[493c477] | 1 | /* emacs edit mode for this file is -*- C++ -*- */ |
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[341696] | 2 | /* $Id$ */ |
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[9bab9f] | 3 | |
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[e4fe2b] | 4 | #include "config.h" |
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[ab4548f] | 5 | |
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[650f2d8] | 6 | #include "cf_assert.h" |
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[93b061] | 7 | #include "debug.h" |
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| 8 | |
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[9bab9f] | 9 | #include "cf_defs.h" |
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| 10 | #include "canonicalform.h" |
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| 11 | #include "cf_iter.h" |
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| 12 | #include "cf_reval.h" |
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[edb4893] | 13 | #include "cf_primes.h" |
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[fbefc9] | 14 | #include "cf_algorithm.h" |
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[2072126] | 15 | #include "cf_factory.h" |
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[f63dbca] | 16 | #include "fac_util.h" |
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[6db552] | 17 | #include "templates/ftmpl_functions.h" |
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[bb82f0] | 18 | #include "algext.h" |
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[10af64] | 19 | #include "cf_gcd_smallp.h" |
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[6e2ef0e] | 20 | #include "cf_map_ext.h" |
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| 21 | #include "cf_util.h" |
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[d990001] | 22 | #include "gfops.h" |
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[edb4893] | 23 | |
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[f11d7b] | 24 | #ifdef HAVE_NTL |
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[034eec] | 25 | #include <NTL/ZZX.h> |
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[f11d7b] | 26 | #include "NTLconvert.h" |
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[a7ec94] | 27 | bool isPurePoly(const CanonicalForm & ); |
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| 28 | static CanonicalForm gcd_univar_ntl0( const CanonicalForm &, const CanonicalForm & ); |
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[c4f4fd] | 29 | static CanonicalForm gcd_univar_ntlp( const CanonicalForm &, const CanonicalForm & ); |
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[f11d7b] | 30 | #endif |
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| 31 | |
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[7e8c9e] | 32 | #ifdef HAVE_FLINT |
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| 33 | #include "FLINTconvert.h" |
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| 34 | static CanonicalForm gcd_univar_flint0 (const CanonicalForm &, const CanonicalForm &); |
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| 35 | static CanonicalForm gcd_univar_flintp (const CanonicalForm &, const CanonicalForm &); |
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| 36 | #endif |
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| 37 | |
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[a7ec94] | 38 | static CanonicalForm cf_content ( const CanonicalForm &, const CanonicalForm & ); |
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| 39 | static void cf_prepgcd( const CanonicalForm &, const CanonicalForm &, int &, int &, int & ); |
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[edb4893] | 40 | |
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[27bb97f] | 41 | void out_cf(const char *s1,const CanonicalForm &f,const char *s2); |
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[6f62c3] | 42 | |
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[110718] | 43 | CanonicalForm chinrem_gcd(const CanonicalForm & FF,const CanonicalForm & GG); |
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[f4b180] | 44 | |
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[f63dbca] | 45 | bool |
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| 46 | gcd_test_one ( const CanonicalForm & f, const CanonicalForm & g, bool swap ) |
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[9bab9f] | 47 | { |
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| 48 | int count = 0; |
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| 49 | // assume polys have same level; |
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[6e2ef0e] | 50 | |
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| 51 | Variable v= Variable (1); |
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| 52 | bool algExtension= (hasFirstAlgVar (f, v) || hasFirstAlgVar (g, v)); |
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[f63dbca] | 53 | CanonicalForm lcf, lcg; |
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[6f62c3] | 54 | if ( swap ) |
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| 55 | { |
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[150dc8] | 56 | lcf = swapvar( LC( f ), Variable(1), f.mvar() ); |
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| 57 | lcg = swapvar( LC( g ), Variable(1), f.mvar() ); |
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[f63dbca] | 58 | } |
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[6f62c3] | 59 | else |
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| 60 | { |
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[150dc8] | 61 | lcf = LC( f, Variable(1) ); |
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| 62 | lcg = LC( g, Variable(1) ); |
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[f63dbca] | 63 | } |
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[6e2ef0e] | 64 | |
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[f63dbca] | 65 | CanonicalForm F, G; |
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[6f62c3] | 66 | if ( swap ) |
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| 67 | { |
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[150dc8] | 68 | F=swapvar( f, Variable(1), f.mvar() ); |
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| 69 | G=swapvar( g, Variable(1), g.mvar() ); |
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[f63dbca] | 70 | } |
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[6f62c3] | 71 | else |
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| 72 | { |
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[150dc8] | 73 | F = f; |
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| 74 | G = g; |
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[f63dbca] | 75 | } |
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[6e2ef0e] | 76 | |
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| 77 | #define TEST_ONE_MAX 50 |
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| 78 | int p= getCharacteristic(); |
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| 79 | bool passToGF= false; |
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| 80 | int k= 1; |
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| 81 | if (p > 0 && p < TEST_ONE_MAX && CFFactory::gettype() != GaloisFieldDomain && !algExtension) |
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| 82 | { |
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| 83 | if (p == 2) |
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| 84 | setCharacteristic (2, 6, 'Z'); |
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| 85 | else if (p == 3) |
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| 86 | setCharacteristic (3, 4, 'Z'); |
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| 87 | else if (p == 5 || p == 7) |
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| 88 | setCharacteristic (p, 3, 'Z'); |
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| 89 | else |
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| 90 | setCharacteristic (p, 2, 'Z'); |
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| 91 | passToGF= true; |
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| 92 | } |
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| 93 | else if (p > 0 && CFFactory::gettype() == GaloisFieldDomain && ipower (p , getGFDegree()) < TEST_ONE_MAX) |
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| 94 | { |
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| 95 | k= getGFDegree(); |
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| 96 | if (ipower (p, 2*k) > TEST_ONE_MAX) |
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| 97 | setCharacteristic (p, 2*k, gf_name); |
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| 98 | else |
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| 99 | setCharacteristic (p, 3*k, gf_name); |
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| 100 | F= GFMapUp (F, k); |
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| 101 | G= GFMapUp (G, k); |
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| 102 | lcf= GFMapUp (lcf, k); |
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| 103 | lcg= GFMapUp (lcg, k); |
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| 104 | } |
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| 105 | else if (p > 0 && p < TEST_ONE_MAX && algExtension) |
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| 106 | { |
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| 107 | bool extOfExt= false; |
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[d990001] | 108 | #ifdef HAVE_NTL |
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[6e2ef0e] | 109 | int d= degree (getMipo (v)); |
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| 110 | CFList source, dest; |
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| 111 | Variable v2; |
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| 112 | CanonicalForm primElem, imPrimElem; |
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| 113 | if (p == 2 && d < 6) |
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| 114 | { |
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| 115 | zz_p::init (p); |
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| 116 | bool primFail= false; |
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| 117 | Variable vBuf; |
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| 118 | primElem= primitiveElement (v, vBuf, primFail); |
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| 119 | ASSERT (!primFail, "failure in integer factorizer"); |
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| 120 | if (d < 3) |
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| 121 | { |
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| 122 | zz_pX NTLIrredpoly; |
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| 123 | BuildIrred (NTLIrredpoly, d*3); |
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| 124 | CanonicalForm newMipo= convertNTLzzpX2CF (NTLIrredpoly, Variable (1)); |
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| 125 | v2= rootOf (newMipo); |
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| 126 | } |
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| 127 | else |
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| 128 | { |
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| 129 | zz_pX NTLIrredpoly; |
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| 130 | BuildIrred (NTLIrredpoly, d*2); |
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| 131 | CanonicalForm newMipo= convertNTLzzpX2CF (NTLIrredpoly, Variable (1)); |
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| 132 | v2= rootOf (newMipo); |
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| 133 | } |
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| 134 | imPrimElem= mapPrimElem (primElem, v, v2); |
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| 135 | extOfExt= true; |
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| 136 | } |
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| 137 | else if ((p == 3 && d < 4) || ((p == 5 || p == 7) && d < 3)) |
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| 138 | { |
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| 139 | zz_p::init (p); |
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| 140 | bool primFail= false; |
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| 141 | Variable vBuf; |
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| 142 | primElem= primitiveElement (v, vBuf, primFail); |
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| 143 | ASSERT (!primFail, "failure in integer factorizer"); |
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| 144 | zz_pX NTLIrredpoly; |
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| 145 | BuildIrred (NTLIrredpoly, d*2); |
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| 146 | CanonicalForm newMipo= convertNTLzzpX2CF (NTLIrredpoly, Variable (1)); |
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| 147 | v2= rootOf (newMipo); |
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| 148 | imPrimElem= mapPrimElem (primElem, v, v2); |
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| 149 | extOfExt= true; |
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| 150 | } |
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| 151 | if (extOfExt) |
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| 152 | { |
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| 153 | F= mapUp (F, v, v2, primElem, imPrimElem, source, dest); |
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| 154 | G= mapUp (G, v, v2, primElem, imPrimElem, source, dest); |
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| 155 | lcf= mapUp (lcf, v, v2, primElem, imPrimElem, source, dest); |
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| 156 | lcg= mapUp (lcg, v, v2, primElem, imPrimElem, source, dest); |
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| 157 | v= v2; |
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| 158 | } |
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[d990001] | 159 | #endif |
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[6e2ef0e] | 160 | } |
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| 161 | |
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| 162 | CFRandom * sample; |
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| 163 | if ((!algExtension && p > 0) || p == 0) |
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| 164 | sample = CFRandomFactory::generate(); |
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| 165 | else |
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| 166 | sample = AlgExtRandomF (v).clone(); |
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| 167 | |
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| 168 | REvaluation e( 2, tmax( f.level(), g.level() ), *sample ); |
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| 169 | delete sample; |
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| 170 | |
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| 171 | if (passToGF) |
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| 172 | { |
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| 173 | lcf= lcf.mapinto(); |
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| 174 | lcg= lcg.mapinto(); |
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| 175 | } |
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| 176 | |
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| 177 | CanonicalForm eval1, eval2; |
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| 178 | if (passToGF) |
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| 179 | { |
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| 180 | eval1= e (lcf); |
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| 181 | eval2= e (lcg); |
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| 182 | } |
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| 183 | else |
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| 184 | { |
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| 185 | eval1= e (lcf); |
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| 186 | eval2= e (lcg); |
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| 187 | } |
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| 188 | |
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| 189 | while ( ( eval1.isZero() || eval2.isZero() ) && count < TEST_ONE_MAX ) |
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| 190 | { |
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| 191 | e.nextpoint(); |
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| 192 | count++; |
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| 193 | eval1= e (lcf); |
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| 194 | eval2= e (lcg); |
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| 195 | } |
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| 196 | if ( count >= TEST_ONE_MAX ) |
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| 197 | { |
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| 198 | if (passToGF) |
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| 199 | setCharacteristic (p); |
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| 200 | if (k > 1) |
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| 201 | setCharacteristic (p, k, gf_name); |
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| 202 | return false; |
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| 203 | } |
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| 204 | |
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| 205 | |
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| 206 | if (passToGF) |
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| 207 | { |
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| 208 | F= F.mapinto(); |
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| 209 | G= G.mapinto(); |
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| 210 | eval1= e (F); |
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| 211 | eval2= e (G); |
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| 212 | } |
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| 213 | else |
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| 214 | { |
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| 215 | eval1= e (F); |
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| 216 | eval2= e (G); |
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| 217 | } |
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| 218 | |
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| 219 | if ( eval1.taildegree() > 0 && eval2.taildegree() > 0 ) |
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| 220 | { |
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| 221 | if (passToGF) |
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| 222 | setCharacteristic (p); |
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| 223 | if (k > 1) |
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| 224 | setCharacteristic (p, k, gf_name); |
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[150dc8] | 225 | return false; |
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[6e2ef0e] | 226 | } |
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| 227 | |
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| 228 | CanonicalForm c= gcd (eval1, eval2); |
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| 229 | bool result= c.degree() < 1; |
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| 230 | |
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| 231 | if (passToGF) |
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| 232 | setCharacteristic (p); |
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| 233 | if (k > 1) |
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| 234 | setCharacteristic (p, k, gf_name); |
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| 235 | return result; |
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[9bab9f] | 236 | } |
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| 237 | |
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[dd3e561] | 238 | //{{{ static CanonicalForm icontent ( const CanonicalForm & f, const CanonicalForm & c ) |
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| 239 | //{{{ docu |
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| 240 | // |
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| 241 | // icontent() - return gcd of c and all coefficients of f which |
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| 242 | // are in a coefficient domain. |
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| 243 | // |
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| 244 | // Used by icontent(). |
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| 245 | // |
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| 246 | //}}} |
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[9bab9f] | 247 | static CanonicalForm |
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| 248 | icontent ( const CanonicalForm & f, const CanonicalForm & c ) |
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| 249 | { |
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[c30347] | 250 | if ( f.inBaseDomain() ) |
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| 251 | { |
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| 252 | if (c.isZero()) return abs(f); |
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| 253 | return bgcd( f, c ); |
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| 254 | } |
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[ef20c7] | 255 | //else if ( f.inCoeffDomain() ) |
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| 256 | // return gcd(f,c); |
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[c30347] | 257 | else |
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| 258 | { |
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[150dc8] | 259 | CanonicalForm g = c; |
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| 260 | for ( CFIterator i = f; i.hasTerms() && ! g.isOne(); i++ ) |
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| 261 | g = icontent( i.coeff(), g ); |
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| 262 | return g; |
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[9bab9f] | 263 | } |
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| 264 | } |
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[dd3e561] | 265 | //}}} |
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[9bab9f] | 266 | |
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[dd3e561] | 267 | //{{{ CanonicalForm icontent ( const CanonicalForm & f ) |
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| 268 | //{{{ docu |
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| 269 | // |
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| 270 | // icontent() - return gcd over all coefficients of f which are |
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| 271 | // in a coefficient domain. |
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| 272 | // |
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| 273 | //}}} |
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[9bab9f] | 274 | CanonicalForm |
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| 275 | icontent ( const CanonicalForm & f ) |
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| 276 | { |
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| 277 | return icontent( f, 0 ); |
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| 278 | } |
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[dd3e561] | 279 | //}}} |
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[9bab9f] | 280 | |
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[dd3e561] | 281 | //{{{ CanonicalForm extgcd ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & a, CanonicalForm & b ) |
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| 282 | //{{{ docu |
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| 283 | // |
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| 284 | // extgcd() - returns polynomial extended gcd of f and g. |
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| 285 | // |
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| 286 | // Returns gcd(f, g) and a and b sucht that f*a+g*b=gcd(f, g). |
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| 287 | // The gcd is calculated using an extended euclidean polynomial |
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| 288 | // remainder sequence, so f and g should be polynomials over an |
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| 289 | // euclidean domain. Normalizes result. |
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| 290 | // |
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| 291 | // Note: be sure that f and g have the same level! |
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| 292 | // |
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| 293 | //}}} |
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[9bab9f] | 294 | CanonicalForm |
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| 295 | extgcd ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & a, CanonicalForm & b ) |
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| 296 | { |
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[e9a5b62] | 297 | if (f.isZero()) |
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| 298 | { |
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| 299 | a= 0; |
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| 300 | b= 1; |
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| 301 | return g; |
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| 302 | } |
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| 303 | else if (g.isZero()) |
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| 304 | { |
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| 305 | a= 1; |
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| 306 | b= 0; |
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| 307 | return f; |
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| 308 | } |
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[034eec] | 309 | #ifdef HAVE_NTL |
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[963057] | 310 | if (isOn(SW_USE_NTL_GCD_P) && ( getCharacteristic() > 0 ) && (CFFactory::gettype() != GaloisFieldDomain) |
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[a86cda] | 311 | && (f.level()==g.level()) && isPurePoly(f) && isPurePoly(g)) |
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[034eec] | 312 | { |
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[c6eecb] | 313 | if (fac_NTL_char!=getCharacteristic()) |
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| 314 | { |
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| 315 | fac_NTL_char=getCharacteristic(); |
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| 316 | #ifdef NTL_ZZ |
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| 317 | ZZ r; |
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| 318 | r=getCharacteristic(); |
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| 319 | ZZ_pContext ccc(r); |
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| 320 | #else |
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| 321 | zz_pContext ccc(getCharacteristic()); |
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| 322 | #endif |
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| 323 | ccc.restore(); |
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| 324 | #ifdef NTL_ZZ |
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| 325 | ZZ_p::init(r); |
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| 326 | #else |
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| 327 | zz_p::init(getCharacteristic()); |
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| 328 | #endif |
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| 329 | } |
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| 330 | #ifdef NTL_ZZ |
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| 331 | ZZ_pX F1=convertFacCF2NTLZZpX(f); |
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| 332 | ZZ_pX G1=convertFacCF2NTLZZpX(g); |
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| 333 | ZZ_pX R; |
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| 334 | ZZ_pX A,B; |
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| 335 | XGCD(R,A,B,F1,G1); |
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| 336 | a=convertNTLZZpX2CF(A,f.mvar()); |
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| 337 | b=convertNTLZZpX2CF(B,f.mvar()); |
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| 338 | return convertNTLZZpX2CF(R,f.mvar()); |
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| 339 | #else |
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[034eec] | 340 | zz_pX F1=convertFacCF2NTLzzpX(f); |
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| 341 | zz_pX G1=convertFacCF2NTLzzpX(g); |
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| 342 | zz_pX R; |
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| 343 | zz_pX A,B; |
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| 344 | XGCD(R,A,B,F1,G1); |
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| 345 | a=convertNTLzzpX2CF(A,f.mvar()); |
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| 346 | b=convertNTLzzpX2CF(B,f.mvar()); |
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| 347 | return convertNTLzzpX2CF(R,f.mvar()); |
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[c6eecb] | 348 | #endif |
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[034eec] | 349 | } |
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[a86cda] | 350 | if (isOn(SW_USE_NTL_GCD_0) && ( getCharacteristic() ==0) |
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| 351 | && (f.level()==g.level()) && isPurePoly(f) && isPurePoly(g)) |
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| 352 | { |
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| 353 | CanonicalForm fc=bCommonDen(f); |
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| 354 | CanonicalForm gc=bCommonDen(g); |
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| 355 | ZZX F1=convertFacCF2NTLZZX(f*fc); |
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| 356 | ZZX G1=convertFacCF2NTLZZX(g*gc); |
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| 357 | ZZX R=GCD(F1,G1); |
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| 358 | CanonicalForm r=convertNTLZZX2CF(R,f.mvar()); |
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| 359 | ZZ RR; |
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| 360 | ZZX A,B; |
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| 361 | if (r.inCoeffDomain()) |
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| 362 | { |
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| 363 | XGCD(RR,A,B,F1,G1,1); |
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| 364 | CanonicalForm rr=convertZZ2CF(RR); |
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| 365 | ASSERT (!rr.isZero(), "NTL:XGCD failed"); |
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| 366 | a=convertNTLZZX2CF(A,f.mvar())*fc/rr; |
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| 367 | b=convertNTLZZX2CF(B,f.mvar())*gc/rr; |
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| 368 | return CanonicalForm(1); |
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| 369 | } |
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| 370 | else |
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| 371 | { |
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| 372 | fc=bCommonDen(f); |
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| 373 | gc=bCommonDen(g); |
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| 374 | F1=convertFacCF2NTLZZX(f*fc/r); |
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| 375 | G1=convertFacCF2NTLZZX(g*gc/r); |
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| 376 | XGCD(RR,A,B,F1,G1,1); |
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| 377 | a=convertNTLZZX2CF(A,f.mvar())*fc; |
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| 378 | b=convertNTLZZX2CF(B,f.mvar())*gc; |
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| 379 | CanonicalForm rr=convertZZ2CF(RR); |
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| 380 | ASSERT (!rr.isZero(), "NTL:XGCD failed"); |
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| 381 | r*=rr; |
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| 382 | if ( r.sign() < 0 ) { r= -r; a= -a; b= -b; } |
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| 383 | return r; |
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| 384 | } |
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| 385 | } |
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[034eec] | 386 | #endif |
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[a86cda] | 387 | // may contain bug in the co-factors, see track 107 |
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[034eec] | 388 | CanonicalForm contf = content( f ); |
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| 389 | CanonicalForm contg = content( g ); |
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[9bab9f] | 390 | |
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[034eec] | 391 | CanonicalForm p0 = f / contf, p1 = g / contg; |
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| 392 | CanonicalForm f0 = 1, f1 = 0, g0 = 0, g1 = 1, q, r; |
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[9bab9f] | 393 | |
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[c6eecb] | 394 | while ( ! p1.isZero() ) |
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| 395 | { |
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[034eec] | 396 | divrem( p0, p1, q, r ); |
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| 397 | p0 = p1; p1 = r; |
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| 398 | r = g0 - g1 * q; |
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| 399 | g0 = g1; g1 = r; |
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| 400 | r = f0 - f1 * q; |
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| 401 | f0 = f1; f1 = r; |
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| 402 | } |
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| 403 | CanonicalForm contp0 = content( p0 ); |
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| 404 | a = f0 / ( contf * contp0 ); |
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| 405 | b = g0 / ( contg * contp0 ); |
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| 406 | p0 /= contp0; |
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[c6eecb] | 407 | if ( p0.sign() < 0 ) |
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| 408 | { |
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[034eec] | 409 | p0 = -p0; |
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| 410 | a = -a; |
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| 411 | b = -b; |
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| 412 | } |
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| 413 | return p0; |
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[9bab9f] | 414 | } |
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[dd3e561] | 415 | //}}} |
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[9bab9f] | 416 | |
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[a7ec94] | 417 | //{{{ static CanonicalForm balance ( const CanonicalForm & f, const CanonicalForm & q ) |
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| 418 | //{{{ docu |
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| 419 | // |
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| 420 | // balance() - map f from positive to symmetric representation |
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| 421 | // mod q. |
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| 422 | // |
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| 423 | // This makes sense for univariate polynomials over Z only. |
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| 424 | // q should be an integer. |
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| 425 | // |
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| 426 | // Used by gcd_poly_univar0(). |
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| 427 | // |
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| 428 | //}}} |
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[edb4893] | 429 | static CanonicalForm |
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[a7ec94] | 430 | balance ( const CanonicalForm & f, const CanonicalForm & q ) |
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[edb4893] | 431 | { |
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[a7ec94] | 432 | Variable x = f.mvar(); |
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| 433 | CanonicalForm result = 0, qh = q / 2; |
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| 434 | CanonicalForm c; |
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| 435 | CFIterator i; |
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| 436 | for ( i = f; i.hasTerms(); i++ ) { |
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| 437 | c = mod( i.coeff(), q ); |
---|
| 438 | if ( c > qh ) |
---|
| 439 | result += power( x, i.exp() ) * (c - q); |
---|
| 440 | else |
---|
| 441 | result += power( x, i.exp() ) * c; |
---|
[edb4893] | 442 | } |
---|
[a7ec94] | 443 | return result; |
---|
| 444 | } |
---|
| 445 | //}}} |
---|
| 446 | |
---|
[01e8874] | 447 | static CanonicalForm gcd_poly_univar0( const CanonicalForm & F, const CanonicalForm & G, bool primitive ) |
---|
[a7ec94] | 448 | { |
---|
[f11d7b] | 449 | CanonicalForm f, g, c, cg, cl, BB, B, M, q, Dp, newD, D, newq; |
---|
[01e8874] | 450 | int p, i; |
---|
[f11d7b] | 451 | |
---|
| 452 | if ( primitive ) |
---|
| 453 | { |
---|
| 454 | f = F; |
---|
| 455 | g = G; |
---|
| 456 | c = 1; |
---|
| 457 | } |
---|
| 458 | else |
---|
| 459 | { |
---|
| 460 | CanonicalForm cF = content( F ), cG = content( G ); |
---|
| 461 | f = F / cF; |
---|
| 462 | g = G / cG; |
---|
| 463 | c = bgcd( cF, cG ); |
---|
| 464 | } |
---|
| 465 | cg = gcd( f.lc(), g.lc() ); |
---|
| 466 | cl = ( f.lc() / cg ) * g.lc(); |
---|
[93b061] | 467 | // B = 2 * cg * tmin( |
---|
[150dc8] | 468 | // maxnorm(f)*power(CanonicalForm(2),f.degree())*isqrt(f.degree()+1), |
---|
| 469 | // maxnorm(g)*power(CanonicalForm(2),g.degree())*isqrt(g.degree()+1) |
---|
| 470 | // )+1; |
---|
[f11d7b] | 471 | M = tmin( maxNorm(f), maxNorm(g) ); |
---|
| 472 | BB = power(CanonicalForm(2),tmin(f.degree(),g.degree()))*M; |
---|
| 473 | q = 0; |
---|
| 474 | i = cf_getNumSmallPrimes() - 1; |
---|
| 475 | while ( true ) |
---|
| 476 | { |
---|
| 477 | B = BB; |
---|
| 478 | while ( i >= 0 && q < B ) |
---|
| 479 | { |
---|
| 480 | p = cf_getSmallPrime( i ); |
---|
| 481 | i--; |
---|
| 482 | while ( i >= 0 && mod( cl, p ) == 0 ) |
---|
| 483 | { |
---|
| 484 | p = cf_getSmallPrime( i ); |
---|
| 485 | i--; |
---|
| 486 | } |
---|
| 487 | setCharacteristic( p ); |
---|
| 488 | Dp = gcd( mapinto( f ), mapinto( g ) ); |
---|
| 489 | Dp = ( Dp / Dp.lc() ) * mapinto( cg ); |
---|
| 490 | setCharacteristic( 0 ); |
---|
| 491 | if ( Dp.degree() == 0 ) |
---|
| 492 | return c; |
---|
| 493 | if ( q.isZero() ) |
---|
| 494 | { |
---|
| 495 | D = mapinto( Dp ); |
---|
| 496 | q = p; |
---|
| 497 | B = power(CanonicalForm(2),D.degree())*M+1; |
---|
| 498 | } |
---|
| 499 | else |
---|
| 500 | { |
---|
| 501 | if ( Dp.degree() == D.degree() ) |
---|
| 502 | { |
---|
| 503 | chineseRemainder( D, q, mapinto( Dp ), p, newD, newq ); |
---|
| 504 | q = newq; |
---|
| 505 | D = newD; |
---|
[150dc8] | 506 | } |
---|
[f11d7b] | 507 | else if ( Dp.degree() < D.degree() ) |
---|
| 508 | { |
---|
| 509 | // all previous p's are bad primes |
---|
| 510 | q = p; |
---|
| 511 | D = mapinto( Dp ); |
---|
| 512 | B = power(CanonicalForm(2),D.degree())*M+1; |
---|
[150dc8] | 513 | } |
---|
[f11d7b] | 514 | // else p is a bad prime |
---|
| 515 | } |
---|
| 516 | } |
---|
| 517 | if ( i >= 0 ) |
---|
| 518 | { |
---|
| 519 | // now balance D mod q |
---|
| 520 | D = pp( balance( D, q ) ); |
---|
[ebc602] | 521 | if ( fdivides( D, f ) && fdivides( D, g ) ) |
---|
[f11d7b] | 522 | return D * c; |
---|
| 523 | else |
---|
| 524 | q = 0; |
---|
[edb4893] | 525 | } |
---|
[f11d7b] | 526 | else |
---|
[a7ec94] | 527 | return gcd_poly( F, G ); |
---|
[f11d7b] | 528 | DEBOUTLN( cerr, "another try ..." ); |
---|
| 529 | } |
---|
[edb4893] | 530 | } |
---|
| 531 | |
---|
[c4f4fd] | 532 | static CanonicalForm |
---|
| 533 | gcd_poly_p( const CanonicalForm & f, const CanonicalForm & g ) |
---|
| 534 | { |
---|
| 535 | CanonicalForm pi, pi1; |
---|
| 536 | CanonicalForm C, Ci, Ci1, Hi, bi, pi2; |
---|
| 537 | bool bpure; |
---|
| 538 | int delta = degree( f ) - degree( g ); |
---|
| 539 | |
---|
| 540 | if ( delta >= 0 ) |
---|
| 541 | { |
---|
| 542 | pi = f; pi1 = g; |
---|
| 543 | } |
---|
| 544 | else |
---|
| 545 | { |
---|
| 546 | pi = g; pi1 = f; delta = -delta; |
---|
| 547 | } |
---|
| 548 | Ci = content( pi ); Ci1 = content( pi1 ); |
---|
| 549 | pi1 = pi1 / Ci1; pi = pi / Ci; |
---|
| 550 | C = gcd( Ci, Ci1 ); |
---|
| 551 | if ( !( pi.isUnivariate() && pi1.isUnivariate() ) ) |
---|
| 552 | { |
---|
| 553 | if ( gcd_test_one( pi1, pi, true ) ) |
---|
| 554 | { |
---|
| 555 | C=abs(C); |
---|
| 556 | //out_cf("GCD:",C,"\n"); |
---|
| 557 | return C; |
---|
| 558 | } |
---|
| 559 | bpure = false; |
---|
| 560 | } |
---|
| 561 | else |
---|
| 562 | { |
---|
| 563 | bpure = isPurePoly(pi) && isPurePoly(pi1); |
---|
[7e8c9e] | 564 | #ifdef HAVE_FLINT |
---|
| 565 | if (bpure && (CFFactory::gettype() != GaloisFieldDomain)) |
---|
| 566 | return gcd_univar_flintp(pi,pi1)*C; |
---|
[7cb5590] | 567 | #else |
---|
| 568 | #ifdef HAVE_NTL |
---|
[c4f4fd] | 569 | if ( isOn(SW_USE_NTL_GCD_P) && bpure && (CFFactory::gettype() != GaloisFieldDomain)) |
---|
| 570 | return gcd_univar_ntlp(pi, pi1 ) * C; |
---|
[7cb5590] | 571 | #endif |
---|
[c4f4fd] | 572 | #endif |
---|
| 573 | } |
---|
| 574 | Variable v = f.mvar(); |
---|
| 575 | Hi = power( LC( pi1, v ), delta ); |
---|
| 576 | if ( (delta+1) % 2 ) |
---|
| 577 | bi = 1; |
---|
| 578 | else |
---|
| 579 | bi = -1; |
---|
[6e2ef0e] | 580 | int maxNumVars= tmax (getNumVars (pi), getNumVars (pi1)); |
---|
| 581 | CanonicalForm oldPi= pi, oldPi1= pi1; |
---|
[c4f4fd] | 582 | while ( degree( pi1, v ) > 0 ) |
---|
| 583 | { |
---|
[6e2ef0e] | 584 | if (!(pi.isUnivariate() && pi1.isUnivariate())) |
---|
| 585 | { |
---|
| 586 | if (size (pi)/maxNumVars > 500 || size (pi1)/maxNumVars > 500) |
---|
| 587 | { |
---|
| 588 | On (SW_USE_FF_MOD_GCD); |
---|
| 589 | C *= gcd (oldPi, oldPi1); |
---|
| 590 | Off (SW_USE_FF_MOD_GCD); |
---|
| 591 | return C; |
---|
| 592 | } |
---|
| 593 | } |
---|
[c4f4fd] | 594 | pi2 = psr( pi, pi1, v ); |
---|
| 595 | pi2 = pi2 / bi; |
---|
| 596 | pi = pi1; pi1 = pi2; |
---|
[6e2ef0e] | 597 | maxNumVars= tmax (getNumVars (pi), getNumVars (pi1)); |
---|
[c4f4fd] | 598 | if ( degree( pi1, v ) > 0 ) |
---|
| 599 | { |
---|
| 600 | delta = degree( pi, v ) - degree( pi1, v ); |
---|
| 601 | if ( (delta+1) % 2 ) |
---|
| 602 | bi = LC( pi, v ) * power( Hi, delta ); |
---|
| 603 | else |
---|
| 604 | bi = -LC( pi, v ) * power( Hi, delta ); |
---|
| 605 | Hi = power( LC( pi1, v ), delta ) / power( Hi, delta-1 ); |
---|
| 606 | } |
---|
| 607 | } |
---|
| 608 | if ( degree( pi1, v ) == 0 ) |
---|
| 609 | { |
---|
| 610 | C=abs(C); |
---|
| 611 | //out_cf("GCD:",C,"\n"); |
---|
| 612 | return C; |
---|
| 613 | } |
---|
| 614 | pi /= content( pi ); |
---|
| 615 | if ( bpure ) |
---|
| 616 | pi /= pi.lc(); |
---|
| 617 | C=abs(C*pi); |
---|
| 618 | //out_cf("GCD:",C,"\n"); |
---|
| 619 | return C; |
---|
| 620 | } |
---|
| 621 | |
---|
[a7ec94] | 622 | static CanonicalForm |
---|
| 623 | gcd_poly_0( const CanonicalForm & f, const CanonicalForm & g ) |
---|
| 624 | { |
---|
| 625 | CanonicalForm pi, pi1; |
---|
[df497a] | 626 | CanonicalForm C, Ci, Ci1, Hi, bi, pi2; |
---|
[a7ec94] | 627 | int delta = degree( f ) - degree( g ); |
---|
| 628 | |
---|
| 629 | if ( delta >= 0 ) |
---|
| 630 | { |
---|
| 631 | pi = f; pi1 = g; |
---|
| 632 | } |
---|
| 633 | else |
---|
| 634 | { |
---|
| 635 | pi = g; pi1 = f; delta = -delta; |
---|
| 636 | } |
---|
[9bab9f] | 637 | Ci = content( pi ); Ci1 = content( pi1 ); |
---|
| 638 | pi1 = pi1 / Ci1; pi = pi / Ci; |
---|
[df497a] | 639 | C = gcd( Ci, Ci1 ); |
---|
[034eec] | 640 | if ( pi.isUnivariate() && pi1.isUnivariate() ) |
---|
| 641 | { |
---|
[c53fdc] | 642 | /*#ifdef HAVE_FLINT |
---|
[7e8c9e] | 643 | if (isPurePoly(pi) && isPurePoly(pi1) ) |
---|
| 644 | return gcd_univar_flint0(pi, pi1 ) * C; |
---|
[c53fdc] | 645 | #else*/ |
---|
[7cb5590] | 646 | #ifdef HAVE_NTL |
---|
[a7ec94] | 647 | if ( isOn(SW_USE_NTL_GCD_0) && isPurePoly(pi) && isPurePoly(pi1) ) |
---|
| 648 | return gcd_univar_ntl0(pi, pi1 ) * C; |
---|
[7cb5590] | 649 | #endif |
---|
[c53fdc] | 650 | //#endif |
---|
[a7ec94] | 651 | return gcd_poly_univar0( pi, pi1, true ) * C; |
---|
[edb4893] | 652 | } |
---|
[034eec] | 653 | else if ( gcd_test_one( pi1, pi, true ) ) |
---|
| 654 | return C; |
---|
[a7ec94] | 655 | Variable v = f.mvar(); |
---|
[9bab9f] | 656 | Hi = power( LC( pi1, v ), delta ); |
---|
| 657 | if ( (delta+1) % 2 ) |
---|
[150dc8] | 658 | bi = 1; |
---|
[9bab9f] | 659 | else |
---|
[150dc8] | 660 | bi = -1; |
---|
[6f62c3] | 661 | while ( degree( pi1, v ) > 0 ) |
---|
| 662 | { |
---|
[150dc8] | 663 | pi2 = psr( pi, pi1, v ); |
---|
| 664 | pi2 = pi2 / bi; |
---|
| 665 | pi = pi1; pi1 = pi2; |
---|
[6f62c3] | 666 | if ( degree( pi1, v ) > 0 ) |
---|
| 667 | { |
---|
[150dc8] | 668 | delta = degree( pi, v ) - degree( pi1, v ); |
---|
| 669 | if ( (delta+1) % 2 ) |
---|
| 670 | bi = LC( pi, v ) * power( Hi, delta ); |
---|
| 671 | else |
---|
| 672 | bi = -LC( pi, v ) * power( Hi, delta ); |
---|
| 673 | Hi = power( LC( pi1, v ), delta ) / power( Hi, delta-1 ); |
---|
| 674 | } |
---|
[9bab9f] | 675 | } |
---|
| 676 | if ( degree( pi1, v ) == 0 ) |
---|
[150dc8] | 677 | return C; |
---|
[df497a] | 678 | else |
---|
[150dc8] | 679 | return C * pp( pi ); |
---|
[9bab9f] | 680 | } |
---|
| 681 | |
---|
[b809a8] | 682 | //{{{ CanonicalForm gcd_poly ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
[dd3e561] | 683 | //{{{ docu |
---|
| 684 | // |
---|
| 685 | // gcd_poly() - calculate polynomial gcd. |
---|
| 686 | // |
---|
| 687 | // This is the dispatcher for polynomial gcd calculation. We call either |
---|
| 688 | // ezgcd(), sparsemod() or gcd_poly1() in dependecy on the current |
---|
[e88604] | 689 | // characteristic and settings of SW_USE_EZGCD. |
---|
[dd3e561] | 690 | // |
---|
| 691 | // Used by gcd() and gcd_poly_univar0(). |
---|
| 692 | // |
---|
| 693 | //}}} |
---|
[0b6919] | 694 | #if 0 |
---|
[bfc606] | 695 | int si_factor_reminder=1; |
---|
[0b6919] | 696 | #endif |
---|
[b809a8] | 697 | CanonicalForm gcd_poly ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
[f63dbca] | 698 | { |
---|
[110718] | 699 | CanonicalForm fc, gc, d1; |
---|
| 700 | int mp, cc, p1, pe; |
---|
| 701 | mp = f.level()+1; |
---|
[ed9927] | 702 | bool fc_isUnivariate=f.isUnivariate(); |
---|
| 703 | bool gc_isUnivariate=g.isUnivariate(); |
---|
| 704 | bool fc_and_gc_Univariate=fc_isUnivariate && gc_isUnivariate; |
---|
| 705 | cf_prepgcd( f, g, cc, p1, pe); |
---|
| 706 | if ( cc != 0 ) |
---|
[110718] | 707 | { |
---|
[ed9927] | 708 | if ( cc > 0 ) |
---|
[abfc3b] | 709 | { |
---|
[ed9927] | 710 | fc = replacevar( f, Variable(cc), Variable(mp) ); |
---|
| 711 | gc = g; |
---|
[e074407] | 712 | } |
---|
[ed9927] | 713 | else |
---|
[110718] | 714 | { |
---|
[ed9927] | 715 | fc = replacevar( g, Variable(-cc), Variable(mp) ); |
---|
| 716 | gc = f; |
---|
[110718] | 717 | } |
---|
[ed9927] | 718 | return cf_content( fc, gc ); |
---|
| 719 | } |
---|
| 720 | // now each appearing variable is in f and g |
---|
| 721 | fc = f; |
---|
| 722 | gc = g; |
---|
| 723 | if ( getCharacteristic() != 0 ) |
---|
| 724 | { |
---|
[2072126] | 725 | #ifdef HAVE_NTL |
---|
[e16f7d] | 726 | if ((!fc_and_gc_Univariate) && (isOn( SW_USE_EZGCD_P ))) |
---|
[49f1f45] | 727 | { |
---|
[08daea] | 728 | fc= EZGCD_P (fc, gc); |
---|
[c30347] | 729 | } |
---|
[10af64] | 730 | else if (isOn(SW_USE_FF_MOD_GCD) && !fc_and_gc_Univariate) |
---|
| 731 | { |
---|
| 732 | Variable a; |
---|
| 733 | if (hasFirstAlgVar (fc, a) || hasFirstAlgVar (gc, a)) |
---|
| 734 | fc=GCD_Fp_extension (fc, gc, a); |
---|
[b5c084] | 735 | else if (CFFactory::gettype() == GaloisFieldDomain) |
---|
[10af64] | 736 | fc=GCD_GF (fc, gc); |
---|
[b5c084] | 737 | else |
---|
| 738 | fc=GCD_small_p (fc, gc); |
---|
[10af64] | 739 | } |
---|
[efcd2dc] | 740 | else |
---|
[2072126] | 741 | #endif |
---|
[efcd2dc] | 742 | if ( p1 == fc.level() ) |
---|
[ed9927] | 743 | fc = gcd_poly_p( fc, gc ); |
---|
| 744 | else |
---|
| 745 | { |
---|
| 746 | fc = replacevar( fc, Variable(p1), Variable(mp) ); |
---|
| 747 | gc = replacevar( gc, Variable(p1), Variable(mp) ); |
---|
| 748 | fc = replacevar( gcd_poly_p( fc, gc ), Variable(mp), Variable(p1) ); |
---|
| 749 | } |
---|
[110718] | 750 | } |
---|
[c30347] | 751 | else if (!fc_and_gc_Univariate) |
---|
[110718] | 752 | { |
---|
[f7a4e9] | 753 | if ( isOn( SW_USE_EZGCD ) ) |
---|
| 754 | { |
---|
| 755 | fc= ezgcd (fc, gc); |
---|
| 756 | /*if ( pe == 1 ) |
---|
| 757 | fc = ezgcd( fc, gc ); |
---|
| 758 | else if ( pe > 0 )// no variable at position 1 |
---|
| 759 | { |
---|
| 760 | fc = replacevar( fc, Variable(pe), Variable(1) ); |
---|
| 761 | gc = replacevar( gc, Variable(pe), Variable(1) ); |
---|
| 762 | fc = replacevar( ezgcd( fc, gc ), Variable(1), Variable(pe) ); |
---|
| 763 | } |
---|
| 764 | else |
---|
| 765 | { |
---|
| 766 | pe = -pe; |
---|
| 767 | fc = swapvar( fc, Variable(pe), Variable(1) ); |
---|
| 768 | gc = swapvar( gc, Variable(pe), Variable(1) ); |
---|
| 769 | fc = swapvar( ezgcd( fc, gc ), Variable(1), Variable(pe) ); |
---|
| 770 | }*/ |
---|
| 771 | } |
---|
| 772 | else if ( |
---|
[c30347] | 773 | isOn(SW_USE_CHINREM_GCD) |
---|
[ed9927] | 774 | && (isPurePoly_m(fc)) && (isPurePoly_m(gc)) |
---|
[c30347] | 775 | ) |
---|
| 776 | { |
---|
[ed9927] | 777 | #if 0 |
---|
| 778 | if ( p1 == fc.level() ) |
---|
| 779 | fc = chinrem_gcd( fc, gc ); |
---|
| 780 | else |
---|
| 781 | { |
---|
| 782 | fc = replacevar( fc, Variable(p1), Variable(mp) ); |
---|
| 783 | gc = replacevar( gc, Variable(p1), Variable(mp) ); |
---|
| 784 | fc = replacevar( chinrem_gcd( fc, gc ), Variable(mp), Variable(p1) ); |
---|
| 785 | } |
---|
| 786 | #else |
---|
| 787 | fc = chinrem_gcd( fc, gc); |
---|
| 788 | #endif |
---|
[c30347] | 789 | } |
---|
| 790 | else |
---|
| 791 | { |
---|
[ed9927] | 792 | fc = gcd_poly_0( fc, gc ); |
---|
[c30347] | 793 | } |
---|
[110718] | 794 | } |
---|
| 795 | else |
---|
| 796 | { |
---|
| 797 | fc = gcd_poly_0( fc, gc ); |
---|
| 798 | } |
---|
| 799 | if ( d1.degree() > 0 ) |
---|
| 800 | fc *= d1; |
---|
| 801 | return fc; |
---|
[f63dbca] | 802 | } |
---|
[dd3e561] | 803 | //}}} |
---|
[93b061] | 804 | |
---|
[dd3e561] | 805 | //{{{ static CanonicalForm cf_content ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
| 806 | //{{{ docu |
---|
| 807 | // |
---|
| 808 | // cf_content() - return gcd(g, content(f)). |
---|
| 809 | // |
---|
| 810 | // content(f) is calculated with respect to f's main variable. |
---|
| 811 | // |
---|
| 812 | // Used by gcd(), content(), content( CF, Variable ). |
---|
| 813 | // |
---|
| 814 | //}}} |
---|
[9bab9f] | 815 | static CanonicalForm |
---|
| 816 | cf_content ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
| 817 | { |
---|
[6f62c3] | 818 | if ( f.inPolyDomain() || ( f.inExtension() && ! getReduce( f.mvar() ) ) ) |
---|
| 819 | { |
---|
[150dc8] | 820 | CFIterator i = f; |
---|
| 821 | CanonicalForm result = g; |
---|
[6f62c3] | 822 | while ( i.hasTerms() && ! result.isOne() ) |
---|
| 823 | { |
---|
[a7ec94] | 824 | result = gcd( i.coeff(), result ); |
---|
[150dc8] | 825 | i++; |
---|
| 826 | } |
---|
| 827 | return result; |
---|
[9bab9f] | 828 | } |
---|
| 829 | else |
---|
[a7ec94] | 830 | return abs( f ); |
---|
[9bab9f] | 831 | } |
---|
[dd3e561] | 832 | //}}} |
---|
[9bab9f] | 833 | |
---|
[4ea0ab] | 834 | //{{{ CanonicalForm content ( const CanonicalForm & f ) |
---|
| 835 | //{{{ docu |
---|
| 836 | // |
---|
| 837 | // content() - return content(f) with respect to main variable. |
---|
| 838 | // |
---|
[dd3e561] | 839 | // Normalizes result. |
---|
| 840 | // |
---|
[4ea0ab] | 841 | //}}} |
---|
[9bab9f] | 842 | CanonicalForm |
---|
| 843 | content ( const CanonicalForm & f ) |
---|
| 844 | { |
---|
[6f62c3] | 845 | if ( f.inPolyDomain() || ( f.inExtension() && ! getReduce( f.mvar() ) ) ) |
---|
| 846 | { |
---|
[a7ec94] | 847 | CFIterator i = f; |
---|
| 848 | CanonicalForm result = abs( i.coeff() ); |
---|
| 849 | i++; |
---|
[6f62c3] | 850 | while ( i.hasTerms() && ! result.isOne() ) |
---|
| 851 | { |
---|
[a7ec94] | 852 | result = gcd( i.coeff(), result ); |
---|
| 853 | i++; |
---|
| 854 | } |
---|
| 855 | return result; |
---|
| 856 | } |
---|
| 857 | else |
---|
| 858 | return abs( f ); |
---|
[9bab9f] | 859 | } |
---|
[4ea0ab] | 860 | //}}} |
---|
[9bab9f] | 861 | |
---|
[dd3e561] | 862 | //{{{ CanonicalForm content ( const CanonicalForm & f, const Variable & x ) |
---|
| 863 | //{{{ docu |
---|
| 864 | // |
---|
| 865 | // content() - return content(f) with respect to x. |
---|
| 866 | // |
---|
| 867 | // x should be a polynomial variable. |
---|
| 868 | // |
---|
| 869 | //}}} |
---|
[9bab9f] | 870 | CanonicalForm |
---|
| 871 | content ( const CanonicalForm & f, const Variable & x ) |
---|
| 872 | { |
---|
[dd3e561] | 873 | ASSERT( x.level() > 0, "cannot calculate content with respect to algebraic variable" ); |
---|
| 874 | Variable y = f.mvar(); |
---|
| 875 | |
---|
| 876 | if ( y == x ) |
---|
[150dc8] | 877 | return cf_content( f, 0 ); |
---|
[dd3e561] | 878 | else if ( y < x ) |
---|
[150dc8] | 879 | return f; |
---|
[9bab9f] | 880 | else |
---|
[150dc8] | 881 | return swapvar( content( swapvar( f, y, x ), y ), y, x ); |
---|
[9bab9f] | 882 | } |
---|
[dd3e561] | 883 | //}}} |
---|
[9bab9f] | 884 | |
---|
[dd3e561] | 885 | //{{{ CanonicalForm vcontent ( const CanonicalForm & f, const Variable & x ) |
---|
| 886 | //{{{ docu |
---|
| 887 | // |
---|
| 888 | // vcontent() - return content of f with repect to variables >= x. |
---|
| 889 | // |
---|
| 890 | // The content is recursively calculated over all coefficients in |
---|
| 891 | // f having level less than x. x should be a polynomial |
---|
| 892 | // variable. |
---|
| 893 | // |
---|
| 894 | //}}} |
---|
[9bab9f] | 895 | CanonicalForm |
---|
| 896 | vcontent ( const CanonicalForm & f, const Variable & x ) |
---|
| 897 | { |
---|
[dd3e561] | 898 | ASSERT( x.level() > 0, "cannot calculate vcontent with respect to algebraic variable" ); |
---|
| 899 | |
---|
[9bab9f] | 900 | if ( f.mvar() <= x ) |
---|
[150dc8] | 901 | return content( f, x ); |
---|
[9bab9f] | 902 | else { |
---|
[150dc8] | 903 | CFIterator i; |
---|
| 904 | CanonicalForm d = 0; |
---|
| 905 | for ( i = f; i.hasTerms() && ! d.isOne(); i++ ) |
---|
| 906 | d = gcd( d, vcontent( i.coeff(), x ) ); |
---|
| 907 | return d; |
---|
[9bab9f] | 908 | } |
---|
| 909 | } |
---|
[dd3e561] | 910 | //}}} |
---|
[9bab9f] | 911 | |
---|
[4ea0ab] | 912 | //{{{ CanonicalForm pp ( const CanonicalForm & f ) |
---|
| 913 | //{{{ docu |
---|
| 914 | // |
---|
| 915 | // pp() - return primitive part of f. |
---|
| 916 | // |
---|
[dd3e561] | 917 | // Returns zero if f equals zero, otherwise f / content(f). |
---|
| 918 | // |
---|
[4ea0ab] | 919 | //}}} |
---|
[9bab9f] | 920 | CanonicalForm |
---|
| 921 | pp ( const CanonicalForm & f ) |
---|
| 922 | { |
---|
| 923 | if ( f.isZero() ) |
---|
[150dc8] | 924 | return f; |
---|
[9bab9f] | 925 | else |
---|
[150dc8] | 926 | return f / content( f ); |
---|
[9bab9f] | 927 | } |
---|
[4ea0ab] | 928 | //}}} |
---|
[9bab9f] | 929 | |
---|
[ff6222] | 930 | CanonicalForm |
---|
[9bab9f] | 931 | gcd ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
| 932 | { |
---|
[a7ec94] | 933 | bool b = f.isZero(); |
---|
| 934 | if ( b || g.isZero() ) |
---|
| 935 | { |
---|
| 936 | if ( b ) |
---|
| 937 | return abs( g ); |
---|
[abfc3b] | 938 | else |
---|
[a7ec94] | 939 | return abs( f ); |
---|
| 940 | } |
---|
| 941 | if ( f.inPolyDomain() || g.inPolyDomain() ) |
---|
| 942 | { |
---|
| 943 | if ( f.mvar() != g.mvar() ) |
---|
| 944 | { |
---|
| 945 | if ( f.mvar() > g.mvar() ) |
---|
| 946 | return cf_content( f, g ); |
---|
| 947 | else |
---|
| 948 | return cf_content( g, f ); |
---|
| 949 | } |
---|
[bb82f0] | 950 | if (isOn(SW_USE_QGCD)) |
---|
| 951 | { |
---|
| 952 | Variable m; |
---|
[fc9f44] | 953 | if ( |
---|
| 954 | (getCharacteristic() == 0) && |
---|
[e6f7ee1] | 955 | (hasFirstAlgVar(f,m) || hasFirstAlgVar(g,m)) |
---|
[bb82f0] | 956 | ) |
---|
[fc31bce] | 957 | { |
---|
[713bdb] | 958 | bool on_rational = isOn(SW_RATIONAL); |
---|
| 959 | CanonicalForm r=QGCD(f,g); |
---|
[f06059] | 960 | On(SW_RATIONAL); |
---|
[713bdb] | 961 | CanonicalForm cdF = bCommonDen( r ); |
---|
| 962 | if (!on_rational) Off(SW_RATIONAL); |
---|
| 963 | return cdF*r; |
---|
[fc31bce] | 964 | } |
---|
[bb82f0] | 965 | } |
---|
[713bdb] | 966 | |
---|
[150dc8] | 967 | if ( f.inExtension() && getReduce( f.mvar() ) ) |
---|
[bb82f0] | 968 | return CanonicalForm(1); |
---|
[a7ec94] | 969 | else |
---|
| 970 | { |
---|
[ebc602] | 971 | if ( fdivides( f, g ) ) |
---|
[a7ec94] | 972 | return abs( f ); |
---|
[ebc602] | 973 | else if ( fdivides( g, f ) ) |
---|
[a7ec94] | 974 | return abs( g ); |
---|
| 975 | if ( !( getCharacteristic() == 0 && isOn( SW_RATIONAL ) ) ) |
---|
| 976 | { |
---|
| 977 | CanonicalForm d; |
---|
[64a501] | 978 | d = gcd_poly( f, g ); |
---|
[a7ec94] | 979 | return abs( d ); |
---|
| 980 | } |
---|
| 981 | else |
---|
| 982 | { |
---|
[56d3c6] | 983 | //printf ("here\n"); |
---|
[150dc8] | 984 | CanonicalForm cdF = bCommonDen( f ); |
---|
| 985 | CanonicalForm cdG = bCommonDen( g ); |
---|
| 986 | Off( SW_RATIONAL ); |
---|
| 987 | CanonicalForm l = lcm( cdF, cdG ); |
---|
| 988 | On( SW_RATIONAL ); |
---|
| 989 | CanonicalForm F = f * l, G = g * l; |
---|
| 990 | Off( SW_RATIONAL ); |
---|
[64a501] | 991 | l = gcd_poly( F, G ); |
---|
[150dc8] | 992 | On( SW_RATIONAL ); |
---|
[a7ec94] | 993 | return abs( l ); |
---|
[150dc8] | 994 | } |
---|
| 995 | } |
---|
[a7ec94] | 996 | } |
---|
| 997 | if ( f.inBaseDomain() && g.inBaseDomain() ) |
---|
| 998 | return bgcd( f, g ); |
---|
[9bab9f] | 999 | else |
---|
[a7ec94] | 1000 | return 1; |
---|
[9bab9f] | 1001 | } |
---|
| 1002 | |
---|
[dd3e561] | 1003 | //{{{ CanonicalForm lcm ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
| 1004 | //{{{ docu |
---|
| 1005 | // |
---|
| 1006 | // lcm() - return least common multiple of f and g. |
---|
| 1007 | // |
---|
| 1008 | // The lcm is calculated using the formula lcm(f, g) = f * g / gcd(f, g). |
---|
| 1009 | // |
---|
| 1010 | // Returns zero if one of f or g equals zero. |
---|
| 1011 | // |
---|
| 1012 | //}}} |
---|
[9bab9f] | 1013 | CanonicalForm |
---|
| 1014 | lcm ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
| 1015 | { |
---|
[dd3e561] | 1016 | if ( f.isZero() || g.isZero() ) |
---|
[a7ec94] | 1017 | return 0; |
---|
[dd3e561] | 1018 | else |
---|
[150dc8] | 1019 | return ( f / gcd( f, g ) ) * g; |
---|
[9bab9f] | 1020 | } |
---|
[dd3e561] | 1021 | //}}} |
---|
[a7ec94] | 1022 | |
---|
| 1023 | #ifdef HAVE_NTL |
---|
| 1024 | |
---|
| 1025 | static CanonicalForm |
---|
| 1026 | gcd_univar_ntl0( const CanonicalForm & F, const CanonicalForm & G ) |
---|
| 1027 | { |
---|
| 1028 | ZZX F1=convertFacCF2NTLZZX(F); |
---|
| 1029 | ZZX G1=convertFacCF2NTLZZX(G); |
---|
| 1030 | ZZX R=GCD(F1,G1); |
---|
| 1031 | return convertNTLZZX2CF(R,F.mvar()); |
---|
| 1032 | } |
---|
| 1033 | |
---|
[c4f4fd] | 1034 | static CanonicalForm |
---|
| 1035 | gcd_univar_ntlp( const CanonicalForm & F, const CanonicalForm & G ) |
---|
| 1036 | { |
---|
| 1037 | if (fac_NTL_char!=getCharacteristic()) |
---|
| 1038 | { |
---|
| 1039 | fac_NTL_char=getCharacteristic(); |
---|
| 1040 | #ifdef NTL_ZZ |
---|
| 1041 | ZZ r; |
---|
| 1042 | r=getCharacteristic(); |
---|
| 1043 | ZZ_pContext ccc(r); |
---|
| 1044 | #else |
---|
| 1045 | zz_pContext ccc(getCharacteristic()); |
---|
| 1046 | #endif |
---|
| 1047 | ccc.restore(); |
---|
| 1048 | #ifdef NTL_ZZ |
---|
| 1049 | ZZ_p::init(r); |
---|
| 1050 | #else |
---|
| 1051 | zz_p::init(getCharacteristic()); |
---|
| 1052 | #endif |
---|
| 1053 | } |
---|
| 1054 | #ifdef NTL_ZZ |
---|
| 1055 | ZZ_pX F1=convertFacCF2NTLZZpX(F); |
---|
| 1056 | ZZ_pX G1=convertFacCF2NTLZZpX(G); |
---|
| 1057 | ZZ_pX R=GCD(F1,G1); |
---|
| 1058 | return convertNTLZZpX2CF(R,F.mvar()); |
---|
| 1059 | #else |
---|
| 1060 | zz_pX F1=convertFacCF2NTLzzpX(F); |
---|
| 1061 | zz_pX G1=convertFacCF2NTLzzpX(G); |
---|
| 1062 | zz_pX R=GCD(F1,G1); |
---|
| 1063 | return convertNTLzzpX2CF(R,F.mvar()); |
---|
| 1064 | #endif |
---|
| 1065 | } |
---|
| 1066 | |
---|
[a7ec94] | 1067 | #endif |
---|
| 1068 | |
---|
[7e8c9e] | 1069 | #ifdef HAVE_FLINT |
---|
| 1070 | static CanonicalForm |
---|
| 1071 | gcd_univar_flintp (const CanonicalForm& F, const CanonicalForm& G) |
---|
| 1072 | { |
---|
| 1073 | nmod_poly_t F1, G1; |
---|
| 1074 | convertFacCF2nmod_poly_t (F1, F); |
---|
| 1075 | convertFacCF2nmod_poly_t (G1, G); |
---|
| 1076 | nmod_poly_gcd (F1, F1, G1); |
---|
| 1077 | CanonicalForm result= convertnmod_poly_t2FacCF (F1, F.mvar()); |
---|
| 1078 | nmod_poly_clear (F1); |
---|
| 1079 | nmod_poly_clear (G1); |
---|
| 1080 | return result; |
---|
| 1081 | } |
---|
| 1082 | |
---|
| 1083 | static CanonicalForm |
---|
| 1084 | gcd_univar_flint0( const CanonicalForm & F, const CanonicalForm & G ) |
---|
| 1085 | { |
---|
| 1086 | fmpz_poly_t F1, G1; |
---|
| 1087 | convertFacCF2Fmpz_poly_t(F1, F); |
---|
| 1088 | convertFacCF2Fmpz_poly_t(G1, G); |
---|
| 1089 | fmpz_poly_gcd (F1, F1, G1); |
---|
| 1090 | CanonicalForm result= convertFmpz_poly_t2FacCF (F1, F.mvar()); |
---|
| 1091 | fmpz_poly_clear (F1); |
---|
| 1092 | fmpz_poly_clear (G1); |
---|
| 1093 | return result; |
---|
| 1094 | } |
---|
| 1095 | #endif |
---|
| 1096 | |
---|
| 1097 | |
---|
[a7ec94] | 1098 | /* |
---|
| 1099 | * compute positions p1 and pe of optimal variables: |
---|
| 1100 | * pe is used in "ezgcd" and |
---|
| 1101 | * p1 in "gcd_poly1" |
---|
| 1102 | */ |
---|
| 1103 | static |
---|
| 1104 | void optvalues ( const int * df, const int * dg, const int n, int & p1, int &pe ) |
---|
| 1105 | { |
---|
| 1106 | int i, o1, oe; |
---|
| 1107 | if ( df[n] > dg[n] ) |
---|
| 1108 | { |
---|
| 1109 | o1 = df[n]; oe = dg[n]; |
---|
| 1110 | } |
---|
| 1111 | else |
---|
| 1112 | { |
---|
| 1113 | o1 = dg[n]; oe = df[n]; |
---|
| 1114 | } |
---|
| 1115 | i = n-1; |
---|
| 1116 | while ( i > 0 ) |
---|
| 1117 | { |
---|
| 1118 | if ( df[i] != 0 ) |
---|
| 1119 | { |
---|
| 1120 | if ( df[i] > dg[i] ) |
---|
| 1121 | { |
---|
| 1122 | if ( o1 > df[i]) { o1 = df[i]; p1 = i; } |
---|
| 1123 | if ( oe <= dg[i]) { oe = dg[i]; pe = i; } |
---|
| 1124 | } |
---|
| 1125 | else |
---|
| 1126 | { |
---|
| 1127 | if ( o1 > dg[i]) { o1 = dg[i]; p1 = i; } |
---|
| 1128 | if ( oe <= df[i]) { oe = df[i]; pe = i; } |
---|
| 1129 | } |
---|
| 1130 | } |
---|
| 1131 | i--; |
---|
| 1132 | } |
---|
| 1133 | } |
---|
| 1134 | |
---|
| 1135 | /* |
---|
| 1136 | * make some changes of variables, see optvalues |
---|
| 1137 | */ |
---|
| 1138 | static void |
---|
| 1139 | cf_prepgcd( const CanonicalForm & f, const CanonicalForm & g, int & cc, int & p1, int &pe ) |
---|
| 1140 | { |
---|
| 1141 | int i, k, n; |
---|
| 1142 | n = f.level(); |
---|
| 1143 | cc = 0; |
---|
| 1144 | p1 = pe = n; |
---|
| 1145 | if ( n == 1 ) |
---|
| 1146 | return; |
---|
| 1147 | int * degsf = new int[n+1]; |
---|
| 1148 | int * degsg = new int[n+1]; |
---|
| 1149 | for ( i = n; i > 0; i-- ) |
---|
| 1150 | { |
---|
| 1151 | degsf[i] = degsg[i] = 0; |
---|
| 1152 | } |
---|
| 1153 | degsf = degrees( f, degsf ); |
---|
| 1154 | degsg = degrees( g, degsg ); |
---|
| 1155 | |
---|
| 1156 | k = 0; |
---|
| 1157 | for ( i = n-1; i > 0; i-- ) |
---|
| 1158 | { |
---|
[f4b180] | 1159 | if ( degsf[i] == 0 ) |
---|
[a7ec94] | 1160 | { |
---|
| 1161 | if ( degsg[i] != 0 ) |
---|
| 1162 | { |
---|
| 1163 | cc = -i; |
---|
| 1164 | break; |
---|
| 1165 | } |
---|
| 1166 | } |
---|
| 1167 | else |
---|
| 1168 | { |
---|
| 1169 | if ( degsg[i] == 0 ) |
---|
| 1170 | { |
---|
| 1171 | cc = i; |
---|
| 1172 | break; |
---|
| 1173 | } |
---|
| 1174 | else k++; |
---|
| 1175 | } |
---|
| 1176 | } |
---|
| 1177 | |
---|
| 1178 | if ( ( cc == 0 ) && ( k != 0 ) ) |
---|
| 1179 | optvalues( degsf, degsg, n, p1, pe ); |
---|
| 1180 | if ( ( pe != 1 ) && ( degsf[1] != 0 ) ) |
---|
| 1181 | pe = -pe; |
---|
[f4b180] | 1182 | |
---|
[a7ec94] | 1183 | delete [] degsf; |
---|
| 1184 | delete [] degsg; |
---|
| 1185 | } |
---|
[6f62c3] | 1186 | |
---|
| 1187 | |
---|
| 1188 | static CanonicalForm |
---|
| 1189 | balance_p ( const CanonicalForm & f, const CanonicalForm & q ) |
---|
| 1190 | { |
---|
| 1191 | Variable x = f.mvar(); |
---|
| 1192 | CanonicalForm result = 0, qh = q / 2; |
---|
| 1193 | CanonicalForm c; |
---|
| 1194 | CFIterator i; |
---|
| 1195 | for ( i = f; i.hasTerms(); i++ ) |
---|
| 1196 | { |
---|
| 1197 | c = i.coeff(); |
---|
| 1198 | if ( c.inCoeffDomain()) |
---|
| 1199 | { |
---|
| 1200 | if ( c > qh ) |
---|
| 1201 | result += power( x, i.exp() ) * (c - q); |
---|
| 1202 | else |
---|
| 1203 | result += power( x, i.exp() ) * c; |
---|
| 1204 | } |
---|
[f4b180] | 1205 | else |
---|
[6f62c3] | 1206 | result += power( x, i.exp() ) * balance_p(c,q); |
---|
| 1207 | } |
---|
| 1208 | return result; |
---|
| 1209 | } |
---|
| 1210 | |
---|
| 1211 | CanonicalForm chinrem_gcd ( const CanonicalForm & FF, const CanonicalForm & GG ) |
---|
| 1212 | { |
---|
[297e92] | 1213 | CanonicalForm f, g, cl, q(0), Dp, newD, D, newq; |
---|
[c1b9927] | 1214 | int p, i, dp_deg, d_deg=-1; |
---|
[6f62c3] | 1215 | |
---|
[01e8874] | 1216 | CanonicalForm cd ( bCommonDen( FF )); |
---|
[6f62c3] | 1217 | f=cd*FF; |
---|
[297e92] | 1218 | Variable x= Variable (1); |
---|
| 1219 | CanonicalForm cf, cg; |
---|
| 1220 | cf= icontent (f); |
---|
| 1221 | f /= cf; |
---|
[08a6ebb] | 1222 | //cd = bCommonDen( f ); f *=cd; |
---|
| 1223 | //f /=vcontent(f,Variable(1)); |
---|
[6f62c3] | 1224 | |
---|
| 1225 | cd = bCommonDen( GG ); |
---|
| 1226 | g=cd*GG; |
---|
[297e92] | 1227 | cg= icontent (g); |
---|
| 1228 | g /= cg; |
---|
[08a6ebb] | 1229 | //cd = bCommonDen( g ); g *=cd; |
---|
| 1230 | //g /=vcontent(g,Variable(1)); |
---|
[6f62c3] | 1231 | |
---|
[297e92] | 1232 | CanonicalForm Dn, test= 0; |
---|
[4704674] | 1233 | cl = gcd (f.lc(),g.lc()); |
---|
[597783] | 1234 | CanonicalForm gcdcfcg= gcd (cf, cg); |
---|
[2488dc3] | 1235 | CanonicalForm fp, gp; |
---|
[4704674] | 1236 | CanonicalForm b= 1; |
---|
| 1237 | int minCommonDeg= 0; |
---|
| 1238 | for (i= tmax (f.level(), g.level()); i > 0; i--) |
---|
| 1239 | { |
---|
| 1240 | if (degree (f, i) <= 0 || degree (g, i) <= 0) |
---|
| 1241 | continue; |
---|
| 1242 | else |
---|
| 1243 | { |
---|
| 1244 | minCommonDeg= tmin (degree (g, i), degree (f, i)); |
---|
| 1245 | break; |
---|
| 1246 | } |
---|
| 1247 | } |
---|
| 1248 | if (i == 0) |
---|
| 1249 | return gcdcfcg; |
---|
| 1250 | for (; i > 0; i--) |
---|
| 1251 | { |
---|
| 1252 | if (degree (f, i) <= 0 || degree (g, i) <= 0) |
---|
| 1253 | continue; |
---|
| 1254 | else |
---|
| 1255 | minCommonDeg= tmin (minCommonDeg, tmin (degree (g, i), degree (f, i))); |
---|
| 1256 | } |
---|
[cb7827] | 1257 | b= 2*tmin (maxNorm (f), maxNorm (g))*abs (cl)* |
---|
| 1258 | power (CanonicalForm (2), minCommonDeg); |
---|
[297e92] | 1259 | bool equal= false; |
---|
[6f62c3] | 1260 | i = cf_getNumBigPrimes() - 1; |
---|
| 1261 | |
---|
[cb7827] | 1262 | CanonicalForm cof, cog, cofp, cogp, newCof, newCog, cofn, cogn; |
---|
[2488dc3] | 1263 | int maxNumVars= tmax (getNumVars (f), getNumVars (g)); |
---|
[297e92] | 1264 | //Off (SW_RATIONAL); |
---|
[6f62c3] | 1265 | while ( true ) |
---|
| 1266 | { |
---|
| 1267 | p = cf_getBigPrime( i ); |
---|
| 1268 | i--; |
---|
[597783] | 1269 | while ( i >= 0 && mod( cl*(lc(f)/cl)*(lc(g)/cl), p ) == 0 ) |
---|
[6f62c3] | 1270 | { |
---|
| 1271 | p = cf_getBigPrime( i ); |
---|
| 1272 | i--; |
---|
| 1273 | } |
---|
[c30347] | 1274 | //printf("try p=%d\n",p); |
---|
[6f62c3] | 1275 | setCharacteristic( p ); |
---|
[2488dc3] | 1276 | fp= mapinto (f); |
---|
| 1277 | gp= mapinto (g); |
---|
[517530] | 1278 | #ifdef HAVE_NTL |
---|
[2488dc3] | 1279 | if (size (fp)/maxNumVars > 500 && size (gp)/maxNumVars > 500) |
---|
| 1280 | Dp = GCD_small_p (fp, gp, cofp, cogp); |
---|
| 1281 | else |
---|
| 1282 | { |
---|
| 1283 | Dp= gcd_poly (fp, gp); |
---|
| 1284 | cofp= fp/Dp; |
---|
| 1285 | cogp= gp/Dp; |
---|
| 1286 | } |
---|
[517530] | 1287 | #else |
---|
[2488dc3] | 1288 | Dp= gcd_poly (fp, gp); |
---|
| 1289 | cofp= fp/Dp; |
---|
| 1290 | cogp= gp/Dp; |
---|
[517530] | 1291 | #endif |
---|
[08a6ebb] | 1292 | Dp /=Dp.lc(); |
---|
[cb7827] | 1293 | cofp /= lc (cofp); |
---|
| 1294 | cogp /= lc (cogp); |
---|
[6f62c3] | 1295 | setCharacteristic( 0 ); |
---|
| 1296 | dp_deg=totaldegree(Dp); |
---|
| 1297 | if ( dp_deg == 0 ) |
---|
[c30347] | 1298 | { |
---|
| 1299 | //printf(" -> 1\n"); |
---|
[297e92] | 1300 | return CanonicalForm(gcdcfcg); |
---|
[c30347] | 1301 | } |
---|
[6f62c3] | 1302 | if ( q.isZero() ) |
---|
| 1303 | { |
---|
| 1304 | D = mapinto( Dp ); |
---|
[cb7827] | 1305 | cof= mapinto (cofp); |
---|
| 1306 | cog= mapinto (cogp); |
---|
[6f62c3] | 1307 | d_deg=dp_deg; |
---|
| 1308 | q = p; |
---|
| 1309 | } |
---|
| 1310 | else |
---|
| 1311 | { |
---|
| 1312 | if ( dp_deg == d_deg ) |
---|
| 1313 | { |
---|
| 1314 | chineseRemainder( D, q, mapinto( Dp ), p, newD, newq ); |
---|
[cb7827] | 1315 | chineseRemainder( cof, q, mapinto (cofp), p, newCof, newq); |
---|
| 1316 | chineseRemainder( cog, q, mapinto (cogp), p, newCog, newq); |
---|
| 1317 | cof= newCof; |
---|
| 1318 | cog= newCog; |
---|
[6f62c3] | 1319 | q = newq; |
---|
| 1320 | D = newD; |
---|
| 1321 | } |
---|
[f4b180] | 1322 | else if ( dp_deg < d_deg ) |
---|
[6f62c3] | 1323 | { |
---|
[c30347] | 1324 | //printf(" were all bad, try more\n"); |
---|
[6f62c3] | 1325 | // all previous p's are bad primes |
---|
| 1326 | q = p; |
---|
| 1327 | D = mapinto( Dp ); |
---|
[cb7827] | 1328 | cof= mapinto (cof); |
---|
| 1329 | cog= mapinto (cog); |
---|
[6f62c3] | 1330 | d_deg=dp_deg; |
---|
[297e92] | 1331 | test= 0; |
---|
| 1332 | equal= false; |
---|
[6f62c3] | 1333 | } |
---|
[c30347] | 1334 | else |
---|
| 1335 | { |
---|
| 1336 | //printf(" was bad, try more\n"); |
---|
| 1337 | } |
---|
[f4b180] | 1338 | //else dp_deg > d_deg: bad prime |
---|
[6f62c3] | 1339 | } |
---|
[08a6ebb] | 1340 | if ( i >= 0 ) |
---|
[6f62c3] | 1341 | { |
---|
[297e92] | 1342 | Dn= Farey(D,q); |
---|
[cb7827] | 1343 | cofn= Farey(cof,q); |
---|
| 1344 | cogn= Farey(cog,q); |
---|
[297e92] | 1345 | int is_rat= isOn (SW_RATIONAL); |
---|
[56d3c6] | 1346 | On (SW_RATIONAL); |
---|
[297e92] | 1347 | cd = bCommonDen( Dn ); // we need On(SW_RATIONAL) |
---|
[cb7827] | 1348 | cofn *= bCommonDen (cofn); |
---|
| 1349 | cogn *= bCommonDen (cogn); |
---|
[297e92] | 1350 | if (!is_rat) |
---|
| 1351 | Off (SW_RATIONAL); |
---|
[c992ec1] | 1352 | Dn *=cd; |
---|
[297e92] | 1353 | if (test != Dn) |
---|
| 1354 | test= Dn; |
---|
| 1355 | else |
---|
| 1356 | equal= true; |
---|
[c992ec1] | 1357 | //Dn /=vcontent(Dn,Variable(1)); |
---|
[1e4b53] | 1358 | if ((terminationTest (f,g, cofn, cogn, Dn)) || |
---|
[597783] | 1359 | ((equal || q > b) && fdivides (Dn, f) && fdivides (Dn, g))) |
---|
[6f62c3] | 1360 | { |
---|
[c30347] | 1361 | //printf(" -> success\n"); |
---|
[297e92] | 1362 | return Dn*gcdcfcg; |
---|
[6f62c3] | 1363 | } |
---|
[297e92] | 1364 | equal= false; |
---|
[c992ec1] | 1365 | //else: try more primes |
---|
[6f62c3] | 1366 | } |
---|
| 1367 | else |
---|
[c992ec1] | 1368 | { // try other method |
---|
[c30347] | 1369 | //printf("try other gcd\n"); |
---|
[6f62c3] | 1370 | Off(SW_USE_CHINREM_GCD); |
---|
| 1371 | D=gcd_poly( f, g ); |
---|
| 1372 | On(SW_USE_CHINREM_GCD); |
---|
[297e92] | 1373 | return D*gcdcfcg; |
---|
[6f62c3] | 1374 | } |
---|
| 1375 | } |
---|
| 1376 | } |
---|
[c5d0aed] | 1377 | |
---|
[cb7827] | 1378 | |
---|