[4dfcb1] | 1 | #include "factoryconf.h" |
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| 2 | |
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| 3 | #ifdef HAVE_CSTDIO |
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| 4 | #include <cstdio> |
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| 5 | #else |
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[c99b6b] | 6 | #include <stdio.h> |
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[4dfcb1] | 7 | #endif |
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[4dd2c4] | 8 | #ifndef NOSTREAMIO |
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[4dfcb1] | 9 | #ifdef HAVE_IOSTREAM_H |
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[c99b6b] | 10 | #include <iostream.h> |
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[4dd2c4] | 11 | #elif defined(HAVE_IOSTREAM) |
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[4dfcb1] | 12 | #include <iostream> |
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| 13 | #endif |
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[4dd2c4] | 14 | #endif |
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[c99b6b] | 15 | |
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[fe2d4c] | 16 | #include "templates/ftmpl_functions.h" |
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[c99b6b] | 17 | #include "cf_defs.h" |
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| 18 | #include "canonicalform.h" |
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| 19 | #include "cf_iter.h" |
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| 20 | #include "cf_primes.h" |
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| 21 | #include "cf_algorithm.h" |
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| 22 | #include "algext.h" |
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[359d742] | 23 | #include "fieldGCD.h" |
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| 24 | #include "cf_map.h" |
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| 25 | #include "cf_generator.h" |
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[c99b6b] | 26 | |
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[fe2d4c] | 27 | /// compressing two polynomials F and G, M is used for compressing, |
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| 28 | /// N to reverse the compression |
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| 29 | static |
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| 30 | int myCompress (const CanonicalForm& F, const CanonicalForm& G, CFMap & M, |
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| 31 | CFMap & N, bool topLevel) |
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| 32 | { |
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| 33 | int n= tmax (F.level(), G.level()); |
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| 34 | int * degsf= new int [n + 1]; |
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| 35 | int * degsg= new int [n + 1]; |
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| 36 | |
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| 37 | for (int i = 0; i <= n; i++) |
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| 38 | degsf[i]= degsg[i]= 0; |
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| 39 | |
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| 40 | degsf= degrees (F, degsf); |
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| 41 | degsg= degrees (G, degsg); |
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| 42 | |
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| 43 | int both_non_zero= 0; |
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| 44 | int f_zero= 0; |
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| 45 | int g_zero= 0; |
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| 46 | int both_zero= 0; |
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| 47 | |
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| 48 | if (topLevel) |
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| 49 | { |
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| 50 | for (int i= 1; i <= n; i++) |
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| 51 | { |
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| 52 | if (degsf[i] != 0 && degsg[i] != 0) |
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| 53 | { |
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| 54 | both_non_zero++; |
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| 55 | continue; |
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| 56 | } |
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| 57 | if (degsf[i] == 0 && degsg[i] != 0 && i <= G.level()) |
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| 58 | { |
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| 59 | f_zero++; |
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| 60 | continue; |
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| 61 | } |
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| 62 | if (degsg[i] == 0 && degsf[i] && i <= F.level()) |
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| 63 | { |
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| 64 | g_zero++; |
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| 65 | continue; |
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| 66 | } |
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| 67 | } |
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| 68 | |
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| 69 | if (both_non_zero == 0) |
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| 70 | { |
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| 71 | delete [] degsf; |
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| 72 | delete [] degsg; |
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| 73 | return 0; |
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| 74 | } |
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| 75 | |
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| 76 | // map Variables which do not occur in both polynomials to higher levels |
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| 77 | int k= 1; |
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| 78 | int l= 1; |
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| 79 | for (int i= 1; i <= n; i++) |
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| 80 | { |
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| 81 | if (degsf[i] != 0 && degsg[i] == 0 && i <= F.level()) |
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| 82 | { |
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| 83 | if (k + both_non_zero != i) |
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| 84 | { |
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| 85 | M.newpair (Variable (i), Variable (k + both_non_zero)); |
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| 86 | N.newpair (Variable (k + both_non_zero), Variable (i)); |
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| 87 | } |
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| 88 | k++; |
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| 89 | } |
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| 90 | if (degsf[i] == 0 && degsg[i] != 0 && i <= G.level()) |
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| 91 | { |
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| 92 | if (l + g_zero + both_non_zero != i) |
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| 93 | { |
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| 94 | M.newpair (Variable (i), Variable (l + g_zero + both_non_zero)); |
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| 95 | N.newpair (Variable (l + g_zero + both_non_zero), Variable (i)); |
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| 96 | } |
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| 97 | l++; |
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| 98 | } |
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| 99 | } |
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| 100 | |
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| 101 | // sort Variables x_{i} in increasing order of |
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| 102 | // min(deg_{x_{i}}(f),deg_{x_{i}}(g)) |
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| 103 | int m= tmax (F.level(), G.level()); |
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| 104 | int min_max_deg; |
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| 105 | k= both_non_zero; |
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| 106 | l= 0; |
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| 107 | int i= 1; |
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| 108 | while (k > 0) |
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| 109 | { |
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| 110 | if (degsf [i] != 0 && degsg [i] != 0) |
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| 111 | min_max_deg= tmax (degsf[i], degsg[i]); |
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| 112 | else |
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| 113 | min_max_deg= 0; |
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| 114 | while (min_max_deg == 0) |
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| 115 | { |
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| 116 | i++; |
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| 117 | min_max_deg= tmax (degsf[i], degsg[i]); |
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| 118 | if (degsf [i] != 0 && degsg [i] != 0) |
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| 119 | min_max_deg= tmax (degsf[i], degsg[i]); |
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| 120 | else |
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| 121 | min_max_deg= 0; |
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| 122 | } |
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| 123 | for (int j= i + 1; j <= m; j++) |
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| 124 | { |
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| 125 | if (tmax (degsf[j],degsg[j]) <= min_max_deg && degsf[j] != 0 && degsg [j] != 0) |
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| 126 | { |
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| 127 | min_max_deg= tmax (degsf[j], degsg[j]); |
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| 128 | l= j; |
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| 129 | } |
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| 130 | } |
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| 131 | if (l != 0) |
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| 132 | { |
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| 133 | if (l != k) |
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| 134 | { |
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| 135 | M.newpair (Variable (l), Variable(k)); |
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| 136 | N.newpair (Variable (k), Variable(l)); |
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| 137 | degsf[l]= 0; |
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| 138 | degsg[l]= 0; |
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| 139 | l= 0; |
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| 140 | } |
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| 141 | else |
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| 142 | { |
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| 143 | degsf[l]= 0; |
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| 144 | degsg[l]= 0; |
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| 145 | l= 0; |
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| 146 | } |
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| 147 | } |
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| 148 | else if (l == 0) |
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| 149 | { |
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| 150 | if (i != k) |
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| 151 | { |
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| 152 | M.newpair (Variable (i), Variable (k)); |
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| 153 | N.newpair (Variable (k), Variable (i)); |
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| 154 | degsf[i]= 0; |
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| 155 | degsg[i]= 0; |
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| 156 | } |
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| 157 | else |
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| 158 | { |
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| 159 | degsf[i]= 0; |
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| 160 | degsg[i]= 0; |
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| 161 | } |
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| 162 | i++; |
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| 163 | } |
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| 164 | k--; |
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| 165 | } |
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| 166 | } |
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| 167 | else |
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| 168 | { |
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| 169 | //arrange Variables such that no gaps occur |
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| 170 | for (int i= 1; i <= n; i++) |
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| 171 | { |
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| 172 | if (degsf[i] == 0 && degsg[i] == 0) |
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| 173 | { |
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| 174 | both_zero++; |
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| 175 | continue; |
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| 176 | } |
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| 177 | else |
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| 178 | { |
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| 179 | if (both_zero != 0) |
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| 180 | { |
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| 181 | M.newpair (Variable (i), Variable (i - both_zero)); |
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| 182 | N.newpair (Variable (i - both_zero), Variable (i)); |
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| 183 | } |
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| 184 | } |
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| 185 | } |
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| 186 | } |
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| 187 | |
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| 188 | delete [] degsf; |
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| 189 | delete [] degsg; |
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| 190 | |
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| 191 | return 1; |
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| 192 | } |
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| 193 | |
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[ad8e1b] | 194 | CanonicalForm reduce(const CanonicalForm & f, const CanonicalForm & M) |
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| 195 | { // polynomials in M.mvar() are considered coefficients |
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| 196 | // M univariate monic polynomial |
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| 197 | // the coefficients of f are reduced modulo M |
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| 198 | if(f.inBaseDomain() || f.level() < M.level()) |
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| 199 | return f; |
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| 200 | if(f.level() == M.level()) |
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| 201 | { |
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| 202 | if(f.degree() < M.degree()) |
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| 203 | return f; |
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[a8e8b9] | 204 | CanonicalForm tmp = mod (f, M); |
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[ad8e1b] | 205 | return tmp; |
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| 206 | } |
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| 207 | // here: f.level() > M.level() |
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| 208 | CanonicalForm result = 0; |
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| 209 | for(CFIterator i=f; i.hasTerms(); i++) |
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| 210 | result += reduce(i.coeff(),M) * power(f.mvar(),i.exp()); |
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| 211 | return result; |
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| 212 | } |
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| 213 | |
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| 214 | void tryInvert( const CanonicalForm & F, const CanonicalForm & M, CanonicalForm & inv, bool & fail ) |
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| 215 | { // F, M are required to be "univariate" polynomials in an algebraic variable |
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| 216 | // we try to invert F modulo M |
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| 217 | if(F.inBaseDomain()) |
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| 218 | { |
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| 219 | if(F.isZero()) |
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| 220 | { |
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| 221 | fail = true; |
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| 222 | return; |
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| 223 | } |
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| 224 | inv = 1/F; |
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| 225 | return; |
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| 226 | } |
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| 227 | CanonicalForm b; |
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| 228 | Variable a = M.mvar(); |
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| 229 | Variable x = Variable(1); |
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| 230 | if(!extgcd( replacevar( F, a, x ), replacevar( M, a, x ), inv, b ).isOne()) |
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| 231 | fail = true; |
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| 232 | else |
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| 233 | inv = replacevar( inv, x, a ); // change back to alg var |
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| 234 | } |
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| 235 | |
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[a8e8b9] | 236 | void tryDivrem (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
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| 237 | CanonicalForm& R, CanonicalForm& inv, const CanonicalForm& mipo, |
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| 238 | bool& fail) |
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| 239 | { |
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| 240 | if (F.inCoeffDomain()) |
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| 241 | { |
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| 242 | Q= 0; |
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| 243 | R= F; |
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| 244 | return; |
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| 245 | } |
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| 246 | |
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| 247 | CanonicalForm A, B; |
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| 248 | Variable x= F.mvar(); |
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| 249 | A= F; |
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| 250 | B= G; |
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| 251 | int degA= degree (A, x); |
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| 252 | int degB= degree (B, x); |
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| 253 | |
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| 254 | if (degA < degB) |
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| 255 | { |
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| 256 | R= A; |
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| 257 | Q= 0; |
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| 258 | return; |
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| 259 | } |
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| 260 | |
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| 261 | tryInvert (Lc (B), mipo, inv, fail); |
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| 262 | if (fail) |
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| 263 | return; |
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| 264 | |
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| 265 | R= A; |
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| 266 | Q= 0; |
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| 267 | CanonicalForm Qi; |
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| 268 | for (int i= degA -degB; i >= 0; i--) |
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| 269 | { |
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| 270 | if (degree (R, x) == i + degB) |
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| 271 | { |
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| 272 | Qi= Lc (R)*inv*power (x, i); |
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| 273 | Qi= reduce (Qi, mipo); |
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| 274 | R -= Qi*B; |
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| 275 | R= reduce (R, mipo); |
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| 276 | Q += Qi; |
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| 277 | } |
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| 278 | } |
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| 279 | } |
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| 280 | |
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[ad8e1b] | 281 | void tryEuclid( const CanonicalForm & A, const CanonicalForm & B, const CanonicalForm & M, CanonicalForm & result, bool & fail ) |
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[c99b6b] | 282 | { |
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| 283 | CanonicalForm P; |
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[ad8e1b] | 284 | if(A.inCoeffDomain()) |
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| 285 | { |
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| 286 | tryInvert( A, M, P, fail ); |
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| 287 | if(fail) |
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| 288 | return; |
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| 289 | result = 1; |
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| 290 | return; |
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| 291 | } |
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| 292 | if(B.inCoeffDomain()) |
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| 293 | { |
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| 294 | tryInvert( B, M, P, fail ); |
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| 295 | if(fail) |
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| 296 | return; |
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| 297 | result = 1; |
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| 298 | return; |
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| 299 | } |
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| 300 | // here: both not inCoeffDomain |
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| 301 | if( A.degree() > B.degree() ) |
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[c99b6b] | 302 | { |
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| 303 | P = A; result = B; |
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| 304 | } |
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| 305 | else |
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| 306 | { |
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| 307 | P = B; result = A; |
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| 308 | } |
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| 309 | CanonicalForm inv; |
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| 310 | if( result.isZero() ) |
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| 311 | { |
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| 312 | tryInvert( Lc(P), M, inv, fail ); |
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| 313 | if(fail) |
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| 314 | return; |
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[ad8e1b] | 315 | result = inv*P; // monify result (not reduced, yet) |
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[5df7d0] | 316 | result= reduce (result, M); |
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[c99b6b] | 317 | return; |
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| 318 | } |
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[ad8e1b] | 319 | Variable x = P.mvar(); |
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[a8e8b9] | 320 | CanonicalForm rem, Q; |
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[c99b6b] | 321 | // here: degree(P) >= degree(result) |
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| 322 | while(true) |
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| 323 | { |
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[a8e8b9] | 324 | tryDivrem (P, result, Q, rem, inv, M, fail); |
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| 325 | if (fail) |
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[c99b6b] | 326 | return; |
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| 327 | if( rem.isZero() ) |
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| 328 | { |
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[ad8e1b] | 329 | result *= inv; |
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[5df7d0] | 330 | result= reduce (result, M); |
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[c99b6b] | 331 | return; |
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| 332 | } |
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[ad8e1b] | 333 | if(result.degree(x) >= rem.degree(x)) |
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| 334 | { |
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| 335 | P = result; |
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| 336 | result = rem; |
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| 337 | } |
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| 338 | else |
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| 339 | P = rem; |
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[359d742] | 340 | } |
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[c99b6b] | 341 | } |
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| 342 | |
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| 343 | bool hasFirstAlgVar( const CanonicalForm & f, Variable & a ) |
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| 344 | { |
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| 345 | if( f.inBaseDomain() ) // f has NO alg. variable |
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| 346 | return false; |
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| 347 | if( f.level()<0 ) // f has only alg. vars, so take the first one |
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| 348 | { |
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| 349 | a = f.mvar(); |
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| 350 | return true; |
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| 351 | } |
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| 352 | for(CFIterator i=f; i.hasTerms(); i++) |
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| 353 | if( hasFirstAlgVar( i.coeff(), a )) |
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| 354 | return true; // 'a' is already set |
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| 355 | return false; |
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| 356 | } |
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| 357 | |
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[ad8e1b] | 358 | CanonicalForm QGCD( const CanonicalForm & F, const CanonicalForm & G ); |
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| 359 | int * leadDeg(const CanonicalForm & f, int *degs); |
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| 360 | bool isLess(int *a, int *b, int lower, int upper); |
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| 361 | bool isEqual(int *a, int *b, int lower, int upper); |
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| 362 | CanonicalForm firstLC(const CanonicalForm & f); |
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| 363 | static CanonicalForm trycontent ( const CanonicalForm & f, const Variable & x, const CanonicalForm & M, bool & fail ); |
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| 364 | static CanonicalForm tryvcontent ( const CanonicalForm & f, const Variable & x, const CanonicalForm & M, bool & fail ); |
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| 365 | static CanonicalForm trycf_content ( const CanonicalForm & f, const CanonicalForm & g, const CanonicalForm & M, bool & fail ); |
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| 366 | static void tryDivide( const CanonicalForm & f, const CanonicalForm & g, const CanonicalForm & M, CanonicalForm & result, bool & divides, bool & fail ); |
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[359d742] | 367 | |
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[5df7d0] | 368 | static inline CanonicalForm |
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| 369 | tryNewtonInterp (const CanonicalForm alpha, const CanonicalForm u, |
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| 370 | const CanonicalForm newtonPoly, const CanonicalForm oldInterPoly, |
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| 371 | const Variable & x, const CanonicalForm& M, bool& fail) |
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| 372 | { |
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| 373 | CanonicalForm interPoly; |
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| 374 | |
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| 375 | CanonicalForm inv; |
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| 376 | tryInvert (newtonPoly (alpha, x), M, inv, fail); |
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| 377 | if (fail) |
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| 378 | return 0; |
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| 379 | |
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| 380 | interPoly= oldInterPoly+reduce ((u - oldInterPoly (alpha, x))*inv*newtonPoly, M); |
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| 381 | return interPoly; |
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| 382 | } |
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[359d742] | 383 | |
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[fe2d4c] | 384 | void tryBrownGCD( const CanonicalForm & F, const CanonicalForm & G, const CanonicalForm & M, CanonicalForm & result, bool & fail, bool topLevel ) |
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[ad8e1b] | 385 | { // assume F,G are multivariate polys over Z/p(a) for big prime p, M "univariate" polynomial in an algebraic variable |
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| 386 | // M is assumed to be monic |
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[359d742] | 387 | if(F.isZero()) |
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| 388 | { |
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| 389 | if(G.isZero()) |
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| 390 | { |
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| 391 | result = G; // G is zero |
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| 392 | return; |
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| 393 | } |
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| 394 | if(G.inCoeffDomain()) |
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| 395 | { |
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| 396 | tryInvert(G,M,result,fail); |
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[ad8e1b] | 397 | if(fail) |
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| 398 | return; |
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| 399 | result = 1; |
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[359d742] | 400 | return; |
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| 401 | } |
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| 402 | // try to make G monic modulo M |
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| 403 | CanonicalForm inv; |
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| 404 | tryInvert(Lc(G),M,inv,fail); |
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| 405 | if(fail) |
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| 406 | return; |
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| 407 | result = inv*G; |
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[5df7d0] | 408 | result= reduce (result, M); |
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[359d742] | 409 | return; |
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| 410 | } |
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| 411 | if(G.isZero()) // F is non-zero |
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| 412 | { |
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| 413 | if(F.inCoeffDomain()) |
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| 414 | { |
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| 415 | tryInvert(F,M,result,fail); |
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[ad8e1b] | 416 | if(fail) |
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| 417 | return; |
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| 418 | result = 1; |
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[359d742] | 419 | return; |
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| 420 | } |
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| 421 | // try to make F monic modulo M |
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| 422 | CanonicalForm inv; |
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| 423 | tryInvert(Lc(F),M,inv,fail); |
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| 424 | if(fail) |
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| 425 | return; |
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| 426 | result = inv*F; |
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[5df7d0] | 427 | result= reduce (result, M); |
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[359d742] | 428 | return; |
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| 429 | } |
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[ad8e1b] | 430 | // here: F,G both nonzero |
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[359d742] | 431 | if(F.inCoeffDomain()) |
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| 432 | { |
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| 433 | tryInvert(F,M,result,fail); |
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[ad8e1b] | 434 | if(fail) |
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| 435 | return; |
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| 436 | result = 1; |
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[359d742] | 437 | return; |
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| 438 | } |
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| 439 | if(G.inCoeffDomain()) |
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| 440 | { |
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| 441 | tryInvert(G,M,result,fail); |
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[ad8e1b] | 442 | if(fail) |
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| 443 | return; |
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| 444 | result = 1; |
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[359d742] | 445 | return; |
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| 446 | } |
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| 447 | CFMap MM,NN; |
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[fe2d4c] | 448 | int lev= myCompress (F, G, MM, NN, topLevel); |
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| 449 | if (lev == 0) |
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| 450 | { |
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| 451 | result= 1; |
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| 452 | return; |
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| 453 | } |
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[359d742] | 454 | CanonicalForm f=MM(F); |
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| 455 | CanonicalForm g=MM(G); |
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[ad8e1b] | 456 | // here: f,g are compressed |
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[359d742] | 457 | // compute largest variable in f or g (least one is Variable(1)) |
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| 458 | int mv = f.level(); |
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| 459 | if(g.level() > mv) |
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| 460 | mv = g.level(); |
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| 461 | // here: mv is level of the largest variable in f, g |
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| 462 | if(mv == 1) // f,g univariate |
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| 463 | { |
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| 464 | tryEuclid(f,g,M,result,fail); |
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| 465 | if(fail) |
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| 466 | return; |
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[5df7d0] | 467 | result= NN (reduce (result, M)); // do not forget to map back |
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[359d742] | 468 | return; |
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| 469 | } |
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| 470 | // here: mv > 1 |
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[ad8e1b] | 471 | CanonicalForm cf = tryvcontent(f, Variable(2), M, fail); // cf is univariate poly in var(1) |
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| 472 | if(fail) |
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| 473 | return; |
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| 474 | CanonicalForm cg = tryvcontent(g, Variable(2), M, fail); |
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| 475 | if(fail) |
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| 476 | return; |
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[359d742] | 477 | CanonicalForm c; |
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| 478 | tryEuclid(cf,cg,M,c,fail); |
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| 479 | if(fail) |
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| 480 | return; |
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[ad8e1b] | 481 | bool divides = true; |
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| 482 | CanonicalForm tmp; |
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| 483 | // f /= cf |
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| 484 | tryDivide(f,cf,M,tmp,divides,fail); // 'divides' is not checked |
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| 485 | if(fail) |
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| 486 | return; |
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| 487 | f = tmp; |
---|
| 488 | // g /= cg |
---|
| 489 | tryDivide(g,cg,M,tmp,divides,fail); // 'divides' is not checked |
---|
| 490 | if(fail) |
---|
| 491 | return; |
---|
| 492 | g = tmp; |
---|
[359d742] | 493 | if(f.inCoeffDomain()) |
---|
| 494 | { |
---|
| 495 | tryInvert(f,M,result,fail); |
---|
| 496 | if(fail) |
---|
| 497 | return; |
---|
[ad8e1b] | 498 | result = NN(c); |
---|
[359d742] | 499 | return; |
---|
| 500 | } |
---|
| 501 | if(g.inCoeffDomain()) |
---|
| 502 | { |
---|
| 503 | tryInvert(g,M,result,fail); |
---|
| 504 | if(fail) |
---|
| 505 | return; |
---|
[ad8e1b] | 506 | result = NN(c); |
---|
[359d742] | 507 | return; |
---|
| 508 | } |
---|
| 509 | int *L = new int[mv+1]; // L is addressed by i from 2 to mv |
---|
| 510 | int *N = new int[mv+1]; |
---|
| 511 | for(int i=2; i<=mv; i++) |
---|
| 512 | L[i] = N[i] = 0; |
---|
| 513 | L = leadDeg(f, L); |
---|
| 514 | N = leadDeg(g, N); |
---|
| 515 | CanonicalForm gamma; |
---|
| 516 | tryEuclid( firstLC(f), firstLC(g), M, gamma, fail ); |
---|
| 517 | if(fail) |
---|
| 518 | return; |
---|
[ad8e1b] | 519 | for(int i=2; i<=mv; i++) // entries at i=0,1 not visited |
---|
[359d742] | 520 | if(N[i] < L[i]) |
---|
| 521 | L[i] = N[i]; |
---|
| 522 | // L is now upper bound for degrees of gcd |
---|
| 523 | int *dg_im = new int[mv+1]; // for the degree vector of the image we don't need any entry at i=1 |
---|
| 524 | for(int i=2; i<=mv; i++) |
---|
| 525 | dg_im[i] = 0; // initialize |
---|
| 526 | CanonicalForm gamma_image, m=1; |
---|
| 527 | CanonicalForm gm=0; |
---|
[5df7d0] | 528 | CanonicalForm g_image, alpha, gnew; |
---|
[359d742] | 529 | FFGenerator gen = FFGenerator(); |
---|
[6f08f3] | 530 | Variable x= Variable (1); |
---|
[359d742] | 531 | for(FFGenerator gen = FFGenerator(); gen.hasItems(); gen.next()) |
---|
| 532 | { |
---|
| 533 | alpha = gen.item(); |
---|
[6f08f3] | 534 | gamma_image = reduce(gamma(alpha, x),M); // plug in alpha for var(1) |
---|
[359d742] | 535 | if(gamma_image.isZero()) // skip lc-bad points var(1)-alpha |
---|
| 536 | continue; |
---|
[6f08f3] | 537 | tryBrownGCD( f(alpha, x), g(alpha, x), M, g_image, fail, false ); // recursive call with one var less |
---|
[359d742] | 538 | if(fail) |
---|
| 539 | return; |
---|
[ad8e1b] | 540 | g_image = reduce(g_image, M); |
---|
[359d742] | 541 | if(g_image.inCoeffDomain()) // early termination |
---|
| 542 | { |
---|
| 543 | tryInvert(g_image,M,result,fail); |
---|
| 544 | if(fail) |
---|
| 545 | return; |
---|
| 546 | result = NN(c); |
---|
| 547 | return; |
---|
| 548 | } |
---|
| 549 | for(int i=2; i<=mv; i++) |
---|
| 550 | dg_im[i] = 0; // reset (this is necessary, because some entries may not be updated by call to leadDeg) |
---|
| 551 | dg_im = leadDeg(g_image, dg_im); // dg_im cannot be NIL-pointer |
---|
| 552 | if(isEqual(dg_im, L, 2, mv)) |
---|
| 553 | { |
---|
[5df7d0] | 554 | CanonicalForm inv; |
---|
| 555 | tryInvert (firstLC (g_image), M, inv, fail); |
---|
| 556 | if (fail) |
---|
| 557 | return; |
---|
| 558 | g_image *= inv; |
---|
[359d742] | 559 | g_image *= gamma_image; // multiply by multiple of image lc(gcd) |
---|
[5df7d0] | 560 | g_image= reduce (g_image, M); |
---|
| 561 | gnew= tryNewtonInterp (alpha, g_image, m, gm, x, M, fail); |
---|
[359d742] | 562 | // gnew = gm mod m |
---|
| 563 | // gnew = g_image mod var(1)-alpha |
---|
| 564 | // mnew = m * (var(1)-alpha) |
---|
| 565 | if(fail) |
---|
| 566 | return; |
---|
[5df7d0] | 567 | m *= (x - alpha); |
---|
[359d742] | 568 | if(gnew == gm) // gnew did not change |
---|
| 569 | { |
---|
[ad8e1b] | 570 | cf = tryvcontent(gm, Variable(2), M, fail); |
---|
| 571 | if(fail) |
---|
| 572 | return; |
---|
| 573 | divides = true; |
---|
| 574 | tryDivide(gm,cf,M,g_image,divides,fail); // 'divides' is ignored |
---|
| 575 | if(fail) |
---|
| 576 | return; |
---|
| 577 | tryDivide(f,g_image,M,tmp,divides,fail); // trial division (f) |
---|
| 578 | if(fail) |
---|
[359d742] | 579 | return; |
---|
[ad8e1b] | 580 | if(divides) |
---|
| 581 | { |
---|
| 582 | tryDivide(g,g_image,M,tmp,divides,fail); // trial division (g) |
---|
| 583 | if(fail) |
---|
| 584 | return; |
---|
| 585 | if(divides) |
---|
| 586 | { |
---|
[5df7d0] | 587 | result = NN(reduce (c*g_image, M)); |
---|
[ad8e1b] | 588 | return; |
---|
| 589 | } |
---|
[359d742] | 590 | } |
---|
| 591 | } |
---|
| 592 | gm = gnew; |
---|
| 593 | continue; |
---|
| 594 | } |
---|
| 595 | |
---|
| 596 | if(isLess(L, dg_im, 2, mv)) // dg_im > L --> current point unlucky |
---|
| 597 | continue; |
---|
| 598 | |
---|
[ad8e1b] | 599 | // here: isLess(dg_im, L, 2, mv) --> all previous points were unlucky |
---|
| 600 | m = CanonicalForm(1); // reset |
---|
| 601 | gm = 0; // reset |
---|
| 602 | for(int i=2; i<=mv; i++) // tighten bound |
---|
| 603 | L[i] = dg_im[i]; |
---|
[359d742] | 604 | } |
---|
| 605 | // we are out of evaluation points |
---|
| 606 | fail = true; |
---|
| 607 | } |
---|
| 608 | |
---|
[ad8e1b] | 609 | CanonicalForm QGCD( const CanonicalForm & F, const CanonicalForm & G ) |
---|
| 610 | { // f,g in Q(a)[x1,...,xn] |
---|
| 611 | if(F.isZero()) |
---|
| 612 | { |
---|
| 613 | if(G.isZero()) |
---|
| 614 | return G; // G is zero |
---|
| 615 | if(G.inCoeffDomain()) |
---|
| 616 | return CanonicalForm(1); |
---|
| 617 | return G/Lc(G); // return monic G |
---|
| 618 | } |
---|
| 619 | if(G.isZero()) // F is non-zero |
---|
| 620 | { |
---|
| 621 | if(F.inCoeffDomain()) |
---|
| 622 | return CanonicalForm(1); |
---|
| 623 | return F/Lc(F); // return monic F |
---|
| 624 | } |
---|
| 625 | if(F.inCoeffDomain() || G.inCoeffDomain()) |
---|
| 626 | return CanonicalForm(1); |
---|
| 627 | // here: both NOT inCoeffDomain |
---|
| 628 | CanonicalForm f, g, tmp, M, q, D, Dp, cl, newq, mipo; |
---|
| 629 | int p, i; |
---|
| 630 | int *bound, *other; // degree vectors |
---|
| 631 | bool fail; |
---|
[713bdb] | 632 | bool off_rational=!isOn(SW_RATIONAL); |
---|
[ad8e1b] | 633 | On( SW_RATIONAL ); // needed by bCommonDen |
---|
| 634 | f = F * bCommonDen(F); |
---|
| 635 | g = G * bCommonDen(G); |
---|
| 636 | Variable a, b; |
---|
| 637 | if(hasFirstAlgVar(f,a)) |
---|
| 638 | { |
---|
| 639 | if(hasFirstAlgVar(g,b)) |
---|
| 640 | { |
---|
| 641 | if(b.level() > a.level()) |
---|
| 642 | a = b; |
---|
| 643 | } |
---|
| 644 | } |
---|
| 645 | else |
---|
| 646 | { |
---|
| 647 | if(!hasFirstAlgVar(g,a))// both not in extension |
---|
| 648 | { |
---|
| 649 | Off( SW_RATIONAL ); |
---|
| 650 | Off( SW_USE_QGCD ); |
---|
| 651 | tmp = gcd( F, G ); |
---|
| 652 | On( SW_USE_QGCD ); |
---|
[713bdb] | 653 | if (off_rational) Off(SW_RATIONAL); |
---|
[ad8e1b] | 654 | return tmp; |
---|
| 655 | } |
---|
| 656 | } |
---|
| 657 | // here: a is the biggest alg. var in f and g AND some of f,g is in extension |
---|
| 658 | // (in the sequel b is used to swap alg/poly vars) |
---|
| 659 | setReduce(a,false); // do not reduce expressions modulo mipo |
---|
| 660 | tmp = getMipo(a); |
---|
| 661 | M = tmp * bCommonDen(tmp); |
---|
| 662 | // here: f, g in Z[a][x1,...,xn], M in Z[a] not necessarily monic |
---|
| 663 | Off( SW_RATIONAL ); // needed by mod |
---|
| 664 | // calculate upper bound for degree vector of gcd |
---|
| 665 | int mv = f.level(); i = g.level(); |
---|
| 666 | if(i > mv) |
---|
| 667 | mv = i; |
---|
| 668 | // here: mv is level of the largest variable in f, g |
---|
| 669 | b = Variable(mv+1); |
---|
| 670 | bound = new int[mv+1]; // 'bound' could be indexed from 0 to mv, but we will only use from 1 to mv |
---|
| 671 | other = new int[mv+1]; |
---|
| 672 | for(int i=1; i<=mv; i++) // initialize 'bound', 'other' with zeros |
---|
| 673 | bound[i] = other[i] = 0; |
---|
| 674 | bound = leadDeg(f,bound); // 'bound' is set the leading degree vector of f |
---|
| 675 | other = leadDeg(g,other); |
---|
| 676 | for(int i=1; i<=mv; i++) // entry at i=0 not visited |
---|
| 677 | if(other[i] < bound[i]) |
---|
| 678 | bound[i] = other[i]; |
---|
| 679 | // now 'bound' is the smaller vector |
---|
| 680 | cl = lc(M) * lc(f) * lc(g); |
---|
| 681 | q = 1; |
---|
| 682 | D = 0; |
---|
[fe2d4c] | 683 | CanonicalForm test= 0; |
---|
| 684 | bool equal= false; |
---|
[ad8e1b] | 685 | for( i=cf_getNumBigPrimes()-1; i>-1; i-- ) |
---|
| 686 | { |
---|
| 687 | p = cf_getBigPrime(i); |
---|
| 688 | if( mod( cl, p ).isZero() ) // skip lc-bad primes |
---|
| 689 | continue; |
---|
| 690 | fail = false; |
---|
| 691 | setCharacteristic(p); |
---|
| 692 | mipo = mapinto(M); |
---|
| 693 | mipo /= mipo.lc(); |
---|
| 694 | // here: mipo is monic |
---|
| 695 | tryBrownGCD( mapinto(f), mapinto(g), mipo, Dp, fail ); |
---|
| 696 | setCharacteristic(0); |
---|
| 697 | if( fail ) // mipo splits in char p |
---|
| 698 | continue; |
---|
| 699 | if( Dp.inCoeffDomain() ) // early termination |
---|
| 700 | { |
---|
| 701 | tryInvert(Dp,mipo,tmp,fail); // check if zero divisor |
---|
| 702 | if(fail) |
---|
| 703 | continue; |
---|
| 704 | setReduce(a,true); |
---|
[713bdb] | 705 | if (off_rational) Off(SW_RATIONAL); else On(SW_RATIONAL); |
---|
[ad8e1b] | 706 | return CanonicalForm(1); |
---|
| 707 | } |
---|
| 708 | // here: Dp NOT inCoeffDomain |
---|
| 709 | for(int i=1; i<=mv; i++) |
---|
| 710 | other[i] = 0; // reset (this is necessary, because some entries may not be updated by call to leadDeg) |
---|
| 711 | other = leadDeg(Dp,other); |
---|
[806c18] | 712 | |
---|
[ad8e1b] | 713 | if(isEqual(bound, other, 1, mv)) // equal |
---|
| 714 | { |
---|
| 715 | chineseRemainder( D, q, replacevar( mapinto(Dp), a, b ), p, tmp, newq ); |
---|
| 716 | // tmp = Dp mod p |
---|
| 717 | // tmp = D mod q |
---|
| 718 | // newq = p*q |
---|
| 719 | q = newq; |
---|
| 720 | if( D != tmp ) |
---|
| 721 | D = tmp; |
---|
| 722 | On( SW_RATIONAL ); |
---|
| 723 | tmp = replacevar( Farey( D, q ), b, a ); // Farey and switch back to alg var |
---|
| 724 | setReduce(a,true); // reduce expressions modulo mipo |
---|
| 725 | On( SW_RATIONAL ); // needed by fdivides |
---|
[fe2d4c] | 726 | if (test != tmp) |
---|
| 727 | test= tmp; |
---|
| 728 | else |
---|
| 729 | equal= true; // modular image did not add any new information |
---|
| 730 | if(equal && fdivides( tmp, f ) && fdivides( tmp, g )) // trial division |
---|
[ad8e1b] | 731 | { |
---|
| 732 | Off( SW_RATIONAL ); |
---|
| 733 | setReduce(a,true); |
---|
[713bdb] | 734 | if (off_rational) Off(SW_RATIONAL); else On(SW_RATIONAL); |
---|
[ad8e1b] | 735 | return tmp; |
---|
| 736 | } |
---|
| 737 | Off( SW_RATIONAL ); |
---|
| 738 | setReduce(a,false); // do not reduce expressions modulo mipo |
---|
| 739 | continue; |
---|
| 740 | } |
---|
| 741 | if( isLess(bound, other, 1, mv) ) // current prime unlucky |
---|
| 742 | continue; |
---|
| 743 | // here: isLess(other, bound, 1, mv) ) ==> all previous primes unlucky |
---|
| 744 | q = p; |
---|
| 745 | D = replacevar( mapinto(Dp), a, b ); // shortcut CRA // shortcut CRA |
---|
| 746 | for(int i=1; i<=mv; i++) // tighten bound |
---|
| 747 | bound[i] = other[i]; |
---|
| 748 | } |
---|
| 749 | // hopefully, we never reach this point |
---|
| 750 | setReduce(a,true); |
---|
| 751 | Off( SW_USE_QGCD ); |
---|
| 752 | D = gcd( f, g ); |
---|
| 753 | On( SW_USE_QGCD ); |
---|
[713bdb] | 754 | if (off_rational) Off(SW_RATIONAL); else On(SW_RATIONAL); |
---|
[ad8e1b] | 755 | return D; |
---|
| 756 | } |
---|
| 757 | |
---|
| 758 | |
---|
| 759 | int * leadDeg(const CanonicalForm & f, int *degs) |
---|
| 760 | { // leading degree vector w.r.t. lex. monomial order x(i+1) > x(i) |
---|
| 761 | // if f is in a coeff domain, the zero pointer is returned |
---|
| 762 | // 'a' should point to an array of sufficient size level(f)+1 |
---|
| 763 | if(f.inCoeffDomain()) |
---|
| 764 | return 0; |
---|
| 765 | CanonicalForm tmp = f; |
---|
| 766 | do |
---|
| 767 | { |
---|
| 768 | degs[tmp.level()] = tmp.degree(); |
---|
| 769 | tmp = LC(tmp); |
---|
| 770 | } |
---|
| 771 | while(!tmp.inCoeffDomain()); |
---|
| 772 | return degs; |
---|
| 773 | } |
---|
| 774 | |
---|
| 775 | |
---|
| 776 | bool isLess(int *a, int *b, int lower, int upper) |
---|
| 777 | { // compares the degree vectors a,b on the specified part. Note: x(i+1) > x(i) |
---|
| 778 | for(int i=upper; i>=lower; i--) |
---|
| 779 | if(a[i] == b[i]) |
---|
| 780 | continue; |
---|
| 781 | else |
---|
| 782 | return a[i] < b[i]; |
---|
| 783 | return true; |
---|
| 784 | } |
---|
| 785 | |
---|
| 786 | |
---|
| 787 | bool isEqual(int *a, int *b, int lower, int upper) |
---|
| 788 | { // compares the degree vectors a,b on the specified part. Note: x(i+1) > x(i) |
---|
| 789 | for(int i=lower; i<=upper; i++) |
---|
| 790 | if(a[i] != b[i]) |
---|
| 791 | return false; |
---|
| 792 | return true; |
---|
| 793 | } |
---|
| 794 | |
---|
| 795 | |
---|
| 796 | CanonicalForm firstLC(const CanonicalForm & f) |
---|
| 797 | { // returns the leading coefficient (LC) of level <= 1 |
---|
| 798 | CanonicalForm ret = f; |
---|
| 799 | while(ret.level() > 1) |
---|
| 800 | ret = LC(ret); |
---|
| 801 | return ret; |
---|
| 802 | } |
---|
| 803 | |
---|
| 804 | void tryExtgcd( const CanonicalForm & F, const CanonicalForm & G, const CanonicalForm & M, CanonicalForm & result, CanonicalForm & s, CanonicalForm & t, bool & fail ) |
---|
| 805 | { // F, G are univariate polynomials (i.e. they have exactly one polynomial variable) |
---|
| 806 | // F and G must have the same level AND level > 0 |
---|
| 807 | // we try to calculate gcd(F,G) = s*F + t*G |
---|
| 808 | // if a zero divisor is encontered, 'fail' is set to one |
---|
| 809 | // M is assumed to be monic |
---|
| 810 | CanonicalForm P; |
---|
| 811 | if(F.inCoeffDomain()) |
---|
| 812 | { |
---|
| 813 | tryInvert( F, M, P, fail ); |
---|
| 814 | if(fail) |
---|
| 815 | return; |
---|
| 816 | result = 1; |
---|
| 817 | s = P; t = 0; |
---|
| 818 | return; |
---|
| 819 | } |
---|
| 820 | if(G.inCoeffDomain()) |
---|
| 821 | { |
---|
| 822 | tryInvert( G, M, P, fail ); |
---|
| 823 | if(fail) |
---|
| 824 | return; |
---|
| 825 | result = 1; |
---|
| 826 | s = 0; t = P; |
---|
| 827 | return; |
---|
| 828 | } |
---|
| 829 | // here: both not inCoeffDomain |
---|
| 830 | CanonicalForm inv, rem, tmp, u, v, q, sum=0; |
---|
| 831 | if( F.degree() > G.degree() ) |
---|
| 832 | { |
---|
| 833 | P = F; result = G; s=v=0; t=u=1; |
---|
| 834 | } |
---|
| 835 | else |
---|
| 836 | { |
---|
| 837 | P = G; result = F; s=v=1; t=u=0; |
---|
| 838 | } |
---|
| 839 | Variable x = P.mvar(); |
---|
| 840 | // here: degree(P) >= degree(result) |
---|
| 841 | while(true) |
---|
| 842 | { |
---|
[fe2d4c] | 843 | tryDivrem (P, result, q, rem, inv, M, fail); |
---|
[ad8e1b] | 844 | if(fail) |
---|
| 845 | return; |
---|
| 846 | if( rem.isZero() ) |
---|
| 847 | { |
---|
| 848 | s*=inv; |
---|
| 849 | t*=inv; |
---|
| 850 | result *= inv; // monify result |
---|
| 851 | return; |
---|
| 852 | } |
---|
| 853 | sum += q; |
---|
| 854 | if(result.degree(x) >= rem.degree(x)) |
---|
| 855 | { |
---|
| 856 | P=result; |
---|
| 857 | result=rem; |
---|
| 858 | tmp=u-sum*s; |
---|
| 859 | u=s; |
---|
| 860 | s=tmp; |
---|
| 861 | tmp=v-sum*t; |
---|
| 862 | v=t; |
---|
| 863 | t=tmp; |
---|
| 864 | sum = 0; // reset |
---|
| 865 | } |
---|
| 866 | else |
---|
| 867 | P = rem; |
---|
| 868 | } |
---|
| 869 | } |
---|
| 870 | |
---|
| 871 | |
---|
| 872 | static CanonicalForm trycontent ( const CanonicalForm & f, const Variable & x, const CanonicalForm & M, bool & fail ) |
---|
| 873 | { // as 'content', but takes care of zero divisors |
---|
| 874 | ASSERT( x.level() > 0, "cannot calculate content with respect to algebraic variable" ); |
---|
| 875 | Variable y = f.mvar(); |
---|
| 876 | if ( y == x ) |
---|
| 877 | return trycf_content( f, 0, M, fail ); |
---|
| 878 | if ( y < x ) |
---|
| 879 | return f; |
---|
| 880 | return swapvar( trycontent( swapvar( f, y, x ), y, M, fail ), y, x ); |
---|
| 881 | } |
---|
| 882 | |
---|
| 883 | |
---|
| 884 | static CanonicalForm tryvcontent ( const CanonicalForm & f, const Variable & x, const CanonicalForm & M, bool & fail ) |
---|
| 885 | { // as vcontent, but takes care of zero divisors |
---|
| 886 | ASSERT( x.level() > 0, "cannot calculate vcontent with respect to algebraic variable" ); |
---|
| 887 | if ( f.mvar() <= x ) |
---|
| 888 | return trycontent( f, x, M, fail ); |
---|
| 889 | CFIterator i; |
---|
| 890 | CanonicalForm d = 0, e, ret; |
---|
| 891 | for ( i = f; i.hasTerms() && ! d.isOne() && ! fail; i++ ) |
---|
| 892 | { |
---|
| 893 | e = tryvcontent( i.coeff(), x, M, fail ); |
---|
| 894 | if(fail) |
---|
| 895 | break; |
---|
| 896 | tryBrownGCD( d, e, M, ret, fail ); |
---|
| 897 | d = ret; |
---|
| 898 | } |
---|
| 899 | return d; |
---|
| 900 | } |
---|
| 901 | |
---|
| 902 | |
---|
| 903 | static CanonicalForm trycf_content ( const CanonicalForm & f, const CanonicalForm & g, const CanonicalForm & M, bool & fail ) |
---|
| 904 | { // as cf_content, but takes care of zero divisors |
---|
| 905 | if ( f.inPolyDomain() || ( f.inExtension() && ! getReduce( f.mvar() ) ) ) |
---|
| 906 | { |
---|
| 907 | CFIterator i = f; |
---|
| 908 | CanonicalForm tmp = g, result; |
---|
| 909 | while ( i.hasTerms() && ! tmp.isOne() && ! fail ) |
---|
| 910 | { |
---|
| 911 | tryBrownGCD( i.coeff(), tmp, M, result, fail ); |
---|
| 912 | tmp = result; |
---|
| 913 | i++; |
---|
| 914 | } |
---|
| 915 | return result; |
---|
| 916 | } |
---|
| 917 | return abs( f ); |
---|
| 918 | } |
---|
| 919 | |
---|
| 920 | |
---|
| 921 | static void tryDivide( const CanonicalForm & f, const CanonicalForm & g, const CanonicalForm & M, CanonicalForm & result, bool & divides, bool & fail ) |
---|
| 922 | { // M "univariate" monic polynomial |
---|
| 923 | // f, g polynomials with coeffs modulo M. |
---|
| 924 | // if f is divisible by g, 'divides' is set to 1 and 'result' == f/g mod M coefficientwise. |
---|
| 925 | // 'fail' is set to 1, iff a zero divisor is encountered. |
---|
| 926 | // divides==1 implies fail==0 |
---|
| 927 | // required: getReduce(M.mvar())==0 |
---|
| 928 | if(g.inBaseDomain()) |
---|
| 929 | { |
---|
| 930 | result = f/g; |
---|
| 931 | divides = true; |
---|
| 932 | return; |
---|
| 933 | } |
---|
| 934 | if(g.inCoeffDomain()) |
---|
| 935 | { |
---|
| 936 | tryInvert(g,M,result,fail); |
---|
| 937 | if(fail) |
---|
| 938 | return; |
---|
| 939 | result = reduce(f*result, M); |
---|
| 940 | divides = true; |
---|
| 941 | return; |
---|
| 942 | } |
---|
| 943 | // here: g NOT inCoeffDomain |
---|
| 944 | Variable x = g.mvar(); |
---|
| 945 | if(f.degree(x) < g.degree(x)) |
---|
| 946 | { |
---|
| 947 | divides = false; |
---|
| 948 | return; |
---|
| 949 | } |
---|
| 950 | // here: f.degree(x) > 0 and f.degree(x) >= g.degree(x) |
---|
| 951 | CanonicalForm F = f; |
---|
| 952 | CanonicalForm q, leadG = LC(g); |
---|
| 953 | result = 0; |
---|
| 954 | while(!F.isZero()) |
---|
| 955 | { |
---|
| 956 | tryDivide(F.LC(x),leadG,M,q,divides,fail); |
---|
| 957 | if(fail || !divides) |
---|
| 958 | return; |
---|
| 959 | if(F.degree(x)<g.degree(x)) |
---|
| 960 | { |
---|
| 961 | divides = false; |
---|
| 962 | return; |
---|
| 963 | } |
---|
| 964 | q *= power(x,F.degree(x)-g.degree(x)); |
---|
| 965 | result += q; |
---|
| 966 | F = reduce(F-q*g, M); |
---|
| 967 | } |
---|
| 968 | result = reduce(result, M); |
---|
| 969 | divides = true; |
---|
| 970 | } |
---|
| 971 | |
---|
[359d742] | 972 | void tryCRA( const CanonicalForm & x1, const CanonicalForm & q1, const CanonicalForm & x2, const CanonicalForm & q2, CanonicalForm & xnew, CanonicalForm & qnew, bool & fail ) |
---|
| 973 | { // polys of level <= 1 are considered coefficients. q1,q2 are assumed to be coprime |
---|
| 974 | // xnew = x1 mod q1 (coefficientwise in the above sense) |
---|
| 975 | // xnew = x2 mod q2 |
---|
| 976 | // qnew = q1*q2 |
---|
| 977 | CanonicalForm tmp; |
---|
| 978 | if(x1.level() <= 1 && x2.level() <= 1) // base case |
---|
| 979 | { |
---|
| 980 | tryExtgcd(q1,q2,tmp,xnew,qnew,fail); |
---|
| 981 | if(fail) |
---|
| 982 | return; |
---|
| 983 | xnew = x1 + (x2-x1) * xnew * q1; |
---|
| 984 | qnew = q1*q2; |
---|
| 985 | xnew = mod(xnew,qnew); |
---|
| 986 | return; |
---|
| 987 | } |
---|
| 988 | CanonicalForm tmp2; |
---|
| 989 | xnew = 0; |
---|
| 990 | qnew = q1 * q2; |
---|
| 991 | // here: x1.level() > 1 || x2.level() > 1 |
---|
| 992 | if(x1.level() > x2.level()) |
---|
| 993 | { |
---|
| 994 | for(CFIterator i=x1; i.hasTerms(); i++) |
---|
| 995 | { |
---|
| 996 | if(i.exp() == 0) // const. term |
---|
| 997 | { |
---|
| 998 | tryCRA(i.coeff(),q1,x2,q2,tmp,tmp2,fail); |
---|
| 999 | if(fail) |
---|
| 1000 | return; |
---|
| 1001 | xnew += tmp; |
---|
| 1002 | } |
---|
| 1003 | else |
---|
| 1004 | { |
---|
| 1005 | tryCRA(i.coeff(),q1,0,q2,tmp,tmp2,fail); |
---|
| 1006 | if(fail) |
---|
| 1007 | return; |
---|
| 1008 | xnew += tmp * power(x1.mvar(),i.exp()); |
---|
| 1009 | } |
---|
| 1010 | } |
---|
| 1011 | return; |
---|
| 1012 | } |
---|
| 1013 | // here: x1.level() <= x2.level() && ( x1.level() > 1 || x2.level() > 1 ) |
---|
| 1014 | if(x2.level() > x1.level()) |
---|
| 1015 | { |
---|
| 1016 | for(CFIterator j=x2; j.hasTerms(); j++) |
---|
| 1017 | { |
---|
| 1018 | if(j.exp() == 0) // const. term |
---|
| 1019 | { |
---|
| 1020 | tryCRA(x1,q1,j.coeff(),q2,tmp,tmp2,fail); |
---|
| 1021 | if(fail) |
---|
| 1022 | return; |
---|
| 1023 | xnew += tmp; |
---|
| 1024 | } |
---|
| 1025 | else |
---|
| 1026 | { |
---|
| 1027 | tryCRA(0,q1,j.coeff(),q2,tmp,tmp2,fail); |
---|
| 1028 | if(fail) |
---|
| 1029 | return; |
---|
| 1030 | xnew += tmp * power(x2.mvar(),j.exp()); |
---|
| 1031 | } |
---|
| 1032 | } |
---|
| 1033 | return; |
---|
| 1034 | } |
---|
| 1035 | // here: x1.level() == x2.level() && x1.level() > 1 && x2.level() > 1 |
---|
| 1036 | CFIterator i = x1; |
---|
| 1037 | CFIterator j = x2; |
---|
| 1038 | while(i.hasTerms() || j.hasTerms()) |
---|
| 1039 | { |
---|
| 1040 | if(i.hasTerms()) |
---|
| 1041 | { |
---|
| 1042 | if(j.hasTerms()) |
---|
| 1043 | { |
---|
| 1044 | if(i.exp() == j.exp()) |
---|
| 1045 | { |
---|
| 1046 | tryCRA(i.coeff(),q1,j.coeff(),q2,tmp,tmp2,fail); |
---|
| 1047 | if(fail) |
---|
| 1048 | return; |
---|
| 1049 | xnew += tmp * power(x1.mvar(),i.exp()); |
---|
| 1050 | i++; j++; |
---|
| 1051 | } |
---|
| 1052 | else |
---|
| 1053 | { |
---|
| 1054 | if(i.exp() < j.exp()) |
---|
| 1055 | { |
---|
| 1056 | tryCRA(i.coeff(),q1,0,q2,tmp,tmp2,fail); |
---|
| 1057 | if(fail) |
---|
| 1058 | return; |
---|
| 1059 | xnew += tmp * power(x1.mvar(),i.exp()); |
---|
| 1060 | i++; |
---|
| 1061 | } |
---|
| 1062 | else // i.exp() > j.exp() |
---|
| 1063 | { |
---|
| 1064 | tryCRA(0,q1,j.coeff(),q2,tmp,tmp2,fail); |
---|
| 1065 | if(fail) |
---|
| 1066 | return; |
---|
| 1067 | xnew += tmp * power(x1.mvar(),j.exp()); |
---|
| 1068 | j++; |
---|
| 1069 | } |
---|
| 1070 | } |
---|
| 1071 | } |
---|
| 1072 | else // j is out of terms |
---|
| 1073 | { |
---|
| 1074 | tryCRA(i.coeff(),q1,0,q2,tmp,tmp2,fail); |
---|
| 1075 | if(fail) |
---|
| 1076 | return; |
---|
| 1077 | xnew += tmp * power(x1.mvar(),i.exp()); |
---|
| 1078 | i++; |
---|
| 1079 | } |
---|
| 1080 | } |
---|
| 1081 | else // i is out of terms |
---|
| 1082 | { |
---|
| 1083 | tryCRA(0,q1,j.coeff(),q2,tmp,tmp2,fail); |
---|
| 1084 | if(fail) |
---|
| 1085 | return; |
---|
| 1086 | xnew += tmp * power(x1.mvar(),j.exp()); |
---|
| 1087 | j++; |
---|
| 1088 | } |
---|
| 1089 | } |
---|
| 1090 | } |
---|
| 1091 | |
---|
| 1092 | |
---|
| 1093 | void tryExtgcd( const CanonicalForm & F, const CanonicalForm & G, CanonicalForm & result, CanonicalForm & s, CanonicalForm & t, bool & fail ) |
---|
| 1094 | { |
---|
| 1095 | // F, G are univariate polynomials (i.e. they have exactly one polynomial variable) |
---|
| 1096 | // F and G must have the same level AND level > 0 |
---|
| 1097 | // we try to calculate gcd(f,g) = s*F + t*G |
---|
| 1098 | // if a zero divisor is encontered, 'fail' is set to one |
---|
| 1099 | Variable a, b; |
---|
| 1100 | if( !hasFirstAlgVar(F,a) && !hasFirstAlgVar(G,b) ) // note lazy evaluation |
---|
| 1101 | { |
---|
| 1102 | result = extgcd( F, G, s, t ); // no zero divisors possible |
---|
| 1103 | return; |
---|
| 1104 | } |
---|
| 1105 | if( b.level() > a.level() ) |
---|
| 1106 | a = b; |
---|
| 1107 | // here: a is the biggest alg. var in F and G |
---|
| 1108 | CanonicalForm M = getMipo(a); |
---|
| 1109 | CanonicalForm P; |
---|
| 1110 | if( degree(F) > degree(G) ) |
---|
| 1111 | { |
---|
| 1112 | P=F; result=G; s=0; t=1; |
---|
| 1113 | } |
---|
| 1114 | else |
---|
| 1115 | { |
---|
| 1116 | P=G; result=F; s=1; t=0; |
---|
| 1117 | } |
---|
| 1118 | CanonicalForm inv, rem, q, u, v; |
---|
| 1119 | // here: degree(P) >= degree(result) |
---|
| 1120 | while(true) |
---|
| 1121 | { |
---|
| 1122 | tryInvert( Lc(result), M, inv, fail ); |
---|
| 1123 | if(fail) |
---|
| 1124 | return; |
---|
| 1125 | // here: Lc(result) is invertible modulo M |
---|
| 1126 | q = Lc(P)*inv * power( P.mvar(), degree(P)-degree(result) ); |
---|
| 1127 | rem = P - q*result; |
---|
| 1128 | // here: s*F + t*G = result |
---|
| 1129 | if( rem.isZero() ) |
---|
| 1130 | { |
---|
| 1131 | s*=inv; |
---|
| 1132 | t*=inv; |
---|
| 1133 | result *= inv; // monify result |
---|
| 1134 | return; |
---|
| 1135 | } |
---|
| 1136 | P=result; |
---|
| 1137 | result=rem; |
---|
| 1138 | rem=u-q*s; |
---|
| 1139 | u=s; |
---|
| 1140 | s=rem; |
---|
| 1141 | rem=v-q*t; |
---|
| 1142 | v=t; |
---|
| 1143 | t=rem; |
---|
| 1144 | } |
---|
| 1145 | } |
---|