1 | |
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2 | #include "config.h" |
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3 | |
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4 | |
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5 | #ifndef NOSTREAMIO |
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6 | #ifdef HAVE_CSTDIO |
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7 | #include <cstdio> |
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8 | #else |
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9 | #include <stdio.h> |
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10 | #endif |
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11 | #ifdef HAVE_IOSTREAM_H |
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12 | #include <iostream.h> |
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13 | #elif defined(HAVE_IOSTREAM) |
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14 | #include <iostream> |
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15 | #endif |
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16 | #endif |
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17 | |
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18 | #include "cf_assert.h" |
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19 | #include "timing.h" |
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20 | |
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21 | #include "templates/ftmpl_functions.h" |
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22 | #include "cf_defs.h" |
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23 | #include "canonicalform.h" |
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24 | #include "cf_iter.h" |
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25 | #include "cf_primes.h" |
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26 | #include "cf_algorithm.h" |
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27 | #include "cfGcdAlgExt.h" |
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28 | #include "cfUnivarGcd.h" |
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29 | #include "cf_map.h" |
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30 | #include "cf_generator.h" |
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31 | #include "facMul.h" |
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32 | #include "cfNTLzzpEXGCD.h" |
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33 | |
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34 | #ifdef HAVE_NTL |
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35 | #include "NTLconvert.h" |
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36 | #endif |
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37 | |
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38 | #ifdef HAVE_FLINT |
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39 | #include "FLINTconvert.h" |
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40 | #endif |
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41 | |
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42 | TIMING_DEFINE_PRINT(alg_content_p) |
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43 | TIMING_DEFINE_PRINT(alg_content) |
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44 | TIMING_DEFINE_PRINT(alg_compress) |
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45 | TIMING_DEFINE_PRINT(alg_termination) |
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46 | TIMING_DEFINE_PRINT(alg_termination_p) |
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47 | TIMING_DEFINE_PRINT(alg_reconstruction) |
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48 | TIMING_DEFINE_PRINT(alg_newton_p) |
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49 | TIMING_DEFINE_PRINT(alg_recursion_p) |
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50 | TIMING_DEFINE_PRINT(alg_gcd_p) |
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51 | TIMING_DEFINE_PRINT(alg_euclid_p) |
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52 | |
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53 | /// compressing two polynomials F and G, M is used for compressing, |
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54 | /// N to reverse the compression |
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55 | static int myCompress (const CanonicalForm& F, const CanonicalForm& G, CFMap & M, |
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56 | CFMap & N, bool topLevel) |
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57 | { |
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58 | int n= tmax (F.level(), G.level()); |
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59 | int * degsf= new int [n + 1]; |
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60 | int * degsg= new int [n + 1]; |
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61 | |
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62 | for (int i = 0; i <= n; i++) |
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63 | degsf[i]= degsg[i]= 0; |
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64 | |
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65 | degsf= degrees (F, degsf); |
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66 | degsg= degrees (G, degsg); |
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67 | |
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68 | int both_non_zero= 0; |
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69 | int f_zero= 0; |
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70 | int g_zero= 0; |
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71 | int both_zero= 0; |
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72 | |
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73 | if (topLevel) |
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74 | { |
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75 | for (int i= 1; i <= n; i++) |
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76 | { |
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77 | if (degsf[i] != 0 && degsg[i] != 0) |
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78 | { |
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79 | both_non_zero++; |
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80 | continue; |
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81 | } |
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82 | if (degsf[i] == 0 && degsg[i] != 0 && i <= G.level()) |
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83 | { |
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84 | f_zero++; |
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85 | continue; |
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86 | } |
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87 | if (degsg[i] == 0 && degsf[i] && i <= F.level()) |
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88 | { |
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89 | g_zero++; |
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90 | continue; |
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91 | } |
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92 | } |
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93 | |
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94 | if (both_non_zero == 0) |
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95 | { |
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96 | delete [] degsf; |
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97 | delete [] degsg; |
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98 | return 0; |
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99 | } |
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100 | |
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101 | // map Variables which do not occur in both polynomials to higher levels |
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102 | int k= 1; |
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103 | int l= 1; |
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104 | for (int i= 1; i <= n; i++) |
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105 | { |
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106 | if (degsf[i] != 0 && degsg[i] == 0 && i <= F.level()) |
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107 | { |
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108 | if (k + both_non_zero != i) |
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109 | { |
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110 | M.newpair (Variable (i), Variable (k + both_non_zero)); |
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111 | N.newpair (Variable (k + both_non_zero), Variable (i)); |
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112 | } |
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113 | k++; |
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114 | } |
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115 | if (degsf[i] == 0 && degsg[i] != 0 && i <= G.level()) |
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116 | { |
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117 | if (l + g_zero + both_non_zero != i) |
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118 | { |
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119 | M.newpair (Variable (i), Variable (l + g_zero + both_non_zero)); |
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120 | N.newpair (Variable (l + g_zero + both_non_zero), Variable (i)); |
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121 | } |
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122 | l++; |
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123 | } |
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124 | } |
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125 | |
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126 | // sort Variables x_{i} in increasing order of |
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127 | // min(deg_{x_{i}}(f),deg_{x_{i}}(g)) |
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128 | int m= tmax (F.level(), G.level()); |
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129 | int min_max_deg; |
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130 | k= both_non_zero; |
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131 | l= 0; |
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132 | int i= 1; |
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133 | while (k > 0) |
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134 | { |
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135 | if (degsf [i] != 0 && degsg [i] != 0) |
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136 | min_max_deg= tmax (degsf[i], degsg[i]); |
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137 | else |
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138 | min_max_deg= 0; |
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139 | while (min_max_deg == 0) |
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140 | { |
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141 | i++; |
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142 | min_max_deg= tmax (degsf[i], degsg[i]); |
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143 | if (degsf [i] != 0 && degsg [i] != 0) |
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144 | min_max_deg= tmax (degsf[i], degsg[i]); |
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145 | else |
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146 | min_max_deg= 0; |
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147 | } |
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148 | for (int j= i + 1; j <= m; j++) |
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149 | { |
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150 | if (tmax (degsf[j],degsg[j]) <= min_max_deg && degsf[j] != 0 && degsg [j] != 0) |
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151 | { |
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152 | min_max_deg= tmax (degsf[j], degsg[j]); |
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153 | l= j; |
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154 | } |
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155 | } |
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156 | if (l != 0) |
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157 | { |
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158 | if (l != k) |
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159 | { |
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160 | M.newpair (Variable (l), Variable(k)); |
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161 | N.newpair (Variable (k), Variable(l)); |
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162 | degsf[l]= 0; |
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163 | degsg[l]= 0; |
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164 | l= 0; |
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165 | } |
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166 | else |
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167 | { |
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168 | degsf[l]= 0; |
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169 | degsg[l]= 0; |
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170 | l= 0; |
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171 | } |
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172 | } |
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173 | else if (l == 0) |
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174 | { |
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175 | if (i != k) |
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176 | { |
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177 | M.newpair (Variable (i), Variable (k)); |
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178 | N.newpair (Variable (k), Variable (i)); |
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179 | degsf[i]= 0; |
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180 | degsg[i]= 0; |
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181 | } |
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182 | else |
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183 | { |
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184 | degsf[i]= 0; |
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185 | degsg[i]= 0; |
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186 | } |
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187 | i++; |
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188 | } |
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189 | k--; |
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190 | } |
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191 | } |
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192 | else |
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193 | { |
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194 | //arrange Variables such that no gaps occur |
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195 | for (int i= 1; i <= n; i++) |
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196 | { |
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197 | if (degsf[i] == 0 && degsg[i] == 0) |
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198 | { |
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199 | both_zero++; |
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200 | continue; |
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201 | } |
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202 | else |
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203 | { |
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204 | if (both_zero != 0) |
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205 | { |
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206 | M.newpair (Variable (i), Variable (i - both_zero)); |
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207 | N.newpair (Variable (i - both_zero), Variable (i)); |
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208 | } |
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209 | } |
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210 | } |
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211 | } |
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212 | |
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213 | delete [] degsf; |
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214 | delete [] degsg; |
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215 | |
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216 | return 1; |
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217 | } |
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218 | |
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219 | void tryInvert( const CanonicalForm & F, const CanonicalForm & M, CanonicalForm & inv, bool & fail ) |
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220 | { // F, M are required to be "univariate" polynomials in an algebraic variable |
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221 | // we try to invert F modulo M |
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222 | if(F.inBaseDomain()) |
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223 | { |
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224 | if(F.isZero()) |
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225 | { |
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226 | fail = true; |
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227 | return; |
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228 | } |
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229 | inv = 1/F; |
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230 | return; |
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231 | } |
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232 | CanonicalForm b; |
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233 | Variable a = M.mvar(); |
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234 | Variable x = Variable(1); |
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235 | if(!extgcd( replacevar( F, a, x ), replacevar( M, a, x ), inv, b ).isOne()) |
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236 | fail = true; |
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237 | else |
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238 | inv = replacevar( inv, x, a ); // change back to alg var |
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239 | } |
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240 | |
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241 | #ifndef HAVE_NTL |
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242 | void tryDivrem (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
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243 | CanonicalForm& R, CanonicalForm& inv, const CanonicalForm& mipo, |
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244 | bool& fail) |
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245 | { |
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246 | if (F.inCoeffDomain()) |
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247 | { |
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248 | Q= 0; |
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249 | R= F; |
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250 | return; |
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251 | } |
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252 | |
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253 | CanonicalForm A, B; |
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254 | Variable x= F.mvar(); |
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255 | A= F; |
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256 | B= G; |
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257 | int degA= degree (A, x); |
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258 | int degB= degree (B, x); |
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259 | |
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260 | if (degA < degB) |
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261 | { |
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262 | R= A; |
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263 | Q= 0; |
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264 | return; |
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265 | } |
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266 | |
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267 | tryInvert (Lc (B), mipo, inv, fail); |
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268 | if (fail) |
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269 | return; |
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270 | |
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271 | R= A; |
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272 | Q= 0; |
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273 | CanonicalForm Qi; |
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274 | for (int i= degA -degB; i >= 0; i--) |
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275 | { |
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276 | if (degree (R, x) == i + degB) |
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277 | { |
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278 | Qi= Lc (R)*inv*power (x, i); |
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279 | Qi= reduce (Qi, mipo); |
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280 | R -= Qi*B; |
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281 | R= reduce (R, mipo); |
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282 | Q += Qi; |
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283 | } |
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284 | } |
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285 | } |
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286 | |
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287 | void tryEuclid( const CanonicalForm & A, const CanonicalForm & B, const CanonicalForm & M, CanonicalForm & result, bool & fail ) |
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288 | { |
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289 | CanonicalForm P; |
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290 | if(A.inCoeffDomain()) |
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291 | { |
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292 | tryInvert( A, M, P, fail ); |
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293 | if(fail) |
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294 | return; |
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295 | result = 1; |
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296 | return; |
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297 | } |
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298 | if(B.inCoeffDomain()) |
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299 | { |
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300 | tryInvert( B, M, P, fail ); |
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301 | if(fail) |
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302 | return; |
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303 | result = 1; |
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304 | return; |
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305 | } |
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306 | // here: both not inCoeffDomain |
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307 | if( A.degree() > B.degree() ) |
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308 | { |
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309 | P = A; result = B; |
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310 | } |
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311 | else |
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312 | { |
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313 | P = B; result = A; |
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314 | } |
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315 | CanonicalForm inv; |
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316 | if( result.isZero() ) |
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317 | { |
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318 | tryInvert( Lc(P), M, inv, fail ); |
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319 | if(fail) |
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320 | return; |
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321 | result = inv*P; // monify result (not reduced, yet) |
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322 | result= reduce (result, M); |
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323 | return; |
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324 | } |
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325 | Variable x = P.mvar(); |
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326 | CanonicalForm rem, Q; |
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327 | // here: degree(P) >= degree(result) |
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328 | while(true) |
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329 | { |
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330 | tryDivrem (P, result, Q, rem, inv, M, fail); |
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331 | if (fail) |
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332 | return; |
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333 | if( rem.isZero() ) |
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334 | { |
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335 | result *= inv; |
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336 | result= reduce (result, M); |
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337 | return; |
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338 | } |
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339 | if(result.degree(x) >= rem.degree(x)) |
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340 | { |
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341 | P = result; |
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342 | result = rem; |
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343 | } |
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344 | else |
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345 | P = rem; |
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346 | } |
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347 | } |
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348 | #endif |
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349 | |
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350 | static CanonicalForm trycontent ( const CanonicalForm & f, const Variable & x, const CanonicalForm & M, bool & fail ); |
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351 | static CanonicalForm tryvcontent ( const CanonicalForm & f, const Variable & x, const CanonicalForm & M, bool & fail ); |
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352 | static CanonicalForm trycf_content ( const CanonicalForm & f, const CanonicalForm & g, const CanonicalForm & M, bool & fail ); |
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353 | |
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354 | static inline CanonicalForm |
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355 | tryNewtonInterp (const CanonicalForm & alpha, const CanonicalForm & u, |
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356 | const CanonicalForm & newtonPoly, const CanonicalForm & oldInterPoly, |
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357 | const Variable & x, const CanonicalForm& M, bool& fail) |
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358 | { |
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359 | CanonicalForm interPoly; |
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360 | |
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361 | CanonicalForm inv; |
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362 | tryInvert (newtonPoly (alpha, x), M, inv, fail); |
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363 | if (fail) |
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364 | return 0; |
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365 | |
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366 | interPoly= oldInterPoly+reduce ((u - oldInterPoly (alpha, x))*inv*newtonPoly, M); |
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367 | return interPoly; |
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368 | } |
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369 | |
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370 | void tryBrownGCD( const CanonicalForm & F, const CanonicalForm & G, const CanonicalForm & M, CanonicalForm & result, bool & fail, bool topLevel ) |
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371 | { // 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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372 | // M is assumed to be monic |
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373 | if(F.isZero()) |
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374 | { |
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375 | if(G.isZero()) |
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376 | { |
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377 | result = G; // G is zero |
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378 | return; |
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379 | } |
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380 | if(G.inCoeffDomain()) |
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381 | { |
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382 | tryInvert(G,M,result,fail); |
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383 | if(fail) |
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384 | return; |
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385 | result = 1; |
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386 | return; |
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387 | } |
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388 | // try to make G monic modulo M |
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389 | CanonicalForm inv; |
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390 | tryInvert(Lc(G),M,inv,fail); |
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391 | if(fail) |
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392 | return; |
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393 | result = inv*G; |
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394 | result= reduce (result, M); |
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395 | return; |
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396 | } |
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397 | if(G.isZero()) // F is non-zero |
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398 | { |
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399 | if(F.inCoeffDomain()) |
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400 | { |
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401 | tryInvert(F,M,result,fail); |
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402 | if(fail) |
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403 | return; |
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404 | result = 1; |
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405 | return; |
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406 | } |
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407 | // try to make F monic modulo M |
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408 | CanonicalForm inv; |
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409 | tryInvert(Lc(F),M,inv,fail); |
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410 | if(fail) |
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411 | return; |
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412 | result = inv*F; |
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413 | result= reduce (result, M); |
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414 | return; |
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415 | } |
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416 | // here: F,G both nonzero |
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417 | if(F.inCoeffDomain()) |
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418 | { |
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419 | tryInvert(F,M,result,fail); |
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420 | if(fail) |
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421 | return; |
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422 | result = 1; |
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423 | return; |
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424 | } |
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425 | if(G.inCoeffDomain()) |
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426 | { |
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427 | tryInvert(G,M,result,fail); |
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428 | if(fail) |
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429 | return; |
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430 | result = 1; |
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431 | return; |
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432 | } |
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433 | TIMING_START (alg_compress) |
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434 | CFMap MM,NN; |
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435 | int lev= myCompress (F, G, MM, NN, topLevel); |
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436 | if (lev == 0) |
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437 | { |
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438 | result= 1; |
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439 | return; |
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440 | } |
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441 | CanonicalForm f=MM(F); |
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442 | CanonicalForm g=MM(G); |
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443 | TIMING_END_AND_PRINT (alg_compress, "time to compress in alg gcd: ") |
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444 | // here: f,g are compressed |
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445 | // compute largest variable in f or g (least one is Variable(1)) |
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446 | int mv = f.level(); |
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447 | if(g.level() > mv) |
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448 | mv = g.level(); |
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449 | // here: mv is level of the largest variable in f, g |
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450 | Variable v1= Variable (1); |
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451 | #ifdef HAVE_NTL |
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452 | Variable v= M.mvar(); |
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453 | if (fac_NTL_char != getCharacteristic()) |
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454 | { |
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455 | fac_NTL_char= getCharacteristic(); |
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456 | zz_p::init (getCharacteristic()); |
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457 | } |
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458 | zz_pX NTLMipo= convertFacCF2NTLzzpX (M); |
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459 | zz_pE::init (NTLMipo); |
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460 | zz_pEX NTLResult; |
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461 | zz_pEX NTLF; |
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462 | zz_pEX NTLG; |
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463 | #endif |
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464 | if(mv == 1) // f,g univariate |
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465 | { |
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466 | TIMING_START (alg_euclid_p) |
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467 | #ifdef HAVE_NTL |
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468 | NTLF= convertFacCF2NTLzz_pEX (f, NTLMipo); |
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469 | NTLG= convertFacCF2NTLzz_pEX (g, NTLMipo); |
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470 | tryNTLGCD (NTLResult, NTLF, NTLG, fail); |
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471 | if (fail) |
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472 | return; |
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473 | result= convertNTLzz_pEX2CF (NTLResult, f.mvar(), v); |
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474 | #else |
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475 | tryEuclid(f,g,M,result,fail); |
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476 | if(fail) |
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477 | return; |
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478 | #endif |
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479 | TIMING_END_AND_PRINT (alg_euclid_p, "time for euclidean alg mod p: ") |
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480 | result= NN (reduce (result, M)); // do not forget to map back |
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481 | return; |
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482 | } |
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483 | TIMING_START (alg_content_p) |
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484 | // here: mv > 1 |
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485 | CanonicalForm cf = tryvcontent(f, Variable(2), M, fail); // cf is univariate poly in var(1) |
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486 | if(fail) |
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487 | return; |
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488 | CanonicalForm cg = tryvcontent(g, Variable(2), M, fail); |
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489 | if(fail) |
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490 | return; |
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491 | CanonicalForm c; |
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492 | #ifdef HAVE_NTL |
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493 | NTLF= convertFacCF2NTLzz_pEX (cf, NTLMipo); |
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494 | NTLG= convertFacCF2NTLzz_pEX (cg, NTLMipo); |
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495 | tryNTLGCD (NTLResult, NTLF, NTLG, fail); |
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496 | if (fail) |
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497 | return; |
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498 | c= convertNTLzz_pEX2CF (NTLResult, v1, v); |
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499 | #else |
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500 | tryEuclid(cf,cg,M,c,fail); |
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501 | if(fail) |
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502 | return; |
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503 | #endif |
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504 | // f /= cf |
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505 | f.tryDiv (cf, M, fail); |
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506 | if(fail) |
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507 | return; |
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508 | // g /= cg |
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509 | g.tryDiv (cg, M, fail); |
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510 | if(fail) |
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511 | return; |
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512 | TIMING_END_AND_PRINT (alg_content_p, "time for content in alg gcd mod p: ") |
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513 | if(f.inCoeffDomain()) |
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514 | { |
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515 | tryInvert(f,M,result,fail); |
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516 | if(fail) |
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517 | return; |
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518 | result = NN(c); |
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519 | return; |
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520 | } |
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521 | if(g.inCoeffDomain()) |
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522 | { |
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523 | tryInvert(g,M,result,fail); |
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524 | if(fail) |
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525 | return; |
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526 | result = NN(c); |
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527 | return; |
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528 | } |
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529 | int *L = new int[mv+1]; // L is addressed by i from 2 to mv |
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530 | int *N = new int[mv+1]; |
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531 | for(int i=2; i<=mv; i++) |
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532 | L[i] = N[i] = 0; |
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533 | L = leadDeg(f, L); |
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534 | N = leadDeg(g, N); |
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535 | CanonicalForm gamma; |
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536 | TIMING_START (alg_euclid_p) |
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537 | #ifdef HAVE_NTL |
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538 | NTLF= convertFacCF2NTLzz_pEX (firstLC (f), NTLMipo); |
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539 | NTLG= convertFacCF2NTLzz_pEX (firstLC (g), NTLMipo); |
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540 | tryNTLGCD (NTLResult, NTLF, NTLG, fail); |
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541 | if (fail) |
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542 | return; |
---|
543 | gamma= convertNTLzz_pEX2CF (NTLResult, v1, v); |
---|
544 | #else |
---|
545 | tryEuclid( firstLC(f), firstLC(g), M, gamma, fail ); |
---|
546 | if(fail) |
---|
547 | return; |
---|
548 | #endif |
---|
549 | TIMING_END_AND_PRINT (alg_euclid_p, "time for gcd of lcs in alg mod p: ") |
---|
550 | for(int i=2; i<=mv; i++) // entries at i=0,1 not visited |
---|
551 | if(N[i] < L[i]) |
---|
552 | L[i] = N[i]; |
---|
553 | // L is now upper bound for degrees of gcd |
---|
554 | int *dg_im = new int[mv+1]; // for the degree vector of the image we don't need any entry at i=1 |
---|
555 | for(int i=2; i<=mv; i++) |
---|
556 | dg_im[i] = 0; // initialize |
---|
557 | CanonicalForm gamma_image, m=1; |
---|
558 | CanonicalForm gm=0; |
---|
559 | CanonicalForm g_image, alpha, gnew; |
---|
560 | FFGenerator gen = FFGenerator(); |
---|
561 | Variable x= Variable (1); |
---|
562 | bool divides= true; |
---|
563 | for(FFGenerator gen = FFGenerator(); gen.hasItems(); gen.next()) |
---|
564 | { |
---|
565 | alpha = gen.item(); |
---|
566 | gamma_image = reduce(gamma(alpha, x),M); // plug in alpha for var(1) |
---|
567 | if(gamma_image.isZero()) // skip lc-bad points var(1)-alpha |
---|
568 | continue; |
---|
569 | TIMING_START (alg_recursion_p) |
---|
570 | tryBrownGCD( f(alpha, x), g(alpha, x), M, g_image, fail, false ); // recursive call with one var less |
---|
571 | TIMING_END_AND_PRINT (alg_recursion_p, |
---|
572 | "time for recursive calls in alg gcd mod p: ") |
---|
573 | if(fail) |
---|
574 | return; |
---|
575 | g_image = reduce(g_image, M); |
---|
576 | if(g_image.inCoeffDomain()) // early termination |
---|
577 | { |
---|
578 | tryInvert(g_image,M,result,fail); |
---|
579 | if(fail) |
---|
580 | return; |
---|
581 | result = NN(c); |
---|
582 | return; |
---|
583 | } |
---|
584 | for(int i=2; i<=mv; i++) |
---|
585 | dg_im[i] = 0; // reset (this is necessary, because some entries may not be updated by call to leadDeg) |
---|
586 | dg_im = leadDeg(g_image, dg_im); // dg_im cannot be NIL-pointer |
---|
587 | if(isEqual(dg_im, L, 2, mv)) |
---|
588 | { |
---|
589 | CanonicalForm inv; |
---|
590 | tryInvert (firstLC (g_image), M, inv, fail); |
---|
591 | if (fail) |
---|
592 | return; |
---|
593 | g_image *= inv; |
---|
594 | g_image *= gamma_image; // multiply by multiple of image lc(gcd) |
---|
595 | g_image= reduce (g_image, M); |
---|
596 | TIMING_START (alg_newton_p) |
---|
597 | gnew= tryNewtonInterp (alpha, g_image, m, gm, x, M, fail); |
---|
598 | TIMING_END_AND_PRINT (alg_newton_p, |
---|
599 | "time for Newton interpolation in alg gcd mod p: ") |
---|
600 | // gnew = gm mod m |
---|
601 | // gnew = g_image mod var(1)-alpha |
---|
602 | // mnew = m * (var(1)-alpha) |
---|
603 | if(fail) |
---|
604 | return; |
---|
605 | m *= (x - alpha); |
---|
606 | if((firstLC(gnew) == gamma) || (gnew == gm)) // gnew did not change |
---|
607 | { |
---|
608 | TIMING_START (alg_termination_p) |
---|
609 | cf = tryvcontent(gnew, Variable(2), M, fail); |
---|
610 | if(fail) |
---|
611 | return; |
---|
612 | divides = true; |
---|
613 | g_image= gnew; |
---|
614 | g_image.tryDiv (cf, M, fail); |
---|
615 | if(fail) |
---|
616 | return; |
---|
617 | divides= tryFdivides (g_image,f, M, fail); // trial division (f) |
---|
618 | if(fail) |
---|
619 | return; |
---|
620 | if(divides) |
---|
621 | { |
---|
622 | bool divides2= tryFdivides (g_image,g, M, fail); // trial division (g) |
---|
623 | if(fail) |
---|
624 | return; |
---|
625 | if(divides2) |
---|
626 | { |
---|
627 | result = NN(reduce (c*g_image, M)); |
---|
628 | TIMING_END_AND_PRINT (alg_termination_p, |
---|
629 | "time for successful termination test in alg gcd mod p: ") |
---|
630 | return; |
---|
631 | } |
---|
632 | } |
---|
633 | TIMING_END_AND_PRINT (alg_termination_p, |
---|
634 | "time for unsuccessful termination test in alg gcd mod p: ") |
---|
635 | } |
---|
636 | gm = gnew; |
---|
637 | continue; |
---|
638 | } |
---|
639 | |
---|
640 | if(isLess(L, dg_im, 2, mv)) // dg_im > L --> current point unlucky |
---|
641 | continue; |
---|
642 | |
---|
643 | // here: isLess(dg_im, L, 2, mv) --> all previous points were unlucky |
---|
644 | m = CanonicalForm(1); // reset |
---|
645 | gm = 0; // reset |
---|
646 | for(int i=2; i<=mv; i++) // tighten bound |
---|
647 | L[i] = dg_im[i]; |
---|
648 | } |
---|
649 | // we are out of evaluation points |
---|
650 | fail = true; |
---|
651 | } |
---|
652 | |
---|
653 | static CanonicalForm |
---|
654 | myicontent ( const CanonicalForm & f, const CanonicalForm & c ) |
---|
655 | { |
---|
656 | #ifdef HAVE_NTL |
---|
657 | if (f.isOne() || c.isOne()) |
---|
658 | return 1; |
---|
659 | if ( f.inBaseDomain() && c.inBaseDomain()) |
---|
660 | { |
---|
661 | if (c.isZero()) return abs(f); |
---|
662 | return bgcd( f, c ); |
---|
663 | } |
---|
664 | else if ( (f.inCoeffDomain() && c.inCoeffDomain()) || |
---|
665 | (f.inCoeffDomain() && c.inBaseDomain()) || |
---|
666 | (f.inBaseDomain() && c.inCoeffDomain())) |
---|
667 | { |
---|
668 | if (c.isZero()) return abs (f); |
---|
669 | #ifdef HAVE_FLINT |
---|
670 | fmpz_poly_t FLINTf, FLINTc; |
---|
671 | convertFacCF2Fmpz_poly_t (FLINTf, f); |
---|
672 | convertFacCF2Fmpz_poly_t (FLINTc, c); |
---|
673 | fmpz_poly_gcd (FLINTc, FLINTc, FLINTf); |
---|
674 | CanonicalForm result; |
---|
675 | if (f.inCoeffDomain()) |
---|
676 | result= convertFmpz_poly_t2FacCF (FLINTc, f.mvar()); |
---|
677 | else |
---|
678 | result= convertFmpz_poly_t2FacCF (FLINTc, c.mvar()); |
---|
679 | fmpz_poly_clear (FLINTc); |
---|
680 | fmpz_poly_clear (FLINTf); |
---|
681 | return result; |
---|
682 | #else |
---|
683 | ZZX NTLf= convertFacCF2NTLZZX (f); |
---|
684 | ZZX NTLc= convertFacCF2NTLZZX (c); |
---|
685 | NTLc= GCD (NTLc, NTLf); |
---|
686 | if (f.inCoeffDomain()) |
---|
687 | return convertNTLZZX2CF(NTLc,f.mvar()); |
---|
688 | else |
---|
689 | return convertNTLZZX2CF(NTLc,c.mvar()); |
---|
690 | #endif |
---|
691 | } |
---|
692 | else |
---|
693 | { |
---|
694 | CanonicalForm g = c; |
---|
695 | for ( CFIterator i = f; i.hasTerms() && ! g.isOne(); i++ ) |
---|
696 | g = myicontent( i.coeff(), g ); |
---|
697 | return g; |
---|
698 | } |
---|
699 | #else |
---|
700 | return 1; |
---|
701 | #endif |
---|
702 | } |
---|
703 | |
---|
704 | static CanonicalForm myicontent ( const CanonicalForm & f ) |
---|
705 | { |
---|
706 | #ifdef HAVE_NTL |
---|
707 | return myicontent( f, 0 ); |
---|
708 | #else |
---|
709 | return 1; |
---|
710 | #endif |
---|
711 | } |
---|
712 | |
---|
713 | CanonicalForm QGCD( const CanonicalForm & F, const CanonicalForm & G ) |
---|
714 | { // f,g in Q(a)[x1,...,xn] |
---|
715 | if(F.isZero()) |
---|
716 | { |
---|
717 | if(G.isZero()) |
---|
718 | return G; // G is zero |
---|
719 | if(G.inCoeffDomain()) |
---|
720 | return CanonicalForm(1); |
---|
721 | CanonicalForm lcinv= 1/Lc (G); |
---|
722 | return G*lcinv; // return monic G |
---|
723 | } |
---|
724 | if(G.isZero()) // F is non-zero |
---|
725 | { |
---|
726 | if(F.inCoeffDomain()) |
---|
727 | return CanonicalForm(1); |
---|
728 | CanonicalForm lcinv= 1/Lc (F); |
---|
729 | return F*lcinv; // return monic F |
---|
730 | } |
---|
731 | if(F.inCoeffDomain() || G.inCoeffDomain()) |
---|
732 | return CanonicalForm(1); |
---|
733 | // here: both NOT inCoeffDomain |
---|
734 | CanonicalForm f, g, tmp, M, q, D, Dp, cl, newq, mipo; |
---|
735 | int p, i; |
---|
736 | int *bound, *other; // degree vectors |
---|
737 | bool fail; |
---|
738 | bool off_rational=!isOn(SW_RATIONAL); |
---|
739 | On( SW_RATIONAL ); // needed by bCommonDen |
---|
740 | f = F * bCommonDen(F); |
---|
741 | g = G * bCommonDen(G); |
---|
742 | TIMING_START (alg_content) |
---|
743 | CanonicalForm contf= myicontent (f); |
---|
744 | CanonicalForm contg= myicontent (g); |
---|
745 | f /= contf; |
---|
746 | g /= contg; |
---|
747 | CanonicalForm gcdcfcg= myicontent (contf, contg); |
---|
748 | TIMING_END_AND_PRINT (alg_content, "time for content in alg gcd: ") |
---|
749 | Variable a, b; |
---|
750 | if(hasFirstAlgVar(f,a)) |
---|
751 | { |
---|
752 | if(hasFirstAlgVar(g,b)) |
---|
753 | { |
---|
754 | if(b.level() > a.level()) |
---|
755 | a = b; |
---|
756 | } |
---|
757 | } |
---|
758 | else |
---|
759 | { |
---|
760 | if(!hasFirstAlgVar(g,a))// both not in extension |
---|
761 | { |
---|
762 | Off( SW_RATIONAL ); |
---|
763 | Off( SW_USE_QGCD ); |
---|
764 | tmp = gcdcfcg*gcd( f, g ); |
---|
765 | On( SW_USE_QGCD ); |
---|
766 | if (off_rational) Off(SW_RATIONAL); |
---|
767 | return tmp; |
---|
768 | } |
---|
769 | } |
---|
770 | // here: a is the biggest alg. var in f and g AND some of f,g is in extension |
---|
771 | setReduce(a,false); // do not reduce expressions modulo mipo |
---|
772 | tmp = getMipo(a); |
---|
773 | M = tmp * bCommonDen(tmp); |
---|
774 | // here: f, g in Z[a][x1,...,xn], M in Z[a] not necessarily monic |
---|
775 | Off( SW_RATIONAL ); // needed by mod |
---|
776 | // calculate upper bound for degree vector of gcd |
---|
777 | int mv = f.level(); i = g.level(); |
---|
778 | if(i > mv) |
---|
779 | mv = i; |
---|
780 | // here: mv is level of the largest variable in f, g |
---|
781 | bound = new int[mv+1]; // 'bound' could be indexed from 0 to mv, but we will only use from 1 to mv |
---|
782 | other = new int[mv+1]; |
---|
783 | for(int i=1; i<=mv; i++) // initialize 'bound', 'other' with zeros |
---|
784 | bound[i] = other[i] = 0; |
---|
785 | bound = leadDeg(f,bound); // 'bound' is set the leading degree vector of f |
---|
786 | other = leadDeg(g,other); |
---|
787 | for(int i=1; i<=mv; i++) // entry at i=0 not visited |
---|
788 | if(other[i] < bound[i]) |
---|
789 | bound[i] = other[i]; |
---|
790 | // now 'bound' is the smaller vector |
---|
791 | cl = lc(M) * lc(f) * lc(g); |
---|
792 | q = 1; |
---|
793 | D = 0; |
---|
794 | CanonicalForm test= 0; |
---|
795 | bool equal= false; |
---|
796 | for( i=cf_getNumBigPrimes()-1; i>-1; i-- ) |
---|
797 | { |
---|
798 | p = cf_getBigPrime(i); |
---|
799 | if( mod( cl, p ).isZero() ) // skip lc-bad primes |
---|
800 | continue; |
---|
801 | fail = false; |
---|
802 | setCharacteristic(p); |
---|
803 | mipo = mapinto(M); |
---|
804 | mipo /= mipo.lc(); |
---|
805 | // here: mipo is monic |
---|
806 | TIMING_START (alg_gcd_p) |
---|
807 | tryBrownGCD( mapinto(f), mapinto(g), mipo, Dp, fail ); |
---|
808 | TIMING_END_AND_PRINT (alg_gcd_p, "time for alg gcd mod p: ") |
---|
809 | if( fail ) // mipo splits in char p |
---|
810 | { |
---|
811 | setCharacteristic(0); |
---|
812 | continue; |
---|
813 | } |
---|
814 | if( Dp.inCoeffDomain() ) // early termination |
---|
815 | { |
---|
816 | tryInvert(Dp,mipo,tmp,fail); // check if zero divisor |
---|
817 | setCharacteristic(0); |
---|
818 | if(fail) |
---|
819 | continue; |
---|
820 | setReduce(a,true); |
---|
821 | if (off_rational) Off(SW_RATIONAL); else On(SW_RATIONAL); |
---|
822 | return gcdcfcg; |
---|
823 | } |
---|
824 | setCharacteristic(0); |
---|
825 | // here: Dp NOT inCoeffDomain |
---|
826 | for(int i=1; i<=mv; i++) |
---|
827 | other[i] = 0; // reset (this is necessary, because some entries may not be updated by call to leadDeg) |
---|
828 | other = leadDeg(Dp,other); |
---|
829 | |
---|
830 | if(isEqual(bound, other, 1, mv)) // equal |
---|
831 | { |
---|
832 | chineseRemainder( D, q, mapinto(Dp), p, tmp, newq ); |
---|
833 | // tmp = Dp mod p |
---|
834 | // tmp = D mod q |
---|
835 | // newq = p*q |
---|
836 | q = newq; |
---|
837 | if( D != tmp ) |
---|
838 | D = tmp; |
---|
839 | On( SW_RATIONAL ); |
---|
840 | TIMING_START (alg_reconstruction) |
---|
841 | tmp = Farey( D, q ); // Farey |
---|
842 | tmp *= bCommonDen (tmp); |
---|
843 | TIMING_END_AND_PRINT (alg_reconstruction, |
---|
844 | "time for rational reconstruction in alg gcd: ") |
---|
845 | setReduce(a,true); // reduce expressions modulo mipo |
---|
846 | On( SW_RATIONAL ); // needed by fdivides |
---|
847 | if (test != tmp) |
---|
848 | test= tmp; |
---|
849 | else |
---|
850 | equal= true; // modular image did not add any new information |
---|
851 | TIMING_START (alg_termination) |
---|
852 | #ifdef HAVE_NTL |
---|
853 | #ifdef HAVE_FLINT |
---|
854 | if (equal && tmp.isUnivariate() && f.isUnivariate() && g.isUnivariate() |
---|
855 | && f.level() == tmp.level() && tmp.level() == g.level()) |
---|
856 | { |
---|
857 | CanonicalForm Q, R; |
---|
858 | newtonDivrem (f, tmp, Q, R); |
---|
859 | if (R.isZero()) |
---|
860 | { |
---|
861 | newtonDivrem (g, tmp, Q, R); |
---|
862 | if (R.isZero()) |
---|
863 | { |
---|
864 | Off (SW_RATIONAL); |
---|
865 | setReduce (a,true); |
---|
866 | if (off_rational) Off(SW_RATIONAL); else On(SW_RATIONAL); |
---|
867 | TIMING_END_AND_PRINT (alg_termination, |
---|
868 | "time for successful termination test in alg gcd: ") |
---|
869 | return tmp*gcdcfcg; |
---|
870 | } |
---|
871 | } |
---|
872 | } |
---|
873 | else |
---|
874 | #endif |
---|
875 | #endif |
---|
876 | if(equal && fdivides( tmp, f ) && fdivides( tmp, g )) // trial division |
---|
877 | { |
---|
878 | Off( SW_RATIONAL ); |
---|
879 | setReduce(a,true); |
---|
880 | if (off_rational) Off(SW_RATIONAL); else On(SW_RATIONAL); |
---|
881 | TIMING_END_AND_PRINT (alg_termination, |
---|
882 | "time for successful termination test in alg gcd: ") |
---|
883 | return tmp*gcdcfcg; |
---|
884 | } |
---|
885 | TIMING_END_AND_PRINT (alg_termination, |
---|
886 | "time for unsuccessful termination test in alg gcd: ") |
---|
887 | Off( SW_RATIONAL ); |
---|
888 | setReduce(a,false); // do not reduce expressions modulo mipo |
---|
889 | continue; |
---|
890 | } |
---|
891 | if( isLess(bound, other, 1, mv) ) // current prime unlucky |
---|
892 | continue; |
---|
893 | // here: isLess(other, bound, 1, mv) ) ==> all previous primes unlucky |
---|
894 | q = p; |
---|
895 | D = mapinto(Dp); // shortcut CRA // shortcut CRA |
---|
896 | for(int i=1; i<=mv; i++) // tighten bound |
---|
897 | bound[i] = other[i]; |
---|
898 | } |
---|
899 | // hopefully, we never reach this point |
---|
900 | setReduce(a,true); |
---|
901 | Off( SW_USE_QGCD ); |
---|
902 | D = gcdcfcg*gcd( f, g ); |
---|
903 | On( SW_USE_QGCD ); |
---|
904 | if (off_rational) Off(SW_RATIONAL); else On(SW_RATIONAL); |
---|
905 | return D; |
---|
906 | } |
---|
907 | |
---|
908 | |
---|
909 | int * leadDeg(const CanonicalForm & f, int *degs) |
---|
910 | { // leading degree vector w.r.t. lex. monomial order x(i+1) > x(i) |
---|
911 | // if f is in a coeff domain, the zero pointer is returned |
---|
912 | // 'a' should point to an array of sufficient size level(f)+1 |
---|
913 | if(f.inCoeffDomain()) |
---|
914 | return 0; |
---|
915 | CanonicalForm tmp = f; |
---|
916 | do |
---|
917 | { |
---|
918 | degs[tmp.level()] = tmp.degree(); |
---|
919 | tmp = LC(tmp); |
---|
920 | } |
---|
921 | while(!tmp.inCoeffDomain()); |
---|
922 | return degs; |
---|
923 | } |
---|
924 | |
---|
925 | |
---|
926 | bool isLess(int *a, int *b, int lower, int upper) |
---|
927 | { // compares the degree vectors a,b on the specified part. Note: x(i+1) > x(i) |
---|
928 | for(int i=upper; i>=lower; i--) |
---|
929 | if(a[i] == b[i]) |
---|
930 | continue; |
---|
931 | else |
---|
932 | return a[i] < b[i]; |
---|
933 | return true; |
---|
934 | } |
---|
935 | |
---|
936 | |
---|
937 | bool isEqual(int *a, int *b, int lower, int upper) |
---|
938 | { // compares the degree vectors a,b on the specified part. Note: x(i+1) > x(i) |
---|
939 | for(int i=lower; i<=upper; i++) |
---|
940 | if(a[i] != b[i]) |
---|
941 | return false; |
---|
942 | return true; |
---|
943 | } |
---|
944 | |
---|
945 | |
---|
946 | CanonicalForm firstLC(const CanonicalForm & f) |
---|
947 | { // returns the leading coefficient (LC) of level <= 1 |
---|
948 | CanonicalForm ret = f; |
---|
949 | while(ret.level() > 1) |
---|
950 | ret = LC(ret); |
---|
951 | return ret; |
---|
952 | } |
---|
953 | |
---|
954 | #ifndef HAVE_NTL |
---|
955 | void tryExtgcd( const CanonicalForm & F, const CanonicalForm & G, const CanonicalForm & M, CanonicalForm & result, CanonicalForm & s, CanonicalForm & t, bool & fail ) |
---|
956 | { // F, G are univariate polynomials (i.e. they have exactly one polynomial variable) |
---|
957 | // F and G must have the same level AND level > 0 |
---|
958 | // we try to calculate gcd(F,G) = s*F + t*G |
---|
959 | // if a zero divisor is encountered, 'fail' is set to one |
---|
960 | // M is assumed to be monic |
---|
961 | CanonicalForm P; |
---|
962 | if(F.inCoeffDomain()) |
---|
963 | { |
---|
964 | tryInvert( F, M, P, fail ); |
---|
965 | if(fail) |
---|
966 | return; |
---|
967 | result = 1; |
---|
968 | s = P; t = 0; |
---|
969 | return; |
---|
970 | } |
---|
971 | if(G.inCoeffDomain()) |
---|
972 | { |
---|
973 | tryInvert( G, M, P, fail ); |
---|
974 | if(fail) |
---|
975 | return; |
---|
976 | result = 1; |
---|
977 | s = 0; t = P; |
---|
978 | return; |
---|
979 | } |
---|
980 | // here: both not inCoeffDomain |
---|
981 | CanonicalForm inv, rem, tmp, u, v, q, sum=0; |
---|
982 | if( F.degree() > G.degree() ) |
---|
983 | { |
---|
984 | P = F; result = G; s=v=0; t=u=1; |
---|
985 | } |
---|
986 | else |
---|
987 | { |
---|
988 | P = G; result = F; s=v=1; t=u=0; |
---|
989 | } |
---|
990 | Variable x = P.mvar(); |
---|
991 | // here: degree(P) >= degree(result) |
---|
992 | while(true) |
---|
993 | { |
---|
994 | tryDivrem (P, result, q, rem, inv, M, fail); |
---|
995 | if(fail) |
---|
996 | return; |
---|
997 | if( rem.isZero() ) |
---|
998 | { |
---|
999 | s*=inv; |
---|
1000 | s= reduce (s, M); |
---|
1001 | t*=inv; |
---|
1002 | t= reduce (t, M); |
---|
1003 | result *= inv; // monify result |
---|
1004 | result= reduce (result, M); |
---|
1005 | return; |
---|
1006 | } |
---|
1007 | sum += q; |
---|
1008 | if(result.degree(x) >= rem.degree(x)) |
---|
1009 | { |
---|
1010 | P=result; |
---|
1011 | result=rem; |
---|
1012 | tmp=u-sum*s; |
---|
1013 | u=s; |
---|
1014 | s=tmp; |
---|
1015 | tmp=v-sum*t; |
---|
1016 | v=t; |
---|
1017 | t=tmp; |
---|
1018 | sum = 0; // reset |
---|
1019 | } |
---|
1020 | else |
---|
1021 | P = rem; |
---|
1022 | } |
---|
1023 | } |
---|
1024 | #endif |
---|
1025 | |
---|
1026 | static CanonicalForm trycontent ( const CanonicalForm & f, const Variable & x, const CanonicalForm & M, bool & fail ) |
---|
1027 | { // as 'content', but takes care of zero divisors |
---|
1028 | ASSERT( x.level() > 0, "cannot calculate content with respect to algebraic variable" ); |
---|
1029 | Variable y = f.mvar(); |
---|
1030 | if ( y == x ) |
---|
1031 | return trycf_content( f, 0, M, fail ); |
---|
1032 | if ( y < x ) |
---|
1033 | return f; |
---|
1034 | return swapvar( trycontent( swapvar( f, y, x ), y, M, fail ), y, x ); |
---|
1035 | } |
---|
1036 | |
---|
1037 | |
---|
1038 | static CanonicalForm tryvcontent ( const CanonicalForm & f, const Variable & x, const CanonicalForm & M, bool & fail ) |
---|
1039 | { // as vcontent, but takes care of zero divisors |
---|
1040 | ASSERT( x.level() > 0, "cannot calculate vcontent with respect to algebraic variable" ); |
---|
1041 | if ( f.mvar() <= x ) |
---|
1042 | return trycontent( f, x, M, fail ); |
---|
1043 | CFIterator i; |
---|
1044 | CanonicalForm d = 0, e, ret; |
---|
1045 | for ( i = f; i.hasTerms() && ! d.isOne() && ! fail; i++ ) |
---|
1046 | { |
---|
1047 | e = tryvcontent( i.coeff(), x, M, fail ); |
---|
1048 | if(fail) |
---|
1049 | break; |
---|
1050 | tryBrownGCD( d, e, M, ret, fail ); |
---|
1051 | d = ret; |
---|
1052 | } |
---|
1053 | return d; |
---|
1054 | } |
---|
1055 | |
---|
1056 | |
---|
1057 | static CanonicalForm trycf_content ( const CanonicalForm & f, const CanonicalForm & g, const CanonicalForm & M, bool & fail ) |
---|
1058 | { // as cf_content, but takes care of zero divisors |
---|
1059 | if ( f.inPolyDomain() || ( f.inExtension() && ! getReduce( f.mvar() ) ) ) |
---|
1060 | { |
---|
1061 | CFIterator i = f; |
---|
1062 | CanonicalForm tmp = g, result; |
---|
1063 | while ( i.hasTerms() && ! tmp.isOne() && ! fail ) |
---|
1064 | { |
---|
1065 | tryBrownGCD( i.coeff(), tmp, M, result, fail ); |
---|
1066 | tmp = result; |
---|
1067 | i++; |
---|
1068 | } |
---|
1069 | return result; |
---|
1070 | } |
---|
1071 | return abs( f ); |
---|
1072 | } |
---|
1073 | |
---|