doc/pip.ps

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   2 %%Creator: dvips(k) 5.86f Copyright 2001 Radical Eye Software
   3 %%Title: pip.dvi
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1998 EBFFE04A485AEC1FE0143F15C0147F158002FF6D13C0EF7F804990C7EA0C0094C7FCA349
1999 5AB3A9B812C0A5260003FEC9FCB3B3B3A7805B011F13E0001FB612F8A533797DF82F>I<
2000 F103F8F13FFEDB1FF891B5FC92B5903903FC078002079139E007E01F913D1FF00FF81F80
2001 3FC0913B3FC003FC3E0002FFC7B45A49486E5A49486E5A49486E48EB1F8049486E6CEB06
2002 0097C7FC49486E7E013F82A249486E7EA201FF82A24A80A24884AA6C95C8FCA26E5CA201
2003 7F5EA26D6C4A5AA2011F5E6D6C4A5AA26D6C4A5A6D6C4A5A496C4A5A6E4AC9FC903A0F3F
2004 C003FC903A1E1FF00FF80207B512E0D91C0091CAFC013CEB1FF892CCFCA25BA3137CA413
2005 7EA27F8080EB1FF091B612FE6DEDFFF818FF6D17E06D17F8010017FE010383010F84D93F
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2007 003F727E49170148CBFC737EA200FE85A76C616C616D1701003F616D17036C6C4D5A6C6C
2008 4D5A6C6C4D5A6C6C4D5A6C6C4D5A26007F80DB01FEC7FCD93FE0ED07FCD90FF8ED1FF0D9
2009 03FFEDFFC0010001F8011F90C8FC021FB612F802011580DA000F01F0C9FC4A707DCC4F>
2010 I<131EEB7F80497E487F487FA66C5B6C5B6D5A011EC7FC90C8FCB3A7EB01F0EA07FFB5FC
2011 A51201EA007F133FA2131FB3B3B3A3497EEBFFFEB612FCA51E727AF12A>105
2012 D<EB01F0EA07FFB5FCA51201EA007F133FA2131FB3B3B3B3B3AC497EEBFFFEB612FEA51F
2013 787AF72A>108 D<D901F0902601FFC0ED0FFED807FF021F01FC92B512E0B5027F01FF02
2014 0314F8922601FC0101C090390FE00FFE922803E0003FE090391F0001FFDB0F80D91FF801
2015 7C6D13C04BC7000F4A147FC6023C912707FC01E06E7E013F496E6C48486E7E011F0170DA
2016 01FF496E7E03F0160790260FF1E04CC8FCDAF3C06E019E6F7E4B169C02F717BC92C913B8
2017 02FFDC7FF86F7E4A5FA34A5FA44A5FB3B3A8496C4C6C4B7ED97FFF030301F8031F13C0B7
2018 D88007B600FC013FB612E0A57B4B7ACA86>I<D901F0903803FF80D807FF021F13F8B502
2019 7F13FE923A01FC03FF80923A03E0007FE0DB0F80EB1FF0031EC76C7EC65C013F496E7E01
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2025 A46C6C4C1380A2000F1900A26C6C4C5AA26C6C4C5A0001606D160F6C606D6C4B5A6D6C4B
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2027 270FFC03FFC8FC0203B512FCDA007F13E0DB07FEC9FC444D7BCB4F>I<DB07FEEC038092
2028 387FFFC04AB500F8130791390FFE00FCDA3FF0131FDA7FC0903807800F494848EB03C049
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2033 FC92C9FCB3A74E7F060713F00507B612F0A54C6C7BCA53>113 D<D903E0EB1FE0D80FFF
2034 ECFFFCB501037F923907E01F8092391F807FC092383E00FF4B4813E000011478D8007F13
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2036 A45CA55CB3B3A480133F90B5FCB712C0A5334B7ACA3D>I<DA1FFFEB038049B512E0010F
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2038 7F48163F003E161F127E170F127C12FC1707A46C1603A27E7F6C7E7F6D92C7FCEA3FFC13
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2062 %EndDVIPSBitmapFont
2063 end
2064 %%EndProlog
2065 %%BeginSetup
2066 %%Feature: *Resolution 600dpi
2067 TeXDict begin
2068 %%PaperSize: A4
2069
2070 %%EndSetup
2071 %%Page: 1 1
2072 1 0 bop 350 976 a Fs(Solving)54 b(Systems)d(of)i(A\016ne)f
2073 (\(In\)Equalities:)73 b(PIP's)1528 1183 y(User's)53 b(Guide)1603
2074 1479 y Fr(P)m(aul)37 b(F)-10 b(eautrier)549 1726 y(Additions)37
2075 b(b)m(y)i(Jean-F)-10 b(ran\030)-52 b(cois)37 b(Collard)g(and)h(C)m
2076 (\023)-55 b(edric)37 b(Bastoul)1052 1846 y(F)-10 b(ourth)38
2077 b(V)-10 b(ersion,)38 b(Septem)m(b)s(er)e(22,)i(2001)1770
2078 2177 y Fq(Abstract)638 2309 y Fp(This)f(do)s(cumen)m(t)i(is)f(the)i
2079 (User's)f(Man)m(ual)g(of)h(PIP)-8 b(,)39 b(a)h(soft)m(w)m(are)g(whic)m
2080 (h)e(solv)m(es)501 2421 y(P)m(arametric)h(In)m(teger)g(Programming)f
2081 (problems.)62 b(That)38 b(is,)i(PIP)e(\014nds)e(the)i(lexi-)501
2082 2534 y(cographic)e(minim)m(um)c(of)j(the)h(set)g(of)f(in)m(teger)h(p)s
2083 (oin)m(ts)e(whic)m(h)g(lie)g(inside)f(a)j(con)m(v)m(ex)501
2084 2647 y(p)s(olyhedron,)22 b(when)f(that)h(p)s(olyhedron)e(dep)s(ends)g
2085 (linearly)g(on)i(one)g(or)g(more)g(in)m(tegral)501 2760
2086 y(parameters.)257 3088 y Fo(1)161 b(In)l(tro)t(duction)257
2087 3307 y Fn(The)39 b(seman)m(tic)e(analysis)g(of)g(programs)g(accessing)h
2088 (arra)m(ys)h(often)e(b)s(oils)f(do)m(wn)j(to)e(\014nding)257
2089 3428 y(in)m(teger)31 b(solutions)e(to)h(parametric)f(linear)g
2090 (programming)f(problems.)42 b(This)31 b(is)f(due)h(to)f(t)m(w)m(o)257
2091 3548 y(main)h(phenomena:)403 3724 y Fm(\017)48 b Fn(Arra)m(y)36
2092 b(subscripts)h(are)f(v)m(ery)h(often)e(linear)f(functions)i(of)f
2093 (surrounding)g(lo)s(op)g(coun-)501 3844 y(ters)e(;)403
2094 4038 y Fm(\017)48 b Fn(The)34 b(program's)e(execution)h(order)g
2095 (enforces)g(an)g(order)f(on)h(p)s(ossible)f(solutions.)404
2096 4214 y(Let)g(us)i(consider)f(the)g(follo)m(wing)c(example:)462
2097 4390 y Fl(for)52 b(i:=)h(0)e(to)h(m)f(do)667 4511 y(for)i(j)e(:=)h(0)f
2098 (to)h(n)g(do)718 b({I})873 4631 y(a[2*i+j])53 b(:=)f(i+j;)404
2099 4807 y Fn(After)24 b(completion)f(of)h(execution,)i(for)e(whic)m(h)h(v)
2100 -5 b(alues)24 b(of)g Fk(k)k Fn(is)c Fl(A[)p Fk(k)s Fl(])h
2101 Fn(de\014ned,)i(and)e(whic)m(h)257 4927 y(instances)37
2102 b(of)f(the)g(assignmen)m(t)g(wrote)g(in)m(to)g(this)f(arra)m(y)i
2103 (elemen)m(t?)53 b(W)-8 b(e)37 b(can)f(easily)f(c)m(hec)m(k)257
2104 5048 y(that)28 b(answ)m(ering)g(this)g(question)g(is)f(equiv)-5
2105 b(alen)m(t)28 b(to)f(\014nding)h(the)g(solutions)f(of)g(the)h(follo)m
2106 (wing)257 5168 y(system,)34 b(where)g Fk(i;)17 b(j)38
2107 b Fn(and)33 b Fk(k)j Fn(are)c(the)h(unkno)m(wns:)1702
2108 5356 y(0)28 b Fm(\024)104 b Fk(i)h Fm(\024)28 b Fk(m;)1243
2109 b Fn(\(1\))1702 5501 y(0)28 b Fm(\024)98 b Fk(j)104 b
2110 Fm(\024)28 b Fk(n;)1270 b Fn(\(2\))1608 5646 y(2)p Fk(i)22
2111 b Fn(+)g Fk(j)89 b Fn(=)83 b Fk(k)s(:)1379 b Fn(\(3\))1946
2112 5896 y(1)p eop
2113 %%Page: 2 2
2114 2 1 bop 257 266 a Fj(2)98 b(USING)32 b(THE)i(PIP)f(SOFTW)-11
2115 b(ARE)1879 b Fn(2)404 573 y(Moreo)m(v)m(er,)30 b(if)c(w)m(e)i(w)m(an)m
2116 (t)h(to)e(kno)m(w)h(whic)m(h)g(instance)g(ga)m(v)m(e)g(its)f
2117 Fi(\014nal)g Fn(v)-5 b(alue)26 b(to)h Fl(A[)p Fk(k)s
2118 Fl(])p Fn(,)j(that)257 693 y(is)k(if)f(w)m(e)i(are)f(lo)s(oking)e(for)h
2119 (the)i Fi(last)f Fn(instance)g(writing)e(in)m(to)i Fl(A[)p
2120 Fk(k)s Fl(])p Fn(,)h(then)g(w)m(e)g(ha)m(v)m(e)g(to)f(lo)s(ok)257
2121 814 y(for)h(the)g(maximal)d(v)-5 b(alue)34 b(of)h(v)m(ector)h(\()p
2122 Fk(i;)17 b(j)6 b Fn(\))35 b(according)f(to)h(lexicographic)e(order.)51
2123 b(W)-8 b(e)35 b(th)m(us)257 934 y(consider)e(the)g(follo)m(wing)d
2124 Fi(p)-5 b(olyhe)g(dr)g(on)32 b Fm(F)10 b Fn(\()p Fk(k)s(;)17
2125 b(m;)g(n)p Fn(\):)741 1152 y Fm(F)10 b Fn(\()p Fk(k)s(;)17
2126 b(m;)g(n)p Fn(\))27 b(=)g Fm(f)p Fk(<)h(i;)17 b(j)33
2127 b(>)28 b Fm(j)p Fn(0)f Fm(\024)h Fk(i)g Fm(\024)g Fk(m;)17
2128 b Fn(0)28 b Fm(\024)g Fk(j)34 b Fm(\024)28 b Fk(n;)17
2129 b Fn(2)p Fk(i)22 b Fn(+)g Fk(j)34 b Fn(=)27 b Fk(k)s
2130 Fm(g)p Fk(:)359 b Fn(\(4\))257 1370 y(What)39 b(is)e(the)i
2131 (lexicographical)c(maxim)m(um)h(of)i(the)h(in)m(teger-v)-5
2132 b(alued)37 b(v)m(ectors)i(included)f(in)257 1491 y Fm(F)10
2133 b Fn(\()p Fk(k)s(;)17 b(m;)g(n)p Fn(\)?)52 b(The)37 b(aim)c(of)i(PIP)i
2134 (is)e(to)g(solv)m(e)h(suc)m(h)h(problems.)52 b(The)36
2135 b(reader)g(is)f(refered)i(to)257 1611 y([1])c(for)f(a)g(mathematical)e
2136 (description)i(of)g(the)h(metho)s(d.)257 1900 y Fh(1.1)136
2137 b(General)45 b(form)l(ulation)257 2085 y Fn(Let)33 b
2138 Fm(F)42 b Fn(b)s(e)33 b(a)f(p)s(olyhedron:)1189 2303
2139 y Fm(F)10 b Fn(\()n Fk(~)-47 b(z)t Fn(\))28 b(=)f Fm(f)n
2140 Fk(~)-47 b(x)p Fm(j)n Fk(~)g(x)28 b Fm(\025)g Fn(0)p
2141 Fk(;)17 b Fg(A)n Fk(~)-48 b(x)23 b Fn(+)f Fg(B)n Fk(~)-47
2142 b(z)27 b Fn(+)17 b Fk(~)-44 b(c)27 b Fm(\025)h Fn(0)p
2143 Fm(g)p Fk(:)807 b Fn(\(5\))257 2521 y(In)29 b(this)f(form)m(ula,)d
2144 Fk(~)-47 b(x)29 b Fn(is)e(a)h(v)m(ector)h(with)f Fk(n)g
2145 Fn(en)m(tries:)42 b(the)29 b(v)m(ector)g(of)f(all)e(unkno)m(wns.)43
2146 b Fk(~)-48 b(z)5 b Fn(,)27 b Fk(~)-47 b(z)33 b Fm(\025)28
2147 b Fn(0,)257 2641 y(is)d(the)h(v)m(ector)g(built)e(from)g(parameters)h
2148 (and)g(has)h Fk(p)f Fn(en)m(tries.)41 b(P)m(olyhedron)26
2149 b Fm(F)10 b Fn(\()n Fk(~)-47 b(z)t Fn(\))25 b(is)g(a)g(subset)257
2150 2762 y(of)32 b Fg(R)452 2726 y Ff(n)532 2762 y Fn(and)g(is)g(de\014ned)
2151 i(b)m(y)g Fk(n)22 b Fn(+)g Fk(l)35 b Fn(inequalities:)42
2152 b Fk(n)32 b Fn(inequalities)f(expressing)1850 2980 y
2153 Fk(~)-48 b(x)29 b Fm(\025)f Fn(0)257 3198 y(and)36 b(the)g
2154 Fk(l)i Fn(inequalities)c(correp)s(onding)h(to)h(ro)m(ws)g(of)g(matrix)e
2155 Fg(A)h Fn(of)h(size)g Fk(l)26 b Fm(\002)f Fk(n)p Fn(,)37
2156 b(matrix)d Fg(B)257 3319 y Fn(of)e(size)h Fk(l)25 b Fm(\002)d
2157 Fk(p)p Fn(,)33 b(and)g(constan)m(t)g(v)m(ector)28 b Fk(~)-43
2158 b(c)32 b Fn(of)g(size)h Fk(l)r Fn(.)404 3439 y(Size)f(parameters)h(can)
2159 g(themselv)m(es)g(b)s(e)g(constrained)g(b)m(y)h(a)e(set)h(of)f(a\016ne)
2160 h(inequalities)1699 3657 y Fg(M)n Fk(~)-47 b(z)28 b Fn(+)1969
2161 3631 y Fk(~)1976 3657 y(h)f Fm(\025)i Fn(0)p Fk(;)257
2162 3875 y Fn(whic)m(h)j(is)f(called)g(the)g Fi(c)-5 b(ontext)32
2163 b Fn(of)f(the)h(problem.)41 b Fg(M)32 b Fn(is)f(an)g
2164 Fk(m)21 b Fm(\002)f Fk(p)31 b Fn(matrix)f(and)3248 3849
2165 y Fk(~)3255 3875 y(h)h Fn(a)g(v)m(ector)257 3996 y(of)37
2166 b(dimension)g Fk(m)p Fn(.)58 b(All)36 b(data)h(of)g(a)h(PIP)g(problem:)
2167 52 b(\()p Fg(A)p Fk(;)17 b Fg(B)p Fk(;)g Fg(M)p Fk(;)11
2168 b(~)-43 b(c)o(;)2756 3970 y(~)2763 3996 y(h)p Fn(\))37
2169 b(are)h(assumed)g(to)f(b)s(e)257 4116 y(in)m(teger-v)-5
2170 b(alued.)257 4449 y Fo(2)161 b(Using)54 b(the)f(PIP)h(Soft)l(w)l(are)
2171 257 4697 y Fh(2.1)136 b(W)-11 b(riting)45 b(the)g(Input)g(File)257
2172 4882 y Fn(The)34 b(input)e(text)h(\014le)f(follo)m(ws)g(the)h(follo)m
2173 (wing)d(con)m(text-free)j(grammar:)257 5083 y Fg(Grammar)k(1)49
2174 b Fi(:)257 5285 y Fl(File)258 b(::=)52 b(Problem)h(...)257
2175 5406 y(Problem)105 b(::=)52 b(\()f(Comments)j(Nn)e(Np)f(Nl)h(Nm)g(Bg)g
2176 (Nq)g(Tableau)h(Context)g(\))257 5526 y(Comments)h(::=)e(List)257
2177 5646 y(List)258 b(::=)52 b(Atom)g(|)g(\()f(List)i(...)f(\))p
2178 eop
2179 %%Page: 3 3
2180 3 2 bop 257 266 a Fj(2)98 b(USING)32 b(THE)i(PIP)f(SOFTW)-11
2181 b(ARE)1879 b Fn(3)257 573 y Fl(Tableau)105 b(::=)52 b(\()f(Vector)i
2182 (...)f(\))257 693 y(Context)105 b(::=)52 b(\()f(Vector)i(...)f(\))257
2183 814 y(Vector)156 b(::=)52 b(#[)g(Integer)h(...)f(])257
2184 934 y(Nn)g(::=)g(Integer)257 1054 y(Np)g(::=)g(Integer)257
2185 1175 y(Nl)g(::=)g(Integer)257 1295 y(Nm)g(::=)g(Integer)257
2186 1416 y(Bg)g(::=)g(Integer)257 1536 y(Nq)g(::=)g(0)g(|)f(1)257
2187 1716 y Fn(This)36 b(syn)m(tax)i(w)m(as)e(c)m(hosen)i(so)e(as)g(to)f
2188 (ease)i(the)f(generation)f(of)g(problems)h(b)m(y)g(a)g(Lisp)f(pro-)257
2189 1837 y(gram.)51 b(In)35 b(particular,)g(eac)m(h)h Fl(Problem)h
2190 Fn(is)e(a)g(balanced)g(list,)f(as)i(far)f(as)g(b)s(oth)g(paren)m
2191 (theses)257 1957 y(and)e(brac)m(k)m(ets)i(are)d(concerned.)403
2192 2155 y Fm(\017)48 b Fl(Comments)36 b Fn(are)d(arbitrary)g(lists.)45
2193 b(These)35 b(commen)m(ts)e(are)h(written)f(v)m(erbatim)g(to)g(the)501
2194 2275 y(output)g(\014le,)f(and)h(are)g(useful)f(to)g(k)m(eep)j(trac)m(k)
2195 e(of)f(problems)g(and)g(solutions.)501 2433 y(Note)47
2196 b(that)f(sev)m(eral)h Fl(Problem)p Fn(s)i(can)e(b)s(e)g(giv)m(en)f(to)g
2197 (PIP)i(in)d(the)i(same)g(\014le.)85 b(The)501 2553 y(problems)26
2198 b(ma)m(y)g(b)s(e)g(separated)h(b)m(y)h(an)m(y)f(text)g(that)f(do)s(es)g
2199 (not)g(con)m(tain)g(a)g(paren)m(thesis.)501 2674 y(By)39
2200 b(using)f(Unix)g(FIF)m(O's)g(as)g(input)g(and)g(output)h(\014les,)h(it)
2201 d(is)g(easy)j(to)d(con)m(v)m(ert)j(the)501 2794 y(presen)m(t)34
2202 b(implemen)m(tation)c(of)i(PIP)h(in)m(to)f(a)h(linear)e(programming)e
2203 (serv)m(er.)403 2990 y Fm(\017)48 b Fl(Nn)34 b Fn(is)e(the)i(n)m(um)m
2204 (b)s(er)f(of)g(unkno)m(wns)i(in)d(the)h(program)f(\(whic)m(h)h(w)m(as)h
2205 (denoted)g(b)m(y)g Fk(n)f Fn(in)501 3110 y(the)g(\014rst)g(section\).)
2206 403 3306 y Fm(\017)48 b Fl(Np)33 b Fn(is)f(the)h(n)m(um)m(b)s(er)g(of)f
2207 (\(sym)m(b)s(olic\))g(parameters)g(\()p Fk(p)p Fn(\))403
2208 3502 y Fm(\017)48 b Fl(Nl)33 b Fn(is)f(the)h(n)m(um)m(b)s(er)g(of)f
2209 (inequalities)f(de\014ning)i(the)g(domain)e(of)h(the)h(unkno)m(wns)h
2210 (\()p Fk(l)r Fn(\).)403 3697 y Fm(\017)48 b Fl(Nm)33
2211 b Fn(is)f(the)h(n)m(um)m(b)s(er)g(of)f(inequalities)f(satis\014ed)i(b)m
2212 (y)h(the)f(parameters)f(\()p Fk(m)p Fn(\).)403 3893 y
2213 Fm(\017)48 b Fl(Bg)28 b Fn(is)f(the)h(index)f(of)g(a)g(\\Big")f
2214 (parameter)g(whose)j(v)-5 b(alue)27 b(is)f(assumed)i(to)f(b)s(e)h
2215 (in\014nitely)501 4014 y(large.)61 b(That)39 b(is,)h(if)e(the)h(big)f
2216 (parameter)g(app)s(ears)h(with)f(a)h(p)s(ositiv)m(e)f(co)s(e\016cien)m
2217 (t)h(in)501 4134 y(a)45 b(form)e Fk(\036)p Fn(,)48 b(then)d(w)m(e)h
2218 (can)e(immediately)e(deduce)k(that)f Fk(\036)j(>)g Fn(0.)80
2219 b(If)44 b Fl(Bg)h Fn(is)f(set)i(to)501 4254 y(a)39 b(nonp)s(ositiv)m(e)
2220 g(v)-5 b(alue,)40 b(then)g(there)g(is)f(no)g(big)f(parameter)h(in)g
2221 (the)g(problem)f(to)h(b)s(e)501 4375 y(solv)m(ed.)501
2222 4533 y(Be)30 b(a)m(w)m(are)g(that)g Fl(Bg)g Fn(is)f(the)h(column)e
2223 (rank)i(of)f(the)h(corresp)s(onding)f(parameter)g(in)g(the)501
2224 4653 y Fl(Tableau)p Fn(,)35 b(and)d(that)h(the)g(\014rst)g(v)-5
2225 b(alid)31 b(v)-5 b(alue)32 b(for)g(it)f(is)h Fl(Nn+1)p
2226 Fn(.)403 4849 y Fm(\017)48 b Fl(Nq)29 b Fn(is)e(an)h(in)m(teger)g(but)h
2227 (should)f(b)s(e)g(in)m(terpreted)h(as)f(a)g(b)s(o)s(olean)e(v)-5
2228 b(alue)28 b Fj(\022)-49 b(a)28 b(la)e Fn(C,)j(that)f(is,)501
2229 4969 y(it)f(denotes)h(\\true")g(if)e(its)h(v)-5 b(alue)27
2230 b(is)g(nonzero.)42 b(If)28 b Fl(Nq)g Fn(is)f(true,)i(then)f(an)f(in)m
2231 (teger-v)-5 b(alued)501 5090 y(solution)44 b(is)g(lo)s(ok)m(ed)h(for.)
2232 80 b(Otherwise,)48 b(PIP)e(\014nds)g(the)f(lexicographic)f(minim)m(um)
2233 501 5210 y(rational)30 b(solution)h(to)i(the)g(problem.)403
2234 5406 y Fm(\017)48 b Fl(Tableau)f Fn(stores)f(the)f(set)h(of)e
2235 (inequalities)f(de\014ning)i(the)h(domain)d(of)h(unkno)m(wns.)501
2236 5526 y(Eac)m(h)38 b Fl(Vector)g Fn(represen)m(ts)g(one)f(inequalit)m(y)
2237 -8 b(.)54 b(The)37 b(en)m(tries)g(in)f Fl(Vector)i Fn(are,)f(in)f(this)
2238 501 5646 y(order:)p eop
2239 %%Page: 4 4
2240 4 3 bop 257 266 a Fj(2)98 b(USING)32 b(THE)i(PIP)f(SOFTW)-11
2241 b(ARE)1879 b Fn(4)611 573 y Fg({)49 b Fn(the)33 b(co)s(e\016cien)m(ts)h
2242 (of)e(the)h(unkno)m(wns)h(\(I.e.,)g(a)e(ro)m(w)h(of)f(matrix)f
2243 Fg(A)p Fn(\),)611 721 y Fg({)49 b Fn(the)33 b(\(additiv)m(e\))f
2244 (constan)m(t,)h(\(I.e.,)h(an)e(en)m(try)i(of)e(v)m(ector)c
2245 Fk(~)-44 b(c)p Fn(\),)611 868 y Fg({)49 b Fn(the)33 b(co)s(e\016cien)m
2246 (ts)h(of)e(the)h(parameters)f(\(I.e.,)i(a)e(ro)m(w)h(of)f(matrix)f
2247 Fg(B)p Fn(\))501 1058 y(This)h(notation)e(hea)m(vily)i(dep)s(ends)h(on)
2248 e(the)h(p)s(ositions)f(giv)m(en)g(to)h(unkno)m(wns)h(and)f(pa-)501
2249 1178 y(rameters:)43 b(it)31 b(is)h(the)g(resp)s(onsabilit)m(y)f(of)g
2250 (the)i(user)f(to)g(enforce)g(a)g(coheren)m(t)h(ordering)501
2251 1299 y(of)38 b(co)s(e\016cien)m(ts)h(and)f(to)f(set)i(a)e(co)s
2252 (e\016cien)m(t)i(to)e(zero)i(when)g(the)f(corresp)s(onding)g(un-)501
2253 1419 y(kno)m(wn/parameter)33 b(do)s(es)h(not)e(app)s(ear.)501
2254 1574 y(There)c(are)f Fk(l)i Fn(suc)m(h)g Fl(Vector)p
2255 Fn(s)g(in)d Fl(Tableau)p Fn(,)k(and)d(eac)m(h)h Fl(vector)g
2256 Fn(exactly)f(has)h Fk(n)11 b Fn(+)g(1)g(+)g Fk(p)501
2257 1694 y Fn(en)m(tries.)403 1883 y Fm(\017)48 b Fn(In)39
2258 b(a)f(similar)d(w)m(a)m(y)-8 b(,)41 b Fl(Context)g Fn(is)d(a)g(list)f
2259 (of)h Fl(Vector)p Fn(s.)63 b(Eac)m(h)39 b Fl(Vector)h
2260 Fn(represen)m(ts)h(a)501 2004 y(ro)m(w)30 b(of)f(Matrix)g
2261 Fg(M)h Fn(follo)m(w)m(ed)f(b)m(y)h(the)g(corresp)s(onding)g(en)m(try)h
2262 (in)d(v)m(ector)3191 1978 y Fk(~)3198 2004 y(h)p Fn(.)43
2263 b Fl(Context)501 2124 y Fn(th)m(us)34 b(includes)e Fk(m)h
2264 Fl(Vector)p Fn(s)i(of)d Fk(p)22 b Fn(+)g(1)32 b(en)m(tries.)257
2265 2377 y Fg(2.1.1)113 b(Example)257 2561 y Fn(This)33 b(example)f(is)g
2266 (tak)m(en)i(from)d([2].)44 b(W)-8 b(e)33 b(consider)f(the)h(lo)s(op)f
2267 (nest)h(b)s(elo)m(w:)257 2723 y Fl(for)52 b(i:=)g(0)g(to)g(m)f(do)411
2268 2843 y(for)h(j)g(:=)g(0)f(to)h(n)f(do)719 b({II})565
2269 2963 y(for)52 b(k)g(:=)f(0)h(to)g(i+j)g(do)g(...)257
2270 3124 y Fn(and)42 b(w)m(e)g(wish)g(to)f(rewrite)h(this)f(nest)h(in)f
2271 (the)h(order)g Fl(k,)51 b(j,)h(i)p Fn(.)71 b(The)42 b(b)s(ounds)g(on)g
2272 Fl(k)f Fn(can)257 3245 y(easily)g(b)s(e)h(guessed)h(\(0)g
2273 Fm(\024)g Fk(k)j Fm(\024)e Fk(m)28 b Fn(+)g Fk(n)p Fn(\),)44
2274 b(so)e(let's)f(lo)s(ok)g(for)g(the)h(lo)m(w)m(er)f(b)s(ound)h(on)g
2275 Fl(j)f Fn(in)257 3365 y(the)33 b(rewritten)f(nest.)45
2276 b(This)32 b(lo)m(w)m(er)h(b)s(ound)f(on)g Fl(j)h Fn(can)f(b)s(e)h
2277 (found)f(b)m(y)h(solving)f(the)g(follo)m(wing)257 3486
2278 y(problem:)1047 3606 y Fm(D)1124 3621 y Fe(2)1163 3606
2279 y Fn(\()p Fk(k)s Fn(\))c(=)f Fm(f)17 b Fk(<)11 b(j;)17
2280 b(i)11 b(>)44 b Fm(j)p Fk(i)28 b Fm(\024)g Fk(m;)17 b(j)34
2281 b Fm(\024)28 b Fk(n;)17 b(k)31 b Fm(\024)d Fk(i)22 b
2282 Fn(+)g Fk(j)6 b Fm(g)p Fk(:)404 3759 y Fn(This)37 b(problem)f(is)g(to)h
2283 (b)s(e)g(solv)m(ed)h(in)e(the)h(con)m(text)h Fk(k)h Fm(\024)c
2284 Fk(m)26 b Fn(+)f Fk(n)p Fn(.)57 b(The)38 b(input)e(\014le)h(ma)m(y)257
2285 3880 y(th)m(us)d(lo)s(ok)d(lik)m(e)h(this:)257 4041 y
2286 Fl(\()52 b(\(Lower)h(bound)f(on)g(j)g(after)g(loop)h(inversion)514
2287 4161 y(\(unknowns)g(j)f(i\))514 4282 y(\(parameters)i(k)d(m)h(n\)\))360
2288 4402 y(2)f(3)h(3)f(1)h(-1)g(1)360 4522 y(\()f(#[0)h(-1)g(0)g(0)f(1)h
2289 (0])462 4643 y(#[-1)h(0)e(0)h(0)f(0)h(1])462 4763 y(#[1)g(1)g(0)g(-1)f
2290 (0)h(0])411 4883 y(\))360 5004 y(\()f(#[-1)i(1)e(1)h(0]\))257
2291 5124 y(\))257 5285 y Fn(The)30 b(\014rst)f(sequence)i(of)d(in)m(tegers)
2292 h(should)g(b)s(e)g(read)f(as:)42 b(This)29 b(problem)e(has)i(2)g(unkno)
2293 m(wns)h(\()p Fk(i)257 5406 y Fn(and)k Fk(j)6 b Fn(\))33
2294 b(and)g(3)g(parameters)h(\()p Fk(k)s Fn(,)f Fk(m)h Fn(and)f
2295 Fk(n)p Fn(\).)46 b(The)34 b(domain)e(is)g(de\014ned)j(b)m(y)f(3)f
2296 (inequalities,)257 5526 y(the)d(con)m(text)h(b)m(y)f(1)f(inequalit)m(y)
2297 -8 b(.)41 b(There)31 b(is)e(no)g(\(-1\))f(big)h(parameter)g(and)g(it)f
2298 (is)h(true)h(\(1\))f(that)257 5646 y(w)m(e)34 b(are)f(lo)s(oking)d(for)
2299 i(an)h(in)m(teger)f(solution.)p eop
2300 %%Page: 5 5
2301 5 4 bop 257 266 a Fj(2)98 b(USING)32 b(THE)i(PIP)f(SOFTW)-11
2302 b(ARE)1879 b Fn(5)257 573 y Fh(2.2)136 b(Calling)46 b(PIP)257
2303 758 y Fn(PIP)34 b(is)e(called)f(b)m(y)j(the)f(follo)m(wing)d(command:)
2304 514 961 y Fl(pip)52 b([-s|-v])h([-z])f([input)h([output]])257
2305 1164 y Fn(PIP)39 b(prin)m(ts)f(some)f(information)e(on)i(the)h(screen)i
2306 (after)d(ha)m(ving)g(solv)m(ed)i(a)e(problem.)58 b(The)257
2307 1285 y Fl(-s)29 b Fn(\(silen)m(t)f(mo)s(de\))g(switc)m(hes)i(this)f
2308 (feature)g(o\013.)42 b(On)28 b(the)h(con)m(trary)-8 b(,)30
2309 b(the)f(v)m(erb)s(ose)h Fl(-v)g Fn(option)257 1405 y(tells)39
2310 b(PIP)i(to)f(cop)m(y)-8 b(,)43 b(in)c(a)h(\014le,)h(all)d(the)i(input)g
2311 (data)g(and)g(all)e(the)i(in)m(termediary)f(results.)257
2312 1526 y(The)33 b(name)f(of)f(this)g(\014le)h(is)f(giv)m(en)h(either)g(b)
2313 m(y)h(the)f(v)-5 b(ariable)30 b Fl(DEBUG)j Fn(in)e(the)i(en)m(vironmen)
2314 m(t)f(or)257 1646 y(is)g(built)f(b)m(y)j Fl(mktemp)p
2315 Fn(.)404 1766 y(If)27 b(the)h Fl(-z)h Fn(option)d(is)i(giv)m(en,)g
2316 (then)h(the)f(solution)e(is)h(somewhat)h(simpli\014ed)e(\(see)j(b)s
2317 (elo)m(w\).)404 1887 y Fl(input)d Fn(and)g Fl(output)h
2318 Fn(are)f(the)g(names)f(of)g(the)h(input)f(\(data\))g(and)h(output)g
2319 (\(results\))f(\014les,)257 2007 y(resp)s(ectiv)m(ely)-8
2320 b(.)65 b(If)39 b(no)g Fl(output)i Fn(\()p Fl(input)p
2321 Fn(\))f(\014le)f(is)f(giv)m(en,)j(then)f(the)g(results)f(are)h(prin)m
2322 (ted)f(to)257 2127 y(the)33 b(standard)g(output)g(\(input\).)257
2323 2387 y Fg(2.2.1)113 b(Messages)403 2572 y Fm(\017)48
2324 b Fl(Version)53 b(X.x)p Fn(.)45 b(Curren)m(tly)-8 b(,)33
2325 b Fl(D.1)p Fn(.)403 2775 y Fm(\017)48 b Fl(cross)53 b(:)103
2326 b(<n>,)52 b(alloc)h(:)102 b(<m>)41 b Fn(This)g(message)g(is)f(output)g
2327 (after)g(solving)g(eac)m(h)501 2896 y(problem.)j(The)33
2328 b(v)-5 b(alue)32 b(of)g Fl(<n>)h Fn(giv)m(es)g(an)g(idea)f(of)g(the)h
2329 (complexit)m(y)f(of)g(the)h(problem.)257 3156 y Fg(Errors)k(related)g
2330 (to)g(the)g(input)403 3359 y Fm(\017)48 b Fl(Syntax)53
2331 b(error)p Fn(:)45 b(un)m(balanced)33 b(paren)m(theses)i(in)d(the)h
2332 (input.)403 3562 y Fm(\017)48 b Fl(Your)53 b(computer)g(doesn't)g(have)
2333 f(enough)h(memory)p Fn(:)45 b(self)33 b(explanatory)-8
2334 b(.)257 3822 y Fg(Errors)37 b(related)g(to)g(the)g(solution)403
2335 4026 y Fm(\017)48 b Fl(Integer)53 b(Overflow)p Fn(:)78
2336 b(A)49 b(n)m(um)m(b)s(er)g(has)g(b)s(een)g(generated)g(that)g(is)f(to)s
2337 (o)f(large)h(to)501 4146 y(b)s(e)41 b(accommo)s(dated)e(in)g(a)h(32)g
2338 (bit)f(in)m(teger.)67 b(Chec)m(k)42 b(the)f(input)f(and/or)g(switc)m(h)
2339 h(to)501 4266 y(Zbigniew)32 b(Chamski's)g(in\014nite)g(precision)g(PIP)
2340 -8 b(.)403 4470 y Fm(\017)48 b Fl(The)k(solution)i(is)e(too)g(complex)p
2341 Fn(:)44 b(the)31 b(solution)e(quast)j(has)f(gro)m(wn)g(b)s(ey)m(ond)h
2342 (the)501 4590 y(memory)k(allo)s(cated)f(to)h(it.)55 b(Chec)m(k)38
2343 b(the)f(input)g(and/or)f(c)m(hange)h(the)g(v)-5 b(alue)37
2344 b(of)f(con-)501 4710 y(stan)m(t)d Fl(SOL)p 907 4710 31
2345 4 v 38 w(SIZE)h Fn(in)d(\014le)i Fl(type.h)p Fn(,)h(then)f(rebuild)f
2346 (PIP)-8 b(.)403 4914 y Fm(\017)48 b Fl(Memory)53 b(overflow)p
2347 Fn(:)46 b(self)32 b(explanatory)-8 b(.)403 5117 y Fm(\017)48
2348 b Fl(<file>)53 b(unaccessible)p Fn(:)62 b(one)41 b(of)f(the)h(input,)h
2349 (output)f(or)f(debug)h(\014le)f(cannot)g(b)s(e)501 5238
2350 y(op)s(ened.)p eop
2351 %%Page: 6 6
2352 6 5 bop 257 266 a Fj(2)98 b(USING)32 b(THE)i(PIP)f(SOFTW)-11
2353 b(ARE)1879 b Fn(6)257 573 y Fg(Dimension)36 b(errors)403
2354 766 y Fm(\017)48 b Fl(Too)k(much)h(variables)403 966
2355 y Fm(\017)48 b Fl(Too)k(much)h(parameters)37 b Fn(:)48
2356 b(Chec)m(k)37 b(the)f(input)e(and/or)h(c)m(hange)h(the)f(v)-5
2357 b(alue)34 b(of)h(con-)501 1087 y(stan)m(ts)f Fl(MAXCOL)g
2358 Fn(and)f Fl(MAXPARM)h Fn(in)e(\014le)g Fl(type.h)p Fn(,)i(then)f
2359 (rebuild)f(PIP)-8 b(.)257 1345 y Fg(Implemen)m(tation)42
2360 b(errors)97 b Fn(All)37 b(suc)m(h)j(error)e(messages)h(b)s(egin)f(b)m
2361 (y)h(the)g(w)m(ord)g Fl(Syserr)p Fn(.)257 1465 y(These)i(messages)f
2362 (indicate)d(a)i(bug)g(in)f(the)h(implemen)m(tation.)59
2363 b(Y)-8 b(ou)39 b(should)f(rep)s(ort)h(suc)m(h)257 1585
2364 y(ev)m(en)m(ts)82 b(b)m(y)f(sending)f(a)f(cop)m(y)i(of)e(the)h(input)f
2365 (\014le)g(b)m(y)i(e-mail)76 b(to)k(the)g(author,)257
2366 1706 y Fl(Paul.Feautrier@prism.uvs)q(q.f)q(r)32 b Fn(who)26
2367 b(will)d(endea)m(v)m(or)28 b(to)d(solv)m(e)i(the)f(problem)f(as)h(so)s
2368 (on)257 1826 y(as)33 b(p)s(ossible.)257 2113 y Fh(2.3)136
2369 b(Output)44 b(Data)257 2298 y Fn(The)34 b(output)f(\014le)f(can)h(b)s
2370 (e)f(describ)s(ed)i(b)m(y)f(the)g(follo)m(wing)d(grammar:)257
2371 2491 y Fg(Grammar)37 b(2)49 b Fi(:)257 2684 y Fl(File)k(::=)f(Result)h
2372 (...)257 2805 y(Result)g(::)f(\()g(Comments)h(Solution)g(\))257
2373 2925 y(Solution)h(::=)e(Quast_group)462 3046 y(|)g(void)257
2374 3166 y(Quast_group)j(::=)d(Quast)462 3286 y(|)g(Newparm)h(...)f(Quast)
2375 257 3407 y(Quast)h(::=)f(Form)462 3527 y(|)g(\(if)g(Vector)h
2376 (Quast_group)h(Quast_group\))257 3647 y(Form)f(::=)f(\(list)g(Vector)h
2377 (...\))462 3768 y(|)f(nil)257 3888 y(Newparm)h(::=)g(\(newparm)g
2378 (Integer)g(Vector)g(Integer\))257 4009 y(Vector)g(::=)f(#[)g
2379 (Coefficient)i(...)e(])257 4129 y(Coefficient)j(::=)d(Integer)h(|)e
2380 (Integer)i(/)f(Integer)257 4322 y Fn(The)38 b Fl(Comments)g
2381 Fn(are)f(copied)f(from)g(the)h(input)f(\014le.)55 b(The)37
2382 b Fl(Solution)i Fn(is)d(said)g(to)g(b)s(e)h Fl(void)257
2383 4443 y Fn(when)k(the)g(initial)36 b(con)m(text)41 b(is)e(v)m(oid.)65
2384 b(Otherwise,)42 b(it)d(is)g(giv)m(en)h(as)g(a)g(quast)h(written)e
2385 Fj(\022)-49 b(a)40 b(la)257 4563 y Fn(Lisp.)i(The)30
2386 b(quast)g(ma)m(y)e(p)s(ossibly)h(b)s(e)g(preceded)h(b)m(y)g(the)f
2387 (de\014nition)f(of)g(one)i(or)e(sev)m(eral)i(new)257
2388 4683 y(parameters.)404 4804 y(The)22 b(v)m(ector)h(co)s(e\016cien)m(ts)
2389 g(ma)m(y)f(b)s(e)g(either)f(in)m(tegers)h(or)g(rationals)d(written)j
2390 (as)g Fl(num/denom)p Fn(.)257 4924 y(The)34 b(latter)d(case)j(o)s
2391 (ccurs)f(if)f Fl(Nq)h Fn(had)f(b)s(een)i(set)f(to)f(0)h(in)e(the)i
2392 (input)f(\014le.)404 5045 y(In)24 b(the)h(solution,)f(a)g
2393 Fl(Vector)i Fn(represen)m(ts)h(an)d(a\016ne)h(form;)h(eac)m(h)f(en)m
2394 (try)g(is)f(the)h(co)s(e\016cien)m(t)257 5165 y(of)j(the)h(corresp)s
2395 (onding)f(parameter)f(\(the)i(parameter)e(of)h(the)h(same)f(rank\).)42
2396 b(The)29 b(last)e(en)m(try)257 5285 y(is)32 b(the)h(additiv)m(e)f
2397 (constan)m(t.)404 5406 y(The)39 b(de\014nition)f(of)g(a)h(new)g
2398 (parameter)f(b)s(egins)h(with)f(the)h(k)m(ey-w)m(ord)i
2399 Fl(newparm)p Fn(,)h(then)257 5526 y(a)e(rank)g(n)m(um)m(b)s(er,)i(a)e
2400 (v)m(ector)h(of)e(co)s(e\016cien)m(ts,)k(and)d(a)f(denominator.)64
2401 b(The)41 b(new)g(parame-)257 5646 y(ter)e(is)f(equal)h(to)f(the)h(in)m
2402 (teger)g(division)e(of)h(the)h(v)m(ector)g(b)m(y)h(the)f(denominator.)
2403 60 b(The)40 b(new)p eop
2404 %%Page: 7 7
2405 7 6 bop 257 266 a Fj(2)98 b(USING)32 b(THE)i(PIP)f(SOFTW)-11
2406 b(ARE)1879 b Fn(7)257 573 y(parameter)38 b(can)h(only)f(app)s(ear)g(in)
2407 g(the)h Fl(Quast)h Fn(follo)m(wing)35 b(its)j(de\014nition.)61
2408 b(In)m(tro)s(ducing)38 b(a)257 693 y(new)d(parameter)e(adds)h(one)g(en)
2409 m(try)g(in)f(the)h(list)e(of)h(parameters,)h(so)g(the)g(length)f(of)g
2410 (v)m(ectors)257 814 y(in)40 b(the)g(solution)f(is)g(not)h(constan)m(t.)
2411 67 b(Ho)m(w)m(ev)m(er,)44 b(this)c(length)f(is)h(alw)m(a)m(ys)g(equal)g
2412 (to)g(1)g(plus)257 934 y(the)34 b(n)m(um)m(b)s(er)f(of)g(original)c
2413 (parameters)k(plus)g(the)g(n)m(um)m(b)s(er)h(of)e(new)i(parameters)f
2414 (curren)m(tly)257 1054 y(de\014ned.)404 1175 y(The)26
2415 b(solution)e(is)g(a)h(m)m(ulti-lev)m(el)d(conditional)h(expression)j
2416 (\(a)f(tree)h(of)f(nested)h(condition-)257 1295 y(als.\))62
2417 b(A)39 b(predicate)g(expression)h Fk(p)f Fn(should)g(b)s(e)g(understo)s
2418 (o)s(d)g(as)g(the)h(b)s(o)s(olean)d(expression)257 1416
2419 y Fk(p)46 b Fm(\025)f Fn(0.)75 b(Lea)m(v)m(es)45 b(of)d(the)h
2420 (conditional)e(tree)i(are)g(either)g Fl(nil)p Fn(,)j(meaning)c(that)h
2421 (the)g(input)257 1536 y(problem)28 b(has)h(no)f(solution,)g(or)g(a)g
2422 Fl(Form)p Fn(.)44 b(A)28 b Fl(Form)i Fn(is)e(a)g(list)f(of)h(v)m
2423 (ectors,)j(eac)m(h)f(v)m(ector)f(giving)257 1656 y(the)k(v)-5
2424 b(alue)32 b(of)g(the)h(corresp)s(onding)g(unkno)m(wn.)257
2425 1916 y Fg(2.3.1)113 b(Example)257 2101 y Fn(The)29 b(output)f(of)f(PIP)
2426 h(is)g(not)f(in)m(tended)h(for)f(h)m(uman)h(consumption.)41
2427 b(No)28 b(attempt)f(has)h(b)s(een)257 2221 y(made)38
2428 b(to)g(implemen)m(t)f(a)h(prett)m(y-prin)m(ter.)61 b(In)39
2429 b(the)g(in)m(terest)g(of)f(readabilit)m(y)-8 b(,)38 b(some)g(of)g(the)
2430 257 2342 y(result)31 b(\014les)g(in)e(this)i(pap)s(er)f(ha)m(v)m(e)i(b)
2431 s(een)g(b)s(eauti\014ed)e(b)m(y)i(hand.)43 b(The)31 b(reader)g(should)g
2432 (not)f(b)s(e)257 2462 y(surprised)k(if)d(he)i(gets)g(results)g(with)f
2433 (di\013eren)m(t)h(la)m(y)m(outs)g(when)h(running)e(the)h(examples.)404
2434 2582 y(Here)g(is)f(the)h(output)g(solution)e(\014le)h(for)g(the)h
2435 (example)f(ab)s(o)m(v)m(e)i(\(2.1.1\):)257 2786 y Fl(\()52
2436 b(\(Lower)h(bound)f(on)g(j)g(after)g(loop)h(inversion)565
2437 2906 y(\(unknowns)h(j)d(i\))565 3026 y(\(parameters)j(k)e(m)f(n\))h(1)g
2438 (\)\(if)g(#[)g(-1)f(1)h(0)g(0])257 3147 y(\(list)h(#[)f(0)f(0)h(0)f(0])
2439 257 3267 y(#[)h(1)g(0)f(0)h(0])257 3388 y(\))257 3508
2440 y(\(list)h(#[)f(1)f(-1)h(0)g(0])257 3628 y(#[)g(0)g(1)f(0)h(0])257
2441 3749 y(\))257 3869 y(\))257 3990 y(\))257 4193 y Fn(T)-8
2442 b(o)46 b(express)i(this)e(solution,)h(no)f(new)g(parameter)g(had)g(to)f
2443 (b)s(e)h(in)m(tro)s(duced.)83 b(The)47 b(form)257 4313
2444 y(asso)s(ciated)33 b(to)f(the)h(\014rst)g(conditional)d(is:)1091
2445 4533 y Fm(\000)p Fn(1)22 b Fm(\002)h Fk(k)i Fn(+)d(1)g
2446 Fm(\002)g Fk(m)h Fn(+)f(0)g Fm(\002)g Fk(n)h Fn(+)f(0)g
2447 Fm(\002)g Fn(1)28 b(=)f Fk(m)c Fm(\000)f Fk(k)257 4753
2448 y Fn(so)28 b(the)f(test)h(should)f(b)s(e)g(read)h(as)f
2449 Fk(m)11 b Fm(\000)g Fk(k)31 b Fm(\025)e Fn(0.)41 b(If)27
2450 b(this)g(inequalit)m(y)f(holds,)i(then)f(the)h(solution)257
2451 4874 y(is)k Fk(<)c Fn(0)p Fk(;)17 b(k)30 b(>)p Fn(.)44
2452 b(Otherwise,)33 b(the)g(solution)e(is)h Fk(<)c(m)22 b
2453 Fm(\000)h Fk(k)s(;)17 b(m)27 b(>)p Fn(.)404 4994 y(T)-8
2454 b(o)32 b(sum)h(things)f(up,)h(the)g(lexicographical)d(minim)m(um)f(of)j
2455 Fm(D)2671 5009 y Fe(2)2743 4994 y Fn(is:)462 5198 y Fl(if)52
2456 b(m-k)g(>=)g(0)g(then)g(<0,)g(k>)g(else)g(<m-k,)h(m>.)257
2457 5401 y Fn(Hence)34 b(the)f(lo)m(w)m(er)g(b)s(ound)g(on)f(the)h(\014rst)
2458 g(co)s(ordinate:)462 5604 y Fl(if)52 b(m-k)g(>=)g(0)g(then)g(0)f(else)i
2459 (m-k)p eop
2460 %%Page: 8 8
2461 8 7 bop 257 266 a Fj(2)98 b(USING)32 b(THE)i(PIP)f(SOFTW)-11
2462 b(ARE)1879 b Fn(8)257 573 y Fg(2.3.2)113 b(Simplifying)34
2463 b(the)k(solution)257 758 y Fn(The)32 b(solution)e(of)h(a)g(parametric)e
2464 (problem)h(ma)m(y)h(b)s(e)g(in)g(the)g(form)f(of)h(a)g(quast)h(all)d
2465 (of)h(whose)257 878 y(lea)m(v)m(es)g(are)f(nil.)41 b(This)29
2466 b(means)g(in)f(fact)h(that)f(the)i(original)25 b(p)s(olyhedron)k(is)f
2467 (empt)m(y)i(whatev)m(er)257 998 y(the)j(v)-5 b(alues)32
2468 b(of)f(the)i(parameters.)43 b(An)32 b(example,)g(due)h(to)e(Dirk)g
2469 (Fimmel,)e(is)j(the)g(follo)m(wing:)257 1207 y Fl(\(\(\(i)53
2470 b(j)e(1\)\(m)i(n\)\))309 1328 y(2)e(2)h(7)f(0)h(-1)g(1)309
2471 1448 y(\(#[2)g(6)f(-9)h(0)g(0])360 1568 y(#[5)g(-3)g(0)f(0)h(0])360
2472 1689 y(#[2)g(-10)g(15)g(0)f(0])360 1809 y(#[-2)h(6)g(-3)g(0)f(0])360
2473 1930 y(#[-2)h(-6)g(17)g(0)f(0])360 2050 y(#[0)h(1)f(0)h(-1)g(0])360
2474 2170 y(#[1)g(0)f(0)h(0)g(-1])309 2291 y(\))309 2411 y(\(\))257
2475 2532 y(\))257 2741 y Fn(Without)32 b(the)h Fl(-z)g Fn(option,)f(the)h
2476 (solution)e(is:)257 2930 y Fl(\(\(\(i)53 b(j)e(1\)\(m)i(n\))e(-1)h(\))
2477 309 3050 y(\(if)g(#[)f(-4)h(0)g(5])360 3170 y(\(if)g(#[)g(0)f(-4)h(3])
2478 411 3291 y(\(\))411 3411 y(\(if)g(#[)g(0)g(-2)f(9])462
2479 3531 y(\(if)h(#[)g(0)g(-2)g(3])514 3652 y(\(newparm)h(2)f(\(div)g(#[)g
2480 (0)f(2)h(3])g(6\)\))514 3772 y(\(newparm)h(3)f(\(div)g(#[)g(0)f(2)h(10)
2481 g(7])f(12\)\))514 3893 y(\(newparm)i(4)f(\(div)g(#[)g(0)f(4)h(0)f(2)h
2482 (1])g(6\)\))514 4013 y(\(\))514 4133 y(\(if)g(#[)g(0)f(-2)h(7])565
2483 4254 y(\(newparm)h(2)f(\(div)g(#[)g(0)g(4)f(3])h(6\)\))565
2484 4374 y(\(if)g(#[)g(0)f(-8)h(6)g(11])g(\(\))g(\(\)\))565
2485 4494 y(\(\)\)\))462 4615 y(\(\)\)\))360 4735 y(\(if)g(#[)g(-1)g(0)f(3])
2486 411 4856 y(\(if)h(#[)g(-1)g(0)f(2])462 4976 y(\(if)h(#[)g(10)g(-2)g
2487 (-15])g(\(\)\(\)\))462 5096 y(\(\)\))360 5217 y(\(\)\)\))257
2488 5337 y(\))257 5526 y Fn(Insp)s(ection)27 b(rev)m(eals)h(that)e(all)f
2489 (lea)m(v)m(es)j(are)e Fl(\(\))p Fn(.)42 b(With)26 b(the)h
2490 Fl(-z)h Fn(option,)f(the)g(solution)e(is)h(m)m(uc)m(h)257
2491 5646 y(simpler:)p eop
2492 %%Page: 9 9
2493 9 8 bop 257 266 a Fj(2)98 b(USING)32 b(THE)i(PIP)f(SOFTW)-11
2494 b(ARE)1879 b Fn(9)257 573 y Fl(\(\(\(i)53 b(j)e(1\)\(m)i(n\))e(-1)h
2495 (\)\(\))257 693 y(\))257 981 y Fh(2.4)136 b(The)44 b(P)l(o)l(w)l(er)i
2496 (of)g(PIP)257 1166 y Fn(In)g(the)g(follo)m(wing)d(sections,)49
2497 b(w)m(e)e(explain)e(ho)m(w)h(PIP)g(can)g(b)s(e)g(used)g(to)g(solv)m(e)f
2498 (extended)257 1286 y(classes)34 b(of)e(problems:)403
2499 1485 y Fm(\017)48 b Fn(Problems)32 b(where)i(equalities)d(o)s(ccur.)403
2500 1686 y Fm(\017)48 b Fn(Problems)32 b(where)i(a)e(lexicographical)e
2501 Fi(maximum)i Fn(has)h(to)f(b)s(e)h(found.)403 1888 y
2502 Fm(\017)48 b Fn(Cases)34 b(when)g(linear)d(cost)i(functions)g(are)f(to)
2503 g(b)s(e)h(optimized.)403 2090 y Fm(\017)48 b Fn(Problems)32
2504 b(where)i(unkno)m(wns)h(and/or)d(parameters)h(ma)m(y)f(b)s(e)h(negativ)
2505 m(e)257 2349 y Fg(2.4.1)113 b(Handling)36 b(Equalities)257
2506 2533 y Fn(When)i(the)f(input)f(problem)g(con)m(tains)h
2507 Fk(r)i Fn(a\016ne)e(equalities)f Fk(f)2584 2548 y Ff(i)2647
2508 2533 y Fn(=)e(0,1)h Fm(\024)g Fk(i)g Fm(\024)g Fk(r)s
2509 Fn(,)j(one)f(ma)m(y)257 2654 y(just)k(write)g Fk(r)i
2510 Fn(inequalities)c Fk(f)1372 2669 y Ff(i)1442 2654 y Fm(\025)i
2511 Fn(0)g(and)g Fk(r)i Fn(inequalties)d Fk(f)2477 2669 y
2512 Ff(i)2546 2654 y Fm(\024)i Fn(0,)h(th)m(us)e(satisfying)f(PIP's)257
2513 2774 y(input)c(syn)m(tax.)55 b(Ho)m(w)m(ev)m(er,)39 b(one)d(ma)m(y)g
2514 (notice)g(that)f(only)h Fk(r)27 b Fn(+)d(1)36 b(inequalities)e(are)i
2515 (needed:)257 2894 y Fk(f)305 2909 y Ff(i)361 2894 y Fm(\025)28
2516 b Fn(0,)33 b(1)27 b Fm(\024)h Fk(i)g Fm(\024)h Fk(r)s
2517 Fn(,)j(and)h(the)g(follo)m(wing)c(inequalit)m(y:)1802
2518 3053 y Ff(r)1759 3078 y Fd(X)1762 3260 y Ff(i)p Fe(=1)1896
2519 3161 y Fk(f)1944 3176 y Ff(i)2000 3161 y Fm(\024)f Fn(0)p
2520 Fk(:)257 3484 y Fg(2.4.2)113 b(The)38 b(bigparm)e(tric)m(k)257
2521 3669 y Fn(In)30 b(some)f(cases,)j(it)c(is)h(useful)g(to)g(supp)s(ose)i
2522 (that)e(one)h(parameter)f(in)f(a)h(PIP)i(problem)d(gro)m(ws)257
2523 3789 y(\\v)m(ery)40 b(large".)60 b(Some)39 b(examples)f(will)e(b)s(e)j
2524 (giv)m(en)g(in)f(the)h(follo)m(wing)d(sections.)62 b(Let)39
2525 b Fk(B)k Fn(b)s(e)257 3910 y(the)36 b(name)e(of)g(this)h(parameter.)49
2526 b(Supp)s(ose)36 b(that)e(in)g(the)i(solution,)d(one)i(of)g(the)g
2527 (predicates)257 4030 y(is:)1720 4150 y Fk(aB)27 b Fn(+)22
2528 b Fk(b)28 b Fm(\025)g Fn(0)p Fk(;)257 4322 y Fn(where)35
2529 b Fk(b)f Fn(ma)m(y)g(dep)s(end)g(on)g(all)d(other)j(parameters.)47
2530 b(F)-8 b(or)33 b Fk(B)38 b Fn(large)33 b(enough,)h(if)f
2531 Fk(a)c(>)h Fn(0)j(then)257 4443 y(the)k(predicate)f(is)g(true,)i(and)e
2532 (if)f Fk(a)g(<)e Fn(0)j(then)h(the)g(predicate)f(is)g(false.)54
2533 b(One)37 b(can)f(\014nd)h(the)257 4563 y(limit)e(shap)s(e)j(of)g(the)g
2534 (solution)f(b)m(y)i(remo)m(ving)e(suc)m(h)i(tests)h(and)e(replacing)e
2535 (them)i(b)m(y)h(their)257 4683 y(true)32 b(of)f(false)g(branc)m(h,)h
2536 (as)g(appropriate.)42 b(This)32 b(can)f(b)s(e)h(done)f
2537 Fj(a)g(p)s(osteriori)k Fn(on)c(the)h(results)257 4804
2538 y(of)c(PIP)-8 b(,)28 b(or)g(PIP)g(can)g(do)g(it)e(\\on)i(the)g(\015y")g
2539 (while)f(solving)f(the)j(problem.)40 b(This)28 b(last)f(metho)s(d)257
2540 4924 y(is)32 b(more)g(e\016cien)m(t,)i(since)f(it)e(tends)j(to)e
2541 (simplify)e(the)j(solution.)404 5045 y(PIP)e(is)g(noti\014ed)f(of)g
2542 (the)i(presence)g(of)f(a)f(big)g(parameter)g(b)m(y)i(setting)e(the)i
2543 Fl(Bg)f Fn(argumen)m(t)257 5165 y(to)37 b(a)f(p)s(ositiv)m(e)g(v)-5
2544 b(alue.)56 b(This)37 b(v)-5 b(alue)36 b(is)h(the)g(rank)g(of)f(the)h
2545 (big)f(parameter)g(in)h(the)g(problem)257 5285 y(tableau.)43
2546 b(Hence,)34 b(the)f(lo)m(w)m(est)g(admissible)e(v)-5
2547 b(alue)32 b(for)g Fl(Bg)h Fn(is)f Fl(Nn)52 b(+)f(1)p
2548 Fn(.)404 5406 y(The)38 b(reader)f(should)g(con)m(vince)h(himself)d
2549 (that)i(in)f(the)h(presence)j(of)c(t)m(w)m(o)i(big)e(parame-)257
2550 5526 y(ters,)e(no)e(suc)m(h)i(simpli\014cations)29 b(are)j(p)s(ossible)
2551 g(unless)h(one)g(has)f(some)h(information)c(on)j(the)257
2552 5646 y(relativ)m(e)f(size)h(of)f(the)g(parameters.)43
2553 b(Suc)m(h)33 b(situations)d(should)i(b)s(e)f(handled)h(b)m(y)g(giving)e
2554 (PIP)p eop
2555 %%Page: 10 10
2556 10 9 bop 257 266 a Fj(2)98 b(USING)32 b(THE)i(PIP)f(SOFTW)-11
2557 b(ARE)1830 b Fn(10)257 573 y(ordinary)34 b(parameters,)g(and)g(doing)f
2558 (the)i(simpli\014cation)30 b(on)k(the)h(solution)d(in)h(the)i(ligh)m(t)
2559 d(of)257 693 y(extra)h(kno)m(wledge.)257 949 y Fg(2.4.3)113
2560 b(Computing)36 b(Lexicographical)h(Maxima)257 1134 y
2561 Fn(T)-8 b(o)33 b(get)f(the)g(maxim)m(um)e(of)i(an)g(unkno)m(wn)i
2562 Fk(x)p Fn(,)f(minimize)c Fk(B)d Fm(\000)c Fk(x)p Fn(,)33
2563 b(where)g(B)f(is)g(a)g(new)h("big")257 1254 y(parameter.)63
2564 b(Adding)39 b(a)g(parameter)f(just)i(adds)g(one)f(column)f(in)h(the)g
2565 (problem)f(tableau.)257 1375 y(The)c(fact)e(that)g(this)f(column)h
2566 (corresp)s(onds)h(to)f(a)g(Big)f(parameter)h(is)g(sp)s(eci\014ed)h(b)m
2567 (y)g(setting)257 1495 y(the)e(5-th)f(switc)m(h)h(to)f(a)f(p)s(ositiv)m
2568 (e)h(v)-5 b(alue,)30 b(this)g(v)-5 b(alue)30 b(b)s(eing)f(the)i(p)s
2569 (osition)d(of)i(the)h(column)e(of)257 1616 y(B)k(in)f(the)h(problem)e
2570 (tableau.)404 1736 y(These)f(cases)f(can)f(b)s(e)h(handled)f
2571 (systematically)e(in)i(the)g(follo)m(wing)d(w)m(a)m(y)-8
2572 b(.)44 b(Supp)s(ose)29 b(that)257 1856 y(w)m(e)34 b(are)f(ask)m(ed)h
2573 (for)e(the)h(in)m(teger)f(maxim)m(um)e(of)i(the)h(p)s(olyhedron:)1684
2574 2052 y Fk(x)84 b Fm(\025)f Fn(0)p Fk(;)1689 2197 y(y)j
2575 Fm(\025)d Fn(0)p Fk(;)1499 b Fn(\(6\))1640 2342 y(3)p
2576 Fk(y)86 b Fm(\024)d Fk(x)23 b Fn(+)f(12)p Fk(;)1689 2487
2577 y(y)86 b Fm(\025)d Fn(2)p Fk(x)23 b Fm(\000)f Fn(3)p
2578 Fk(:)257 2683 y Fn(Let)33 b(us)g(in)m(tro)s(duce)g(the)g(new)g(unkno)m
2579 (wns:)1445 2878 y Fk(x)1500 2837 y Fc(0)1551 2878 y Fn(=)28
2580 b Fk(B)f Fm(\000)c Fk(x;)72 b(y)2062 2837 y Fc(0)2112
2581 2878 y Fn(=)28 b Fk(B)f Fm(\000)c Fk(y)t(;)257 3073 y
2582 Fn(where)34 b Fk(B)k Fn(is)32 b(the)h(big)f(parameter.)43
2583 b(System)33 b(\(6\))f(translates)g(to:)1880 3268 y Fm(\000)p
2584 Fk(x)2012 3227 y Fc(0)2058 3268 y Fn(+)22 b Fk(B)88 b
2585 Fm(\025)c Fn(0)p Fk(;)1884 3414 y Fm(\000)p Fk(y)2013
2586 3372 y Fc(0)2058 3414 y Fn(+)22 b Fk(B)88 b Fm(\025)c
2587 Fn(0)p Fk(;)1368 3559 y Fm(\000)p Fk(x)1500 3518 y Fc(0)1547
2588 3559 y Fn(+)22 b(3)p Fk(y)1746 3518 y Fc(0)1790 3559
2589 y Fn(+)g(12)g Fm(\000)h Fn(2)p Fk(B)87 b Fm(\025)d Fn(0)p
2590 Fk(;)1542 3704 y Fn(2)p Fk(x)1646 3663 y Fc(0)1692 3704
2591 y Fm(\000)22 b Fk(y)1843 3663 y Fc(0)1888 3704 y Fn(+)g(3)g
2592 Fm(\000)g Fk(B)88 b Fm(\025)c Fn(0)p Fk(:)257 3899 y
2593 Fn(Finding)26 b(the)i(maxim)m(um)d(of)i(\()p Fk(x;)17
2594 b(y)t Fn(\))1560 3863 y Ff(T)1642 3899 y Fn(is)27 b(equiv)-5
2595 b(alen)m(t)27 b(to)g(\014nding)g(the)h(minim)m(um)d(of)i(\()p
2596 Fk(x)3421 3863 y Fc(0)3444 3899 y Fk(;)17 b(y)3540 3863
2597 y Fc(0)3562 3899 y Fn(\))3600 3863 y Ff(T)3655 3899 y
2598 Fn(,)257 4020 y(pro)m(vided)33 b Fk(B)38 b Fn(is)32 b(large)g(enough.)
2599 44 b(The)33 b(solution)e(of)h(the)h(ab)s(o)m(v)m(e)h(problem)d(is:)257
2600 4202 y Fl(\(\(a)52 b(maximization)j(problem)e(1)f(\))309
2601 4322 y(\(if)g(#[)f(-1)h(6])360 4443 y(\(if)g(#[)g(-1)g(3])411
2602 4563 y(\(list)h(#[)f(0)f(0])719 4683 y(#[)h(0)f(0]\))411
2603 4804 y(\(if)h(#[)g(-5)g(27])462 4924 y(\(newparm)i(1)d(\(div)i(#[)e(1)h
2604 (1])g(2\)\))462 5045 y(\(list)h(#[)f(1)f(-1)h(-1])770
2605 5165 y(#[)g(0)f(0)h(0]\))462 5285 y(\(list)h(#[)f(1)f(-4])770
2606 5406 y(#[)h(1)f(-5]\)\)\))360 5526 y(\(list)h(#[)g(1)g(-4])667
2607 5646 y(#[)g(1)g(-5]\)\)\))p eop
2608 %%Page: 11 11
2609 11 10 bop 257 266 a Fj(2)98 b(USING)32 b(THE)i(PIP)f(SOFTW)-11
2610 b(ARE)1830 b Fn(11)404 573 y(Supp)s(ose)33 b(w)m(e)h(tell)d(PIP)i(that)
2611 g Fk(B)k Fn(is)32 b(a)h(large)e(parameter.)43 b(The)34
2612 b(input)e(\014le)g(is)g(no)m(w:)257 764 y Fl(\(\(a)52
2613 b(maximization)j(problem\))309 884 y(2)c(1)h(4)f(0)h(3)f(1)309
2614 1005 y(\(#[-1)h(0)g(0)f(1])360 1125 y(#[0)h(-1)g(0)f(1])360
2615 1245 y(#[-1)h(3)g(12)g(-2])360 1366 y(#[2)g(-1)g(3)f(-1])309
2616 1486 y(\))309 1607 y(\(\))257 1727 y(\))257 1918 y Fn(and)33
2617 b(the)g(solution)e(is)h(m)m(uc)m(h)h(simpler:)257 2109
2618 y Fl(\(\(a)52 b(maximization)j(problem)e(1)f(\))309 2229
2619 y(\(list)g(#[)g(1)g(-4])616 2350 y(#[)g(1)g(-5]\)\))257
2620 2541 y Fn(The)39 b(reader)f(ma)m(y)g(care)g(to)f(c)m(hec)m(k)j(that)d
2621 (this)h(result)f(is)g(equiv)-5 b(alen)m(t)38 b(to)f(the)h(previous)g
2622 (one)257 2661 y(as)k(so)s(on)f(as)g Fk(B)47 b(>)42 b
2623 Fn(5.)69 b(The)42 b(p)s(osition)d(of)i(the)h(minim)m(um)37
2624 b(is:)61 b Fk(x)2671 2625 y Fc(0)2736 2661 y Fn(=)43
2625 b Fk(B)33 b Fm(\000)28 b Fn(4)p Fk(;)17 b(y)3212 2625
2626 y Fc(0)3276 2661 y Fn(=)42 b Fk(B)33 b Fm(\000)c Fn(5,)257
2627 2781 y(from)35 b(whic)m(h)g(w)m(e)i(deduce:)51 b Fk(x)33
2628 b Fn(=)f(4)p Fk(;)17 b(y)35 b Fn(=)d(5.)52 b(As)36 b(exp)s(ected,)i
2629 Fk(B)j Fn(has)35 b(disapp)s(eared)h(from)e(the)257 2902
2630 y(solution.)56 b(If)37 b(this)g(do)s(es)h(not)f(happ)s(en,)i(w)m(e)f
2631 (observ)m(e)h(\014rst)e(that)g Fk(B)43 b Fn(m)m(ust)37
2632 b(ha)m(v)m(e)h(a)f(p)s(ositiv)m(e)257 3022 y(co)s(e\016cien)m(t)c(in)f
2633 (the)h(result)f(\(if)f(not,)i(one)f(of)g(the)h(inequalities)e
2634 Fk(x;)17 b(y)31 b Fm(\025)d Fn(0)k(w)m(ould)g(b)s(e)h(violated)257
2635 3142 y(for)39 b Fk(B)44 b Fn(large)37 b(enough\).)63
2636 b(This)39 b(means)g(that)f(the)h(original)d(p)s(olyhedron)j(is)f(not)h
2637 (b)s(ounded,)257 3263 y(since,)k(whatev)m(er)f Fk(B)5
2638 b Fn(,)43 b(it)c(con)m(tains)h(a)g(p)s(oin)m(t)g(whose)h(co)s
2639 (ordinates)f(are)g Fk(O)s Fn(\()p Fk(B)5 b Fn(\),)42
2640 b(and)e(hence)257 3383 y(has)33 b(no)g(maxim)m(um.)257
2641 3641 y Fg(2.4.4)113 b(Optimizing)34 b(Linear)k(Cost)f(F)-9
2642 b(unctions)257 3826 y Fn(The)36 b(problem)d(here)j(is)e(to)g(compute)g
2643 (the)h(minim)m(um)d(of)i(a)g(linear)f(function)h Fk(cx)h
2644 Fn(in)f(a)g(p)s(oly-)257 3946 y(hedron)k Fk(P)14 b Fn(,)38
2645 b(where)h Fk(c)e Fn(is)g(a)g(v)m(ector)h(with)f(in)m(teger)g(co)s
2646 (e\016cien)m(ts.)59 b(Let)37 b(us)h(in)m(tro)s(duce)g(a)f(new)257
2647 4066 y(unkno)m(wn)e Fk(y)t Fn(.)45 b(Solv)m(e)33 b(the)h(linear)e
2648 (programming)e(problem)i(obtained)h(b)m(y)h(adding)e(the)i(con-)257
2649 4187 y(strain)m(t)e Fk(y)f Fm(\025)d Fk(cx)33 b Fn(to)f(the)g
2650 (de\014ning)g(constrain)m(ts)h(of)f Fk(P)14 b Fn(.)42
2651 b Fk(y)36 b Fn(should)c(b)s(e)g(the)h(\014rst)f(unkno)m(wn)i(in)257
2652 4307 y(the)28 b(lexicographic)d(ordering.)41 b(Let)27
2653 b Fk(y)1640 4322 y Ff(s)1676 4307 y Fk(;)17 b(x)1775
2654 4322 y Ff(s)1839 4307 y Fn(b)s(e)27 b(the)h(solution.)40
2655 b(Supp)s(ose)28 b(that)e(the)i(minim)m(um)257 4428 y(of)j
2656 Fk(cx)g Fn(in)g Fk(P)44 b Fn(is)30 b(obtained)h(at)g
2657 Fk(x)1384 4443 y Ff(m)1482 4428 y Fn(and)g(set)g Fk(y)1868
2658 4443 y Ff(m)1962 4428 y Fn(=)d Fk(cx)2163 4443 y Ff(m)2230
2659 4428 y Fn(.)43 b(Since)31 b Fk(x)2608 4443 y Ff(s)2676
2660 4428 y Fn(is)g(in)f Fk(P)14 b Fn(,)31 b(and)g Fk(y)3256
2661 4443 y Ff(s)3320 4428 y Fm(\025)d Fk(cx)3522 4443 y Ff(s)3559
2662 4428 y Fn(,)k(it)257 4548 y(is)26 b(clear)f(that)g Fk(y)827
2663 4563 y Ff(s)891 4548 y Fm(\025)j Fk(y)1044 4563 y Ff(m)1110
2664 4548 y Fn(.)42 b(Con)m(v)m(ersely)-8 b(,)29 b(\()p Fk(y)1772
2665 4563 y Ff(m)1838 4548 y Fk(;)17 b(x)1937 4563 y Ff(m)2003
2666 4548 y Fn(\))26 b(satis\014es)g(the)g(constrain)m(ts)g(of)g(the)g
2667 (problem)257 4668 y(of)33 b(whic)m(h)h(\()p Fk(y)735
2668 4683 y Ff(s)771 4668 y Fk(;)17 b(x)870 4683 y Ff(m)937
2669 4668 y Fn(\))33 b(is)f(the)i(lexicographic)d(minim)m(um.)42
2670 b(Hence)34 b(\()p Fk(y)2715 4683 y Ff(s)2752 4668 y Fk(;)17
2671 b(x)2851 4683 y Ff(s)2888 4668 y Fn(\))28 b Fm(\034)g
2672 Fn(\()p Fk(y)3168 4683 y Ff(m)3234 4668 y Fk(;)17 b(x)3333
2673 4683 y Ff(m)3400 4668 y Fn(\),)33 b(and,)257 4789 y(since)38
2674 b Fk(y)j Fn(is)c(the)h(\014rst)g(unkno)m(wn,)i Fk(y)1570
2675 4804 y Ff(s)1643 4789 y Fm(\024)c Fk(y)1804 4804 y Ff(m)1871
2676 4789 y Fn(.)58 b(Hence,)40 b Fk(y)2327 4804 y Ff(m)2429
2677 4789 y Fn(=)c Fk(y)2589 4804 y Ff(s)2625 4789 y Fn(.)59
2678 b(There)38 b(is)f(no)h(guaran)m(tee,)257 4909 y(ho)m(w)m(ev)m(er,)d
2679 (that)e Fk(x)925 4924 y Ff(s)990 4909 y Fn(=)27 b Fk(x)1148
2680 4924 y Ff(m)1215 4909 y Fn(.)257 5167 y Fg(2.4.5)113
2681 b(Negativ)m(e)36 b(Unkno)m(wns)j(and)f(P)m(arameters)257
2682 5351 y Fn(Supp)s(ose)32 b(w)m(e)f(w)m(an)m(t)g(to)f(\014nd)h(the)f
2683 (minim)m(um)d(of)j Fk(f)11 b Fn(\()p Fk(i;)17 b(j)6 b
2684 Fn(\))27 b(=)h Fk(i)17 b Fm(\000)h Fn(2)p Fk(j)36 b Fn(o)m(v)m(er)31
2685 b(the)g(square)g(domain)257 5472 y(represen)m(ted)k(in)d(Figure)g(1)
2686 1246 5436 y Fe(1)1285 5472 y Fn(.)p 257 5555 1371 4 v
2687 370 5616 a Fb(1)407 5646 y Fa(This)27 b(example)h(w)n(as)e(prop)r(osed)
2688 h(and)g(solv)n(ed)g(b)n(y)g(Pierre)f(Boulet.)p eop
2689 %%Page: 12 12
2690 12 11 bop 257 266 a Fj(2)98 b(USING)32 b(THE)i(PIP)f(SOFTW)-11
2691 b(ARE)1830 b Fn(12)p 257 477 3426 4 v 952 2012 a @beginspecial
2692 0 @llx 0 @lly 362 @urx 273 @ury 1842 @rhi @setspecial
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2801 322 333 m
2802 gs 1 -1 scale (n+5) col-1 show gr
2803 $F2psEnd
2804
2805 %%EndDocument
2806  @endspecial 1404 2215 a(Figure)31 b(1:)43 b(Problem)32
2807 b(domain)404 2499 y(As)i(ab)s(o)m(v)m(e,)g(w)m(e)h(in)m(tro)s(duce)e(a)
2808 h(new)g(unkno)m(wn)h Fk(f)45 b Fn(and)33 b(the)h(inequalit)m(y)f
2809 Fk(f)g Fm(\000)23 b Fk(i)g Fn(+)g(2)p Fk(j)35 b Fm(\025)30
2810 b Fn(0.)257 2620 y(Since)j(w)m(e)h(w)m(an)m(t)f(to)f(optimize)f
2811 Fk(f)11 b Fn(,)32 b Fk(f)44 b Fn(will)30 b(app)s(ear)i(as)h(the)g
2812 (\014rst)g(unkno)m(wn.)404 2740 y(The)24 b(tric)m(k)f(for)f(solving)g
2813 (the)i(problem)d(in)i Fk(Z)29 b Fn(is)23 b(to)g(in)m(tro)s(duce)g(the)g
2814 (follo)m(wing)d(parameters:)403 2941 y Fm(\017)48 b Fk(G)28
2815 b Fm(\025)g Fk(max)p Fn(\(0)p Fk(;)17 b Fm(\000)p Fk(i;)g
2816 Fm(\000)p Fk(j;)g Fm(\000)p Fk(f)11 b Fn(\).)403 3144
2817 y Fm(\017)48 b Fk(P)41 b Fn(=)28 b Fk(max)p Fn(\(0)p
2818 Fk(;)17 b Fm(\000)p Fk(n)p Fn(\).)257 3346 y(This)33
2819 b(c)m(hoice)g(insure)g(that)f(the)h(new)h(v)-5 b(ariables)31
2820 b(and)i(parameters:)1680 3563 y Fk(f)1739 3522 y Fc(0)1845
2821 3563 y Fn(=)83 b Fk(G)22 b Fn(+)g Fk(f)1706 3709 y(i)1739
2822 3667 y Fc(0)1845 3709 y Fn(=)83 b Fk(G)22 b Fn(+)g Fk(i)1693
2823 3854 y(j)1739 3813 y Fc(0)1845 3854 y Fn(=)83 b Fk(G)22
2824 b Fn(+)g Fk(j)1681 3999 y(n)1739 3958 y Fc(0)1845 3999
2825 y Fn(=)83 b Fk(P)36 b Fn(+)22 b Fk(n)257 4217 y Fn(are)41
2826 b(all)d(p)s(ositiv)m(e.)66 b(This)41 b(prop)s(ert)m(y)g(sta)m(ys)g
2827 (true)g(if)e Fk(G)h Fn(gro)m(ws.)68 b(Hence,)44 b Fk(G)c
2828 Fn(is)g(again)e(a)i(big)257 4337 y(parameter.)78 b(Ho)m(w)m(ev)m(er,)50
2829 b Fk(P)57 b Fn(m)m(ust)45 b(b)s(e)f(considered)h(as)g(an)f(ordinary)f
2830 (parameter.)78 b(After)257 4458 y(replacemen)m(t)39 b(of)e
2831 Fk(i;)17 b(j;)g(n;)g(f)49 b Fn(b)m(y)40 b(the)e(new)h(v)-5
2832 b(ariables)37 b Fk(i)2248 4422 y Fc(0)2272 4458 y Fk(;)17
2833 b(j)2362 4422 y Fc(0)2385 4458 y Fk(;)g(n)2487 4422 y
2834 Fc(0)2510 4458 y Fk(;)g(f)2613 4422 y Fc(0)2636 4458
2835 y Fn(,)39 b(w)m(e)h(get)e(a)g(system)h(whic)m(h)257 4578
2836 y(corresp)s(onds)34 b(to)e(the)h(follo)m(wing)d(input:)257
2837 4804 y Fl(\()309 4924 y(\()51 b(Solving)i(MIN\(i-2.j\))h(under)f(the)f
2838 (following)i(constraints:)411 5045 y(Unknowns)g(may)e(be)f(negative.)
2839 411 5165 y(Order:)411 5285 y(f')h(i')g(j')g(constant)h(G)e(P)h(n')360
2840 5406 y(\))309 5526 y(3)f(3)h(5)f(0)h(4)f(1)309 5646 y(\()p
2841 eop
2842 %%Page: 13 13
2843 13 12 bop 257 266 a Fj(2)98 b(USING)32 b(THE)i(PIP)f(SOFTW)-11
2844 b(ARE)1830 b Fn(13)360 573 y Fl(#[)52 b(0)f(1)h(1)f(20)h(-2)g(-4)g(4)f
2845 (])360 693 y(#[)h(1)f(-1)h(2)g(0)f(-2)h(0)f(0)h(])360
2846 814 y(#[)g(0)f(-1)h(-1)g(0)f(2)h(0)f(0)h(])360 934 y(#[)g(0)f(1)h(-1)g
2847 (10)f(0)h(-2)g(2)f(])360 1054 y(#[)h(0)f(-1)h(1)g(10)f(0)h(-2)g(2)f(])
2848 411 1175 y(\))309 1295 y(\()g(\)\))404 1496 y Fn(The)33
2849 b(result)g(is:)257 1679 y Fl(\()309 1800 y(\()51 b(Solving)i
2850 (MIN\(i-2.j\))h(under)f(the)f(following)i(constraints:)411
2851 1920 y(Unknowns)g(may)e(be)f(negative.)411 2040 y(Order:)411
2852 2161 y(f')h(i')g(j')g(constant)h(G)e(P)h(n')411 2281
2853 y(-1)g(\)\(if)g(#[)g(0)g(-1)f(1)h(5])257 2402 y(\(list)h(#[)f(1)f(3)h
2854 (-3)g(-15])257 2522 y(#[)g(1)g(1)f(-1)h(-5])257 2642
2855 y(#[)g(1)g(-1)f(1)h(5])257 2763 y(\))257 2883 y(\(\))257
2856 3004 y(\))257 3124 y(\))257 3307 y Fn(whic)m(h)33 b(should)g(b)s(e)g
2857 (read)f(as:)422 3503 y(\()p Fk(f)519 3462 y Fc(0)542
2858 3503 y Fk(;)17 b(i)619 3462 y Fc(0)642 3503 y Fk(;)g(j)732
2859 3462 y Fc(0)755 3503 y Fn(\))83 b(=)g Fl(if)50 b Fm(\000)23
2860 b Fk(P)36 b Fn(+)22 b Fk(n)1542 3462 y Fc(0)1587 3503
2861 y Fm(\000)h Fn(5)k Fm(\025)i Fn(0)1035 3648 y Fl(then)g
2862 Fn(\()p Fk(G)22 b Fn(+)g(3)p Fk(P)35 b Fm(\000)23 b Fn(3)p
2863 Fk(n)1857 3607 y Fc(0)1902 3648 y Fm(\000)g Fn(15)p Fk(;)17
2864 b(G)k Fn(+)h Fk(P)36 b Fm(\000)22 b Fk(n)2596 3607 y
2865 Fc(0)2642 3648 y Fm(\000)h Fn(5)p Fk(;)17 b(G)k Fm(\000)i
2866 Fk(P)35 b Fn(+)22 b Fk(n)3287 3607 y Fc(0)3333 3648 y
2867 Fn(+)g(5\))1035 3794 y Fl(else)29 b Fm(?)257 3990 y Fn(That)k(is,)f(in)
2868 g(the)h(original)d(co)s(ordinate)h(system:)722 4186 y(\()p
2869 Fk(f)5 b(;)17 b(i;)g(j)6 b Fn(\))28 b(=)f Fl(if)h Fk(n)g
2870 Fm(\025)g Fn(5)g Fl(then)h Fn(\()p Fm(\000)p Fn(3)p Fk(n)22
2871 b Fm(\000)h Fn(15)p Fk(;)17 b Fm(\000)p Fk(n)22 b Fm(\000)g
2872 Fn(5)p Fk(;)17 b(n)22 b Fn(+)g(5\))28 b Fl(else)g Fm(?)257
2873 4382 y Fn(I.e.,)35 b(the)g(minim)m(um)30 b(v)-5 b(alue)34
2874 b(for)f(function)g Fk(f)45 b Fn(is)33 b Fm(\000)p Fn(3)p
2875 Fk(n)24 b Fm(\000)f Fn(15,)34 b(and)g(this)g(v)-5 b(alue)33
2876 b(is)h(reac)m(hed)h(at)257 4503 y(p)s(oin)m(t)c(\()p
2877 Fm(\000)p Fk(n)19 b Fm(\000)h Fn(5)p Fk(;)d(n)i Fn(+)g(5\).)42
2878 b(This)31 b(minim)m(um)d(exists)k(only)f(if)e Fk(n)f
2879 Fm(\025)h Fn(5;)i(otherwise,)h(the)f(feasible)257 4623
2880 y(set)j(is)e(empt)m(y)-8 b(.)257 4879 y Fg(2.4.6)113
2881 b(Mixed)37 b(Programming)257 5064 y Fn(A)43 b(mixed)e(program)g(is)h(a)
2882 g(program)f(in)g(whic)m(h)i(some)f(v)-5 b(ariables)41
2883 b(are)h(constrained)g(to)g(b)s(e)257 5185 y(in)m(tegers)28
2884 b(while)f(others)h(ma)m(y)g(tak)m(e)g(rational)d(v)-5
2885 b(alues.)42 b(Supp)s(ose)28 b(for)f(instance)h(that)g(w)m(e)g(ha)m(v)m
2886 (e)257 5305 y(to)33 b(solv)m(e:)1441 5501 y Fk(S)89 b
2887 Fn(=)83 b(min)15 b Fk(ax)23 b Fn(+)f Fk(by)t(;)1749 5646
2888 y(Ax)h Fn(+)f Fk(B)5 b(y)25 b Fn(+)d Fk(c)28 b Fm(\025)g
2889 Fn(0)p Fk(;)p eop
2890 %%Page: 14 14
2891 14 13 bop 257 266 a Fj(3)98 b(USING)32 b(THE)i(PIP)f(LIBRAR)-8
2892 b(Y)1947 b Fn(14)257 573 y(where)34 b Fk(y)i Fn(is)c(the)h(v)m(ector)h
2893 (of)e(the)h(in)m(teger)f(v)-5 b(ariables.)42 b(First,)32
2894 b(solv)m(e)1439 902 y Fk(T)96 b Fn(=)83 b(min)16 b Fk(ax;)1751
2895 1047 y(Ax)23 b Fn(+)f Fk(B)5 b(y)25 b Fn(+)e Fk(c)k Fm(\025)h
2896 Fn(0)p Fk(;)257 1255 y Fn(in)22 b(rational,)h(with)f
2897 Fk(y)k Fn(as)e(parameters.)40 b(The)24 b(result)e(is)h(a)f(quast.)42
2898 b(T)-8 b(o)22 b(eac)m(h)i(leaf)e Fk(i)h Fn(is)g(asso)s(ciated)257
2899 1376 y(a)40 b(linear)e(function)i Fk(f)1062 1391 y Ff(i)1090
2900 1376 y Fn(\()p Fk(y)t Fn(\))f(and)h(a)g(set)h(of)e(inequalities)f
2901 Fk(C)2411 1391 y Ff(i)2439 1376 y Fk(y)30 b Fn(+)d Fk(d)2671
2902 1391 y Ff(i)2740 1376 y Fm(\025)40 b Fn(0.)66 b Fk(T)53
2903 b Fn(is)40 b(equal)g(to)f Fk(f)3654 1391 y Ff(i)257 1496
2904 y Fn(when)33 b Fk(y)h Fn(is)c(suc)m(h)j(that)d(the)i(corresp)s(onding)f
2905 (inequalities)e(are)i(satis\014ed.)43 b(F)-8 b(or)31
2906 b(eac)m(h)g Fk(i)p Fn(,)h(solv)m(e)257 1616 y(the)h(problem:)1485
2907 1825 y Fk(S)1545 1840 y Ff(i)1657 1825 y Fn(=)83 b(min)15
2908 b Fk(f)2043 1840 y Ff(i)2071 1825 y Fn(\()p Fk(y)t Fn(\))21
2909 b(+)h Fk(by)t(;)1816 1970 y(C)1886 1985 y Ff(i)1914 1970
2910 y Fk(y)j Fn(+)d Fk(d)2136 1985 y Ff(i)2191 1970 y Fm(\025)29
2911 b Fn(0)p Fk(;)257 2178 y Fn(in)41 b(in)m(tegers.)69 b(The)43
2912 b(\014nal)d(result)h(is)g(the)g(minim)m(um)d(of)j(all)e
2913 Fk(S)2571 2193 y Ff(i)2599 2178 y Fn(.)69 b(Ob)m(viously)-8
2914 b(,)44 b(the)d(metho)s(d)257 2299 y(can)34 b(accomo)s(date)e
2915 (parameters)g(in)h(the)g(constrain)m(ts.)45 b(The)33
2916 b Fk(S)2554 2314 y Ff(i)2615 2299 y Fn(will)e(b)s(e)i(functions)g(of)f
2917 (these)257 2419 y(parameters,)h(and)g(the)g(minim)m(um)c(m)m(ust)k(b)s
2918 (e)f(computed)h(sym)m(b)s(olically)-8 b(.)257 2750 y
2919 Fo(3)161 b(Using)54 b(the)f(PIP)h(Library)257 2969 y
2920 Fn(The)42 b(PIP)f(Library)e(\(PipLib)g(for)h(short\))h(w)m(as)g
2921 (implemen)m(ted)e(to)h(allo)m(w)e(the)j(user)g(to)f(call)257
2922 3090 y(PIP)f(directly)d(from)h(his)g(programs,)g(without)g(\014le)g
2923 (accesses)j(or)d(system)i(calls.)56 b(The)39 b(user)257
2924 3210 y(only)i(needs)h(to)f(link)e(his)i(programs)f(with)h(C)g
2925 (libraries.)66 b(The)42 b(PipLib)e(mainly)f(pro)m(vides)257
2926 3330 y(one)29 b(function)f(whic)m(h)g(tak)m(es)i(as)e(input)g(the)h
2927 (problem)e(description)g(and)i(some)f(options,)g(and)257
2928 3451 y(returns)43 b(a)f Fl(Quast)h Fn(\(see)g(grammar)d(2)h(in)h
2929 (section)g(2.3\))f(corresp)s(onding)h(to)g(the)g(solution.)257
2930 3571 y(Some)c(other)g(functions)f(are)h(pro)m(vided)h(for)e(con)m(v)m
2931 (enience)j(reasons)f(;)h(they)f(are)f(describ)s(ed)257
2932 3691 y(in)k(section)h(3.2.)73 b(Most)43 b(of)f(them)g(require)h(some)g
2933 (sp)s(eci\014c)g(structures)h(to)f(represen)m(t)h(the)257
2934 3812 y(problem)32 b(or)g(the)h(solution)e(;)i(these)g(structures)i(are)
2935 d(describ)s(ed)i(in)e(section)g(3.1.)257 4099 y Fh(3.1)136
2936 b(PipLib)44 b(data)i(structures)f(description)257 4284
2937 y Fg(3.1.1)113 b(PipMatrix)36 b(structure)257 4468 y
2938 Fl(struct)53 b(pipmatrix)257 4589 y({)f(unsigned)h(NbRows,)g(NbColumns)
2939 h(;)360 4709 y(Entier)f(**)f(p)f(;)360 4830 y(Entier)i(*)e(p_Init)i(;)
2940 257 4950 y(})f(;)257 5070 y(typedef)h(struct)g(pipmatrix)h(PipMatrix)g
2941 (;)257 5285 y Fn(The)49 b Fl(PipMatrix)h Fn(structure)f(is)f(dev)m
2942 (oted)h(to)e(represen)m(t)j(a)e(constrain)m(ts)g(matrix)e(in)h(the)257
2943 5406 y(P)m(olyLib)41 b(shap)s(e)h([3)o(].)70 b(The)42
2944 b(whole)f(matrix)e(is)i(arranged)g(ro)m(w)h(after)e(ro)m(w)i(at)f(the)g
2945 Fl(p)p 3446 5406 31 4 v 38 w(Init)257 5526 y Fn(adress.)k
2946 Fl(p)33 b Fn(is)f(an)h(arra)m(y)g(of)f(p)s(oin)m(ters)g(in)g(whic)m(h)h
2947 Fl(p[i])h Fn(p)s(oin)m(ts)e(to)h(the)g(b)s(eginning)e(of)h(the)h(i)3613
2948 5490 y Ff(th)257 5646 y Fn(ro)m(w.)44 b Fl(NbRows)31
2949 b Fn(and)g Fl(NbColumns)i Fn(are)d(resp)s(ectiv)m(ely)h(the)g(n)m(um)m
2950 (b)s(er)g(of)f(ro)m(ws)h(and)f(columns)g(of)p eop
2951 %%Page: 15 15
2952 15 14 bop 257 266 a Fj(3)98 b(USING)32 b(THE)i(PIP)f(LIBRAR)-8
2953 b(Y)1947 b Fn(15)257 573 y(the)38 b(matrix.)56 b(W)-8
2954 b(e)37 b(use)h(this)f(structure)i(to)d(carry)i(p)s(olyhedrons.)58
2955 b(Eac)m(h)38 b(ro)m(w)f(corresp)s(onds)257 693 y(to)c(a)f(constrain)m
2956 (t)h(whic)m(h)g(the)g(p)s(olyhedron)g(m)m(ust)g(satisfy)-8
2957 b(.)44 b(The)33 b(constrain)m(t)g(is)f(an)h(equalit)m(y)257
2958 814 y(if)e(the)i(\014rst)g(elemen)m(t)f(is)f(0,)h(an)g(inequalit)m(y)g
2959 Fk(p)p Fn(\()p Fk(x)p Fn(\))27 b Fm(\025)i Fn(0)i(if)h(the)g(\014rst)h
2960 (elemen)m(t)f(is)f(1.)43 b(The)34 b(next)257 934 y(elemen)m(ts)47
2961 b(are)e(the)i(unkno)m(wn)g(co)s(e\016cien)m(ts,)k(follo)m(w)m(ed)45
2962 b(b)m(y)i(the)f(parameter)f(co)s(e\016cien)m(ts.)257
2963 1054 y(The)38 b(last)d(elemen)m(t)h(is)g(the)g(constan)m(t)i(factor.)54
2964 b(F)-8 b(or)35 b(instance,)i(in)f(the)h(problem)e(of)g(section)257
2965 1175 y(2.1.1)d(the)h(domain)e(is)h(de\014ned)i(b)m(y)g(3)e(constrain)m
2966 (ts:)1608 1302 y Fd(8)1608 1377 y(>)1608 1402 y(<)1608
2967 1552 y(>)1608 1576 y(:)1723 1380 y Fm(\000)p Fk(i)23
2968 b Fn(+)f Fk(m)28 b Fm(\025)h Fn(0)1723 1501 y Fm(\000)p
2969 Fk(j)g Fn(+)22 b Fk(n)28 b Fm(\025)g Fn(0)1723 1621 y
2970 Fk(j)h Fn(+)22 b Fk(i)g Fm(\000)h Fk(k)30 b Fm(\025)f
2971 Fn(0)257 1829 y(the)k(ro)m(ws)h(corresp)s(onding)e(to)h(these)g
2972 (constrain)m(ts)g(w)m(ould)g(b)s(e:)257 2027 y Fl(#)52
2973 b(eq/in)104 b(i)154 b(j)g(k)g(m)g(n)g(cst)462 2147 y(1)206
2974 b(0)103 b(-1)154 b(0)g(1)g(0)205 b(0)462 2268 y(1)154
2975 b(-1)h(0)f(0)g(0)g(1)205 b(0)462 2388 y(1)h(1)154 b(1)103
2976 b(-1)154 b(0)g(0)205 b(0)257 2587 y Fn(The)34 b(con)m(text)g(is)e
2977 (de\014ned)i(b)m(y)f(one)g(constrain)m(t:)1547 2704 y
2978 Fd(n)1644 2800 y Fm(\000)p Fk(k)26 b Fn(+)c Fk(m)g Fn(+)g
2979 Fk(n)28 b Fm(\025)g Fn(0)257 3015 y(the)33 b(ro)m(w)g(corresp)s(onding)
2980 g(to)f(this)g(constrain)m(t)h(w)m(ould)f(b)s(e:)257 3213
2981 y Fl(#)52 b(eq/in)104 b(k)154 b(m)g(n)g(cst)462 3334
2982 y(1)g(-1)h(1)f(1)205 b(0)257 3593 y Fg(3.1.2)113 b(PipV)-9
2983 b(ector)35 b(structure)257 3777 y Fl(struct)53 b(pipvector)257
2984 3898 y({)f(int)g(nb_elements)i(;)360 4018 y(Entier)f(*)e(the_vector)j
2985 (;)360 4138 y(Entier)f(*)e(the_deno)j(;)257 4259 y(})e(;)257
2986 4379 y(typedef)h(struct)g(pipvector)h(PipVector)g(;)257
2987 4601 y Fn(The)34 b Fl(PipVector)g Fn(structure)g(represen)m(ts)h(a)d
2988 Fl(Vector)i Fn(as)e(describ)s(ed)i(in)d(grammar)f(2)i(in)g(sec-)257
2989 4721 y(tion)h(2.3.)45 b Fl(nb)p 764 4721 31 4 v 37 w(elements)36
2990 b Fn(is)c(the)i(n)m(um)m(b)s(er)g(of)f(v)m(ector)h(elemen)m(ts,)g
2991 Fl(the)p 2853 4721 V 38 w(vector)g Fn(is)f(an)g(arra)m(y)257
2992 4842 y(whic)m(h)e(con)m(tains)f(the)h(n)m(umerators)f(of)f(these)j
2993 (elemen)m(ts)e(and)g Fl(the)p 2688 4842 V 38 w(deno)h
2994 Fn(is)f(an)g(arra)m(y)g(whic)m(h)257 4962 y(con)m(tains)j(their)f
2995 (denominators:)42 b(the)33 b(i)1714 4926 y Ff(th)1816
2996 4962 y Fn(elemen)m(t)g(is)f Fl(the)p 2434 4962 V 38 w(vector[i]/the)p
2997 3135 4962 V 40 w(deno[i])p Fn(.)257 5221 y Fg(3.1.3)113
2998 b(PipNewparm)36 b(structure)257 5406 y Fl(struct)53 b(pipnewparm)257
2999 5526 y({)f(int)g(rank)g(;)360 5646 y(PipVector)i(*)d(vector)i(;)p
3000 eop
3001 %%Page: 16 16
3002 16 15 bop 257 266 a Fj(3)98 b(USING)32 b(THE)i(PIP)f(LIBRAR)-8
3003 b(Y)1947 b Fn(16)360 573 y Fl(Entier)53 b(deno)f(;)360
3004 693 y(struct)h(pipnewparm)h(*)d(next)i(;)257 814 y(})f(;)257
3005 934 y(typedef)h(struct)g(pipnewparm)h(PipNewparm)g(;)257
3006 1162 y Fn(The)29 b Fl(PipNewparm)h Fn(structure)f(represen)m(ts)h(a)d
3007 Fl(NULL)i Fn(terminated)d(link)m(ed)i(list)e(of)h Fl(Newparm)i
3008 Fn(as)257 1283 y(describ)s(ed)36 b(in)f(grammar)e(2)i(in)f(section)h
3009 (2.3.)51 b(F)-8 b(or)35 b(eac)m(h)h Fl(Newparm)p Fn(,)h(the)f(rank)f
3010 (is)g Fl(rank)p Fn(,)i(the)257 1403 y(v)m(ector)32 b(of)d(co)s
3011 (e\016cien)m(ts)j(is)d(p)s(oin)m(ted)h(b)m(y)h Fl(vector)p
3012 Fn(,)i(and)d(the)h(denominator)d(is)i Fl(deno)p Fn(.)44
3013 b Fl(next)31 b Fn(is)257 1523 y(a)i(p)s(oin)m(ter)f(to)g(the)h(next)g
3014 Fl(PipNewparm)j Fn(structure.)257 1783 y Fg(3.1.4)113
3015 b(PipList)35 b(structure)257 1968 y Fl(struct)53 b(piplist)257
3016 2088 y({)f(PipVector)i(*)d(vector)i(;)360 2209 y(struct)g(piplist)g(*)e
3017 (next)i(;)257 2329 y(})f(;)257 2449 y(typedef)h(struct)g(piplist)g
3018 (PipList)h(;)257 2678 y Fn(The)48 b Fl(PipList)g Fn(structure)f
3019 (represen)m(ts)i(a)d Fl(NULL)h Fn(terminated)e(link)m(ed)h(list)f(of)h
3020 Fl(Vector)h Fn(as)257 2798 y(describ)s(ed)38 b(in)d(grammar)f(2)i(in)g
3021 (section)h(2.3.)54 b Fl(vector)38 b Fn(is)e(a)g(p)s(oin)m(ter)g(to)g
3022 (the)h(v)m(ector)g(of)f(the)257 2919 y(curren)m(t)e(no)s(de)f(and)g
3023 Fl(next)g Fn(is)f(a)g(p)s(oin)m(ter)g(to)h(the)g(next)g
3024 Fl(PipList)i Fn(structure.)257 3178 y Fg(3.1.5)113 b(PipQuast)37
3025 b(structure)257 3363 y Fl(struct)53 b(pipquast)257 3483
3026 y({)f(PipNewparm)i(*)d(newparm)i(;)360 3604 y(PipList)g(*)f(list)g(;)
3027 360 3724 y(PipVector)i(*)d(condition)j(;)360 3845 y(struct)f(pipquast)g
3028 (*)f(next_then)h(;)360 3965 y(struct)g(pipquast)g(*)f(next_else)h(;)360
3029 4085 y(struct)g(pipquast)g(*)f(father)h(;)257 4206 y(})f(;)257
3030 4326 y(typedef)h(struct)g(pipquast)h(PipQuast)f(;)257
3031 4554 y Fn(The)29 b Fl(PipQuast)i Fn(represen)m(ts)f(a)e
3032 Fl(Quast)h Fn(as)g(describ)s(ed)f(in)g(grammar)e(2)h(in)h(section)g
3033 (2.3.)42 b(Eac)m(h)257 4675 y Fl(Quast)24 b Fn(has)f(a)g(tree)g
3034 (structure)h(and)f(b)s(egins)f(with)h(a)f(list)f(of)i
3035 Fl(Newparm)h Fn(\(\014eld)e Fl(newparm)p Fn(\).)42 b(If)23
3036 b(the)257 4795 y(p)s(oin)m(ter)37 b Fl(condition)j Fn(is)d(not)g
3037 Fl(NULL)p Fn(,)i(the)e(list)f(of)h Fl(Newparm)i Fn(is)e(follo)m(w)m(ed)
3038 g(b)m(y)h(a)f(conditional)257 4916 y(structure)50 b(:)76
3039 b(if)47 b(the)i(condition)e(p)s(oin)m(ted)h(b)m(y)i Fl(condition)h
3040 Fn(is)d(true,)53 b(then)c(the)g(solution)257 5036 y(con)m(tin)m(ues)28
3041 b(in)d(the)i Fl(Quast)h Fn(p)s(oin)m(ted)e(b)m(y)h Fl(next)p
3042 1917 5036 31 4 v 38 w(then)p Fn(,)h(in)e(the)h Fl(Quast)g
3043 Fn(p)s(oin)m(ted)f(b)m(y)i Fl(next)p 3446 5036 V 38 w(else)257
3044 5156 y Fn(otherwise.)61 b(If)38 b(the)h(p)s(oin)m(ter)e
3045 Fl(condition)k Fn(is)d Fl(NULL)p Fn(,)h(the)f(list)f(of)h
3046 Fl(Newparm)i Fn(is)d(follo)m(w)m(ed)h(b)m(y)257 5277
3047 y(a)e(list)f(of)h(v)m(ectors)h(\(\014eld)f Fl(list)p
3048 Fn(\).)55 b(F)-8 b(or)36 b Fl(Quast)h Fn(manipulation)c(con)m(v)m
3049 (enience,)39 b(a)d(p)s(oin)m(ter)g(to)257 5397 y(the)e(father)e(in)g
3050 (the)i(tree)f(is)f(pro)m(vided)h(\(\014eld)g Fl(father)p
3051 Fn(\),)h(ob)m(viously)f(the)g(father)g(of)f(the)h(ro)s(ot)257
3052 5518 y(is)f Fl(NULL)p Fn(.)p eop
3053 %%Page: 17 17
3054 17 16 bop 257 266 a Fj(3)98 b(USING)32 b(THE)i(PIP)f(LIBRAR)-8
3055 b(Y)1947 b Fn(17)257 573 y Fh(3.2)136 b(PipLib)44 b(functions)h
3056 (description)257 758 y Fg(3.2.1)113 b(pip)p 763 758 34
3057 4 v 40 w(solv)m(e)37 b(function)257 942 y Fl(PipQuast)54
3058 b(*)d(pip_solve)257 1063 y(\()h(PipMatrix)i(*)d(domain,)360
3059 1183 y(PipMatrix)j(*)d(context,)360 1303 y(int)h(Bg,)360
3060 1424 y(int)g(Nq,)360 1544 y(int)g(Verbose,)360 1665 y(int)g(Simplify,)
3061 360 1785 y(int)g(Max)257 1905 y(\))g(;)257 2134 y Fn(The)47
3062 b Fl(pip)p 630 2134 31 4 v 37 w(solve)g Fn(function)e(solv)m(es)h(a)f
3063 (linear)f(problem)g(pro)m(vided)i(as)f(input.)82 b(The)46
3064 b(\014rst)257 2254 y(three)d(parameters)g(describ)s(e)f(the)h(problem)e
3065 (that)h(the)h(user)g(w)m(an)m(ts)g(to)f(solv)m(e.)73
3066 b(The)43 b(last)257 2374 y(four)33 b(parameters)g(describ)s(e)h(the)f
3067 (options)g(that)g(the)g(user)h(has)f(to)g(set.)46 b(These)35
3068 b(parameters)257 2495 y(are:)377 2698 y(1.)48 b Fl(domain)p
3069 Fn(:)c(a)29 b(p)s(oin)m(ter)f(to)h(the)h(equations)f(and)h(inequations)
3070 e(system)j(whic)m(h)e(describ)s(es)501 2819 y(the)k(unkno)m(wn)i
3071 (domain)30 b(in)i(the)h(P)m(olyLib)f(constrain)m(ts)h(matrix)e(shap)s
3072 (e,)377 3022 y(2.)48 b Fl(context)p Fn(:)64 b(a)41 b(p)s(oin)m(ter)h
3073 (to)f(the)h(equations)g(and)g(inequations)g(system)g(satis\014ed)h(b)m
3074 (y)501 3142 y(the)34 b(parameters)g(con)m(text)h(in)e(the)h(P)m(olyLib)
3075 f(constrain)m(ts)h(matrix)e(shap)s(e)i(\(it)e(can)i(b)s(e)501
3076 3263 y(NULL)f(if)e(there)j(is)e(no)g(con)m(text\),)377
3077 3466 y(3.)48 b Fl(Bg)p Fn(:)i(the)36 b(column)e(rank)i(of)f(the)g(bign)
3078 m(um)g(\(\014rst)g(column)g(rank)g(is)g(0\),)h(or)f(a)g(negativ)m(e)501
3079 3587 y(v)-5 b(alue)32 b(if)g(there)h(is)f(no)h(big)e(parameter)h(in)g
3080 (the)h(problem)e(to)i(b)s(e)f(solv)m(ed,)377 3790 y(4.)48
3081 b Fl(Nq)p Fn(:)c(a)33 b(b)s(o)s(olean)e(set)i(to)f(1)g(if)g(an)g(in)m
3082 (teger)h(solution)e(is)h(needed,)i(0)e(otherwise,)377
3083 3993 y(5.)48 b Fl(Verbose)p Fn(:)d(a)31 b(b)s(o)s(olean)e(set)j(to)e(1)
3084 h(if)f(debug)i(informations)c(are)j(needed)i(\(in)d(this)h(case,)501
3085 4114 y(most)i(of)h(the)g(pro)s(cessing)g(will)e(b)s(e)i(prin)m(ted)g
3086 (in)f(a)g(\014le.)47 b(The)35 b(\014le)e(name)g(is)h(generated)501
3087 4234 y(at)e(random)g(\(b)m(y)i Fi(mkstemp)6 b Fn(\))31
3088 b(or)h(set)i(with)e(constan)m(t)h(DEBUG\),)g(0)f(if)f(not,)377
3089 4438 y(6.)48 b Fl(Simplify)p Fn(:)42 b(a)26 b(b)s(o)s(olean)e(set)i(to)
3090 g(1)f(if)g(some)g(trivial)e(quast)k(simpli\014cations)22
3091 b(are)k(needed)501 4558 y(\(recursiv)m(e)40 b(elimination)35
3092 b(of)j(degenerated)h(patterns)h(lik)m(e)e Fl(if)51 b(#[...])104
3093 b(\(\))52 b(\(\))p Fn(\),)41 b(0)501 4678 y(otherwise,)377
3094 4882 y(7.)48 b Fl(Max)p Fn(:)c(not)33 b(in)f(use)h(y)m(et,)h(please)f
3095 (set)g(it)f(to)g(0)g(for)g(future)h(compatibilit)m(y)-8
3096 b(.)257 5085 y(This)33 b(function)f(returns)i(a)e(p)s(oin)m(ter)g(to)g
3097 (a)g Fl(PipQuast)j Fn(structure)f(con)m(taining)d(the)i(solution,)257
3098 5205 y(it)f(will)e(b)s(e)j Fl(NULL)h Fn(if)d(the)i(con)m(text)h(is)e
3099 Fl(void)p Fn(.)p eop
3100 %%Page: 18 18
3101 18 17 bop 257 266 a Fj(3)98 b(USING)32 b(THE)i(PIP)f(LIBRAR)-8
3102 b(Y)1947 b Fn(18)257 573 y Fg(3.2.2)113 b(pip)p 763 573
3103 34 4 v 40 w(matrix)p 1132 573 V 39 w(allo)s(c)36 b(function)257
3104 758 y Fl(PipMatrix)54 b(*)e(pip_matrix_alloc)257 878
3105 y(\()g(unsigned)h(nb_rows,)360 998 y(unsigned)g(nb_columns)257
3106 1119 y(\))f(;)257 1309 y Fn(The)c Fl(pip)p 631 1309 31
3107 4 v 37 w(matrix)p 974 1309 V 39 w(alloc)f Fn(function)f(allo)s(cates)e
3108 (the)j(memory)e(space)j(for)d(a)h Fl(PipMatrix)257 1429
3109 y Fn(structure)53 b(with)d Fl(nb)p 1042 1429 V 37 w(rows)i
3110 Fn(ro)m(ws)g(and)f Fl(nb)p 1889 1429 V 38 w(columns)h
3111 Fn(columns.)98 b(It)51 b(\014lls)f(the)h Fl(Nb)p 3419
3112 1429 V 37 w(Rows)p Fn(,)257 1549 y Fl(Nb)p 365 1549 V
3113 38 w(Columns)35 b Fn(and)f Fl(p)g Fn(\014elds)g(and)g(initializes)d
3114 (the)j(matrix)e(en)m(tries)i(to)g(0,)g(then)g(it)f(returns)i(a)257
3115 1670 y(p)s(oin)m(ter)d(to)h(this)f(structure.)257 1925
3116 y Fg(3.2.3)113 b(pip)p 763 1925 34 4 v 40 w(matrix)p
3117 1132 1925 V 39 w(read)38 b(function)257 2109 y Fl(PipMatrix)54
3118 b(*)e(pip_matrix_read\(FILE)k(*\))c(;)257 2299 y Fn(The)45
3119 b Fl(pip)p 628 2299 31 4 v 38 w(matrix)p 972 2299 V 38
3120 w(read)g Fn(function)e(read)g(a)g(matrix)f(from)h(a)g(\014le.)76
3121 b(It)43 b(tak)m(es)i(as)f(input)f(a)257 2420 y(p)s(oin)m(ter)g(to)f
3122 (the)i(\014le)e(it)g(has)i(to)e(read)i(\(p)s(ossibly)e
3123 Fl(stdin)p Fn(\),)47 b(and)c(returns)h(a)f(p)s(oin)m(ter)f(to)h(a)257
3124 2540 y Fl(PipMatrix)35 b Fn(structure.)45 b(The)34 b(input)e(has)h(the)
3125 g(follo)m(wing)d(syn)m(tax:)403 2715 y Fm(\017)48 b Fn(some)33
3126 b(optional)d(commen)m(t)i(lines)g(whic)m(h)h(b)s(egin)f(with)g
3127 Fl(#)p Fn(,)403 2909 y Fm(\017)48 b Fn(the)34 b(ro)m(w)f(n)m(um)m(b)s
3128 (ers)h(and)f(column)f(n)m(um)m(b)s(ers,)j(p)s(ossibly)d(follo)m(w)m(ed)
3129 g(b)m(y)i(commen)m(ts,)g(on)501 3029 y(a)f(single)e(line,)403
3130 3223 y Fm(\017)48 b Fn(the)32 b(matrix)e(ro)m(ws,)i(eac)m(h)h(ro)m(w)e
3131 (m)m(ust)h(b)s(e)g(on)f(a)g(single)f(line)g(and)i(is)f(p)s(ossibly)f
3132 (follo)m(w)m(ed)501 3343 y(b)m(y)k(commen)m(ts.)257 3518
3133 y(F)-8 b(or)32 b(instance,)h(in)f(the)h(problem)e(of)h(section)h(2.1.1)
3134 f(the)h(domain)e(is)h(de\014ned)i(as)f(follo)m(ws)257
3135 3693 y Fl(#)52 b(This)g(is)g(the)g(domain)257 3813 y(3)g(7)872
3136 b(#)51 b(3)h(lines)g(and)g(7)g(columns)257 3933 y(1)103
3137 b(0)52 b(-1)103 b(0)g(1)f(0)h(0)52 b(#)f(-i)h(+)g(m)f(>=)h(0)257
3138 4054 y(1)g(-1)103 b(0)g(0)g(0)f(1)h(0)52 b(#)f(-j)h(+)g(n)f(>=)h(0)257
3139 4174 y(1)103 b(1)g(1)51 b(-1)104 b(0)e(0)h(0)52 b(#)f(j)h(+)f(i)h(-)f
3140 (k)h(>=)g(0)257 4429 y Fg(3.2.4)113 b(Prin)m(ting)35
3141 b(F)-9 b(unctions)257 4614 y Fl(void)53 b(pip_matrix_print\(FILE)k(*,)
3142 51 b(PipMatrix)j(*\))e(;)257 4734 y(void)h(pip_vector_print\(FILE)k(*,)
3143 51 b(PipVector)j(*\))e(;)257 4855 y(void)h(pip_newparm_print\(FILE)k
3144 (*,)52 b(PipNewparm)i(*,)d(int)h(indent\))i(;)257 4975
3145 y(void)f(pip_list_print\(FILE)j(*,)c(PipList)h(*,)f(int)g(indent\))h(;)
3146 257 5095 y(void)g(pip_quast_print\(FILE)j(*,)c(PipQuast)h(*,)f(int)g
3147 (indent\))h(;)257 5285 y Fn(There)40 b(is)f(a)f(prin)m(ting)f(function)
3148 i(for)f(eac)m(h)i(structure)g(of)e(the)h(PipLib.)61 b(They)40
3149 b(all)d(tak)m(e)i(as)257 5406 y(input)32 b(a)g(p)s(oin)m(ter)g(to)g(a)g
3150 (\014le)g(\(p)s(ossibly)g Fl(stdout)p Fn(\))i(and)e(a)g(p)s(oin)m(ter)g
3151 (to)g(a)g(structure.)45 b(Some)32 b(of)257 5526 y(them)g(tak)m(es)h(as)
3152 f(input)f(an)h(inden)m(t)g(step.)44 b(They)33 b(prin)m(t)f(the)g
3153 (structure)h(con)m(ten)m(ts)g(to)f(the)g(\014le)257 5646
3154 y(without)g(inden)m(t)h(if)f Fl(indent)i Fk(<)27 b Fn(0,)33
3155 b(with)f(an)g(inden)m(tation)g(step)h(of)g Fl(indent)h
3156 Fn(otherwise.)p eop
3157 %%Page: 19 19
3158 19 18 bop 257 266 a Fj(3)98 b(USING)32 b(THE)i(PIP)f(LIBRAR)-8
3159 b(Y)1947 b Fn(19)257 573 y Fg(3.2.5)113 b(Memory)37 b(Deallo)s(cation)f
3160 (F)-9 b(unctions)257 758 y Fl(void)53 b(pip_matrix_free\(PipMatrix)58
3161 b(*\))51 b(;)257 878 y(void)i(pip_vector_free\(PipVector)58
3162 b(*\))51 b(;)257 998 y(void)i(pip_newparm_free\(PipNewpar)q(m)k(*\))52
3163 b(;)257 1119 y(void)h(pip_list_free\(PipList)k(*\))51
3164 b(;)257 1239 y(void)i(pip_quast_free\(PipQuast)k(*\))52
3165 b(;)257 1467 y Fn(There)35 b(is)e(a)h(memory)f(deallo)s(cation)d
3166 (function)k(for)f(eac)m(h)h(structure)h(of)e(the)h(PipLib.)46
3167 b(They)257 1588 y(free)33 b(the)g(allo)s(cated)e(memory)h(for)g(the)h
3168 (structure.)257 1877 y Fh(3.3)136 b(Example)257 2061
3169 y Fn(Here)42 b(is)e(a)h(simple)e(example)h(sho)m(wing)h(ho)m(w)g(one)g
3170 (can)g(use)h(the)f(PipLib,)h(assuming)e(that)257 2182
3171 y(a)c(basic)g(installation)c(w)m(as)37 b(done.)53 b(The)37
3172 b(follo)m(wing)c(C)j(program)f(reads)h(a)g(domain)e(and)i(its)257
3173 2302 y(con)m(text)i(on)e(the)h(standard)f(input)g(then)h(prin)m(ts)f
3174 (the)h(solution)e(on)h(the)h(standard)f(output.)257 2423
3175 y(Options)47 b(are)g(preselected)i(:)73 b(there)48 b(is)f(no)g(bign)m
3176 (um,)j(w)m(e)f(are)e(lo)s(oking)e(for)i(an)g(in)m(tegral)257
3177 2543 y(solution)31 b(without)i(simpli\014cation)c(and)j(w)m(e)i(don't)f
3178 (w)m(an)m(t)g(debug)g(informations.)257 2746 y Fl(/*)52
3179 b(example.c)i(*/)257 2867 y(#)e(include)h(<stdio.h>)257
3180 2987 y(#)f(include)h(<piplib/piplib64.h>)257 3228 y(int)f(main\(\))257
3181 3348 y({)g(PipMatrix)i(*)d(domain,)i(*)f(context)104
3182 b(;)360 3469 y(PipQuast)53 b(*)f(solution)h(;)360 3709
3183 y(domain)g(=)e(pip_matrix_read\(stdin\))57 b(;)360 3830
3184 y(context)c(=)f(pip_matrix_read\(stdin\))57 b(;)360 4070
3185 y(solution)c(=)f(pip_solve\(domain,context,-)q(1,1)q(,0,0)q(,0\))58
3186 b(;)360 4311 y(pip_matrix_free\(domain\))f(;)360 4432
3187 y(pip_matrix_free\(context\))h(;)360 4672 y(pip_quast_print\(stdout,so)
3188 q(luti)q(on,0)q(\))f(;)257 4793 y(})257 4996 y Fn(The)34
3189 b(compilation)29 b(command)i(could)i(b)s(e)f(:)257 5200
3190 y Fl(gcc)52 b(example.c)i(-lpiplib64)g(-o)e(example)257
3191 5403 y Fn(Supp)s(osing)30 b(that)g(the)g(user)h(w)m(an)m(ts)g(to)e
3192 (solv)m(e)i(the)f(problem)e(of)i(section)g(2.1.1,)g(he)g(will)d(t)m(yp)
3193 s(e:)p eop
3194 %%Page: 20 20
3195 20 19 bop 257 266 a Fj(4)98 b(INST)-8 b(ALLING)33 b(PIP)2377
3196 b Fn(20)257 573 y Fl(3)52 b(7)257 693 y(1)103 b(0)52
3197 b(-1)103 b(0)g(1)f(0)h(0)257 814 y(1)52 b(-1)103 b(0)g(0)g(0)f(1)h(0)
3198 257 934 y(1)g(1)g(1)51 b(-1)104 b(0)e(0)h(0)257 1054
3199 y(1)52 b(5)257 1175 y(1)g(-1)103 b(1)g(1)g(0)257 1351
3200 y Fn(And)33 b(the)g(program)f(will)e(prin)m(t)i(:)257
3201 1527 y Fl(\(if)52 b(#[)g(-1)g(1)g(0)f(0])309 1648 y(\(list)360
3202 1768 y(#[)h(0)f(0)h(0)f(0])360 1889 y(#[)h(1)f(0)h(0)f(0])309
3203 2009 y(\))309 2129 y(\(list)360 2250 y(#[)h(1)f(-1)h(0)g(0])360
3204 2370 y(#[)g(0)f(1)h(0)f(0])309 2490 y(\))257 2611 y(\))257
3205 2939 y Fo(4)161 b(Installing)56 b(PIP)257 3187 y Fh(4.1)136
3206 b(Implemen)l(tation)47 b(Notes)257 3372 y Fn(The)28 b(main)c(problem)h
3207 (with)h(an)m(y)h(in)m(teger)f(programming)d(soft)m(w)m(are)k(is)f
3208 (that,)h(since)g(one)f(m)m(ust)257 3492 y(distinguish)c(b)s(et)m(w)m
3209 (een)k(in)m(teger)d(and)h(rationals,)f(all)e(computations)h(are)i(to)f
3210 (b)s(e)g(done)h(exactly)-8 b(.)257 3613 y(Rationals)41
3211 b(m)m(ust)h(b)s(e)h(represen)m(ted)h(as)f(quotien)m(ts)g(with)f(an)g
3212 (in)m(teger)g(n)m(umerator)g(and)h(an)257 3733 y(in)m(teger)36
3213 b(denominator.)53 b(In)36 b(the)g(preceding)h(v)m(ersion,)g(there)g(w)m
3214 (as)g(only)f(one)g(denominator)257 3853 y(for)46 b(the)h(whole)g
3215 (tableau.)85 b(The)47 b(consequence)j(w)m(as)d(that)g
3216 (simpli\014cations)c(where)48 b(most)257 3974 y(unlik)m(ely)-8
3217 b(,)25 b(and)e(that)g(the)g(in)m(tegers)h(in)e(the)h(tableau)g(w)m(ere)
3218 h(gro)m(wing)e(un)m(til)g(o)m(v)m(er\015o)m(ws)k(o)s(ccured.)404
3219 4094 y(In)c(the)h(presen)m(t)h(v)m(ersion,)h(there)e(is)f(one)g
3220 (denominator)f(p)s(er)i(ro)m(w)f(of)g(the)h(tableau.)39
3221 b(Reduc-)257 4214 y(tion)30 b(to)h(lo)m(w)m(er)g(terms)g(o)s(ccurs)h
3222 (frequen)m(tly)-8 b(,)32 b(and)f(the)g(gro)m(wth)h(of)e(n)m(um)m(b)s
3223 (ers)i(in)e(the)i(problem)257 4335 y(tableau)i(is)f(limited.)44
3224 b(As)35 b(a)e(consequence,)38 b(m)m(uc)m(h)c(larger)f(problems)g(can)h
3225 (b)s(e)g(solv)m(ed.)48 b(This)257 4455 y(has)37 b(had)g(the)g
3226 (unfortunate)f(consequence)k(that)c(sev)m(eral)h(bugs,)i(whic)m(h)e(w)m
3227 (ere)g(b)s(ey)m(ond)h(the)257 4576 y(domain)c(of)i(the)g(old)f(v)m
3228 (ersion,)i(ha)m(v)m(e)h(no)m(w)e(surfaced.)54 b(These)38
3229 b(bugs)e(ha)m(v)m(e)i(b)s(een)e(corrected.)257 4696 y(As)g(far)e(as)g
3230 (the)i(author)e(can)h(tell,)f(these)h(bugs)h(mainly)c(ga)m(v)m(e)k
3231 (correct)f(results)g(whic)m(h)g(w)m(ere)257 4816 y(not)f(in)e(simplest)
3232 h(form:)44 b(the)34 b(quast)g(had)f(extraneous)i(lea)m(v)m(es.)47
3233 b(In)34 b(some)g(cases,)h(the)f(result)257 4937 y(w)m(as)h(wrong)f(but)
3234 g(the)g(error)g(w)m(as)h(self)e(eviden)m(t:)47 b(for)33
3235 b(instance,)i(there)f(w)m(ere)i(denominators)257 5057
3236 y(in)c(in)m(teger)h(results.)257 5341 y Fh(4.2)136 b(License)257
3237 5526 y Fn(This)44 b(program)e(is)h(free)h(soft)m(w)m(are;)50
3238 b(y)m(ou)44 b(can)g(redistribute)f(it)f(and/or)h(mo)s(dify)f(it)h
3239 (under)257 5646 y(the)32 b(terms)g(of)f(the)g(GNU)h(General)e(Public)h
3240 (License)h(v)m(ersion)g(2)f(as)h(published)f(b)m(y)i(the)e(F)-8
3241 b(ree)p eop
3242 %%Page: 21 21
3243 21 20 bop 257 266 a Fj(4)98 b(INST)-8 b(ALLING)33 b(PIP)2377
3244 b Fn(21)257 573 y(Soft)m(w)m(are)34 b(F)-8 b(oundation.)404
3245 693 y(This)28 b(program)e(is)i(distributed)f(in)h(the)g(hop)s(e)g(that)
3246 g(it)f(will)f(b)s(e)i(useful,)h(but)f(WITHOUT)257 814
3247 y(ANY)39 b(W)-11 b(ARRANTY;)39 b(without)f(ev)m(en)i(the)f(implied)d(w)
3248 m(arran)m(t)m(y)k(of)e(MER)m(CHANT)-8 b(ABIL-)257 934
3249 y(ITY)49 b(or)f(FITNESS)h(F)m(OR)e(A)h(P)-8 b(AR)g(TICULAR)49
3250 b(PURPOSE.)g(See)g(the)g(GNU)e(General)257 1054 y(Public)32
3251 b(License)h(for)g(more)e(details.)43 b Fl(http://www.gnu.org/copylef)q
3252 (t/g)q(pl.h)q(tml)257 1340 y Fh(4.3)136 b(Building)45
3253 b(the)g(Executable)h(and)f(the)g(Library)257 1525 y Fn(T)-8
3254 b(o)40 b(build)e(PIP)-8 b(,)41 b(\014rst)f(cop)m(y)h(the)f(ab)s(o)m(v)m
3255 (e)g(tar\014le)f(to)g(an)m(y)h(con)m(v)m(enien)m(t)i(directory)-8
3256 b(.)64 b(Expand)257 1645 y(the)33 b(tar\014le)f(using:)257
3257 1829 y Fl(zcat)53 b(pip.tar.Z)g(|)f(tar)g(xvf)g(-)257
3258 2013 y Fn(Y)-8 b(ou)33 b(should)f(obtain)g(the)h(follo)m(wing)d
3259 (\014les:)403 2198 y Fm(\017)48 b Fn(header)34 b(\014les)e(in)g(the)h
3260 Fl(include)h Fn(directory)-8 b(,)403 2395 y Fm(\017)48
3261 b Fn(C)33 b(co)s(de)g(\014les)g(in)f(the)h Fl(source)h
3262 Fn(directory)-8 b(,)403 2592 y Fm(\017)48 b Fn(a)33 b(lot)f(of)g(data)h
3263 (\014les)g Fl(*.dat)h Fn(and)f(of)g(result)g(\014les)g
3264 Fl(*.ll)h Fn(in)e(the)i Fl(test)g Fn(directory)f(\(y)m(ou)501
3265 2712 y(should)e(then)g(run)f(at)h(least)f(some)g(of)g(the)h
3266 Fl(*.dat)h Fn(\014les)e(and)h(compare)f(the)h(results)g(to)501
3267 2832 y(the)i(corresp)s(onding)g Fl(*.ll)g Fn(\014le\),)403
3268 3029 y Fm(\017)48 b Fn(a)31 b(simple)f(example)h(sho)m(wing)h(ho)m(w)g
3269 (to)f(use)i(the)e(PipLib)g(in)f(the)i Fl(example)i Fn(directory)-8
3270 b(,)403 3226 y Fm(\017)48 b Fn(a)33 b(p)s(ostscript)f(v)m(ersion)h(of)f
3271 (the)h(presen)m(t)i(do)s(cumen)m(t,)e Fl(pip.ps)h Fn(in)d(the)i
3272 Fl(doc)h Fn(directory)-8 b(,)403 3423 y Fm(\017)48 b
3273 Fn(\014les)33 b(needed)h(for)e(compilation)d(and)k(installation)c(in)j
3274 (the)h(PIP)g(ro)s(ot)f(directory)-8 b(.)257 3680 y Fg(4.3.1)113
3275 b(basic)37 b(installation)257 3864 y Fn(The)f Fl(configure)i
3276 Fn(shell)c(script)h(attempts)g(to)f(guess)j(correct)e(v)-5
3277 b(alues)35 b(for)g(v)-5 b(arious)34 b(system-)257 3985
3278 y(dep)s(enden)m(t)45 b(v)-5 b(ariables)42 b(used)i(during)e
3279 (compilation.)72 b(It)43 b(uses)h(those)g(v)-5 b(alues)43
3280 b(to)f(create)i(a)257 4105 y Fl(Makefile)p Fn(.)79 b(The)45
3281 b(\014le)e Fl(configure.in)k Fn(is)c(used)i(to)e(create)h
3282 Fl(configure)i Fn(b)m(y)f(a)e(program)257 4226 y(called)28
3283 b Fl(autoconf)p Fn(.)44 b(Y)-8 b(ou)28 b(only)g(need)h
3284 Fl(configure.in)j Fn(if)27 b(y)m(ou)i(w)m(an)m(t)g(to)f(c)m(hange)h(it)
3285 e(or)h(regen-)257 4346 y(erate)33 b Fl(configure)i Fn(using)d(a)h(new)m
3286 (er)h(v)m(ersion)f(of)f Fl(autoconf)p Fn(.)404 4466 y(The)h(simplest)f
3287 (w)m(a)m(y)h(to)g(compile)d(this)j(pac)m(k)-5 b(age)33
3288 b(is:)403 4651 y Fm(\017)48 b Fl(cd)71 b Fn(to)e(the)i(directory)e(con)
3289 m(taining)g(the)h(pac)m(k)-5 b(age's)71 b(source)g(co)s(de)g(and)f(t)m
3290 (yp)s(e)501 4771 y Fl(./configure)55 b Fn(to)c(con\014gure)i(the)f(pac)
3291 m(k)-5 b(age)52 b(for)f(y)m(our)h(system)h(\(while)e(running,)501
3292 4891 y Fl(configure)35 b Fn(prin)m(ts)e(some)f(messages)i(telling)c
3293 (whic)m(h)j(features)g(it)f(is)g(c)m(hec)m(king)i(for\),)403
3294 5088 y Fm(\017)48 b Fn(t)m(yp)s(e)34 b Fl(make)f Fn(to)g(compile)d(the)
3295 j(pac)m(k)-5 b(age)33 b(and)g(install)d(the)j(program)e(and)i(the)g
3296 (library)-8 b(,)403 5285 y Fm(\017)48 b Fn(y)m(ou)33
3297 b(can)f(remo)m(v)m(e)h(the)f(program)f(binaries)g(and)h(ob)5
3298 b(ject)33 b(\014les)f(from)f(the)h(source)h(co)s(de)501
3299 5406 y(directory)j(b)m(y)g(t)m(yping)g Fl(make)52 b(clean)p
3300 Fn(.)i(T)-8 b(o)35 b(also)g(remo)m(v)m(e)h(the)g(\014les)g(that)f
3301 Fl(configure)501 5526 y Fn(created)i(\(so)g(y)m(ou)g(can)g(compile)d
3302 (the)j(pac)m(k)-5 b(age)37 b(for)f(a)g(di\013eren)m(t)g(kind)h(of)f
3303 (computer\))501 5646 y(t)m(yp)s(e)e Fl(make)52 b(distclean)p
3304 Fn(.)p eop
3305 %%Page: 22 22
3306 22 21 bop 257 266 a Fj(REFERENCES)2659 b Fn(22)404 573
3307 y(PIP)27 b(and)g(the)h(PipLib)d(ha)m(v)m(e)k(b)s(een)e(successfully)i
3308 (compiled)c(on)h(the)i(follo)m(wing)c(systems:)403 776
3309 y Fm(\017)48 b Fn(PC's)34 b(under)g(Lin)m(ux,)e(with)h(the)g
3310 Fl(gcc)g Fn(compiler,)403 980 y Fm(\017)48 b Fn(PC's)26
3311 b(under)g(Windo)m(ws)f(\(Cygwin\),)i(with)e(the)g Fl(gcc)g
3312 Fn(compiler)e(\(but)i(b)s(ecause)h(of)f(some)501 1100
3313 y(Cygwin)31 b(limitations,)26 b(only)j(32)g(bits)h(v)m(ersion)g(will)e
3314 (w)m(ork)j(and)f(user)h(ma)m(y)e(exp)s(erience)501 1220
3315 y(some)k(problems)e(when)j(linking)d(with)h(PipLib\),)403
3316 1424 y Fm(\017)48 b Fn(SparcStations,)33 b(with)f(the)h
3317 Fl(gcc)g Fn(compiler.)257 1684 y Fg(4.3.2)113 b(optionnal)36
3318 b(features)403 1868 y Fm(\017)48 b Fn(By)67 b(default,)73
3319 b Fl(make)67 b Fn(will)c(install)h(the)i(pac)m(k)-5 b(age's)67
3320 b(\014les)f(in)f Fl(/usr/local/bin)p Fn(,)501 1989 y
3321 Fl(/usr/local/lib)p Fn(,)52 b(etc.)81 b(Y)-8 b(ou)44
3322 b(can)h(sp)s(ecify)h(an)e(installation)e(pre\014x)k(other)e(than)501
3323 2109 y Fl(/usr/local)35 b Fn(b)m(y)f(giving)d Fl(configure)k
3324 Fn(the)e(option)e Fl(--prefix=PATH)p Fn(.)403 2313 y
3325 Fm(\017)48 b Fn(By)37 b(default,)g(b)s(oth)g(PIP)g(and)g(the)g(PipLib)e
3326 (are)i(compiled)d(and)j(installed.)54 b(By)37 b(giv-)501
3327 2433 y(ing)31 b Fl(configure)i Fn(the)f(option)e Fl(--without-pip)35
3328 b Fn(y)m(ou)d(disable)e(the)i(compilation)27 b(and)501
3329 2553 y(installation)38 b(of)j(PIP)-8 b(.)43 b(By)f(giving)e
3330 Fl(configure)k Fn(the)e(option)f Fl(--without-lib)k Fn(y)m(ou)501
3331 2674 y(disable)32 b(the)h(compilation)c(and)k(installation)c(of)j(the)h
3332 (PipLib.)403 2877 y Fm(\017)48 b Fn(By)32 b(default,)f(b)s(oth)g(in)m
3333 (t)f(\(32)h(bits\))g(and)g(long)f(long)g(in)m(t)g(\(64)h(bits\))g(v)m
3334 (ersions)h(are)f(built.)501 2997 y(By)f(giving)d Fl(configure)k
3335 Fn(the)e(option)f Fl(--enable-int)k Fn(y)m(ou)d(ask)h(for)e(in)m(t)h(v)
3336 m(ersion)g(only)501 3118 y(and)g(b)m(y)g(giving)e Fl(configure)k
3337 Fn(the)e(option)f Fl(--enable-llint)k Fn(y)m(ou)d(ask)g(for)f(long)g
3338 (long)501 3238 y(in)m(t)k(v)m(ersion)h(only)-8 b(.)257
3339 3498 y Fg(4.3.3)113 b(uninstallation)257 3683 y Fn(Y)-8
3340 b(ou)44 b(can)g(easily)f(remo)m(v)m(e)h(PIP)g(and)g(the)g(PipLib)e
3341 (from)h(y)m(our)h(system)h(b)m(y)f(t)m(yping)f Fl(make)257
3342 3803 y(uninstall)p Fn(.)404 4042 y(Rep)s(ort)32 b(all)e(bugs,)k
3343 (problems,)e(inaccuracies)g(in)g(the)h(do)s(cumen)m(tation)e(to:)404
3344 4162 y Fl(Paul.Feautrier@prism.uvsq.)q(fr)404 4282 y
3345 Fn(Praise)h(is)g(also)g(appreciated.)404 4403 y(Let)g(the)h(p)s(o)m(w)m
3346 (er)h(of)e(parametric)f(in)m(teger)i(programming)c(b)s(e)k(with)f(y)m
3347 (ou.)257 4736 y Fo(References)257 4955 y Fn([1])49 b(P)m(aul)67
3348 b(F)-8 b(eautrier.)147 b(P)m(arametric)67 b(in)m(teger)g(programming.)
3349 145 b Fi(RAIR)n(O)67 b(R)-5 b(e)g(cher)g(che)409 5075
3350 y(Op)n(\023)-47 b(er)-5 b(ationnel)5 b(le)p Fn(,)30 b(22:243{268,)h
3351 (Septem)m(b)s(er)i(1988.)257 5278 y([2])49 b(P)m(aul)g(F)-8
3352 b(eautrier.)94 b(Seman)m(tical)48 b(analysis)h(and)g(mathematical)e
3353 (programming;)55 b(ap-)409 5399 y(plication)45 b(to)j(parallelization)
3354 43 b(and)48 b(v)m(ectorization.)89 b(In)48 b(M.)g(Cosnard,)53
3355 b(Y.)48 b(Rob)s(ert,)409 5519 y(P)-8 b(.)41 b(Quin)m(ton,)i(and)e(M.)h
3356 (Ra)m(ynal,)g(editors,)h Fi(Workshop)f(on)h(Par)-5 b(al)5
3357 b(lel)42 b(and)g(Distribute)-5 b(d)409 5639 y(A)n(lgorithms,)34
3358 b(Bonas)p Fn(,)e(pages)h(309{320.)e(North)h(Holland,)f(1989.)p
3359 eop
3360 %%Page: 23 23
3361 23 22 bop 257 266 a Fj(REFERENCES)2659 b Fn(23)257 573
3362 y([3])49 b(Doran)28 b(K.)h(Wilde.)36 b(A)28 b(library)g(for)g(doing)f
3363 (p)s(olyhedral)h(op)s(erations.)36 b(T)-8 b(ec)m(hnical)29
3364 b(Rep)s(ort)409 693 y(785,)j(IRISA,)h(Rennes,)h(F)-8
3365 b(rance,)33 b(1993.)p eop
3366 %%Trailer
3367 end
3368 userdict /end-hook known{end-hook}if
3369 %%EOF