| Austin Schuh | 405fa6c | 2015-09-06 18:13:55 -0700 | [diff] [blame] | 1 | %!PS-Adobe-2.0 |
| 2 | %%Creator: dvips(k) 5.96 Copyright 2005 Radical Eye Software |
| 3 | %%Title: cddlibman.dvi |
| 4 | %%CreationDate: Sun Mar 23 15:43:43 2008 |
| 5 | %%Pages: 19 |
| 6 | %%PageOrder: Ascend |
| 7 | %%BoundingBox: 0 0 595 842 |
| 8 | %%DocumentFonts: CMR17 CMR12 CMBX12 CMBX10 CMR10 CMTT10 CMMI10 CMSY10 |
| 9 | %%+ CMR8 CMMI8 CMTI10 CMMI6 CMSY8 CMTT12 MSBM10 |
| 10 | %%DocumentPaperSizes: a4 |
| 11 | %%EndComments |
| 12 | %DVIPSWebPage: (www.radicaleye.com) |
| 13 | %DVIPSCommandLine: /usr/local/texlive/2007/bin/powerpc-darwin/dvips -R |
| 14 | %+ -Poutline -o /tmp/altpdflatex.82070-1206283419/cddlibman.ps |
| 15 | %+ cddlibman.dvi |
| 16 | %DVIPSParameters: dpi=600 |
| 17 | %DVIPSSource: TeX output 2008.03.23:1543 |
| 18 | %%BeginProcSet: tex.pro 0 0 |
| 19 | %! |
| 20 | /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S |
| 21 | N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 |
| 22 | mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 |
| 23 | 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ |
| 24 | landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize |
| 25 | mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ |
| 26 | matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round |
| 27 | exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ |
| 28 | statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] |
| 29 | N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin |
| 30 | /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array |
| 31 | /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 |
| 32 | array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N |
| 33 | df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A |
| 34 | definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get |
| 35 | }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} |
| 36 | B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr |
| 37 | 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S |
| 38 | /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy |
| 39 | setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask |
| 40 | restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn |
| 41 | /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put |
| 42 | }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ |
| 43 | bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A |
| 44 | mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ |
| 45 | SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ |
| 46 | userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X |
| 47 | 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 |
| 48 | index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N |
| 49 | /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ |
| 50 | /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) |
| 51 | (LaserWriter 16/600)]{A length product length le{A length product exch 0 |
| 52 | exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse |
| 53 | end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask |
| 54 | grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} |
| 55 | imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round |
| 56 | exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto |
| 57 | fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p |
| 58 | delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} |
| 59 | B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ |
| 60 | p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S |
| 61 | rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end |
| 62 | |
| 63 | %%EndProcSet |
| 64 | %%BeginProcSet: texps.pro 0 0 |
| 65 | %! |
| 66 | TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 |
| 67 | index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll |
| 68 | exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0 |
| 69 | ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{ |
| 70 | pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get |
| 71 | div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type |
| 72 | /nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end |
| 73 | definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup |
| 74 | sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll |
| 75 | mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[ |
| 76 | exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if} |
| 77 | forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def |
| 78 | end |
| 79 | |
| 80 | %%EndProcSet |
| 81 | %%BeginFont: MSBM10 |
| 82 | %!PS-AdobeFont-1.1: MSBM10 2.1 |
| 83 | %%CreationDate: 1993 Sep 17 11:10:37 |
| 84 | % Math Symbol fonts were designed by the American Mathematical Society. |
| 85 | % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 86 | 11 dict begin |
| 87 | /FontInfo 7 dict dup begin |
| 88 | /version (2.1) readonly def |
| 89 | /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| 90 | /FullName (MSBM10) readonly def |
| 91 | /FamilyName (Euler) readonly def |
| 92 | /Weight (Medium) readonly def |
| 93 | /ItalicAngle 0 def |
| 94 | /isFixedPitch false def |
| 95 | end readonly def |
| 96 | /FontName /MSBM10 def |
| 97 | /PaintType 0 def |
| 98 | /FontType 1 def |
| 99 | /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| 100 | /Encoding 256 array |
| 101 | 0 1 255 {1 index exch /.notdef put} for |
| 102 | dup 82 /R put |
| 103 | readonly def |
| 104 | /FontBBox{-55 -420 2343 920}readonly def |
| 105 | currentdict end |
| 106 | currentfile eexec |
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| 154 | cleartomark |
| 155 | %%EndFont |
| 156 | %%BeginFont: CMTT12 |
| 157 | %!PS-AdobeFont-1.1: CMTT12 1.0 |
| 158 | %%CreationDate: 1991 Aug 20 16:45:46 |
| 159 | % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 160 | 11 dict begin |
| 161 | /FontInfo 7 dict dup begin |
| 162 | /version (1.0) readonly def |
| 163 | /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| 164 | /FullName (CMTT12) readonly def |
| 165 | /FamilyName (Computer Modern) readonly def |
| 166 | /Weight (Medium) readonly def |
| 167 | /ItalicAngle 0 def |
| 168 | /isFixedPitch true def |
| 169 | end readonly def |
| 170 | /FontName /CMTT12 def |
| 171 | /PaintType 0 def |
| 172 | /FontType 1 def |
| 173 | /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| 174 | /Encoding 256 array |
| 175 | 0 1 255 {1 index exch /.notdef put} for |
| 176 | dup 101 /e put |
| 177 | dup 111 /o put |
| 178 | dup 112 /p put |
| 179 | dup 114 /r put |
| 180 | dup 115 /s put |
| 181 | dup 116 /t put |
| 182 | readonly def |
| 183 | /FontBBox{-1 -234 524 695}readonly def |
| 184 | currentdict end |
| 185 | currentfile eexec |
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| 250 | cleartomark |
| 251 | %%EndFont |
| 252 | %%BeginFont: CMSY8 |
| 253 | %!PS-AdobeFont-1.1: CMSY8 1.0 |
| 254 | %%CreationDate: 1991 Aug 15 07:22:10 |
| 255 | % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 256 | 11 dict begin |
| 257 | /FontInfo 7 dict dup begin |
| 258 | /version (1.0) readonly def |
| 259 | /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| 260 | /FullName (CMSY8) readonly def |
| 261 | /FamilyName (Computer Modern) readonly def |
| 262 | /Weight (Medium) readonly def |
| 263 | /ItalicAngle -14.035 def |
| 264 | /isFixedPitch false def |
| 265 | end readonly def |
| 266 | /FontName /CMSY8 def |
| 267 | /PaintType 0 def |
| 268 | /FontType 1 def |
| 269 | /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| 270 | /Encoding 256 array |
| 271 | 0 1 255 {1 index exch /.notdef put} for |
| 272 | dup 0 /minus put |
| 273 | readonly def |
| 274 | /FontBBox{-30 -955 1185 779}readonly def |
| 275 | currentdict end |
| 276 | currentfile eexec |
| 277 | D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 |
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| 293 | D8A90A56EC2B520060869AE646FC671A833C92A49961EAFF92500196526B5CE2 |
| 294 | 3792E028EC16CD1402D86FB8AD69679F70A14AD2276AD0A1CE82302ABBAE8D54 |
| 295 | 308988AD |
| 296 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 297 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 298 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 299 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 300 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 301 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 302 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 303 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 304 | cleartomark |
| 305 | %%EndFont |
| 306 | %%BeginFont: CMBX12 |
| 307 | %!PS-AdobeFont-1.1: CMBX12 1.0 |
| 308 | %%CreationDate: 1991 Aug 20 16:34:54 |
| 309 | % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 310 | 11 dict begin |
| 311 | /FontInfo 7 dict dup begin |
| 312 | /version (1.0) readonly def |
| 313 | /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| 314 | /FullName (CMBX12) readonly def |
| 315 | /FamilyName (Computer Modern) readonly def |
| 316 | /Weight (Bold) readonly def |
| 317 | /ItalicAngle 0 def |
| 318 | /isFixedPitch false def |
| 319 | end readonly def |
| 320 | /FontName /CMBX12 def |
| 321 | /PaintType 0 def |
| 322 | /FontType 1 def |
| 323 | /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| 324 | /Encoding 256 array |
| 325 | 0 1 255 {1 index exch /.notdef put} for |
| 326 | dup 40 /parenleft put |
| 327 | dup 41 /parenright put |
| 328 | dup 45 /hyphen put |
| 329 | dup 46 /period put |
| 330 | dup 47 /slash put |
| 331 | dup 49 /one put |
| 332 | dup 50 /two put |
| 333 | dup 51 /three put |
| 334 | dup 52 /four put |
| 335 | dup 53 /five put |
| 336 | dup 54 /six put |
| 337 | dup 55 /seven put |
| 338 | dup 56 /eight put |
| 339 | dup 57 /nine put |
| 340 | dup 65 /A put |
| 341 | dup 66 /B put |
| 342 | dup 67 /C put |
| 343 | dup 68 /D put |
| 344 | dup 69 /E put |
| 345 | dup 70 /F put |
| 346 | dup 71 /G put |
| 347 | dup 72 /H put |
| 348 | dup 73 /I put |
| 349 | dup 76 /L put |
| 350 | dup 77 /M put |
| 351 | dup 78 /N put |
| 352 | dup 79 /O put |
| 353 | dup 80 /P put |
| 354 | dup 82 /R put |
| 355 | dup 83 /S put |
| 356 | dup 84 /T put |
| 357 | dup 85 /U put |
| 358 | dup 86 /V put |
| 359 | dup 97 /a put |
| 360 | dup 98 /b put |
| 361 | dup 99 /c put |
| 362 | dup 100 /d put |
| 363 | dup 101 /e put |
| 364 | dup 102 /f put |
| 365 | dup 103 /g put |
| 366 | dup 104 /h put |
| 367 | dup 105 /i put |
| 368 | dup 106 /j put |
| 369 | dup 107 /k put |
| 370 | dup 108 /l put |
| 371 | dup 109 /m put |
| 372 | dup 110 /n put |
| 373 | dup 111 /o put |
| 374 | dup 112 /p put |
| 375 | dup 114 /r put |
| 376 | dup 115 /s put |
| 377 | dup 116 /t put |
| 378 | dup 117 /u put |
| 379 | dup 119 /w put |
| 380 | dup 120 /x put |
| 381 | dup 121 /y put |
| 382 | dup 122 /z put |
| 383 | readonly def |
| 384 | /FontBBox{-53 -251 1139 750}readonly def |
| 385 | currentdict end |
| 386 | currentfile eexec |
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| 683 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 684 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 685 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 686 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 687 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 688 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 689 | cleartomark |
| 690 | %%EndFont |
| 691 | %%BeginFont: CMMI6 |
| 692 | %!PS-AdobeFont-1.1: CMMI6 1.100 |
| 693 | %%CreationDate: 1996 Jul 23 07:53:52 |
| 694 | % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 695 | 11 dict begin |
| 696 | /FontInfo 7 dict dup begin |
| 697 | /version (1.100) readonly def |
| 698 | /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| 699 | /FullName (CMMI6) readonly def |
| 700 | /FamilyName (Computer Modern) readonly def |
| 701 | /Weight (Medium) readonly def |
| 702 | /ItalicAngle -14.04 def |
| 703 | /isFixedPitch false def |
| 704 | end readonly def |
| 705 | /FontName /CMMI6 def |
| 706 | /PaintType 0 def |
| 707 | /FontType 1 def |
| 708 | /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| 709 | /Encoding 256 array |
| 710 | 0 1 255 {1 index exch /.notdef put} for |
| 711 | dup 106 /j put |
| 712 | readonly def |
| 713 | /FontBBox{11 -250 1241 750}readonly def |
| 714 | currentdict end |
| 715 | currentfile eexec |
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| 758 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 759 | cleartomark |
| 760 | %%EndFont |
| 761 | %%BeginFont: CMTI10 |
| 762 | %!PS-AdobeFont-1.1: CMTI10 1.00B |
| 763 | %%CreationDate: 1992 Feb 19 19:56:16 |
| 764 | % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 765 | 11 dict begin |
| 766 | /FontInfo 7 dict dup begin |
| 767 | /version (1.00B) readonly def |
| 768 | /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| 769 | /FullName (CMTI10) readonly def |
| 770 | /FamilyName (Computer Modern) readonly def |
| 771 | /Weight (Medium) readonly def |
| 772 | /ItalicAngle -14.04 def |
| 773 | /isFixedPitch false def |
| 774 | end readonly def |
| 775 | /FontName /CMTI10 def |
| 776 | /PaintType 0 def |
| 777 | /FontType 1 def |
| 778 | /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| 779 | /Encoding 256 array |
| 780 | 0 1 255 {1 index exch /.notdef put} for |
| 781 | dup 12 /fi put |
| 782 | dup 39 /quoteright put |
| 783 | dup 40 /parenleft put |
| 784 | dup 41 /parenright put |
| 785 | dup 44 /comma put |
| 786 | dup 45 /hyphen put |
| 787 | dup 46 /period put |
| 788 | dup 48 /zero put |
| 789 | dup 49 /one put |
| 790 | dup 50 /two put |
| 791 | dup 51 /three put |
| 792 | dup 58 /colon put |
| 793 | dup 65 /A put |
| 794 | dup 67 /C put |
| 795 | dup 68 /D put |
| 796 | dup 69 /E put |
| 797 | dup 71 /G put |
| 798 | dup 72 /H put |
| 799 | dup 73 /I put |
| 800 | dup 74 /J put |
| 801 | dup 76 /L put |
| 802 | dup 77 /M put |
| 803 | dup 78 /N put |
| 804 | dup 79 /O put |
| 805 | dup 80 /P put |
| 806 | dup 82 /R put |
| 807 | dup 83 /S put |
| 808 | dup 84 /T put |
| 809 | dup 85 /U put |
| 810 | dup 86 /V put |
| 811 | dup 97 /a put |
| 812 | dup 98 /b put |
| 813 | dup 99 /c put |
| 814 | dup 100 /d put |
| 815 | dup 101 /e put |
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| 833 | dup 120 /x put |
| 834 | dup 121 /y put |
| 835 | dup 122 /z put |
| 836 | dup 123 /endash put |
| 837 | readonly def |
| 838 | /FontBBox{-163 -250 1146 969}readonly def |
| 839 | currentdict end |
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| 1163 | 934A72674BA63CE844EAFD1F80C2CAA9F012F414433DC9D009BC7B40E4BF5DC9 |
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| 1172 | C2F80158A15EB130B08D3739C235B9EEABD6455DA67246ABFD7A3089AA4C8218 |
| 1173 | D42C1FDC1216D19890886097A933FEC71DDF2C22E7B46C7A415B2455E660A596 |
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| 1182 | E8E60EFF9971CCB2DDA77733800335C0BE493BE94950887D3DA58789C6A3B9BF |
| 1183 | 968A6DE73B9439A84CCE41448F69F47C8B841682D3A2A3839EC8B56EF981778B |
| 1184 | 343CCE18E3886951B83882038F0A5BD8DE6BB6A0EED6064008D5F29392573173 |
| 1185 | 62BC23254C65C49404F8F1C58E8CFC0AFF6B09E9567D050415F6C5B7073F06FF |
| 1186 | 61C4FB63CA877EEF8EF97292CF20D3EAE1A5CEB3E5B9185EA13FCD175F64B87B |
| 1187 | 37616CB5CF6996B72E7A0FD0B7FC11D75DDE00FB53FCDB852FF78EBCFE27AA0F |
| 1188 | F146E3CAE775F2636C4BBB2AECB109CFEAA15B41F7ABB89E3B78C88E51D72B05 |
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| 1198 | 2CA276014E1D9DF75908A803FE6B2413945F5EF522FB4547F86E1C11E042559D |
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| 1202 | 2101F38E9166FC7E3A629C3B1C6E95FDE05771313F469C1B6E40F8A2BD42ACB0 |
| 1203 | 51DCDE9F73F74FF5CE2056D32EF1FCDD8CD3A4EBEDCBB8D2B9C7C4967EBC1889 |
| 1204 | CC89C7BB39B15177CA47B048250ACF0CEC47D4090A2EA3BE1C321B071394A8BA |
| 1205 | 6FB07B2915813ECD28776735827D3BDDD34D69C00D87AE9D88A32C0C1C217461 |
| 1206 | E98C9D7FEF42D19DA4039DB56C4616CACFB519CF7B970F35D4CBAED7003FC0EA |
| 1207 | 703B527A2F6A8F7E0F7A7E647E3757BF08A31523D3E95FDE71F064A4648B1D59 |
| 1208 | 6DCC4CBDE6981C199B35AEF86F2A2EC44F3D6AD7DDC2E80A43EF586481530D2D |
| 1209 | D4923D542A738AA3EB370BB6BD076223D126AC66524389E772AF831B607A71DE |
| 1210 | CCA8B822D3EEDEA16A462D07ED61C27874FFE3D9770354D660ECDF58126636BB |
| 1211 | EA3D68FCA7842E4A6219F8ED487767B276F1C3D8F61316D7ADE6EC4217072A4B |
| 1212 | 28152B404B7DBC9C22A07D6509D562C1503FD15E176EEDB7B9C973E96049DBFC |
| 1213 | 348DC2AC2F99DDD4FD42353E7261A5D876FD3B91EBC607F843DCC1D5EA7F7C2E |
| 1214 | 04BF867A9B928BEA994B85EEDE98C16A06669AE114E2E7651A90E436C1265E2F |
| 1215 | 7D9C121D22B5AE7E147A28D928D5C721C170390E |
| 1216 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1217 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1218 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1219 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1220 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1221 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1222 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1223 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1224 | cleartomark |
| 1225 | %%EndFont |
| 1226 | %%BeginFont: CMMI8 |
| 1227 | %!PS-AdobeFont-1.1: CMMI8 1.100 |
| 1228 | %%CreationDate: 1996 Jul 23 07:53:54 |
| 1229 | % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 1230 | 11 dict begin |
| 1231 | /FontInfo 7 dict dup begin |
| 1232 | /version (1.100) readonly def |
| 1233 | /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| 1234 | /FullName (CMMI8) readonly def |
| 1235 | /FamilyName (Computer Modern) readonly def |
| 1236 | /Weight (Medium) readonly def |
| 1237 | /ItalicAngle -14.04 def |
| 1238 | /isFixedPitch false def |
| 1239 | end readonly def |
| 1240 | /FontName /CMMI8 def |
| 1241 | /PaintType 0 def |
| 1242 | /FontType 1 def |
| 1243 | /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| 1244 | /Encoding 256 array |
| 1245 | 0 1 255 {1 index exch /.notdef put} for |
| 1246 | dup 84 /T put |
| 1247 | dup 100 /d put |
| 1248 | dup 105 /i put |
| 1249 | dup 106 /j put |
| 1250 | dup 107 /k put |
| 1251 | dup 110 /n put |
| 1252 | dup 114 /r put |
| 1253 | dup 115 /s put |
| 1254 | dup 116 /t put |
| 1255 | readonly def |
| 1256 | /FontBBox{-24 -250 1110 750}readonly def |
| 1257 | currentdict end |
| 1258 | currentfile eexec |
| 1259 | D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE |
| 1260 | 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B |
| 1261 | 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 |
| 1262 | B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B |
| 1263 | 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE |
| 1264 | D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 |
| 1265 | 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC |
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| 1268 | C36F49B3C2E5E62AFB56DCEE87445A12A942C14AE618D1FE1B11A9CF9FAA1F32 |
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| 1276 | A304BAF752BEEC36C4440B556AEFECF454BA7CBBA7537BCB10EBC21047333A89 |
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| 1280 | AE05F9E35B2B2CB9C7874C114F57A185C8563C0DCCA93F8096384D71A2994748 |
| 1281 | A3C7C8B8AF54961A8838AD279441D9A5EB6C1FE26C98BD025F353124DA68A827 |
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| 1283 | 85B385124F47443FC116DD7EA23AF95CAFE257D2D685037EBE3F423D7BC0C630 |
| 1284 | B8B1210BCED93CC1C8C6D694F4DC523323FCD3C6015B1EBB5306760C6BC63822 |
| 1285 | 272D60A1368B72B356043E29574434B0238E581D1C5BE338E4CFCF3F2FFFAC29 |
| 1286 | BB70C40CA278611595869383B55FD12D75AA8708DB1DFA1D3F16DE89BF706476 |
| 1287 | 70AEB3AD4E871A1C3201ABA5FB85FDDD62B2D2C3BF3E57FF873523B087AA87D1 |
| 1288 | FA4927C2402003302C349F75A9F7C2D0AE27DA37A0C382D945DD6DBF39EB42BA |
| 1289 | 6343B9CA57FC8E9F6D90255F47B71FAA258C99D6E5E062DCE30DFCDDF2DAF194 |
| 1290 | 3C39A4BFFDC4DEB2125BCC857B90E404225B0AEEF3B85D6051238C777979F884 |
| 1291 | 9CA686FB051C4403AC260118327025D7BCF59690630847E970AC9BC4462E9FAD |
| 1292 | C502D6BCCF7814443D20CCDA9636D8FBEE9C0BB699A606C217FEA184A8035545 |
| 1293 | D05B7F02409291457CC05291241E4EF85E1A72514C84D350E24223E067FB4FCF |
| 1294 | B27EA2B9195014D2B0C3E92C325AA7D880EB59FB2C50E7A5ED176B3F323B0531 |
| 1295 | 1946A616772981A1C30C9A70ED9ADF8706D62B5A02E8AB3600454DFE01869486 |
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| 1298 | 1EFF1FB577093A5B754DA03F30FA2DE01DF51F2AFFD53BB643CE1BA4A3888168 |
| 1299 | A5E62D3906560CF360ECF3ECF76976B288483DEFCDC2A605F9840811BEF57327 |
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| 1303 | 78840078650AEC83207ECFE0FE2DFAB0CE55A4BFB091CDAE615EAC92EEF89E8F |
| 1304 | 0B37511C9BE09A9E2467620AFBFC1E51F853B2E07DBD9B5D39E6AEEF5A4ACCAB |
| 1305 | B56B495CD093A208064B9032A18A24712797EF3A404BAF9F7D93317FD5681594 |
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| 1307 | 31CBB320101FF9586BD6ADFC020BC04F7CB8BD8A9A7CA240B6B97A037261EF64 |
| 1308 | C996503F353E729418FAEB1AC6EDB28436F7DC878A274372CDCF3E1FD12D11BD |
| 1309 | F682A41BD83B9A8D9AA8EDC8A633592FD6F6C29BF1E9F1000447DCAD5541D722 |
| 1310 | 0F9C097A9990002C21D10D2745AEBA3A7B588245F88E1803786A17DFC7B55AFE |
| 1311 | FCE0A4EBDA32D76B24A168C0C9E682815172906284D02D6A51D0A64A83EB18DC |
| 1312 | C69837DDAE75C123BA245AC72F670CCF65F7DA55805DAB17FFA4963CD45C5F68 |
| 1313 | D28E6B5F768E725534A51C6729408F0F1B50CF78AD095DFB191F46949CD15A0B |
| 1314 | BE1AE986153E69AC08656EA3BE2D4CF0A6B43AB66A71DDDEC8815D5ACBD06FE7 |
| 1315 | 3DB2A361BFEF5AC79F23F8AF888166F8277B6B694B4D1478D1F6D48D1EDF57E9 |
| 1316 | 39BCE550C9EE8584A9143C4704F91440032FE3DC3D4430815ECDB10DEF215CB3 |
| 1317 | 303D91824414067E95CEC005161F771E9A332B5C726370AABFB732A4B046F8DE |
| 1318 | 207FF4E857F281DD79C77BBCC5CC991292C528482D59F417645218C000C6E3E5 |
| 1319 | 83374A8163E976E2A4246898D713770DF55DB98A442639B4F90506344BF1400B |
| 1320 | F1314312A720092C436DB6E854219168F699EF6BEDC561201AAC6286EB2E7969 |
| 1321 | 1F13D807BEFFB838A926A7973EE6DC6AFA65879BF86B4325C689E89D98F77CD1 |
| 1322 | 5FA98C930E6E5F88F839EBB598CD3E9CE185A3A93E8D265A279F533CFD79554A |
| 1323 | 6DAABB2D698C217886FBA5F1D33DCF0531A0319793F2F08DAA16FB43C5FF40F4 |
| 1324 | 710F91201D1FEFBF55D46CA9BB3322974BD9113D229CBEA68A70A9D481D13E47 |
| 1325 | F7B849EC2403BF7D67C504D3287B73ABAD948AFAFEB66B5466E401FC9090192F |
| 1326 | 43C5C68FDDB1B2F25973BD5C8EA437B761B0CFCD2BF8590CF66CAEE24845D933 |
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| 1328 | 8BC391B20100024E33563B5A8E139E6D9EDBAFD111029F83D7D4958DD0348DBA |
| 1329 | 36FC54A394D375655FF1A8DD5FD407EA9AFDF58A66B13499DBA3361E8B3D2E96 |
| 1330 | A6804461871BEDC6698EA0479FDB73DDBEE54D70F3D8D53DBAB8E86BECAFD953 |
| 1331 | 21647381DE82FB2D66686026F757A5FC6FE4F97669D8F316C6CBBA1A31C3C2E8 |
| 1332 | 5E32A4E544394520EF3E51D092CB4C2A7369748EBDEFEB99DB9F9EAE5E752FA8 |
| 1333 | 9D1AC7B023A3660F34A20EB36571839BB94642236245BF873B6896F47E746702 |
| 1334 | E2CFAE0EAAFD0B74D41BFEC2F3CDD9BA71F937D24E0A76264F631BEA7DF5E4D1 |
| 1335 | 4F706E9158550EE0CBDA7DA04B539D2E50036D48F256BC36C4B70C19BEC110A3 |
| 1336 | 4A88530D59E17921BA7CB6D0D7ADF2F45B1AE5B56BFC790CACB0F9CD2D6141AB |
| 1337 | 3855692E313DCC811CFB201899AEA7951F5914AA3AC666CA3AFD9020847F3E40 |
| 1338 | D892BD0765CE2438326A22C970A29E3EC6C66AE949F020E3B15D52B257A58E13 |
| 1339 | 613B775E5B50E36BD4BE10C2C4D25A3DA527E23CFE4648952EA0754FDC2332B7 |
| 1340 | 85437AB3F1F4084CEC936CFD8997B57317E05AF13F5049F758C34FD82CA8849D |
| 1341 | 6CBD39151E8342C52F758D85A4D5664DA4A34EBF7939442930A410831FAFADDA |
| 1342 | 1C98A9B7A9B8D5332AD743E6A8D9D717BBEF970B244FA55C85485D472A2BB026 |
| 1343 | 319BEBCCE731AAB6D1DD5A031110FCD6BF0E9AD39F25A09B98BC3C10EC07694A |
| 1344 | 036E62C7A497B787CA0C4E5E571924EAD5380EB34672A5020409255FCB39E69D |
| 1345 | CE3AFFB58CEEFD9D1107100FD51254585CFF2506CE3B08C58C31E95CA5567F7A |
| 1346 | 762CFEF828C5B8261EABAEF839EEB3761697CA |
| 1347 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1348 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1349 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1350 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1351 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1352 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1353 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1354 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1355 | cleartomark |
| 1356 | %%EndFont |
| 1357 | %%BeginFont: CMR8 |
| 1358 | %!PS-AdobeFont-1.1: CMR8 1.0 |
| 1359 | %%CreationDate: 1991 Aug 20 16:39:40 |
| 1360 | % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 1361 | 11 dict begin |
| 1362 | /FontInfo 7 dict dup begin |
| 1363 | /version (1.0) readonly def |
| 1364 | /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| 1365 | /FullName (CMR8) readonly def |
| 1366 | /FamilyName (Computer Modern) readonly def |
| 1367 | /Weight (Medium) readonly def |
| 1368 | /ItalicAngle 0 def |
| 1369 | /isFixedPitch false def |
| 1370 | end readonly def |
| 1371 | /FontName /CMR8 def |
| 1372 | /PaintType 0 def |
| 1373 | /FontType 1 def |
| 1374 | /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| 1375 | /Encoding 256 array |
| 1376 | 0 1 255 {1 index exch /.notdef put} for |
| 1377 | dup 43 /plus put |
| 1378 | dup 49 /one put |
| 1379 | dup 50 /two put |
| 1380 | dup 51 /three put |
| 1381 | dup 55 /seven put |
| 1382 | readonly def |
| 1383 | /FontBBox{-36 -250 1070 750}readonly def |
| 1384 | currentdict end |
| 1385 | currentfile eexec |
| 1386 | D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 |
| 1387 | 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 |
| 1388 | 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F |
| 1389 | D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 |
| 1390 | 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 |
| 1391 | 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C |
| 1392 | 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 |
| 1393 | 3645B82392D5CAE11A7CB49D7E2E82DCD485CBA1772CE422BB1D7283AD675B65 |
| 1394 | 48A7EA0069A883EC1DAA3E1F9ECE7586D6CF0A128CD557C7E5D7AA3EA97EBAD3 |
| 1395 | 9619D1BFCF4A6D64768741EDEA0A5B0EFBBF347CDCBE2E03D756967A16B613DB |
| 1396 | 0FC45FA2A3312E0C46A5FD0466AB097C58FFEEC40601B8395E52775D0AFCD7DB |
| 1397 | 8AB317333110531E5C44A4CB4B5ACD571A1A60960B15E450948A5EEA14DD330F |
| 1398 | EA209265DB8E1A1FC80DCD3860323FD26C113B041A88C88A21655878680A4466 |
| 1399 | FA10403D24BB97152A49B842C180E4D258C9D48F21D057782D90623116830BA3 |
| 1400 | 9902B3C5F2F2DD01433B0D7099C07DBDE268D0FFED5169BCD03D48B2F058AD62 |
| 1401 | D8678C626DC7A3F352152C99BA963EF95F8AD11DB8B0D351210A17E4C2C55AD8 |
| 1402 | 9EB64172935D3C20A398F3EEEEC31551966A7438EF3FEE422C6D4E05337620D5 |
| 1403 | ACC7B52BED984BFAAD36EF9D20748B05D07BE4414A63975125D272FAD83F76E6 |
| 1404 | 10FFF8363014BE526D580873C5A42B70FA911EC7B86905F13AFE55EB0273F582 |
| 1405 | 83158793B8CC296B8DE1DCCF1250FD57CB0E035C7EDA3B0092ED940D37A05493 |
| 1406 | 2EC54E09B984FCA4AB7D2EA182BCF1263AA244B07EC0EA901C077A059F709F30 |
| 1407 | 4384CB5FA748F2054FAD9A7A43D4EA427918BD414F766531136B60C3477C6632 |
| 1408 | BEFE3897B58C19276A301926C2AEF2756B367319772C9B201C49B4D935A8267B |
| 1409 | 041D6F1783B6AEA4DAC4F5B3507D7032AA640AAB12E343A4E9BDCF419C04A721 |
| 1410 | 3888B25AF4E293AACED9A6BDC78E61DA1C424C6503CC1885F762BD779B0C3709 |
| 1411 | 4DF9CB65F0648B72373A300B521D8280D79BA483342B304C7FCD0887A49D9CE1 |
| 1412 | 192E8AE36AB004FFB7CE0A42F4EA48B04B505F2C0D88EB09CF93D6FFCDB7AD23 |
| 1413 | D03BF93916F6D7E71637553DC8AC311C856BE481C14947F7E6353B3171FDFEB9 |
| 1414 | 652488E0592D5A181B538AF6A8C5BDFFA421891E6B7D8BFA7DAEE72A40F93D72 |
| 1415 | 3808E7BA8BF96F3E842579DE45B64E1DF88FC79ACF0DD8A1F4ADF3B4C2C3B9CD |
| 1416 | CE5DA82925DF5913168831DA588564E14032513910DB1EA1E7F8304FAD58E635 |
| 1417 | 4E43575BC260A964ACFEECA8AFACE164E654C9924FB6431C8C791F9C1BB2BDBF |
| 1418 | 6516014072E620D137219D1078D928C735B1B356A4FB7C8AE44E5689AA1C8268 |
| 1419 | 0C9D72C2B7BA568A05DA601156B2A5FCE941B07F4052242244BF79AC217C84A5 |
| 1420 | 5EB1A7FB851212D37DD5E4220FAC2138D60119E9DA87F5156CD298576E5ED59F |
| 1421 | DB192E59D3DDD92C1A4CAD15D2A8AD4E24AA1644FA1C47F5F559DD4727E5FE02 |
| 1422 | 01F45337263E57F4F464A8E36AC54E410C3E61225C3739546D6A0475C0183144 |
| 1423 | 5CA242D40A7B48A09A36C057D819797B2A30B9FA5934F40DDAAEE9855750392E |
| 1424 | A41AFD637143440853BF0315CAC4AF1782248AE28B42C2D3B4FD293754E69512 |
| 1425 | AFB3467CC9CDE4E44BBBC1FB928EA0E686DF898ABB65BEF8E60C2F08760A95FC |
| 1426 | 3D0298321B315CE0633CF6ED222BB17976ABFF3E9CC413C0327E0498B8462D1E |
| 1427 | DADE882D15C4CE194D835C7E26E247D049CAF861CC5A79BFD79610F32FBA3A49 |
| 1428 | 3D3BE7FF757DC36BFD0F715121E0C0BE2E22497E9CB46284CAAA347063385A52 |
| 1429 | 775028C5C7E4F8970A1BF68A1F7CBDD546C6A396095AF5CF37EF9B7F74781986 |
| 1430 | B4995A7554DE578ADF2F3FF5184DADE2A349A261E31A2D8E4DB02086B03C3619 |
| 1431 | 44C90FCF04C00CF4662A16134A0E713C0DC9EA5E97711708EDA217B3FBA37CE4 |
| 1432 | 014BBB0D9C4EEB4206132A744A1402D3DAEF05E029C1A21F430323643A47A764 |
| 1433 | B52D1225ED92B2F1792E41D5E4919344596FD630BFF045A59D4318A01E081654 |
| 1434 | DF63DD3970656D455EB4CC71433F2AD0E4800674CB858434A0187AAB2847247C |
| 1435 | 86A67466164E8D48841DDE6574758567246EFA5902C4A2D5FDE6B4190AD0B84F |
| 1436 | 836DB28929B2E4C595F12F6D7F6CF983F4A577336A62174990FB5FDD8E926EC0 |
| 1437 | CD78150BFA466CB10179 |
| 1438 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1439 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1440 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1441 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1442 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1443 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1444 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1445 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1446 | cleartomark |
| 1447 | %%EndFont |
| 1448 | %%BeginFont: CMSY10 |
| 1449 | %!PS-AdobeFont-1.1: CMSY10 1.0 |
| 1450 | %%CreationDate: 1991 Aug 15 07:20:57 |
| 1451 | % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 1452 | 11 dict begin |
| 1453 | /FontInfo 7 dict dup begin |
| 1454 | /version (1.0) readonly def |
| 1455 | /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| 1456 | /FullName (CMSY10) readonly def |
| 1457 | /FamilyName (Computer Modern) readonly def |
| 1458 | /Weight (Medium) readonly def |
| 1459 | /ItalicAngle -14.035 def |
| 1460 | /isFixedPitch false def |
| 1461 | end readonly def |
| 1462 | /FontName /CMSY10 def |
| 1463 | /PaintType 0 def |
| 1464 | /FontType 1 def |
| 1465 | /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| 1466 | /Encoding 256 array |
| 1467 | 0 1 255 {1 index exch /.notdef put} for |
| 1468 | dup 0 /minus put |
| 1469 | dup 2 /multiply put |
| 1470 | dup 20 /lessequal put |
| 1471 | dup 21 /greaterequal put |
| 1472 | dup 50 /element put |
| 1473 | dup 56 /universal put |
| 1474 | dup 57 /existential put |
| 1475 | dup 91 /union put |
| 1476 | dup 102 /braceleft put |
| 1477 | dup 103 /braceright put |
| 1478 | dup 106 /bar put |
| 1479 | dup 110 /backslash put |
| 1480 | readonly def |
| 1481 | /FontBBox{-29 -960 1116 775}readonly def |
| 1482 | currentdict end |
| 1483 | currentfile eexec |
| 1484 | D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 |
| 1485 | 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 |
| 1486 | A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 |
| 1487 | E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A |
| 1488 | 221A37D9A807DD01161779DDE7D31FF2B87F97C73D63EECDDA4C49501773468A |
| 1489 | 27D1663E0B62F461F6E40A5D6676D1D12B51E641C1D4E8E2771864FC104F8CBF |
| 1490 | 5B78EC1D88228725F1C453A678F58A7E1B7BD7CA700717D288EB8DA1F57C4F09 |
| 1491 | 0ABF1D42C5DDD0C384C7E22F8F8047BE1D4C1CC8E33368FB1AC82B4E96146730 |
| 1492 | DE3302B2E6B819CB6AE455B1AF3187FFE8071AA57EF8A6616B9CB7941D44EC7A |
| 1493 | 71A7BB3DF755178D7D2E4BB69859EFA4BBC30BD6BB1531133FD4D9438FF99F09 |
| 1494 | 4ECC068A324D75B5F696B8688EEB2F17E5ED34CCD6D047A4E3806D000C199D7C |
| 1495 | 515DB70A8D4F6146FE068DC1E5DE8BC57034F5E81DEF84AB9E5382F3E6B5715F |
| 1496 | DF7C70F5A0FBFCD627713545CD2F7BF03BE000B484FBC808AE5D71F2419857EC |
| 1497 | B194BF5AC776B9AE3C54BE1E828E0E65149BEFFF4D6D90076101D9F63FE3E917 |
| 1498 | 886E40F2678A1544FE310DE90354B85323413329E9C28005293E70E54EAA0755 |
| 1499 | 29028DB44DA905896AC25CEC38EC175D03321FC1B83403417BB61AAAFC8DCE3F |
| 1500 | A6914D079987E5B87C83D44D74F649617796F6F306D492DB24F76D28FFE9645D |
| 1501 | 843F62529794974BE830FDC1F58F11FFB104EFEA3C40993C10F128EEF1B5FBE8 |
| 1502 | 6F442B5BD3A377AD5A457878AA37D623E9D0176109A94D4593126A7D1A5F081E |
| 1503 | 5704353CFB81F182083A6A431DEC94B2CA4681B766D31124970919F603609655 |
| 1504 | 77CEFEBBDDEC59DD36A1040DBE5FCBA23F74239A2C226C15B9FF071DB31542EA |
| 1505 | 2D1BCF96BF85423A29C64396AB052DCEBA874740EAF985507F580AD435EBBDA3 |
| 1506 | BFE3F588569DE4B6D081CC210C82AE6F0BA28FF1EED9E89399FA3BD3D0750FF3 |
| 1507 | 1F5FBB680DF78510278DB853838581CA837CF64E4DD772C969A8F26489CF7988 |
| 1508 | E6FF1B4BD122E4651B856FD5732B3049A267F0EEBFDAF16036E58A4D8F39BA36 |
| 1509 | 40658E8ED7419E8166869A7D2EEBE2A0F456AF9428B4DCF2BB36290668E06404 |
| 1510 | 4EC497CFA46C6484C7F762314A28CFE77DA97DD68F6BCAFF90D98E7650E400D3 |
| 1511 | FD4DD94847269BC42FB9F573235646B430CD2CE1E56EEE4EA44F1B73EA2A451F |
| 1512 | 2A285E1EAFFAFC32A5FD8426689DF320711E6059D20454FA0D7DDCCA5B64691D |
| 1513 | 3D5E891EB06363541CE1667A02A96414F12D721C92C0A7B0A7D24FC9C83E0B45 |
| 1514 | 3F69F35DFC531AA3CA912E990C7CD8CE626406A1160A6B5B82A33F2745F39CCD |
| 1515 | DEF54E26B044F0DBF61785149B0536F9B77889B96BB7C7A5245667DFCA9C4D51 |
| 1516 | 5B527A848A58FC60A42D722AD913DD1DF87EC624AA643A648FBC66CA95529DF1 |
| 1517 | D29FE249D82D5D7654D606B1D6BD9A22342D13B3F873BABAE88B8C42C074E8F4 |
| 1518 | CF25CC7B759B8FA07367FAE74ADA0D7A596F77EF712B32FF99ECF5C33A950E7C |
| 1519 | 4B01FC4FA16504745B768C59B4C53A9199158948A585965448C07701B7DB19F5 |
| 1520 | 0501D243E2338C15773D936345A643E8E5B59C31D228DF8C079B66F813CC1291 |
| 1521 | EC11E8E18504AC9A455FA364E40FA9DDAECCD3376EF38EC9CD871B7A0ABEE362 |
| 1522 | FD4EABC96B0F01C8AAA33FCFA8D055E565ECE1476B90DE4314F843CF01758DA2 |
| 1523 | 2463024D1E2FF5424B8694876E771975B30283C9F71DBB23549F07B0E5D19C22 |
| 1524 | 33066511B04A55AD044EB16E7607313B1A04B457E51DD4BF75B54C0063EA8276 |
| 1525 | 3176A25FB0FBE8472A21081843F5D191CAACB68BFF0A86A3D5708CD42A5481D0 |
| 1526 | 4B7B3A554E22438834127C22A9CFC62CC0A313E1AFE075A794299CF46ADD0D91 |
| 1527 | 11825248D40BC1C13ADB74D7A2D9DF2D3209ED5E4925CF344777CCAD44085B1B |
| 1528 | A2D3F3F75E2AB018429E551C4DAE8B3532D1EA018B3AE46065C107DEED0C8403 |
| 1529 | D6E956E11A3D357E29C9069C0203F104E02E877A08F3779BC3EC42A1D9A24E2E |
| 1530 | 67D4C6F25F0C92378598B8BD8F40AA1B2671A98DAD75E24FB4C0F10A680A3698 |
| 1531 | 08E0A46FC7E2C54DFA58CDEDEBEAF3BC88CF64090F14B1E5C9C5819012EE4F2B |
| 1532 | 40B057BE38761FDAACF40EE1EA1ABAD242E67503C7A1ABB321E140F67654B9AF |
| 1533 | 80A608D5E637D22072B27993013C8C67AC539229320C1EA25A6455F042CC6C58 |
| 1534 | 971DAACA05402A378B0537CD43CB909A64562CD69CF92C138C76DE19463718D0 |
| 1535 | 6FF778E7C49011A101DDFC99AD76F400A60835FDCD3E1DA360AB5B088FB1CA66 |
| 1536 | A15FA1FBCA1D7199715ADB8079E1A904D10CA5158CDF92629AC4C2D7A1D8C288 |
| 1537 | 424491151FD408FA56054E2A9DE19CF688A08B6501E42CB50AF52574DAF19110 |
| 1538 | 0D6DC81FE5A281525A20568E80B3322F9D9ECD2F9C6A050FB9E45527C9BA9365 |
| 1539 | 75C4D006248BDE9FD0D5B7EE112AEB36E734592B33EED6CEE8D95D1B7B60279B |
| 1540 | 68B8846FF13DCF6865954F793030CB9A3855AAC01ECD5B0861596CCD8907E867 |
| 1541 | 4313465D47A0D32733B6A8E4A6F387D6069C8CA2E300B4CF6995B6DED3A3B1AD |
| 1542 | 389B56A89E0E900F8C38A1DD073AE23A8C957BED37241B4426451D9BB592249E |
| 1543 | DF7D065010F53FDDF14C9A78565481CCE8AD1F05BFCB65530B3146BF5F53245C |
| 1544 | FAD0F87785C491B3109B8438D779A224149C5CBD6489888EFF798C38EB7D1342 |
| 1545 | 523D2EA9D47936F2C332CE2FB25CD3B4A4AC552F1BD79DE9FCD64AA62D6A181C |
| 1546 | 305BA813D85D2DB54BE476F990709808CA3DEB53C94918038106F6FCC62A1559 |
| 1547 | 610DA7F9EF3534D3BC2C279246DA5C3AE391CBD5A821857B4D699DD92AC4A430 |
| 1548 | 4F9248452C9C95010C3FD1909891476C5AB54C5125FAE5C89CB4B92CCA049940 |
| 1549 | F1E7AC052A4CA7127553898BC84CE9B29287206B32D1940D8B90B263 |
| 1550 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1551 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1552 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1553 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1554 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1555 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1556 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1557 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 1558 | cleartomark |
| 1559 | %%EndFont |
| 1560 | %%BeginFont: CMMI10 |
| 1561 | %!PS-AdobeFont-1.1: CMMI10 1.100 |
| 1562 | %%CreationDate: 1996 Jul 23 07:53:57 |
| 1563 | % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 1564 | 11 dict begin |
| 1565 | /FontInfo 7 dict dup begin |
| 1566 | /version (1.100) readonly def |
| 1567 | /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| 1568 | /FullName (CMMI10) readonly def |
| 1569 | /FamilyName (Computer Modern) readonly def |
| 1570 | /Weight (Medium) readonly def |
| 1571 | /ItalicAngle -14.04 def |
| 1572 | /isFixedPitch false def |
| 1573 | end readonly def |
| 1574 | /FontName /CMMI10 def |
| 1575 | /PaintType 0 def |
| 1576 | /FontType 1 def |
| 1577 | /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| 1578 | /Encoding 256 array |
| 1579 | 0 1 255 {1 index exch /.notdef put} for |
| 1580 | dup 58 /period put |
| 1581 | dup 59 /comma put |
| 1582 | dup 61 /slash put |
| 1583 | dup 62 /greater put |
| 1584 | dup 65 /A put |
| 1585 | dup 75 /K put |
| 1586 | dup 76 /L put |
| 1587 | dup 80 /P put |
| 1588 | dup 82 /R put |
| 1589 | dup 83 /S put |
| 1590 | dup 84 /T put |
| 1591 | dup 98 /b put |
| 1592 | dup 99 /c put |
| 1593 | dup 100 /d put |
| 1594 | dup 101 /e put |
| 1595 | dup 103 /g put |
| 1596 | dup 105 /i put |
| 1597 | dup 106 /j put |
| 1598 | dup 107 /k put |
| 1599 | dup 109 /m put |
| 1600 | dup 110 /n put |
| 1601 | dup 111 /o put |
| 1602 | dup 112 /p put |
| 1603 | dup 113 /q put |
| 1604 | dup 114 /r put |
| 1605 | dup 115 /s put |
| 1606 | dup 116 /t put |
| 1607 | dup 118 /v put |
| 1608 | dup 120 /x put |
| 1609 | dup 122 /z put |
| 1610 | readonly def |
| 1611 | /FontBBox{-32 -250 1048 750}readonly def |
| 1612 | currentdict end |
| 1613 | currentfile eexec |
| 1614 | D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE |
| 1615 | 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B |
| 1616 | 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 |
| 1617 | B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B |
| 1618 | 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE |
| 1619 | D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 |
| 1620 | 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 |
| 1621 | 990CD249827D9648574955F61BAAA11263A91B6C3D47A5190165B0C25ABF6D3E |
| 1622 | 6EC187E4B05182126BB0D0323D943170B795255260F9FD25F2248D04F45DFBFB |
| 1623 | DEF7FF8B19BFEF637B210018AE02572B389B3F76282BEB29CC301905D388C721 |
| 1624 | 59616893E774413F48DE0B408BC66DCE3FE17CB9F84D205839D58014D6A88823 |
| 1625 | D9320AE93AF96D97A02C4D5A2BB2B8C7925C4578003959C46E3CE1A2F0EAC4BF |
| 1626 | 8B9B325E46435BDE60BC54D72BC8ACB5C0A34413AC87045DC7B84646A324B808 |
| 1627 | 6FD8E34217213E131C3B1510415CE45420688ED9C1D27890EC68BD7C1235FAF9 |
| 1628 | 1DAB3A369DD2FC3BE5CF9655C7B7EDA7361D7E05E5831B6B8E2EEC542A7B38EE |
| 1629 | 03BE4BAC6079D038ACB3C7C916279764547C2D51976BABA94BA9866D79F13909 |
| 1630 | 95AA39B0F03103A07CBDF441B8C5669F729020AF284B7FF52A29C6255FCAACF1 |
| 1631 | 74109050FBA2602E72593FBCBFC26E726EE4AEF97B7632BC4F5F353B5C67FED2 |
| 1632 | 3EA752A4A57B8F7FEFF1D7341D895F0A3A0BE1D8E3391970457A967EFF84F6D8 |
| 1633 | 47750B1145B8CC5BD96EE7AA99DDC9E06939E383BDA41175233D58AD263EBF19 |
| 1634 | AFC27E4A7E07D09FB08355F6EA74E530B0743143F2A871732D62D80F35B19FD2 |
| 1635 | C7FDF08105847F13D50934419AC647CBA71DF74F4531DC02BBDA22AEEA3FBBBB |
| 1636 | 407E0ACC52BDC60D01A29407CC4F93EB8BF6D4813E9BA858D54F38918AC82720 |
| 1637 | 4956D50291F0546E50FCAFA6DBD0099123F5ECD4AB338DB310DB4CAE11337A89 |
| 1638 | 8ED99B6F483940C97544F888EAF0CBEB11094A13C073D0061808662A04A82BA0 |
| 1639 | AD35E8782F854AF66C20C0FEF18D0ECDD1646321B93D327E53D88CA0E825FA95 |
| 1640 | 05AA57BD719A0FA14B38EC39164AD228A5F496C48EE1FCF6D7412E372C9B81E5 |
| 1641 | 98C9C34E00C6FF27D711277BD06F13279AEEDDD39489406FAA129D28084D1201 |
| 1642 | 0F381B7130D1278ADFF30212AF2F33E651255C8665D5076E0CBD9F5D5C5D0A81 |
| 1643 | 81C113CF16966411D8738FD33B179FCD58753583ED6B95AAB7C69ABFD9C0A57B |
| 1644 | 8AEC7A803CBD4686C4C103E7F75883E756865F355473D6852599B498E8B4F0E7 |
| 1645 | 28776667059E17CEE485CBB169C637F8A3A2C96A9661ACC154459C622D3748C3 |
| 1646 | 2CC0B8608057500C303FC1F79F30D8CF39288FDF3819460482E8AC115833BCD5 |
| 1647 | 82434238B0215EFC24428E5D658D6D4F22189412EE4113D0B937934213A11B82 |
| 1648 | 745C1D8A62BE022F04AC17FCDA3B8CC1F513D74046A0D0FCB0B6023C2BDCDAE0 |
| 1649 | 53913C074E99C05EA8A1D416DA1EA10860FF73E77744F6FEBAF1FBF8457D484E |
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| 1838 | %!PS-AdobeFont-1.1: CMTT10 1.00B |
| 1839 | %%CreationDate: 1992 Apr 26 10:42:42 |
| 1840 | % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 1841 | 11 dict begin |
| 1842 | /FontInfo 7 dict dup begin |
| 1843 | /version (1.00B) readonly def |
| 1844 | /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| 1845 | /FullName (CMTT10) readonly def |
| 1846 | /FamilyName (Computer Modern) readonly def |
| 1847 | /Weight (Medium) readonly def |
| 1848 | /ItalicAngle 0 def |
| 1849 | /isFixedPitch true def |
| 1850 | end readonly def |
| 1851 | /FontName /CMTT10 def |
| 1852 | /PaintType 0 def |
| 1853 | /FontType 1 def |
| 1854 | /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| 1855 | /Encoding 256 array |
| 1856 | 0 1 255 {1 index exch /.notdef put} for |
| 1857 | dup 35 /numbersign put |
| 1858 | dup 38 /ampersand put |
| 1859 | dup 40 /parenleft put |
| 1860 | dup 41 /parenright put |
| 1861 | dup 42 /asterisk put |
| 1862 | dup 44 /comma put |
| 1863 | dup 45 /hyphen put |
| 1864 | dup 46 /period put |
| 1865 | dup 47 /slash put |
| 1866 | dup 48 /zero put |
| 1867 | dup 49 /one put |
| 1868 | dup 50 /two put |
| 1869 | dup 51 /three put |
| 1870 | dup 52 /four put |
| 1871 | dup 53 /five put |
| 1872 | dup 55 /seven put |
| 1873 | dup 59 /semicolon put |
| 1874 | dup 61 /equal put |
| 1875 | dup 62 /greater put |
| 1876 | dup 65 /A put |
| 1877 | dup 66 /B put |
| 1878 | dup 67 /C put |
| 1879 | dup 68 /D put |
| 1880 | dup 69 /E put |
| 1881 | dup 70 /F put |
| 1882 | dup 71 /G put |
| 1883 | dup 72 /H put |
| 1884 | dup 73 /I put |
| 1885 | dup 76 /L put |
| 1886 | dup 77 /M put |
| 1887 | dup 78 /N put |
| 1888 | dup 79 /O put |
| 1889 | dup 80 /P put |
| 1890 | dup 82 /R put |
| 1891 | dup 83 /S put |
| 1892 | dup 84 /T put |
| 1893 | dup 85 /U put |
| 1894 | dup 86 /V put |
| 1895 | dup 87 /W put |
| 1896 | dup 90 /Z put |
| 1897 | dup 91 /bracketleft put |
| 1898 | dup 93 /bracketright put |
| 1899 | dup 95 /underscore put |
| 1900 | dup 97 /a put |
| 1901 | dup 98 /b put |
| 1902 | dup 99 /c put |
| 1903 | dup 100 /d put |
| 1904 | dup 101 /e put |
| 1905 | dup 102 /f put |
| 1906 | dup 103 /g put |
| 1907 | dup 104 /h put |
| 1908 | dup 105 /i put |
| 1909 | dup 106 /j put |
| 1910 | dup 107 /k put |
| 1911 | dup 108 /l put |
| 1912 | dup 109 /m put |
| 1913 | dup 110 /n put |
| 1914 | dup 111 /o put |
| 1915 | dup 112 /p put |
| 1916 | dup 113 /q put |
| 1917 | dup 114 /r put |
| 1918 | dup 115 /s put |
| 1919 | dup 116 /t put |
| 1920 | dup 117 /u put |
| 1921 | dup 118 /v put |
| 1922 | dup 119 /w put |
| 1923 | dup 120 /x put |
| 1924 | dup 121 /y put |
| 1925 | dup 122 /z put |
| 1926 | dup 123 /braceleft put |
| 1927 | dup 125 /braceright put |
| 1928 | readonly def |
| 1929 | /FontBBox{-4 -235 731 800}readonly def |
| 1930 | currentdict end |
| 1931 | currentfile eexec |
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| 2304 | 9147 |
| 2305 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 2306 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 2307 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 2308 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 2309 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 2310 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 2311 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 2312 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 2313 | cleartomark |
| 2314 | %%EndFont |
| 2315 | %%BeginFont: CMR10 |
| 2316 | %!PS-AdobeFont-1.1: CMR10 1.00B |
| 2317 | %%CreationDate: 1992 Feb 19 19:54:52 |
| 2318 | % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 2319 | 11 dict begin |
| 2320 | /FontInfo 7 dict dup begin |
| 2321 | /version (1.00B) readonly def |
| 2322 | /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| 2323 | /FullName (CMR10) readonly def |
| 2324 | /FamilyName (Computer Modern) readonly def |
| 2325 | /Weight (Medium) readonly def |
| 2326 | /ItalicAngle 0 def |
| 2327 | /isFixedPitch false def |
| 2328 | end readonly def |
| 2329 | /FontName /CMR10 def |
| 2330 | /PaintType 0 def |
| 2331 | /FontType 1 def |
| 2332 | /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| 2333 | /Encoding 256 array |
| 2334 | 0 1 255 {1 index exch /.notdef put} for |
| 2335 | dup 11 /ff put |
| 2336 | dup 12 /fi put |
| 2337 | dup 13 /fl put |
| 2338 | dup 14 /ffi put |
| 2339 | dup 34 /quotedblright put |
| 2340 | dup 37 /percent put |
| 2341 | dup 39 /quoteright put |
| 2342 | dup 40 /parenleft put |
| 2343 | dup 41 /parenright put |
| 2344 | dup 42 /asterisk put |
| 2345 | dup 43 /plus put |
| 2346 | dup 44 /comma put |
| 2347 | dup 45 /hyphen put |
| 2348 | dup 46 /period put |
| 2349 | dup 47 /slash put |
| 2350 | dup 48 /zero put |
| 2351 | dup 49 /one put |
| 2352 | dup 50 /two put |
| 2353 | dup 51 /three put |
| 2354 | dup 52 /four put |
| 2355 | dup 53 /five put |
| 2356 | dup 54 /six put |
| 2357 | dup 55 /seven put |
| 2358 | dup 56 /eight put |
| 2359 | dup 57 /nine put |
| 2360 | dup 58 /colon put |
| 2361 | dup 59 /semicolon put |
| 2362 | dup 61 /equal put |
| 2363 | dup 63 /question put |
| 2364 | dup 65 /A put |
| 2365 | dup 66 /B put |
| 2366 | dup 67 /C put |
| 2367 | dup 68 /D put |
| 2368 | dup 69 /E put |
| 2369 | dup 70 /F put |
| 2370 | dup 71 /G put |
| 2371 | dup 72 /H put |
| 2372 | dup 73 /I put |
| 2373 | dup 74 /J put |
| 2374 | dup 75 /K put |
| 2375 | dup 76 /L put |
| 2376 | dup 77 /M put |
| 2377 | dup 78 /N put |
| 2378 | dup 79 /O put |
| 2379 | dup 80 /P put |
| 2380 | dup 81 /Q put |
| 2381 | dup 82 /R put |
| 2382 | dup 83 /S put |
| 2383 | dup 84 /T put |
| 2384 | dup 85 /U put |
| 2385 | dup 86 /V put |
| 2386 | dup 87 /W put |
| 2387 | dup 89 /Y put |
| 2388 | dup 90 /Z put |
| 2389 | dup 91 /bracketleft put |
| 2390 | dup 92 /quotedblleft put |
| 2391 | dup 93 /bracketright put |
| 2392 | dup 97 /a put |
| 2393 | dup 98 /b put |
| 2394 | dup 99 /c put |
| 2395 | dup 100 /d put |
| 2396 | dup 101 /e put |
| 2397 | dup 102 /f put |
| 2398 | dup 103 /g put |
| 2399 | dup 104 /h put |
| 2400 | dup 105 /i put |
| 2401 | dup 106 /j put |
| 2402 | dup 107 /k put |
| 2403 | dup 108 /l put |
| 2404 | dup 109 /m put |
| 2405 | dup 110 /n put |
| 2406 | dup 111 /o put |
| 2407 | dup 112 /p put |
| 2408 | dup 113 /q put |
| 2409 | dup 114 /r put |
| 2410 | dup 115 /s put |
| 2411 | dup 116 /t put |
| 2412 | dup 117 /u put |
| 2413 | dup 118 /v put |
| 2414 | dup 119 /w put |
| 2415 | dup 120 /x put |
| 2416 | dup 121 /y put |
| 2417 | dup 122 /z put |
| 2418 | dup 123 /endash put |
| 2419 | dup 126 /tilde put |
| 2420 | dup 127 /dieresis put |
| 2421 | readonly def |
| 2422 | /FontBBox{-251 -250 1009 969}readonly def |
| 2423 | currentdict end |
| 2424 | currentfile eexec |
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| 2917 | 45 |
| 2918 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 2919 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 2920 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 2921 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 2922 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 2923 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 2924 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 2925 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 2926 | cleartomark |
| 2927 | %%EndFont |
| 2928 | %%BeginFont: CMBX10 |
| 2929 | %!PS-AdobeFont-1.1: CMBX10 1.00B |
| 2930 | %%CreationDate: 1992 Feb 19 19:54:06 |
| 2931 | % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 2932 | 11 dict begin |
| 2933 | /FontInfo 7 dict dup begin |
| 2934 | /version (1.00B) readonly def |
| 2935 | /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| 2936 | /FullName (CMBX10) readonly def |
| 2937 | /FamilyName (Computer Modern) readonly def |
| 2938 | /Weight (Bold) readonly def |
| 2939 | /ItalicAngle 0 def |
| 2940 | /isFixedPitch false def |
| 2941 | end readonly def |
| 2942 | /FontName /CMBX10 def |
| 2943 | /PaintType 0 def |
| 2944 | /FontType 1 def |
| 2945 | /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| 2946 | /Encoding 256 array |
| 2947 | 0 1 255 {1 index exch /.notdef put} for |
| 2948 | dup 12 /fi put |
| 2949 | dup 40 /parenleft put |
| 2950 | dup 41 /parenright put |
| 2951 | dup 45 /hyphen put |
| 2952 | dup 46 /period put |
| 2953 | dup 48 /zero put |
| 2954 | dup 49 /one put |
| 2955 | dup 50 /two put |
| 2956 | dup 51 /three put |
| 2957 | dup 52 /four put |
| 2958 | dup 53 /five put |
| 2959 | dup 54 /six put |
| 2960 | dup 55 /seven put |
| 2961 | dup 56 /eight put |
| 2962 | dup 57 /nine put |
| 2963 | dup 65 /A put |
| 2964 | dup 66 /B put |
| 2965 | dup 67 /C put |
| 2966 | dup 68 /D put |
| 2967 | dup 69 /E put |
| 2968 | dup 70 /F put |
| 2969 | dup 71 /G put |
| 2970 | dup 72 /H put |
| 2971 | dup 73 /I put |
| 2972 | dup 76 /L put |
| 2973 | dup 77 /M put |
| 2974 | dup 78 /N put |
| 2975 | dup 79 /O put |
| 2976 | dup 80 /P put |
| 2977 | dup 83 /S put |
| 2978 | dup 84 /T put |
| 2979 | dup 85 /U put |
| 2980 | dup 86 /V put |
| 2981 | dup 97 /a put |
| 2982 | dup 98 /b put |
| 2983 | dup 99 /c put |
| 2984 | dup 100 /d put |
| 2985 | dup 101 /e put |
| 2986 | dup 102 /f put |
| 2987 | dup 103 /g put |
| 2988 | dup 104 /h put |
| 2989 | dup 105 /i put |
| 2990 | dup 106 /j put |
| 2991 | dup 108 /l put |
| 2992 | dup 109 /m put |
| 2993 | dup 110 /n put |
| 2994 | dup 111 /o put |
| 2995 | dup 112 /p put |
| 2996 | dup 113 /q put |
| 2997 | dup 114 /r put |
| 2998 | dup 115 /s put |
| 2999 | dup 116 /t put |
| 3000 | dup 117 /u put |
| 3001 | dup 118 /v put |
| 3002 | dup 120 /x put |
| 3003 | dup 121 /y put |
| 3004 | readonly def |
| 3005 | /FontBBox{-301 -250 1164 946}readonly def |
| 3006 | currentdict end |
| 3007 | currentfile eexec |
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| 3341 | 89903EDF186B4ADCBD360EBA0B |
| 3342 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 3343 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 3344 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 3345 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 3346 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 3347 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 3348 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 3349 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 3350 | cleartomark |
| 3351 | %%EndFont |
| 3352 | %%BeginFont: CMR12 |
| 3353 | %!PS-AdobeFont-1.1: CMR12 1.0 |
| 3354 | %%CreationDate: 1991 Aug 20 16:38:05 |
| 3355 | % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 3356 | 11 dict begin |
| 3357 | /FontInfo 7 dict dup begin |
| 3358 | /version (1.0) readonly def |
| 3359 | /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| 3360 | /FullName (CMR12) readonly def |
| 3361 | /FamilyName (Computer Modern) readonly def |
| 3362 | /Weight (Medium) readonly def |
| 3363 | /ItalicAngle 0 def |
| 3364 | /isFixedPitch false def |
| 3365 | end readonly def |
| 3366 | /FontName /CMR12 def |
| 3367 | /PaintType 0 def |
| 3368 | /FontType 1 def |
| 3369 | /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| 3370 | /Encoding 256 array |
| 3371 | 0 1 255 {1 index exch /.notdef put} for |
| 3372 | dup 40 /parenleft put |
| 3373 | dup 41 /parenright put |
| 3374 | dup 44 /comma put |
| 3375 | dup 45 /hyphen put |
| 3376 | dup 46 /period put |
| 3377 | dup 48 /zero put |
| 3378 | dup 50 /two put |
| 3379 | dup 52 /four put |
| 3380 | dup 55 /seven put |
| 3381 | dup 56 /eight put |
| 3382 | dup 57 /nine put |
| 3383 | dup 67 /C put |
| 3384 | dup 69 /E put |
| 3385 | dup 70 /F put |
| 3386 | dup 72 /H put |
| 3387 | dup 73 /I put |
| 3388 | dup 75 /K put |
| 3389 | dup 79 /O put |
| 3390 | dup 82 /R put |
| 3391 | dup 83 /S put |
| 3392 | dup 84 /T put |
| 3393 | dup 90 /Z put |
| 3394 | dup 97 /a put |
| 3395 | dup 98 /b put |
| 3396 | dup 99 /c put |
| 3397 | dup 100 /d put |
| 3398 | dup 101 /e put |
| 3399 | dup 102 /f put |
| 3400 | dup 104 /h put |
| 3401 | dup 105 /i put |
| 3402 | dup 107 /k put |
| 3403 | dup 108 /l put |
| 3404 | dup 109 /m put |
| 3405 | dup 110 /n put |
| 3406 | dup 111 /o put |
| 3407 | dup 112 /p put |
| 3408 | dup 114 /r put |
| 3409 | dup 115 /s put |
| 3410 | dup 116 /t put |
| 3411 | dup 117 /u put |
| 3412 | dup 118 /v put |
| 3413 | dup 119 /w put |
| 3414 | dup 121 /y put |
| 3415 | dup 122 /z put |
| 3416 | readonly def |
| 3417 | /FontBBox{-34 -251 988 750}readonly def |
| 3418 | currentdict end |
| 3419 | currentfile eexec |
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| 3660 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 3661 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 3662 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 3663 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 3664 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 3665 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 3666 | 0000000000000000000000000000000000000000000000000000000000000000 |
| 3667 | cleartomark |
| 3668 | %%EndFont |
| 3669 | %%BeginFont: CMR17 |
| 3670 | %!PS-AdobeFont-1.1: CMR17 1.0 |
| 3671 | %%CreationDate: 1991 Aug 20 16:38:24 |
| 3672 | % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
| 3673 | 11 dict begin |
| 3674 | /FontInfo 7 dict dup begin |
| 3675 | /version (1.0) readonly def |
| 3676 | /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
| 3677 | /FullName (CMR17) readonly def |
| 3678 | /FamilyName (Computer Modern) readonly def |
| 3679 | /Weight (Medium) readonly def |
| 3680 | /ItalicAngle 0 def |
| 3681 | /isFixedPitch false def |
| 3682 | end readonly def |
| 3683 | /FontName /CMR17 def |
| 3684 | /PaintType 0 def |
| 3685 | /FontType 1 def |
| 3686 | /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
| 3687 | /Encoding 256 array |
| 3688 | 0 1 255 {1 index exch /.notdef put} for |
| 3689 | dup 77 /M put |
| 3690 | dup 82 /R put |
| 3691 | dup 97 /a put |
| 3692 | dup 98 /b put |
| 3693 | dup 99 /c put |
| 3694 | dup 100 /d put |
| 3695 | dup 101 /e put |
| 3696 | dup 102 /f put |
| 3697 | dup 105 /i put |
| 3698 | dup 108 /l put |
| 3699 | dup 110 /n put |
| 3700 | dup 114 /r put |
| 3701 | dup 117 /u put |
| 3702 | readonly def |
| 3703 | /FontBBox{-33 -250 945 749}readonly def |
| 3704 | currentdict end |
| 3705 | currentfile eexec |
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| 3960 | Fj(x)1530 1764 y Fh(1)1570 1750 y Fj(;)15 b(x)1662 1764 |
| 3961 | y Fh(2)1701 1750 y Fj(;)g(:)g(:)g(:)i(;)e(x)1955 1765 |
| 3962 | y Fg(d)1996 1750 y Fo(\))2031 1712 y Fg(T)2111 1750 y |
| 3963 | Fi(2)25 b Fj(R)2267 1712 y Fg(d)2333 1750 y Fo(:)g Fj(b)20 |
| 3964 | b Fi(\000)g Fj(Ax)26 b Fi(\025)f Fo(0)p Fi(g)0 1944 y |
| 3965 | Fo(where)37 b Fj(A)h Fo(is)f(a)h(giv)m(en)h Fj(m)24 b |
| 3966 | Fi(\002)h Fj(d)38 b Fo(real)g(matrix)g(and)f Fj(b)g Fo(is)h(a)f(giv)m |
| 3967 | (en)i(real)f Fj(m)p Fo(-v)m(ector.)64 b(In)37 b(the)g(mathematical)j |
| 3968 | (lan-)0 2057 y(guage,)e(the)e(computation)g(is)g(the)f(transformation)h |
| 3969 | (of)g(an)f Ff(H-r)-5 b(epr)g(esentation)45 b Fo(of)35 |
| 3970 | b(a)h(con)m(v)m(ex)h(p)s(olytop)s(e)e(to)i(an)0 2170 |
| 3971 | y Ff(V-r)-5 b(epr)g(esentation)p Fo(.)141 2283 y(cddlib)38 |
| 3972 | b(is)h(a)h(C-library)e(v)m(ersion)h(of)h(the)f(previously)f(released)i |
| 3973 | (C-co)s(de)f(cdd/cdd+.)65 b(In)39 b(order)f(to)i(mak)m(e)0 |
| 3974 | 2396 y(this)31 b(library)g(v)m(ersion,)i(a)e(large)i(part)e(of)h(the)f |
| 3975 | (cdd)g(source)g(\(V)-8 b(ersion)33 b(0.61\))g(has)e(b)s(een)g |
| 3976 | (rewritten.)43 b(This)31 b(library)0 2509 y(v)m(ersion)j(is)f(more)g |
| 3977 | (\015exible)g(since)h(it)f(can)g(b)s(e)g(called)h(from)f(other)g |
| 3978 | (programs)g(in)f(C/C++.)48 b(Unlik)m(e)34 b(cdd/cdd+,)0 |
| 3979 | 2622 y(cddlib)c(can)h(handle)f(an)m(y)h(general)h(input)e(and)g(is)h |
| 3980 | (more)f(general.)43 b(F)-8 b(urtthermore,)31 b(additional)h(functions)e |
| 3981 | (ha)m(v)m(e)0 2735 y(b)s(een)g(written)g(to)h(extend)g(its)f |
| 3982 | (functionalit)m(y)-8 b(.)141 2848 y(One)31 b(useful)f(feature)h(of)g |
| 3983 | (cddlib/cdd/cdd+)e(is)i(its)h(capabilit)m(y)g(of)f(handling)f(the)h |
| 3984 | (dual)g(\(rev)m(erse\))h(problem)0 2961 y(without)38 |
| 3985 | b(an)m(y)g(transformation)h(of)f(data.)64 b(The)37 b(dual)h |
| 3986 | (transformation)g(problem)g(of)g(a)g(V-represen)m(tation)i(to)0 |
| 3987 | 3073 y(a)e(minimal)g(H-represen)m(tation)h(and)e(is)h(often)g(called)h |
| 3988 | (the)f Ff(\(c)-5 b(onvex\))40 b(hul)5 b(l)40 b(pr)-5 |
| 3989 | b(oblem)7 b Fo(.)64 b(More)38 b(explicitly)-8 b(,)42 |
| 3990 | b(is)c(to)0 3186 y(obtain)27 b(a)f(linear)h(inequalit)m(y)g(represen)m |
| 3991 | (tation)h(of)e(a)h(con)m(v)m(ex)g(p)s(olyhedron)e(giv)m(en)j(as)e(the)g |
| 3992 | (Mink)m(o)m(wski)i(sum)d(of)h(the)0 3299 y(con)m(v)m(ex)32 |
| 3993 | b(h)m(ull)e(of)h(a)g(\014nite)f(set)h(of)f(p)s(oin)m(ts)h(and)e(the)i |
| 3994 | (nonnegativ)m(e)h(h)m(ull)e(of)h(a)g(\014nite)f(set)h(of)f(p)s(oin)m |
| 3995 | (ts)h(in)f Fj(R)3532 3266 y Fg(d)3572 3299 y Fo(:)1007 |
| 3996 | 3494 y Fj(P)38 b Fo(=)25 b Fj(conv)s Fo(\()p Fj(v)1463 |
| 3997 | 3508 y Fh(1)1503 3494 y Fj(;)15 b(:)g(:)g(:)i(;)e(v)1749 |
| 3998 | 3508 y Fg(n)1796 3494 y Fo(\))21 b(+)f Fj(nonneg)s Fo(\()p |
| 3999 | Fj(r)2316 3508 y Fg(n)p Fh(+1)2453 3494 y Fj(;)15 b(:)g(:)g(:)h(;)f(r) |
| 4000 | 2695 3508 y Fg(n)p Fh(+)p Fg(s)2830 3494 y Fo(\))p Fj(;)0 |
| 4001 | 3689 y Fo(where)30 b(the)g Ff(Minkowski)j(sum)g(of)g(two)g(subsets)g |
| 4002 | Fj(S)k Ff(and)d Fj(T)43 b Fo(of)31 b Fj(R)2184 3656 y |
| 4003 | Fg(d)2254 3689 y Fo(is)g(de\014ned)e(as)1277 3884 y Fj(S)c |
| 4004 | Fo(+)20 b Fj(T)38 b Fo(=)25 b Fi(f)p Fj(s)20 b Fo(+)g |
| 4005 | Fj(t)25 b Fi(j)p Fj(s)g Fi(2)g Fj(S)36 b Fo(and)29 b |
| 4006 | Fj(t)c Fi(2)g Fj(T)13 b Fi(g)p Fj(:)0 4078 y Fo(As)38 |
| 4007 | b(w)m(e)h(see)g(in)f(this)g(man)m(ual,)j(the)e(computation)g(can)f(b)s |
| 4008 | (e)g(done)g(in)g(straigh)m(tforw)m(ard)h(manner.)64 b(Unlik)m(e)39 |
| 4009 | b(the)0 4191 y(earlier)29 b(v)m(ersions)f(of)g(cdd/cdd+)f(that)h |
| 4010 | (assume)g(certain)h(regularit)m(y)g(conditions)f(for)g(input,)g(cddlib) |
| 4011 | f(is)h(designed)0 4304 y(to)40 b(do)f(a)g(correct)i(transformation)e |
| 4012 | (for)g(an)m(y)h(general)g(input.)66 b(The)39 b(user)f(m)m(ust)h(b)s(e)g |
| 4013 | (a)m(w)m(are)h(of)g(the)f(fact)h(that)0 4417 y(in)g(certain)i(cases)g |
| 4014 | (the)e(transformation)h(is)g(not)g(unique)f(and)g(there)h(are)g(p)s |
| 4015 | (olyhedra)e(with)i(in\014nitely)f(man)m(y)0 4530 y(represen)m(tations.) |
| 4016 | 54 b(F)-8 b(or)35 b(example,)i(a)e(line)g(segmen)m(t)g(\(1-dimensional) |
| 4017 | h(p)s(olytop)s(e\))f(in)f Fj(R)3055 4497 y Fh(3)3129 |
| 4018 | 4530 y Fo(has)g(in\014nitely)g(man)m(y)0 4643 y(minimal)43 |
| 4019 | b(H-represen)m(tations,)48 b(and)43 b(a)g(halfspace)h(in)f(the)g(same)h |
| 4020 | (space)g(has)e(in\014nitely)i(man)m(y)f(minimal)g(V-)0 |
| 4021 | 4756 y(represen)m(tations.)f(cddlib)29 b(generates)j(merely)f(one)g |
| 4022 | (minimal)f(represen)m(tation.)141 4869 y(cddlib)g(comes)h(with)e(an)h |
| 4023 | (LP)g(co)s(de)h(to)g(solv)m(e)g(the)f(general)i(linear)e(programming)g |
| 4024 | (\(LP\))g(problem)g(to)h(maxi-)0 4982 y(mize)g(\(or)g(minimize\))g(a)g |
| 4025 | (linear)f(function)h(o)m(v)m(er)g(p)s(olyhedron)e Fj(P)13 |
| 4026 | b Fo(.)41 b(It)30 b(is)h(useful)e(mainly)i(for)f(solving)h(dense)f |
| 4027 | (LP's)0 5094 y(with)35 b(large)i Fj(m)f Fo(\(sa)m(y)-8 |
| 4028 | b(,)39 b(up)34 b(to)j(few)e(h)m(undred)f(thousands\))h(and)g(small)i |
| 4029 | Fj(d)f Fo(\(sa)m(y)-8 b(,)38 b(up)d(to)i(100\).)58 b(It)36 |
| 4030 | b(implemen)m(ts)g(a)0 5207 y(revised)30 b(dual)g(simplex)h(metho)s(d)e |
| 4031 | (that)i(up)s(dates)f(\()p Fj(d)21 b Fo(+)f(1\))h Fi(\002)f |
| 4032 | Fo(\()p Fj(d)g Fo(+)g(1\))31 b(matrix)g(for)f(a)h(piv)m(ot)g(op)s |
| 4033 | (eration.)141 5320 y(The)k(program)f(cddlib)h(has)f(an)h(I/O)g |
| 4034 | (routines)g(that)g(read)g(and)g(write)g(\014les)g(in)f |
| 4035 | Ff(Polyhe)-5 b(dr)g(a)40 b(format)45 b Fo(whic)m(h)0 |
| 4036 | 5433 y(w)m(as)32 b(de\014ned)f(b)m(y)h(Da)m(vid)h(Avis)f(and)f(the)h |
| 4037 | (author)g(in)f(1993,)k(and)c(has)h(b)s(een)f(up)s(dated)f(in)i(1997)i |
| 4038 | (and)d(1999.)47 b(The)1926 5682 y(2)p eop end |
| 4039 | %%Page: 3 3 |
| 4040 | TeXDict begin 3 2 bop 0 44 a Fo(program)40 b(called)i(lrs)e(and)f |
| 4041 | (lrslib)i([2])g(dev)m(elop)s(ed)f(b)m(y)h(Da)m(vid)g(Avis)g(is)f(a)h |
| 4042 | (C-implemen)m(tation)h(of)e(the)h(rev)m(erse)0 157 y(searc)m(h)d |
| 4043 | (algorithm)g([4])f(for)g(the)g(same)h(en)m(umeration)f(purp)s(ose,)g |
| 4044 | (and)g(it)g(conforms)g(to)h(P)m(olyhedra)f(format)g(as)0 |
| 4045 | 270 y(w)m(ell.)k(Hop)s(efully)-8 b(,)30 b(this)g(compatibilit)m(y)h(of) |
| 4046 | f(the)f(t)m(w)m(o)i(programs)e(enables)h(users)e(to)i(use)g(b)s(oth)e |
| 4047 | (programs)h(for)g(the)0 383 y(same)35 b(input)e(\014les)h(and)g(to)h(c) |
| 4048 | m(ho)s(ose)h(whic)m(hev)m(er)e(is)h(useful)e(for)i(their)f(purp)s |
| 4049 | (oses.)51 b(F)-8 b(rom)35 b(our)e(exp)s(eriences)i(with)0 |
| 4050 | 496 y(relativ)m(ely)40 b(large)f(problems,)h(the)e(t)m(w)m(o)h(metho)s |
| 4051 | (ds)f(are)g(b)s(oth)f(useful)h(and)f(p)s(erhaps)f(complemen)m(tary)k |
| 4052 | (to)e(eac)m(h)0 609 y(other.)61 b(In)36 b(general,)k(the)d(program)g |
| 4053 | (cddlib)g(tends)f(to)i(b)s(e)e(e\016cien)m(t)j(for)e(highly)g |
| 4054 | (degenerate)h(inputs)e(and)g(the)0 722 y(program)30 b(rs)g(tends)g(to)h |
| 4055 | (b)s(e)f(e\016cien)m(t)i(for)e(nondegenerate)h(or)f(sligh)m(tly)i |
| 4056 | (degenerate)g(problems.)141 835 y(Although)39 b(the)h(program)f(can)g |
| 4057 | (b)s(e)g(used)f(for)h(nondegenerate)h(inputs,)h(it)e(migh)m(t)h(not)g |
| 4058 | (b)s(e)e(v)m(ery)i(e\016cien)m(t.)0 947 y(F)-8 b(or)38 |
| 4059 | b(nondegenerate)f(inputs,)h(other)f(a)m(v)-5 b(ailable)40 |
| 4060 | b(programs,)e(suc)m(h)f(as)g(the)g(rev)m(erse)h(searc)m(h)f(co)s(de)g |
| 4061 | (lrs)g(or)g(qh)m(ull)0 1060 y(\(dev)m(elop)s(ed)c(b)m(y)g(the)g |
| 4062 | (Geometry)g(Cen)m(ter\),)i(migh)m(t)e(b)s(e)f(more)h(e\016cien)m(t.)49 |
| 4063 | b(See)32 b(Section)i(8)f(for)f(p)s(oin)m(ters)h(to)g(these)0 |
| 4064 | 1173 y(co)s(des.)38 b(The)22 b(pap)s(er)g([3])h(con)m(tains)h(man)m(y)f |
| 4065 | (in)m(teresting)h(results)e(on)g(p)s(olyhedral)g(computation)i(and)e |
| 4066 | (exp)s(erimen)m(tal)0 1286 y(results)30 b(on)g(cdd+,)g(lrs,)g(qh)m(ull) |
| 4067 | h(and)e(p)s(orta.)141 1399 y(This)d(program)h(can)h(b)s(e)e |
| 4068 | (distributed)g(freely)i(under)d(the)j(GNU)f(GENERAL)h(PUBLIC)e |
| 4069 | (LICENSE.)g(Please)0 1512 y(read)k(the)h(\014le)f(COPYING)g(carefully)h |
| 4070 | (b)s(efore)f(using.)141 1625 y(I)e(will)g(not)g(tak)m(e)h(an)m(y)f |
| 4071 | (resp)s(onsibilit)m(y)g(of)g(an)m(y)g(problems)f(y)m(ou)h(migh)m(t)g |
| 4072 | (ha)m(v)m(e)h(with)f(this)f(program.)40 b(But)28 b(I)f(will)0 |
| 4073 | 1738 y(b)s(e)37 b(glad)h(to)g(receiv)m(e)h(bug)d(rep)s(orts)h(or)g |
| 4074 | (suggestions)h(at)g(the)g(e-mail)g(addresses)f(ab)s(o)m(v)m(e.)63 |
| 4075 | b(If)37 b(cddlib)f(turns)g(out)0 1851 y(to)d(b)s(e)f(useful,)g(please)h |
| 4076 | (kindly)f(inform)g(me)g(of)h(what)f(purp)s(oses)f(cdd)h(has)g(b)s(een)f |
| 4077 | (used)h(for.)46 b(I)33 b(will)f(b)s(e)g(happ)m(y)g(to)0 |
| 4078 | 1964 y(include)h(a)h(list)g(of)f(applications)i(in)e(future)f |
| 4079 | (distribution)h(if)g(I)g(receiv)m(e)j(enough)d(replies.)49 |
| 4080 | b(The)33 b(most)h(p)s(o)m(w)m(erful)0 2077 y(supp)s(ort)29 |
| 4081 | b(for)h(free)g(soft)m(w)m(are)i(dev)m(elopmen)m(t)g(is)e(user's)g |
| 4082 | (appreciation)i(and)d(collab)s(oration.)0 2359 y Fq(2)135 |
| 4083 | b(P)l(olyhedra)45 b(H-)g(and)g(V-F)-11 b(ormats)44 b(\(V)-11 |
| 4084 | b(ersion)46 b(1999\))0 2645 y Fo(Ev)m(ery)34 b(con)m(v)m(ex)h(p)s |
| 4085 | (olyhedron)d(has)h(t)m(w)m(o)i(represen)m(tations,)g(one)f(as)g(the)f |
| 4086 | (in)m(tersection)i(of)f(\014nite)f(halfspaces)h(and)0 |
| 4087 | 2757 y(the)h(other)g(as)g(Mink)m(o)m(wski)h(sum)e(of)i(the)f(con)m(v)m |
| 4088 | (ex)h(h)m(ull)f(of)g(\014nite)g(p)s(oin)m(ts)g(and)f(the)h(nonnegativ)m |
| 4089 | (e)i(h)m(ull)e(of)g(\014nite)0 2870 y(directions.)41 |
| 4090 | b(These)30 b(are)h(called)h(H-represen)m(tation)g(and)d(V-represen)m |
| 4091 | (tation,)k(resp)s(ectiv)m(ely)-8 b(.)141 2983 y(Naturally)32 |
| 4092 | b(there)g(are)f(t)m(w)m(o)i(basic)e(P)m(olyhedra)h(formats,)g(H-format) |
| 4093 | g(for)e(H-represen)m(tation)j(and)e(V-format)0 3096 y(for)h(V-represen) |
| 4094 | m(tation.)49 b(These)33 b(t)m(w)m(o)h(formats)f(are)g(designed)f(to)h |
| 4095 | (b)s(e)f(almost)i(indistinguishable,)f(and)f(in)g(fact,)0 |
| 4096 | 3209 y(one)h(can)h(almost)g(pretend)e(one)h(for)g(the)g(other.)49 |
| 4097 | b(There)32 b(is)h(some)h(asymmetry)f(arising)g(from)f(the)h(asymmetry)0 |
| 4098 | 3322 y(of)e(t)m(w)m(o)g(represen)m(tations.)141 3435 |
| 4099 | y(First)37 b(w)m(e)g(start)g(with)f(the)h(H-represen)m(tation.)60 |
| 4100 | b(Let)37 b Fj(A)g Fo(b)s(e)f(an)g Fj(m)24 b Fi(\002)g |
| 4101 | Fj(d)37 b Fo(matrix,)h(and)e(let)i Fj(b)e Fo(b)s(e)g(a)h(column)0 |
| 4102 | 3548 y Fj(m)p Fo(-v)m(ector.)60 b(The)36 b(P)m(olyhedra)g(format)h(\()p |
| 4103 | Ff(H-format)g Fo(\))g(of)f(the)g(system)72 b Fj(b)24 |
| 4104 | b Fi(\000)f Fj(Ax)36 b Fi(\025)e Fp(0)72 b Fo(of)36 b |
| 4105 | Fj(m)g Fo(inequalities)h(in)f Fj(d)0 3661 y Fo(v)-5 b(ariables)31 |
| 4106 | b Fj(x)25 b Fo(=)g(\()p Fj(x)637 3675 y Fh(1)677 3661 |
| 4107 | y Fj(;)15 b(x)769 3675 y Fh(2)809 3661 y Fj(;)g(:)g(:)g(:)h(;)f(x)1062 |
| 4108 | 3676 y Fg(d)1103 3661 y Fo(\))1138 3628 y Fg(T)1224 3661 |
| 4109 | y Fo(is)p 141 3808 1178 4 v 191 3887 a(v)-5 b(arious)30 |
| 4110 | b(commen)m(ts)191 4000 y Fp(H-represen)m(tation)191 4113 |
| 4111 | y(\(linearit)m(y)k Fj(t)60 b(i)770 4127 y Fh(1)870 4113 |
| 4112 | y Fj(i)901 4127 y Fh(2)1001 4113 y Fj(:)15 b(:)g(:)61 |
| 4113 | b(i)1198 4127 y Fg(t)1228 4113 y Fp(\))191 4226 y(b)s(egin)191 |
| 4114 | 4339 y Fj(m)99 b(d)21 b Fo(+)f(1)100 b(n)m(um)m(b)s(ert)m(yp)s(e)211 |
| 4115 | 4452 y Fj(b)153 b Fi(\000)p Fj(A)191 4564 y Fp(end)191 |
| 4116 | 4677 y Fo(v)-5 b(arious)30 b(options)p 141 4715 V 0 4869 |
| 4117 | a(where)38 b(n)m(um)m(b)s(ert)m(yp)s(e)f(can)i(b)s(e)f(one)h(of)g(in)m |
| 4118 | (teger,)j(rational)e(or)e(real.)66 b(When)38 b(rational)i(t)m(yp)s(e)f |
| 4119 | (is)f(selected,)k(eac)m(h)0 4982 y(comp)s(onen)m(t)26 |
| 4120 | b(of)h Fj(b)f Fo(and)f Fj(A)h Fo(can)h(b)s(e)e(sp)s(eci\014ed)h(b)m(y)g |
| 4121 | (the)g(usual)g(in)m(teger)h(expression)f(or)g(b)m(y)g(the)g(rational)i |
| 4122 | (expression)0 5094 y(\\)p Fj(p=q)s Fo(")41 b(or)e(\\)p |
| 4123 | Fi(\000)p Fj(p=q)s Fo(")i(where)e Fj(p)h Fo(and)f Fj(q)j |
| 4124 | Fo(are)f(arbitrary)e(long)i(p)s(ositiv)m(e)f(in)m(tegers)h(\(see)g(the) |
| 4125 | f(example)h(input)d(\014le)0 5207 y(rational.ine\).)75 |
| 4126 | b(In)40 b(the)i(1997)h(format,)h(w)m(e)e(in)m(tro)s(duced)e |
| 4127 | (\\H-represen)m(tation")k(whic)m(h)c(m)m(ust)h(app)s(ear)g(b)s(efore)0 |
| 4128 | 5320 y(\\b)s(egin".)f(There)27 b(w)m(as)g(one)g(restriction)h(in)f(the) |
| 4129 | g(old)g(p)s(olyhedra)f(format)h(\(b)s(efore)g(1997\):)42 |
| 4130 | b(the)27 b(last)h Fj(d)f Fo(ro)m(ws)g(m)m(ust)0 5433 |
| 4131 | y(determine)j(a)h(v)m(ertex)h(of)e Fj(P)13 b Fo(.)41 |
| 4132 | b(This)30 b(is)g(obsolete)i(no)m(w.)1926 5682 y(3)p eop |
| 4133 | end |
| 4134 | %%Page: 4 4 |
| 4135 | TeXDict begin 4 3 bop 141 44 a Fo(In)30 b(the)i(new)e(1999)j(format,)f |
| 4136 | (w)m(e)f(added)f(the)i(p)s(ossibilit)m(y)f(of)g(sp)s(ecifying)g |
| 4137 | Fp(linearit)m(y)q Fo(.)43 b(This)30 b(means)h(that)h(for)0 |
| 4138 | 157 y(H-represen)m(tation,)40 b(some)d(of)f(the)h(input)f(ro)m(ws)g |
| 4139 | (can)h(b)s(e)f(sp)s(eci\014ed)g(as)h Fp(equalities)p |
| 4140 | Fo(:)52 b Fj(b)3036 171 y Fg(i)3060 181 y Fe(j)3121 157 |
| 4141 | y Fi(\000)24 b Fj(A)3284 171 y Fg(i)3308 181 y Fe(j)3345 |
| 4142 | 157 y Fj(x)36 b Fo(=)f(0)72 b(for)37 b(all)0 270 y Fj(j)31 |
| 4143 | b Fo(=)25 b(1)p Fj(;)15 b Fo(2)p Fj(;)g(:)g(:)g(:)j(;)d(t)p |
| 4144 | Fo(.)41 b(The)30 b(linearit)m(y)h(line)g(ma)m(y)g(b)s(e)f(omitted)h(if) |
| 4145 | g(there)f(are)h(no)f(equalities.)141 383 y(Option)h(lines)g(can)g(b)s |
| 4146 | (e)g(used)f(to)i(con)m(trol)g(computation)g(of)g(a)f(sp)s(eci\014c)g |
| 4147 | (program.)42 b(In)31 b(particular)g(b)s(oth)f(cdd)0 496 |
| 4148 | y(and)39 b(lrs)g(use)g(the)h(option)g(lines)g(to)g(represen)m(t)f(a)h |
| 4149 | (linear)g(ob)5 b(jectiv)m(e)42 b(function.)67 b(See)40 |
| 4150 | b(the)g(attac)m(hed)h(LP)e(\014les,)0 609 y(samplelp*.ine.)141 |
| 4151 | 813 y(Next)25 b(w)m(e)f(de\014ne)f(P)m(olyhedra)h Ff(V-format)p |
| 4152 | Fo(.)39 b(Let)24 b Fj(P)37 b Fo(b)s(e)23 b(represen)m(ted)h(b)m(y)f |
| 4153 | Fj(n)g Fo(gerating)i(p)s(oin)m(ts)f(and)f Fj(s)g Fo(generating)0 |
| 4154 | 926 y(directions)30 b(\(ra)m(ys\))h(as)f Fj(P)39 b Fo(=)24 |
| 4155 | b Fj(conv)s Fo(\()p Fj(v)1244 940 y Fh(1)1285 926 y Fj(;)15 |
| 4156 | b(:)g(:)g(:)h(;)f(v)1530 940 y Fg(n)1578 926 y Fo(\))k(+)g |
| 4157 | Fj(nonneg)s Fo(\()p Fj(r)2095 940 y Fg(n)p Fh(+1)2232 |
| 4158 | 926 y Fj(;)c(:)g(:)g(:)i(;)e(r)2475 940 y Fg(n)p Fh(+)p |
| 4159 | Fg(s)2610 926 y Fo(\).)41 b(Then)28 b(the)i(P)m(olyhedra)h(V-format)0 |
| 4160 | 1039 y(for)f Fj(P)43 b Fo(is)p 141 1163 1202 4 v 191 |
| 4161 | 1242 a(v)-5 b(arious)30 b(commen)m(ts)191 1355 y Fp(V-represen)m |
| 4162 | (tation)191 1468 y Fo(\()p Fp(linearit)m(y)35 b Fj(t)60 |
| 4163 | b(i)765 1482 y Fh(1)865 1468 y Fj(i)896 1482 y Fh(2)996 |
| 4164 | 1468 y Fj(:)15 b(:)g(:)61 b(i)1193 1482 y Fg(t)1258 1468 |
| 4165 | y Fo(\))191 1581 y Fp(b)s(egin)191 1694 y Fj(n)20 b Fo(+)g |
| 4166 | Fj(s)99 b(d)20 b Fo(+)g(1)100 b(n)m(um)m(b)s(ert)m(yp)s(e)272 |
| 4167 | 1807 y(1)182 b Fj(v)543 1821 y Fh(1)282 1900 y Fo(.)282 |
| 4168 | 1934 y(.)282 1967 y(.)499 1900 y(.)499 1934 y(.)499 1967 |
| 4169 | y(.)272 2080 y(1)g Fj(v)543 2094 y Fg(n)272 2193 y Fo(0)g |
| 4170 | Fj(r)540 2207 y Fg(n)p Fh(+1)282 2286 y Fo(.)282 2319 |
| 4171 | y(.)282 2352 y(.)499 2286 y(.)499 2319 y(.)499 2352 y(.)272 |
| 4172 | 2465 y(0)g Fj(r)540 2479 y Fg(n)p Fh(+)p Fg(s)191 2578 |
| 4173 | y Fp(end)191 2691 y Fo(v)-5 b(arious)30 b(options)p 141 |
| 4174 | 2728 V 0 2891 a(Here)42 b(w)m(e)h(do)e(not)h(require)g(that)g(v)m |
| 4175 | (ertices)i(and)d(ra)m(ys)h(are)g(listed)h(separately;)49 |
| 4176 | b(they)42 b(can)g(app)s(ear)f(mixed)h(in)0 3004 y(arbitrary)30 |
| 4177 | b(order.)141 3117 y(Linearit)m(y)i(for)f(V-represen)m(tation)h(sp)s |
| 4178 | (eci\014es)f(a)g(subset)g(of)g(generators)h(whose)e(co)s(e\016cien)m |
| 4179 | (ts)j(are)e(relaxed)h(to)0 3230 y(b)s(e)f Fp(free)p Fo(:)44 |
| 4180 | b(for)32 b(all)g Fj(j)i Fo(=)27 b(1)p Fj(;)15 b Fo(2)p |
| 4181 | Fj(;)g(:)g(:)g(:)j(;)d(t)p Fo(,)33 b(the)f Fj(k)f Fo(=)d |
| 4182 | Fj(i)1592 3244 y Fg(j)1629 3230 y Fo(th)j(generator)i(\()p |
| 4183 | Fj(v)2231 3245 y Fg(k)2306 3230 y Fo(or)f Fj(r)2460 3245 |
| 4184 | y Fg(k)2535 3230 y Fo(whic)m(hev)m(er)g(is)g(the)h Fj(i)3241 |
| 4185 | 3244 y Fg(j)3277 3230 y Fo(th)f(generator\))i(is)0 3343 |
| 4186 | y(a)c(free)h(generator.)41 b(This)30 b(means)g(for)f(eac)m(h)j(suc)m(h) |
| 4187 | d(a)i(ra)m(y)f Fj(r)1995 3358 y Fg(k)2038 3343 y Fo(,)g(the)h(line)f |
| 4188 | (generated)h(b)m(y)f Fj(r)3002 3358 y Fg(k)3075 3343 |
| 4189 | y Fo(is)g(in)f(the)i(p)s(olyhedron,)0 3456 y(and)26 b(for)h(eac)m(h)h |
| 4190 | (suc)m(h)e(a)h(v)m(ertex)h Fj(v)1099 3471 y Fg(k)1142 |
| 4191 | 3456 y Fo(,)g(its)f(co)s(e\016cien)m(t)i(is)d(no)h(longer)g(nonnegativ) |
| 4192 | m(e)i(but)d(still)i(the)f(co)s(e\016cien)m(ts)h(for)f(all)0 |
| 4193 | 3569 y Fj(v)44 3583 y Fg(i)72 3569 y Fo('s)i(m)m(ust)g(sum)f(up)g(to)i |
| 4194 | (one.)41 b(It)29 b(is)g(highly)g(unlik)m(ely)h(that)f(one)h(needs)e(to) |
| 4195 | i(use)f(linearit)m(y)h(for)f(v)m(ertex)h(generators,)0 |
| 4196 | 3681 y(and)g(it)h(is)f(de\014ned)f(mostly)i(for)f(formalit)m(y)-8 |
| 4197 | b(.)141 3794 y(When)33 b(the)h(represen)m(tation)g(statemen)m(t,)i |
| 4198 | (either)e(\\H-represen)m(tation")h(or)f(\\V-represen)m(tation",)i(is)e |
| 4199 | (omit-)0 3907 y(ted,)d(the)f(former)g(\\H-represen)m(tation")j(is)e |
| 4200 | (assumed.)141 4020 y(It)g(is)f(strongly)h(suggested)g(to)g(use)f(the)h |
| 4201 | (follo)m(wing)g(rule)g(for)f(naming)g(H-format)h(\014les)f(and)g |
| 4202 | (V-format)h(\014les:)0 4198 y Fp(\(a\))45 b Fo(use)30 |
| 4203 | b(the)g(\014lename)h(extension)g(\\.ine")g(for)g(H-\014les)f(\(where)h |
| 4204 | (ine)f(stands)g(for)g(inequalities\),)i(and)0 4381 y |
| 4205 | Fp(\(b\))45 b Fo(use)30 b(the)g(\014lename)h(extension)g(\\.ext")h(for) |
| 4206 | e(V-\014les)h(\(where)f(ext)i(stands)d(for)h(extreme)i(p)s(oin)m(ts/ra) |
| 4207 | m(ys\).)0 4666 y Fq(3)135 b(Basic)45 b(Ob)7 b(ject)45 |
| 4208 | b(T)l(yp)t(es)g(\(Structures\))g(in)g(cddlib)0 4869 y |
| 4209 | Fo(Here)g(are)f(the)g(t)m(yp)s(es)g(\(de\014ned)f(in)h(cddt)m(yp)s |
| 4210 | (es.h\))g(that)g(are)h(imp)s(ortan)m(t)f(for)g(the)g(cddlib)g(user.)81 |
| 4211 | b(The)43 b(most)0 4982 y(imp)s(ortan)m(t)36 b(one,)i |
| 4212 | Fn(dd)p 736 4982 29 4 v 34 w(MatrixType)p Fo(,)d(is)h(to)g(store)h(a)f |
| 4213 | (P)m(olyhedra)h(data)f(in)g(a)g(straigh)m(tforw)m(ard)h(manner.)57 |
| 4214 | b(Once)0 5094 y(the)32 b(user)f(sets)i(up)e(a)h(\(p)s(oin)m(ter)g(to\)) |
| 4215 | i Fn(dd)p 1343 5094 V 33 w(MatrixType)c Fo(data,)j(he/she)f(can)g(load) |
| 4216 | h(the)f(data)h(to)f(an)g(in)m(ternal)h(data)0 5207 y(t)m(yp)s(e)h(\()p |
| 4217 | Fn(dd)p 345 5207 V 34 w(PolyhedraType)p Fo(\))c(b)m(y)j(using)h |
| 4218 | (functions)f(describ)s(ed)f(in)h(the)h(next)g(section,)h(and)e(apply)g |
| 4219 | (the)h(double)0 5320 y(descrition)j(metho)s(d)e(to)i(get)g(another)f |
| 4220 | (represen)m(tation.)59 b(As)36 b(an)g(option)h Fn(dd)p |
| 4221 | 2680 5320 V 33 w(MatrixType)d Fo(can)i(sa)m(v)m(e)i(a)e(linear)0 |
| 4222 | 5433 y(ob)5 b(jectiv)m(e)32 b(function)e(to)i(b)s(e)d(used)h(b)m(y)g(a) |
| 4223 | h(linear)g(programming)f(solv)m(er.)1926 5682 y(4)p eop |
| 4224 | end |
| 4225 | %%Page: 5 5 |
| 4226 | TeXDict begin 5 4 bop 141 44 a Fo(The)40 b(t)m(w)m(o)h(dimensional)f |
| 4227 | (arra)m(y)h(data)g(in)f(the)g(structure)g Fn(dd)p 2278 |
| 4228 | 44 29 4 v 33 w(MatrixType)e Fo(is)i Fn(dd)p 3026 44 V |
| 4229 | 34 w(Amatrix)e Fo(whose)i(com-)0 157 y(p)s(onen)m(ts)c(are)h(of)f(t)m |
| 4230 | (yp)s(e)g Fn(mytype)p Fo(.)57 b(The)36 b(t)m(yp)s(e)h(m)m(yt)m(yp)s(e)f |
| 4231 | (is)h(set)f(to)h(b)s(e)f(either)h(the)f(rational)i(t)m(yp)s(e)e |
| 4232 | Fn(mpq)p 3550 157 V 34 w(t)g Fo(of)g(the)0 270 y(GNU)f(MP)f(Library)f |
| 4233 | (or)h(the)h(C)f(double)f(arra)m(y)i(of)f(size)h(1.)53 |
| 4234 | b(This)33 b(abstract)i(t)m(yp)s(e)f(allo)m(ws)i(us)d(to)i(write)f(a)h |
| 4235 | (single)0 383 y(program)f(that)h(can)g(b)s(e)e(compiled)i(with)f(the)g |
| 4236 | (t)m(w)m(o)i(or)e(more)h(di\013eren)m(t)f(arithmetics,)j(see)e(example) |
| 4237 | g(programs)0 496 y(suc)m(h)d(as)h(simplecdd.c,)g(testlp*.c)h(and)e |
| 4238 | (testcdd*.c)i(in)e(the)h Fn(src)e Fo(and)h Fn(src-gmp)e |
| 4239 | Fo(sub)s(directories)i(of)h(the)f(source)0 609 y(distribution.)141 |
| 4240 | 722 y(There)42 b(is)h(another)g(data)g(t)m(yp)s(e)g(that)g(is)g(used)f |
| 4241 | (v)m(ery)h(often,)j Fn(dd)p 2419 722 V 34 w(SetFamilyType)p |
| 4242 | Fo(.)74 b(This)42 b(is)h(to)g(store)g(a)0 835 y(family)35 |
| 4243 | b(of)h(subsets)e(of)h(a)g(\014nite)g(set.)55 b(Suc)m(h)35 |
| 4244 | b(a)g(family)h(can)f(represen)m(t)g(the)g(incidence)h(relations)g(b)s |
| 4245 | (et)m(w)m(een)f(the)0 947 y(set)f(of)g(extreme)h(p)s(oin)m(ts)e(and)g |
| 4246 | (the)h(set)g(of)g(facets)h(of)f(a)g(p)s(olyhedron.)49 |
| 4247 | b(Also,)36 b(it)e(can)g(represen)m(t)f(a)h(graph)g(struc-)0 |
| 4248 | 1060 y(ture)g(b)m(y)g(listing)i(the)e(set)h(of)f(v)m(ertices)i(adjacen) |
| 4249 | m(t)g(to)f(eac)m(h)h(v)m(ertex)f(\(i.e.)54 b(the)34 b(adjacency)i |
| 4250 | (list\).)53 b(T)-8 b(o)35 b(implemen)m(t)0 1173 y Fn(dd)p |
| 4251 | 102 1173 V 34 w(SetFamilyType)p Fo(,)26 b(w)m(e)j(use)f(a)h(separate)h |
| 4252 | (set)g(library)e(called)i Fn(setoper)p Fo(,)d(that)j(handles)e(the)h |
| 4253 | (basic)g(set)g(op)s(era-)0 1286 y(tions,)i(This)f(library)g(is)g |
| 4254 | (brie\015y)f(in)m(tro)s(duced)h(in)g(Section)i(4.6.)0 |
| 4255 | 1594 y Fn(#define)46 b(dd_FALSE)f(0)0 1707 y(#define)h(dd_TRUE)g(1)0 |
| 4256 | 1933 y(typedef)g(long)g(dd_rowrange;)0 2046 y(typedef)g(long)g |
| 4257 | (dd_colrange;)0 2159 y(typedef)g(long)g(dd_bigrange;)0 |
| 4258 | 2385 y(typedef)g(set_type)f(dd_rowset;)141 b(/*)47 b(set_type)f |
| 4259 | (defined)f(in)j(setoper.h)d(*/)0 2498 y(typedef)h(set_type)f |
| 4260 | (dd_colset;)0 2610 y(typedef)h(long)g(*dd_rowindex;)0 |
| 4261 | 2723 y(typedef)g(int)h(*dd_rowflag;)0 2836 y(typedef)f(long)g |
| 4262 | (*dd_colindex;)0 2949 y(typedef)g(mytype)g(**dd_Amatrix;)92 |
| 4263 | b(/*)47 b(mytype)f(is)h(either)f(GMP)h(mpq_t)g(or)g(1-dim)f(double)g |
| 4264 | (array.)g(*/)0 3062 y(typedef)g(mytype)g(*dd_Arow;)0 |
| 4265 | 3175 y(typedef)g(set_type)f(*dd_SetVector;)0 3401 y(typedef)h(enum)g({) |
| 4266 | 95 3514 y(dd_Real,)g(dd_Rational,)e(dd_Integer,)h(dd_Unknown)0 |
| 4267 | 3627 y(})i(dd_NumberType;)0 3852 y(typedef)f(enum)g({)95 |
| 4268 | 3965 y(dd_Inequality,)e(dd_Generator,)h(dd_Unspecified)0 |
| 4269 | 4078 y(})i(dd_RepresentationType;)0 4304 y(typedef)f(enum)g({)95 |
| 4270 | 4417 y(dd_MaxIndex,)f(dd_MinIndex,)f(dd_MinCutoff,)h(dd_MaxCutoff,)f |
| 4271 | (dd_MixCutoff,)143 4530 y(dd_LexMin,)h(dd_LexMax,)g(dd_RandomRow)0 |
| 4272 | 4643 y(})i(dd_RowOrderType;)0 4869 y(typedef)f(enum)g({)95 |
| 4273 | 4982 y(dd_InProgress,)e(dd_AllFound,)h(dd_RegionEmpty)0 |
| 4274 | 5094 y(})i(dd_CompStatusType;)0 5320 y(typedef)f(enum)g({)95 |
| 4275 | 5433 y(dd_DimensionTooLarge,)c(dd_ImproperInputFormat,)1926 |
| 4276 | 5682 y Fo(5)p eop end |
| 4277 | %%Page: 6 6 |
| 4278 | TeXDict begin 6 5 bop 95 44 a Fn(dd_NegativeMatrixSize,)42 |
| 4279 | b(dd_EmptyVrepresentation,)95 157 y(dd_IFileNotFound,)i |
| 4280 | (dd_OFileNotOpen,)f(dd_NoLPObjective,)95 270 y(dd_NoRealNumberSupport,) |
| 4281 | f(dd_NoError)0 383 y(})47 b(dd_ErrorType;)0 609 y(typedef)f(enum)g({)95 |
| 4282 | 722 y(dd_LPnone=0,)f(dd_LPmax,)g(dd_LPmin)0 835 y(})i |
| 4283 | (dd_LPObjectiveType;)0 1060 y(typedef)f(enum)g({)95 1173 |
| 4284 | y(dd_LPSundecided,)e(dd_Optimal,)h(dd_Inconsistent,)e |
| 4285 | (dd_DualInconsistent,)95 1286 y(dd_StrucInconsistent,)f |
| 4286 | (dd_StrucDualInconsistent,)95 1399 y(dd_Unbounded,)j(dd_DualUnbounded)0 |
| 4287 | 1512 y(})i(dd_LPStatusType;)0 1738 y(typedef)f(struct)g(matrixdata)f |
| 4288 | (*dd_MatrixPtr;)0 1851 y(typedef)h(struct)g(matrixdata)f({)95 |
| 4289 | 1964 y(dd_rowrange)g(rowsize;)95 2077 y(dd_rowset)h(linset;)191 |
| 4290 | 2189 y(/*)95 b(a)47 b(subset)f(of)i(rows)e(of)h(linearity)f(\(ie,)g |
| 4291 | (generators)f(of)382 2302 y(linearity)g(space)h(for)h |
| 4292 | (V-representation,)c(and)k(equations)382 2415 y(for)g |
| 4293 | (H-representation.)c(*/)95 2528 y(dd_colrange)i(colsize;)95 |
| 4294 | 2641 y(dd_RepresentationType)d(representation;)95 2754 |
| 4295 | y(dd_NumberType)j(numbtype;)95 2867 y(dd_Amatrix)g(matrix;)95 |
| 4296 | 2980 y(dd_LPObjectiveType)e(objective;)95 3093 y(dd_Arow)j(rowvec;)0 |
| 4297 | 3206 y(})95 b(dd_MatrixType;)0 3431 y(typedef)46 b(struct)g(setfamily)f |
| 4298 | (*dd_SetFamilyPtr;)0 3544 y(typedef)h(struct)g(setfamily)f({)95 |
| 4299 | 3657 y(dd_bigrange)g(famsize;)95 3770 y(dd_bigrange)g(setsize;)95 |
| 4300 | 3883 y(dd_SetVector)g(set;)0 3996 y(})i(dd_SetFamilyType;)0 |
| 4301 | 4222 y(typedef)f(struct)g(lpsolution)f(*dd_LPSolutionPtr;)0 |
| 4302 | 4335 y(typedef)h(struct)g(lpsolution)f({)95 4448 y(dd_DataFileType)f |
| 4303 | (filename;)95 4561 y(dd_LPObjectiveType)f(objective;)95 |
| 4304 | 4673 y(dd_LPSolverType)h(solver;)95 4786 y(dd_rowrange)h(m;)95 |
| 4305 | 4899 y(dd_colrange)g(d;)95 5012 y(dd_NumberType)g(numbtype;)95 |
| 4306 | 5238 y(dd_LPStatusType)f(LPS;)94 b(/*)48 b(the)f(current)e(solution)h |
| 4307 | (status)g(*/)95 5351 y(mytype)h(optvalue;)93 b(/*)47 |
| 4308 | b(optimal)f(value)g(*/)1926 5682 y Fo(6)p eop end |
| 4309 | %%Page: 7 7 |
| 4310 | TeXDict begin 7 6 bop 95 44 a Fn(dd_Arow)46 b(sol;)142 |
| 4311 | b(/*)48 b(primal)e(solution)f(*/)95 157 y(dd_Arow)h(dsol;)94 |
| 4312 | b(/*)48 b(dual)e(solution)g(*/)95 270 y(dd_colindex)f(nbindex;)93 |
| 4313 | b(/*)48 b(current)d(basis)i(represented)e(by)i(nonbasic)e(indices)h(*/) |
| 4314 | 95 383 y(dd_rowrange)f(re;)95 b(/*)47 b(row)g(index)f(as)h(a)h |
| 4315 | (certificate)d(in)i(the)g(case)f(of)i(inconsistency)c(*/)95 |
| 4316 | 496 y(dd_colrange)h(se;)95 b(/*)47 b(col)g(index)f(as)h(a)h |
| 4317 | (certificate)d(in)i(the)g(case)f(of)i(dual)e(inconsistency)e(*/)95 |
| 4318 | 609 y(long)j(pivots[5];)143 722 y(/*)g(pivots[0]=setup)d(\(to)j(find)g |
| 4319 | (a)g(basis\),)f(pivots[1]=PhaseI)d(or)k(Criss-Cross,)286 |
| 4320 | 835 y(pivots[2]=Phase)d(II,)j(pivots[3]=Anticycling,)42 |
| 4321 | b(pivots[4]=GMP)i(postopt)93 b(*/)95 947 y(long)47 b(total_pivots;)0 |
| 4322 | 1060 y(})g(dd_LPSolutionType;)0 1455 y Fq(4)135 b(Library)45 |
| 4323 | b(F)-11 b(unctions)0 1658 y Fo(Here)24 b(w)m(e)g(list)g(some)f(of)h |
| 4324 | (the)f(most)h(imp)s(ortan)m(t)f(library)g(functions/pro)s(cedures.)37 |
| 4325 | b(W)-8 b(e)25 b(use)e(the)g(follo)m(wing)i(con)m(v)m(en-)0 |
| 4326 | 1771 y(tion:)42 b Fn(poly)29 b Fo(is)i(of)g(t)m(yp)s(e)g |
| 4327 | Fn(dd)p 947 1771 29 4 v 34 w(PolyhedraPtr)p Fo(,)c Fn(matrix)p |
| 4328 | Fo(,)j Fn(matrix1)e Fo(and)i Fn(matrix2)f Fo(are)i(of)g(t)m(yp)s(e)g |
| 4329 | Fn(dd)p 3415 1771 V 34 w(MatrixPtr)p Fo(,)0 1884 y Fn(matrixP)p |
| 4330 | Fo(,)40 b(of)h(t)m(yp)s(e)g Fn(dd)p 832 1884 V 34 w(MatrixPtr*)p |
| 4331 | Fo(,)h Fn(err)e Fo(is)h(of)h(t)m(yp)s(e)f Fn(dd)p 2125 |
| 4332 | 1884 V 34 w(ErrorType*)p Fo(,)g Fn(ifile)f Fo(and)h Fn(ofile)e |
| 4333 | Fo(are)j(of)f(t)m(yp)s(e)0 1996 y Fn(char*)p Fo(,)29 |
| 4334 | b Fn(A)h Fo(is)g(of)g(t)m(yp)s(e)g Fn(dd)p 872 1996 V |
| 4335 | 34 w(Amatrix)p Fo(,)e Fn(point)h Fo(and)g Fn(vector)f |
| 4336 | Fo(are)j(of)f(t)m(yp)s(e)g Fn(dd)p 2611 1996 V 34 w(Arow)p |
| 4337 | Fo(,)f Fn(d)g Fo(is)h(of)h(t)m(yp)s(e)f Fn(dd)p 3463 |
| 4338 | 1996 V 34 w(colrange)p Fo(,)0 2109 y Fn(m)37 b Fo(and)h |
| 4339 | Fn(i)f Fo(are)h(of)g(t)m(yp)s(e)g Fn(dd)p 939 2109 V |
| 4340 | 34 w(rowrange)p Fo(,)g Fn(x)f Fo(is)h(of)g(t)m(yp)s(e)g |
| 4341 | Fn(mytype)p Fo(,)g Fn(a)g Fo(is)f(of)h(t)m(yp)s(e)g Fn(signed)47 |
| 4342 | b(long)f(integer)p Fo(,)38 b Fn(b)g Fo(is)0 2222 y(of)33 |
| 4343 | 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 |
| 4344 | Fn(set)p 1388 2222 V 34 w(type)p Fo(.)45 b(Also,)34 b |
| 4345 | Fn(setfam)d Fo(is)i(of)f(t)m(yp)s(e)h Fn(dd)p 2738 2222 |
| 4346 | V 34 w(SetFamilyPtr)p Fo(,)c Fn(lp)j Fo(is)h(of)f(t)m(yp)s(e)0 |
| 4347 | 2335 y Fn(dd)p 102 2335 V 34 w(LPPtr)p Fo(,)h Fn(lps)g |
| 4348 | Fo(is)g(of)h(t)m(yp)s(e)g Fn(dd)p 1116 2335 V 34 w(LPSolutionPtr)p |
| 4349 | Fo(,)d Fn(solver)g Fo(is)j(of)g(t)m(yp)s(e)f Fn(dd)p |
| 4350 | 2654 2335 V 34 w(LPSolverType)p Fo(,)f Fn(roworder)f |
| 4351 | Fo(is)i(of)0 2448 y(t)m(yp)s(e)e Fn(dd)p 307 2448 V 33 |
| 4352 | w(RowOrderType)p Fo(.)0 2687 y Fd(4.1)112 b(Library)39 |
| 4353 | b(Initialization)0 2858 y Fn(void)47 b(dd)p 341 2858 |
| 4354 | V 33 w(set)p 518 2858 V 34 w(global)p 840 2858 V 33 w |
| 4355 | (constants\(void\))41 b Fo(:)227 2971 y(This)31 b(is)g(to)h(set)g(the)f |
| 4356 | (global)h(constan)m(ts)h(suc)m(h)e(as)g Fn(dd)p 2032 |
| 4357 | 2971 V 34 w(zero)p Fo(,)f Fn(dd)p 2409 2971 V 34 w(purezero)f |
| 4358 | Fo(and)i Fn(dd)p 3130 2971 V 34 w(one)f Fo(for)h(sign)g(recog-)227 |
| 4359 | 3084 y(nition)h(and)g(basic)g(arithmetic)h(op)s(erations.)46 |
| 4360 | b(Ev)m(ery)32 b(program)g(to)h(use)f(cddlib)f(m)m(ust)h(call)h(this)f |
| 4361 | (function)227 3197 y(b)s(efore)37 b(doing)g(an)m(y)h(computation.)62 |
| 4362 | b(Just)36 b(call)j(this)e(once.)61 b(See)38 b(Section)g(4.3.3)h(for)d |
| 4363 | (the)i(de\014nitions)e(of)227 3310 y(constan)m(ts.)0 |
| 4364 | 3487 y Fn(void)47 b(dd)p 341 3487 V 33 w(free)p 566 3487 |
| 4365 | V 34 w(global)p 888 3487 V 32 w(constants\(void\))42 |
| 4366 | b Fo(:)227 3600 y(This)e(is)g(to)h(free)f(the)g(global)i(constan)m(ts.) |
| 4367 | 71 b(This)39 b(should)h(b)s(e)f(called)i(when)f(one)g(do)s(es)g(not)g |
| 4368 | (use)g(cddlib)227 3713 y(functions)30 b(an)m(ymore.)0 |
| 4369 | 3951 y Fd(4.2)112 b(Core)38 b(F)-9 b(unctions)0 4123 |
| 4370 | 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 |
| 4371 | (in)f(cddlib.)51 b(The)33 b(\014rst)g(t)m(yp)s(e)h(runs)f(the)h(double) |
| 4372 | f(description)h(\(DD\))0 4236 y(algorithm)28 b(and)f(do)s(es)g(a)h |
| 4373 | (represen)m(tation)h(con)m(v)m(ersion)f(of)g(a)g(sp)s(eci\014ed)f(p)s |
| 4374 | (olyhedron.)38 b(The)27 b(standard)g(header)g(for)0 4349 |
| 4375 | y(this)32 b(t)m(yp)s(e)g(is)g Fn(dd)p 580 4349 V 34 w(DD*)p |
| 4376 | Fo(.)45 b(The)32 b(second)g(t)m(yp)s(e)g(solv)m(es)h(one)g(or)f(more)g |
| 4377 | (linear)h(programs)e(with)h(no)g(sp)s(ecial)h(headers.)0 |
| 4378 | 4462 y(Both)k(t)m(yp)s(es)f(of)g(computations)h(are)f(non)m(trivial)h |
| 4379 | (and)f(the)g(users)f(\(esp)s(ecially)i(for)f(the)g(DD)h(algorithm\))h |
| 4380 | (m)m(ust)0 4575 y(kno)m(w)e(that)i(there)e(is)h(a)g(serous)f(limit)h |
| 4381 | (in)f(the)h(sizes)g(of)g(problems)e(that)j(can)e(b)s(e)g(practically)i |
| 4382 | (solv)m(ed.)60 b(Please)0 4687 y(c)m(hec)m(k)32 b(*.ext)g(and)d(*.ine)j |
| 4383 | (\014les)e(that)h(come)g(with)f(cddlib)g(to)h(get)h(ideas)e(of)h |
| 4384 | (tractable)h(problems.)141 4800 y(In)k(addition)h(to)g(previously)g |
| 4385 | (de\014ned)e(ob)5 b(jects,)39 b(the)e(sym)m(b)s(ol)f |
| 4386 | Fn(roworder)f Fo(is)h(of)h Fn(dd)p 3056 4800 V 34 w(RowOrderType)p |
| 4387 | Fo(.)56 b(The)0 4913 y(sym)m(b)s(ol)32 b Fn(matrixP)f |
| 4388 | Fo(is)h(a)h(p)s(oin)m(ter)g(to)g Fp(dd)p 1402 4913 32 |
| 4389 | 4 v 38 w(MatrixT)m(yp)s(e)p Fo(.)46 b(the)33 b(argumen)m(ts)g |
| 4390 | Fn(impl)p 2859 4913 29 4 v 33 w(lin)f Fo(and)g Fn(redset)e |
| 4391 | Fo(are)j(b)s(oth)0 5026 y(a)e(p)s(oin)m(ter)f(to)h Fn(dd)p |
| 4392 | 602 5026 V 34 w(rowset)e Fo(t)m(yp)s(e,)i(and)e Fn(newpos)g |
| 4393 | Fo(is)h(a)h(p)s(oin)m(ter)g(to)g Fn(dd)p 2364 5026 V |
| 4394 | 33 w(rowindex)e Fo(t)m(yp)s(e.)0 5207 y Fn(dd)p 102 5207 |
| 4395 | V 34 w(PolyhedraPtr)44 b(dd)p 852 5207 V 34 w(DDMatrix2Poly\(matrix,)e |
| 4396 | (err\))i Fo(:)227 5320 y(Store)32 b(the)g(represen)m(tation)g(giv)m(en) |
| 4397 | h(b)m(y)e Fn(matrix)f Fo(in)h(a)h(p)s(olyhedra)e(data,)j(and)e |
| 4398 | (generate)i(the)e(second)h(rep-)227 5433 y(resen)m(tation)46 |
| 4399 | b(of)e Fn(*poly)p Fo(.)80 b(It)44 b(returns)e(a)j(p)s(oin)m(ter)f(to)g |
| 4400 | (the)g(data.)82 b Fn(*err)43 b Fo(returns)g Fn(dd)p 3269 |
| 4401 | 5433 V 34 w(NoError)f Fo(if)h(the)1926 5682 y(7)p eop |
| 4402 | end |
| 4403 | %%Page: 8 8 |
| 4404 | TeXDict begin 8 7 bop 227 44 a Fo(computation)38 b(terminates)g |
| 4405 | (normally)-8 b(.)60 b(Otherwise,)38 b(it)g(returns)d(a)i(v)-5 |
| 4406 | b(alue)37 b(according)h(to)g(the)f(error)f(o)s(c-)227 |
| 4407 | 157 y(cured.)0 345 y Fn(dd)p 102 345 29 4 v 34 w(PolyhedraPtr)44 |
| 4408 | b(dd)p 852 345 V 34 w(DDMatrix2Poly2\(matrix,)e(roworder,)j(err\))f |
| 4409 | Fo(:)227 458 y(This)f(is)h(the)f(same)h(function)f(as)h |
| 4410 | Fn(dd)p 1559 458 V 34 w(DDMatrix2Poly)c Fo(except)k(that)g(the)g |
| 4411 | (insertion)g(order)f(is)g(sp)s(eci-)227 571 y(\014ed)34 |
| 4412 | b(b)m(y)h(the)g(user.)53 b(The)34 b(argumen)m(t)h Fn(roworder)e |
| 4413 | Fo(is)i(of)f Fn(dd)p 2260 571 V 34 w(RowOrderType)e Fo(and)i(tak)m(es)i |
| 4414 | (one)f(of)g(the)g(v)-5 b(al-)227 683 y(ues:)55 b Fn(dd)p |
| 4415 | 536 683 V 33 w(MaxIndex)p Fo(,)37 b Fn(dd)p 1111 683 |
| 4416 | V 34 w(MinIndex)p Fo(,)g Fn(dd)p 1687 683 V 34 w(MinCutoff)p |
| 4417 | Fo(,)g Fn(dd)p 2311 683 V 34 w(MaxCutoff)p Fo(,)g Fn(dd)p |
| 4418 | 2935 683 V 33 w(MixCutoff)p Fo(,)g Fn(dd)p 3558 683 V |
| 4419 | 34 w(LexMin)p Fo(,)227 796 y Fn(dd)p 329 796 V 34 w(LexMax)p |
| 4420 | Fo(,)d Fn(dd)p 806 796 V 34 w(RandomRow)p Fo(.)51 b(In)33 |
| 4421 | b(general,)k Fn(dd)p 1905 796 V 34 w(LexMin)c Fo(is)h(the)h(b)s(est)f |
| 4422 | (c)m(hoice)j(whic)m(h)d(is)g(in)h(fact)g(c)m(hosen)227 |
| 4423 | 909 y(in)29 b Fn(dd)p 434 909 V 34 w(DDMatrix2Poly)p |
| 4424 | Fo(.)36 b(If)28 b(y)m(ou)i(kno)m(w)e(that)i(the)f(input)e(is)i(already) |
| 4425 | h(sorted)e(in)h(the)g(order)f(y)m(ou)h(lik)m(e,)i(use)227 |
| 4426 | 1022 y Fn(dd)p 329 1022 V 34 w(MinIndex)k Fo(or)i Fn(dd)p |
| 4427 | 996 1022 V 34 w(MaxIndex)p Fo(.)59 b(If)37 b(the)h(input)e(con)m(tains) |
| 4428 | i(man)m(y)g(redundan)m(t)e(ro)m(ws)h(\(sa)m(y)h(more)g(than)227 |
| 4429 | 1135 y(80\045)j(redundan)m(t\),)h(y)m(ou)e(migh)m(t)h(w)m(an)m(t)g(to)g |
| 4430 | (try)f Fn(dd)p 2009 1135 V 34 w(MaxCutoff)d Fo(whic)m(h)j(migh)m(t)h |
| 4431 | (result)f(in)g(m)m(uc)m(h)g(faster)227 1248 y(termination,)32 |
| 4432 | b(see)f([3,)g(16)q(])0 1436 y Fn(boolean)46 b(dd)p 484 |
| 4433 | 1436 V 34 w(DDInputAppend\(poly,)c(matrix,)k(err\))e |
| 4434 | Fo(:)227 1549 y(Mo)s(dify)27 b(the)h(input)e(represen)m(tation)i(in)f |
| 4435 | Fn(*poly)f Fo(b)m(y)h(app)s(ending)f(the)h(matrix)h(of)f |
| 4436 | Fn(*matrix)p Fo(,)g(and)f(compute)227 1661 y(the)36 b(second)f |
| 4437 | (represen)m(tation.)56 b(The)34 b(n)m(um)m(b)s(er)g(of)h(columns)g(in)g |
| 4438 | Fn(*matrix)e Fo(m)m(ust)i(b)s(e)g(equal)g(to)h(the)g(input)227 |
| 4439 | 1774 y(represen)m(tation.)0 1962 y Fn(boolean)46 b(dd)p |
| 4440 | 484 1962 V 34 w(LPSolve\(lp,)e(solver,)i(err\))f Fo(:)227 |
| 4441 | 2075 y(Solv)m(e)28 b Fn(lp)e Fo(b)m(y)h(the)g(algorithm)h |
| 4442 | Fn(solver)d Fo(and)i(sa)m(v)m(e)h(the)f(solututions)g(in)g |
| 4443 | Fn(*lp)p Fo(.)39 b(Unlik)m(e)27 b(the)g(earlier)h(v)m(ersions)227 |
| 4444 | 2188 y(\(dplex,)d(cdd+\),)g(it)f(can)f(deal)h(with)f(equations)h(and)f |
| 4445 | (totally)i(zero)g(righ)m(t)f(hand)e(sides.)38 b(It)24 |
| 4446 | b(is)f(recommended)227 2301 y(that)32 b Fn(solver)d Fo(is)i |
| 4447 | Fn(dd)p 936 2301 V 33 w(DualSimplex)p Fo(,)d(the)j(revised)g(dual)g |
| 4448 | (simplex)f(metho)s(d)h(that)g(up)s(dates)f(a)h Fj(d)21 |
| 4449 | b Fi(\002)f Fj(d)31 b Fo(dual)227 2414 y(basis)g(matrix)f(in)g(eac)m(h) |
| 4450 | i(piv)m(ot)f(\(where)f Fj(d)h Fo(is)f(the)h(column)f(size)h(of)g(lp\).) |
| 4451 | 227 2564 y(The)39 b(revised)f(dual)h(simplex)f(metho)s(d)g(is)h(ideal)h |
| 4452 | (for)e(dense)h(LPs)f(in)g(small)h(n)m(um)m(b)s(er)f(of)h(v)-5 |
| 4453 | b(ariables)39 b(\(i.e.)227 2677 y(small)e(column)g(size,)i(t)m |
| 4454 | (ypically)g(less)e(than)f(100\))i(and)e(man)m(y)h(inequalit)m(y)h |
| 4455 | (constrain)m(ts)g(\(i.e.)61 b(large)37 b(ro)m(w)227 2790 |
| 4456 | y(size,)f(can)e(b)s(e)f(a)h(few)g(ten)g(thousands\).)50 |
| 4457 | b(If)34 b(y)m(our)f(LP)h(has)f(man)m(y)h(v)-5 b(ariables)35 |
| 4458 | b(but)e(only)h(few)f(constrain)m(ts,)227 2903 y(solv)m(e)f(the)f(dual)f |
| 4459 | (LP)g(b)m(y)g(this)g(function.)227 3053 y(When)g(it)h(is)f(compiled)g |
| 4460 | (for)g(GMP)g(rational)h(arithmetic,)h(it)f(\014rst)e(tries)h(to)h(solv) |
| 4461 | m(e)g(an)f(LP)g(with)f(C)h(double)227 3166 y(\015oating-p)s(oin)m(t)d |
| 4462 | (arithmetic)h(and)d(v)m(eri\014es)h(whether)f(the)h(output)g(basis)f |
| 4463 | (is)h(correct)h(with)f(GMP)-8 b(.)27 b(If)e(so,)i(the)227 |
| 4464 | 3279 y(correct)38 b(solution)f(is)f(computed)h(with)f(GMP)-8 |
| 4465 | b(.)37 b(Otherwise,)h(it)f(\(re\)solv)m(es)i(the)d(LP)g(from)g(scratc)m |
| 4466 | (h)i(with)227 3392 y(GMP)-8 b(.)32 b(This)e(is)g(newly)h(implemen)m |
| 4467 | (ted)g(in)f(the)h(v)m(ersion)g(093.)42 b(The)30 b(original)i |
| 4468 | (\(non-crosso)m(v)m(er\))g(v)m(ersion)f(of)227 3505 y(the)g(same)g |
| 4469 | (function)f(is)g(still)h(a)m(v)-5 b(ailable)33 b(as)e |
| 4470 | Fn(boolean)46 b(dd)p 2206 3505 V 33 w(LPSolve0)p Fo(.)0 |
| 4471 | 3692 y Fn(dd)p 102 3692 V 34 w(boolean)g(dd)p 614 3692 |
| 4472 | V 33 w(Redundant\(matrix,)e(i,)j(point,)f(err\))e Fo(:)227 |
| 4473 | 3805 y(Chec)m(k)28 b(whether)g Fj(i)p Fo(th)g(data)g(in)g |
| 4474 | Fn(matrix)e Fo(is)h(redundan)m(t)g(for)h(the)g(represen)m(tation.)41 |
| 4475 | b(If)27 b(it)h(is)g(nonredundan)m(t,)227 3918 y(it)d(returns)e(a)h |
| 4476 | (certi\014cate.)41 b(F)-8 b(or)25 b(H-represen)m(tation,)i(it)e(is)f(a) |
| 4477 | g Fn(point)f Fo(in)h Fj(R)2684 3885 y Fg(d)2748 3918 |
| 4478 | y Fo(whic)m(h)g(satis\014es)g(all)h(inequalities)227 |
| 4479 | 4031 y(except)31 b(for)f(the)g Fj(i)p Fo(th)g(inequalit)m(y)-8 |
| 4480 | b(.)42 b(If)29 b Fj(i)h Fo(is)g(a)g(linearit)m(y)-8 b(,)32 |
| 4481 | b(it)f(do)s(es)e(nothing)h(and)f(alw)m(a)m(ys)i(returns)e |
| 4482 | Fn(dd)p 3606 4031 V 34 w(FALSE)p Fo(.)0 4219 y Fn(dd)p |
| 4483 | 102 4219 V 34 w(rowset)46 b(dd)p 566 4219 V 34 w |
| 4484 | (RedundantRows\(matrix,)c(err\))i Fo(:)227 4331 y(Returns)24 |
| 4485 | b(a)h(maximal)h(set)f(of)g(ro)m(w)g(indices)f(suc)m(h)h(that)g(the)g |
| 4486 | (asso)s(ciated)h(ro)m(ws)f(can)g(b)s(e)f(eliminated)i(without)227 |
| 4487 | 4444 y(c)m(hanging)32 b(the)e(p)s(olyhedron.)40 b(The)29 |
| 4488 | b(function)h(w)m(orks)h(for)f(b)s(oth)g(V-)g(and)g(H-represen)m |
| 4489 | (tations.)0 4632 y Fn(dd)p 102 4632 V 34 w(boolean)46 |
| 4490 | b(dd)p 614 4632 V 33 w(SRedundant\(matrix,)d(i,)k(point,)g(err\))d |
| 4491 | Fo(:)227 4745 y(Chec)m(k)32 b(whether)e Fj(i)p Fo(th)i(data)g(in)e |
| 4492 | Fn(matrix)g Fo(is)h(strongly)h(redundan)m(t)e(for)h(the)g(represen)m |
| 4493 | (tation.)44 b(If)31 b Fj(i)h Fo(is)f(a)g(lin-)227 4858 |
| 4494 | y(earit)m(y)-8 b(,)32 b(it)d(do)s(es)f(nothing)h(and)f(alw)m(a)m(ys)j |
| 4495 | (returns)c Fn(dd)p 2005 4858 V 34 w(FALSE)p Fo(.)h(Here,)h |
| 4496 | Fj(i)p Fo(th)g(inequalit)m(y)i(in)d(H-represen)m(tation)227 |
| 4497 | 4971 y(is)h Ff(str)-5 b(ongly)33 b(r)-5 b(e)g(dundant)39 |
| 4498 | 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 |
| 4499 | (t)h(in)f(the)h(p)s(olyhedron)e(satisfying)i(the)227 |
| 4500 | 5084 y(inequalit)m(y)c(with)e(equalit)m(y)-8 b(.)41 b(In)22 |
| 4501 | b(V-represen)m(tation,)27 b Fj(i)p Fo(th)d(p)s(oin)m(t)f(is)g |
| 4502 | Ff(str)-5 b(ongly)28 b(r)-5 b(e)g(dundant)35 b Fo(if)23 |
| 4503 | b(it)h(is)g(redundan)m(t)227 5197 y(and)31 b(it)g(is)g(in)f(the)i |
| 4504 | (relativ)m(e)h(in)m(terior)e(of)g(the)g(p)s(olyhedron.)41 |
| 4505 | b(If)31 b(it)g(is)g(not)g(strongly)g(redundan)m(t,)f(it)i(returns)227 |
| 4506 | 5309 y(a)f(certi\014cate.)1926 5682 y(8)p eop end |
| 4507 | %%Page: 9 9 |
| 4508 | TeXDict begin 9 8 bop 0 44 a Fn(dd)p 102 44 29 4 v 34 |
| 4509 | w(boolean)46 b(dd)p 614 44 V 33 w(ImplicitLinearity\(matrix,)41 |
| 4510 | b(i,)48 b(err\))c Fo(:)227 157 y(Chec)m(k)33 b(whether)e |
| 4511 | 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 |
| 4512 | (linearit)m(y)j(\(equalit)m(y)g(for)d(H-represen)m(tation\).)227 |
| 4513 | 270 y(If)f Fj(i)h Fo(is)f(linearit)m(y)i(itself,)g(it)e(do)s(es)g |
| 4514 | (nothing)h(and)f(alw)m(a)m(ys)h(returns)f Fn(dd)p 2589 |
| 4515 | 270 V 33 w(FALSE)p Fo(.)0 454 y Fn(dd)p 102 454 V 34 |
| 4516 | w(rowset)46 b(dd)p 566 454 V 34 w(ImplicitLinearityRows\(m)o(atri)o(x,) |
| 4517 | c(err\))i Fo(:)227 567 y(Returns)39 b(the)g(set)h(of)g(indices)g(of)f |
| 4518 | (ro)m(ws)h(that)g(are)g(implicitly)g(linearit)m(y)-8 |
| 4519 | b(.)70 b(It)40 b(simply)f(calls)h(the)g(library)227 680 |
| 4520 | y(function)e Fn(dd)p 693 680 V 34 w(ImplicitLinearity)33 |
| 4521 | b Fo(for)38 b(eac)m(h)h(inequalit)m(y)g(and)f(collects)i(the)e(ro)m(w)g |
| 4522 | (indices)g(for)g(whic)m(h)227 793 y(the)31 b(answ)m(er)f(is)h |
| 4523 | Fn(dd)p 879 793 V 33 w(TRUE)p Fo(.)0 977 y Fn(dd)p 102 |
| 4524 | 977 V 34 w(boolean)46 b(dd)p 614 977 V 33 w |
| 4525 | (MatrixCanonicalize\(matrixP)o(,)c(impl)p 2176 977 V |
| 4526 | 33 w(lin,)47 b(redset,)f(newpos,)f(err\))g Fo(:)227 1090 |
| 4527 | y(The)30 b(input)f(is)h(a)h(p)s(oin)m(ter)f Fn(matrixP)e |
| 4528 | Fo(to)j(a)f(matrix)h(and)e(the)i(function)f(mo)s(di\014es)f(the)h |
| 4529 | (matrix)h(b)m(y)f(putting)227 1203 y(a)40 b(maximally)h(linear)f(indep) |
| 4530 | s(enden)m(t)e(linearities)j(\(basis\))g(at)f(the)g(top)g(of)f(the)h |
| 4531 | (matrix,)j(and)c(remo)m(ving)227 1316 y(all)33 b(redundan)m(t)e(data.) |
| 4532 | 46 b(All)33 b(implicit)g(linearities)g(and)e(all)i(\(remo)m(v)m(ed\))h |
| 4533 | (redundan)m(t)d(ro)m(ws)h(in)f(the)h(original)227 1429 |
| 4534 | y(matrix)c(will)f(b)s(e)f(returned)g(in)g(the)i(corresp)s(onding)e(ro)m |
| 4535 | (w)h(sets.)40 b(The)26 b(new)g(p)s(ositions)h(of)g(the)g(original)i(ro) |
| 4536 | m(ws)227 1542 y(are)i(returned)e(b)m(y)h(the)h(arra)m(y)g |
| 4537 | Fn(newpos)p Fo(.)227 1690 y(The)c(cardinalit)m(y)i(of)f(the)g(new)f |
| 4538 | (linearit)m(y)i(set)f Fn(\(*matrixP\)->linset)22 b Fo(is)28 |
| 4539 | b(the)g(co)s(dimension)f(of)h(the)g(p)s(oly-)227 1803 |
| 4540 | y(hedron)i(if)g(it)h(is)f(H-p)s(olyhedron,)g(and)g(is)g(the)h |
| 4541 | (dimension)e(of)i(linearit)m(y)h(space)f(if)f(it)h(is)f(V-p)s |
| 4542 | (olyhedron.)227 1952 y(Note)e(that)e(the)h(presen)m(t)f(v)m(ersion)g |
| 4543 | (should)g(not)g(b)s(e)f(called)j(a)e(canonicalization)k(b)s(ecause)c |
| 4544 | (it)g(ma)m(y)h(generate)227 2065 y(t)m(w)m(o)j(di\013eren)m(t)e |
| 4545 | (represen)m(tations)h(of)f(the)h(same)f(p)s(olyhedron.)39 |
| 4546 | b(In)27 b(the)h(future,)g(this)g(function)g(is)g(exp)s(ected)227 |
| 4547 | 2178 y(to)j(b)s(e)f(correctly)i(implemen)m(ted.)0 2362 |
| 4548 | y Fn(dd)p 102 2362 V 34 w(boolean)46 b(dd)p 614 2362 |
| 4549 | V 33 w(MatrixCanonicalizeLinearit)o(y\(ma)o(tri)o(xP,)41 |
| 4550 | b(impl)p 2605 2362 V 34 w(linset,)k(newpos.)94 b(err\))44 |
| 4551 | b Fo(:)227 2475 y(It)g(do)s(es)g(only)g(the)g(\014rst)f(half)h(of)g |
| 4552 | Fn(dd)p 1554 2475 V 34 w(boolean)i(dd)p 2066 2475 V 33 |
| 4553 | w(MatrixCanonicalize)p Fo(,)d(namely)-8 b(,)48 b(it)d(detects)g(all)227 |
| 4554 | 2588 y(implicit)27 b(linearities)g(and)f(puts)f(a)h(maximally)h(indep)s |
| 4555 | (enden)m(t)d(linearities)k(at)e(the)g(top)g(of)g(the)g(matrix.)40 |
| 4556 | b(F)-8 b(or)227 2701 y(example,)32 b(this)e(function)g(can)h(b)s(e)e |
| 4557 | (used)h(to)h(detect)h(the)e(dimension)g(of)h(an)f(H-p)s(olyhedron.)0 |
| 4558 | 2885 y Fn(dd)p 102 2885 V 34 w(boolean)46 b(dd)p 614 |
| 4559 | 2885 V 33 w(MatrixRedundancyRemove\(mat)o(rixP)o(,)c(redset,)k(newpos,) |
| 4560 | f(err\))g Fo(:)227 2998 y(It)25 b(do)s(es)e(essen)m(tially)j(the)f |
| 4561 | (second)f(half)g(of)g Fn(dd)p 1757 2998 V 34 w(boolean)46 |
| 4562 | b(dd)p 2269 2998 V 34 w(MatrixCanonicalize)p Fo(,)20 |
| 4563 | b(namely)-8 b(,)27 b(it)d(detects)227 3111 y(all)30 b(redundancies.)39 |
| 4564 | b(This)27 b(function)h(should)g(b)s(e)f(used)h(after)h |
| 4565 | Fn(dd)p 2421 3111 V 34 w(MatrixCanonicalizeLinear)o(ity)22 |
| 4566 | b Fo(has)227 3224 y(b)s(een)30 b(called.)0 3408 y Fn(dd)p |
| 4567 | 102 3408 V 34 w(boolean)46 b(dd)p 614 3408 V 33 w |
| 4568 | (FindRelativeInterior\(matri)o(x,)c(impl)p 2224 3408 |
| 4569 | V 33 w(lin,)k(lin)p 2639 3408 V 34 w(basis,)g(lps,)h(err\))d |
| 4570 | Fo(:)227 3521 y(Computes)39 b(a)h(p)s(oin)m(t)g(in)f(the)h(relativ)m(e) |
| 4571 | h(in)m(terior)f(of)g(an)g(H-p)s(olyhedron)e(giv)m(en)j(b)m(y)e(matrix,) |
| 4572 | 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 |
| 4573 | (will)g(b)s(e)f(returned)f(b)m(y)h Fn(lps)p Fo(.)44 b(See)31 |
| 4574 | b(the)h(sample)g(program)f(allfaces.c)j(that)e(generates)227 |
| 4575 | 3747 y(all)47 b(nonempt)m(y)e(faces)h(of)g(an)f(H-p)s(olyhedron)g(and)g |
| 4576 | (a)h(relativ)m(e)h(in)m(terior)g(p)s(oin)m(t)e(for)g(eac)m(h)i(face.)87 |
| 4577 | b(The)227 3860 y(former)35 b(returns)f(all)i(implicit)g(linearit)m(y)g |
| 4578 | (ro)m(ws)g(\(implicit)g(equations\))g(and)f(the)g(latter)h(returns)e(a) |
| 4579 | i(basis)227 3973 y(of)31 b(the)g(union)e(of)i(linearit)m(y)h(ro)m(ws)f |
| 4580 | (and)f(implicit)h(linearit)m(y)h(ro)m(ws.)41 b(This)30 |
| 4581 | b(means)h(that)g(the)f(cardinalit)m(y)i(of)227 4086 y |
| 4582 | Fn(*lin)p 425 4086 V 34 w(basis)c Fo(is)j(the)f(co)s(dimension)h(of)f |
| 4583 | (the)h(p)s(olyhedron.)0 4270 y Fn(dd)p 102 4270 V 34 |
| 4584 | w(boolean)46 b(dd)p 614 4270 V 33 w(ExistsRestrictedFace\(matri)o(x,)c |
| 4585 | (R,)47 b(S,)g(err\))d Fo(:)227 4383 y(Returns)26 b(the)i(answ)m(er)f |
| 4586 | (to)g(the)h(F)-8 b(ark)j(as')28 b(t)m(yp)s(e)f(decision)h(problem)e(as) |
| 4587 | i(to)f(whether)g(there)g(is)g(a)h(p)s(oin)m(t)f(in)f(the)227 |
| 4588 | 4496 y(p)s(olyhedron)h(giv)m(en)h(b)m(y)g(matrix)g(satisfying)g(all)h |
| 4589 | (constrain)m(ts)f(in)g Fn(R)f Fo(with)g(equalit)m(y)j(and)d(all)h |
| 4590 | (constrain)m(ts)h(in)227 4609 y Fn(S)h Fo(with)g(strict)i(inequalit)m |
| 4591 | (y)-8 b(.)42 b(More)31 b(precisely)-8 b(,)32 b(it)f(is)f(the)h(linear)f |
| 4592 | (feasibilit)m(y)j(problem:)1111 4795 y Fi(9)p Fo(?)82 |
| 4593 | b Fj(x)113 b Fo(satisfying)h Fj(b)1979 4809 y Fg(r)2038 |
| 4594 | 4795 y Fi(\000)19 b Fj(A)2196 4809 y Fg(r)2235 4795 y |
| 4595 | Fj(x)83 b Fo(=)25 b(0)p Fj(;)41 b Fi(8)p Fj(r)27 b Fi(2)e |
| 4596 | Fj(R)c Fi([)f Fj(L)1940 4908 y(b)1979 4922 y Fg(s)2036 |
| 4597 | 4908 y Fi(\000)g Fj(A)2195 4922 y Fg(s)2232 4908 y Fj(x)86 |
| 4598 | b(>)25 b Fo(0)p Fj(;)41 b Fi(8)p Fj(s)24 b Fi(2)h Fj(S)1940 |
| 4599 | 5021 y(b)1979 5035 y Fg(t)2029 5021 y Fi(\000)20 b Fj(A)2188 |
| 4600 | 5035 y Fg(t)2218 5021 y Fj(x)100 b Fi(\025)25 b Fo(0)p |
| 4601 | Fj(;)41 b Fi(8)p Fj(t)24 b Fi(2)h Fj(T)8 b(;)227 5207 |
| 4602 | y Fo(where)31 b Fj(L)g Fo(is)g(the)g(set)h(of)f(linearit)m(y)i(ro)m(ws) |
| 4603 | e(of)g Fn(matrix)p Fo(,)f(and)g Fj(T)44 b Fo(represen)m(ts)31 |
| 4604 | b(the)g(set)h(of)f(ro)m(ws)g(that)h(are)f(not)227 5320 |
| 4605 | y(in)e Fj(R)17 b Fi([)f Fj(L)h Fi([)f Fj(S)5 b Fo(.)40 |
| 4606 | b(Both)29 b Fn(R)g Fo(and)f Fn(S)g Fo(are)h(of)g Fn(dd)p |
| 4607 | 1683 5320 V 33 w(rowset)e Fo(t)m(yp)s(e.)41 b(The)28 |
| 4608 | b(set)h Fj(S)k Fo(is)c(supp)s(osed)e(to)i(b)s(e)f(disjoin)m(t)h(from) |
| 4609 | 227 5433 y(b)s(oth)h Fj(R)h Fo(and)f Fj(L)p Fo(.)40 b(If)30 |
| 4610 | b(it)h(is)f(not)h(the)g(case,)g(the)g(set)g Fj(S)k Fo(will)c(b)s(e)f |
| 4611 | (considered)g(as)g Fj(S)c Fi(n)20 b Fo(\()p Fj(R)i Fi([)d |
| 4612 | Fj(L)p Fo(\).)1926 5682 y(9)p eop end |
| 4613 | %%Page: 10 10 |
| 4614 | TeXDict begin 10 9 bop 227 44 a Fo(This)32 b(function)h(ignores)g |
| 4615 | Fn(matrix->representation)p Fo(,)28 b(and)k(th)m(us)h(ev)m(en)g(if)g |
| 4616 | (it)g(is)g(set)h(to)f Fn(dd)p 3440 44 29 4 v 34 w(Generator)227 |
| 4617 | 157 y Fo(or)e Fn(dd)p 441 157 V 33 w(Unspecified)p Fo(,)d(it)j(treats)g |
| 4618 | (the)g(matrix)g(as)f(if)g(it)h(w)m(ere)g(inequalit)m(y)h(represen)m |
| 4619 | (tation.)0 337 y Fn(dd)p 102 337 V 34 w(boolean)46 b(dd)p |
| 4620 | 614 337 V 33 w(ExistsRestrictedFace2\(matr)o(ix,)41 b(R,)47 |
| 4621 | b(S,)h(lps,)e(err\))f Fo(:)227 450 y(It)27 b(is)h(the)f(same)g(as)h |
| 4622 | (the)f(function)g Fn(dd)p 1505 450 V 34 w(ExistsRestrictedFace)21 |
| 4623 | b Fo(except)28 b(that)g(it)f(returns)f(also)i(a)g(certi\014-)227 |
| 4624 | 562 y(cate)k(for)e(the)h(answ)m(er.)41 b(The)29 b(certi\014cate)k(is)d |
| 4625 | (a)h(solution)g(to)g(the)g(b)s(ounded)d(LP:)891 717 y(\(P\))83 |
| 4626 | b(max)15 b Fj(z)118 b Fo(sub)5 b(ject)30 b(to)114 b Fj(b)1999 |
| 4627 | 731 y Fg(r)2058 717 y Fi(\000)19 b Fj(A)2216 731 y Fg(r)2255 |
| 4628 | 717 y Fj(x)283 b Fo(=)25 b(0)p Fj(;)41 b Fi(8)p Fj(r)27 |
| 4629 | b Fi(2)e Fj(R)c Fi([)f Fj(L)1960 830 y(b)1999 844 y Fg(s)2056 |
| 4630 | 830 y Fi(\000)g Fj(A)2215 844 y Fg(s)2252 830 y Fj(x)86 |
| 4631 | b Fi(\000)p Fj(z)h Fi(\025)25 b Fo(0)p Fj(;)41 b Fi(8)p |
| 4632 | Fj(s)24 b Fi(2)h Fj(S)1960 943 y(b)1999 957 y Fg(t)2049 |
| 4633 | 943 y Fi(\000)20 b Fj(A)2208 957 y Fg(t)2238 943 y Fj(x)300 |
| 4634 | b Fi(\025)25 b Fo(0)p Fj(;)41 b Fi(8)p Fj(t)24 b Fi(2)h |
| 4635 | Fj(T)1960 1056 y Fo(1)385 b Fi(\000)p Fj(z)87 b Fi(\025)25 |
| 4636 | b Fo(0)p Fj(;)227 1228 y Fo(where)31 b Fj(L)g Fo(is)g(the)g(set)h(of)f |
| 4637 | (linearit)m(y)i(ro)m(ws)e(of)g Fn(matrix)p Fo(,)f(and)g |
| 4638 | Fj(T)44 b Fo(represen)m(ts)31 b(the)g(set)h(of)f(ro)m(ws)g(that)h(are)f |
| 4639 | (not)227 1341 y(in)j Fj(R)23 b Fi([)f Fj(L)g Fi([)g Fj(S)5 |
| 4640 | b Fo(.)51 b(The)33 b(answ)m(er)h(for)f(the)h(decision)g(problem)f(is)h |
| 4641 | (YES)f(if)h(and)f(only)h(if)g(the)f(LP)h(attains)h(an)227 |
| 4642 | 1454 y(optimal)c(and)d(the)i(optimal)g(v)-5 b(alue)30 |
| 4643 | b(is)g(p)s(ositiv)m(e.)41 b(The)29 b(dual)g(solution)h(\(either)g(an)f |
| 4644 | (optimal)i(solution)f(or)f(a)227 1567 y(dual)d(un)m(b)s(ounded)e |
| 4645 | (direction\))j(can)g(b)s(e)e(considered)h(as)h(a)g(certi\014cate)h(for) |
| 4646 | e(the)h(NO)f(answ)m(er,)h(if)f(the)h(answ)m(er)227 1679 |
| 4647 | y(is)k(NO.)227 1826 y(This)h(function)h(ignores)g Fn |
| 4648 | (matrix->representation)p Fo(,)28 b(and)k(th)m(us)h(ev)m(en)g(if)g(it)g |
| 4649 | (is)g(set)h(to)f Fn(dd)p 3440 1826 V 34 w(Generator)227 |
| 4650 | 1939 y Fo(or)e Fn(dd)p 441 1939 V 33 w(Unspecified)p |
| 4651 | Fo(,)d(it)j(treats)g(the)g(matrix)g(as)f(if)g(it)h(w)m(ere)g(inequalit) |
| 4652 | m(y)h(represen)m(tation.)0 2118 y Fn(dd)p 102 2118 V |
| 4653 | 34 w(SetFamilyPtr)44 b(dd)p 852 2118 V 34 w(Matrix2Adjacency\(matrix,)d |
| 4654 | (err\))k Fo(:)227 2231 y(Computes)e(the)g(adjacency)g(list)h(of)f |
| 4655 | (input)f(ro)m(ws)g(using)h(the)g(LP)f(solv)m(er)i(and)e(without)h |
| 4656 | (running)e(the)227 2344 y(represen)m(tation)27 b(con)m(v)m(ersion.)41 |
| 4657 | b(When)26 b(the)g(input)f(is)h(H-represen)m(tation,)j(it)d(giv)m(es)h |
| 4658 | (the)g(facet)g(graph)e(of)h(the)227 2457 y(p)s(olyhedron.)37 |
| 4659 | b(F)-8 b(or)24 b(V-represen)m(tation,)j(it)c(giv)m(es)i(the)e(\(v)m |
| 4660 | (ertex\))i(graph)e(of)g(the)h(p)s(olyhedron.)37 b(It)23 |
| 4661 | b(is)g(required)227 2570 y(that)28 b(the)f(input)g(matrix)g(is)g(a)h |
| 4662 | (minimal)f(represen)m(tation.)41 b(Run)26 b(redundancy)g(remo)m(v)-5 |
| 4663 | b(al)28 b(functions)f(b)s(efore)227 2683 y(calling)32 |
| 4664 | b(this)e(function,)h(see)g(the)f(sample)h(co)s(de)f(adjacency)-8 |
| 4665 | b(.c.)0 2862 y Fn(dd)p 102 2862 V 34 w(SetFamilyPtr)44 |
| 4666 | b(dd)p 852 2862 V 34 w(Matrix2WeakAdjacency\(matr)o(ix,)d(err\))k |
| 4667 | Fo(:)227 2975 y(Computes)37 b(the)h(w)m(eak)h(adjacency)g(list)f(of)g |
| 4668 | (input)f(ro)m(ws)h(using)f(the)h(LP)g(solv)m(er)g(and)f(without)h |
| 4669 | (running)227 3088 y(the)31 b(represen)m(tation)h(con)m(v)m(ersion.)44 |
| 4670 | b(When)30 b(the)h(input)f(is)h(H-represen)m(tation,)i(it)e(giv)m(es)i |
| 4671 | (the)e(graph)f(where)227 3201 y(its)g(no)s(des)f(are)h(the)g(facets)h |
| 4672 | (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) |
| 4673 | g(asso)s(ciated)h(facets)f(ha)m(v)m(e)h(some)227 3314 |
| 4674 | y(in)m(tersection.)41 b(F)-8 b(or)26 b(V-represen)m(tation,)j(it)d(giv) |
| 4675 | m(es)h(the)f(graph)f(where)g(its)h(no)s(des)e(are)i(the)g(v)m(ertices)h |
| 4676 | (and)e(t)m(w)m(o)227 3427 y(no)s(des)i(are)g(adjacen)m(t)i(if)e(and)g |
| 4677 | (only)g(if)g(the)h(asso)s(ciated)h(v)m(ertices)g(are)e(on)g(a)h(common) |
| 4678 | g(facet.)41 b(It)27 b(is)g(required)227 3540 y(that)h(the)f(input)g |
| 4679 | (matrix)g(is)g(a)h(minimal)f(represen)m(tation.)41 b(Run)26 |
| 4680 | b(redundancy)g(remo)m(v)-5 b(al)28 b(functions)f(b)s(efore)227 |
| 4681 | 3653 y(calling)32 b(this)e(function,)h(see)g(the)f(sample)h(co)s(de)f |
| 4682 | (adjacency)-8 b(.c.)0 3832 y Fn(dd)p 102 3832 V 34 w(MatrixPtr)45 |
| 4683 | b(dd)p 709 3832 V 34 w(FourierElimination\(matri)o(x,)d(err\))i |
| 4684 | Fo(:)227 3945 y(Eliminate)32 b(the)f(last)h(v)-5 b(ariable)32 |
| 4685 | b(from)f(a)g(system)g(of)g(linear)h(inequalities)g(giv)m(en)g(b)m(y)f |
| 4686 | (matrix)g(b)m(y)g(using)f(the)227 4058 y(F)-8 b(ourier's)21 |
| 4687 | b(Elimination.)38 b(If)20 b(the)h(input)e(matrix)i(is)f(V-represen)m |
| 4688 | (tation,)k Fn(*err)19 b Fo(returns)g Fn(dd)p 3275 4058 |
| 4689 | V 34 w(NotAvailForV)p Fo(.)227 4171 y(This)32 b(function)h(do)s(es)f |
| 4690 | (not)h(remo)m(v)m(e)i(redundancy)c(and)h(one)h(migh)m(t)h(w)m(an)m(t)f |
| 4691 | (to)h(call)g(redundancy)d(remo)m(v)-5 b(al)227 4284 y(functions)30 |
| 4692 | b(afterw)m(ards.)41 b(See)31 b(the)f(sample)h(co)s(de)f(fourier.c.)0 |
| 4693 | 4463 y Fn(dd)p 102 4463 V 34 w(MatrixPtr)45 b(dd)p 709 |
| 4694 | 4463 V 34 w(BlockElimination\(matrix,)c(set,)47 b(err\))d |
| 4695 | Fo(:)227 4576 y(Eliminate)d(a)f(set)g(of)f(v)-5 b(ariables)41 |
| 4696 | b(from)e(a)h(system)f(of)h(linear)g(inequalities)h(giv)m(en)f(b)m(y)g |
| 4697 | (matrix)g(b)m(y)f(using)227 4689 y(the)33 b(extreme)g(ra)m(ys)g(of)g |
| 4698 | (the)g(dual)f(linear)h(system.)48 b(See)33 b(commen)m(ts)g(in)f(the)h |
| 4699 | (co)s(de)g(cddpro)5 b(j.c)32 b(for)g(details.)227 4802 |
| 4700 | 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) |
| 4701 | h(v)-5 b(ariables)28 b(than)e(the)i(rep)s(eated)f(F)-8 |
| 4702 | b(ourierElimination)29 b(when)227 4915 y(the)35 b(n)m(um)m(b)s(er)e(of) |
| 4703 | h(v)-5 b(ariables)35 b(to)g(eliminate)g(is)g(large.)53 |
| 4704 | b(If)33 b(the)i(input)e(matrix)i(is)f(V-represen)m(tation,)j |
| 4705 | Fn(*err)227 5028 y Fo(returns)26 b Fn(dd)p 640 5028 V |
| 4706 | 34 w(NotAvailForV)p Fo(.)d(This)k(function)f(do)s(es)h(not)g(remo)m(v)m |
| 4707 | (e)h(redundancy)d(and)h(one)i(migh)m(t)f(w)m(an)m(t)h(to)227 |
| 4708 | 5141 y(call)k(redundancy)d(remo)m(v)-5 b(al)31 b(functions)f(afterw)m |
| 4709 | (ards.)41 b(See)31 b(the)f(sample)h(co)s(de)f(pro)5 b(jection.c.)0 |
| 4710 | 5320 y Fn(dd)p 102 5320 V 34 w(rowrange)45 b(dd)p 661 |
| 4711 | 5320 V 34 w(RayShooting\(matrix,)e(point,)j(vector\))d |
| 4712 | Fo(:)227 5433 y(Finds)32 b(the)g(index)g(of)g(a)h(halfspace)f(\014rst)g |
| 4713 | (left)g(b)m(y)g(the)h(ra)m(y)f(starting)h(from)f Fn(point)f |
| 4714 | Fo(to)m(w)m(ard)i(the)f(direction)1903 5682 y(10)p eop |
| 4715 | end |
| 4716 | %%Page: 11 11 |
| 4717 | TeXDict begin 11 10 bop 227 44 a Fn(vector)p Fo(.)39 |
| 4718 | b(It)28 b(resolv)m(es)i(tie)f(b)m(y)f(a)h(lexicographic)h(p)s |
| 4719 | (erturbation.)39 b(Those)28 b(inequalities)i(violated)g(b)m(y)e |
| 4720 | Fn(point)227 157 y Fo(will)j(b)s(e)f(simply)g(ignored.)0 |
| 4721 | 401 y Fd(4.3)112 b(Data)38 b(Manipulations)0 572 y Fp(4.3.1)105 |
| 4722 | b(Num)m(b)s(er)35 b(Assignmen)m(ts)0 744 y Fo(F)-8 b(or)31 |
| 4723 | b(n)m(um)m(b)s(er)d(assignmen)m(ts,)j(one)f(cannot)h(use)e(suc)m(h)h |
| 4724 | (expressions)f(as)h Fn(x=\(mytype\)a)p Fo(.)38 b(This)29 |
| 4725 | b(is)g(b)s(ecause)h(cddlib)0 857 y(uses)37 b(an)g(abstract)h(n)m(um)m |
| 4726 | (b)s(er)e(t)m(yp)s(e)h(\()p Fn(mytype)p Fo(\))f(so)i(that)f(it)h(can)f |
| 4727 | (compute)h(with)f(v)-5 b(arious)37 b(n)m(um)m(b)s(er)f(t)m(yp)s(es)h |
| 4728 | (suc)m(h)0 970 y(as)c(C)f(double)g(and)h(GMP)g(rational.)49 |
| 4729 | 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 |
| 4730 | (e)i(b)m(y)g(rede\014ning)e(arithmetic)0 1082 y(op)s(erations)g(in)f |
| 4731 | (cddmp.h)e(and)i(cddmp.c.)0 1295 y Fn(void)47 b(dd)p |
| 4732 | 341 1295 29 4 v 33 w(init\(x\))d Fo(:)227 1408 y(This)36 |
| 4733 | b(is)g(to)h(initialize)h(a)f Fn(mytype)d Fo(v)-5 b(ariable)37 |
| 4734 | b Fn(x)f Fo(and)g(to)h(set)f(it)h(to)g(zero.)59 b(This)35 |
| 4735 | b(initialization)k(has)d(to)h(b)s(e)227 1521 y(called)32 |
| 4736 | b(b)s(efore)e(an)m(y)h Fn(mytype)d Fo(v)-5 b(ariable)31 |
| 4737 | b(to)g(b)s(e)f(used.)0 1708 y Fn(void)47 b(dd)p 341 1708 |
| 4738 | V 33 w(clear\(x\))d Fo(:)227 1821 y(This)30 b(is)g(to)h(free)g(the)f |
| 4739 | (space)h(allo)s(cated)i(to)e(a)g Fn(mytype)d Fo(v)-5 |
| 4740 | b(ariable)31 b Fn(x)p Fo(.)0 2009 y Fn(void)47 b(dd)p |
| 4741 | 341 2009 V 33 w(set)p 518 2009 V 34 w(si\(x,)f(a\))f |
| 4742 | Fo(:)227 2122 y(This)30 b(is)g(to)h(set)g(a)g Fn(mytype)e |
| 4743 | Fo(v)-5 b(ariable)31 b Fn(x)f Fo(to)h(the)g(v)-5 b(alue)30 |
| 4744 | b(of)h(signed)f(long)h(in)m(teger)h Fn(a)p Fo(.)0 2310 |
| 4745 | y Fn(void)47 b(dd)p 341 2310 V 33 w(set)p 518 2310 V |
| 4746 | 34 w(si2\(x,)f(a,)h(b\))e Fo(:)227 2422 y(This)24 b(is)g(to)i(set)f(a)f |
| 4747 | Fn(mytype)f Fo(v)-5 b(ariable)25 b Fn(x)f Fo(to)i(the)e(v)-5 |
| 4748 | b(alue)25 b(of)g(the)f(rational)i(expression)e Fn(a/b)p |
| 4749 | Fo(,)h(where)f Fn(a)g Fo(is)h(signed)227 2535 y(long)31 |
| 4750 | b(and)f Fn(b)g Fo(is)g(unsigned)g(long)h(in)m(tegers.)0 |
| 4751 | 2723 y Fn(void)47 b(dd)p 341 2723 V 33 w(set)p 518 2723 |
| 4752 | V 34 w(d\(x,)g(b\))e Fo(:)227 2836 y(This)33 b(is)g(to)h(set)f(a)h |
| 4753 | Fn(mytype)d Fo(v)-5 b(ariable)34 b Fn(x)f Fo(to)h(the)f(v)-5 |
| 4754 | b(alue)34 b(of)f(double)g Fn(b)p Fo(.)49 b(This)32 b(is)h(a)m(v)-5 |
| 4755 | b(ailable)36 b(only)d(when)f(the)227 2949 y(library)e(is)h(compiled)f |
| 4756 | (without)h Fn(-DGMPRATIONAL)c Fo(compiler)k(option.)0 |
| 4757 | 3189 y Fp(4.3.2)105 b(Arithmetic)35 b(Op)s(erations)g(for)g |
| 4758 | Fn(mytype)e Fp(Num)m(b)s(ers)0 3361 y Fo(Belo)m(w)f Fn(x)p |
| 4759 | Fo(,)e Fn(y)p Fo(,)h Fn(z)f Fo(are)g(of)h(t)m(yp)s(e)f |
| 4760 | Fn(mytype)p Fo(.)0 3573 y Fn(void)47 b(dd)p 341 3573 |
| 4761 | V 33 w(add\(x,)g(y,)g(z\))e Fo(:)227 3686 y(Set)31 b |
| 4762 | Fn(x)f Fo(to)h(b)s(e)f(the)g(sum)g(of)g Fn(y)g Fo(and)g |
| 4763 | Fn(z)p Fo(.)0 3874 y Fn(void)47 b(dd)p 341 3874 V 33 |
| 4764 | w(sub\(x,)g(y,)g(z\))e Fo(:)227 3987 y(Set)31 b Fn(x)f |
| 4765 | Fo(to)h(b)s(e)f(the)g(substraction)h(of)f Fn(z)g Fo(from)g |
| 4766 | Fn(y)p Fo(.)0 4174 y Fn(void)47 b(dd)p 341 4174 V 33 |
| 4767 | w(mul\(x,)g(y,)g(z\))e Fo(:)227 4287 y(Set)31 b Fn(x)f |
| 4768 | Fo(to)h(b)s(e)f(the)g(m)m(ultiplication)j(of)d Fn(y)g |
| 4769 | Fo(and)g Fn(z)p Fo(.)0 4475 y Fn(void)47 b(dd)p 341 4475 |
| 4770 | V 33 w(div\(x,)g(y,)g(z\))e Fo(:)227 4588 y(Set)31 b |
| 4771 | 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 |
| 4772 | Fn(z)p Fo(.)0 4775 y Fn(void)47 b(dd)p 341 4775 V 33 |
| 4773 | 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 |
| 4774 | (reciplo)s(cal)i(of)f Fn(y)p Fo(.)1903 5682 y(11)p eop |
| 4775 | end |
| 4776 | %%Page: 12 12 |
| 4777 | TeXDict begin 12 11 bop 0 44 a Fp(4.3.3)105 b(Prede\014ned)36 |
| 4778 | b(Constan)m(ts)0 216 y Fo(There)24 b(are)h(sev)m(eral)g |
| 4779 | Fn(mytype)e Fo(constan)m(ts)i(de\014ned)f(when)f Fn(dd)p |
| 4780 | 2043 216 29 4 v 34 w(set)p 2221 216 V 33 w(global)p 2542 |
| 4781 | 216 V 33 w(constants\(void\))d Fo(is)25 b(called.)40 |
| 4782 | b(Some)0 329 y(constan)m(ts)35 b(dep)s(end)d(on)i(the)g(double)g |
| 4783 | (constan)m(t)h Fn(dd)p 1784 329 V 34 w(almostzero)c Fo(whic)m(h)i(is)h |
| 4784 | (normally)g(set)h(to)g(10)3417 296 y Fc(\000)p Fh(7)3546 |
| 4785 | 329 y Fo(in)e(cdd.h.)0 442 y(This)28 b(v)-5 b(alue)29 |
| 4786 | b(can)f(b)s(e)g(mo)s(di\014ed)f(dep)s(ending)g(on)i(ho)m(w)f(n)m |
| 4787 | (umerically)h(delicate)i(y)m(our)d(problems)g(are)h(but)f(an)g(extra)0 |
| 4788 | 555 y(caution)j(should)f(b)s(e)f(tak)m(en.)0 767 y Fn(mytype)46 |
| 4789 | b(dd)p 436 767 V 34 w(purezero)d Fo(:)227 880 y(This)30 |
| 4790 | b(represen)m(ts)g(the)h(mathematical)h(zero)g(0.)0 1068 |
| 4791 | y Fn(mytype)46 b(dd)p 436 1068 V 34 w(zero)e Fo(:)227 |
| 4792 | 1181 y(This)36 b(represen)m(ts)g(the)h(largest)h(p)s(ositiv)m(e)f(n)m |
| 4793 | (um)m(b)s(er)e(whic)m(h)h(should)g(b)s(e)g(considered)g(to)h(b)s(e)f |
| 4794 | (zero.)60 b(In)35 b(the)227 1293 y(GMPRA)-8 b(TIONAL)34 |
| 4795 | b(mo)s(de,)h(it)f(is)g(equal)g(to)g Fn(dd)p 1890 1293 |
| 4796 | V 34 w(purezero)p Fo(.)49 b(In)33 b(the)h(C)f(double)g(mo)s(de,)i(it)f |
| 4797 | (is)g(set)g(to)g(the)227 1406 y(v)-5 b(alue)31 b(of)g |
| 4798 | Fn(dd)p 668 1406 V 33 w(almostzero)p Fo(.)0 1594 y Fn(mytype)46 |
| 4799 | b(dd)p 436 1594 V 34 w(minuszero)d Fo(:)227 1707 y(The)30 |
| 4800 | b(negativ)m(e)j(of)d Fn(dd)p 978 1707 V 34 w(zero)p Fo(.)0 |
| 4801 | 1895 y Fn(mytype)46 b(dd)p 436 1895 V 34 w(one)f Fo(:)227 |
| 4802 | 2007 y(This)30 b(represen)m(ts)g(the)h(mathematical)h(one)f(1.)0 |
| 4803 | 2248 y Fp(4.3.4)105 b(Sign)36 b(Ev)-6 b(aluation)34 b(and)h(Comparison) |
| 4804 | h(for)f Fn(mytype)e Fp(Num)m(b)s(ers)0 2419 y Fo(Belo)m(w)f |
| 4805 | Fn(x)p Fo(,)e Fn(y)p Fo(,)h Fn(z)f Fo(are)g(of)h(t)m(yp)s(e)f |
| 4806 | Fn(mytype)p Fo(.)0 2632 y Fn(dd)p 102 2632 V 34 w(boolean)46 |
| 4807 | b(dd)p 614 2632 V 33 w(Positive\(x\))d Fo(:)227 2745 |
| 4808 | y(Returns)23 b Fn(dd)p 668 2745 V 34 w(TRUE)f Fo(if)h |
| 4809 | Fn(x)g Fo(is)g(considered)h(p)s(ositiv)m(e,)i(and)c Fn(dd)p |
| 4810 | 2207 2745 V 34 w(FALSE)g Fo(otherwise.)39 b(In)23 b(the)g(GMPRA)-8 |
| 4811 | b(TIONAL)227 2858 y(mo)s(de,)23 b(the)d(p)s(ositivit)m(y)i(recognition) |
| 4812 | g(is)f(exact.)39 b(In)20 b(the)h(C)f(double)g(mo)s(de,)i(this)f(means)f |
| 4813 | (the)h(v)-5 b(alue)21 b(is)g(strictly)227 2970 y(larger)31 |
| 4814 | b(than)f Fn(dd)p 799 2970 V 34 w(zero)p Fo(.)227 3121 |
| 4815 | y Fn(dd)p 329 3121 V 34 w(boolean)46 b(dd)p 841 3121 |
| 4816 | V 34 w(Negative\(x\))27 b Fo(w)m(orks)j(similarly)-8 |
| 4817 | b(.)0 3308 y Fn(dd)p 102 3308 V 34 w(boolean)46 b(dd)p |
| 4818 | 614 3308 V 33 w(Nonpositive\(x\))c Fo(:)227 3421 y(Returns)30 |
| 4819 | b(the)g(negation)i(of)f Fn(dd)p 1304 3421 V 33 w(Positive\(x\))p |
| 4820 | Fo(.)38 b Fn(dd)p 2024 3421 V 34 w(Nonnegative\(x\))26 |
| 4821 | b Fo(w)m(orks)31 b(similarly)-8 b(.)0 3609 y Fn(dd)p |
| 4822 | 102 3609 V 34 w(boolean)46 b(dd)p 614 3609 V 33 w(EqualToZero\(x\))c |
| 4823 | Fo(:)227 3722 y(Returns)34 b Fn(dd)p 679 3722 V 34 w(TRUE)g |
| 4824 | Fo(if)h Fn(x)g Fo(is)g(considered)g(zero,)i(and)d Fn(dd)p |
| 4825 | 2156 3722 V 34 w(FALSE)g Fo(otherwise.)55 b(In)34 b(the)h(GMPRA)-8 |
| 4826 | b(TIONAL)227 3835 y(mo)s(de,)28 b(the)f(zero)h(recognition)h(is)f |
| 4827 | (exact.)41 b(In)26 b(the)i(C)e(double)h(mo)s(de,)h(this)f(means)g(the)h |
| 4828 | (v)-5 b(alue)27 b(is)h(in)m(b)s(et)m(w)m(een)227 3948 |
| 4829 | y Fn(dd)p 329 3948 V 34 w(minuszero)g Fo(and)i Fn(dd)p |
| 4830 | 1096 3948 V 34 w(zero)f Fo(inclusiv)m(e.)0 4135 y Fn(dd)p |
| 4831 | 102 4135 V 34 w(boolean)46 b(dd)p 614 4135 V 33 w(Larger\(x,)g(y\))f |
| 4832 | Fo(:)227 4248 y(Returns)31 b Fn(dd)p 676 4248 V 33 w(TRUE)f |
| 4833 | Fo(if)i Fn(x)f Fo(is)g(strictly)h(larger)g(than)f Fn(y)p |
| 4834 | Fo(,)g(and)g Fn(dd)p 2349 4248 V 34 w(FALSE)f Fo(otherwise.)43 |
| 4835 | b(This)31 b(is)g(implemen)m(ted)227 4361 y(as)f(dd)p |
| 4836 | 445 4361 28 4 v 32 w(P)m(ositiv)m(e\(z\))j(where)d Fn(z)f |
| 4837 | Fo(is)h(the)g(subtraction)g(of)g Fn(y)g Fo(from)f Fn(x)p |
| 4838 | Fo(.)40 b Fn(dd)p 2602 4361 29 4 v 34 w(Smaller\(x,)45 |
| 4839 | b(y\))29 b Fo(w)m(orks)h(similarly)-8 b(.)0 4549 y Fn(dd)p |
| 4840 | 102 4549 V 34 w(boolean)46 b(dd)p 614 4549 V 33 w(Equal\(x,)g(y\))f |
| 4841 | Fo(:)227 4662 y(Returns)30 b Fn(dd)p 675 4662 V 33 w(TRUE)g |
| 4842 | Fo(if)g Fn(x)g Fo(is)g(considered)g(equal)h(to)g Fn(y)p |
| 4843 | Fo(,)g(and)e Fn(dd)p 2354 4662 V 34 w(FALSE)g Fo(otherwise.)41 |
| 4844 | b(This)30 b(is)g(implemen)m(ted)227 4774 y(as)h(dd)p |
| 4845 | 446 4774 28 4 v 32 w(EqualT)-8 b(oZero\(z\))32 b(where)e |
| 4846 | Fn(z)g Fo(is)g(the)h(subtraction)f(of)h Fn(y)f Fo(from)g |
| 4847 | Fn(x)p Fo(.)0 5015 y Fp(4.3.5)105 b(P)m(olyhedra)36 b(Data)f |
| 4848 | (Manipulation)0 5186 y Fn(dd)p 102 5186 29 4 v 34 w(MatrixPtr)45 |
| 4849 | b(dd)p 709 5186 V 34 w(PolyFile2Matrix)f(\(f,)i(err\))f |
| 4850 | Fo(:)227 5299 y(Read)28 b(a)f(P)m(olyhedra)h(data)g(from)e(stream)i |
| 4851 | Fn(f)f Fo(and)f(store)i(it)g(in)f Fn(matrixdata)d Fo(and)j(return)f(a)h |
| 4852 | (p)s(oin)m(ter)g(to)h(the)227 5412 y(data.)1903 5682 |
| 4853 | y(12)p eop end |
| 4854 | %%Page: 13 13 |
| 4855 | TeXDict begin 13 12 bop 0 44 a Fn(dd)p 102 44 29 4 v |
| 4856 | 34 w(MatrixPtr)45 b(dd)p 709 44 V 34 w(CopyInequalities\(poly\))39 |
| 4857 | b Fo(:)227 157 y(Cop)m(y)26 b(the)f(inequalit)m(y)i(represen)m(tation)g |
| 4858 | (p)s(oin)m(ted)e(b)m(y)g(p)s(oly)g(to)h Fn(matrixdata)d |
| 4859 | Fo(and)i(return)f Fn(dd)p 3415 157 V 34 w(MatrixPtr)p |
| 4860 | Fo(.)0 345 y Fn(dd)p 102 345 V 34 w(MatrixPtr)45 b(dd)p |
| 4861 | 709 345 V 34 w(CopyGenerators\(poly\))40 b Fo(:)227 458 |
| 4862 | y(Cop)m(y)27 b(the)g(generator)i(represen)m(tation)f(p)s(oin)m(ted)e(b) |
| 4863 | m(y)h(p)s(oly)g(to)h Fn(matrixdata)c Fo(and)i(return)g |
| 4864 | Fn(dd)p 3415 458 V 34 w(MatrixPtr)p Fo(.)0 645 y Fn(dd)p |
| 4865 | 102 645 V 34 w(SetFamilyPtr)44 b(dd)p 852 645 V 34 w |
| 4866 | (CopyIncidence\(poly\))c Fo(:)227 758 y(Cop)m(y)21 b(the)f(incidence)h |
| 4867 | (represen)m(tation)g(of)g(the)f(computed)g(represen)m(tation)i(p)s(oin) |
| 4868 | m(ted)e(b)m(y)g(p)s(oly)g(to)h Fn(setfamily)227 871 y |
| 4869 | Fo(and)38 b(return)f Fn(dd)p 800 871 V 34 w(SetFamilyPtr)p |
| 4870 | Fo(.)62 b(The)37 b(computed)i(represen)m(tation)g(is)g |
| 4871 | Fn(Inequality)c Fo(if)k(the)f(input)g(is)227 984 y Fn(Generator)p |
| 4872 | Fo(,)28 b(and)i(the)h(vice)g(visa.)0 1172 y Fn(dd)p 102 |
| 4873 | 1172 V 34 w(SetFamilyPtr)44 b(dd)p 852 1172 V 34 w |
| 4874 | (CopyAdjacency\(poly\))c Fo(:)227 1285 y(Cop)m(y)21 b(the)f(adjacency)h |
| 4875 | (represen)m(tation)g(of)g(the)f(computed)g(represen)m(tation)i(p)s(oin) |
| 4876 | m(ted)e(b)m(y)g(p)s(oly)g(to)h Fn(setfamily)227 1397 |
| 4877 | y Fo(and)38 b(return)f Fn(dd)p 800 1397 V 34 w(SetFamilyPtr)p |
| 4878 | Fo(.)62 b(The)37 b(computed)i(represen)m(tation)g(is)g |
| 4879 | Fn(Inequality)c Fo(if)k(the)f(input)g(is)227 1510 y Fn(Generator)p |
| 4880 | Fo(,)28 b(and)i(the)h(vice)g(visa.)0 1698 y Fn(dd)p 102 |
| 4881 | 1698 V 34 w(SetFamilyPtr)44 b(dd)p 852 1698 V 34 w |
| 4882 | (CopyInputIncidence\(poly\))39 b Fo(:)227 1811 y(Cop)m(y)30 |
| 4883 | b(the)g(incidence)h(represen)m(tation)g(of)f(the)g(input)f(represen)m |
| 4884 | (tation)h(p)s(oin)m(ted)g(b)m(y)g(p)s(oly)f(to)i Fn(setfamily)227 |
| 4885 | 1924 y Fo(and)f(return)f Fn(d)p 736 1924 V 34 w(SetFamilyPtr)p |
| 4886 | Fo(.)0 2111 y Fn(dd)p 102 2111 V 34 w(SetFamilyPtr)44 |
| 4887 | b(dd)p 852 2111 V 34 w(CopyInputAdjacency\(poly\))39 |
| 4888 | b Fo(:)227 2224 y(Cop)m(y)28 b(the)g(adjacency)h(represen)m(tation)f |
| 4889 | (of)g(the)g(input)f(represen)m(tation)i(p)s(oin)m(ted)f(b)m(y)f(p)s |
| 4890 | (oly)h(to)g Fn(setfamily)227 2337 y Fo(and)i(return)f |
| 4891 | Fn(d)p 736 2337 V 34 w(SetFamilyPtr)p Fo(.)0 2525 y Fn(void)47 |
| 4892 | b(dd)p 341 2525 V 33 w(FreePolyhedra\(poly\))41 b Fo(:)227 |
| 4893 | 2638 y(F)-8 b(ree)32 b(memory)e(allo)s(cated)i(to)f Fn(poly)p |
| 4894 | Fo(.)0 2878 y Fp(4.3.6)105 b(LP)36 b(Data)e(Manipulation)0 |
| 4895 | 3050 y Fn(dd)p 102 3050 V 34 w(LPPtr)46 b(dd)p 518 3050 |
| 4896 | V 34 w(MakeLPforInteriorFinding)o(\(lp\))39 b Fo(:)227 |
| 4897 | 3162 y(Set)29 b(up)f(an)g(LP)g(to)i(\014nd)d(an)h(in)m(terior)i(p)s |
| 4898 | (oin)m(t)e(of)h(the)g(feasible)g(region)g(of)g Fn(lp)f |
| 4899 | Fo(and)g(return)f(a)i(p)s(oin)m(ter)g(to)g(the)227 3275 |
| 4900 | y(LP)-8 b(.)36 b(The)f(new)g(LP)g(has)h(one)g(new)f(v)-5 |
| 4901 | b(ariable)36 b Fj(x)1867 3290 y Fg(d)p Fh(+1)2033 3275 |
| 4902 | y Fo(and)f(one)h(more)g(constrain)m(t:)52 b(max)15 b |
| 4903 | Fj(x)3329 3290 y Fg(d)p Fh(+1)3495 3275 y Fo(sub)5 b(ject)36 |
| 4904 | b(to)227 3388 y Fj(b)20 b Fi(\000)g Fj(Ax)h Fi(\000)f |
| 4905 | Fj(x)661 3403 y Fg(d)p Fh(+1)816 3388 y Fi(\025)25 b |
| 4906 | Fo(0)31 b(and)f Fj(x)1217 3403 y Fg(d)p Fh(+1)1373 3388 |
| 4907 | y Fi(\024)25 b Fj(K)7 b Fo(,)30 b(where)g Fj(K)37 b Fo(is)30 |
| 4908 | b(a)h(p)s(ositiv)m(e)g(constan)m(t.)0 3576 y Fn(dd)p |
| 4909 | 102 3576 V 34 w(LPPtr)46 b(dd)p 518 3576 V 34 w(Matrix2LP\(matrix,)d |
| 4910 | (err\))h Fo(:)227 3689 y(Load)31 b Fn(matrix)d Fo(to)k |
| 4911 | Fn(lpdata)c Fo(and)i(return)f(a)i(p)s(oin)m(ter)f(to)i(the)e(data.)0 |
| 4912 | 3876 y Fn(dd)p 102 3876 V 34 w(LPSolutionPtr)44 b(dd)p |
| 4913 | 900 3876 V 34 w(CopyLPSolution\(lp\))c Fo(:)227 3989 |
| 4914 | y(Load)32 b(the)g(solutions)g(of)g Fn(lp)f Fo(to)i Fn(lpsolution)c |
| 4915 | Fo(and)i(return)g(a)h(p)s(oin)m(ter)g(to)g(the)g(data.)46 |
| 4916 | b(This)31 b(replaces)h(the)227 4102 y(old)f(name)f Fn(dd)p |
| 4917 | 723 4102 V 34 w(LPSolutionLoad\(lp\))p Fo(.)0 4290 y |
| 4918 | Fn(void)47 b(dd)p 341 4290 V 33 w(FreeLPData\(lp\))42 |
| 4919 | b Fo(:)227 4403 y(F)-8 b(ree)32 b(memory)e(allo)s(cated)i(as)f(an)f(LP) |
| 4920 | g(data)h(p)s(oin)m(ted)g(b)m(y)f Fn(lp)p Fo(.)0 4590 |
| 4921 | y Fn(void)47 b(dd)p 341 4590 V 33 w(FreeLPSolution\(lps\))41 |
| 4922 | b Fo(:)227 4703 y(F)-8 b(ree)32 b(memory)e(allo)s(cated)i(as)f(an)f(LP) |
| 4923 | g(solution)h(data)g(p)s(oin)m(ted)g(b)m(y)f Fn(lps)p |
| 4924 | Fo(.)0 4943 y Fp(4.3.7)105 b(Matrix)35 b(Manipulation)0 |
| 4925 | 5115 y Fn(dd)p 102 5115 V 34 w(MatrixPtr)45 b(dd)p 709 |
| 4926 | 5115 V 34 w(CopyMatrix\(matrix\))40 b Fo(:)227 5228 y(Mak)m(e)32 |
| 4927 | b(a)f(cop)m(y)g(of)g(matrixdata)g(p)s(oin)m(ted)f(b)m(y)h |
| 4928 | Fn(matrix)d Fo(and)i(return)f(a)i(p)s(oin)m(ter)g(to)g(the)f(cop)m(y)-8 |
| 4929 | b(.)1903 5682 y(13)p eop end |
| 4930 | %%Page: 14 14 |
| 4931 | TeXDict begin 14 13 bop 0 44 a Fn(dd)p 102 44 29 4 v |
| 4932 | 34 w(MatrixPtr)45 b(dd)p 709 44 V 34 w(AppendMatrix\(matrix1,)d |
| 4933 | (matrix2\))h Fo(:)227 157 y(Mak)m(e)32 b(a)e(matrixdata)h(b)m(y)e(cop)m |
| 4934 | (ying)i Fn(*matrix1)c Fo(and)i(app)s(ending)g(the)h(matrix)g(in)f |
| 4935 | Fn(*matrix2)f Fo(and)h(return)227 270 y(a)39 b(p)s(oin)m(ter)g(to)h |
| 4936 | (the)f(data.)67 b(The)38 b(colsize)j(m)m(ust)d(b)s(e)h(equal)g(in)g |
| 4937 | (the)g(t)m(w)m(o)h(input)e(matrices.)67 b(It)39 b(returns)e(a)227 |
| 4938 | 383 y Fn(NULL)j Fo(p)s(oin)m(ter)g(if)h(the)g(input)e(matrices)j(are)f |
| 4939 | (not)g(appropriate.)71 b(Its)41 b Fn(rowsize)e Fo(is)h(set)i(to)f(the)g |
| 4940 | (sum)e(of)227 496 y(the)k(ro)m(wsizes)h(of)e Fn(matrix1)f |
| 4941 | Fo(and)h Fn(matrix2)p Fo(.)75 b(The)42 b(new)g(matrixdata)i(inherits)e |
| 4942 | (ev)m(erything)i(else)f(\(i.e.)227 609 y(n)m(um)m(b)s(ert)m(yp)s(e,)30 |
| 4943 | b(represen)m(tation,)i(etc\))f(from)f(the)h(\014rst)e(matrix.)0 |
| 4944 | 794 y Fn(int)47 b(dd)p 293 794 V 34 w(MatrixAppendTo\(&)c(matrix1,)j |
| 4945 | (matrix2\))d Fo(:)227 907 y(Same)31 b(as)f Fn(dd)p 683 |
| 4946 | 907 V 34 w(AppendMatrix)d Fo(except)32 b(that)e(the)h(\014rst)f(matrix) |
| 4947 | g(is)h(mo)s(di\014ed)e(to)i(tak)m(e)h(the)f(result.)0 |
| 4948 | 1092 y Fn(int)47 b(dd)p 293 1092 V 34 w(MatrixRowRemove\(&)c(matrix,)j |
| 4949 | (i\))f Fo(:)227 1205 y(Remo)m(v)m(e)32 b(the)f Fj(i)p |
| 4950 | Fo(th)f(ro)m(w)h(of)g Fn(matrix)p Fo(.)0 1391 y Fn(dd)p |
| 4951 | 102 1391 V 34 w(MatrixPtr)45 b(dd)p 709 1391 V 34 w |
| 4952 | (MatrixSubmatrix\(matrix,)c(set\))k Fo(:)227 1504 y(Generate)36 |
| 4953 | b(the)f(submatrix)f(of)h Fn(matrix)e Fo(b)m(y)i(remo)m(ving)h(the)f(ro) |
| 4954 | m(ws)f(indexed)h(b)m(y)f Fn(set)g Fo(and)g(return)g(a)h(ma-)227 |
| 4955 | 1617 y(trixdata)d(p)s(oin)m(ter.)0 1802 y Fn(dd)p 102 |
| 4956 | 1802 V 34 w(SetFamilyPtr)44 b(dd)p 852 1802 V 34 w |
| 4957 | (Matrix2Adjacency\(matrix,)d(err\))k Fo(:)227 1915 y(Return)25 |
| 4958 | b(the)i(adjacency)f(list)h(of)f(the)g(represen)m(tation)h(giv)m(en)g(b) |
| 4959 | m(y)f Fn(matrix)p Fo(.)38 b(The)25 b(computation)i(is)f(done)g(b)m(y) |
| 4960 | 227 2028 y(the)h(built-in)g(LP)f(solv)m(er.)41 b(The)26 |
| 4961 | b(represen)m(tation)i(should)e(b)s(e)g(free)h(of)g(redundancy)e(when)g |
| 4962 | (this)i(function)g(is)227 2141 y(called.)40 b(See)22 |
| 4963 | b(the)h(function)f Fn(dd)p 1270 2141 V 33 w(rowset)47 |
| 4964 | b(dd)p 1734 2141 V 33 w(RedundantRows)19 b Fo(and)j(the)g(example)h |
| 4965 | (program)f(adjacency)-8 b(.c.)0 2383 y Fd(4.4)112 b(Input/Output)38 |
| 4966 | b(F)-9 b(unctions)0 2555 y Fn(dd)p 102 2555 V 34 w(MatrixPtr)45 |
| 4967 | b(dd)p 709 2555 V 34 w(PolyFile2Matrix)f(\(f,)i(err\))f |
| 4968 | Fo(:)227 2668 y(Read)28 b(a)f(P)m(olyhedra)h(data)g(from)e(stream)i |
| 4969 | Fn(f)f Fo(and)f(store)i(it)g(in)f Fn(matrixdata)d Fo(and)j(return)f(a)h |
| 4970 | (p)s(oin)m(ter)g(to)h(the)227 2780 y(data.)0 2966 y Fn(boolean)46 |
| 4971 | b(dd)p 484 2966 V 34 w(DDFile2File\(ifile,)c(ofile,)47 |
| 4972 | b(err\))d Fo(:)227 3079 y(Compute)33 b(the)g(represen)m(tation)h(con)m |
| 4973 | (v)m(ersion)g(for)e(a)i(p)s(olyhedron)d(giv)m(en)j(b)m(y)f(a)g(P)m |
| 4974 | (olyhedra)g(\014le)g(i\014le,)h(and)227 3192 y(write)d(the)g(other)f |
| 4975 | (represen)m(tation)i(in)e(a)h(P)m(olyhedra)g(\014le)f(o\014le.)42 |
| 4976 | b Fn(*err)29 b Fo(returns)g Fn(dd)p 3073 3192 V 34 w(NoError)g |
| 4977 | Fo(if)h(the)h(com-)227 3305 y(putation)g(terminates)g(normally)-8 |
| 4978 | b(.)42 b(Otherwise,)30 b(it)h(returns)e(a)i(v)-5 b(alue)31 |
| 4979 | b(according)g(to)g(the)g(error)f(o)s(ccured.)0 3490 y |
| 4980 | Fn(void)47 b(dd)p 341 3490 V 33 w(WriteMatrix\(f,)d(matrix\))g |
| 4981 | Fo(:)227 3603 y(W)-8 b(rite)32 b Fn(matrix)d Fo(to)i(stream)g |
| 4982 | Fn(f)p Fo(.)0 3788 y Fn(void)47 b(dd)p 341 3788 V 33 |
| 4983 | w(WriteNumber\(f,)d(x\))h Fo(:)227 3901 y(W)-8 b(rite)27 |
| 4984 | b Fn(x)e Fo(to)h(stream)f Fn(f)p Fo(.)39 b(If)25 b Fn(x)f |
| 4985 | Fo(is)i(of)f(GMP)h(mp)s(q)p 1827 3901 28 4 v 31 w(t)g(rational)g |
| 4986 | Fj(p=q)s Fo(,)h(the)e(output)g(is)g Fj(p=q)s Fo(.)39 |
| 4987 | b(If)25 b(it)g(is)h(of)f(C)g(double,)227 4014 y(it)31 |
| 4988 | b(is)g(formated)f(as)h(a)g(double)f(\015oat)h(with)f(a)g(decimal)i(p)s |
| 4989 | (oin)m(t.)0 4200 y Fn(void)47 b(dd)p 341 4200 29 4 v |
| 4990 | 33 w(WritePolyFile\(f,)d(poly\))g Fo(:)227 4312 y(W)-8 |
| 4991 | b(rite)32 b(tt)f(p)s(oly)f(to)h(stream)g Fn(f)f Fo(in)g(P)m(olyhedra)h |
| 4992 | (format.)0 4498 y Fn(void)47 b(dd)p 341 4498 V 33 w |
| 4993 | (WriteErrorMessages\(f,)42 b(err\))j Fo(:)227 4611 y(W)-8 |
| 4994 | b(rite)32 b(error)e(messages)h(giv)m(en)h(b)m(y)e Fn(err)g |
| 4995 | Fo(to)h(stream)f Fn(f)p Fo(.)0 4796 y Fn(void)47 b(dd)p |
| 4996 | 341 4796 V 33 w(WriteSetFamily\(f,)d(setfam\))f Fo(:)227 |
| 4997 | 4909 y(W)-8 b(rite)37 b(the)e(set)h(family)g(p)s(oin)m(ted)f(b)m(y)g |
| 4998 | Fn(setfam)e Fo(to)j(stream)g Fn(f)p Fo(.)55 b(F)-8 b(or)35 |
| 4999 | b(eac)m(h)i(set,)g(it)f(outputs)e(its)i(index,)g(its)227 |
| 5000 | 5022 y(cardinalit)m(y)-8 b(,)33 b(a)e(colon)g(\\:")41 |
| 5001 | b(and)30 b(a)h(ordered)f(list)h(of)f(its)h(elemen)m(ts.)0 |
| 5002 | 5207 y Fn(void)47 b(dd)p 341 5207 V 33 w(WriteSetFamilyCompressed\(f)o |
| 5003 | (,)42 b(setfam\))h Fo(:)227 5320 y(W)-8 b(rite)37 b(the)e(set)h(family) |
| 5004 | g(p)s(oin)m(ted)f(b)m(y)g Fn(setfam)e Fo(to)j(stream)g |
| 5005 | Fn(f)p Fo(.)55 b(F)-8 b(or)35 b(eac)m(h)i(set,)g(it)f(outputs)e(its)i |
| 5006 | (index,)g(its)227 5433 y(cardinalit)m(y)g(or)e(the)g(negativ)m(e)j(of)d |
| 5007 | (the)g(cardinalit)m(y)-8 b(,)38 b(a)c(colon)h(\\:")50 |
| 5008 | b(and)33 b(the)i(elemen)m(ts)g(in)f(the)g(set)h(or)f(its)1903 |
| 5009 | 5682 y(14)p eop end |
| 5010 | %%Page: 15 15 |
| 5011 | TeXDict begin 15 14 bop 227 44 a Fo(complemen)m(ts)40 |
| 5012 | b(whic)m(hev)m(er)g(is)f(smaller.)67 b(Whenev)m(er)40 |
| 5013 | b(it)g(outputs)e(the)h(complemen)m(ts,)k(the)c(cardinalit)m(y)227 |
| 5014 | 157 y(is)i(negated)h(so)g(that)g(there)f(is)g(no)g(am)m(biguit)m(y)-8 |
| 5015 | b(.)75 b(This)40 b(will)h(b)s(e)g(considered)g(standard)f(for)h |
| 5016 | (outputing)227 270 y(incidence)32 b(\(*.icd,)i(*ecd\))e(and)f |
| 5017 | (adjacency)h(\(*.iad,)i(*.ead\))e(data)h(in)e(cddlib.)43 |
| 5018 | b(But)32 b(there)g(is)f(some)h(minor)227 383 y(incompatibilit)m(y)h |
| 5019 | (with)d(cdd/cdd+)f(standalone)i(co)s(des.)0 571 y Fn(void)47 |
| 5020 | b(dd)p 341 571 29 4 v 33 w(WriteProgramDescription\(f\))39 |
| 5021 | b Fo(:)227 683 y(W)-8 b(rite)32 b(the)f(cddlib)e(v)m(ersion)i |
| 5022 | (information)g(to)g(stream)g Fn(f)p Fo(.)0 871 y Fn(void)47 |
| 5023 | b(dd)p 341 871 V 33 w(WriteDDTimes\(f,)d(poly\))g Fo(:)227 |
| 5024 | 984 y(W)-8 b(rite)32 b(the)f(represen)m(tation)g(con)m(v)m(ersion)h |
| 5025 | (time)f(information)f(on)h Fn(poly)e Fo(to)i(stream)g |
| 5026 | Fn(f)p Fo(.)0 1227 y Fd(4.5)112 b(Obsolete)38 b(F)-9 |
| 5027 | b(unctions)0 1399 y Fn(boolean)46 b(dd)p 484 1399 V 34 |
| 5028 | w(DoubleDescription\(poly,)41 b(err\))j Fo(:)d(\(remo)m(v)m(ed)32 |
| 5029 | b(in)e(V)-8 b(ersion)31 b(0.90c\))227 1512 y(The)e(new)f(function)g |
| 5030 | Fn(dd)p 1054 1512 V 34 w(DDMatrix2Poly\(matrix,)42 b(err\))28 |
| 5031 | b Fo(\(see)h(Section)h(4.2\))g(replaces)f(\(and)g(actually)227 |
| 5032 | 1625 y(com)m(bines\))j(b)s(oth)d(this)h(and)g Fn(dd)p |
| 5033 | 1325 1625 V 34 w(Matrix2Poly\(matrix,)43 b(err\))p Fo(.)0 |
| 5034 | 1812 y Fn(dd)p 102 1812 V 34 w(PolyhedraPtr)h(dd)p 852 |
| 5035 | 1812 V 34 w(Matrix2Poly\(matrix,)f(err\))h Fo(:)d(\(remo)m(v)m(ed)31 |
| 5036 | b(in)f(V)-8 b(ersion)31 b(0.90c\))227 1925 y(See)g(ab)s(o)m(v)m(e)g |
| 5037 | (for)g(the)f(reason)h(for)f(remo)m(v)-5 b(al.)0 2113 |
| 5038 | y Fn(dd)p 102 2113 V 34 w(LPSolutionPtr)44 b(dd)p 900 |
| 5039 | 2113 V 34 w(LPSolutionLoad\(lp\))c Fo(:)h(\(renamed)30 |
| 5040 | b(in)g(V)-8 b(ersion)31 b(0.90c\))227 2226 y(This)f(function)g(is)g(no) |
| 5041 | m(w)h(called)g Fn(dd)p 1431 2226 V 34 w(CopyLPSolution\(lp\))p |
| 5042 | Fo(.)0 2469 y Fd(4.6)112 b(Set)38 b(F)-9 b(unctions)38 |
| 5043 | b(in)f Fb(setoper)j Fd(library)0 2641 y Fo(The)30 b(cddlib)g(comes)i |
| 5044 | (with)e(a)h(simple)g(set)g(op)s(eration)g(library)g Fn(setoper)p |
| 5045 | Fo(.)39 b(The)30 b(k)m(ey)i(t)m(yp)s(e)f(de\014ned)e(is)i |
| 5046 | Fn(set)p 3654 2641 V 33 w(type)p Fo(.)0 2754 y(A)g(set)h(is)f(represen) |
| 5047 | m(ted)g(b)m(y)g(a)g(\014xed)g(length)g(binary)f(strings.)43 |
| 5048 | 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) |
| 5049 | s(e)0 2867 y(declared)g(when)e(it)i(is)g(initialized.)141 |
| 5050 | 2980 y(Belo)m(w)e(the)f(sym)m(b)s(ols)g Fn(a)p Fo(,)g |
| 5051 | Fn(b)p Fo(,)g Fn(c)g Fo(are)g(of)g(t)m(yp)s(e)g Fn(set)p |
| 5052 | 1786 2980 V 33 w(type)p Fo(.)39 b(The)27 b(sym)m(b)s(ols)h |
| 5053 | Fn(aP)f Fo(is)h(a)g(p)s(oin)m(ter)f(to)i(t)m(yp)s(e)f |
| 5054 | Fn(set)p 3654 2980 V 33 w(type)p Fo(,)0 3093 y(and)j |
| 5055 | Fn(s)p Fo(,)i Fn(t)f Fo(are)g(of)g(t)m(yp)s(e)h Fn(long)p |
| 5056 | Fo(.)44 b(Here)33 b(are)g(some)f(of)g(the)h(functions)e(de\014ned.)45 |
| 5057 | b(See)32 b Fn(setoper.h)e Fo(for)i(a)g(complete)0 3206 |
| 5058 | y(listing.)0 3418 y Fn(void)47 b(set)p 389 3418 V 33 |
| 5059 | w(initialize\(aP,)d(s\))h Fo(:)227 3531 y(Allo)s(cate)35 |
| 5060 | b(a)e Fn(set)p 816 3531 V 33 w(type)f Fo(space)h(of)f(maxim)m(um)h |
| 5061 | (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) |
| 5062 | g Fn(aP)p Fo(.)h(The)f(set)h(is)227 3644 y(initialized)f(as)f(empt)m(y) |
| 5063 | g(set.)0 3832 y Fn(void)47 b(set)p 389 3832 V 33 w(free\(a\))d |
| 5064 | Fo(:)227 3944 y(F)-8 b(ree)32 b(the)e Fn(set)p 732 3944 |
| 5065 | V 34 w(type)f Fo(space)i(allo)s(cated)h(for)f Fn(a)p |
| 5066 | Fo(.)0 4132 y Fn(void)47 b(set)p 389 4132 V 33 w(copy\(a,)f(b\)\))f |
| 5067 | Fo(:)227 4245 y(Set)31 b Fn(a)g Fo(to)g(b)s(e)f Fn(b)p |
| 5068 | Fo(.)42 b(The)30 b(set)h Fn(a)g Fo(m)m(ust)f(b)s(e)g(pre-initialized)j |
| 5069 | (with)d(the)h(same)g(maxim)m(um)g(cardinalit)m(y)h(as)f(that)227 |
| 5070 | 4358 y(of)g Fn(b)p Fo(.)0 4546 y Fn(void)47 b(set)p 389 |
| 5071 | 4546 V 33 w(addelem\(a,)e(t\)\))g Fo(:)227 4658 y(Add)30 |
| 5072 | b(an)g(elemen)m(t)i Fn(t)e Fo(to)h(a)g(set)g Fn(a)p Fo(.)40 |
| 5073 | b(The)30 b(set)h Fn(a)f Fo(sta)m(ys)h(unc)m(hanged)f(if)h(it)f(con)m |
| 5074 | (tains)i(the)f(elemen)m(t)g Fn(t)p Fo(.)0 4846 y Fn(long)47 |
| 5075 | b(set)p 389 4846 V 33 w(card\(a\)\))c Fo(:)227 4959 y(Return)30 |
| 5076 | b(the)g(cardinalit)m(y)i(of)f(set)g Fn(a)p Fo(.)0 5147 |
| 5077 | y Fn(int)47 b(set)p 341 5147 V 33 w(member\(t,)f(a\)\))e |
| 5078 | 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 |
| 5079 | (of)g(set)h Fn(a)p Fo(,)g(and)e(0)i(otherwise.)1903 5682 |
| 5080 | y(15)p eop end |
| 5081 | %%Page: 16 16 |
| 5082 | TeXDict begin 16 15 bop 0 44 a Fn(void)47 b(set)p 389 |
| 5083 | 44 29 4 v 33 w(write\(a\)\))c Fo(:)227 157 y(Prin)m(t)28 |
| 5084 | b(out)g(the)g(elemen)m(ts)h(of)f(set)h Fn(a)e Fo(to)i |
| 5085 | Fn(stdout)p Fo(.)38 b(The)27 b(function)h Fn(void)47 |
| 5086 | b(set)p 2846 157 V 33 w(fwrite\(f,)e(a\)\))27 b Fo(output)h(to)227 |
| 5087 | 270 y(stream)j Fn(f)p Fo(.)0 556 y Fq(5)135 b(An)44 b(Extension)i(of)f |
| 5088 | (the)g(CDD)g(Library)g(in)g(GMP)f(mo)t(de)0 759 y Fo(Starting)35 |
| 5089 | b(from)e(the)i(v)m(ersion)g(093,)h(the)f(GMP)f(v)m(ersion)h(of)g |
| 5090 | (cddlib,)f Fn(libcddgmp.a)p Fo(,)f(con)m(tains)i(all)g(cdd)f(library)0 |
| 5091 | 872 y(functions)j(in)g(t)m(w)m(o)i(arithmetics.)63 b(All)38 |
| 5092 | b(functions)f(with)g(the)h(standard)f(pre\014x)f Fn(dd)p |
| 5093 | 2907 872 V 71 w Fo(are)i(computed)f(with)g(the)0 985 |
| 5094 | y(GMP)g(rational)h(arithmetic)g(as)e(b)s(efore.)59 b(The)36 |
| 5095 | b(same)h(fuctions)f(with)h(the)f(new)g(pre\014x)g Fn(ddf)p |
| 5096 | 3243 985 V 70 w Fo(are)h(no)m(w)f(added)0 1098 y(to)d(the)g(library)f |
| 5097 | Fn(libcddgmp.a)d Fo(that)k(are)g(based)f(on)g(the)g(C)g(double)g |
| 5098 | (\015oating-p)s(oin)m(t)i(arithmetic.)48 b(Th)m(us)31 |
| 5099 | b(these)0 1211 y(functions)38 b(are)g(equiv)-5 b(alen)m(t)39 |
| 5100 | b(to)g Fn(libcdd.a)d Fo(functions,)k(except)f(that)f(all)h(functions)f |
| 5101 | (and)f(v)-5 b(ariable)39 b(t)m(yp)s(es)f(are)0 1324 y(with)30 |
| 5102 | b(pre\014x)f Fn(ddf)p 612 1324 V 64 w Fo(and)h(the)h(v)-5 |
| 5103 | b(ariable)31 b(t)m(yp)s(e)f Fn(mytype)f Fo(is)h(replaced)h(b)m(y)f |
| 5104 | Fn(myfloat)p Fo(.)141 1437 y(In)21 b(this)h(sense,)i |
| 5105 | Fn(libcdd.a)19 b Fo(is)j(a)h(prop)s(er)d(subset)h(of)h |
| 5106 | Fn(libcddgmp.a)d Fo(and)j(in)f(principle)h(one)g(can)g(do)g(ev)m |
| 5107 | (erything)0 1550 y(with)30 b Fn(libcddgmp.a)p Fo(.)38 |
| 5108 | b(See)30 b(ho)m(w)h(the)f(new)g Fn(dd)p 1594 1550 V 34 |
| 5109 | w(LPSolve)e Fo(is)j(written)f(in)g(cddlp.c.)0 1836 y |
| 5110 | Fq(6)135 b(Examples)0 2039 y Fo(See)27 b(example)g(co)s(des)g(suc)m(h)g |
| 5111 | (as)g(testcdd*.c)h(,)g(testlp*.c,)h(redc)m(hec)m(k.c,)h(adjacency)-8 |
| 5112 | b(.c,)29 b(allfaces,c)g(and)e(simplecdd.c)0 2152 y(in)j(the)h |
| 5113 | Fn(src)e Fo(and)h Fn(src-gmp)e Fo(sub)s(directories)i(of)h(the)f |
| 5114 | (source)h(distribution.)0 2438 y Fq(7)135 b(Numerical)46 |
| 5115 | b(Accuracy)0 2641 y Fo(A)37 b(little)i(caution)g(is)e(in)g(order.)61 |
| 5116 | b(Man)m(y)38 b(p)s(eople)f(ha)m(v)m(e)h(observ)m(ed)g(n)m(umerical)g |
| 5117 | (problems)e(of)h(cddlib)g(when)f(the)0 2754 y(\015oating)k(v)m(ersion)f |
| 5118 | (of)f(cddlib)h(is)f(used.)65 b(As)39 b(w)m(e)g(all)g(kno)m(w,)i |
| 5119 | (\015oating-p)s(oin)m(t)f(computation)g(migh)m(t)f(not)g(giv)m(e)h(a)0 |
| 5120 | 2867 y(correct)27 b(answ)m(er,)g(esp)s(ecially)f(when)f(an)h(input)e |
| 5121 | (data)j(is)e(v)m(ery)h(sensitiv)m(e)h(to)f(a)h(small)f(p)s |
| 5122 | (erturbation.)38 b(When)25 b(some)0 2980 y(strange)j(b)s(eha)m(vior)g |
| 5123 | (is)g(observ)m(ed,)g(it)g(is)g(alw)m(a)m(ys)h(wise)f(to)g(create)h(a)f |
| 5124 | (rationalization)j(of)c(the)h(input)f(\(for)h(example,)0 |
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| 5232 | end |
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| 5310 | (are/gfan/gfan.h)m(tml.)0 5320 y([18])47 b(A.)30 b(Marzetta.)43 |
| 5311 | b Ff(p)-5 b(d)34 b({)f(C-implementation)h(of)f(the)g(primal-dual)h |
| 5312 | (algoirithm)p Fo(,)f(1997.)42 b(co)s(de)30 b(a)m(v)-5 |
| 5313 | b(ailable)33 b(from)187 5433 y(h)m(ttp://www.cs.un)m |
| 5314 | (b.ca/profs/bremner/p)s(d/.)1903 5682 y(18)p eop end |
| 5315 | %%Page: 19 19 |
| 5316 | TeXDict begin 19 18 bop 0 44 a Fo([19])47 b(T.S.)34 b(Motzkin,)j(H.)e |
| 5317 | (Rai\013a,)i(GL.)f(Thompson,)e(and)g(R.M.)i(Thrall.)53 |
| 5318 | b(The)34 b(double)g(description)h(metho)s(d.)187 157 |
| 5319 | y(In)c(H.W.)i(Kuhn)d(and)h(A.W.T)-8 b(uc)m(k)m(er,)35 |
| 5320 | b(editors,)d Ff(Contributions)k(to)e(the)-5 b(ory)36 |
| 5321 | b(of)e(games,)h(V)-7 b(ol.)34 b(2)p Fo(.)e(Princeton)187 |
| 5322 | 270 y(Univ)m(ersit)m(y)f(Press,)g(Princeton,)g(RI,)f(1953.)0 |
| 5323 | 458 y([20])47 b(K.)37 b(Mulm)m(uley)-8 b(.)63 b Ff(Computational)43 |
| 5324 | b(Ge)-5 b(ometry,)42 b(An)d(Intr)-5 b(o)g(duction)42 |
| 5325 | b(Thr)-5 b(ough)41 b(Randamize)-5 b(d)42 b(Algorithms)p |
| 5326 | Fo(.)187 571 y(Pren)m(tice-Hall,)33 b(1994.)0 758 y([21])47 |
| 5327 | b(J.)f(Ram)m(bau.)87 b Ff(TOPCOM,)46 b(a)h(p)-5 b(ackage)48 |
| 5328 | b(for)f(c)-5 b(omputing)49 b(T)-7 b(riangulations)49 |
| 5329 | b(Of)d(Point)h(Con\014gur)-5 b(ations)187 871 y(and)53 |
| 5330 | b(Oriente)-5 b(d)53 b(Matr)-5 b(oids)p Fo(.)106 b(Univ)m(ersit)m(y)53 |
| 5331 | b(of)f(Ba)m(yreuth,)58 b(2005.)106 b(a)m(v)-5 b(ailable)54 |
| 5332 | b(from)d(h)m(ttp://www.uni-)187 984 y(ba)m(yreuth.de/departmen)m |
| 5333 | (ts/wirtsc)m(haftsmathematik/ram)m(bau/TOPCOM/.)0 1172 |
| 5334 | y([22])c(C.)27 b(W)-8 b(eib)s(el.)37 b Ff(Minksum)30 |
| 5335 | b(version)g(1.1)p Fo(.)37 b(Mathematics)30 b(Institute,)e(EPF)g |
| 5336 | (Lausanne,)f(2005.)38 b(a)m(v)-5 b(ailable)30 b(from)187 |
| 5337 | 1285 y(h)m(ttp://roso.ep\015.c)m(h/cw/p)s(oly/public.php.)0 |
| 5338 | 1472 y([23])47 b(D.K.)30 b(Wilde.)39 b(A)29 b(library)g(for)f(doing)i |
| 5339 | (p)s(olyhedral)e(op)s(erations.)39 b(Master's)30 b(thesis,)g(Oregon)f |
| 5340 | (State)h(Univ)m(er-)187 1585 y(sit)m(y)-8 b(,)37 b(Corv)-5 |
| 5341 | b(allis,)36 b(Oregon,)f(Dec)g(1993.)54 b(Also)35 b(published)d(in)i |
| 5342 | (IRISA)g(tec)m(hnical)i(rep)s(ort)d(PI)h(785,)j(Rennes,)187 |
| 5343 | 1698 y(F)-8 b(rance;)32 b(Dec,)f(1993.)1903 5682 y(19)p |
| 5344 | eop end |
| 5345 | %%Trailer |
| 5346 | |
| 5347 | userdict /end-hook known{end-hook}if |
| 5348 | %%EOF |