[17b1f3] | 1 | /*****************************************************************************\ |
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| 2 | * Computer Algebra System SINGULAR |
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| 3 | \*****************************************************************************/ |
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| 4 | /** @file cfModResultant.cc |
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| 5 | * |
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| 6 | * modular resultant algorithm |
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| 7 | * |
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| 8 | * @author Martin Lee |
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| 9 | * |
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| 10 | **/ |
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| 11 | /*****************************************************************************/ |
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| 12 | |
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[e4fe2b] | 13 | #include "config.h" |
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[17b1f3] | 14 | |
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[e4fe2b] | 15 | #include "cf_assert.h" |
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[17b1f3] | 16 | |
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| 17 | #include "cfModResultant.h" |
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| 18 | #include "cf_primes.h" |
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| 19 | #include "templates/ftmpl_functions.h" |
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| 20 | #include "cf_map.h" |
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| 21 | #include "cf_algorithm.h" |
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[e4fe2b] | 22 | #include "cf_iter.h" |
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[17b1f3] | 23 | |
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| 24 | #ifdef HAVE_NTL |
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| 25 | #include "NTLconvert.h" |
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| 26 | #endif |
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| 27 | |
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[4604b84] | 28 | #ifdef HAVE_FLINT |
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| 29 | #include "FLINTconvert.h" |
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| 30 | #endif |
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| 31 | |
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[17b1f3] | 32 | //TODO arrange by bound= deg (F,xlevel)*deg (G,i)+deg (G,xlevel)*deg (F, i) |
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| 33 | static inline |
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| 34 | void myCompress (const CanonicalForm& F, const CanonicalForm& G, CFMap & M, |
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| 35 | CFMap & N, const Variable& x) |
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| 36 | { |
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| 37 | int n= tmax (F.level(), G.level()); |
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| 38 | int * degsf= new int [n + 1]; |
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| 39 | int * degsg= new int [n + 1]; |
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| 40 | |
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| 41 | for (int i = 0; i <= n; i++) |
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| 42 | degsf[i]= degsg[i]= 0; |
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| 43 | |
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| 44 | degsf= degrees (F, degsf); |
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| 45 | degsg= degrees (G, degsg); |
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| 46 | |
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| 47 | int both_non_zero= 0; |
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| 48 | int f_zero= 0; |
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| 49 | int g_zero= 0; |
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| 50 | int both_zero= 0; |
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| 51 | int degsfx, degsgx; |
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| 52 | |
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| 53 | if (x.level() != 1) |
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| 54 | { |
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| 55 | int xlevel= x.level(); |
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| 56 | |
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| 57 | for (int i= 1; i <= n; i++) |
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| 58 | { |
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| 59 | if (degsf[i] != 0 && degsg[i] != 0) |
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| 60 | { |
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| 61 | both_non_zero++; |
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| 62 | continue; |
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| 63 | } |
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| 64 | if (degsf[i] == 0 && degsg[i] != 0 && i <= G.level()) |
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| 65 | { |
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| 66 | f_zero++; |
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| 67 | continue; |
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| 68 | } |
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| 69 | if (degsg[i] == 0 && degsf[i] && i <= F.level()) |
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| 70 | { |
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| 71 | g_zero++; |
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| 72 | continue; |
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| 73 | } |
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| 74 | } |
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| 75 | |
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| 76 | M.newpair (Variable (xlevel), Variable (1)); |
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| 77 | N.newpair (Variable (1), Variable (xlevel)); |
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| 78 | degsfx= degsf [xlevel]; |
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| 79 | degsgx= degsg [xlevel]; |
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| 80 | degsf [xlevel]= 0; |
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| 81 | degsg [xlevel]= 0; |
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| 82 | if (getNumVars (F) == 2 || getNumVars (G) == 2) |
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| 83 | { |
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| 84 | int pos= 2; |
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| 85 | for (int i= 1; i <= n; i++) |
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| 86 | { |
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| 87 | if (i != xlevel) |
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| 88 | { |
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| 89 | if (i != pos && (degsf[i] != 0 || degsg [i] != 0)) |
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| 90 | { |
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| 91 | M.newpair (Variable (i), Variable (pos)); |
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| 92 | N.newpair (Variable (pos), Variable (i)); |
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| 93 | pos++; |
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| 94 | } |
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| 95 | } |
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| 96 | } |
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| 97 | delete [] degsf; |
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| 98 | delete [] degsg; |
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| 99 | return; |
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| 100 | } |
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| 101 | |
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| 102 | if (both_non_zero == 0) |
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| 103 | { |
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| 104 | delete [] degsf; |
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| 105 | delete [] degsg; |
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| 106 | return; |
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| 107 | } |
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| 108 | |
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| 109 | // map Variables which do not occur in both polynomials to higher levels |
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| 110 | int k= 1; |
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| 111 | int l= 1; |
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| 112 | for (int i= 1; i <= n; i++) |
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| 113 | { |
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| 114 | if (i == xlevel) |
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| 115 | continue; |
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| 116 | if (degsf[i] != 0 && degsg[i] == 0 && i <= F.level()) |
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| 117 | { |
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| 118 | if (k + both_non_zero != i) |
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| 119 | { |
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| 120 | M.newpair (Variable (i), Variable (k + both_non_zero)); |
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| 121 | N.newpair (Variable (k + both_non_zero), Variable (i)); |
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| 122 | } |
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| 123 | k++; |
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| 124 | } |
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| 125 | if (degsf[i] == 0 && degsg[i] != 0 && i <= G.level()) |
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| 126 | { |
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| 127 | if (l + g_zero + both_non_zero != i) |
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| 128 | { |
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| 129 | M.newpair (Variable (i), Variable (l + g_zero + both_non_zero)); |
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| 130 | N.newpair (Variable (l + g_zero + both_non_zero), Variable (i)); |
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| 131 | } |
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| 132 | l++; |
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| 133 | } |
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| 134 | } |
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| 135 | |
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| 136 | int m= tmax (F.level(), G.level()); |
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| 137 | int min_max_deg; |
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| 138 | k= both_non_zero; |
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| 139 | l= 0; |
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| 140 | int i= 1; |
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| 141 | while (k > 0) |
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| 142 | { |
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| 143 | if (degsf [i] != 0 && degsg [i] != 0) |
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| 144 | min_max_deg= degsgx*degsf[i] + degsfx*degsg[i]; |
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| 145 | else |
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| 146 | min_max_deg= 0; |
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| 147 | while (min_max_deg == 0) |
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| 148 | { |
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| 149 | i++; |
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| 150 | if (degsf [i] != 0 && degsg [i] != 0) |
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| 151 | min_max_deg= degsgx*degsf[i] + degsfx*degsg[i]; |
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| 152 | else |
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| 153 | min_max_deg= 0; |
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| 154 | } |
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| 155 | for (int j= i + 1; j <= m; j++) |
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| 156 | { |
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| 157 | if (degsgx*degsf[j] + degsfx*degsg[j] <= min_max_deg && |
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| 158 | degsf[j] != 0 && degsg [j] != 0) |
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| 159 | { |
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| 160 | min_max_deg= degsgx*degsf[j] + degsfx*degsg[j]; |
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| 161 | l= j; |
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| 162 | } |
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| 163 | } |
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| 164 | if (l != 0) |
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| 165 | { |
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| 166 | if (l != k && l != xlevel && k != 1) |
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| 167 | { |
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| 168 | M.newpair (Variable (l), Variable(k)); |
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| 169 | N.newpair (Variable (k), Variable(l)); |
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| 170 | degsf[l]= 0; |
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| 171 | degsg[l]= 0; |
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| 172 | l= 0; |
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| 173 | } |
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| 174 | else |
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| 175 | { |
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| 176 | degsf[l]= 0; |
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| 177 | degsg[l]= 0; |
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| 178 | l= 0; |
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| 179 | } |
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| 180 | } |
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| 181 | else if (l == 0) |
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| 182 | { |
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| 183 | if (i != k && i != xlevel && k != 1) |
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| 184 | { |
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| 185 | M.newpair (Variable (i), Variable (k)); |
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| 186 | N.newpair (Variable (k), Variable (i)); |
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| 187 | degsf[i]= 0; |
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| 188 | degsg[i]= 0; |
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| 189 | } |
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| 190 | else |
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| 191 | { |
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| 192 | degsf[i]= 0; |
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| 193 | degsg[i]= 0; |
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| 194 | } |
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| 195 | i++; |
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| 196 | } |
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| 197 | k--; |
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| 198 | } |
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| 199 | } |
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| 200 | else |
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| 201 | { |
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| 202 | //arrange Variables such that no gaps occur |
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| 203 | for (int i= 1; i <= n; i++) |
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| 204 | { |
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| 205 | if (degsf[i] == 0 && degsg[i] == 0) |
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| 206 | { |
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| 207 | both_zero++; |
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| 208 | continue; |
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| 209 | } |
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| 210 | else |
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| 211 | { |
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| 212 | if (both_zero != 0 && i != 1) |
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| 213 | { |
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| 214 | M.newpair (Variable (i), Variable (i - both_zero)); |
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| 215 | N.newpair (Variable (i - both_zero), Variable (i)); |
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| 216 | } |
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| 217 | } |
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| 218 | } |
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| 219 | } |
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| 220 | |
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| 221 | delete [] degsf; |
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| 222 | delete [] degsg; |
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| 223 | } |
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| 224 | |
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| 225 | static inline |
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| 226 | CanonicalForm oneNorm (const CanonicalForm& F) |
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| 227 | { |
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| 228 | if (F.inZ()) |
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| 229 | return abs (F); |
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| 230 | |
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| 231 | CanonicalForm result= 0; |
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| 232 | for (CFIterator i= F; i.hasTerms(); i++) |
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| 233 | result += oneNorm (i.coeff()); |
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| 234 | return result; |
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| 235 | } |
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| 236 | |
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| 237 | // if F and G are both non constant, make sure their level is equal |
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| 238 | static inline |
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| 239 | CanonicalForm uniResultant (const CanonicalForm& F, const CanonicalForm& G) |
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| 240 | { |
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| 241 | #ifdef HAVE_NTL |
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| 242 | ASSERT (getCharacteristic() > 0, "characteristic > 0 expected"); |
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| 243 | if (F.inCoeffDomain() && G.inCoeffDomain()) |
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| 244 | return 1; |
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| 245 | if (F.isZero() || G.isZero()) |
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| 246 | return 0; |
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| 247 | |
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[4604b84] | 248 | #ifdef HAVE_FLINT |
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| 249 | nmod_poly_t FLINTF, FLINTG; |
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| 250 | convertFacCF2nmod_poly_t (FLINTF, F); |
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| 251 | convertFacCF2nmod_poly_t (FLINTG, G); |
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| 252 | mp_limb_t FLINTresult= nmod_poly_resultant (FLINTF, FLINTG); |
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| 253 | nmod_poly_clear (FLINTF); |
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| 254 | nmod_poly_clear (FLINTG); |
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| 255 | return CanonicalForm ((long) FLINTresult); |
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| 256 | #else |
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[35eb6c] | 257 | zz_pBak bak; |
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| 258 | bak.save(); |
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[17b1f3] | 259 | zz_p::init (getCharacteristic()); |
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| 260 | zz_pX NTLF= convertFacCF2NTLzzpX (F); |
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| 261 | zz_pX NTLG= convertFacCF2NTLzzpX (G); |
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| 262 | |
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| 263 | zz_p NTLResult= resultant (NTLF, NTLG); |
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| 264 | |
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[35eb6c] | 265 | bak.restore(); |
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[17b1f3] | 266 | return CanonicalForm (to_long (rep (NTLResult))); |
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[4604b84] | 267 | #endif |
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[17b1f3] | 268 | #else |
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| 269 | return resultant (F, G, F.mvar()); |
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| 270 | #endif |
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| 271 | } |
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| 272 | |
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| 273 | static inline |
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| 274 | void evalPoint (const CanonicalForm& F, const CanonicalForm& G, |
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| 275 | CanonicalForm& FEval, CanonicalForm& GEval, int& evalPoint) |
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| 276 | { |
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| 277 | int degF, degG; |
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| 278 | Variable x= Variable (1); |
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| 279 | degF= degree (F, x); |
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| 280 | degG= degree (G, x); |
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| 281 | do |
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| 282 | { |
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| 283 | evalPoint++; |
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| 284 | if (evalPoint >= getCharacteristic()) |
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| 285 | break; |
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| 286 | FEval= F (evalPoint, 2); |
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| 287 | GEval= G (evalPoint, 2); |
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| 288 | if (degree (FEval, 1) < degF || degree (GEval, 1) < degG) |
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| 289 | continue; |
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| 290 | else |
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| 291 | return; |
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| 292 | } |
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| 293 | while (evalPoint < getCharacteristic()); |
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| 294 | } |
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| 295 | |
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| 296 | static inline CanonicalForm |
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| 297 | newtonInterp (const CanonicalForm alpha, const CanonicalForm u, |
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| 298 | const CanonicalForm newtonPoly, const CanonicalForm oldInterPoly, |
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| 299 | const Variable & x) |
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| 300 | { |
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| 301 | CanonicalForm interPoly; |
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| 302 | |
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| 303 | interPoly= oldInterPoly+((u - oldInterPoly (alpha, x))/newtonPoly (alpha, x)) |
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| 304 | *newtonPoly; |
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| 305 | return interPoly; |
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| 306 | } |
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| 307 | |
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| 308 | CanonicalForm |
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| 309 | resultantFp (const CanonicalForm& A, const CanonicalForm& B, const Variable& x, |
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| 310 | bool prob) |
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| 311 | { |
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| 312 | ASSERT (getCharacteristic() > 0, "characteristic > 0 expected"); |
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| 313 | |
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| 314 | if (A.isZero() || B.isZero()) |
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| 315 | return 0; |
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| 316 | |
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| 317 | int degAx= degree (A, x); |
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| 318 | int degBx= degree (B, x); |
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| 319 | if (A.level() < x.level()) |
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| 320 | return power (A, degBx); |
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| 321 | if (B.level() < x.level()) |
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| 322 | return power (B, degAx); |
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| 323 | |
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| 324 | if (degAx == 0) |
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| 325 | return power (A, degBx); |
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| 326 | else if (degBx == 0) |
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| 327 | return power (B, degAx); |
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| 328 | |
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| 329 | CanonicalForm F= A; |
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| 330 | CanonicalForm G= B; |
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| 331 | |
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| 332 | CFMap M, N; |
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| 333 | myCompress (F, G, M, N, x); |
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| 334 | |
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| 335 | F= M (F); |
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| 336 | G= M (G); |
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| 337 | |
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| 338 | Variable y= Variable (2); |
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| 339 | |
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| 340 | if (F.isUnivariate() && G.isUnivariate() && F.level() == G.level()) |
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| 341 | return N(uniResultant (F, G)); |
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| 342 | |
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| 343 | int i= -1; |
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| 344 | CanonicalForm GEval, FEval, recResult, H; |
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| 345 | CanonicalForm newtonPoly= 1; |
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| 346 | CanonicalForm modResult= 0; |
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| 347 | |
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| 348 | Variable z= Variable (1); |
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| 349 | int bound= degAx*degree (G, 2) + degree (F, 2)*degBx; |
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| 350 | |
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| 351 | int count= 0; |
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| 352 | int equalCount= 0; |
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| 353 | do |
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| 354 | { |
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| 355 | evalPoint (F, G, FEval, GEval, i); |
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| 356 | |
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| 357 | ASSERT (i < getCharacteristic(), "ran out of points"); |
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| 358 | |
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| 359 | recResult= resultantFp (FEval, GEval, z); |
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| 360 | |
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| 361 | H= newtonInterp (i, recResult, newtonPoly, modResult, y); |
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| 362 | |
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| 363 | if (H == modResult) |
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| 364 | equalCount++; |
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| 365 | |
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| 366 | count++; |
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| 367 | if (count > bound || (prob && equalCount == 2)) |
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| 368 | break; |
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| 369 | |
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| 370 | modResult= H; |
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| 371 | newtonPoly *= (y - i); |
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| 372 | } while (1); |
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| 373 | |
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| 374 | return N (H); |
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| 375 | } |
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| 376 | |
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| 377 | static inline |
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| 378 | CanonicalForm |
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| 379 | balanceUni ( const CanonicalForm & f, const CanonicalForm & q ) |
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| 380 | { |
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| 381 | Variable x = f.mvar(); |
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| 382 | CanonicalForm result = 0, qh = q / 2; |
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| 383 | CanonicalForm c; |
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| 384 | CFIterator i; |
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| 385 | for ( i = f; i.hasTerms(); i++ ) { |
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| 386 | c = mod( i.coeff(), q ); |
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| 387 | if ( c > qh ) |
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| 388 | result += power( x, i.exp() ) * (c - q); |
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| 389 | else |
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| 390 | result += power( x, i.exp() ) * c; |
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| 391 | } |
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| 392 | return result; |
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| 393 | } |
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| 394 | |
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| 395 | static inline |
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| 396 | CanonicalForm |
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| 397 | symmetricRemainder (const CanonicalForm& f, const CanonicalForm& q) |
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| 398 | { |
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| 399 | CanonicalForm result= 0; |
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| 400 | if (f.isUnivariate() || f.inCoeffDomain()) |
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| 401 | return balanceUni (f, q); |
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| 402 | else |
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| 403 | { |
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| 404 | Variable x= f.mvar(); |
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| 405 | for (CFIterator i= f; i.hasTerms(); i++) |
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| 406 | result += power (x, i.exp())*symmetricRemainder (i.coeff(), q); |
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| 407 | } |
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| 408 | return result; |
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| 409 | } |
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| 410 | |
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| 411 | CanonicalForm |
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| 412 | resultantZ (const CanonicalForm& A, const CanonicalForm& B, const Variable& x, |
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| 413 | bool prob) |
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| 414 | { |
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| 415 | ASSERT (getCharacteristic() == 0, "characteristic > 0 expected"); |
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| 416 | |
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| 417 | int degAx= degree (A, x); |
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| 418 | int degBx= degree (B, x); |
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| 419 | if (A.level() < x.level()) |
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| 420 | return power (A, degBx); |
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| 421 | if (B.level() < x.level()) |
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| 422 | return power (B, degAx); |
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| 423 | |
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| 424 | if (degAx == 0) |
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| 425 | return power (A, degBx); |
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| 426 | else if (degBx == 0) |
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| 427 | return power (B, degAx); |
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| 428 | |
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| 429 | CanonicalForm F= A; |
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| 430 | CanonicalForm G= B; |
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| 431 | |
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| 432 | Variable X= x; |
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| 433 | if (F.level() != x.level() || G.level() != x.level()) |
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| 434 | { |
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| 435 | if (F.level() > G.level()) |
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| 436 | X= F.mvar(); |
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| 437 | else |
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| 438 | X= G.mvar(); |
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| 439 | F= swapvar (F, X, x); |
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| 440 | G= swapvar (G, X, x); |
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| 441 | } |
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| 442 | |
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| 443 | // now X is the main variable |
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| 444 | |
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| 445 | CanonicalForm d= 0; |
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| 446 | CanonicalForm dd= 0; |
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| 447 | CanonicalForm buf; |
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| 448 | for (CFIterator i= F; i.hasTerms(); i++) |
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| 449 | { |
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| 450 | buf= oneNorm (i.coeff()); |
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| 451 | d= (buf > d) ? buf : d; |
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| 452 | } |
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| 453 | CanonicalForm e= 0, ee= 0; |
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| 454 | for (CFIterator i= G; i.hasTerms(); i++) |
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| 455 | { |
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| 456 | buf= oneNorm (i.coeff()); |
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| 457 | e= (buf > e) ? buf : e; |
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| 458 | } |
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| 459 | d= power (d, degBx); |
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| 460 | e= power (e, degAx); |
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| 461 | CanonicalForm bound= 1; |
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| 462 | for (int i= degBx + degAx; i > 1; i--) |
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| 463 | bound *= i; |
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| 464 | bound *= d*e; |
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| 465 | bound *= 2; |
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| 466 | |
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| 467 | bool onRational= isOn (SW_RATIONAL); |
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| 468 | if (onRational) |
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| 469 | Off (SW_RATIONAL); |
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| 470 | int i = cf_getNumBigPrimes() - 1; |
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| 471 | int p; |
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| 472 | CanonicalForm l= lc (F)*lc(G); |
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| 473 | CanonicalForm resultModP, q (0), newResult, newQ; |
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| 474 | CanonicalForm result; |
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| 475 | do |
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| 476 | { |
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| 477 | p = cf_getBigPrime( i ); |
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| 478 | i--; |
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| 479 | while ( i >= 0 && mod( l, p ) == 0) |
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| 480 | { |
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| 481 | p = cf_getBigPrime( i ); |
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| 482 | i--; |
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| 483 | } |
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| 484 | |
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| 485 | ASSERT (i >= 0, "ran out of primes"); //sic |
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| 486 | |
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| 487 | setCharacteristic (p); |
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| 488 | |
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| 489 | resultModP= resultantFp (mapinto (F), mapinto (G), X, prob); |
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| 490 | |
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| 491 | setCharacteristic (0); |
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| 492 | |
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| 493 | if ( q.isZero() ) |
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| 494 | { |
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| 495 | result= mapinto(resultModP); |
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| 496 | q= p; |
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| 497 | } |
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| 498 | else |
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| 499 | { |
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| 500 | chineseRemainder( result, q, mapinto (resultModP), p, newResult, newQ ); |
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| 501 | q= newQ; |
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| 502 | result= newResult; |
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| 503 | if (newQ > bound) |
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| 504 | { |
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| 505 | result= symmetricRemainder (result, q); |
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| 506 | break; |
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| 507 | } |
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| 508 | } |
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| 509 | } while (1); |
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| 510 | |
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| 511 | if (onRational) |
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| 512 | On (SW_RATIONAL); |
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| 513 | return swapvar (result, X, x); |
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| 514 | } |
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| 515 | |
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