[0d5627] | 1 | #include "config.h" |
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[4dfcb1] | 2 | |
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[9c6887] | 3 | #ifndef NOSTREAMIO |
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[4dfcb1] | 4 | #ifdef HAVE_CSTDIO |
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| 5 | #include <cstdio> |
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| 6 | #else |
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[c99b6b] | 7 | #include <stdio.h> |
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[4dfcb1] | 8 | #endif |
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| 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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[517530] | 16 | #include "cf_assert.h" |
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[2a95b2] | 17 | #include "timing.h" |
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[517530] | 18 | |
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[fe2d4c] | 19 | #include "templates/ftmpl_functions.h" |
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[c99b6b] | 20 | #include "cf_defs.h" |
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| 21 | #include "canonicalform.h" |
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| 22 | #include "cf_iter.h" |
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| 23 | #include "cf_primes.h" |
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| 24 | #include "cf_algorithm.h" |
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| 25 | #include "algext.h" |
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[359d742] | 26 | #include "cf_map.h" |
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| 27 | #include "cf_generator.h" |
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[1e5c50] | 28 | #include "facMul.h" |
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[c99b6b] | 29 | |
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[2156ec] | 30 | #ifdef HAVE_NTL |
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| 31 | #include "NTLconvert.h" |
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| 32 | #endif |
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| 33 | |
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[4782bc] | 34 | #ifdef HAVE_FLINT |
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| 35 | #include "FLINTconvert.h" |
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| 36 | #endif |
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| 37 | |
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[2a95b2] | 38 | TIMING_DEFINE_PRINT(alg_content_p) |
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| 39 | TIMING_DEFINE_PRINT(alg_content) |
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| 40 | TIMING_DEFINE_PRINT(alg_compress) |
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| 41 | TIMING_DEFINE_PRINT(alg_termination) |
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| 42 | TIMING_DEFINE_PRINT(alg_termination_p) |
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| 43 | TIMING_DEFINE_PRINT(alg_reconstruction) |
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| 44 | TIMING_DEFINE_PRINT(alg_newton_p) |
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| 45 | TIMING_DEFINE_PRINT(alg_recursion_p) |
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| 46 | TIMING_DEFINE_PRINT(alg_gcd_p) |
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| 47 | TIMING_DEFINE_PRINT(alg_euclid_p) |
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| 48 | |
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[fe2d4c] | 49 | /// compressing two polynomials F and G, M is used for compressing, |
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| 50 | /// N to reverse the compression |
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| 51 | static |
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| 52 | int myCompress (const CanonicalForm& F, const CanonicalForm& G, CFMap & M, |
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| 53 | CFMap & N, bool topLevel) |
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| 54 | { |
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| 55 | int n= tmax (F.level(), G.level()); |
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| 56 | int * degsf= new int [n + 1]; |
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| 57 | int * degsg= new int [n + 1]; |
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| 58 | |
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| 59 | for (int i = 0; i <= n; i++) |
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| 60 | degsf[i]= degsg[i]= 0; |
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| 61 | |
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| 62 | degsf= degrees (F, degsf); |
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| 63 | degsg= degrees (G, degsg); |
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| 64 | |
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| 65 | int both_non_zero= 0; |
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| 66 | int f_zero= 0; |
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| 67 | int g_zero= 0; |
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| 68 | int both_zero= 0; |
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| 69 | |
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| 70 | if (topLevel) |
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| 71 | { |
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| 72 | for (int i= 1; i <= n; i++) |
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| 73 | { |
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| 74 | if (degsf[i] != 0 && degsg[i] != 0) |
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| 75 | { |
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| 76 | both_non_zero++; |
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| 77 | continue; |
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| 78 | } |
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| 79 | if (degsf[i] == 0 && degsg[i] != 0 && i <= G.level()) |
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| 80 | { |
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| 81 | f_zero++; |
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| 82 | continue; |
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| 83 | } |
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| 84 | if (degsg[i] == 0 && degsf[i] && i <= F.level()) |
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| 85 | { |
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| 86 | g_zero++; |
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| 87 | continue; |
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| 88 | } |
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| 89 | } |
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| 90 | |
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| 91 | if (both_non_zero == 0) |
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| 92 | { |
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| 93 | delete [] degsf; |
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| 94 | delete [] degsg; |
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| 95 | return 0; |
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| 96 | } |
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| 97 | |
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| 98 | // map Variables which do not occur in both polynomials to higher levels |
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| 99 | int k= 1; |
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| 100 | int l= 1; |
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| 101 | for (int i= 1; i <= n; i++) |
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| 102 | { |
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| 103 | if (degsf[i] != 0 && degsg[i] == 0 && i <= F.level()) |
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| 104 | { |
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| 105 | if (k + both_non_zero != i) |
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| 106 | { |
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| 107 | M.newpair (Variable (i), Variable (k + both_non_zero)); |
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| 108 | N.newpair (Variable (k + both_non_zero), Variable (i)); |
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| 109 | } |
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| 110 | k++; |
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| 111 | } |
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| 112 | if (degsf[i] == 0 && degsg[i] != 0 && i <= G.level()) |
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| 113 | { |
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| 114 | if (l + g_zero + both_non_zero != i) |
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| 115 | { |
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| 116 | M.newpair (Variable (i), Variable (l + g_zero + both_non_zero)); |
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| 117 | N.newpair (Variable (l + g_zero + both_non_zero), Variable (i)); |
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| 118 | } |
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| 119 | l++; |
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| 120 | } |
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| 121 | } |
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| 122 | |
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| 123 | // sort Variables x_{i} in increasing order of |
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| 124 | // min(deg_{x_{i}}(f),deg_{x_{i}}(g)) |
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| 125 | int m= tmax (F.level(), G.level()); |
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| 126 | int min_max_deg; |
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| 127 | k= both_non_zero; |
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| 128 | l= 0; |
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| 129 | int i= 1; |
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| 130 | while (k > 0) |
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| 131 | { |
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| 132 | if (degsf [i] != 0 && degsg [i] != 0) |
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| 133 | min_max_deg= tmax (degsf[i], degsg[i]); |
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| 134 | else |
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| 135 | min_max_deg= 0; |
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| 136 | while (min_max_deg == 0) |
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| 137 | { |
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| 138 | i++; |
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| 139 | min_max_deg= tmax (degsf[i], degsg[i]); |
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| 140 | if (degsf [i] != 0 && degsg [i] != 0) |
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| 141 | min_max_deg= tmax (degsf[i], degsg[i]); |
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| 142 | else |
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| 143 | min_max_deg= 0; |
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| 144 | } |
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| 145 | for (int j= i + 1; j <= m; j++) |
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| 146 | { |
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| 147 | if (tmax (degsf[j],degsg[j]) <= min_max_deg && degsf[j] != 0 && degsg [j] != 0) |
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| 148 | { |
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| 149 | min_max_deg= tmax (degsf[j], degsg[j]); |
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| 150 | l= j; |
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| 151 | } |
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| 152 | } |
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| 153 | if (l != 0) |
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| 154 | { |
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| 155 | if (l != k) |
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| 156 | { |
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| 157 | M.newpair (Variable (l), Variable(k)); |
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| 158 | N.newpair (Variable (k), Variable(l)); |
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| 159 | degsf[l]= 0; |
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| 160 | degsg[l]= 0; |
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| 161 | l= 0; |
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| 162 | } |
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| 163 | else |
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| 164 | { |
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| 165 | degsf[l]= 0; |
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| 166 | degsg[l]= 0; |
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| 167 | l= 0; |
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| 168 | } |
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| 169 | } |
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| 170 | else if (l == 0) |
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| 171 | { |
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| 172 | if (i != k) |
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| 173 | { |
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| 174 | M.newpair (Variable (i), Variable (k)); |
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| 175 | N.newpair (Variable (k), Variable (i)); |
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| 176 | degsf[i]= 0; |
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| 177 | degsg[i]= 0; |
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| 178 | } |
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| 179 | else |
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| 180 | { |
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| 181 | degsf[i]= 0; |
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| 182 | degsg[i]= 0; |
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| 183 | } |
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| 184 | i++; |
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| 185 | } |
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| 186 | k--; |
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| 187 | } |
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| 188 | } |
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| 189 | else |
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| 190 | { |
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| 191 | //arrange Variables such that no gaps occur |
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| 192 | for (int i= 1; i <= n; i++) |
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| 193 | { |
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| 194 | if (degsf[i] == 0 && degsg[i] == 0) |
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| 195 | { |
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| 196 | both_zero++; |
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| 197 | continue; |
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| 198 | } |
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| 199 | else |
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| 200 | { |
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| 201 | if (both_zero != 0) |
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| 202 | { |
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| 203 | M.newpair (Variable (i), Variable (i - both_zero)); |
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| 204 | N.newpair (Variable (i - both_zero), Variable (i)); |
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| 205 | } |
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| 206 | } |
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| 207 | } |
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| 208 | } |
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| 209 | |
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| 210 | delete [] degsf; |
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| 211 | delete [] degsg; |
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| 212 | |
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| 213 | return 1; |
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| 214 | } |
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| 215 | |
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[ad8e1b] | 216 | void tryInvert( const CanonicalForm & F, const CanonicalForm & M, CanonicalForm & inv, bool & fail ) |
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| 217 | { // F, M are required to be "univariate" polynomials in an algebraic variable |
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| 218 | // we try to invert F modulo M |
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| 219 | if(F.inBaseDomain()) |
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| 220 | { |
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| 221 | if(F.isZero()) |
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| 222 | { |
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| 223 | fail = true; |
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| 224 | return; |
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| 225 | } |
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| 226 | inv = 1/F; |
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| 227 | return; |
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| 228 | } |
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| 229 | CanonicalForm b; |
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| 230 | Variable a = M.mvar(); |
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| 231 | Variable x = Variable(1); |
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| 232 | if(!extgcd( replacevar( F, a, x ), replacevar( M, a, x ), inv, b ).isOne()) |
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| 233 | fail = true; |
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| 234 | else |
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| 235 | inv = replacevar( inv, x, a ); // change back to alg var |
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| 236 | } |
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| 237 | |
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[a8e8b9] | 238 | void tryDivrem (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
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| 239 | CanonicalForm& R, CanonicalForm& inv, const CanonicalForm& mipo, |
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| 240 | bool& fail) |
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| 241 | { |
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| 242 | if (F.inCoeffDomain()) |
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| 243 | { |
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| 244 | Q= 0; |
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| 245 | R= F; |
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| 246 | return; |
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| 247 | } |
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| 248 | |
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| 249 | CanonicalForm A, B; |
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| 250 | Variable x= F.mvar(); |
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| 251 | A= F; |
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| 252 | B= G; |
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| 253 | int degA= degree (A, x); |
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| 254 | int degB= degree (B, x); |
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| 255 | |
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| 256 | if (degA < degB) |
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| 257 | { |
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| 258 | R= A; |
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| 259 | Q= 0; |
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| 260 | return; |
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| 261 | } |
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| 262 | |
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| 263 | tryInvert (Lc (B), mipo, inv, fail); |
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| 264 | if (fail) |
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| 265 | return; |
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| 266 | |
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| 267 | R= A; |
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| 268 | Q= 0; |
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| 269 | CanonicalForm Qi; |
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| 270 | for (int i= degA -degB; i >= 0; i--) |
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| 271 | { |
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| 272 | if (degree (R, x) == i + degB) |
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| 273 | { |
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| 274 | Qi= Lc (R)*inv*power (x, i); |
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| 275 | Qi= reduce (Qi, mipo); |
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| 276 | R -= Qi*B; |
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| 277 | R= reduce (R, mipo); |
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| 278 | Q += Qi; |
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| 279 | } |
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| 280 | } |
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| 281 | } |
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| 282 | |
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[ad8e1b] | 283 | void tryEuclid( const CanonicalForm & A, const CanonicalForm & B, const CanonicalForm & M, CanonicalForm & result, bool & fail ) |
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[c99b6b] | 284 | { |
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| 285 | CanonicalForm P; |
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[ad8e1b] | 286 | if(A.inCoeffDomain()) |
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| 287 | { |
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| 288 | tryInvert( A, M, P, fail ); |
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| 289 | if(fail) |
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| 290 | return; |
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| 291 | result = 1; |
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| 292 | return; |
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| 293 | } |
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| 294 | if(B.inCoeffDomain()) |
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| 295 | { |
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| 296 | tryInvert( B, M, P, fail ); |
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| 297 | if(fail) |
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| 298 | return; |
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| 299 | result = 1; |
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| 300 | return; |
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| 301 | } |
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| 302 | // here: both not inCoeffDomain |
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| 303 | if( A.degree() > B.degree() ) |
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[c99b6b] | 304 | { |
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| 305 | P = A; result = B; |
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| 306 | } |
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| 307 | else |
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| 308 | { |
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| 309 | P = B; result = A; |
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| 310 | } |
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| 311 | CanonicalForm inv; |
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| 312 | if( result.isZero() ) |
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| 313 | { |
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| 314 | tryInvert( Lc(P), M, inv, fail ); |
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| 315 | if(fail) |
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| 316 | return; |
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[ad8e1b] | 317 | result = inv*P; // monify result (not reduced, yet) |
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[5df7d0] | 318 | result= reduce (result, M); |
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[c99b6b] | 319 | return; |
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| 320 | } |
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[ad8e1b] | 321 | Variable x = P.mvar(); |
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[a8e8b9] | 322 | CanonicalForm rem, Q; |
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[c99b6b] | 323 | // here: degree(P) >= degree(result) |
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| 324 | while(true) |
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| 325 | { |
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[a8e8b9] | 326 | tryDivrem (P, result, Q, rem, inv, M, fail); |
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| 327 | if (fail) |
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[c99b6b] | 328 | return; |
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| 329 | if( rem.isZero() ) |
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| 330 | { |
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[ad8e1b] | 331 | result *= inv; |
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[5df7d0] | 332 | result= reduce (result, M); |
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[c99b6b] | 333 | return; |
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| 334 | } |
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[ad8e1b] | 335 | if(result.degree(x) >= rem.degree(x)) |
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| 336 | { |
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| 337 | P = result; |
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| 338 | result = rem; |
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| 339 | } |
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| 340 | else |
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| 341 | P = rem; |
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[359d742] | 342 | } |
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[c99b6b] | 343 | } |
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| 344 | |
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| 345 | bool hasFirstAlgVar( const CanonicalForm & f, Variable & a ) |
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| 346 | { |
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| 347 | if( f.inBaseDomain() ) // f has NO alg. variable |
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| 348 | return false; |
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| 349 | if( f.level()<0 ) // f has only alg. vars, so take the first one |
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| 350 | { |
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| 351 | a = f.mvar(); |
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| 352 | return true; |
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| 353 | } |
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| 354 | for(CFIterator i=f; i.hasTerms(); i++) |
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| 355 | if( hasFirstAlgVar( i.coeff(), a )) |
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| 356 | return true; // 'a' is already set |
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| 357 | return false; |
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| 358 | } |
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| 359 | |
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[ad8e1b] | 360 | CanonicalForm QGCD( const CanonicalForm & F, const CanonicalForm & G ); |
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| 361 | int * leadDeg(const CanonicalForm & f, int *degs); |
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| 362 | bool isLess(int *a, int *b, int lower, int upper); |
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| 363 | bool isEqual(int *a, int *b, int lower, int upper); |
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| 364 | CanonicalForm firstLC(const CanonicalForm & f); |
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| 365 | static CanonicalForm trycontent ( const CanonicalForm & f, const Variable & x, const CanonicalForm & M, bool & fail ); |
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| 366 | static CanonicalForm tryvcontent ( const CanonicalForm & f, const Variable & x, const CanonicalForm & M, bool & fail ); |
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| 367 | static CanonicalForm trycf_content ( const CanonicalForm & f, const CanonicalForm & g, const CanonicalForm & M, bool & fail ); |
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[359d742] | 368 | |
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[5df7d0] | 369 | static inline CanonicalForm |
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| 370 | tryNewtonInterp (const CanonicalForm alpha, const CanonicalForm u, |
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| 371 | const CanonicalForm newtonPoly, const CanonicalForm oldInterPoly, |
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| 372 | const Variable & x, const CanonicalForm& M, bool& fail) |
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| 373 | { |
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| 374 | CanonicalForm interPoly; |
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| 375 | |
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| 376 | CanonicalForm inv; |
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| 377 | tryInvert (newtonPoly (alpha, x), M, inv, fail); |
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| 378 | if (fail) |
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| 379 | return 0; |
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| 380 | |
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| 381 | interPoly= oldInterPoly+reduce ((u - oldInterPoly (alpha, x))*inv*newtonPoly, M); |
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| 382 | return interPoly; |
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| 383 | } |
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[359d742] | 384 | |
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[fe2d4c] | 385 | void tryBrownGCD( const CanonicalForm & F, const CanonicalForm & G, const CanonicalForm & M, CanonicalForm & result, bool & fail, bool topLevel ) |
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[ad8e1b] | 386 | { // 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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| 387 | // M is assumed to be monic |
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[359d742] | 388 | if(F.isZero()) |
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| 389 | { |
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| 390 | if(G.isZero()) |
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| 391 | { |
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| 392 | result = G; // G is zero |
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| 393 | return; |
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| 394 | } |
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| 395 | if(G.inCoeffDomain()) |
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| 396 | { |
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| 397 | tryInvert(G,M,result,fail); |
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[ad8e1b] | 398 | if(fail) |
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| 399 | return; |
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| 400 | result = 1; |
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[359d742] | 401 | return; |
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| 402 | } |
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| 403 | // try to make G monic modulo M |
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| 404 | CanonicalForm inv; |
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| 405 | tryInvert(Lc(G),M,inv,fail); |
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| 406 | if(fail) |
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| 407 | return; |
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| 408 | result = inv*G; |
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[5df7d0] | 409 | result= reduce (result, M); |
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[359d742] | 410 | return; |
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| 411 | } |
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| 412 | if(G.isZero()) // F is non-zero |
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| 413 | { |
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| 414 | if(F.inCoeffDomain()) |
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| 415 | { |
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| 416 | tryInvert(F,M,result,fail); |
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[ad8e1b] | 417 | if(fail) |
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| 418 | return; |
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| 419 | result = 1; |
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[359d742] | 420 | return; |
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| 421 | } |
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| 422 | // try to make F monic modulo M |
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| 423 | CanonicalForm inv; |
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| 424 | tryInvert(Lc(F),M,inv,fail); |
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| 425 | if(fail) |
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| 426 | return; |
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| 427 | result = inv*F; |
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[5df7d0] | 428 | result= reduce (result, M); |
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[359d742] | 429 | return; |
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| 430 | } |
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[ad8e1b] | 431 | // here: F,G both nonzero |
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[359d742] | 432 | if(F.inCoeffDomain()) |
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| 433 | { |
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| 434 | tryInvert(F,M,result,fail); |
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[ad8e1b] | 435 | if(fail) |
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| 436 | return; |
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| 437 | result = 1; |
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[359d742] | 438 | return; |
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| 439 | } |
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| 440 | if(G.inCoeffDomain()) |
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| 441 | { |
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| 442 | tryInvert(G,M,result,fail); |
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[ad8e1b] | 443 | if(fail) |
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| 444 | return; |
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| 445 | result = 1; |
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[359d742] | 446 | return; |
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| 447 | } |
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[2a95b2] | 448 | TIMING_START (alg_compress) |
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[359d742] | 449 | CFMap MM,NN; |
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[fe2d4c] | 450 | int lev= myCompress (F, G, MM, NN, topLevel); |
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| 451 | if (lev == 0) |
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| 452 | { |
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| 453 | result= 1; |
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| 454 | return; |
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| 455 | } |
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[359d742] | 456 | CanonicalForm f=MM(F); |
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| 457 | CanonicalForm g=MM(G); |
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[2a95b2] | 458 | TIMING_END_AND_PRINT (alg_compress, "time to compress in alg gcd: ") |
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[ad8e1b] | 459 | // here: f,g are compressed |
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[359d742] | 460 | // compute largest variable in f or g (least one is Variable(1)) |
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| 461 | int mv = f.level(); |
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| 462 | if(g.level() > mv) |
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| 463 | mv = g.level(); |
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| 464 | // here: mv is level of the largest variable in f, g |
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| 465 | if(mv == 1) // f,g univariate |
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| 466 | { |
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[2a95b2] | 467 | TIMING_START (alg_euclid_p) |
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[359d742] | 468 | tryEuclid(f,g,M,result,fail); |
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[2a95b2] | 469 | TIMING_END_AND_PRINT (alg_euclid_p, "time for euclidean alg mod p: ") |
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[359d742] | 470 | if(fail) |
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| 471 | return; |
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[5df7d0] | 472 | result= NN (reduce (result, M)); // do not forget to map back |
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[359d742] | 473 | return; |
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| 474 | } |
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[2a95b2] | 475 | TIMING_START (alg_content_p) |
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[359d742] | 476 | // here: mv > 1 |
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[ad8e1b] | 477 | CanonicalForm cf = tryvcontent(f, Variable(2), M, fail); // cf is univariate poly in var(1) |
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| 478 | if(fail) |
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| 479 | return; |
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| 480 | CanonicalForm cg = tryvcontent(g, Variable(2), M, fail); |
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| 481 | if(fail) |
---|
| 482 | return; |
---|
[359d742] | 483 | CanonicalForm c; |
---|
| 484 | tryEuclid(cf,cg,M,c,fail); |
---|
| 485 | if(fail) |
---|
| 486 | return; |
---|
[ad8e1b] | 487 | // f /= cf |
---|
[13f494] | 488 | f.tryDiv (cf, M, fail); |
---|
[ad8e1b] | 489 | if(fail) |
---|
| 490 | return; |
---|
| 491 | // g /= cg |
---|
[13f494] | 492 | g.tryDiv (cg, M, fail); |
---|
[ad8e1b] | 493 | if(fail) |
---|
| 494 | return; |
---|
[2a95b2] | 495 | TIMING_END_AND_PRINT (alg_content_p, "time for content in alg gcd mod p: ") |
---|
[359d742] | 496 | if(f.inCoeffDomain()) |
---|
| 497 | { |
---|
| 498 | tryInvert(f,M,result,fail); |
---|
| 499 | if(fail) |
---|
| 500 | return; |
---|
[ad8e1b] | 501 | result = NN(c); |
---|
[359d742] | 502 | return; |
---|
| 503 | } |
---|
| 504 | if(g.inCoeffDomain()) |
---|
| 505 | { |
---|
| 506 | tryInvert(g,M,result,fail); |
---|
| 507 | if(fail) |
---|
| 508 | return; |
---|
[ad8e1b] | 509 | result = NN(c); |
---|
[359d742] | 510 | return; |
---|
| 511 | } |
---|
| 512 | int *L = new int[mv+1]; // L is addressed by i from 2 to mv |
---|
| 513 | int *N = new int[mv+1]; |
---|
| 514 | for(int i=2; i<=mv; i++) |
---|
| 515 | L[i] = N[i] = 0; |
---|
| 516 | L = leadDeg(f, L); |
---|
| 517 | N = leadDeg(g, N); |
---|
| 518 | CanonicalForm gamma; |
---|
[2a95b2] | 519 | TIMING_START (alg_euclid_p) |
---|
[359d742] | 520 | tryEuclid( firstLC(f), firstLC(g), M, gamma, fail ); |
---|
[2a95b2] | 521 | TIMING_END_AND_PRINT (alg_euclid_p, "time for gcd of lcs in alg mod p: ") |
---|
[359d742] | 522 | if(fail) |
---|
| 523 | return; |
---|
[ad8e1b] | 524 | for(int i=2; i<=mv; i++) // entries at i=0,1 not visited |
---|
[359d742] | 525 | if(N[i] < L[i]) |
---|
| 526 | L[i] = N[i]; |
---|
| 527 | // L is now upper bound for degrees of gcd |
---|
| 528 | int *dg_im = new int[mv+1]; // for the degree vector of the image we don't need any entry at i=1 |
---|
| 529 | for(int i=2; i<=mv; i++) |
---|
| 530 | dg_im[i] = 0; // initialize |
---|
| 531 | CanonicalForm gamma_image, m=1; |
---|
| 532 | CanonicalForm gm=0; |
---|
[5df7d0] | 533 | CanonicalForm g_image, alpha, gnew; |
---|
[359d742] | 534 | FFGenerator gen = FFGenerator(); |
---|
[6f08f3] | 535 | Variable x= Variable (1); |
---|
[13f494] | 536 | bool divides= true; |
---|
[359d742] | 537 | for(FFGenerator gen = FFGenerator(); gen.hasItems(); gen.next()) |
---|
| 538 | { |
---|
| 539 | alpha = gen.item(); |
---|
[6f08f3] | 540 | gamma_image = reduce(gamma(alpha, x),M); // plug in alpha for var(1) |
---|
[359d742] | 541 | if(gamma_image.isZero()) // skip lc-bad points var(1)-alpha |
---|
| 542 | continue; |
---|
[2a95b2] | 543 | TIMING_START (alg_recursion_p) |
---|
[6f08f3] | 544 | tryBrownGCD( f(alpha, x), g(alpha, x), M, g_image, fail, false ); // recursive call with one var less |
---|
[2a95b2] | 545 | TIMING_END_AND_PRINT (alg_recursion_p, |
---|
| 546 | "time for recursive calls in alg gcd mod p: ") |
---|
[359d742] | 547 | if(fail) |
---|
| 548 | return; |
---|
[ad8e1b] | 549 | g_image = reduce(g_image, M); |
---|
[359d742] | 550 | if(g_image.inCoeffDomain()) // early termination |
---|
| 551 | { |
---|
| 552 | tryInvert(g_image,M,result,fail); |
---|
| 553 | if(fail) |
---|
| 554 | return; |
---|
| 555 | result = NN(c); |
---|
| 556 | return; |
---|
| 557 | } |
---|
| 558 | for(int i=2; i<=mv; i++) |
---|
| 559 | dg_im[i] = 0; // reset (this is necessary, because some entries may not be updated by call to leadDeg) |
---|
| 560 | dg_im = leadDeg(g_image, dg_im); // dg_im cannot be NIL-pointer |
---|
| 561 | if(isEqual(dg_im, L, 2, mv)) |
---|
| 562 | { |
---|
[5df7d0] | 563 | CanonicalForm inv; |
---|
| 564 | tryInvert (firstLC (g_image), M, inv, fail); |
---|
| 565 | if (fail) |
---|
| 566 | return; |
---|
| 567 | g_image *= inv; |
---|
[359d742] | 568 | g_image *= gamma_image; // multiply by multiple of image lc(gcd) |
---|
[5df7d0] | 569 | g_image= reduce (g_image, M); |
---|
[2a95b2] | 570 | TIMING_START (alg_newton_p) |
---|
[5df7d0] | 571 | gnew= tryNewtonInterp (alpha, g_image, m, gm, x, M, fail); |
---|
[2a95b2] | 572 | TIMING_END_AND_PRINT (alg_newton_p, |
---|
| 573 | "time for Newton interpolation in alg gcd mod p: ") |
---|
[359d742] | 574 | // gnew = gm mod m |
---|
| 575 | // gnew = g_image mod var(1)-alpha |
---|
| 576 | // mnew = m * (var(1)-alpha) |
---|
| 577 | if(fail) |
---|
| 578 | return; |
---|
[5df7d0] | 579 | m *= (x - alpha); |
---|
[6bbe94] | 580 | if((firstLC(gnew) == gamma) || (gnew == gm)) // gnew did not change |
---|
[359d742] | 581 | { |
---|
[2a95b2] | 582 | TIMING_START (alg_termination_p) |
---|
[6bbe94] | 583 | cf = tryvcontent(gnew, Variable(2), M, fail); |
---|
[ad8e1b] | 584 | if(fail) |
---|
| 585 | return; |
---|
| 586 | divides = true; |
---|
[6bbe94] | 587 | g_image= gnew; |
---|
[13f494] | 588 | g_image.tryDiv (cf, M, fail); |
---|
[ad8e1b] | 589 | if(fail) |
---|
| 590 | return; |
---|
[13f494] | 591 | divides= tryFdivides (g_image,f, M, fail); // trial division (f) |
---|
[ad8e1b] | 592 | if(fail) |
---|
[359d742] | 593 | return; |
---|
[ad8e1b] | 594 | if(divides) |
---|
| 595 | { |
---|
[13f494] | 596 | bool divides2= tryFdivides (g_image,g, M, fail); // trial division (g) |
---|
[ad8e1b] | 597 | if(fail) |
---|
| 598 | return; |
---|
[13f494] | 599 | if(divides2) |
---|
[ad8e1b] | 600 | { |
---|
[5df7d0] | 601 | result = NN(reduce (c*g_image, M)); |
---|
[2a95b2] | 602 | TIMING_END_AND_PRINT (alg_termination_p, |
---|
| 603 | "time for successful termination test in alg gcd mod p: ") |
---|
[ad8e1b] | 604 | return; |
---|
| 605 | } |
---|
[359d742] | 606 | } |
---|
[2a95b2] | 607 | TIMING_END_AND_PRINT (alg_termination_p, |
---|
| 608 | "time for unsuccessful termination test in alg gcd mod p: ") |
---|
[359d742] | 609 | } |
---|
| 610 | gm = gnew; |
---|
| 611 | continue; |
---|
| 612 | } |
---|
| 613 | |
---|
| 614 | if(isLess(L, dg_im, 2, mv)) // dg_im > L --> current point unlucky |
---|
| 615 | continue; |
---|
| 616 | |
---|
[ad8e1b] | 617 | // here: isLess(dg_im, L, 2, mv) --> all previous points were unlucky |
---|
| 618 | m = CanonicalForm(1); // reset |
---|
| 619 | gm = 0; // reset |
---|
| 620 | for(int i=2; i<=mv; i++) // tighten bound |
---|
| 621 | L[i] = dg_im[i]; |
---|
[359d742] | 622 | } |
---|
| 623 | // we are out of evaluation points |
---|
| 624 | fail = true; |
---|
| 625 | } |
---|
| 626 | |
---|
[2156ec] | 627 | static CanonicalForm |
---|
| 628 | myicontent ( const CanonicalForm & f, const CanonicalForm & c ) |
---|
| 629 | { |
---|
[517530] | 630 | #ifdef HAVE_NTL |
---|
[2156ec] | 631 | if (f.isOne() || c.isOne()) |
---|
| 632 | return 1; |
---|
| 633 | if ( f.inBaseDomain() && c.inBaseDomain()) |
---|
| 634 | { |
---|
| 635 | if (c.isZero()) return abs(f); |
---|
| 636 | return bgcd( f, c ); |
---|
| 637 | } |
---|
| 638 | else if ( (f.inCoeffDomain() && c.inCoeffDomain()) || |
---|
| 639 | (f.inCoeffDomain() && c.inBaseDomain()) || |
---|
| 640 | (f.inBaseDomain() && c.inCoeffDomain())) |
---|
| 641 | { |
---|
| 642 | if (c.isZero()) return abs (f); |
---|
[4782bc] | 643 | #ifdef HAVE_FLINT |
---|
| 644 | fmpz_poly_t FLINTf, FLINTc; |
---|
| 645 | convertFacCF2Fmpz_poly_t (FLINTf, f); |
---|
| 646 | convertFacCF2Fmpz_poly_t (FLINTc, c); |
---|
| 647 | fmpz_poly_gcd (FLINTc, FLINTc, FLINTf); |
---|
| 648 | CanonicalForm result; |
---|
| 649 | if (f.inCoeffDomain()) |
---|
| 650 | result= convertFmpz_poly_t2FacCF (FLINTc, f.mvar()); |
---|
| 651 | else |
---|
| 652 | result= convertFmpz_poly_t2FacCF (FLINTc, c.mvar()); |
---|
| 653 | fmpz_poly_clear (FLINTc); |
---|
| 654 | fmpz_poly_clear (FLINTf); |
---|
| 655 | return result; |
---|
| 656 | #else |
---|
[2156ec] | 657 | ZZX NTLf= convertFacCF2NTLZZX (f); |
---|
| 658 | ZZX NTLc= convertFacCF2NTLZZX (c); |
---|
| 659 | NTLc= GCD (NTLc, NTLf); |
---|
| 660 | if (f.inCoeffDomain()) |
---|
| 661 | return convertNTLZZX2CF(NTLc,f.mvar()); |
---|
| 662 | else |
---|
| 663 | return convertNTLZZX2CF(NTLc,c.mvar()); |
---|
[4782bc] | 664 | #endif |
---|
[2156ec] | 665 | } |
---|
| 666 | else |
---|
| 667 | { |
---|
| 668 | CanonicalForm g = c; |
---|
| 669 | for ( CFIterator i = f; i.hasTerms() && ! g.isOne(); i++ ) |
---|
| 670 | g = myicontent( i.coeff(), g ); |
---|
| 671 | return g; |
---|
| 672 | } |
---|
[517530] | 673 | #else |
---|
| 674 | return 1; |
---|
[2156ec] | 675 | #endif |
---|
[517530] | 676 | } |
---|
[2156ec] | 677 | |
---|
| 678 | CanonicalForm |
---|
| 679 | myicontent ( const CanonicalForm & f ) |
---|
| 680 | { |
---|
| 681 | #ifdef HAVE_NTL |
---|
| 682 | return myicontent( f, 0 ); |
---|
| 683 | #else |
---|
| 684 | return 1; |
---|
| 685 | #endif |
---|
| 686 | } |
---|
| 687 | |
---|
[ad8e1b] | 688 | CanonicalForm QGCD( const CanonicalForm & F, const CanonicalForm & G ) |
---|
| 689 | { // f,g in Q(a)[x1,...,xn] |
---|
| 690 | if(F.isZero()) |
---|
| 691 | { |
---|
| 692 | if(G.isZero()) |
---|
| 693 | return G; // G is zero |
---|
| 694 | if(G.inCoeffDomain()) |
---|
| 695 | return CanonicalForm(1); |
---|
[1682691] | 696 | CanonicalForm lcinv= 1/Lc (G); |
---|
| 697 | return G*lcinv; // return monic G |
---|
[ad8e1b] | 698 | } |
---|
| 699 | if(G.isZero()) // F is non-zero |
---|
| 700 | { |
---|
| 701 | if(F.inCoeffDomain()) |
---|
| 702 | return CanonicalForm(1); |
---|
[1682691] | 703 | CanonicalForm lcinv= 1/Lc (F); |
---|
| 704 | return F*lcinv; // return monic F |
---|
[ad8e1b] | 705 | } |
---|
| 706 | if(F.inCoeffDomain() || G.inCoeffDomain()) |
---|
| 707 | return CanonicalForm(1); |
---|
| 708 | // here: both NOT inCoeffDomain |
---|
| 709 | CanonicalForm f, g, tmp, M, q, D, Dp, cl, newq, mipo; |
---|
| 710 | int p, i; |
---|
| 711 | int *bound, *other; // degree vectors |
---|
| 712 | bool fail; |
---|
[713bdb] | 713 | bool off_rational=!isOn(SW_RATIONAL); |
---|
[ad8e1b] | 714 | On( SW_RATIONAL ); // needed by bCommonDen |
---|
| 715 | f = F * bCommonDen(F); |
---|
| 716 | g = G * bCommonDen(G); |
---|
[2a95b2] | 717 | TIMING_START (alg_content) |
---|
[2156ec] | 718 | CanonicalForm contf= myicontent (f); |
---|
| 719 | CanonicalForm contg= myicontent (g); |
---|
| 720 | f /= contf; |
---|
| 721 | g /= contg; |
---|
| 722 | CanonicalForm gcdcfcg= myicontent (contf, contg); |
---|
[2a95b2] | 723 | TIMING_END_AND_PRINT (alg_content, "time for content in alg gcd: ") |
---|
[ad8e1b] | 724 | Variable a, b; |
---|
| 725 | if(hasFirstAlgVar(f,a)) |
---|
| 726 | { |
---|
| 727 | if(hasFirstAlgVar(g,b)) |
---|
| 728 | { |
---|
| 729 | if(b.level() > a.level()) |
---|
| 730 | a = b; |
---|
| 731 | } |
---|
| 732 | } |
---|
| 733 | else |
---|
| 734 | { |
---|
| 735 | if(!hasFirstAlgVar(g,a))// both not in extension |
---|
| 736 | { |
---|
| 737 | Off( SW_RATIONAL ); |
---|
| 738 | Off( SW_USE_QGCD ); |
---|
[2156ec] | 739 | tmp = gcdcfcg*gcd( f, g ); |
---|
[ad8e1b] | 740 | On( SW_USE_QGCD ); |
---|
[713bdb] | 741 | if (off_rational) Off(SW_RATIONAL); |
---|
[ad8e1b] | 742 | return tmp; |
---|
| 743 | } |
---|
| 744 | } |
---|
| 745 | // here: a is the biggest alg. var in f and g AND some of f,g is in extension |
---|
| 746 | setReduce(a,false); // do not reduce expressions modulo mipo |
---|
| 747 | tmp = getMipo(a); |
---|
| 748 | M = tmp * bCommonDen(tmp); |
---|
| 749 | // here: f, g in Z[a][x1,...,xn], M in Z[a] not necessarily monic |
---|
| 750 | Off( SW_RATIONAL ); // needed by mod |
---|
| 751 | // calculate upper bound for degree vector of gcd |
---|
| 752 | int mv = f.level(); i = g.level(); |
---|
| 753 | if(i > mv) |
---|
| 754 | mv = i; |
---|
| 755 | // here: mv is level of the largest variable in f, g |
---|
| 756 | bound = new int[mv+1]; // 'bound' could be indexed from 0 to mv, but we will only use from 1 to mv |
---|
| 757 | other = new int[mv+1]; |
---|
| 758 | for(int i=1; i<=mv; i++) // initialize 'bound', 'other' with zeros |
---|
| 759 | bound[i] = other[i] = 0; |
---|
| 760 | bound = leadDeg(f,bound); // 'bound' is set the leading degree vector of f |
---|
| 761 | other = leadDeg(g,other); |
---|
| 762 | for(int i=1; i<=mv; i++) // entry at i=0 not visited |
---|
| 763 | if(other[i] < bound[i]) |
---|
| 764 | bound[i] = other[i]; |
---|
| 765 | // now 'bound' is the smaller vector |
---|
| 766 | cl = lc(M) * lc(f) * lc(g); |
---|
| 767 | q = 1; |
---|
| 768 | D = 0; |
---|
[fe2d4c] | 769 | CanonicalForm test= 0; |
---|
| 770 | bool equal= false; |
---|
[ad8e1b] | 771 | for( i=cf_getNumBigPrimes()-1; i>-1; i-- ) |
---|
| 772 | { |
---|
| 773 | p = cf_getBigPrime(i); |
---|
| 774 | if( mod( cl, p ).isZero() ) // skip lc-bad primes |
---|
| 775 | continue; |
---|
| 776 | fail = false; |
---|
| 777 | setCharacteristic(p); |
---|
| 778 | mipo = mapinto(M); |
---|
| 779 | mipo /= mipo.lc(); |
---|
| 780 | // here: mipo is monic |
---|
[2a95b2] | 781 | TIMING_START (alg_gcd_p) |
---|
[ad8e1b] | 782 | tryBrownGCD( mapinto(f), mapinto(g), mipo, Dp, fail ); |
---|
[2a95b2] | 783 | TIMING_END_AND_PRINT (alg_gcd_p, "time for alg gcd mod p: ") |
---|
[ad8e1b] | 784 | if( fail ) // mipo splits in char p |
---|
| 785 | continue; |
---|
| 786 | if( Dp.inCoeffDomain() ) // early termination |
---|
| 787 | { |
---|
| 788 | tryInvert(Dp,mipo,tmp,fail); // check if zero divisor |
---|
| 789 | if(fail) |
---|
| 790 | continue; |
---|
| 791 | setReduce(a,true); |
---|
[713bdb] | 792 | if (off_rational) Off(SW_RATIONAL); else On(SW_RATIONAL); |
---|
[0a7d0ca] | 793 | setCharacteristic(0); |
---|
[2156ec] | 794 | return gcdcfcg; |
---|
[ad8e1b] | 795 | } |
---|
[0a7d0ca] | 796 | setCharacteristic(0); |
---|
[ad8e1b] | 797 | // here: Dp NOT inCoeffDomain |
---|
| 798 | for(int i=1; i<=mv; i++) |
---|
| 799 | other[i] = 0; // reset (this is necessary, because some entries may not be updated by call to leadDeg) |
---|
| 800 | other = leadDeg(Dp,other); |
---|
[806c18] | 801 | |
---|
[ad8e1b] | 802 | if(isEqual(bound, other, 1, mv)) // equal |
---|
| 803 | { |
---|
[6bbe94] | 804 | chineseRemainder( D, q, mapinto(Dp), p, tmp, newq ); |
---|
[ad8e1b] | 805 | // tmp = Dp mod p |
---|
| 806 | // tmp = D mod q |
---|
| 807 | // newq = p*q |
---|
| 808 | q = newq; |
---|
| 809 | if( D != tmp ) |
---|
| 810 | D = tmp; |
---|
| 811 | On( SW_RATIONAL ); |
---|
[2a95b2] | 812 | TIMING_START (alg_reconstruction) |
---|
[6bbe94] | 813 | tmp = Farey( D, q ); // Farey |
---|
| 814 | tmp *= bCommonDen (tmp); |
---|
[2a95b2] | 815 | TIMING_END_AND_PRINT (alg_reconstruction, |
---|
| 816 | "time for rational reconstruction in alg gcd: ") |
---|
[ad8e1b] | 817 | setReduce(a,true); // reduce expressions modulo mipo |
---|
| 818 | On( SW_RATIONAL ); // needed by fdivides |
---|
[fe2d4c] | 819 | if (test != tmp) |
---|
| 820 | test= tmp; |
---|
| 821 | else |
---|
| 822 | equal= true; // modular image did not add any new information |
---|
[2a95b2] | 823 | TIMING_START (alg_termination) |
---|
[1e5c50] | 824 | #ifdef HAVE_FLINT |
---|
| 825 | if (equal && tmp.isUnivariate() && f.isUnivariate() && g.isUnivariate() |
---|
| 826 | && f.level() == tmp.level() && tmp.level() == g.level()) |
---|
| 827 | { |
---|
| 828 | CanonicalForm Q, R, sf, sg, stmp; |
---|
| 829 | Variable x= Variable (1); |
---|
| 830 | sf= swapvar (f, f.mvar(), x); |
---|
| 831 | sg= swapvar (g, f.mvar(), x); |
---|
| 832 | stmp= swapvar (tmp, f.mvar(), x); |
---|
| 833 | newtonDivrem (sf, stmp, Q, R); |
---|
| 834 | if (R.isZero()) |
---|
| 835 | { |
---|
| 836 | newtonDivrem (sg, stmp, Q, R); |
---|
| 837 | if (R.isZero()) |
---|
| 838 | { |
---|
| 839 | Off (SW_RATIONAL); |
---|
| 840 | setReduce (a,true); |
---|
| 841 | if (off_rational) Off(SW_RATIONAL); else On(SW_RATIONAL); |
---|
| 842 | TIMING_END_AND_PRINT (alg_termination, |
---|
| 843 | "time for successful termination test in alg gcd: ") |
---|
| 844 | return tmp*gcdcfcg; |
---|
| 845 | } |
---|
| 846 | } |
---|
| 847 | } |
---|
| 848 | else |
---|
| 849 | #endif |
---|
[fe2d4c] | 850 | if(equal && fdivides( tmp, f ) && fdivides( tmp, g )) // trial division |
---|
[ad8e1b] | 851 | { |
---|
| 852 | Off( SW_RATIONAL ); |
---|
| 853 | setReduce(a,true); |
---|
[713bdb] | 854 | if (off_rational) Off(SW_RATIONAL); else On(SW_RATIONAL); |
---|
[2a95b2] | 855 | TIMING_END_AND_PRINT (alg_termination, |
---|
| 856 | "time for successful termination test in alg gcd: ") |
---|
[2156ec] | 857 | return tmp*gcdcfcg; |
---|
[ad8e1b] | 858 | } |
---|
[2a95b2] | 859 | TIMING_END_AND_PRINT (alg_termination, |
---|
| 860 | "time for unsuccessful termination test in alg gcd: ") |
---|
[ad8e1b] | 861 | Off( SW_RATIONAL ); |
---|
| 862 | setReduce(a,false); // do not reduce expressions modulo mipo |
---|
| 863 | continue; |
---|
| 864 | } |
---|
| 865 | if( isLess(bound, other, 1, mv) ) // current prime unlucky |
---|
| 866 | continue; |
---|
| 867 | // here: isLess(other, bound, 1, mv) ) ==> all previous primes unlucky |
---|
| 868 | q = p; |
---|
[6bbe94] | 869 | D = mapinto(Dp); // shortcut CRA // shortcut CRA |
---|
[ad8e1b] | 870 | for(int i=1; i<=mv; i++) // tighten bound |
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| 871 | bound[i] = other[i]; |
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| 872 | } |
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| 873 | // hopefully, we never reach this point |
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| 874 | setReduce(a,true); |
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| 875 | Off( SW_USE_QGCD ); |
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[2156ec] | 876 | D = gcdcfcg*gcd( f, g ); |
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[ad8e1b] | 877 | On( SW_USE_QGCD ); |
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[713bdb] | 878 | if (off_rational) Off(SW_RATIONAL); else On(SW_RATIONAL); |
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[ad8e1b] | 879 | return D; |
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| 880 | } |
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| 881 | |
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| 882 | |
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| 883 | int * leadDeg(const CanonicalForm & f, int *degs) |
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| 884 | { // leading degree vector w.r.t. lex. monomial order x(i+1) > x(i) |
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| 885 | // if f is in a coeff domain, the zero pointer is returned |
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| 886 | // 'a' should point to an array of sufficient size level(f)+1 |
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| 887 | if(f.inCoeffDomain()) |
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| 888 | return 0; |
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| 889 | CanonicalForm tmp = f; |
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| 890 | do |
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| 891 | { |
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| 892 | degs[tmp.level()] = tmp.degree(); |
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| 893 | tmp = LC(tmp); |
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| 894 | } |
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| 895 | while(!tmp.inCoeffDomain()); |
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| 896 | return degs; |
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| 897 | } |
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| 898 | |
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| 899 | |
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| 900 | bool isLess(int *a, int *b, int lower, int upper) |
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| 901 | { // compares the degree vectors a,b on the specified part. Note: x(i+1) > x(i) |
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| 902 | for(int i=upper; i>=lower; i--) |
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| 903 | if(a[i] == b[i]) |
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| 904 | continue; |
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| 905 | else |
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| 906 | return a[i] < b[i]; |
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| 907 | return true; |
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| 908 | } |
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| 909 | |
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| 910 | |
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| 911 | bool isEqual(int *a, int *b, int lower, int upper) |
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| 912 | { // compares the degree vectors a,b on the specified part. Note: x(i+1) > x(i) |
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| 913 | for(int i=lower; i<=upper; i++) |
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| 914 | if(a[i] != b[i]) |
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| 915 | return false; |
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| 916 | return true; |
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| 917 | } |
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| 918 | |
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| 919 | |
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| 920 | CanonicalForm firstLC(const CanonicalForm & f) |
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| 921 | { // returns the leading coefficient (LC) of level <= 1 |
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| 922 | CanonicalForm ret = f; |
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| 923 | while(ret.level() > 1) |
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| 924 | ret = LC(ret); |
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| 925 | return ret; |
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| 926 | } |
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| 927 | |
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| 928 | void tryExtgcd( const CanonicalForm & F, const CanonicalForm & G, const CanonicalForm & M, CanonicalForm & result, CanonicalForm & s, CanonicalForm & t, bool & fail ) |
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| 929 | { // F, G are univariate polynomials (i.e. they have exactly one polynomial variable) |
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| 930 | // F and G must have the same level AND level > 0 |
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| 931 | // we try to calculate gcd(F,G) = s*F + t*G |
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| 932 | // if a zero divisor is encontered, 'fail' is set to one |
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| 933 | // M is assumed to be monic |
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| 934 | CanonicalForm P; |
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| 935 | if(F.inCoeffDomain()) |
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| 936 | { |
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| 937 | tryInvert( F, M, P, fail ); |
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| 938 | if(fail) |
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| 939 | return; |
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| 940 | result = 1; |
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| 941 | s = P; t = 0; |
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| 942 | return; |
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| 943 | } |
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| 944 | if(G.inCoeffDomain()) |
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| 945 | { |
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| 946 | tryInvert( G, M, P, fail ); |
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| 947 | if(fail) |
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| 948 | return; |
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| 949 | result = 1; |
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| 950 | s = 0; t = P; |
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| 951 | return; |
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| 952 | } |
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| 953 | // here: both not inCoeffDomain |
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| 954 | CanonicalForm inv, rem, tmp, u, v, q, sum=0; |
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| 955 | if( F.degree() > G.degree() ) |
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| 956 | { |
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| 957 | P = F; result = G; s=v=0; t=u=1; |
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| 958 | } |
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| 959 | else |
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| 960 | { |
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| 961 | P = G; result = F; s=v=1; t=u=0; |
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| 962 | } |
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| 963 | Variable x = P.mvar(); |
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| 964 | // here: degree(P) >= degree(result) |
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| 965 | while(true) |
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| 966 | { |
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[fe2d4c] | 967 | tryDivrem (P, result, q, rem, inv, M, fail); |
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[ad8e1b] | 968 | if(fail) |
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| 969 | return; |
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| 970 | if( rem.isZero() ) |
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| 971 | { |
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| 972 | s*=inv; |
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[4a05ed] | 973 | s= reduce (s, M); |
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[ad8e1b] | 974 | t*=inv; |
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[4a05ed] | 975 | t= reduce (t, M); |
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[ad8e1b] | 976 | result *= inv; // monify result |
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[4a05ed] | 977 | result= reduce (result, M); |
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[ad8e1b] | 978 | return; |
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| 979 | } |
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| 980 | sum += q; |
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| 981 | if(result.degree(x) >= rem.degree(x)) |
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| 982 | { |
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| 983 | P=result; |
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| 984 | result=rem; |
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| 985 | tmp=u-sum*s; |
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| 986 | u=s; |
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| 987 | s=tmp; |
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| 988 | tmp=v-sum*t; |
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| 989 | v=t; |
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| 990 | t=tmp; |
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| 991 | sum = 0; // reset |
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| 992 | } |
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| 993 | else |
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| 994 | P = rem; |
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| 995 | } |
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| 996 | } |
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| 997 | |
---|
| 998 | |
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| 999 | static CanonicalForm trycontent ( const CanonicalForm & f, const Variable & x, const CanonicalForm & M, bool & fail ) |
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| 1000 | { // as 'content', but takes care of zero divisors |
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| 1001 | ASSERT( x.level() > 0, "cannot calculate content with respect to algebraic variable" ); |
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| 1002 | Variable y = f.mvar(); |
---|
| 1003 | if ( y == x ) |
---|
| 1004 | return trycf_content( f, 0, M, fail ); |
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| 1005 | if ( y < x ) |
---|
| 1006 | return f; |
---|
| 1007 | return swapvar( trycontent( swapvar( f, y, x ), y, M, fail ), y, x ); |
---|
| 1008 | } |
---|
| 1009 | |
---|
| 1010 | |
---|
| 1011 | static CanonicalForm tryvcontent ( const CanonicalForm & f, const Variable & x, const CanonicalForm & M, bool & fail ) |
---|
| 1012 | { // as vcontent, but takes care of zero divisors |
---|
| 1013 | ASSERT( x.level() > 0, "cannot calculate vcontent with respect to algebraic variable" ); |
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| 1014 | if ( f.mvar() <= x ) |
---|
| 1015 | return trycontent( f, x, M, fail ); |
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| 1016 | CFIterator i; |
---|
| 1017 | CanonicalForm d = 0, e, ret; |
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| 1018 | for ( i = f; i.hasTerms() && ! d.isOne() && ! fail; i++ ) |
---|
| 1019 | { |
---|
| 1020 | e = tryvcontent( i.coeff(), x, M, fail ); |
---|
| 1021 | if(fail) |
---|
| 1022 | break; |
---|
| 1023 | tryBrownGCD( d, e, M, ret, fail ); |
---|
| 1024 | d = ret; |
---|
| 1025 | } |
---|
| 1026 | return d; |
---|
| 1027 | } |
---|
| 1028 | |
---|
| 1029 | |
---|
| 1030 | static CanonicalForm trycf_content ( const CanonicalForm & f, const CanonicalForm & g, const CanonicalForm & M, bool & fail ) |
---|
| 1031 | { // as cf_content, but takes care of zero divisors |
---|
| 1032 | if ( f.inPolyDomain() || ( f.inExtension() && ! getReduce( f.mvar() ) ) ) |
---|
| 1033 | { |
---|
| 1034 | CFIterator i = f; |
---|
| 1035 | CanonicalForm tmp = g, result; |
---|
| 1036 | while ( i.hasTerms() && ! tmp.isOne() && ! fail ) |
---|
| 1037 | { |
---|
| 1038 | tryBrownGCD( i.coeff(), tmp, M, result, fail ); |
---|
| 1039 | tmp = result; |
---|
| 1040 | i++; |
---|
| 1041 | } |
---|
| 1042 | return result; |
---|
| 1043 | } |
---|
| 1044 | return abs( f ); |
---|
| 1045 | } |
---|
| 1046 | |
---|
[4a05ed] | 1047 | void tryExtgcd( const CanonicalForm & F, const CanonicalForm & G, CanonicalForm & result, CanonicalForm & s, CanonicalForm & t, bool & fail ) |
---|
| 1048 | { |
---|
| 1049 | // F, G are univariate polynomials (i.e. they have exactly one polynomial variable) |
---|
| 1050 | // F and G must have the same level AND level > 0 |
---|
| 1051 | // we try to calculate gcd(f,g) = s*F + t*G |
---|
| 1052 | // if a zero divisor is encontered, 'fail' is set to one |
---|
| 1053 | Variable a, b; |
---|
| 1054 | if( !hasFirstAlgVar(F,a) && !hasFirstAlgVar(G,b) ) // note lazy evaluation |
---|
| 1055 | { |
---|
| 1056 | result = extgcd( F, G, s, t ); // no zero divisors possible |
---|
| 1057 | return; |
---|
| 1058 | } |
---|
| 1059 | if( b.level() > a.level() ) |
---|
| 1060 | a = b; |
---|
| 1061 | // here: a is the biggest alg. var in F and G |
---|
| 1062 | CanonicalForm M = getMipo(a); |
---|
| 1063 | CanonicalForm P; |
---|
| 1064 | if( degree(F) > degree(G) ) |
---|
| 1065 | { |
---|
| 1066 | P=F; result=G; s=0; t=1; |
---|
| 1067 | } |
---|
| 1068 | else |
---|
| 1069 | { |
---|
| 1070 | P=G; result=F; s=1; t=0; |
---|
| 1071 | } |
---|
| 1072 | CanonicalForm inv, rem, q, u, v; |
---|
| 1073 | // here: degree(P) >= degree(result) |
---|
| 1074 | while(true) |
---|
| 1075 | { |
---|
| 1076 | tryInvert( Lc(result), M, inv, fail ); |
---|
| 1077 | if(fail) |
---|
| 1078 | return; |
---|
| 1079 | // here: Lc(result) is invertible modulo M |
---|
| 1080 | q = Lc(P)*inv * power( P.mvar(), degree(P)-degree(result) ); |
---|
| 1081 | rem = P - q*result; |
---|
| 1082 | // here: s*F + t*G = result |
---|
| 1083 | if( rem.isZero() ) |
---|
| 1084 | { |
---|
| 1085 | s*=inv; |
---|
| 1086 | t*=inv; |
---|
| 1087 | result *= inv; // monify result |
---|
| 1088 | return; |
---|
| 1089 | } |
---|
| 1090 | P=result; |
---|
| 1091 | result=rem; |
---|
| 1092 | rem=u-q*s; |
---|
| 1093 | u=s; |
---|
| 1094 | s=rem; |
---|
| 1095 | rem=v-q*t; |
---|
| 1096 | v=t; |
---|
| 1097 | t=rem; |
---|
| 1098 | } |
---|
| 1099 | } |
---|
| 1100 | |
---|
[359d742] | 1101 | void tryCRA( const CanonicalForm & x1, const CanonicalForm & q1, const CanonicalForm & x2, const CanonicalForm & q2, CanonicalForm & xnew, CanonicalForm & qnew, bool & fail ) |
---|
| 1102 | { // polys of level <= 1 are considered coefficients. q1,q2 are assumed to be coprime |
---|
| 1103 | // xnew = x1 mod q1 (coefficientwise in the above sense) |
---|
| 1104 | // xnew = x2 mod q2 |
---|
| 1105 | // qnew = q1*q2 |
---|
| 1106 | CanonicalForm tmp; |
---|
| 1107 | if(x1.level() <= 1 && x2.level() <= 1) // base case |
---|
| 1108 | { |
---|
| 1109 | tryExtgcd(q1,q2,tmp,xnew,qnew,fail); |
---|
| 1110 | if(fail) |
---|
| 1111 | return; |
---|
| 1112 | xnew = x1 + (x2-x1) * xnew * q1; |
---|
| 1113 | qnew = q1*q2; |
---|
| 1114 | xnew = mod(xnew,qnew); |
---|
| 1115 | return; |
---|
| 1116 | } |
---|
| 1117 | CanonicalForm tmp2; |
---|
| 1118 | xnew = 0; |
---|
| 1119 | qnew = q1 * q2; |
---|
| 1120 | // here: x1.level() > 1 || x2.level() > 1 |
---|
| 1121 | if(x1.level() > x2.level()) |
---|
| 1122 | { |
---|
| 1123 | for(CFIterator i=x1; i.hasTerms(); i++) |
---|
| 1124 | { |
---|
| 1125 | if(i.exp() == 0) // const. term |
---|
| 1126 | { |
---|
| 1127 | tryCRA(i.coeff(),q1,x2,q2,tmp,tmp2,fail); |
---|
| 1128 | if(fail) |
---|
| 1129 | return; |
---|
| 1130 | xnew += tmp; |
---|
| 1131 | } |
---|
| 1132 | else |
---|
| 1133 | { |
---|
| 1134 | tryCRA(i.coeff(),q1,0,q2,tmp,tmp2,fail); |
---|
| 1135 | if(fail) |
---|
| 1136 | return; |
---|
| 1137 | xnew += tmp * power(x1.mvar(),i.exp()); |
---|
| 1138 | } |
---|
| 1139 | } |
---|
| 1140 | return; |
---|
| 1141 | } |
---|
| 1142 | // here: x1.level() <= x2.level() && ( x1.level() > 1 || x2.level() > 1 ) |
---|
| 1143 | if(x2.level() > x1.level()) |
---|
| 1144 | { |
---|
| 1145 | for(CFIterator j=x2; j.hasTerms(); j++) |
---|
| 1146 | { |
---|
| 1147 | if(j.exp() == 0) // const. term |
---|
| 1148 | { |
---|
| 1149 | tryCRA(x1,q1,j.coeff(),q2,tmp,tmp2,fail); |
---|
| 1150 | if(fail) |
---|
| 1151 | return; |
---|
| 1152 | xnew += tmp; |
---|
| 1153 | } |
---|
| 1154 | else |
---|
| 1155 | { |
---|
| 1156 | tryCRA(0,q1,j.coeff(),q2,tmp,tmp2,fail); |
---|
| 1157 | if(fail) |
---|
| 1158 | return; |
---|
| 1159 | xnew += tmp * power(x2.mvar(),j.exp()); |
---|
| 1160 | } |
---|
| 1161 | } |
---|
| 1162 | return; |
---|
| 1163 | } |
---|
| 1164 | // here: x1.level() == x2.level() && x1.level() > 1 && x2.level() > 1 |
---|
| 1165 | CFIterator i = x1; |
---|
| 1166 | CFIterator j = x2; |
---|
| 1167 | while(i.hasTerms() || j.hasTerms()) |
---|
| 1168 | { |
---|
| 1169 | if(i.hasTerms()) |
---|
| 1170 | { |
---|
| 1171 | if(j.hasTerms()) |
---|
| 1172 | { |
---|
| 1173 | if(i.exp() == j.exp()) |
---|
| 1174 | { |
---|
| 1175 | tryCRA(i.coeff(),q1,j.coeff(),q2,tmp,tmp2,fail); |
---|
| 1176 | if(fail) |
---|
| 1177 | return; |
---|
| 1178 | xnew += tmp * power(x1.mvar(),i.exp()); |
---|
| 1179 | i++; j++; |
---|
| 1180 | } |
---|
| 1181 | else |
---|
| 1182 | { |
---|
| 1183 | if(i.exp() < j.exp()) |
---|
| 1184 | { |
---|
| 1185 | tryCRA(i.coeff(),q1,0,q2,tmp,tmp2,fail); |
---|
| 1186 | if(fail) |
---|
| 1187 | return; |
---|
| 1188 | xnew += tmp * power(x1.mvar(),i.exp()); |
---|
| 1189 | i++; |
---|
| 1190 | } |
---|
| 1191 | else // i.exp() > j.exp() |
---|
| 1192 | { |
---|
| 1193 | tryCRA(0,q1,j.coeff(),q2,tmp,tmp2,fail); |
---|
| 1194 | if(fail) |
---|
| 1195 | return; |
---|
| 1196 | xnew += tmp * power(x1.mvar(),j.exp()); |
---|
| 1197 | j++; |
---|
| 1198 | } |
---|
| 1199 | } |
---|
| 1200 | } |
---|
| 1201 | else // j is out of terms |
---|
| 1202 | { |
---|
| 1203 | tryCRA(i.coeff(),q1,0,q2,tmp,tmp2,fail); |
---|
| 1204 | if(fail) |
---|
| 1205 | return; |
---|
| 1206 | xnew += tmp * power(x1.mvar(),i.exp()); |
---|
| 1207 | i++; |
---|
| 1208 | } |
---|
| 1209 | } |
---|
| 1210 | else // i is out of terms |
---|
| 1211 | { |
---|
| 1212 | tryCRA(0,q1,j.coeff(),q2,tmp,tmp2,fail); |
---|
| 1213 | if(fail) |
---|
| 1214 | return; |
---|
| 1215 | xnew += tmp * power(x1.mvar(),j.exp()); |
---|
| 1216 | j++; |
---|
| 1217 | } |
---|
| 1218 | } |
---|
| 1219 | } |
---|
| 1220 | |
---|