1 | // ideal.h |
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2 | |
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3 | // This class implements a saturated binomial ideal. Such an ideal is |
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4 | // given by its generators. |
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5 | // If SUPPORT_DRIVEN_METHODS_EXTENDED are enabled, the generators are not |
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6 | // stored in a single list, but classified according to their support. This |
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7 | // representation, together with the subset_tree data structure, allows a |
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8 | // faster search for reducers. |
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9 | |
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10 | // The entries of the two-dimensional array "subsets_of_support" are to be |
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11 | // read as bit vectors with List_Support_Variables components (the constant |
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12 | // List_Support_Variables is defined in globals.h). So all entries |
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13 | // are in the range between 0 and 2^List_Support_Variables -1. |
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14 | // subsets_of_support[i] is an array that contains all binary vectors whose |
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15 | // support is a subset of that of i, i read as a binary vector. |
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16 | // number_of_subsets[i] is the number of these subsets, i.e. the length of |
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17 | // the array subsets_of_support[i]. |
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18 | |
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19 | // If, for example, List_Support_Variables=8, the datastructure subset_tree |
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20 | // has 6561=8^3 entries. |
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21 | |
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22 | |
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23 | |
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24 | #ifndef IDEAL_H |
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25 | #define IDEAL_H |
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26 | |
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27 | |
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28 | |
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29 | #include "list.h" |
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30 | #include "matrix.h" |
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31 | |
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32 | |
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33 | |
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34 | |
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35 | const short Number_of_Lists=1<<List_Support_Variables; |
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36 | |
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37 | |
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38 | |
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39 | |
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40 | ////////////////////// struct subset_tree //////////////////////////////////// |
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41 | |
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42 | typedef struct |
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43 | { |
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44 | short* subsets_of_support[Number_of_Lists]; |
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45 | short number_of_subsets[Number_of_Lists]; |
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46 | } subset_tree; |
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47 | |
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48 | //////////////////////// class ideal ///////////////////////////////////////// |
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49 | |
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50 | class ideal |
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51 | { |
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52 | |
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53 | private: |
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54 | |
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55 | // generator lists |
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56 | |
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57 | #ifdef SUPPORT_DRIVEN_METHODS_EXTENDED |
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58 | |
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59 | subset_tree S; |
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60 | |
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61 | list generators[Number_of_Lists]; |
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62 | |
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63 | list new_generators[Number_of_Lists]; |
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64 | // Only needed in some special versions of BuchbergerŽs algorithm to |
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65 | // store newly found generators. |
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66 | |
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67 | #endif // SUPPORT_DRIVEN_METHODS_EXTENDED |
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68 | |
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69 | |
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70 | #ifdef NO_SUPPORT_DRIVEN_METHODS_EXTENDED |
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71 | |
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72 | list generators; |
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73 | |
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74 | list new_generators; |
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75 | // Only needed in some special versions of BuchbergerŽs algorithm to |
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76 | // store newly found generators. |
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77 | |
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78 | #endif // NO_SUPPORT_DRIVEN_METHODS_EXTENDED |
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79 | |
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80 | // flags for the use of the S-pair criteria and the autoreduction |
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81 | |
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82 | short rel_primeness; |
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83 | short M_criterion; |
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84 | short F_criterion; |
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85 | short B_criterion; |
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86 | short second_criterion; |
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87 | // When BuchbergerŽs algorithm is called, we only use one argument which |
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88 | // describes the combination of the criteria to be used (see in globals.h). |
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89 | // But we use five flags instead of one here because this is a little more |
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90 | // efficient. |
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91 | // The standard setting enables the relative primeness criterion, the M- and |
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92 | // the B-criterion. |
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93 | |
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94 | float interreduction_percentage; |
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95 | // To determine the autoreduction frequency, i.e. the percentage of new |
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96 | // generators (i.e. generators found since the last autoreduction) with |
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97 | // respect to the total size of an ideal that must be reached to cause an |
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98 | // interreduction. |
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99 | // If Interreduction_Percentage==0, interreduction is done after each |
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100 | // S-Pair computation. If Interreduction_Percentage<0, interreduction is |
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101 | // only done once at the end of the algorithm. |
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102 | // The standard setting is 12 (=12%). |
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103 | |
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104 | // further members |
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105 | |
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106 | term_ordering w; |
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107 | // For technical reasons, the term ordering is taken as a member. |
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108 | // So we do not need to pass it as an argument to each ideal function, |
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109 | // and the management of the generator lists is easier and safer when |
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110 | // using SUPPORT_DRIVEN_METHODS_EXTENDED. |
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111 | |
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112 | list aux_list; |
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113 | // an auxiliary list for keeping (for example) S-pairs |
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114 | |
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115 | long size; |
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116 | // the actual number of generators of the ideal |
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117 | // Also used as an "error flag"; a negative size means that an error has |
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118 | // occurred: |
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119 | // -1 indicates a "semantic" error (which occurs e.g. if some constructor |
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120 | // argument is out of range) |
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121 | // -2 indicates an error occured when reading from a file |
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122 | // -3 indicates an overflow of an integer type variable |
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123 | |
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124 | |
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125 | long number_of_new_binomials; |
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126 | // the number of newly found generators |
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127 | |
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128 | // private member functions |
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129 | |
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130 | // subroutines for building and deleting the subset_tree data structure |
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131 | // (implemented in ideal.cc) |
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132 | |
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133 | #ifdef SUPPORT_DRIVEN_METHODS_EXTENDED |
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134 | |
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135 | void create_subset_tree(); |
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136 | void destroy_subset_tree(); |
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137 | |
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138 | #endif // SUPPORT_DRIVEN_METHODS_EXTENDED |
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139 | |
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140 | // subroutines for BuchbergerŽs algorithm (implemented in Buchberger.cc except |
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141 | // from the first two implemeted in ideal.cc) |
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142 | |
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143 | // Some of these procedures do not interact with all versions of BuchbergerŽs |
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144 | // algorithm. Generally, they cannot be combined as one likes. This header |
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145 | // file only gives a brief overview of their features. For more detailed |
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146 | // comments see the implementation. |
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147 | |
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148 | ideal& add_new_generator(binomial& bin); |
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149 | ideal& add_generator(binomial& bin); |
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150 | // Inserts a (new) generator in the appropriate list. |
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151 | |
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152 | short add_new_generators(); |
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153 | // Moves the new generators to the generator lists. |
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154 | |
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155 | // S-pair computation |
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156 | |
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157 | BOOLEAN unnecessary_S_pair(list_iterator&, list_iterator&) const; |
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158 | // Checks if the S_Pair of the binomials referenced by the iterators can be |
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159 | // discarded (by the criteria chosen in globals.h). |
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160 | |
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161 | ideal& compute_actual_S_pairs_1(); |
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162 | ideal& compute_actual_S_pairs_1a(); |
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163 | ideal& compute_actual_S_pairs_2(); |
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164 | ideal& compute_actual_S_pairs_3(); |
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165 | // different versions for computing the S-binomials of the actual generators |
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166 | // They differ for example in the following points: |
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167 | // - They insert the S-binomials in the order in which they are computed, in |
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168 | // the order given by the term ordering w or according to their support. |
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169 | // - Some reduce an S-binomial directly after its computation by the ideal |
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170 | // generators, others reduce it later (after having computed more |
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171 | // S-binomials, hence more possible reducers). |
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172 | // - Some reduce the ideal generators immediately by a newly computed |
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173 | // S-binomial, others donŽt. |
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174 | // In each case, the use of the list flag "done" guarantees that we perform |
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175 | // the S-pair computation only once for each binomial pair. |
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176 | |
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177 | // minimalization / autoreduction |
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178 | |
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179 | ideal& reduce_by(const binomial&, list_iterator&, list_iterator&); |
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180 | // Reduces the heads of the ideal generators by the given binomial |
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181 | // (used by some versions of the S-pair computation). |
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182 | |
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183 | ideal& minimalize_S_pairs(); |
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184 | ideal& minimalize_new_generators(); |
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185 | ideal& minimalize(); |
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186 | // Performs an autoreduction of the binomials stored in aux_list |
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187 | // respectively in the list(s) new_generators respectively in the |
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188 | // generators. |
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189 | // The respective list(s) is (are) reduced to a minimal set (not |
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190 | // necessarily to a reduced set); this means that only the heads of the |
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191 | // binomials are reduced, not their tails. This strategy showed to be |
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192 | // a little more efficient than a complete autoreduction. |
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193 | // The last two procedures use the "head_reduced" flag of the list class |
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194 | // to avoid unnecessary tests for interreduction. |
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195 | |
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196 | ideal& final_reduce(); |
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197 | // final interreduction of the ideal generators |
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198 | // It seems to be slightly more efficient to perform a complete |
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199 | // interreduction only at the end of Buchberger's algorithm and |
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200 | // to replace the intermediate interreductions by minimalizations. |
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201 | // For efficiency reasons, this routine is designed for reducing a |
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202 | // Groebner basis of an saturated ideal. Reducing another generating |
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203 | // set with it may cause inconsistencies (cf. comment in the implementation). |
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204 | |
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205 | // constructor subroutines for the handling of the different algorithms |
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206 | // (implemented in ideal.cc) |
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207 | // The "Conti-Traverso constructors" create a toric ideal directly from the |
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208 | // input matrix and the term ordering given by the input weight vector. |
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209 | // The other constructors create a toric ideal from a lattice basis of |
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210 | // the kernel of the input matrix. |
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211 | |
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212 | ideal& Conti_Traverso_ideal(matrix&, const term_ordering&); |
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213 | ideal& Positive_Conti_Traverso_ideal(matrix&, const term_ordering&); |
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214 | ideal& Pottier_ideal(matrix&, const term_ordering&); |
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215 | ideal& Hosten_Sturmfels_ideal(matrix&, const term_ordering&); |
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216 | ideal& DiBiase_Urbanke_ideal(matrix&, const term_ordering&); |
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217 | ideal& Bigatti_LaScala_Robbiano_ideal(matrix&, const term_ordering&); |
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218 | |
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219 | public: |
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220 | |
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221 | // constructors and destructor (implemented in ideal.cc) |
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222 | |
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223 | ideal(matrix&, const term_ordering&, const short& algorithm); |
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224 | // Creates a binomial ideal from the given matrix using the given algorithm |
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225 | // (see in globals.h). |
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226 | // The arguments are checked for consistency as far as possible. |
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227 | // The term ordering is needed to determine the leading terms of the |
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228 | // binomials in the resulting generating set. Neither is the generated ideal |
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229 | // saturated nor is it given in form of a Groebner basis. |
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230 | // Such computations must be explicitely demanded. |
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231 | // The argument matrix cannot be declared as const because the constructor |
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232 | // may call the LLL-algorithm to compute the matrix kernel (if this hasnŽt |
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233 | // been done yet). This algorithm changes some matrix members. But the |
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234 | // real matrix remains, of course, unchanged. |
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235 | |
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236 | ideal(const ideal&); |
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237 | // copy-constructor |
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238 | // It might be useful to keep several Groebner bases of the same ideal |
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239 | // (or of an ideal and its elimination ideal). |
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240 | |
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241 | ideal(ifstream&, const term_ordering&, const short& number_of_generators); |
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242 | // Reads an ideal from a given ifstream in the following way: |
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243 | // A block of integers is converted into a binomial |
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244 | // that is stored in the generator list(s) with respect to the given |
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245 | // term ordering. The size of such a block is the size of the given |
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246 | // term_ordering, i.e. the number of variables for which it was constructed. |
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247 | // This is done number_of_generators times. |
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248 | |
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249 | ~ideal(); |
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250 | // destructor |
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251 | |
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252 | // object information (implemented in ideal.cc) |
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253 | |
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254 | long number_of_generators() const; |
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255 | // Returns the actual number of generators. |
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256 | |
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257 | short error_status() const; |
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258 | // Returns -1 if an error has occurred (i.e. size<0), else 0. |
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259 | |
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260 | // Buchberger stuff (implemented in Buchberger.cc) |
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261 | |
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262 | ideal& reduced_Groebner_basis_1(const short& S_pair_criteria=11, |
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263 | const float& interred_percentage=12.0); |
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264 | ideal& reduced_Groebner_basis_1a(const short& S_pair_criteria=11, |
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265 | const float& interred_percentage=12.0); |
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266 | ideal& reduced_Groebner_basis_2(const short& S_pair_criteria=11, |
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267 | const float& interred_percentage=12.0); |
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268 | ideal& reduced_Groebner_basis_3(const short& S_pair_criteria=11, |
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269 | const float& interred_percentage=12.0); |
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270 | // Several different versions of BuchbergerŽs algorithm for computing |
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271 | // the reduced Groebner basis of the actual ideal. They differ in the |
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272 | // steering of the algorithm (i.e. in the way in which the S-pair |
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273 | // computation and reduction is organized). Further variants can be |
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274 | // obtained by setting the flags in globals.h (autoreduction frequency, |
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275 | // use of the S-pair criteria and the support information). |
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276 | // Except from the variant 1a (which orders the S-pairs with respect to w |
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277 | // and which showed to be quite slow), the efficiency of these algorithms |
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278 | // does not differ too much. |
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279 | // The first argument indicates which criteria will be used to discard |
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280 | // unnecessary S-pairs. For an explaination of how this works, see in |
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281 | // globals.h. The default value 11 means that we use the criterion of |
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282 | // relatively prime leading terms as well as the M- and the B-criterion. |
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283 | // The second argument specifies the interreduction frequency; see the |
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284 | // comment for the member interreduction_percentage above. The standard |
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285 | // value is 12%. |
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286 | // ATTENTION: In spite of the different argument type you should never |
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287 | // try to use these functions with one default argument (either two or |
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288 | // none). When writing |
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289 | // reduced_Groebner_basis_1(10.5), |
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290 | // the GNU C++ compiler casts 10.5 into an integer and takes it as the |
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291 | // argument S_pair_criteria! |
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292 | // ATTENTION: If the input ideal is not saturated, the computed reduced |
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293 | // Groebner basis is not that of the input ideal, but that of an ideal |
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294 | // "between" the input ideal and its saturation. |
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295 | |
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296 | ideal& reduced_Groebner_basis(const short& version=1, |
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297 | const short& S_pair_criteria=11, |
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298 | const float& interred_percentage=12.0); |
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299 | // Takes the version of BuchbergerŽs algorithm as above as an argument |
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300 | // (to allow a call of different versions of our IP-algorithms from the |
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301 | // command line). To call version 1a, the first argument has to be zero. |
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302 | |
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303 | binomial& reduce(binomial& bin, BOOLEAN complete=TRUE) const; |
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304 | // Reduces a binomial by the actual generators. |
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305 | // To reduce a binomial by the ideal (outside of BuchbergerŽs algorithm), |
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306 | // be sure to have computed the reduced Groebner basis before. |
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307 | // The flag "complete" indicates if the binomial is reduced completely |
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308 | // (head and tail); if it is set to FALSE, only the tail will be reduced. |
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309 | // Using this flag in a clever manner allows to improve the performance |
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310 | // of Buchberger's algorithm by up to three percent. |
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311 | |
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312 | // special features needed by our IP-algorithms |
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313 | // (implemented in ideal_stuff.cc) |
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314 | |
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315 | ideal& eliminate(); |
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316 | // Eliminates the generators involving elimination variables |
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317 | // (with respect to the term ordering w). |
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318 | // The reduced Groebner basis with respect to w should have been computed |
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319 | // before calling this routine. |
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320 | |
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321 | ideal& pseudo_eliminate(); |
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322 | // Eliminates the generators involving the last weighted variable - |
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323 | // a special routine needed by the algorithm of Bigatti, La Scala and |
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324 | // Robbiano. |
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325 | |
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326 | ideal& change_term_ordering_to(const term_ordering& _w); |
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327 | // Replaces the actual term ordering by the argument ordering. |
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328 | // Their "compatibility" (number of variables) is checked, head and tail |
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329 | // of the generators are adapted to the new ordering. |
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330 | // This may especially involve a rebuilding of the list structure if |
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331 | // SUPPORT_DRIVEN_METHODS_EXTENDED are enabled. |
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332 | |
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333 | ideal& swap_variables(const short& i, const short& j); |
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334 | // Swaps the i-th and the j-th variable in all generators as well as the |
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335 | // corresponding components of the term ordering's weight vector. |
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336 | // If SUPPORT_DRIVEN_METHODS are enabled, the list structure is rebuilt |
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337 | // according to the new order on the variables. |
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338 | |
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339 | ideal& swap_variables_unsafe(const short& i, const short& j); |
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340 | // Swaps the i-th and the j-th variable in all generators as well as the |
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341 | // corresponding components of the term ordering's weight vector. |
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342 | // DANGER: The head/tail structure is not rebuilt! |
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343 | |
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344 | ideal& flip_variable_unsafe(const short& i); |
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345 | // Inverts the sign of the i-th variable in all generators. |
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346 | // DANGER: The list structure is not rebuilt! |
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347 | |
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348 | // output (implemented in ideal.cc) |
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349 | |
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350 | void print() const; |
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351 | void print_all() const; |
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352 | // Writes the ideal to the standard output medium. |
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353 | // The second routine also prints the S-pair flags. |
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354 | |
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355 | void print(FILE*) const; |
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356 | void print_all(FILE*) const; |
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357 | // Writes the ideal to the referenced file which has to be opened |
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358 | // for writing before. |
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359 | |
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360 | void print(ofstream&) const; |
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361 | void print_all(ofstream &) const; |
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362 | // Writes the ideal to the given ofstream. |
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363 | |
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364 | void format_print(ofstream&) const; |
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365 | // Writes the ideal generators to the given ofstream in a format |
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366 | // that allows them to be reread by the ideal constructor from an ifstream. |
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367 | |
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368 | }; |
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369 | |
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370 | #endif // IDEAL_H |
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