oat)h(with)f(a)g(decimal)i(p)s |
| (oin)m(t.)0 4200 y Fn(void)47 b(dd)p 341 4200 29 4 v |
| 33 w(WritePolyFile\(f,)d(poly\))g Fo(:)227 4312 y(W)-8 |
| b(rite)32 b(tt)f(p)s(oly)f(to)h(stream)g Fn(f)f Fo(in)g(P)m(olyhedra)h |
| (format.)0 4498 y Fn(void)47 b(dd)p 341 4498 V 33 w |
| (WriteErrorMessages\(f,)42 b(err\))j Fo(:)227 4611 y(W)-8 |
| b(rite)32 b(error)e(messages)h(giv)m(en)h(b)m(y)e Fn(err)g |
| Fo(to)h(stream)f Fn(f)p Fo(.)0 4796 y Fn(void)47 b(dd)p |
| 341 4796 V 33 w(WriteSetFamily\(f,)d(setfam\))f Fo(:)227 |
| 4909 y(W)-8 b(rite)37 b(the)e(set)h(family)g(p)s(oin)m(ted)f(b)m(y)g |
| Fn(setfam)e Fo(to)j(stream)g Fn(f)p Fo(.)55 b(F)-8 b(or)35 |
| b(eac)m(h)i(set,)g(it)f(outputs)e(its)i(index,)g(its)227 |
| 5022 y(cardinalit)m(y)-8 b(,)33 b(a)e(colon)g(\\:")41 |
| b(and)30 b(a)h(ordered)f(list)h(of)f(its)h(elemen)m(ts.)0 |
| 5207 y Fn(void)47 b(dd)p 341 5207 V 33 w(WriteSetFamilyCompressed\(f)o |
| (,)42 b(setfam\))h Fo(:)227 5320 y(W)-8 b(rite)37 b(the)e(set)h(family) |
| g(p)s(oin)m(ted)f(b)m(y)g Fn(setfam)e Fo(to)j(stream)g |
| Fn(f)p Fo(.)55 b(F)-8 b(or)35 b(eac)m(h)i(set,)g(it)f(outputs)e(its)i |
| (index,)g(its)227 5433 y(cardinalit)m(y)g(or)e(the)g(negativ)m(e)j(of)d |
| (the)g(cardinalit)m(y)-8 b(,)38 b(a)c(colon)h(\\:")50 |
| b(and)33 b(the)i(elemen)m(ts)g(in)f(the)g(set)h(or)f(its)1903 |
| 5682 y(14)p eop end |
| %%Page: 15 15 |
| TeXDict begin 15 14 bop 227 44 a Fo(complemen)m(ts)40 |
| b(whic)m(hev)m(er)g(is)f(smaller.)67 b(Whenev)m(er)40 |
| b(it)g(outputs)e(the)h(complemen)m(ts,)k(the)c(cardinalit)m(y)227 |
| 157 y(is)i(negated)h(so)g(that)g(there)f(is)g(no)g(am)m(biguit)m(y)-8 |
| b(.)75 b(This)40 b(will)h(b)s(e)g(considered)g(standard)f(for)h |
| (outputing)227 270 y(incidence)32 b(\(*.icd,)i(*ecd\))e(and)f |
| (adjacency)h(\(*.iad,)i(*.ead\))e(data)h(in)e(cddlib.)43 |
| b(But)32 b(there)g(is)f(some)h(minor)227 383 y(incompatibilit)m(y)h |
| (with)d(cdd/cdd+)f(standalone)i(co)s(des.)0 571 y Fn(void)47 |
| b(dd)p 341 571 29 4 v 33 w(WriteProgramDescription\(f\))39 |
| b Fo(:)227 683 y(W)-8 b(rite)32 b(the)f(cddlib)e(v)m(ersion)i |
| (information)g(to)g(stream)g Fn(f)p Fo(.)0 871 y Fn(void)47 |
| b(dd)p 341 871 V 33 w(WriteDDTimes\(f,)d(poly\))g Fo(:)227 |
| 984 y(W)-8 b(rite)32 b(the)f(represen)m(tation)g(con)m(v)m(ersion)h |
| (time)f(information)f(on)h Fn(poly)e Fo(to)i(stream)g |
| Fn(f)p Fo(.)0 1227 y Fd(4.5)112 b(Obsolete)38 b(F)-9 |
| b(unctions)0 1399 y Fn(boolean)46 b(dd)p 484 1399 V 34 |
| w(DoubleDescription\(poly,)41 b(err\))j Fo(:)d(\(remo)m(v)m(ed)32 |
| b(in)e(V)-8 b(ersion)31 b(0.90c\))227 1512 y(The)e(new)f(function)g |
| Fn(dd)p 1054 1512 V 34 w(DDMatrix2Poly\(matrix,)42 b(err\))28 |
| b Fo(\(see)h(Section)h(4.2\))g(replaces)f(\(and)g(actually)227 |
| 1625 y(com)m(bines\))j(b)s(oth)d(this)h(and)g Fn(dd)p |
| 1325 1625 V 34 w(Matrix2Poly\(matrix,)43 b(err\))p Fo(.)0 |
| 1812 y Fn(dd)p 102 1812 V 34 w(PolyhedraPtr)h(dd)p 852 |
| 1812 V 34 w(Matrix2Poly\(matrix,)f(err\))h Fo(:)d(\(remo)m(v)m(ed)31 |
| b(in)f(V)-8 b(ersion)31 b(0.90c\))227 1925 y(See)g(ab)s(o)m(v)m(e)g |
| (for)g(the)f(reason)h(for)f(remo)m(v)-5 b(al.)0 2113 |
| y Fn(dd)p 102 2113 V 34 w(LPSolutionPtr)44 b(dd)p 900 |
| 2113 V 34 w(LPSolutionLoad\(lp\))c Fo(:)h(\(renamed)30 |
| b(in)g(V)-8 b(ersion)31 b(0.90c\))227 2226 y(This)f(function)g(is)g(no) |
| m(w)h(called)g Fn(dd)p 1431 2226 V 34 w(CopyLPSolution\(lp\))p |
| Fo(.)0 2469 y Fd(4.6)112 b(Set)38 b(F)-9 b(unctions)38 |
| b(in)f Fb(setoper)j Fd(library)0 2641 y Fo(The)30 b(cddlib)g(comes)i |
| (with)e(a)h(simple)g(set)g(op)s(eration)g(library)g Fn(setoper)p |
| Fo(.)39 b(The)30 b(k)m(ey)i(t)m(yp)s(e)f(de\014ned)e(is)i |
| Fn(set)p 3654 2641 V 33 w(type)p Fo(.)0 2754 y(A)g(set)h(is)f(represen) |
| m(ted)g(b)m(y)g(a)g(\014xed)g(length)g(binary)f(strings.)43 |
| b(Th)m(us,)30 b(the)i(maxim)m(um)f(length)g(of)g(a)h(set)f(m)m(ust)g(b) |
| s(e)0 2867 y(declared)g(when)e(it)i(is)g(initialized.)141 |
| 2980 y(Belo)m(w)e(the)f(sym)m(b)s(ols)g Fn(a)p Fo(,)g |
| Fn(b)p Fo(,)g Fn(c)g Fo(are)g(of)g(t)m(yp)s(e)g Fn(set)p |
| 1786 2980 V 33 w(type)p Fo(.)39 b(The)27 b(sym)m(b)s(ols)h |
| Fn(aP)f Fo(is)h(a)g(p)s(oin)m(ter)f(to)i(t)m(yp)s(e)f |
| Fn(set)p 3654 2980 V 33 w(type)p Fo(,)0 3093 y(and)j |
| Fn(s)p Fo(,)i Fn(t)f Fo(are)g(of)g(t)m(yp)s(e)h Fn(long)p |
| Fo(.)44 b(Here)33 b(are)g(some)f(of)g(the)h(functions)e(de\014ned.)45 |
| b(See)32 b Fn(setoper.h)e Fo(for)i(a)g(complete)0 3206 |
| y(listing.)0 3418 y Fn(void)47 b(set)p 389 3418 V 33 |
| w(initialize\(aP,)d(s\))h Fo(:)227 3531 y(Allo)s(cate)35 |
| b(a)e Fn(set)p 816 3531 V 33 w(type)f Fo(space)h(of)f(maxim)m(um)h |
| (cardinalit)m(y)g Fn(s)g Fo(and)e(mak)m(e)j(it)f(p)s(oin)m(ted)f(b)m(y) |
| g Fn(aP)p Fo(.)h(The)f(set)h(is)227 3644 y(initialized)f(as)f(empt)m(y) |
| g(set.)0 3832 y Fn(void)47 b(set)p 389 3832 V 33 w(free\(a\))d |
| Fo(:)227 3944 y(F)-8 b(ree)32 b(the)e Fn(set)p 732 3944 |
| V 34 w(type)f Fo(space)i(allo)s(cated)h(for)f Fn(a)p |
| Fo(.)0 4132 y Fn(void)47 b(set)p 389 4132 V 33 w(copy\(a,)f(b\)\))f |
| Fo(:)227 4245 y(Set)31 b Fn(a)g Fo(to)g(b)s(e)f Fn(b)p |
| Fo(.)42 b(The)30 b(set)h Fn(a)g Fo(m)m(ust)f(b)s(e)g(pre-initialized)j |
| (with)d(the)h(same)g(maxim)m(um)g(cardinalit)m(y)h(as)f(that)227 |
| 4358 y(of)g Fn(b)p Fo(.)0 4546 y Fn(void)47 b(set)p 389 |
| 4546 V 33 w(addelem\(a,)e(t\)\))g Fo(:)227 4658 y(Add)30 |
| b(an)g(elemen)m(t)i Fn(t)e Fo(to)h(a)g(set)g Fn(a)p Fo(.)40 |
| b(The)30 b(set)h Fn(a)f Fo(sta)m(ys)h(unc)m(hanged)f(if)h(it)f(con)m |
| (tains)i(the)f(elemen)m(t)g Fn(t)p Fo(.)0 4846 y Fn(long)47 |
| b(set)p 389 4846 V 33 w(card\(a\)\))c Fo(:)227 4959 y(Return)30 |
| b(the)g(cardinalit)m(y)i(of)f(set)g Fn(a)p Fo(.)0 5147 |
| y Fn(int)47 b(set)p 341 5147 V 33 w(member\(t,)f(a\)\))e |
| Fo(:)227 5260 y(Return)30 b(1)h(if)f Fn(t)g Fo(is)g(a)h(mem)m(b)s(er)f |
| (of)g(set)h Fn(a)p Fo(,)g(and)e(0)i(otherwise.)1903 5682 |
| y(15)p eop end |
| %%Page: 16 16 |
| TeXDict begin 16 15 bop 0 44 a Fn(void)47 b(set)p 389 |
| 44 29 4 v 33 w(write\(a\)\))c Fo(:)227 157 y(Prin)m(t)28 |
| b(out)g(the)g(elemen)m(ts)h(of)f(set)h Fn(a)e Fo(to)i |
| Fn(stdout)p Fo(.)38 b(The)27 b(function)h Fn(void)47 |
| b(set)p 2846 157 V 33 w(fwrite\(f,)e(a\)\))27 b Fo(output)h(to)227 |
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| %%Page: 17 17 |
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| end |
| %%Page: 18 18 |
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| %%Page: 19 19 |
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| eop end |
| %%Trailer |
| |
| userdict /end-hook known{end-hook}if |
| %%EOF |