1 | /*****************************************************************************\ |
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2 | * Computer Algebra System SINGULAR |
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3 | \*****************************************************************************/ |
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4 | /** @file facFqFactorize.h |
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5 | * |
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6 | * This file provides functions for factorizing a multivariate polynomial over |
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7 | * \f$ F_{p} \f$ , \f$ F_{p}(\alpha ) \f$ or GF. |
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8 | * |
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9 | * @author Martin Lee |
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10 | * |
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11 | **/ |
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12 | /*****************************************************************************/ |
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13 | |
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14 | #ifndef FAC_FQ_FACTORIZE_H |
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15 | #define FAC_FQ_FACTORIZE_H |
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16 | |
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17 | // #include "config.h" |
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18 | #include "timing.h" |
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19 | |
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20 | #include "facFqBivar.h" |
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21 | #include "DegreePattern.h" |
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22 | #include "ExtensionInfo.h" |
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23 | #include "cf_util.h" |
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24 | #include "facFqSquarefree.h" |
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25 | #include "facFqBivarUtil.h" |
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26 | |
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27 | |
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28 | TIMING_DEFINE_PRINT (fac_fq_squarefree) |
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29 | TIMING_DEFINE_PRINT (fac_fq_factor_squarefree) |
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30 | |
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31 | /// Factorization over a finite field |
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32 | /// |
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33 | /// @return @a multiFactorize returns a factorization of F |
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34 | /// @sa biFactorize(), extFactorize() |
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35 | CFList |
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36 | multiFactorize (const CanonicalForm& F, ///< [in] poly to be factored |
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37 | const ExtensionInfo& info ///< [in] info about extension |
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38 | ); |
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39 | |
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40 | /// factorize a squarefree multivariate polynomial over \f$ F_{p} \f$ |
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41 | /// |
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42 | /// @return @a FpSqrfFactorize returns a list of monic factors, the first |
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43 | /// element is the leading coefficient. |
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44 | /// @sa FqSqrfFactorize(), GFSqrfFactorize() |
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45 | #ifdef HAVE_NTL |
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46 | inline |
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47 | CFList FpSqrfFactorize (const CanonicalForm & F ///< [in] a multivariate poly |
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48 | ) |
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49 | { |
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50 | if (getNumVars (F) == 2) |
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51 | return FpBiSqrfFactorize (F); |
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52 | ExtensionInfo info= ExtensionInfo (false); |
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53 | CFList result= multiFactorize (F, info); |
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54 | result.insert (Lc(F)); |
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55 | return result; |
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56 | } |
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57 | |
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58 | /// factorize a squarefree multivariate polynomial over \f$ F_{p} (\alpha ) \f$ |
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59 | /// |
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60 | /// @return @a FqSqrfFactorize returns a list of monic factors, the first |
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61 | /// element is the leading coefficient. |
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62 | /// @sa FpSqrfFactorize(), GFSqrfFactorize() |
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63 | inline |
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64 | CFList FqSqrfFactorize (const CanonicalForm & F, ///< [in] a multivariate poly |
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65 | const Variable& alpha ///< [in] algebraic variable |
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66 | ) |
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67 | { |
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68 | if (getNumVars (F) == 2) |
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69 | return FqBiSqrfFactorize (F, alpha); |
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70 | ExtensionInfo info= ExtensionInfo (alpha, false); |
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71 | CFList result= multiFactorize (F, info); |
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72 | result.insert (Lc(F)); |
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73 | return result; |
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74 | } |
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75 | |
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76 | /// factorize a squarefree multivariate polynomial over GF |
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77 | /// |
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78 | /// @return @a GFSqrfFactorize returns a list of monic factors, the first |
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79 | /// element is the leading coefficient. |
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80 | /// @sa FpSqrfFactorize(), FqSqrfFactorize() |
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81 | inline |
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82 | CFList GFSqrfFactorize (const CanonicalForm & F ///< [in] a multivariate poly |
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83 | ) |
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84 | { |
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85 | ASSERT (CFFactory::gettype() == GaloisFieldDomain, |
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86 | "GF as base field expected"); |
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87 | if (getNumVars (F) == 2) |
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88 | return GFBiSqrfFactorize (F); |
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89 | ExtensionInfo info= ExtensionInfo (getGFDegree(), gf_name, false); |
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90 | CFList result= multiFactorize (F, info); |
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91 | result.insert (Lc(F)); |
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92 | return result; |
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93 | } |
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94 | |
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95 | /// factorize a multivariate polynomial over \f$ F_{p} \f$ |
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96 | /// |
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97 | /// @return @a FpFactorize returns a list of monic factors with |
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98 | /// multiplicity, the first element is the leading coefficient. |
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99 | /// @sa FqFactorize(), GFFactorize() |
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100 | inline |
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101 | CFFList FpFactorize (const CanonicalForm& G,///< [in] a multivariate poly |
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102 | bool substCheck= true ///< [in] enables substitute check |
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103 | ) |
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104 | { |
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105 | if (getNumVars (G) == 2) |
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106 | return FpBiFactorize (G, substCheck); |
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107 | |
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108 | CanonicalForm F= G; |
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109 | if (substCheck) |
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110 | { |
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111 | bool foundOne= false; |
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112 | int * substDegree= new int [F.level()]; |
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113 | for (int i= 1; i <= F.level(); i++) |
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114 | { |
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115 | if (degree (F, i) > 0) |
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116 | { |
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117 | substDegree[i-1]= substituteCheck (F, Variable (i)); |
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118 | if (substDegree [i-1] > 1) |
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119 | { |
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120 | foundOne= true; |
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121 | subst (F, F, substDegree[i-1], Variable (i)); |
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122 | } |
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123 | } |
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124 | else |
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125 | substDegree[i-1]= -1; |
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126 | } |
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127 | if (foundOne) |
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128 | { |
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129 | CFFList result= FpFactorize (F, false); |
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130 | CFFList newResult, tmp; |
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131 | CanonicalForm tmp2; |
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132 | newResult.insert (result.getFirst()); |
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133 | result.removeFirst(); |
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134 | for (CFFListIterator i= result; i.hasItem(); i++) |
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135 | { |
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136 | tmp2= i.getItem().factor(); |
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137 | for (int j= 1; j <= G.level(); j++) |
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138 | { |
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139 | if (substDegree[j-1] > 1) |
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140 | tmp2= reverseSubst (tmp2, substDegree[j-1], Variable (j)); |
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141 | } |
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142 | tmp= FpFactorize (tmp2, false); |
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143 | tmp.removeFirst(); |
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144 | for (CFFListIterator j= tmp; j.hasItem(); j++) |
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145 | newResult.append (CFFactor (j.getItem().factor(), |
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146 | j.getItem().exp()*i.getItem().exp())); |
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147 | } |
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148 | delete [] substDegree; |
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149 | return newResult; |
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150 | } |
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151 | delete [] substDegree; |
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152 | } |
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153 | |
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154 | ExtensionInfo info= ExtensionInfo (false); |
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155 | Variable a= Variable (1); |
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156 | CanonicalForm LcF= Lc (F); |
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157 | TIMING_START (fac_fq_squarefree); |
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158 | CFFList sqrf= FpSqrf (F, false); |
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159 | TIMING_END_AND_PRINT (fac_fq_squarefree, |
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160 | "time for squarefree factorization over Fq: "); |
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161 | CFFList result; |
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162 | CFList bufResult; |
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163 | sqrf.removeFirst(); |
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164 | CFListIterator i; |
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165 | for (CFFListIterator iter= sqrf; iter.hasItem(); iter++) |
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166 | { |
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167 | TIMING_START (fac_fq_factor_squarefree); |
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168 | bufResult= multiFactorize (iter.getItem().factor(), info); |
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169 | TIMING_END_AND_PRINT (fac_fq_factor_squarefree, |
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170 | "time to factorize sqrfree factor over Fq: "); |
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171 | for (i= bufResult; i.hasItem(); i++) |
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172 | result.append (CFFactor (i.getItem(), iter.getItem().exp())); |
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173 | } |
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174 | result.insert (CFFactor (LcF, 1)); |
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175 | return result; |
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176 | } |
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177 | |
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178 | /// factorize a multivariate polynomial over \f$ F_{p} (\alpha ) \f$ |
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179 | /// |
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180 | /// @return @a FqFactorize returns a list of monic factors with |
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181 | /// multiplicity, the first element is the leading coefficient. |
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182 | /// @sa FpFactorize(), GFFactorize() |
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183 | inline |
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184 | CFFList FqFactorize (const CanonicalForm& G, ///< [in] a multivariate poly |
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185 | const Variable& alpha, ///< [in] algebraic variable |
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186 | bool substCheck= true ///< [in] enables substitute check |
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187 | ) |
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188 | { |
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189 | if (getNumVars (G) == 2) |
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190 | return FqBiFactorize (G, alpha, substCheck); |
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191 | |
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192 | CanonicalForm F= G; |
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193 | if (substCheck) |
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194 | { |
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195 | bool foundOne= false; |
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196 | int * substDegree= new int [F.level()]; |
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197 | for (int i= 1; i <= F.level(); i++) |
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198 | { |
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199 | if (degree (F, i) > 0) |
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200 | { |
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201 | substDegree[i-1]= substituteCheck (F, Variable (i)); |
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202 | if (substDegree [i-1] > 1) |
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203 | { |
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204 | foundOne= true; |
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205 | subst (F, F, substDegree[i-1], Variable (i)); |
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206 | } |
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207 | } |
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208 | else |
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209 | substDegree[i-1]= -1; |
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210 | } |
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211 | if (foundOne) |
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212 | { |
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213 | CFFList result= FqFactorize (F, alpha, false); |
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214 | CFFList newResult, tmp; |
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215 | CanonicalForm tmp2; |
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216 | newResult.insert (result.getFirst()); |
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217 | result.removeFirst(); |
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218 | for (CFFListIterator i= result; i.hasItem(); i++) |
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219 | { |
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220 | tmp2= i.getItem().factor(); |
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221 | for (int j= 1; j <= G.level(); j++) |
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222 | { |
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223 | if (substDegree[j-1] > 1) |
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224 | tmp2= reverseSubst (tmp2, substDegree[j-1], Variable (j)); |
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225 | } |
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226 | tmp= FqFactorize (tmp2, alpha, false); |
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227 | tmp.removeFirst(); |
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228 | for (CFFListIterator j= tmp; j.hasItem(); j++) |
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229 | newResult.append (CFFactor (j.getItem().factor(), |
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230 | j.getItem().exp()*i.getItem().exp())); |
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231 | } |
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232 | delete [] substDegree; |
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233 | return newResult; |
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234 | } |
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235 | delete [] substDegree; |
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236 | } |
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237 | |
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238 | ExtensionInfo info= ExtensionInfo (alpha, false); |
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239 | CanonicalForm LcF= Lc (F); |
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240 | TIMING_START (fac_fq_squarefree); |
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241 | CFFList sqrf= FqSqrf (F, alpha, false); |
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242 | TIMING_END_AND_PRINT (fac_fq_squarefree, |
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243 | "time for squarefree factorization over Fq: "); |
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244 | CFFList result; |
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245 | CFList bufResult; |
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246 | sqrf.removeFirst(); |
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247 | CFListIterator i; |
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248 | for (CFFListIterator iter= sqrf; iter.hasItem(); iter++) |
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249 | { |
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250 | TIMING_START (fac_fq_factor_squarefree); |
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251 | bufResult= multiFactorize (iter.getItem().factor(), info); |
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252 | TIMING_END_AND_PRINT (fac_fq_factor_squarefree, |
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253 | "time to factorize sqrfree factor over Fq: "); |
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254 | for (i= bufResult; i.hasItem(); i++) |
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255 | result.append (CFFactor (i.getItem(), iter.getItem().exp())); |
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256 | } |
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257 | result.insert (CFFactor (LcF, 1)); |
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258 | return result; |
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259 | } |
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260 | |
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261 | /// factorize a multivariate polynomial over GF |
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262 | /// |
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263 | /// @return @a GFFactorize returns a list of monic factors with |
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264 | /// multiplicity, the first element is the leading coefficient. |
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265 | /// @sa FpFactorize(), FqFactorize() |
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266 | inline |
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267 | CFFList GFFactorize (const CanonicalForm& G, ///< [in] a multivariate poly |
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268 | bool substCheck= true ///< [in] enables substitute check |
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269 | ) |
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270 | { |
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271 | ASSERT (CFFactory::gettype() == GaloisFieldDomain, |
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272 | "GF as base field expected"); |
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273 | if (getNumVars (G) == 2) |
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274 | return GFBiFactorize (G, substCheck); |
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275 | |
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276 | CanonicalForm F= G; |
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277 | if (substCheck) |
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278 | { |
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279 | bool foundOne= false; |
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280 | int * substDegree= new int [F.level()]; |
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281 | for (int i= 1; i <= F.level(); i++) |
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282 | { |
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283 | if (degree (F, i) > 0) |
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284 | { |
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285 | substDegree[i-1]= substituteCheck (F, Variable (i)); |
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286 | if (substDegree [i-1] > 1) |
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287 | { |
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288 | foundOne= true; |
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289 | subst (F, F, substDegree[i-1], Variable (i)); |
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290 | } |
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291 | } |
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292 | else |
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293 | substDegree[i-1]= -1; |
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294 | } |
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295 | if (foundOne) |
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296 | { |
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297 | CFFList result= GFFactorize (F, false); |
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298 | CFFList newResult, tmp; |
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299 | CanonicalForm tmp2; |
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300 | newResult.insert (result.getFirst()); |
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301 | result.removeFirst(); |
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302 | for (CFFListIterator i= result; i.hasItem(); i++) |
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303 | { |
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304 | tmp2= i.getItem().factor(); |
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305 | for (int j= 1; j <= G.level(); j++) |
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306 | { |
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307 | if (substDegree[j-1] > 1) |
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308 | tmp2= reverseSubst (tmp2, substDegree[j-1], Variable (j)); |
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309 | } |
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310 | tmp= GFFactorize (tmp2, false); |
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311 | tmp.removeFirst(); |
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312 | for (CFFListIterator j= tmp; j.hasItem(); j++) |
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313 | newResult.append (CFFactor (j.getItem().factor(), |
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314 | j.getItem().exp()*i.getItem().exp())); |
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315 | } |
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316 | delete [] substDegree; |
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317 | return newResult; |
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318 | } |
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319 | delete [] substDegree; |
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320 | } |
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321 | |
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322 | Variable a= Variable (1); |
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323 | ExtensionInfo info= ExtensionInfo (getGFDegree(), gf_name, false); |
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324 | CanonicalForm LcF= Lc (F); |
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325 | CFFList sqrf= GFSqrf (F, false); |
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326 | CFFList result; |
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327 | CFList bufResult; |
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328 | sqrf.removeFirst(); |
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329 | CFListIterator i; |
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330 | for (CFFListIterator iter= sqrf; iter.hasItem(); iter++) |
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331 | { |
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332 | bufResult= multiFactorize (iter.getItem().factor(), info); |
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333 | for (i= bufResult; i.hasItem(); i++) |
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334 | result.append (CFFactor (i.getItem(), iter.getItem().exp())); |
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335 | } |
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336 | result.insert (CFFactor (LcF, 1)); |
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337 | return result; |
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338 | } |
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339 | |
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340 | #endif |
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341 | |
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342 | /// Naive factor recombination for multivariate factorization over an extension |
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343 | /// of the initial field. No precomputed is used to exclude combinations. |
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344 | /// |
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345 | /// @return @a extFactorRecombination returns a list of factors of @a F, whose |
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346 | /// shift to zero is reversed. |
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347 | /// @sa factorRecombination() |
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348 | CFList |
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349 | extFactorRecombination ( |
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350 | const CFList& factors, ///< [in] list of lifted factors |
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351 | ///< that are monic wrt Variable (1) |
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352 | const CanonicalForm& F, ///< [in] poly to be factored |
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353 | const CFList& M, ///< [in] a list of powers of |
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354 | ///< Variables |
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355 | const ExtensionInfo& info, ///< [in] info about extension |
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356 | const CFList& evaluation ///< [in] evaluation point |
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357 | ); |
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358 | |
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359 | /// Naive factor recombination for multivariate factorization. |
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360 | /// No precomputed is used to exclude combinations. |
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361 | /// |
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362 | /// @return @a factorRecombination returns a list of factors of @a F |
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363 | /// @sa extFactorRecombination() |
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364 | CFList |
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365 | factorRecombination (const CanonicalForm& F,///< [in] poly to be factored |
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366 | const CFList& factors, ///< [in] list of lifted factors |
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367 | ///< that are monic wrt Variable (1) |
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368 | const CFList& M ///< [in] a list of powers of |
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369 | ///< Variables |
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370 | ); |
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371 | |
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372 | /// recombination of bivariate factors @a factors1 s. t. the result evaluated |
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373 | /// at @a evalPoint coincides with @a factors2 |
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374 | CFList |
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375 | recombination (const CFList& factors1, ///<[in] list of bivariate factors |
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376 | const CFList& factors2, ///<[in] list univariate factors |
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377 | int s, ///<[in] algorithm starts checking |
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378 | ///< subsets of size s |
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379 | int thres, ///<[in] threshold for the size of |
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380 | ///< subsets which are checked |
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381 | const CanonicalForm& evalPoint,///<[in] evaluation point |
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382 | const Variable& x ///<[in] second variable of |
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383 | ///< bivariate factors |
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384 | ); |
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385 | |
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386 | /// Lift bound adaption. Essentially an early factor detection but only the lift |
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387 | /// bound is adapted. |
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388 | /// |
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389 | /// @return @a liftBoundAdaption returns an adapted lift bound. |
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390 | /// @sa earlyFactorDetect(), earlyFactorDetection() |
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391 | int |
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392 | liftBoundAdaption (const CanonicalForm& F, ///< [in] a poly |
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393 | const CFList& factors, ///< [in] list of list of lifted |
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394 | ///< factors that are monic wrt |
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395 | ///< Variable (1) |
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396 | bool& success, ///< [in,out] indicates that no |
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397 | ///< further lifting is necessary |
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398 | const int deg, ///< [in] stage of Hensel lifting |
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399 | const CFList& MOD, ///< [in] a list of powers of |
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400 | ///< Variables |
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401 | const int bound ///< [in] initial lift bound |
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402 | ); |
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403 | |
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404 | /// Lift bound adaption over an extension of the initial field. Essentially an |
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405 | ///early factor detection but only the lift bound is adapted. |
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406 | /// |
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407 | /// @return @a liftBoundAdaption returns an adapted lift bound. |
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408 | /// @sa earlyFactorDetect(), earlyFactorDetection() |
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409 | int |
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410 | extLiftBoundAdaption ( |
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411 | const CanonicalForm& F, ///< [in] a poly |
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412 | const CFList& factors, ///< [in] list of list of lifted |
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413 | ///< factors that are monic wrt |
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414 | bool& success, ///< [in,out] indicates that no further |
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415 | ///< lifting is necessary |
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416 | const ExtensionInfo& info, ///< [in] info about extension |
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417 | const CFList& eval, ///< [in] evaluation point |
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418 | const int deg, ///< [in] stage of Hensel lifting |
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419 | const CFList& MOD, ///< [in] a list of powers of |
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420 | ///< Variables |
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421 | const int bound ///< [in] initial lift bound |
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422 | ); |
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423 | |
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424 | /// detects factors of @a F at stage @a deg of Hensel lifting. |
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425 | /// No combinations of more than one factor are tested. Lift bound is adapted. |
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426 | /// |
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427 | /// @return @a earlyFactorDetect returns a list of factors of F (possibly |
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428 | /// incomplete), in case of success. Otherwise an empty list. |
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429 | /// @sa factorRecombination(), extEarlyFactorDetect() |
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430 | CFList |
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431 | earlyFactorDetect ( |
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432 | CanonicalForm& F, ///< [in,out] poly to be factored, |
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433 | ///< returns poly divided by detected |
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434 | ///< factors in case of success |
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435 | CFList& factors, ///< [in,out] list of factors lifted up |
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436 | ///< to @a deg, returns a list of factors |
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437 | ///< without detected factors |
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438 | int& adaptedLiftBound, ///< [in,out] adapted lift bound |
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439 | bool& success, ///< [in,out] indicating success |
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440 | const int deg, ///< [in] stage of Hensel lifting |
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441 | const CFList& MOD, ///< [in] a list of powers of |
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442 | ///< Variables |
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443 | const int bound ///< [in] initial lift bound |
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444 | ); |
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445 | |
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446 | /// detects factors of @a F at stage @a deg of Hensel lifting. |
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447 | /// No combinations of more than one factor are tested. Lift bound is adapted. |
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448 | /// |
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449 | /// @return @a extEarlyFactorDetect returns a list of factors of F (possibly |
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450 | /// incomplete), whose shift to zero is reversed, in case of success. |
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451 | /// Otherwise an empty list. |
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452 | /// @sa factorRecombination(), earlyFactorDetection() |
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453 | CFList |
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454 | extEarlyFactorDetect ( |
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455 | CanonicalForm& F, ///< [in,out] poly to be factored, |
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456 | ///< returns poly divided by detected |
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457 | ///< factors in case of success |
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458 | CFList& factors, ///< [in,out] list of factors lifted up |
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459 | ///< to @a deg, returns a list of factors |
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460 | ///< without detected factors |
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461 | int& adaptedLiftBound, ///< [in,out] adapted lift bound |
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462 | bool& success, ///< [in,out] indicating succes |
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463 | const ExtensionInfo& info, ///< [in] info about extension |
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464 | const CFList& eval, ///< [in] evaluation point |
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465 | const int deg, ///< [in] stage of Hensel lifting |
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466 | const CFList& MOD, ///< [in] a list of powers of Variables |
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467 | const int bound ///< [in] initial lift bound |
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468 | ); |
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469 | |
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470 | /// evaluation point search for multivariate factorization, |
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471 | /// looks for a (F.level() - 1)-tuple such that the resulting univariate |
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472 | /// polynomial has main variable Variable (1), is squarefree and its degree |
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473 | /// coincides with degree(F) and the bivariate one is primitive wrt. |
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474 | /// Variable(1), and successively evaluated polynomials have the same degree in |
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475 | /// their main variable as F has, fails if there are no valid evaluation points, |
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476 | /// eval contains the intermediate evaluated polynomials. |
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477 | /// |
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478 | /// @return @a evalPoints returns an evaluation point, which is valid if and |
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479 | /// only if fail == false. |
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480 | CFList |
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481 | evalPoints (const CanonicalForm& F, ///< [in] a compressed poly |
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482 | CFList & eval, ///< [in,out] an empty list, returns @a F |
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483 | ///< successive evaluated |
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484 | const Variable& alpha, ///< [in] algebraic variable |
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485 | CFList& list, ///< [in,out] a list of points already |
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486 | ///< considered, a point is encoded as a |
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487 | ///< poly of degree F.level()-1 in |
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488 | ///< Variable(1) |
---|
489 | const bool& GF, ///< [in] GF? |
---|
490 | bool& fail ///< [in,out] indicates failure |
---|
491 | ); |
---|
492 | |
---|
493 | /// hensel Lifting and early factor detection |
---|
494 | /// |
---|
495 | /// @return @a henselLiftAndEarly returns monic (wrt Variable (1)) lifted |
---|
496 | /// factors without factors which have been detected at an early stage |
---|
497 | /// of Hensel lifting |
---|
498 | /// @sa earlyFactorDetectn(), extEarlyFactorDetect() |
---|
499 | CFList |
---|
500 | henselLiftAndEarly ( |
---|
501 | CanonicalForm& A, ///< [in,out] poly to be factored, |
---|
502 | ///< returns poly divided by detected |
---|
503 | ///< factors, in case of success |
---|
504 | CFList& MOD, ///< [in,out] a list of powers of |
---|
505 | ///< Variables |
---|
506 | int*& liftBounds, ///< [in,out] initial lift bounds, returns |
---|
507 | ///< adapted lift bounds |
---|
508 | bool& earlySuccess, ///< [in,out] indicating success |
---|
509 | CFList& earlyFactors, ///< [in,out] early factors |
---|
510 | const CFList& Aeval, ///< [in] @a A successively evaluated at |
---|
511 | ///< elements of @a evaluation |
---|
512 | const CFList& biFactors, ///< [in] bivariate factors |
---|
513 | const CFList& evaluation, ///< [in] evaluation point |
---|
514 | const ExtensionInfo& info ///< [in] info about extension |
---|
515 | ); |
---|
516 | |
---|
517 | /// Factorization over an extension of initial field |
---|
518 | /// |
---|
519 | /// @return @a extFactorize returns factorization of F over initial field |
---|
520 | /// @sa extBiFactorize(), multiFactorize() |
---|
521 | CFList |
---|
522 | extFactorize (const CanonicalForm& F, ///< [in] poly to be factored |
---|
523 | const ExtensionInfo& info ///< [in] info about extension |
---|
524 | ); |
---|
525 | |
---|
526 | /// compute the LCM of the contents of @a A wrt to each variable occuring in @a |
---|
527 | /// A. |
---|
528 | /// |
---|
529 | /// @return @a lcmContent returns the LCM of the contents of @a A wrt to each |
---|
530 | /// variable occuring in @a A. |
---|
531 | CanonicalForm |
---|
532 | lcmContent (const CanonicalForm& A, ///< [in] a compressed multivariate poly |
---|
533 | CFList& contentAi ///< [in,out] an empty list, returns a list |
---|
534 | ///< of the contents of @a A wrt to each |
---|
535 | ///< variable occuring in @a A starting from |
---|
536 | ///< @a A.mvar(). |
---|
537 | ); |
---|
538 | |
---|
539 | /// compress a polynomial s.t. \f$ deg_{x_{i}} (F) >= deg_{x_{i+1}} (F) \f$ and |
---|
540 | /// no gaps between the variables occur |
---|
541 | /// |
---|
542 | /// @return a compressed poly with the above properties |
---|
543 | CanonicalForm myCompress (const CanonicalForm& F, ///< [in] a poly |
---|
544 | CFMap& N ///< [in,out] a map to |
---|
545 | ///< decompress |
---|
546 | ); |
---|
547 | |
---|
548 | /// evaluate a poly A with main variable at level 1 at an evaluation point in |
---|
549 | /// K^(n-1) wrt different second variables. If this evaluation is valid (see |
---|
550 | /// evalPoints) then Aeval contains A successively evaluated at this point, |
---|
551 | /// otherwise this entry is empty |
---|
552 | void |
---|
553 | evaluationWRTDifferentSecondVars ( |
---|
554 | CFList*& Aeval, ///<[in,out] an array of length n-2 |
---|
555 | ///< if variable at level i > 2 |
---|
556 | ///< admits a valid evaluation |
---|
557 | ///< this entry contains A |
---|
558 | ///< successively evaluated at this |
---|
559 | ///< point otherwise an empty list |
---|
560 | const CFList& evaluation,///<[in] a valid evaluation point |
---|
561 | ///< for main variable at level 1 |
---|
562 | ///< and second variable at level 2 |
---|
563 | const CanonicalForm& A ///<[in] some poly |
---|
564 | ); |
---|
565 | |
---|
566 | /// refine a bivariate factorization of A with l factors to one with |
---|
567 | /// minFactorsLength |
---|
568 | void |
---|
569 | refineBiFactors (const CanonicalForm& A, ///< [in] some poly |
---|
570 | CFList& biFactors, ///< [in,out] list of bivariate to be |
---|
571 | ///< refined, returns refined factors |
---|
572 | CFList* const& factors, ///< [in] list of bivariate |
---|
573 | ///< factorizations of A wrt different |
---|
574 | ///< second variables |
---|
575 | const CFList& evaluation,///< [in] the evaluation point |
---|
576 | int minFactorsLength ///< [in] the minimal number of factors |
---|
577 | ); |
---|
578 | |
---|
579 | /// plug in evalPoint for y in a list of polys |
---|
580 | /// |
---|
581 | /// @return returns a list of the evaluated polys, these evaluated polys are |
---|
582 | /// made monic |
---|
583 | CFList |
---|
584 | buildUniFactors (const CFList& biFactors, ///< [in] a list of polys |
---|
585 | const CanonicalForm& evalPoint,///< [in] some evaluation point |
---|
586 | const Variable& y ///< [in] some variable |
---|
587 | ); |
---|
588 | |
---|
589 | |
---|
590 | /// sort bivariate factors in Aeval such that their corresponding univariate |
---|
591 | /// factors coincide with uniFactors |
---|
592 | void sortByUniFactors (CFList*& Aeval, ///< [in,out] array of bivariate |
---|
593 | ///< factors |
---|
594 | int AevalLength, ///< [in] length of Aeval |
---|
595 | const CFList& uniFactors,///< [in] univariate factors |
---|
596 | const CFList& evaluation ///< [in] evaluation point |
---|
597 | ); |
---|
598 | |
---|
599 | /// extract leading coefficients wrt Variable(1) from bivariate factors obtained |
---|
600 | /// from factorizations of A wrt different second variables |
---|
601 | void |
---|
602 | getLeadingCoeffs (const CanonicalForm& A, ///< [in] some poly |
---|
603 | CFList*& Aeval ///< [in,out] array of bivariate |
---|
604 | ///< factors, returns the leading |
---|
605 | ///< coefficients of these factors |
---|
606 | ); |
---|
607 | |
---|
608 | /// normalize precomputed leading coefficients such that leading coefficients |
---|
609 | /// evaluated at @a evaluation in K^(n-2) equal the leading coeffs wrt |
---|
610 | /// Variable(1) of bivariate factors and change @a A and @a Aeval accordingly |
---|
611 | void |
---|
612 | prepareLeadingCoeffs (CFList*& LCs, ///<[in,out] |
---|
613 | CanonicalForm& A, ///<[in,out] |
---|
614 | CFList& Aeval, ///<[in,out] |
---|
615 | int n, ///<[in] level of poly to be |
---|
616 | ///< factored |
---|
617 | const CFList& leadingCoeffs,///<[in] precomputed leading |
---|
618 | ///< coeffs |
---|
619 | const CFList& biFactors, ///<[in] bivariate factors |
---|
620 | const CFList& evaluation ///<[in] evaluation point |
---|
621 | ); |
---|
622 | |
---|
623 | /// obtain factors of F by reconstructing their leading coeffs |
---|
624 | /// |
---|
625 | /// @return returns the reconstructed factors |
---|
626 | /// @sa factorRecombination() |
---|
627 | CFList |
---|
628 | leadingCoeffReconstruction (const CanonicalForm& F,///<[in] poly to be factored |
---|
629 | const CFList& factors, ///<[in] factors of f monic |
---|
630 | ///< wrt Variable (1) |
---|
631 | const CFList& M ///<[in] a list of powers of |
---|
632 | ///< Variables |
---|
633 | ); |
---|
634 | |
---|
635 | /// distribute content |
---|
636 | /// |
---|
637 | /// @return returns a list result of polys such that prod (result)= prod (L) |
---|
638 | /// but the first entry of L may be (partially) factorized and these factors |
---|
639 | /// are distributed onto other entries in L |
---|
640 | CFList |
---|
641 | distributeContent ( |
---|
642 | const CFList& L, ///<[in] list of polys, first |
---|
643 | ///< entry the content to be |
---|
644 | ///< distributed |
---|
645 | const CFList* differentSecondVarFactors,///<[in] factorization wrt |
---|
646 | ///< different second vars |
---|
647 | int length ///<[in] length of |
---|
648 | ///<differentSecondVarFactors |
---|
649 | ); |
---|
650 | |
---|
651 | /// gcd free basis of two lists of factors |
---|
652 | void |
---|
653 | gcdFreeBasis (CFFList& factors1, ///< [in,out] list of factors, returns gcd free |
---|
654 | ///< factors |
---|
655 | CFFList& factors2 ///< [in,out] list of factors, returns gcd free |
---|
656 | ///< factors |
---|
657 | ); |
---|
658 | |
---|
659 | /// computes a list l of length length(LCFFactors)+1 of polynomials such that |
---|
660 | /// prod (l)=LCF, note that the first entry of l may be non constant. Intended |
---|
661 | /// to be used to precompute coefficients of a polynomial f from its bivariate |
---|
662 | /// factorizations. |
---|
663 | /// |
---|
664 | /// @return see above |
---|
665 | CFList |
---|
666 | precomputeLeadingCoeff (const CanonicalForm& LCF, ///<[in] a multivariate |
---|
667 | ///< poly |
---|
668 | const CFList& LCFFactors, ///<[in] a list of |
---|
669 | ///< univariate factors |
---|
670 | ///< of LCF of level 2 |
---|
671 | const Variable& alpha, ///<[in] algebraic var. |
---|
672 | const CFList& evaluation, ///<[in] an evaluation |
---|
673 | ///< point having |
---|
674 | ///< lSecondVarLCs+1 |
---|
675 | ///< components |
---|
676 | CFList* & differentSecondVarLCs,///<[in] LCs of factors |
---|
677 | ///< of f wrt different |
---|
678 | ///< second variables |
---|
679 | int lSecondVarLCs, ///<[in] length of the |
---|
680 | ///< above |
---|
681 | Variable& y ///<[in,out] if y.level() |
---|
682 | ///< is not 1 on output |
---|
683 | ///< the second variable |
---|
684 | ///< has been changed to |
---|
685 | ///< y |
---|
686 | ); |
---|
687 | |
---|
688 | /// changes the second variable to be @a w and updates all relevant data |
---|
689 | void |
---|
690 | changeSecondVariable (CanonicalForm& A, ///<[in,out] a multivariate poly |
---|
691 | CFList& biFactors, ///<[in,out] bivariate factors |
---|
692 | CFList& evaluation, ///<[in,out] evaluation point |
---|
693 | CFList*& oldAeval, ///<[in,out] old bivariate factors |
---|
694 | ///< wrt. different second vars |
---|
695 | int lengthAeval2, ///<[in] length of oldAeval |
---|
696 | const CFList& uniFactors,///<[in] univariate factors |
---|
697 | const Variable& w ///<[in] some variable |
---|
698 | ); |
---|
699 | |
---|
700 | /// distributes a divisor LCmultiplier of LC(A,1) on the bivariate factors and |
---|
701 | /// the precomputed leading coefficients |
---|
702 | void |
---|
703 | distributeLCmultiplier (CanonicalForm& A, ///<[in,out] some poly |
---|
704 | CFList& leadingCoeffs, ///<[in,out] leading |
---|
705 | ///< coefficients |
---|
706 | CFList& biFactors, ///<[in,out] bivariate |
---|
707 | ///< factors |
---|
708 | const CFList& evaluation, ///<[in] eval. point |
---|
709 | const CanonicalForm& LCmultipler///<[in] multiplier |
---|
710 | ); |
---|
711 | |
---|
712 | /// heuristic to distribute @a LCmultiplier onto factors based on the variables |
---|
713 | /// that occur in @a LCmultiplier and in the leading coeffs of bivariate factors |
---|
714 | void |
---|
715 | LCHeuristic (CanonicalForm& A, ///<[in,out] a poly |
---|
716 | const CanonicalForm& LCmultiplier,///<[in,out] divisor of LC (A,1) |
---|
717 | CFList& biFactors, ///<[in,out] bivariate factors |
---|
718 | CFList*& leadingCoeffs, ///<[in,out] leading coeffs |
---|
719 | const CFList* oldAeval, ///<[in] bivariate factors wrt. |
---|
720 | ///< different second vars |
---|
721 | int lengthAeval, ///<[in] length of oldAeval |
---|
722 | const CFList& evaluation, ///<[in] evaluation point |
---|
723 | const CFList& oldBiFactors ///<[in] bivariate factors |
---|
724 | ///< without LCmultiplier |
---|
725 | ///< distributed on them |
---|
726 | ); |
---|
727 | |
---|
728 | /// checks if prod(LCs)==LC (oldA,1) and if so divides elements of leadingCoeffs |
---|
729 | /// by elements in contents, sets A to oldA and sets foundTrueMultiplier to true |
---|
730 | void |
---|
731 | LCHeuristicCheck (const CFList& LCs, ///<[in] leading coeffs computed |
---|
732 | const CFList& contents, ///<[in] content of factors |
---|
733 | CanonicalForm& A, ///<[in,out] oldA*LCmultiplier^m |
---|
734 | const CanonicalForm& oldA,///<[in] some poly |
---|
735 | CFList& leadingCoeffs, ///<[in,out] leading coefficients |
---|
736 | bool& foundTrueMultiplier ///<[in,out] success? |
---|
737 | ); |
---|
738 | |
---|
739 | /// heuristic to distribute @a LCmultiplier onto factors based on the contents |
---|
740 | /// of @a factors. @a factors are assumed to come from LucksWangSparseHeuristic. |
---|
741 | /// If not successful @a contents will contain the content of each element of @a |
---|
742 | /// factors and @a LCs will contain the LC of each element of @a factors divided |
---|
743 | /// by its content |
---|
744 | void |
---|
745 | LCHeuristic2 (const CanonicalForm& LCmultiplier,///<[in] divisor of LC (A,1) |
---|
746 | const CFList& factors, ///<[in] result of |
---|
747 | ///< LucksWangSparseHeuristic |
---|
748 | CFList& leadingCoeffs, ///<[in,out] leading coeffs |
---|
749 | CFList& contents, ///<[in,out] content of factors |
---|
750 | CFList& LCs, ///<[in,out] LC of factors |
---|
751 | ///< divided by content of |
---|
752 | ///< factors |
---|
753 | bool& foundTrueMultiplier ///<[in,out] success? |
---|
754 | ); |
---|
755 | |
---|
756 | /// heuristic to remove @a LCmultiplier from a factor based on the contents |
---|
757 | /// of @a factors. @a factors are assumed to come from LucksWangSparseHeuristic. |
---|
758 | void |
---|
759 | LCHeuristic3 (const CanonicalForm& LCmultiplier,///<[in] divisor of LC (A,1) |
---|
760 | const CFList& factors, ///<[in] result of |
---|
761 | ///< LucksWangSparseHeuristic |
---|
762 | const CFList& oldBiFactors, ///<[in] bivariate factors |
---|
763 | ///< without LCmultiplier |
---|
764 | ///< distributed on them |
---|
765 | const CFList& contents, ///<[in] content of factors |
---|
766 | const CFList* oldAeval, ///<[in] bivariate factors wrt. |
---|
767 | ///< different second vars |
---|
768 | CanonicalForm& A, ///<[in,out] poly |
---|
769 | CFList*& leadingCoeffs, ///<[in,out] leading coeffs |
---|
770 | int lengthAeval, ///<[in] length of oldAeval |
---|
771 | bool& foundMultiplier ///<[in,out] success? |
---|
772 | ); |
---|
773 | |
---|
774 | /// heuristic to remove factors of @a LCmultiplier from @a factors. |
---|
775 | /// More precisely checks if elements of @a contents divide @a LCmultiplier. |
---|
776 | /// Assumes LCHeuristic3 is run before it and was successful. |
---|
777 | void |
---|
778 | LCHeuristic4 (const CFList& oldBiFactors, ///<[in] bivariate factors |
---|
779 | ///< without LCmultiplier |
---|
780 | ///< distributed on them |
---|
781 | const CFList* oldAeval, ///<[in] bivariate factors wrt. |
---|
782 | ///< different second vars |
---|
783 | const CFList& contents, ///<[in] content of factors |
---|
784 | const CFList& factors, ///<[in] result of |
---|
785 | ///< LucksWangSparseHeuristic |
---|
786 | const CanonicalForm& testVars,///<[in] product of second vars that |
---|
787 | ///< occur among oldAeval |
---|
788 | int lengthAeval, ///<[in] length of oldAeval |
---|
789 | CFList*& leadingCoeffs, ///<[in,out] leading coeffs |
---|
790 | CanonicalForm& A, ///<[in,out] poly |
---|
791 | CanonicalForm& LCmultiplier, ///<[in,out] divisor of LC (A,1) |
---|
792 | bool& foundMultiplier ///<[in] success? |
---|
793 | ); |
---|
794 | |
---|
795 | #endif |
---|
796 | /* FAC_FQ_FACTORIZE_H */ |
---|
797 | |
---|