1 | // -*- Mode: C++; tab-width: 2; indent-tabs-mode: nil; c-basic-offset: 2 -*- |
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2 | /*****************************************************************************\ |
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3 | * Computer Algebra System SINGULAR |
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4 | \*****************************************************************************/ |
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5 | /** @file syzextra.h |
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6 | * |
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7 | * Computation of Syzygies |
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8 | * |
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9 | * ABSTRACT: Computation of Syzygies due to Schreyer |
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10 | * |
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11 | * @author Oleksandr Motsak |
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12 | * |
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13 | **/ |
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14 | /*****************************************************************************/ |
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15 | |
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16 | #ifndef SYZEXTRA_H |
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17 | #define SYZEXTRA_H |
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18 | |
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19 | #include <vector> |
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20 | #include <map> |
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21 | #include <string.h> |
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22 | #include <stack> |
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23 | |
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24 | // include basic definitions |
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25 | #include "singularxx_defs.h" |
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26 | #include "kernel/ideals.h" |
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27 | |
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28 | class idrec; typedef idrec * idhdl; |
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29 | |
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30 | class kBucket; typedef kBucket* kBucket_pt; |
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31 | |
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32 | #ifndef NOPRODUCT |
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33 | # define NOPRODUCT 1 |
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34 | #endif |
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35 | |
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36 | // set to 1 if all leading coeffs are assumed to be all =1... |
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37 | // note the use of simplify on input in SSinit! |
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38 | #ifndef NODIVISION |
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39 | # define NODIVISION 1 |
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40 | #endif |
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41 | |
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42 | poly leadmonom(const poly p, const ring r, const bool bSetZeroComp = true); |
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43 | |
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44 | /// return the tail of a given polynomial or vector |
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45 | /// returns NULL if input is NULL, otherwise |
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46 | /// the result is a new polynomial/vector in the ring r |
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47 | poly p_Tail(const poly p, const ring r); |
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48 | |
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49 | |
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50 | /// return the tail of a given ideal or module |
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51 | /// returns NULL if input is NULL, otherwise |
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52 | /// the result is a new ideal/module in the ring r |
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53 | /// NOTE: the resulting rank is autocorrected |
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54 | ideal id_Tail(const ideal id, const ring r); |
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55 | |
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56 | /// inplace sorting of the module (ideal) id wrt <_(c,ds) |
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57 | void Sort_c_ds(const ideal id, const ring r); |
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58 | |
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59 | |
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60 | class sBucket; typedef sBucket* sBucket_pt; |
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61 | |
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62 | /** @class SBucketFactory syzextra.h |
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63 | * |
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64 | * sBucket Factory |
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65 | * |
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66 | * Cleate/store/reuse buckets |
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67 | * |
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68 | */ |
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69 | class SBucketFactory: private std::stack <sBucket_pt> |
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70 | { |
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71 | private: |
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72 | typedef std::stack <sBucket_pt> Base; |
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73 | // typedef std::vector<Bucket> Memory; |
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74 | // typedef std::deque <Bucket> Memory; |
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75 | // typedef std::stack <Bucket, Memory > Base; |
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76 | |
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77 | public: |
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78 | typedef Base::value_type Bucket; |
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79 | |
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80 | SBucketFactory(const ring r) |
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81 | #ifndef SING_NDEBUG |
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82 | : m_ring(r) |
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83 | #endif |
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84 | { |
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85 | push ( _CreateBucket(r) ); // start with at least one sBucket...? |
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86 | assume( top() != NULL ); |
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87 | }; |
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88 | |
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89 | ~SBucketFactory() |
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90 | { |
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91 | while( !empty() ) |
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92 | { |
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93 | _DestroyBucket( top() ); |
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94 | pop(); |
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95 | } |
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96 | } |
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97 | |
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98 | Bucket getBucket(const ring r, const bool remove = true) |
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99 | { |
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100 | assume( r == m_ring ); |
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101 | |
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102 | Bucket bt = NULL; |
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103 | |
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104 | if( !empty() ) |
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105 | { |
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106 | bt = top(); |
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107 | |
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108 | if( remove ) |
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109 | pop(); |
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110 | } |
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111 | else |
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112 | { |
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113 | bt = _CreateBucket(r); |
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114 | |
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115 | if( !remove ) |
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116 | { |
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117 | push(bt); |
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118 | assume( bt == top() ); |
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119 | } |
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120 | } |
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121 | |
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122 | assume( bt != NULL ); |
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123 | assume( _IsBucketEmpty(bt) ); |
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124 | assume( r == _GetBucketRing(bt) ); |
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125 | |
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126 | return bt; |
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127 | } |
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128 | |
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129 | // TODO: this may be spared if we give-out a smart Bucket (which returns here upon its destructor!) |
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130 | void putBucket(const Bucket & bt, const bool replace = false) |
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131 | { |
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132 | assume( bt != NULL ); |
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133 | assume( _IsBucketEmpty(bt) ); |
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134 | assume( m_ring == _GetBucketRing(bt) ); |
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135 | |
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136 | if( empty() ) |
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137 | push( bt ); |
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138 | else |
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139 | { |
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140 | if( replace ) |
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141 | top() = bt; |
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142 | else |
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143 | { |
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144 | if( bt != top() ) |
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145 | push( bt ); |
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146 | } |
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147 | } |
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148 | |
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149 | assume( bt == top() ); |
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150 | } |
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151 | |
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152 | private: |
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153 | |
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154 | #ifndef SING_NDEBUG |
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155 | const ring m_ring; ///< For debugging: all buckets are over the same ring... right?! |
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156 | |
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157 | /// get bucket ring |
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158 | static ring _GetBucketRing(const Bucket& bt); |
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159 | |
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160 | static bool _IsBucketEmpty(const Bucket& bt); |
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161 | #endif |
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162 | |
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163 | /// inital allocation for new buckets |
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164 | static Bucket _CreateBucket(const ring r); |
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165 | |
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166 | /// we only expect empty buckets to be left at the end for destructor |
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167 | /// bt will be set to NULL |
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168 | static void _DestroyBucket(Bucket & bt); |
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169 | |
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170 | private: |
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171 | SBucketFactory(); |
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172 | SBucketFactory(const SBucketFactory&); |
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173 | void operator=(const SBucketFactory&); |
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174 | |
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175 | }; |
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176 | |
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177 | |
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178 | |
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179 | |
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180 | |
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181 | |
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182 | |
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183 | /// Computation attribute storage |
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184 | struct SchreyerSyzygyComputationFlags |
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185 | { |
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186 | SchreyerSyzygyComputationFlags(idhdl rootRingHdl); |
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187 | |
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188 | SchreyerSyzygyComputationFlags(const SchreyerSyzygyComputationFlags& attr): |
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189 | OPT__DEBUG(attr.OPT__DEBUG), |
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190 | OPT__LEAD2SYZ(attr.OPT__LEAD2SYZ), OPT__TAILREDSYZ(attr.OPT__TAILREDSYZ), |
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191 | OPT__HYBRIDNF(attr.OPT__HYBRIDNF), OPT__IGNORETAILS(attr.OPT__IGNORETAILS), |
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192 | OPT__SYZNUMBER(attr.OPT__SYZNUMBER), OPT__TREEOUTPUT(attr.OPT__TREEOUTPUT), |
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193 | OPT__SYZCHECK(attr.OPT__SYZCHECK), OPT__PROT(attr.OPT__PROT), |
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194 | OPT__NOCACHING(attr.OPT__NOCACHING), |
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195 | m_rBaseRing(attr.m_rBaseRing) |
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196 | {} |
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197 | |
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198 | /// output all the intermediate states |
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199 | const int OPT__DEBUG; // DebugOutput; |
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200 | |
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201 | /// ? |
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202 | const int OPT__LEAD2SYZ; // TwoLeadingSyzygyTerms; |
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203 | |
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204 | /// Reduce syzygy tails wrt the leading syzygy terms |
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205 | const int OPT__TAILREDSYZ; // TailReducedSyzygies; |
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206 | |
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207 | /// Use the usual NF's S-poly reduction while dropping lower order terms |
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208 | /// 2 means - smart selection! |
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209 | const int OPT__HYBRIDNF; // UseHybridNF |
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210 | |
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211 | |
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212 | /// ignore tails and compute the pure Schreyer frame |
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213 | const int OPT__IGNORETAILS; // @IGNORETAILS |
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214 | |
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215 | /// Syzygy level (within a resolution) |
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216 | mutable int OPT__SYZNUMBER; |
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217 | |
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218 | inline void nextSyzygyLayer() const |
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219 | { |
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220 | OPT__SYZNUMBER++; |
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221 | } |
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222 | |
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223 | /// output lifting tree |
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224 | const int OPT__TREEOUTPUT; |
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225 | |
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226 | /// CheckSyzygyProperty: TODO |
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227 | const int OPT__SYZCHECK; |
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228 | |
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229 | /// TEST_OPT_PROT |
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230 | const bool OPT__PROT; |
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231 | |
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232 | /// no caching/stores/lookups |
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233 | const int OPT__NOCACHING; |
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234 | |
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235 | /// global base ring |
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236 | const ring m_rBaseRing; |
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237 | }; |
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238 | |
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239 | class SchreyerSyzygyComputation; |
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240 | |
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241 | class CLCM: public SchreyerSyzygyComputationFlags, public std::vector<bool> |
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242 | { |
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243 | public: |
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244 | CLCM(const ideal& L, const SchreyerSyzygyComputationFlags& flags); |
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245 | |
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246 | bool Check(const poly m) const; |
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247 | |
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248 | private: |
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249 | bool m_compute; |
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250 | |
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251 | const unsigned int m_N; ///< number of ring variables |
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252 | }; |
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253 | |
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254 | |
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255 | class CLeadingTerm |
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256 | { |
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257 | public: |
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258 | CLeadingTerm(unsigned int label, const poly lt, const ring); |
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259 | |
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260 | #ifndef SING_NDEBUG |
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261 | ~CLeadingTerm(); |
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262 | #endif |
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263 | |
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264 | #if NOPRODUCT |
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265 | bool DivisibilityCheck(const poly multiplier, const poly t, const unsigned long not_sev, const ring r) const; |
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266 | #endif |
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267 | bool DivisibilityCheck(const poly product, const unsigned long not_sev, const ring r) const; |
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268 | |
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269 | bool CheckLT( const ideal & L ) const; |
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270 | |
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271 | #ifndef SING_NDEBUG |
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272 | poly lt() const; |
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273 | unsigned long sev() const; |
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274 | unsigned int label() const; |
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275 | #else |
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276 | inline poly lt() const { return m_lt; }; |
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277 | inline unsigned long sev() const { return m_sev; }; |
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278 | inline unsigned int label() const { return m_label; }; |
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279 | #endif |
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280 | |
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281 | private: |
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282 | const unsigned long m_sev; ///< not short exp. vector |
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283 | |
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284 | // NOTE/TODO: either of the following should be enough: |
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285 | const unsigned int m_label; ///< index in the main L[] + 1 |
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286 | |
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287 | const poly m_lt; ///< the leading term itself L[label-1] |
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288 | |
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289 | #ifndef SING_NDEBUG |
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290 | const ring _R; |
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291 | |
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292 | const poly m_lt_copy; ///< original copy of LEAD(lt) (only for debug!!!) |
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293 | #endif |
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294 | |
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295 | // disable the following: |
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296 | CLeadingTerm(); |
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297 | CLeadingTerm(const CLeadingTerm&); |
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298 | void operator=(const CLeadingTerm&); |
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299 | }; |
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300 | |
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301 | |
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302 | // TODO: needs a specialized variant without a component (hash!) |
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303 | class CReducerFinder: public SchreyerSyzygyComputationFlags |
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304 | { |
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305 | #if NOPRODUCT |
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306 | friend class CDivisorEnumerator2; |
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307 | #endif |
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308 | friend class CDivisorEnumerator; |
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309 | |
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310 | public: |
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311 | typedef long TComponentKey; |
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312 | typedef std::vector<const CLeadingTerm*> TReducers; |
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313 | |
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314 | private: |
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315 | typedef std::map< TComponentKey, TReducers> CReducersHash; |
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316 | |
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317 | public: |
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318 | /// goes over all leading terms |
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319 | CReducerFinder(const ideal L, const SchreyerSyzygyComputationFlags& flags); |
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320 | |
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321 | void Initialize(const ideal L); |
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322 | |
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323 | ~CReducerFinder(); |
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324 | |
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325 | |
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326 | #if NOPRODUCT |
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327 | poly |
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328 | FindReducer(const poly multiplier, const poly monom, const poly syzterm, const CReducerFinder& checker) const; |
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329 | |
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330 | #endif |
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331 | // TODO: save shortcut (syz: |-.->) LM(LM(m) * "t") -> syz? |
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332 | poly // const_iterator // TODO: return const_iterator it, s.th: it->m_lt is the needed |
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333 | FindReducer(const poly product, const poly syzterm, const CReducerFinder& checker) const; |
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334 | |
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335 | bool IsDivisible(const poly q) const; |
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336 | |
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337 | |
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338 | inline bool IsNonempty() const { return !m_hash.empty(); } |
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339 | |
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340 | /// is the term to be "preprocessed" as lower order term or lead to only reducible syzygies... |
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341 | int PreProcessTerm(const poly t, CReducerFinder& syzChecker) const; |
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342 | |
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343 | #ifndef SING_NDEBUG |
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344 | void DebugPrint() const; |
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345 | void Verify() const; |
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346 | #endif |
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347 | |
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348 | private: |
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349 | ideal m_L; ///< only for debug |
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350 | |
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351 | CReducersHash m_hash; // can also be replaced with a vector indexed by components |
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352 | |
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353 | private: |
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354 | CReducerFinder(const CReducerFinder&); |
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355 | void operator=(const CReducerFinder&); |
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356 | }; |
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357 | |
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358 | bool my_p_LmCmp (poly, poly, const ring); |
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359 | |
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360 | typedef poly TCacheKey; |
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361 | typedef poly TCacheValue; |
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362 | |
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363 | struct CCacheCompare |
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364 | { |
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365 | const ring & m_ring; |
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366 | |
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367 | CCacheCompare(); |
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368 | |
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369 | CCacheCompare(const ring& r): m_ring(r) { assume(r != NULL); } |
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370 | |
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371 | CCacheCompare(const CCacheCompare& lhs): m_ring(lhs.m_ring) { assume(m_ring != NULL); } |
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372 | CCacheCompare& operator=(const CCacheCompare& lhs) { assume(lhs.m_ring != NULL); return (const_cast<CCacheCompare&>(lhs)); } |
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373 | |
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374 | inline bool operator() (const TCacheKey& l, const TCacheKey& r) const { assume(m_ring != NULL); return my_p_LmCmp(l, r, m_ring); } |
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375 | }; |
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376 | |
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377 | typedef std::map<TCacheKey, TCacheValue, CCacheCompare> TP2PCache; // deallocation??? !!! |
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378 | typedef std::map<int, TP2PCache> TCache; |
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379 | |
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380 | |
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381 | /** @class SchreyerSyzygyComputation syzextra.h |
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382 | * |
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383 | * Computing syzygies after Schreyer |
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384 | * |
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385 | * Storing/accumulating data during the computation requires some global |
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386 | * object, like this class. Ideally the above global functions should not |
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387 | * be used in favour of this class. |
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388 | * |
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389 | * @sa Schreyer Syzygy Computation Paper & Talk & Python prototype |
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390 | */ |
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391 | class SchreyerSyzygyComputation: public SchreyerSyzygyComputationFlags |
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392 | { |
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393 | friend class CLCM; |
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394 | friend class CReducerFinder; |
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395 | |
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396 | public: |
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397 | /// Construct a global object for given input data (separated into leads & tails) |
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398 | SchreyerSyzygyComputation(const ideal idLeads, const ideal idTails, const SchreyerSyzygyComputationFlags setting): |
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399 | SchreyerSyzygyComputationFlags(setting), |
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400 | m_idLeads(idLeads), m_idTails(id_Copy(idTails, setting.m_rBaseRing)), |
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401 | m_syzLeads(NULL), m_syzTails(NULL), |
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402 | m_LS(NULL), m_lcm(m_idLeads, setting), |
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403 | m_div(m_idLeads, setting), m_checker(NULL, setting), m_cache(), |
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404 | m_sum_bucket_factory(setting.m_rBaseRing), |
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405 | m_spoly_bucket(NULL) |
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406 | { |
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407 | if( UNLIKELY(OPT__PROT) ) memset( &m_stat, 0, sizeof(m_stat) ); |
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408 | } |
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409 | |
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410 | /// Construct a global object for given input data (separated into leads & tails) |
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411 | SchreyerSyzygyComputation(const ideal idLeads, const ideal idTails, const ideal syzLeads, const SchreyerSyzygyComputationFlags setting): |
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412 | SchreyerSyzygyComputationFlags(setting), |
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413 | m_idLeads(idLeads), m_idTails(id_Copy(idTails, setting.m_rBaseRing)), |
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414 | m_syzLeads(syzLeads), m_syzTails(NULL), |
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415 | m_LS(syzLeads), m_lcm(m_idLeads, setting), |
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416 | m_div(m_idLeads, setting), m_checker(NULL, setting), m_cache(), |
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417 | m_sum_bucket_factory(setting.m_rBaseRing), |
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418 | m_spoly_bucket(NULL) |
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419 | { |
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420 | if( UNLIKELY(OPT__PROT) ) memset( &m_stat, 0, sizeof(m_stat) ); |
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421 | |
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422 | if( LIKELY(OPT__TAILREDSYZ && !OPT__IGNORETAILS) ) |
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423 | { |
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424 | if (syzLeads != NULL) |
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425 | m_checker.Initialize(syzLeads); |
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426 | // if( idTails != NULL ) |
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427 | // SetUpTailTerms(); |
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428 | } |
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429 | } |
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430 | |
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431 | /// Destructor should not destruct the resulting m_syzLeads, m_syzTails. |
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432 | ~SchreyerSyzygyComputation(){ CleanUp(); } |
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433 | |
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434 | /// Convert the given ideal of tails into the internal representation (with reducers!) |
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435 | /// Preprocess m_idTails as well...? |
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436 | void SetUpTailTerms(); |
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437 | |
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438 | /// print statistics about the used heuristics |
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439 | void PrintStats() const; |
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440 | |
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441 | /// Read off the results while detaching them from this object |
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442 | /// NOTE: no copy! |
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443 | inline void ReadOffResult(ideal& syzL, ideal& syzT) |
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444 | { |
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445 | syzL = m_syzLeads; syzT = m_syzTails; |
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446 | |
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447 | m_syzLeads = m_syzTails = NULL; // m_LS ? |
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448 | |
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449 | if ( UNLIKELY(OPT__PROT) ) |
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450 | PrintStats(); |
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451 | } |
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452 | |
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453 | |
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454 | /// The main driver function: computes |
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455 | void ComputeSyzygy(); |
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456 | |
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457 | /// Computes Syz(leads) or only LEAD of it. |
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458 | /// The result is stored into m_syzLeads |
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459 | void ComputeLeadingSyzygyTerms(bool bComputeSecondTerms = true); |
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460 | |
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461 | |
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462 | |
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463 | /// Main HybridNF == 1: poly reduce + LOT + LCM? |
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464 | poly SchreyerSyzygyNF(const poly syz_lead, poly syz_2 = NULL) const; |
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465 | |
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466 | |
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467 | // Main (HybridNF == 0) Tree Travers + LOT + LCM? |
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468 | poly TraverseNF(const poly syz_lead, const poly syz_2 = NULL) const; |
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469 | |
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470 | /// High level caching function!!! |
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471 | poly TraverseTail(poly multiplier, const int tail) const; |
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472 | |
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473 | // REMOVE? |
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474 | /// called only from above and from outside (for testing) |
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475 | poly TraverseTail(poly multiplier, poly tail) const; |
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476 | |
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477 | /// TODO: save shortcut (syz: |-.->) LM(m) * "t" -> ? ??? |
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478 | poly ReduceTerm(poly multiplier, poly term4reduction, poly syztermCheck) const; |
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479 | |
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480 | /// low level computation... |
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481 | poly ComputeImage(poly multiplier, const int tail) const; |
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482 | |
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483 | |
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484 | |
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485 | public: |
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486 | /// just for testing via the wrapper below |
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487 | inline poly _FindReducer(const poly product, const poly syzterm) const |
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488 | { return m_div.FindReducer(product, syzterm, m_checker); } |
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489 | private: |
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490 | void CleanUp(); |
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491 | protected: |
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492 | |
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493 | |
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494 | /// just leading terms |
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495 | ideal Compute1LeadingSyzygyTerms(); |
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496 | |
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497 | /// leading + second terms |
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498 | ideal Compute2LeadingSyzygyTerms(); |
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499 | |
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500 | |
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501 | |
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502 | private: |
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503 | /// input leading terms |
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504 | const ideal m_idLeads; |
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505 | |
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506 | /// input tails |
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507 | const ideal m_idTails; |
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508 | |
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509 | /// output (syzygy) leading terms (+2nd terms?) |
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510 | ideal m_syzLeads; |
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511 | |
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512 | /// output (syzygy) tails |
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513 | ideal m_syzTails; |
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514 | |
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515 | /*mutable?*/ ideal m_LS; ///< leading syzygy terms used for reducing syzygy tails |
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516 | |
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517 | |
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518 | /// Bitmask for variables occuring in leading terms |
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519 | const CLCM m_lcm; |
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520 | |
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521 | /// Divisor finder |
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522 | const CReducerFinder m_div; |
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523 | |
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524 | /// for checking tail-terms and makeing them irreducible (wrt m_LS!) |
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525 | CReducerFinder m_checker; |
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526 | |
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527 | /* |
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528 | // need more data here: |
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529 | // (m_idLeads : m_tailterm) = (m, pos, compl), s.th: compl * m_tailterm divides m_idLeads[pos] |
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530 | // but resulting sysygy compl * gen(pos) should not be in |
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531 | // Idea: extend CReducerFinder??!! |
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532 | struct CTailTerm |
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533 | { |
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534 | const poly m_tailterm; |
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535 | |
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536 | const CReducerFinder m_reducers; // positions are labels (in m_idLeads)... |
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537 | // compl - to be computed if needed? |
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538 | |
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539 | CTailTerm(const poly tt, const CReducerFinder reds): m_tailterm(tt), m_reducers(reds) {} |
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540 | }; |
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541 | |
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542 | typedef std::vector<const CTailTerm*> TTail; |
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543 | typedef std::vector<TTail> TTailTerms; |
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544 | |
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545 | TTailTerms m_idTailTerms; |
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546 | */ |
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547 | |
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548 | mutable TCache m_cache; // cacher comp + poly -> poly! // mutable??? |
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549 | |
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550 | /// TODO: look into m_idTailTerms!!!!!!!!!!!!!!!!!!!!!!!! map? heaps??? |
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551 | // NOTE/TODO: the following globally shared buckets violate reentrance - they should rather belong to TLS! |
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552 | |
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553 | /// used for simple summing up |
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554 | mutable SBucketFactory m_sum_bucket_factory; // sBucket_pt |
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555 | |
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556 | /// for S-Polynomial reductions |
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557 | mutable kBucket_pt m_spoly_bucket; // only used inside of SchreyerSyzygyNF! destruction by CleanUp()! |
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558 | |
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559 | |
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560 | /// Statistics: |
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561 | /// 0..3: as in SetUpTailTerms()::PreProcessTerm() // TODO!!?? |
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562 | /// 4: number of terms discarded due to LOT heuristics |
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563 | /// 5: number of terms discarded due to LCM heuristics |
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564 | /// 6, 7: lookups without & with rescale, 8: stores |
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565 | mutable unsigned long m_stat[9]; |
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566 | }; |
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567 | |
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568 | // The following wrappers are just for testing separate functions on highest level (within schreyer.lib) |
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569 | |
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570 | static inline void ComputeSyzygy(const ideal L, const ideal T, ideal& LL, ideal& TT, const SchreyerSyzygyComputationFlags A) |
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571 | { |
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572 | SchreyerSyzygyComputation syz(L, T, A); |
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573 | syz.ComputeSyzygy(); |
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574 | syz.ReadOffResult(LL, TT); |
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575 | } |
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576 | |
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577 | static inline ideal ComputeLeadingSyzygyTerms(const ideal& L, const SchreyerSyzygyComputationFlags A) |
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578 | { |
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579 | SchreyerSyzygyComputation syz(L, NULL, A); |
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580 | syz.ComputeLeadingSyzygyTerms(false); |
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581 | ideal LL, TT; |
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582 | syz.ReadOffResult(LL, TT); |
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583 | return LL; // assume TT is NULL! |
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584 | } |
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585 | |
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586 | static inline ideal Compute2LeadingSyzygyTerms(const ideal& L, const SchreyerSyzygyComputationFlags A) |
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587 | { |
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588 | SchreyerSyzygyComputation syz(L, NULL, A); |
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589 | syz.ComputeLeadingSyzygyTerms(true); |
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590 | ideal LL, TT; |
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591 | syz.ReadOffResult(LL, TT); |
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592 | return LL; // assume TT is NULL! |
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593 | } |
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594 | |
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595 | static inline poly FindReducer(poly product, poly syzterm, |
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596 | ideal L, ideal LS, const SchreyerSyzygyComputationFlags A) |
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597 | { |
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598 | SchreyerSyzygyComputation syz(L, NULL, LS, A); |
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599 | return syz._FindReducer(product, syzterm); |
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600 | } |
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601 | |
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602 | static inline poly TraverseTail(poly multiplier, poly tail, |
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603 | ideal L, ideal T, ideal LS, const SchreyerSyzygyComputationFlags A) |
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604 | { |
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605 | SchreyerSyzygyComputation syz(L, T, LS, A); |
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606 | return syz.TraverseTail(multiplier, tail); |
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607 | } |
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608 | |
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609 | static inline poly ReduceTerm(poly multiplier, poly term4reduction, poly syztermCheck, |
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610 | ideal L, ideal T, ideal LS, const SchreyerSyzygyComputationFlags A) |
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611 | { |
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612 | SchreyerSyzygyComputation syz(L, T, LS, A); |
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613 | return syz.ReduceTerm(multiplier, term4reduction, syztermCheck); |
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614 | } |
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615 | |
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616 | |
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617 | static inline poly SchreyerSyzygyNF(poly syz_lead, poly syz_2, |
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618 | ideal L, ideal T, ideal LS, const SchreyerSyzygyComputationFlags A) |
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619 | { |
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620 | SchreyerSyzygyComputation syz(L, T, LS, A); |
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621 | return syz.SchreyerSyzygyNF(syz_lead, syz_2); |
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622 | } |
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623 | |
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624 | #endif |
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625 | /* #ifndef SYZEXTRA_H */ |
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626 | |
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627 | // Vi-modeline: vim: filetype=c:syntax:shiftwidth=2:tabstop=8:textwidth=0:expandtab |
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628 | |
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