1 | /*****************************************************************************\ |
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2 | * Computer Algebra System SINGULAR |
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3 | \*****************************************************************************/ |
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4 | /** @file facHensel.cc |
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5 | * |
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6 | * This file implements functions to lift factors via Hensel lifting and |
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7 | * functions for modular multiplication and division with remainder. |
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8 | * |
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9 | * ABSTRACT: Hensel lifting is described in "Efficient Multivariate |
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10 | * Factorization over Finite Fields" by L. Bernardin & M. Monagon. Division with |
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11 | * remainder is described in "Fast Recursive Division" by C. Burnikel and |
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12 | * J. Ziegler. Karatsuba multiplication is described in "Modern Computer |
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13 | * Algebra" by J. von zur Gathen and J. Gerhard. |
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14 | * |
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15 | * @author Martin Lee |
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16 | * |
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17 | * @internal @version \$Id$ |
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18 | * |
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19 | **/ |
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20 | /*****************************************************************************/ |
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21 | |
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22 | #include "assert.h" |
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23 | #include "debug.h" |
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24 | #include "timing.h" |
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25 | |
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26 | #include "facHensel.h" |
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27 | #include "cf_util.h" |
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28 | #include "fac_util.h" |
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29 | #include "cf_algorithm.h" |
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30 | |
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31 | #ifdef HAVE_NTL |
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32 | #include <NTL/lzz_pEX.h> |
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33 | #include "NTLconvert.h" |
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34 | |
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35 | CanonicalForm |
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36 | mulNTL (const CanonicalForm& F, const CanonicalForm& G) |
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37 | { |
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38 | if (F.inCoeffDomain() || G.inCoeffDomain()) |
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39 | return F*G; |
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40 | ASSERT (F.isUnivariate() && G.isUnivariate(), "expected univariate polys"); |
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41 | ASSERT (F.level() == G.level(), "expected polys of same level"); |
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42 | if (CFFactory::gettype() == GaloisFieldDomain) |
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43 | return F*G; |
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44 | zz_p::init (getCharacteristic()); |
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45 | Variable alpha; |
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46 | CanonicalForm result; |
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47 | if (hasFirstAlgVar (F, alpha) || hasFirstAlgVar (G, alpha)) |
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48 | { |
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49 | zz_pX NTLMipo= convertFacCF2NTLzzpX (getMipo (alpha)); |
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50 | zz_pE::init (NTLMipo); |
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51 | zz_pEX NTLF= convertFacCF2NTLzz_pEX (F, NTLMipo); |
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52 | zz_pEX NTLG= convertFacCF2NTLzz_pEX (G, NTLMipo); |
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53 | mul (NTLF, NTLF, NTLG); |
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54 | result= convertNTLzz_pEX2CF(NTLF, F.mvar(), alpha); |
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55 | } |
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56 | else |
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57 | { |
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58 | zz_pX NTLF= convertFacCF2NTLzzpX (F); |
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59 | zz_pX NTLG= convertFacCF2NTLzzpX (G); |
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60 | mul (NTLF, NTLF, NTLG); |
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61 | result= convertNTLzzpX2CF(NTLF, F.mvar()); |
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62 | } |
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63 | return result; |
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64 | } |
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65 | |
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66 | CanonicalForm |
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67 | modNTL (const CanonicalForm& F, const CanonicalForm& G) |
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68 | { |
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69 | if (F.inCoeffDomain() && G.isUnivariate()) |
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70 | return F; |
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71 | else if (F.inCoeffDomain() && G.inCoeffDomain()) |
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72 | return mod (F, G); |
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73 | else if (F.isUnivariate() && G.inCoeffDomain()) |
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74 | return mod (F,G); |
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75 | |
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76 | ASSERT (F.isUnivariate() && G.isUnivariate(), "expected univariate polys"); |
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77 | ASSERT (F.level() == G.level(), "expected polys of same level"); |
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78 | if (CFFactory::gettype() == GaloisFieldDomain) |
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79 | return mod (F, G); |
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80 | zz_p::init (getCharacteristic()); |
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81 | Variable alpha; |
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82 | CanonicalForm result; |
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83 | if (hasFirstAlgVar (F, alpha) || hasFirstAlgVar (G, alpha)) |
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84 | { |
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85 | zz_pX NTLMipo= convertFacCF2NTLzzpX(getMipo (alpha)); |
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86 | zz_pE::init (NTLMipo); |
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87 | zz_pEX NTLF= convertFacCF2NTLzz_pEX (F, NTLMipo); |
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88 | zz_pEX NTLG= convertFacCF2NTLzz_pEX (G, NTLMipo); |
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89 | rem (NTLF, NTLF, NTLG); |
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90 | result= convertNTLzz_pEX2CF(NTLF, F.mvar(), alpha); |
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91 | } |
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92 | else |
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93 | { |
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94 | zz_pX NTLF= convertFacCF2NTLzzpX (F); |
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95 | zz_pX NTLG= convertFacCF2NTLzzpX (G); |
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96 | rem (NTLF, NTLF, NTLG); |
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97 | result= convertNTLzzpX2CF(NTLF, F.mvar()); |
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98 | } |
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99 | return result; |
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100 | } |
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101 | |
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102 | CanonicalForm |
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103 | divNTL (const CanonicalForm& F, const CanonicalForm& G) |
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104 | { |
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105 | if (F.inCoeffDomain() && G.isUnivariate()) |
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106 | return F; |
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107 | else if (F.inCoeffDomain() && G.inCoeffDomain()) |
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108 | return div (F, G); |
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109 | else if (F.isUnivariate() && G.inCoeffDomain()) |
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110 | return div (F,G); |
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111 | |
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112 | ASSERT (F.isUnivariate() && G.isUnivariate(), "expected univariate polys"); |
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113 | ASSERT (F.level() == G.level(), "expected polys of same level"); |
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114 | if (CFFactory::gettype() == GaloisFieldDomain) |
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115 | return div (F, G); |
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116 | zz_p::init (getCharacteristic()); |
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117 | Variable alpha; |
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118 | CanonicalForm result; |
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119 | if (hasFirstAlgVar (F, alpha) || hasFirstAlgVar (G, alpha)) |
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120 | { |
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121 | zz_pX NTLMipo= convertFacCF2NTLzzpX(getMipo (alpha)); |
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122 | zz_pE::init (NTLMipo); |
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123 | zz_pEX NTLF= convertFacCF2NTLzz_pEX (F, NTLMipo); |
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124 | zz_pEX NTLG= convertFacCF2NTLzz_pEX (G, NTLMipo); |
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125 | div (NTLF, NTLF, NTLG); |
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126 | result= convertNTLzz_pEX2CF(NTLF, F.mvar(), alpha); |
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127 | } |
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128 | else |
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129 | { |
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130 | zz_pX NTLF= convertFacCF2NTLzzpX (F); |
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131 | zz_pX NTLG= convertFacCF2NTLzzpX (G); |
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132 | div (NTLF, NTLF, NTLG); |
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133 | result= convertNTLzzpX2CF(NTLF, F.mvar()); |
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134 | } |
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135 | return result; |
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136 | } |
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137 | |
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138 | /* |
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139 | void |
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140 | divremNTL (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
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141 | CanonicalForm& R) |
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142 | { |
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143 | if (F.inCoeffDomain() && G.isUnivariate()) |
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144 | { |
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145 | R= F; |
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146 | Q= 0; |
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147 | } |
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148 | else if (F.inCoeffDomain() && G.inCoeffDomain()) |
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149 | { |
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150 | divrem (F, G, Q, R); |
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151 | return; |
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152 | } |
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153 | else if (F.isUnivariate() && G.inCoeffDomain()) |
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154 | { |
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155 | divrem (F, G, Q, R); |
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156 | return; |
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157 | } |
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158 | |
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159 | ASSERT (F.isUnivariate() && G.isUnivariate(), "expected univariate polys"); |
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160 | ASSERT (F.level() == G.level(), "expected polys of same level"); |
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161 | if (CFFactory::gettype() == GaloisFieldDomain) |
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162 | { |
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163 | divrem (F, G, Q, R); |
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164 | return; |
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165 | } |
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166 | zz_p::init (getCharacteristic()); |
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167 | Variable alpha; |
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168 | CanonicalForm result; |
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169 | if (hasFirstAlgVar (F, alpha) || hasFirstAlgVar (G, alpha)) |
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170 | { |
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171 | zz_pX NTLMipo= convertFacCF2NTLzzpX(getMipo (alpha)); |
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172 | zz_pE::init (NTLMipo); |
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173 | zz_pEX NTLF= convertFacCF2NTLzz_pEX (F, NTLMipo); |
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174 | zz_pEX NTLG= convertFacCF2NTLzz_pEX (G, NTLMipo); |
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175 | zz_pEX NTLQ; |
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176 | zz_pEX NTLR; |
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177 | DivRem (NTLQ, NTLR, NTLF, NTLG); |
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178 | Q= convertNTLzz_pEX2CF(NTLQ, F.mvar(), alpha); |
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179 | R= convertNTLzz_pEX2CF(NTLR, F.mvar(), alpha); |
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180 | return; |
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181 | } |
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182 | else |
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183 | { |
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184 | zz_pX NTLF= convertFacCF2NTLzzpX (F); |
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185 | zz_pX NTLG= convertFacCF2NTLzzpX (G); |
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186 | zz_pX NTLQ; |
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187 | zz_pX NTLR; |
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188 | DivRem (NTLQ, NTLR, NTLF, NTLG); |
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189 | Q= convertNTLzzpX2CF(NTLQ, F.mvar()); |
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190 | R= convertNTLzzpX2CF(NTLR, F.mvar()); |
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191 | return; |
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192 | } |
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193 | }*/ |
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194 | |
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195 | CanonicalForm mod (const CanonicalForm& F, const CFList& M) |
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196 | { |
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197 | CanonicalForm A= F; |
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198 | for (CFListIterator i= M; i.hasItem(); i++) |
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199 | A= mod (A, i.getItem()); |
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200 | return A; |
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201 | } |
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202 | |
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203 | zz_pX kronSubFp (const CanonicalForm& A, int d) |
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204 | { |
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205 | int degAy= degree (A); |
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206 | zz_pX result; |
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207 | result.rep.SetLength (d*(degAy + 1)); |
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208 | |
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209 | zz_p *resultp; |
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210 | resultp= result.rep.elts(); |
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211 | zz_pX buf; |
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212 | zz_p *bufp; |
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213 | |
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214 | for (CFIterator i= A; i.hasTerms(); i++) |
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215 | { |
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216 | if (i.coeff().inCoeffDomain()) |
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217 | buf= convertFacCF2NTLzzpX (i.coeff()); |
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218 | else |
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219 | buf= convertFacCF2NTLzzpX (i.coeff()); |
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220 | |
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221 | int k= i.exp()*d; |
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222 | bufp= buf.rep.elts(); |
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223 | int bufRepLength= (int) buf.rep.length(); |
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224 | for (int j= 0; j < bufRepLength; j++) |
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225 | resultp [j + k]= bufp [j]; |
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226 | } |
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227 | result.normalize(); |
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228 | |
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229 | return result; |
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230 | } |
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231 | |
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232 | zz_pEX kronSub (const CanonicalForm& A, int d, const Variable& alpha) |
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233 | { |
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234 | int degAy= degree (A); |
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235 | zz_pEX result; |
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236 | result.rep.SetLength (d*(degAy + 1)); |
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237 | |
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238 | Variable v; |
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239 | zz_pE *resultp; |
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240 | resultp= result.rep.elts(); |
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241 | zz_pEX buf1; |
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242 | zz_pE *buf1p; |
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243 | zz_pX buf2; |
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244 | zz_pX NTLMipo= convertFacCF2NTLzzpX (getMipo (alpha)); |
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245 | |
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246 | for (CFIterator i= A; i.hasTerms(); i++) |
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247 | { |
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248 | if (i.coeff().inCoeffDomain()) |
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249 | { |
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250 | buf2= convertFacCF2NTLzzpX (i.coeff()); |
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251 | buf1= to_zz_pEX (to_zz_pE (buf2)); |
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252 | } |
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253 | else |
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254 | buf1= convertFacCF2NTLzz_pEX (i.coeff(), NTLMipo); |
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255 | |
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256 | int k= i.exp()*d; |
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257 | buf1p= buf1.rep.elts(); |
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258 | int buf1RepLength= (int) buf1.rep.length(); |
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259 | for (int j= 0; j < buf1RepLength; j++) |
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260 | resultp [j + k]= buf1p [j]; |
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261 | } |
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262 | result.normalize(); |
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263 | |
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264 | return result; |
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265 | } |
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266 | |
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267 | void |
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268 | kronSubRecipro (zz_pEX& subA1, zz_pEX& subA2,const CanonicalForm& A, int d, |
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269 | const Variable& alpha) |
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270 | { |
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271 | int degAy= degree (A); |
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272 | subA1.rep.SetLength ((long) d*(degAy + 1)); |
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273 | subA2.rep.SetLength ((long) d*(degAy + 1)); |
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274 | |
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275 | Variable v; |
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276 | zz_pE *subA1p; |
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277 | zz_pE *subA2p; |
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278 | subA1p= subA1.rep.elts(); |
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279 | subA2p= subA2.rep.elts(); |
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280 | zz_pEX buf; |
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281 | zz_pE *bufp; |
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282 | zz_pX buf2; |
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283 | zz_pX NTLMipo= convertFacCF2NTLzzpX (getMipo (alpha)); |
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284 | |
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285 | for (CFIterator i= A; i.hasTerms(); i++) |
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286 | { |
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287 | if (i.coeff().inCoeffDomain()) |
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288 | { |
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289 | buf2= convertFacCF2NTLzzpX (i.coeff()); |
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290 | buf= to_zz_pEX (to_zz_pE (buf2)); |
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291 | } |
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292 | else |
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293 | buf= convertFacCF2NTLzz_pEX (i.coeff(), NTLMipo); |
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294 | |
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295 | int k= i.exp()*d; |
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296 | int kk= (degAy - i.exp())*d; |
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297 | bufp= buf.rep.elts(); |
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298 | int bufRepLength= (int) buf.rep.length(); |
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299 | for (int j= 0; j < bufRepLength; j++) |
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300 | { |
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301 | subA1p [j + k]= bufp [j]; |
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302 | subA2p [j + kk]= bufp [j]; |
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303 | } |
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304 | } |
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305 | subA1.normalize(); |
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306 | subA2.normalize(); |
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307 | } |
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308 | |
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309 | void |
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310 | kronSubRecipro (zz_pX& subA1, zz_pX& subA2,const CanonicalForm& A, int d) |
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311 | { |
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312 | int degAy= degree (A); |
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313 | subA1.rep.SetLength ((long) d*(degAy + 1)); |
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314 | subA2.rep.SetLength ((long) d*(degAy + 1)); |
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315 | |
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316 | Variable v; |
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317 | zz_p *subA1p; |
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318 | zz_p *subA2p; |
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319 | subA1p= subA1.rep.elts(); |
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320 | subA2p= subA2.rep.elts(); |
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321 | zz_pX buf; |
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322 | zz_p *bufp; |
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323 | |
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324 | for (CFIterator i= A; i.hasTerms(); i++) |
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325 | { |
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326 | buf= convertFacCF2NTLzzpX (i.coeff()); |
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327 | |
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328 | int k= i.exp()*d; |
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329 | int kk= (degAy - i.exp())*d; |
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330 | bufp= buf.rep.elts(); |
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331 | int bufRepLength= (int) buf.rep.length(); |
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332 | for (int j= 0; j < bufRepLength; j++) |
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333 | { |
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334 | subA1p [j + k]= bufp [j]; |
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335 | subA2p [j + kk]= bufp [j]; |
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336 | } |
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337 | } |
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338 | subA1.normalize(); |
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339 | subA2.normalize(); |
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340 | } |
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341 | |
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342 | CanonicalForm |
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343 | reverseSubst (const zz_pEX& F, const zz_pEX& G, int d, int k, |
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344 | const Variable& alpha) |
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345 | { |
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346 | Variable y= Variable (2); |
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347 | Variable x= Variable (1); |
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348 | |
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349 | zz_pEX f= F; |
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350 | zz_pEX g= G; |
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351 | int degf= deg(f); |
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352 | int degg= deg(g); |
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353 | |
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354 | zz_pEX buf1; |
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355 | zz_pEX buf2; |
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356 | zz_pEX buf3; |
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357 | |
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358 | zz_pE *buf1p; |
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359 | zz_pE *buf2p; |
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360 | zz_pE *buf3p; |
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361 | if (f.rep.length() < (long) d*(k+1)) //zero padding |
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362 | f.rep.SetLength ((long)d*(k+1)); |
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363 | |
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364 | zz_pE *gp= g.rep.elts(); |
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365 | zz_pE *fp= f.rep.elts(); |
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366 | CanonicalForm result= 0; |
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367 | int i= 0; |
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368 | int lf= 0; |
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369 | int lg= d*k; |
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370 | int degfSubLf= degf; |
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371 | int deggSubLg= degg-lg; |
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372 | int repLengthBuf2; |
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373 | int repLengthBuf1; |
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374 | zz_pE zzpEZero= zz_pE(); |
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375 | |
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376 | while (degf >= lf || lg >= 0) |
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377 | { |
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378 | if (degfSubLf >= d) |
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379 | repLengthBuf1= d; |
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380 | else if (degfSubLf < 0) |
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381 | repLengthBuf1= 0; |
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382 | else |
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383 | repLengthBuf1= degfSubLf + 1; |
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384 | buf1.rep.SetLength((long) repLengthBuf1); |
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385 | |
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386 | buf1p= buf1.rep.elts(); |
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387 | for (int ind= 0; ind < repLengthBuf1; ind++) |
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388 | buf1p [ind]= fp [ind + lf]; |
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389 | buf1.normalize(); |
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390 | |
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391 | repLengthBuf1= buf1.rep.length(); |
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392 | |
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393 | if (deggSubLg >= d - 1) |
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394 | repLengthBuf2= d - 1; |
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395 | else if (deggSubLg < 0) |
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396 | repLengthBuf2= 0; |
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397 | else |
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398 | repLengthBuf2= deggSubLg + 1; |
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399 | |
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400 | buf2.rep.SetLength ((long) repLengthBuf2); |
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401 | buf2p= buf2.rep.elts(); |
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402 | for (int ind= 0; ind < repLengthBuf2; ind++) |
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403 | { |
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404 | buf2p [ind]= gp [ind + lg]; |
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405 | } |
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406 | buf2.normalize(); |
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407 | |
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408 | repLengthBuf2= buf2.rep.length(); |
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409 | |
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410 | buf3.rep.SetLength((long) repLengthBuf2 + d); |
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411 | buf3p= buf3.rep.elts(); |
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412 | buf2p= buf2.rep.elts(); |
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413 | buf1p= buf1.rep.elts(); |
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414 | for (int ind= 0; ind < repLengthBuf1; ind++) |
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415 | buf3p [ind]= buf1p [ind]; |
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416 | for (int ind= repLengthBuf1; ind < d; ind++) |
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417 | buf3p [ind]= zzpEZero; |
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418 | for (int ind= 0; ind < repLengthBuf2; ind++) |
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419 | buf3p [ind + d]= buf2p [ind]; |
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420 | buf3.normalize(); |
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421 | |
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422 | result += convertNTLzz_pEX2CF (buf3, x, alpha)*power (y, i); |
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423 | i++; |
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424 | |
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425 | |
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426 | lf= i*d; |
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427 | degfSubLf= degf - lf; |
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428 | |
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429 | lg= d*(k-i); |
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430 | deggSubLg= degg - lg; |
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431 | |
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432 | buf1p= buf1.rep.elts(); |
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433 | |
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434 | if (lg >= 0 && deggSubLg > 0) |
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435 | { |
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436 | if (repLengthBuf2 > degfSubLf + 1) |
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437 | degfSubLf= repLengthBuf2 - 1; |
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438 | int tmp= tmin (repLengthBuf1, deggSubLg); |
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439 | for (int ind= 0; ind < tmp; ind++) |
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440 | gp [ind + lg] -= buf1p [ind]; |
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441 | } |
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442 | |
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443 | if (lg < 0) |
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444 | break; |
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445 | |
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446 | buf2p= buf2.rep.elts(); |
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447 | if (degfSubLf >= 0) |
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448 | { |
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449 | for (int ind= 0; ind < repLengthBuf2; ind++) |
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450 | fp [ind + lf] -= buf2p [ind]; |
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451 | } |
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452 | } |
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453 | |
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454 | return result; |
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455 | } |
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456 | |
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457 | CanonicalForm |
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458 | reverseSubst (const zz_pX& F, const zz_pX& G, int d, int k) |
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459 | { |
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460 | Variable y= Variable (2); |
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461 | Variable x= Variable (1); |
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462 | |
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463 | zz_pX f= F; |
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464 | zz_pX g= G; |
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465 | int degf= deg(f); |
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466 | int degg= deg(g); |
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467 | |
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468 | zz_pX buf1; |
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469 | zz_pX buf2; |
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470 | zz_pX buf3; |
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471 | |
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472 | zz_p *buf1p; |
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473 | zz_p *buf2p; |
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474 | zz_p *buf3p; |
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475 | |
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476 | if (f.rep.length() < (long) d*(k+1)) //zero padding |
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477 | f.rep.SetLength ((long)d*(k+1)); |
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478 | |
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479 | zz_p *gp= g.rep.elts(); |
---|
480 | zz_p *fp= f.rep.elts(); |
---|
481 | CanonicalForm result= 0; |
---|
482 | int i= 0; |
---|
483 | int lf= 0; |
---|
484 | int lg= d*k; |
---|
485 | int degfSubLf= degf; |
---|
486 | int deggSubLg= degg-lg; |
---|
487 | int repLengthBuf2; |
---|
488 | int repLengthBuf1; |
---|
489 | zz_p zzpZero= zz_p(); |
---|
490 | while (degf >= lf || lg >= 0) |
---|
491 | { |
---|
492 | if (degfSubLf >= d) |
---|
493 | repLengthBuf1= d; |
---|
494 | else if (degfSubLf < 0) |
---|
495 | repLengthBuf1= 0; |
---|
496 | else |
---|
497 | repLengthBuf1= degfSubLf + 1; |
---|
498 | buf1.rep.SetLength((long) repLengthBuf1); |
---|
499 | |
---|
500 | buf1p= buf1.rep.elts(); |
---|
501 | for (int ind= 0; ind < repLengthBuf1; ind++) |
---|
502 | buf1p [ind]= fp [ind + lf]; |
---|
503 | buf1.normalize(); |
---|
504 | |
---|
505 | repLengthBuf1= buf1.rep.length(); |
---|
506 | |
---|
507 | if (deggSubLg >= d - 1) |
---|
508 | repLengthBuf2= d - 1; |
---|
509 | else if (deggSubLg < 0) |
---|
510 | repLengthBuf2= 0; |
---|
511 | else |
---|
512 | repLengthBuf2= deggSubLg + 1; |
---|
513 | |
---|
514 | buf2.rep.SetLength ((long) repLengthBuf2); |
---|
515 | buf2p= buf2.rep.elts(); |
---|
516 | for (int ind= 0; ind < repLengthBuf2; ind++) |
---|
517 | buf2p [ind]= gp [ind + lg]; |
---|
518 | |
---|
519 | buf2.normalize(); |
---|
520 | |
---|
521 | repLengthBuf2= buf2.rep.length(); |
---|
522 | |
---|
523 | |
---|
524 | buf3.rep.SetLength((long) repLengthBuf2 + d); |
---|
525 | buf3p= buf3.rep.elts(); |
---|
526 | buf2p= buf2.rep.elts(); |
---|
527 | buf1p= buf1.rep.elts(); |
---|
528 | for (int ind= 0; ind < repLengthBuf1; ind++) |
---|
529 | buf3p [ind]= buf1p [ind]; |
---|
530 | for (int ind= repLengthBuf1; ind < d; ind++) |
---|
531 | buf3p [ind]= zzpZero; |
---|
532 | for (int ind= 0; ind < repLengthBuf2; ind++) |
---|
533 | buf3p [ind + d]= buf2p [ind]; |
---|
534 | buf3.normalize(); |
---|
535 | |
---|
536 | result += convertNTLzzpX2CF (buf3, x)*power (y, i); |
---|
537 | i++; |
---|
538 | |
---|
539 | |
---|
540 | lf= i*d; |
---|
541 | degfSubLf= degf - lf; |
---|
542 | |
---|
543 | lg= d*(k-i); |
---|
544 | deggSubLg= degg - lg; |
---|
545 | |
---|
546 | buf1p= buf1.rep.elts(); |
---|
547 | |
---|
548 | if (lg >= 0 && deggSubLg > 0) |
---|
549 | { |
---|
550 | if (repLengthBuf2 > degfSubLf + 1) |
---|
551 | degfSubLf= repLengthBuf2 - 1; |
---|
552 | int tmp= tmin (repLengthBuf1, deggSubLg); |
---|
553 | for (int ind= 0; ind < tmp; ind++) |
---|
554 | gp [ind + lg] -= buf1p [ind]; |
---|
555 | } |
---|
556 | if (lg < 0) |
---|
557 | break; |
---|
558 | |
---|
559 | buf2p= buf2.rep.elts(); |
---|
560 | if (degfSubLf >= 0) |
---|
561 | { |
---|
562 | for (int ind= 0; ind < repLengthBuf2; ind++) |
---|
563 | fp [ind + lf] -= buf2p [ind]; |
---|
564 | } |
---|
565 | } |
---|
566 | |
---|
567 | return result; |
---|
568 | } |
---|
569 | |
---|
570 | CanonicalForm reverseSubst (const zz_pEX& F, int d, const Variable& alpha) |
---|
571 | { |
---|
572 | Variable y= Variable (2); |
---|
573 | Variable x= Variable (1); |
---|
574 | |
---|
575 | zz_pEX f= F; |
---|
576 | zz_pE *fp= f.rep.elts(); |
---|
577 | |
---|
578 | zz_pEX buf; |
---|
579 | zz_pE *bufp; |
---|
580 | CanonicalForm result= 0; |
---|
581 | int i= 0; |
---|
582 | int degf= deg(f); |
---|
583 | int k= 0; |
---|
584 | int degfSubK; |
---|
585 | int repLength; |
---|
586 | while (degf >= k) |
---|
587 | { |
---|
588 | degfSubK= degf - k; |
---|
589 | if (degfSubK >= d) |
---|
590 | repLength= d; |
---|
591 | else |
---|
592 | repLength= degfSubK + 1; |
---|
593 | |
---|
594 | buf.rep.SetLength ((long) repLength); |
---|
595 | bufp= buf.rep.elts(); |
---|
596 | for (int j= 0; j < repLength; j++) |
---|
597 | bufp [j]= fp [j + k]; |
---|
598 | buf.normalize(); |
---|
599 | |
---|
600 | result += convertNTLzz_pEX2CF (buf, x, alpha)*power (y, i); |
---|
601 | i++; |
---|
602 | k= d*i; |
---|
603 | } |
---|
604 | |
---|
605 | return result; |
---|
606 | } |
---|
607 | |
---|
608 | CanonicalForm reverseSubstFp (const zz_pX& F, int d) |
---|
609 | { |
---|
610 | Variable y= Variable (2); |
---|
611 | Variable x= Variable (1); |
---|
612 | |
---|
613 | zz_pX f= F; |
---|
614 | zz_p *fp= f.rep.elts(); |
---|
615 | |
---|
616 | zz_pX buf; |
---|
617 | zz_p *bufp; |
---|
618 | CanonicalForm result= 0; |
---|
619 | int i= 0; |
---|
620 | int degf= deg(f); |
---|
621 | int k= 0; |
---|
622 | int degfSubK; |
---|
623 | int repLength; |
---|
624 | while (degf >= k) |
---|
625 | { |
---|
626 | degfSubK= degf - k; |
---|
627 | if (degfSubK >= d) |
---|
628 | repLength= d; |
---|
629 | else |
---|
630 | repLength= degfSubK + 1; |
---|
631 | |
---|
632 | buf.rep.SetLength ((long) repLength); |
---|
633 | bufp= buf.rep.elts(); |
---|
634 | for (int j= 0; j < repLength; j++) |
---|
635 | bufp [j]= fp [j + k]; |
---|
636 | buf.normalize(); |
---|
637 | |
---|
638 | result += convertNTLzzpX2CF (buf, x)*power (y, i); |
---|
639 | i++; |
---|
640 | k= d*i; |
---|
641 | } |
---|
642 | |
---|
643 | return result; |
---|
644 | } |
---|
645 | |
---|
646 | // assumes input to be reduced mod M and to be an element of Fq not Fp |
---|
647 | CanonicalForm |
---|
648 | mulMod2NTLFpReci (const CanonicalForm& F, const CanonicalForm& G, const |
---|
649 | CanonicalForm& M) |
---|
650 | { |
---|
651 | int d1= tmax (degree (F, 1), degree (G, 1)) + 1; |
---|
652 | int d2= tmax (degree (F, 2), degree (G, 2)); |
---|
653 | |
---|
654 | zz_pX F1, F2; |
---|
655 | kronSubRecipro (F1, F2, F, d1); |
---|
656 | zz_pX G1, G2; |
---|
657 | kronSubRecipro (G1, G2, G, d1); |
---|
658 | |
---|
659 | int k= d1*degree (M); |
---|
660 | MulTrunc (F1, F1, G1, (long) k); |
---|
661 | |
---|
662 | mul (F2, F2, G2); |
---|
663 | F2 >>= k; |
---|
664 | |
---|
665 | return reverseSubst (F1, F2, d1, d2); |
---|
666 | } |
---|
667 | |
---|
668 | //Kronecker substitution |
---|
669 | CanonicalForm |
---|
670 | mulMod2NTLFp (const CanonicalForm& F, const CanonicalForm& G, const |
---|
671 | CanonicalForm& M) |
---|
672 | { |
---|
673 | CanonicalForm A= F; |
---|
674 | CanonicalForm B= G; |
---|
675 | |
---|
676 | int degAx= degree (A, 1); |
---|
677 | int degAy= degree (A, 2); |
---|
678 | int degBx= degree (B, 1); |
---|
679 | int degBy= degree (B, 2); |
---|
680 | int d1= degAx + 1 + degBx; |
---|
681 | int d2= tmax (degree (A, 2), degree (B, 2)); |
---|
682 | |
---|
683 | if (d1 > 128 && d2 > 160 && (degAy == degBy)) |
---|
684 | return mulMod2NTLFpReci (A, B, M); |
---|
685 | |
---|
686 | zz_pX NTLA= kronSubFp (A, d1); |
---|
687 | zz_pX NTLB= kronSubFp (B, d1); |
---|
688 | |
---|
689 | int k= d1*degree (M); |
---|
690 | MulTrunc (NTLA, NTLA, NTLB, (long) k); |
---|
691 | |
---|
692 | A= reverseSubstFp (NTLA, d1); |
---|
693 | |
---|
694 | return A; |
---|
695 | } |
---|
696 | |
---|
697 | // assumes input to be reduced mod M and to be an element of Fq not Fp |
---|
698 | CanonicalForm |
---|
699 | mulMod2NTLFqReci (const CanonicalForm& F, const CanonicalForm& G, const |
---|
700 | CanonicalForm& M, const Variable& alpha) |
---|
701 | { |
---|
702 | int d1= tmax (degree (F, 1), degree (G, 1)) + 1; |
---|
703 | int d2= tmax (degree (F, 2), degree (G, 2)); |
---|
704 | |
---|
705 | zz_pEX F1, F2; |
---|
706 | kronSubRecipro (F1, F2, F, d1, alpha); |
---|
707 | zz_pEX G1, G2; |
---|
708 | kronSubRecipro (G1, G2, G, d1, alpha); |
---|
709 | |
---|
710 | int k1= d1*degree (M); |
---|
711 | MulTrunc (F1, F1, G1, (long) k1); |
---|
712 | |
---|
713 | mul (F2, F2, G2); |
---|
714 | |
---|
715 | F2 >>= k1; |
---|
716 | |
---|
717 | CanonicalForm result= reverseSubst (F1, F2, d1, d2, alpha); |
---|
718 | |
---|
719 | return result; |
---|
720 | } |
---|
721 | |
---|
722 | CanonicalForm |
---|
723 | mulMod2NTLFq (const CanonicalForm& F, const CanonicalForm& G, const |
---|
724 | CanonicalForm& M) |
---|
725 | { |
---|
726 | Variable alpha; |
---|
727 | CanonicalForm A= F; |
---|
728 | CanonicalForm B= G; |
---|
729 | |
---|
730 | if (hasFirstAlgVar (A, alpha) || hasFirstAlgVar (B, alpha)) |
---|
731 | { |
---|
732 | int degAx= degree (A, 1); |
---|
733 | int degAy= degree (A, 2); |
---|
734 | int degBx= degree (B, 1); |
---|
735 | int degBy= degree (B, 2); |
---|
736 | int d1= degAx + degBx + 1; |
---|
737 | int d2= tmax (degree (A, 2), degree (B, 2)); |
---|
738 | zz_p::init (getCharacteristic()); |
---|
739 | zz_pX NTLMipo= convertFacCF2NTLzzpX (getMipo (alpha)); |
---|
740 | zz_pE::init (NTLMipo); |
---|
741 | |
---|
742 | int degMipo= degree (getMipo (alpha)); |
---|
743 | if ((d1 > 128/degMipo) && (d2 > 160/degMipo) && (degAy == degBy)) |
---|
744 | return mulMod2NTLFqReci (A, B, M, alpha); |
---|
745 | |
---|
746 | zz_pEX NTLA= kronSub (A, d1, alpha); |
---|
747 | zz_pEX NTLB= kronSub (B, d1, alpha); |
---|
748 | |
---|
749 | int k= d1*degree (M); |
---|
750 | |
---|
751 | MulTrunc (NTLA, NTLA, NTLB, (long) k); |
---|
752 | |
---|
753 | A= reverseSubst (NTLA, d1, alpha); |
---|
754 | |
---|
755 | return A; |
---|
756 | } |
---|
757 | else |
---|
758 | return mulMod2NTLFp (A, B, M); |
---|
759 | } |
---|
760 | |
---|
761 | CanonicalForm mulMod2 (const CanonicalForm& A, const CanonicalForm& B, |
---|
762 | const CanonicalForm& M) |
---|
763 | { |
---|
764 | if (A.isZero() || B.isZero()) |
---|
765 | return 0; |
---|
766 | |
---|
767 | ASSERT (M.isUnivariate(), "M must be univariate"); |
---|
768 | |
---|
769 | CanonicalForm F= mod (A, M); |
---|
770 | CanonicalForm G= mod (B, M); |
---|
771 | if (F.inCoeffDomain() || G.inCoeffDomain()) |
---|
772 | return F*G; |
---|
773 | Variable y= M.mvar(); |
---|
774 | int degF= degree (F, y); |
---|
775 | int degG= degree (G, y); |
---|
776 | |
---|
777 | if ((degF < 1 && degG < 1) && (F.isUnivariate() && G.isUnivariate()) && |
---|
778 | (F.level() == G.level())) |
---|
779 | { |
---|
780 | CanonicalForm result= mulNTL (F, G); |
---|
781 | return mod (result, M); |
---|
782 | } |
---|
783 | else if (degF <= 1 && degG <= 1) |
---|
784 | { |
---|
785 | CanonicalForm result= F*G; |
---|
786 | return mod (result, M); |
---|
787 | } |
---|
788 | |
---|
789 | int sizeF= size (F); |
---|
790 | int sizeG= size (G); |
---|
791 | |
---|
792 | int fallBackToNaive= 50; |
---|
793 | if (sizeF < fallBackToNaive || sizeG < fallBackToNaive) |
---|
794 | return mod (F*G, M); |
---|
795 | |
---|
796 | if (CFFactory::gettype() != GaloisFieldDomain && |
---|
797 | (((degF-degG) < 50 && degF > degG) || ((degG-degF) < 50 && degF <= degG))) |
---|
798 | return mulMod2NTLFq (F, G, M); |
---|
799 | |
---|
800 | int m= (int) ceil (degree (M)/2.0); |
---|
801 | if (degF >= m || degG >= m) |
---|
802 | { |
---|
803 | CanonicalForm MLo= power (y, m); |
---|
804 | CanonicalForm MHi= power (y, degree (M) - m); |
---|
805 | CanonicalForm F0= mod (F, MLo); |
---|
806 | CanonicalForm F1= div (F, MLo); |
---|
807 | CanonicalForm G0= mod (G, MLo); |
---|
808 | CanonicalForm G1= div (G, MLo); |
---|
809 | CanonicalForm F0G1= mulMod2 (F0, G1, MHi); |
---|
810 | CanonicalForm F1G0= mulMod2 (F1, G0, MHi); |
---|
811 | CanonicalForm F0G0= mulMod2 (F0, G0, M); |
---|
812 | return F0G0 + MLo*(F0G1 + F1G0); |
---|
813 | } |
---|
814 | else |
---|
815 | { |
---|
816 | m= (int) ceil (tmax (degF, degG)/2.0); |
---|
817 | CanonicalForm yToM= power (y, m); |
---|
818 | CanonicalForm F0= mod (F, yToM); |
---|
819 | CanonicalForm F1= div (F, yToM); |
---|
820 | CanonicalForm G0= mod (G, yToM); |
---|
821 | CanonicalForm G1= div (G, yToM); |
---|
822 | CanonicalForm H00= mulMod2 (F0, G0, M); |
---|
823 | CanonicalForm H11= mulMod2 (F1, G1, M); |
---|
824 | CanonicalForm H01= mulMod2 (F0 + F1, G0 + G1, M); |
---|
825 | return H11*yToM*yToM + (H01 - H11 - H00)*yToM + H00; |
---|
826 | } |
---|
827 | DEBOUTLN (cerr, "fatal end in mulMod2"); |
---|
828 | } |
---|
829 | |
---|
830 | CanonicalForm mulMod (const CanonicalForm& A, const CanonicalForm& B, |
---|
831 | const CFList& MOD) |
---|
832 | { |
---|
833 | if (A.isZero() || B.isZero()) |
---|
834 | return 0; |
---|
835 | |
---|
836 | if (MOD.length() == 1) |
---|
837 | return mulMod2 (A, B, MOD.getLast()); |
---|
838 | |
---|
839 | CanonicalForm M= MOD.getLast(); |
---|
840 | CanonicalForm F= mod (A, M); |
---|
841 | CanonicalForm G= mod (B, M); |
---|
842 | if (F.inCoeffDomain() || G.inCoeffDomain()) |
---|
843 | return F*G; |
---|
844 | Variable y= M.mvar(); |
---|
845 | int degF= degree (F, y); |
---|
846 | int degG= degree (G, y); |
---|
847 | |
---|
848 | if ((degF <= 1 && F.level() <= M.level()) && |
---|
849 | (degG <= 1 && G.level() <= M.level())) |
---|
850 | { |
---|
851 | CFList buf= MOD; |
---|
852 | buf.removeLast(); |
---|
853 | if (degF == 1 && degG == 1) |
---|
854 | { |
---|
855 | CanonicalForm F0= mod (F, y); |
---|
856 | CanonicalForm F1= div (F, y); |
---|
857 | CanonicalForm G0= mod (G, y); |
---|
858 | CanonicalForm G1= div (G, y); |
---|
859 | if (degree (M) > 2) |
---|
860 | { |
---|
861 | CanonicalForm H00= mulMod (F0, G0, buf); |
---|
862 | CanonicalForm H11= mulMod (F1, G1, buf); |
---|
863 | CanonicalForm H01= mulMod (F0 + F1, G0 + G1, buf); |
---|
864 | return H11*y*y + (H01 - H00 - H11)*y + H00; |
---|
865 | } |
---|
866 | else //here degree (M) == 2 |
---|
867 | { |
---|
868 | buf.append (y); |
---|
869 | CanonicalForm F0G1= mulMod (F0, G1, buf); |
---|
870 | CanonicalForm F1G0= mulMod (F1, G0, buf); |
---|
871 | CanonicalForm F0G0= mulMod (F0, G0, MOD); |
---|
872 | CanonicalForm result= F0G0 + y*(F0G1 + F1G0); |
---|
873 | return result; |
---|
874 | } |
---|
875 | } |
---|
876 | else if (degF == 1 && degG == 0) |
---|
877 | return mulMod (div (F, y), G, buf)*y + mulMod (mod (F, y), G, buf); |
---|
878 | else if (degF == 0 && degG == 1) |
---|
879 | return mulMod (div (G, y), F, buf)*y + mulMod (mod (G, y), F, buf); |
---|
880 | else |
---|
881 | return mulMod (F, G, buf); |
---|
882 | } |
---|
883 | int m= (int) ceil (degree (M)/2.0); |
---|
884 | if (degF >= m || degG >= m) |
---|
885 | { |
---|
886 | CanonicalForm MLo= power (y, m); |
---|
887 | CanonicalForm MHi= power (y, degree (M) - m); |
---|
888 | CanonicalForm F0= mod (F, MLo); |
---|
889 | CanonicalForm F1= div (F, MLo); |
---|
890 | CanonicalForm G0= mod (G, MLo); |
---|
891 | CanonicalForm G1= div (G, MLo); |
---|
892 | CFList buf= MOD; |
---|
893 | buf.removeLast(); |
---|
894 | buf.append (MHi); |
---|
895 | CanonicalForm F0G1= mulMod (F0, G1, buf); |
---|
896 | CanonicalForm F1G0= mulMod (F1, G0, buf); |
---|
897 | CanonicalForm F0G0= mulMod (F0, G0, MOD); |
---|
898 | return F0G0 + MLo*(F0G1 + F1G0); |
---|
899 | } |
---|
900 | else |
---|
901 | { |
---|
902 | m= (int) ceil (tmax (degF, degG)/2.0); |
---|
903 | CanonicalForm yToM= power (y, m); |
---|
904 | CanonicalForm F0= mod (F, yToM); |
---|
905 | CanonicalForm F1= div (F, yToM); |
---|
906 | CanonicalForm G0= mod (G, yToM); |
---|
907 | CanonicalForm G1= div (G, yToM); |
---|
908 | CanonicalForm H00= mulMod (F0, G0, MOD); |
---|
909 | CanonicalForm H11= mulMod (F1, G1, MOD); |
---|
910 | CanonicalForm H01= mulMod (F0 + F1, G0 + G1, MOD); |
---|
911 | return H11*yToM*yToM + (H01 - H11 - H00)*yToM + H00; |
---|
912 | } |
---|
913 | DEBOUTLN (cerr, "fatal end in mulMod"); |
---|
914 | } |
---|
915 | |
---|
916 | CanonicalForm prodMod (const CFList& L, const CanonicalForm& M) |
---|
917 | { |
---|
918 | if (L.isEmpty()) |
---|
919 | return 1; |
---|
920 | int l= L.length(); |
---|
921 | if (l == 1) |
---|
922 | return mod (L.getFirst(), M); |
---|
923 | else if (l == 2) { |
---|
924 | CanonicalForm result= mulMod2 (L.getFirst(), L.getLast(), M); |
---|
925 | return result; |
---|
926 | } |
---|
927 | else |
---|
928 | { |
---|
929 | l /= 2; |
---|
930 | CFList tmp1, tmp2; |
---|
931 | CFListIterator i= L; |
---|
932 | CanonicalForm buf1, buf2; |
---|
933 | for (int j= 1; j <= l; j++, i++) |
---|
934 | tmp1.append (i.getItem()); |
---|
935 | tmp2= Difference (L, tmp1); |
---|
936 | buf1= prodMod (tmp1, M); |
---|
937 | buf2= prodMod (tmp2, M); |
---|
938 | CanonicalForm result= mulMod2 (buf1, buf2, M); |
---|
939 | return result; |
---|
940 | } |
---|
941 | } |
---|
942 | |
---|
943 | CanonicalForm prodMod (const CFList& L, const CFList& M) |
---|
944 | { |
---|
945 | if (L.isEmpty()) |
---|
946 | return 1; |
---|
947 | else if (L.length() == 1) |
---|
948 | return L.getFirst(); |
---|
949 | else if (L.length() == 2) |
---|
950 | return mulMod (L.getFirst(), L.getLast(), M); |
---|
951 | else |
---|
952 | { |
---|
953 | int l= L.length()/2; |
---|
954 | CFListIterator i= L; |
---|
955 | CFList tmp1, tmp2; |
---|
956 | CanonicalForm buf1, buf2; |
---|
957 | for (int j= 1; j <= l; j++, i++) |
---|
958 | tmp1.append (i.getItem()); |
---|
959 | tmp2= Difference (L, tmp1); |
---|
960 | buf1= prodMod (tmp1, M); |
---|
961 | buf2= prodMod (tmp2, M); |
---|
962 | return mulMod (buf1, buf2, M); |
---|
963 | } |
---|
964 | } |
---|
965 | |
---|
966 | |
---|
967 | CanonicalForm reverse (const CanonicalForm& F, int d) |
---|
968 | { |
---|
969 | if (d == 0) |
---|
970 | return F; |
---|
971 | CanonicalForm A= F; |
---|
972 | Variable y= Variable (2); |
---|
973 | Variable x= Variable (1); |
---|
974 | if (degree (A, x) > 0) |
---|
975 | { |
---|
976 | A= swapvar (A, x, y); |
---|
977 | CanonicalForm result= 0; |
---|
978 | CFIterator i= A; |
---|
979 | while (d - i.exp() < 0) |
---|
980 | i++; |
---|
981 | |
---|
982 | for (; i.hasTerms() && (d - i.exp() >= 0); i++) |
---|
983 | result += swapvar (i.coeff(),x,y)*power (x, d - i.exp()); |
---|
984 | return result; |
---|
985 | } |
---|
986 | else |
---|
987 | return A*power (x, d); |
---|
988 | } |
---|
989 | |
---|
990 | CanonicalForm |
---|
991 | newtonInverse (const CanonicalForm& F, const int n, const CanonicalForm& M) |
---|
992 | { |
---|
993 | int l= ilog2(n); |
---|
994 | |
---|
995 | CanonicalForm g= mod (F, M)[0] [0]; |
---|
996 | |
---|
997 | ASSERT (!g.isZero(), "expected a unit"); |
---|
998 | |
---|
999 | Variable alpha; |
---|
1000 | |
---|
1001 | if (!g.isOne()) |
---|
1002 | g = 1/g; |
---|
1003 | Variable x= Variable (1); |
---|
1004 | CanonicalForm result; |
---|
1005 | int exp= 0; |
---|
1006 | if (n & 1) |
---|
1007 | { |
---|
1008 | result= g; |
---|
1009 | exp= 1; |
---|
1010 | } |
---|
1011 | CanonicalForm h; |
---|
1012 | |
---|
1013 | for (int i= 1; i <= l; i++) |
---|
1014 | { |
---|
1015 | h= mulMod2 (g, mod (F, power (x, (1 << i))), M); |
---|
1016 | h= mod (h, power (x, (1 << i)) - 1); |
---|
1017 | h= div (h, power (x, (1 << (i - 1)))); |
---|
1018 | h= mod (h, M); |
---|
1019 | g -= power (x, (1 << (i - 1)))* |
---|
1020 | mod (mulMod2 (g, h, M), power (x, (1 << (i - 1)))); |
---|
1021 | |
---|
1022 | if (n & (1 << i)) |
---|
1023 | { |
---|
1024 | if (exp) |
---|
1025 | { |
---|
1026 | h= mulMod2 (result, mod (F, power (x, exp + (1 << i))), M); |
---|
1027 | h= mod (h, power (x, exp + (1 << i)) - 1); |
---|
1028 | h= div (h, power (x, exp)); |
---|
1029 | h= mod (h, M); |
---|
1030 | result -= power(x, exp)*mod (mulMod2 (g, h, M), |
---|
1031 | power (x, (1 << i))); |
---|
1032 | exp += (1 << i); |
---|
1033 | } |
---|
1034 | else |
---|
1035 | { |
---|
1036 | exp= (1 << i); |
---|
1037 | result= g; |
---|
1038 | } |
---|
1039 | } |
---|
1040 | } |
---|
1041 | |
---|
1042 | return result; |
---|
1043 | } |
---|
1044 | |
---|
1045 | CanonicalForm |
---|
1046 | newtonDiv (const CanonicalForm& F, const CanonicalForm& G, const CanonicalForm& |
---|
1047 | M) |
---|
1048 | { |
---|
1049 | ASSERT (getCharacteristic() > 0, "positive characteristic expected"); |
---|
1050 | ASSERT (CFFactory::gettype() != GaloisFieldDomain, "no GF expected"); |
---|
1051 | |
---|
1052 | CanonicalForm A= mod (F, M); |
---|
1053 | CanonicalForm B= mod (G, M); |
---|
1054 | |
---|
1055 | Variable x= Variable (1); |
---|
1056 | int degA= degree (A, x); |
---|
1057 | int degB= degree (B, x); |
---|
1058 | int m= degA - degB; |
---|
1059 | if (m < 0) |
---|
1060 | return 0; |
---|
1061 | |
---|
1062 | CanonicalForm Q; |
---|
1063 | if (degB <= 1 || CFFactory::gettype() == GaloisFieldDomain) |
---|
1064 | { |
---|
1065 | CanonicalForm R; |
---|
1066 | divrem2 (A, B, Q, R, M); |
---|
1067 | } |
---|
1068 | else |
---|
1069 | { |
---|
1070 | CanonicalForm R= reverse (A, degA); |
---|
1071 | CanonicalForm revB= reverse (B, degB); |
---|
1072 | revB= newtonInverse (revB, m + 1, M); |
---|
1073 | Q= mulMod2 (R, revB, M); |
---|
1074 | Q= mod (Q, power (x, m + 1)); |
---|
1075 | Q= reverse (Q, m); |
---|
1076 | } |
---|
1077 | |
---|
1078 | return Q; |
---|
1079 | } |
---|
1080 | |
---|
1081 | void |
---|
1082 | newtonDivrem (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
---|
1083 | CanonicalForm& R, const CanonicalForm& M) |
---|
1084 | { |
---|
1085 | CanonicalForm A= mod (F, M); |
---|
1086 | CanonicalForm B= mod (G, M); |
---|
1087 | Variable x= Variable (1); |
---|
1088 | int degA= degree (A, x); |
---|
1089 | int degB= degree (B, x); |
---|
1090 | int m= degA - degB; |
---|
1091 | |
---|
1092 | if (m < 0) |
---|
1093 | { |
---|
1094 | R= A; |
---|
1095 | Q= 0; |
---|
1096 | return; |
---|
1097 | } |
---|
1098 | |
---|
1099 | if (degB <= 1 || CFFactory::gettype() == GaloisFieldDomain) |
---|
1100 | { |
---|
1101 | divrem2 (A, B, Q, R, M); |
---|
1102 | } |
---|
1103 | else |
---|
1104 | { |
---|
1105 | R= reverse (A, degA); |
---|
1106 | |
---|
1107 | CanonicalForm revB= reverse (B, degB); |
---|
1108 | revB= newtonInverse (revB, m + 1, M); |
---|
1109 | Q= mulMod2 (R, revB, M); |
---|
1110 | |
---|
1111 | Q= mod (Q, power (x, m + 1)); |
---|
1112 | Q= reverse (Q, m); |
---|
1113 | |
---|
1114 | R= A - mulMod2 (Q, B, M); |
---|
1115 | } |
---|
1116 | } |
---|
1117 | |
---|
1118 | static inline |
---|
1119 | CFList split (const CanonicalForm& F, const int m, const Variable& x) |
---|
1120 | { |
---|
1121 | CanonicalForm A= F; |
---|
1122 | CanonicalForm buf= 0; |
---|
1123 | bool swap= false; |
---|
1124 | if (degree (A, x) <= 0) |
---|
1125 | return CFList(A); |
---|
1126 | else if (x.level() != A.level()) |
---|
1127 | { |
---|
1128 | swap= true; |
---|
1129 | A= swapvar (A, x, A.mvar()); |
---|
1130 | } |
---|
1131 | |
---|
1132 | int j= (int) floor ((double) degree (A)/ m); |
---|
1133 | CFList result; |
---|
1134 | CFIterator i= A; |
---|
1135 | for (; j >= 0; j--) |
---|
1136 | { |
---|
1137 | while (i.hasTerms() && i.exp() - j*m >= 0) |
---|
1138 | { |
---|
1139 | if (swap) |
---|
1140 | buf += i.coeff()*power (A.mvar(), i.exp() - j*m); |
---|
1141 | else |
---|
1142 | buf += i.coeff()*power (x, i.exp() - j*m); |
---|
1143 | i++; |
---|
1144 | } |
---|
1145 | if (swap) |
---|
1146 | result.append (swapvar (buf, x, F.mvar())); |
---|
1147 | else |
---|
1148 | result.append (buf); |
---|
1149 | buf= 0; |
---|
1150 | } |
---|
1151 | return result; |
---|
1152 | } |
---|
1153 | |
---|
1154 | static inline |
---|
1155 | void divrem32 (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
---|
1156 | CanonicalForm& R, const CFList& M); |
---|
1157 | |
---|
1158 | static inline |
---|
1159 | void divrem21 (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
---|
1160 | CanonicalForm& R, const CFList& M) |
---|
1161 | { |
---|
1162 | CanonicalForm A= mod (F, M); |
---|
1163 | CanonicalForm B= mod (G, M); |
---|
1164 | Variable x= Variable (1); |
---|
1165 | int degB= degree (B, x); |
---|
1166 | int degA= degree (A, x); |
---|
1167 | if (degA < degB) |
---|
1168 | { |
---|
1169 | Q= 0; |
---|
1170 | R= A; |
---|
1171 | return; |
---|
1172 | } |
---|
1173 | ASSERT (2*degB > degA, "expected degree (F, 1) < 2*degree (G, 1)"); |
---|
1174 | if (degB < 1) |
---|
1175 | { |
---|
1176 | divrem (A, B, Q, R); |
---|
1177 | Q= mod (Q, M); |
---|
1178 | R= mod (R, M); |
---|
1179 | return; |
---|
1180 | } |
---|
1181 | |
---|
1182 | int m= (int) ceil ((double) (degB + 1)/2.0) + 1; |
---|
1183 | CFList splitA= split (A, m, x); |
---|
1184 | if (splitA.length() == 3) |
---|
1185 | splitA.insert (0); |
---|
1186 | if (splitA.length() == 2) |
---|
1187 | { |
---|
1188 | splitA.insert (0); |
---|
1189 | splitA.insert (0); |
---|
1190 | } |
---|
1191 | if (splitA.length() == 1) |
---|
1192 | { |
---|
1193 | splitA.insert (0); |
---|
1194 | splitA.insert (0); |
---|
1195 | splitA.insert (0); |
---|
1196 | } |
---|
1197 | |
---|
1198 | CanonicalForm xToM= power (x, m); |
---|
1199 | |
---|
1200 | CFListIterator i= splitA; |
---|
1201 | CanonicalForm H= i.getItem(); |
---|
1202 | i++; |
---|
1203 | H *= xToM; |
---|
1204 | H += i.getItem(); |
---|
1205 | i++; |
---|
1206 | H *= xToM; |
---|
1207 | H += i.getItem(); |
---|
1208 | i++; |
---|
1209 | |
---|
1210 | divrem32 (H, B, Q, R, M); |
---|
1211 | |
---|
1212 | CFList splitR= split (R, m, x); |
---|
1213 | if (splitR.length() == 1) |
---|
1214 | splitR.insert (0); |
---|
1215 | |
---|
1216 | H= splitR.getFirst(); |
---|
1217 | H *= xToM; |
---|
1218 | H += splitR.getLast(); |
---|
1219 | H *= xToM; |
---|
1220 | H += i.getItem(); |
---|
1221 | |
---|
1222 | CanonicalForm bufQ; |
---|
1223 | divrem32 (H, B, bufQ, R, M); |
---|
1224 | |
---|
1225 | Q *= xToM; |
---|
1226 | Q += bufQ; |
---|
1227 | return; |
---|
1228 | } |
---|
1229 | |
---|
1230 | static inline |
---|
1231 | void divrem32 (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
---|
1232 | CanonicalForm& R, const CFList& M) |
---|
1233 | { |
---|
1234 | CanonicalForm A= mod (F, M); |
---|
1235 | CanonicalForm B= mod (G, M); |
---|
1236 | Variable x= Variable (1); |
---|
1237 | int degB= degree (B, x); |
---|
1238 | int degA= degree (A, x); |
---|
1239 | if (degA < degB) |
---|
1240 | { |
---|
1241 | Q= 0; |
---|
1242 | R= A; |
---|
1243 | return; |
---|
1244 | } |
---|
1245 | ASSERT (3*(degB/2) > degA, "expected degree (F, 1) < 3*(degree (G, 1)/2)"); |
---|
1246 | if (degB < 1) |
---|
1247 | { |
---|
1248 | divrem (A, B, Q, R); |
---|
1249 | Q= mod (Q, M); |
---|
1250 | R= mod (R, M); |
---|
1251 | return; |
---|
1252 | } |
---|
1253 | int m= (int) ceil ((double) (degB + 1)/ 2.0); |
---|
1254 | |
---|
1255 | CFList splitA= split (A, m, x); |
---|
1256 | CFList splitB= split (B, m, x); |
---|
1257 | |
---|
1258 | if (splitA.length() == 2) |
---|
1259 | { |
---|
1260 | splitA.insert (0); |
---|
1261 | } |
---|
1262 | if (splitA.length() == 1) |
---|
1263 | { |
---|
1264 | splitA.insert (0); |
---|
1265 | splitA.insert (0); |
---|
1266 | } |
---|
1267 | CanonicalForm xToM= power (x, m); |
---|
1268 | |
---|
1269 | CanonicalForm H; |
---|
1270 | CFListIterator i= splitA; |
---|
1271 | i++; |
---|
1272 | |
---|
1273 | if (degree (splitA.getFirst(), x) < degree (splitB.getFirst(), x)) |
---|
1274 | { |
---|
1275 | H= splitA.getFirst()*xToM + i.getItem(); |
---|
1276 | divrem21 (H, splitB.getFirst(), Q, R, M); |
---|
1277 | } |
---|
1278 | else |
---|
1279 | { |
---|
1280 | R= splitA.getFirst()*xToM + i.getItem() + splitB.getFirst() - |
---|
1281 | splitB.getFirst()*xToM; |
---|
1282 | Q= xToM - 1; |
---|
1283 | } |
---|
1284 | |
---|
1285 | H= mulMod (Q, splitB.getLast(), M); |
---|
1286 | |
---|
1287 | R= R*xToM + splitA.getLast() - H; |
---|
1288 | |
---|
1289 | while (degree (R, x) >= degB) |
---|
1290 | { |
---|
1291 | xToM= power (x, degree (R, x) - degB); |
---|
1292 | Q += LC (R, x)*xToM; |
---|
1293 | R -= mulMod (LC (R, x), B, M)*xToM; |
---|
1294 | Q= mod (Q, M); |
---|
1295 | R= mod (R, M); |
---|
1296 | } |
---|
1297 | |
---|
1298 | return; |
---|
1299 | } |
---|
1300 | |
---|
1301 | void divrem2 (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
---|
1302 | CanonicalForm& R, const CanonicalForm& M) |
---|
1303 | { |
---|
1304 | CanonicalForm A= mod (F, M); |
---|
1305 | CanonicalForm B= mod (G, M); |
---|
1306 | |
---|
1307 | if (B.inCoeffDomain()) |
---|
1308 | { |
---|
1309 | divrem (A, B, Q, R); |
---|
1310 | return; |
---|
1311 | } |
---|
1312 | if (A.inCoeffDomain() && !B.inCoeffDomain()) |
---|
1313 | { |
---|
1314 | Q= 0; |
---|
1315 | R= A; |
---|
1316 | return; |
---|
1317 | } |
---|
1318 | |
---|
1319 | if (B.level() < A.level()) |
---|
1320 | { |
---|
1321 | divrem (A, B, Q, R); |
---|
1322 | return; |
---|
1323 | } |
---|
1324 | if (A.level() > B.level()) |
---|
1325 | { |
---|
1326 | R= A; |
---|
1327 | Q= 0; |
---|
1328 | return; |
---|
1329 | } |
---|
1330 | if (B.level() == 1 && B.isUnivariate()) |
---|
1331 | { |
---|
1332 | divrem (A, B, Q, R); |
---|
1333 | return; |
---|
1334 | } |
---|
1335 | if (!(B.level() == 1 && B.isUnivariate()) && (A.level() == 1 && A.isUnivariate())) |
---|
1336 | { |
---|
1337 | Q= 0; |
---|
1338 | R= A; |
---|
1339 | return; |
---|
1340 | } |
---|
1341 | |
---|
1342 | Variable x= Variable (1); |
---|
1343 | int degB= degree (B, x); |
---|
1344 | if (degB > degree (A, x)) |
---|
1345 | { |
---|
1346 | Q= 0; |
---|
1347 | R= A; |
---|
1348 | return; |
---|
1349 | } |
---|
1350 | |
---|
1351 | CFList splitA= split (A, degB, x); |
---|
1352 | |
---|
1353 | CanonicalForm xToDegB= power (x, degB); |
---|
1354 | CanonicalForm H, bufQ; |
---|
1355 | Q= 0; |
---|
1356 | CFListIterator i= splitA; |
---|
1357 | H= i.getItem()*xToDegB; |
---|
1358 | i++; |
---|
1359 | H += i.getItem(); |
---|
1360 | CFList buf; |
---|
1361 | while (i.hasItem()) |
---|
1362 | { |
---|
1363 | buf= CFList (M); |
---|
1364 | divrem21 (H, B, bufQ, R, buf); |
---|
1365 | i++; |
---|
1366 | if (i.hasItem()) |
---|
1367 | H= R*xToDegB + i.getItem(); |
---|
1368 | Q *= xToDegB; |
---|
1369 | Q += bufQ; |
---|
1370 | } |
---|
1371 | return; |
---|
1372 | } |
---|
1373 | |
---|
1374 | void divrem (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
---|
1375 | CanonicalForm& R, const CFList& MOD) |
---|
1376 | { |
---|
1377 | CanonicalForm A= mod (F, MOD); |
---|
1378 | CanonicalForm B= mod (G, MOD); |
---|
1379 | Variable x= Variable (1); |
---|
1380 | int degB= degree (B, x); |
---|
1381 | if (degB > degree (A, x)) |
---|
1382 | { |
---|
1383 | Q= 0; |
---|
1384 | R= A; |
---|
1385 | return; |
---|
1386 | } |
---|
1387 | |
---|
1388 | if (degB <= 0) |
---|
1389 | { |
---|
1390 | divrem (A, B, Q, R); |
---|
1391 | Q= mod (Q, MOD); |
---|
1392 | R= mod (R, MOD); |
---|
1393 | return; |
---|
1394 | } |
---|
1395 | CFList splitA= split (A, degB, x); |
---|
1396 | |
---|
1397 | CanonicalForm xToDegB= power (x, degB); |
---|
1398 | CanonicalForm H, bufQ; |
---|
1399 | Q= 0; |
---|
1400 | CFListIterator i= splitA; |
---|
1401 | H= i.getItem()*xToDegB; |
---|
1402 | i++; |
---|
1403 | H += i.getItem(); |
---|
1404 | while (i.hasItem()) |
---|
1405 | { |
---|
1406 | divrem21 (H, B, bufQ, R, MOD); |
---|
1407 | i++; |
---|
1408 | if (i.hasItem()) |
---|
1409 | H= R*xToDegB + i.getItem(); |
---|
1410 | Q *= xToDegB; |
---|
1411 | Q += bufQ; |
---|
1412 | } |
---|
1413 | return; |
---|
1414 | } |
---|
1415 | |
---|
1416 | void sortList (CFList& list, const Variable& x) |
---|
1417 | { |
---|
1418 | int l= 1; |
---|
1419 | int k= 1; |
---|
1420 | CanonicalForm buf; |
---|
1421 | CFListIterator m; |
---|
1422 | for (CFListIterator i= list; l <= list.length(); i++, l++) |
---|
1423 | { |
---|
1424 | for (CFListIterator j= list; k <= list.length() - l; k++) |
---|
1425 | { |
---|
1426 | m= j; |
---|
1427 | m++; |
---|
1428 | if (degree (j.getItem(), x) > degree (m.getItem(), x)) |
---|
1429 | { |
---|
1430 | buf= m.getItem(); |
---|
1431 | m.getItem()= j.getItem(); |
---|
1432 | j.getItem()= buf; |
---|
1433 | j++; |
---|
1434 | j.getItem()= m.getItem(); |
---|
1435 | } |
---|
1436 | else |
---|
1437 | j++; |
---|
1438 | } |
---|
1439 | k= 1; |
---|
1440 | } |
---|
1441 | } |
---|
1442 | |
---|
1443 | static inline |
---|
1444 | CFList diophantine (const CanonicalForm& F, const CFList& factors) |
---|
1445 | { |
---|
1446 | CanonicalForm buf1, buf2, buf3, S, T; |
---|
1447 | CFListIterator i= factors; |
---|
1448 | CFList result; |
---|
1449 | if (i.hasItem()) |
---|
1450 | i++; |
---|
1451 | buf1= F/factors.getFirst(); |
---|
1452 | buf2= divNTL (F, i.getItem()); |
---|
1453 | buf3= extgcd (buf1, buf2, S, T); |
---|
1454 | result.append (S); |
---|
1455 | result.append (T); |
---|
1456 | if (i.hasItem()) |
---|
1457 | i++; |
---|
1458 | for (; i.hasItem(); i++) |
---|
1459 | { |
---|
1460 | buf1= divNTL (F, i.getItem()); |
---|
1461 | buf3= extgcd (buf3, buf1, S, T); |
---|
1462 | CFListIterator k= factors; |
---|
1463 | for (CFListIterator j= result; j.hasItem(); j++, k++) |
---|
1464 | { |
---|
1465 | j.getItem()= mulNTL (j.getItem(), S); |
---|
1466 | j.getItem()= modNTL (j.getItem(), k.getItem()); |
---|
1467 | } |
---|
1468 | result.append (T); |
---|
1469 | } |
---|
1470 | return result; |
---|
1471 | } |
---|
1472 | |
---|
1473 | void |
---|
1474 | henselStep12 (const CanonicalForm& F, const CFList& factors, |
---|
1475 | CFArray& bufFactors, const CFList& diophant, CFMatrix& M, |
---|
1476 | CFArray& Pi, int j) |
---|
1477 | { |
---|
1478 | CanonicalForm E; |
---|
1479 | CanonicalForm xToJ= power (F.mvar(), j); |
---|
1480 | Variable x= F.mvar(); |
---|
1481 | // compute the error |
---|
1482 | if (j == 1) |
---|
1483 | E= F[j]; |
---|
1484 | else |
---|
1485 | { |
---|
1486 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
1487 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
1488 | else |
---|
1489 | E= F[j]; |
---|
1490 | } |
---|
1491 | |
---|
1492 | CFArray buf= CFArray (diophant.length()); |
---|
1493 | bufFactors[0]= mod (factors.getFirst(), power (F.mvar(), j + 1)); |
---|
1494 | int k= 0; |
---|
1495 | CanonicalForm remainder; |
---|
1496 | // actual lifting |
---|
1497 | for (CFListIterator i= diophant; i.hasItem(); i++, k++) |
---|
1498 | { |
---|
1499 | if (degree (bufFactors[k], x) > 0) |
---|
1500 | { |
---|
1501 | if (k > 0) |
---|
1502 | remainder= modNTL (E, bufFactors[k] [0]); |
---|
1503 | else |
---|
1504 | remainder= E; |
---|
1505 | } |
---|
1506 | else |
---|
1507 | remainder= modNTL (E, bufFactors[k]); |
---|
1508 | |
---|
1509 | buf[k]= mulNTL (i.getItem(), remainder); |
---|
1510 | if (degree (bufFactors[k], x) > 0) |
---|
1511 | buf[k]= modNTL (buf[k], bufFactors[k] [0]); |
---|
1512 | else |
---|
1513 | buf[k]= modNTL (buf[k], bufFactors[k]); |
---|
1514 | } |
---|
1515 | for (k= 1; k < factors.length(); k++) |
---|
1516 | bufFactors[k] += xToJ*buf[k]; |
---|
1517 | |
---|
1518 | // update Pi [0] |
---|
1519 | int degBuf0= degree (bufFactors[0], x); |
---|
1520 | int degBuf1= degree (bufFactors[1], x); |
---|
1521 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1522 | M (j + 1, 1)= mulNTL (bufFactors[0] [j], bufFactors[1] [j]); |
---|
1523 | CanonicalForm uIZeroJ; |
---|
1524 | if (j == 1) |
---|
1525 | { |
---|
1526 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1527 | uIZeroJ= mulNTL ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
1528 | (bufFactors[1] [0] + buf[1])) - M(1, 1) - M(j + 1, 1); |
---|
1529 | else if (degBuf0 > 0) |
---|
1530 | uIZeroJ= mulNTL (bufFactors[0] [j], bufFactors[1]); |
---|
1531 | else if (degBuf1 > 0) |
---|
1532 | uIZeroJ= mulNTL (bufFactors[0], buf[1]); |
---|
1533 | else |
---|
1534 | uIZeroJ= 0; |
---|
1535 | Pi [0] += xToJ*uIZeroJ; |
---|
1536 | } |
---|
1537 | else |
---|
1538 | { |
---|
1539 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1540 | uIZeroJ= mulNTL ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
1541 | (bufFactors[1] [0] + buf[1])) - M(1, 1) - M(j + 1, 1); |
---|
1542 | else if (degBuf0 > 0) |
---|
1543 | uIZeroJ= mulNTL (bufFactors[0] [j], bufFactors[1]); |
---|
1544 | else if (degBuf1 > 0) |
---|
1545 | uIZeroJ= mulNTL (bufFactors[0], buf[1]); |
---|
1546 | else |
---|
1547 | uIZeroJ= 0; |
---|
1548 | Pi [0] += xToJ*uIZeroJ; |
---|
1549 | } |
---|
1550 | CFArray tmp= CFArray (factors.length() - 1); |
---|
1551 | for (k= 0; k < factors.length() - 1; k++) |
---|
1552 | tmp[k]= 0; |
---|
1553 | CFIterator one, two; |
---|
1554 | one= bufFactors [0]; |
---|
1555 | two= bufFactors [1]; |
---|
1556 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1557 | { |
---|
1558 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
1559 | { |
---|
1560 | if (k != j - k + 1) |
---|
1561 | { |
---|
1562 | if ((one.hasTerms() && one.exp() == j - k + 1) && (two.hasTerms() && two.exp() == j - k + 1)) |
---|
1563 | { |
---|
1564 | tmp[0] += mulNTL ((bufFactors[0] [k] + one.coeff()), (bufFactors[1] [k] + |
---|
1565 | two.coeff())) - M (k + 1, 1) - M (j - k + 2, 1); |
---|
1566 | one++; |
---|
1567 | two++; |
---|
1568 | } |
---|
1569 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
1570 | { |
---|
1571 | tmp[0] += mulNTL ((bufFactors[0] [k] + one.coeff()), bufFactors[1] [k]) - |
---|
1572 | M (k + 1, 1); |
---|
1573 | one++; |
---|
1574 | } |
---|
1575 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
1576 | { |
---|
1577 | tmp[0] += mulNTL (bufFactors[0] [k], (bufFactors[1] [k] + two.coeff())) - |
---|
1578 | M (k + 1, 1); |
---|
1579 | two++; |
---|
1580 | } |
---|
1581 | } |
---|
1582 | else |
---|
1583 | { |
---|
1584 | tmp[0] += M (k + 1, 1); |
---|
1585 | } |
---|
1586 | } |
---|
1587 | } |
---|
1588 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
1589 | |
---|
1590 | // update Pi [l] |
---|
1591 | int degPi, degBuf; |
---|
1592 | for (int l= 1; l < factors.length() - 1; l++) |
---|
1593 | { |
---|
1594 | degPi= degree (Pi [l - 1], x); |
---|
1595 | degBuf= degree (bufFactors[l + 1], x); |
---|
1596 | if (degPi > 0 && degBuf > 0) |
---|
1597 | M (j + 1, l + 1)= mulNTL (Pi [l - 1] [j], bufFactors[l + 1] [j]); |
---|
1598 | if (j == 1) |
---|
1599 | { |
---|
1600 | if (degPi > 0 && degBuf > 0) |
---|
1601 | Pi [l] += xToJ*(mulNTL (Pi [l - 1] [0] + Pi [l - 1] [j], |
---|
1602 | bufFactors[l + 1] [0] + buf[l + 1]) - M (j + 1, l +1) - |
---|
1603 | M (1, l + 1)); |
---|
1604 | else if (degPi > 0) |
---|
1605 | Pi [l] += xToJ*(mulNTL (Pi [l - 1] [j], bufFactors[l + 1])); |
---|
1606 | else if (degBuf > 0) |
---|
1607 | Pi [l] += xToJ*(mulNTL (Pi [l - 1], buf[l + 1])); |
---|
1608 | } |
---|
1609 | else |
---|
1610 | { |
---|
1611 | if (degPi > 0 && degBuf > 0) |
---|
1612 | { |
---|
1613 | uIZeroJ= mulNTL (uIZeroJ, bufFactors [l + 1] [0]); |
---|
1614 | uIZeroJ += mulNTL (Pi [l - 1] [0], buf [l + 1]); |
---|
1615 | } |
---|
1616 | else if (degPi > 0) |
---|
1617 | uIZeroJ= mulNTL (uIZeroJ, bufFactors [l + 1]); |
---|
1618 | else if (degBuf > 0) |
---|
1619 | { |
---|
1620 | uIZeroJ= mulNTL (uIZeroJ, bufFactors [l + 1] [0]); |
---|
1621 | uIZeroJ += mulNTL (Pi [l - 1], buf[l + 1]); |
---|
1622 | } |
---|
1623 | Pi[l] += xToJ*uIZeroJ; |
---|
1624 | } |
---|
1625 | one= bufFactors [l + 1]; |
---|
1626 | two= Pi [l - 1]; |
---|
1627 | if (two.hasTerms() && two.exp() == j + 1) |
---|
1628 | { |
---|
1629 | if (degBuf > 0 && degPi > 0) |
---|
1630 | { |
---|
1631 | tmp[l] += mulNTL (two.coeff(), bufFactors[l + 1][0]); |
---|
1632 | two++; |
---|
1633 | } |
---|
1634 | else if (degPi > 0) |
---|
1635 | { |
---|
1636 | tmp[l] += mulNTL (two.coeff(), bufFactors[l + 1]); |
---|
1637 | two++; |
---|
1638 | } |
---|
1639 | } |
---|
1640 | if (degBuf > 0 && degPi > 0) |
---|
1641 | { |
---|
1642 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
1643 | { |
---|
1644 | if (k != j - k + 1) |
---|
1645 | { |
---|
1646 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
1647 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
1648 | { |
---|
1649 | tmp[l] += mulNTL ((bufFactors[l + 1] [k] + one.coeff()), (Pi[l - 1] [k] + |
---|
1650 | two.coeff())) - M (k + 1, l + 1) - M (j - k + 2, l + 1); |
---|
1651 | one++; |
---|
1652 | two++; |
---|
1653 | } |
---|
1654 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
1655 | { |
---|
1656 | tmp[l] += mulNTL ((bufFactors[l + 1] [k] + one.coeff()), Pi[l - 1] [k]) - |
---|
1657 | M (k + 1, l + 1); |
---|
1658 | one++; |
---|
1659 | } |
---|
1660 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
1661 | { |
---|
1662 | tmp[l] += mulNTL (bufFactors[l + 1] [k], (Pi[l - 1] [k] + two.coeff())) - |
---|
1663 | M (k + 1, l + 1); |
---|
1664 | two++; |
---|
1665 | } |
---|
1666 | } |
---|
1667 | else |
---|
1668 | tmp[l] += M (k + 1, l + 1); |
---|
1669 | } |
---|
1670 | } |
---|
1671 | Pi[l] += tmp[l]*xToJ*F.mvar(); |
---|
1672 | } |
---|
1673 | return; |
---|
1674 | } |
---|
1675 | |
---|
1676 | void |
---|
1677 | henselLift12 (const CanonicalForm& F, CFList& factors, int l, CFArray& Pi, |
---|
1678 | CFList& diophant, CFMatrix& M, bool sort) |
---|
1679 | { |
---|
1680 | if (sort) |
---|
1681 | sortList (factors, Variable (1)); |
---|
1682 | Pi= CFArray (factors.length() - 1); |
---|
1683 | CFListIterator j= factors; |
---|
1684 | diophant= diophantine (F[0], factors); |
---|
1685 | DEBOUTLN (cerr, "diophant= " << diophant); |
---|
1686 | j++; |
---|
1687 | Pi [0]= mulNTL (j.getItem(), mod (factors.getFirst(), F.mvar())); |
---|
1688 | M (1, 1)= Pi [0]; |
---|
1689 | int i= 1; |
---|
1690 | if (j.hasItem()) |
---|
1691 | j++; |
---|
1692 | for (; j.hasItem(); j++, i++) |
---|
1693 | { |
---|
1694 | Pi [i]= mulNTL (Pi [i - 1], j.getItem()); |
---|
1695 | M (1, i + 1)= Pi [i]; |
---|
1696 | } |
---|
1697 | CFArray bufFactors= CFArray (factors.length()); |
---|
1698 | i= 0; |
---|
1699 | for (CFListIterator k= factors; k.hasItem(); i++, k++) |
---|
1700 | { |
---|
1701 | if (i == 0) |
---|
1702 | bufFactors[i]= mod (k.getItem(), F.mvar()); |
---|
1703 | else |
---|
1704 | bufFactors[i]= k.getItem(); |
---|
1705 | } |
---|
1706 | for (i= 1; i < l; i++) |
---|
1707 | henselStep12 (F, factors, bufFactors, diophant, M, Pi, i); |
---|
1708 | |
---|
1709 | CFListIterator k= factors; |
---|
1710 | for (i= 0; i < factors.length (); i++, k++) |
---|
1711 | k.getItem()= bufFactors[i]; |
---|
1712 | factors.removeFirst(); |
---|
1713 | return; |
---|
1714 | } |
---|
1715 | |
---|
1716 | void |
---|
1717 | henselLiftResume12 (const CanonicalForm& F, CFList& factors, int start, int |
---|
1718 | end, CFArray& Pi, const CFList& diophant, CFMatrix& M) |
---|
1719 | { |
---|
1720 | CFArray bufFactors= CFArray (factors.length()); |
---|
1721 | int i= 0; |
---|
1722 | CanonicalForm xToStart= power (F.mvar(), start); |
---|
1723 | for (CFListIterator k= factors; k.hasItem(); k++, i++) |
---|
1724 | { |
---|
1725 | if (i == 0) |
---|
1726 | bufFactors[i]= mod (k.getItem(), xToStart); |
---|
1727 | else |
---|
1728 | bufFactors[i]= k.getItem(); |
---|
1729 | } |
---|
1730 | for (i= start; i < end; i++) |
---|
1731 | henselStep12 (F, factors, bufFactors, diophant, M, Pi, i); |
---|
1732 | |
---|
1733 | CFListIterator k= factors; |
---|
1734 | for (i= 0; i < factors.length(); k++, i++) |
---|
1735 | k.getItem()= bufFactors [i]; |
---|
1736 | factors.removeFirst(); |
---|
1737 | return; |
---|
1738 | } |
---|
1739 | |
---|
1740 | static inline |
---|
1741 | CFList |
---|
1742 | biDiophantine (const CanonicalForm& F, const CFList& factors, const int d) |
---|
1743 | { |
---|
1744 | Variable y= F.mvar(); |
---|
1745 | CFList result; |
---|
1746 | if (y.level() == 1) |
---|
1747 | { |
---|
1748 | result= diophantine (F, factors); |
---|
1749 | return result; |
---|
1750 | } |
---|
1751 | else |
---|
1752 | { |
---|
1753 | CFList buf= factors; |
---|
1754 | for (CFListIterator i= buf; i.hasItem(); i++) |
---|
1755 | i.getItem()= mod (i.getItem(), y); |
---|
1756 | CanonicalForm A= mod (F, y); |
---|
1757 | int bufD= 1; |
---|
1758 | CFList recResult= biDiophantine (A, buf, bufD); |
---|
1759 | CanonicalForm e= 1; |
---|
1760 | CFList p; |
---|
1761 | CFArray bufFactors= CFArray (factors.length()); |
---|
1762 | CanonicalForm yToD= power (y, d); |
---|
1763 | int k= 0; |
---|
1764 | for (CFListIterator i= factors; i.hasItem(); i++, k++) |
---|
1765 | { |
---|
1766 | bufFactors [k]= i.getItem(); |
---|
1767 | } |
---|
1768 | CanonicalForm b; |
---|
1769 | for (k= 0; k < factors.length(); k++) //TODO compute b's faster |
---|
1770 | { |
---|
1771 | b= 1; |
---|
1772 | if (fdivides (bufFactors[k], F)) |
---|
1773 | b= F/bufFactors[k]; |
---|
1774 | else |
---|
1775 | { |
---|
1776 | for (int l= 0; l < factors.length(); l++) |
---|
1777 | { |
---|
1778 | if (l == k) |
---|
1779 | continue; |
---|
1780 | else |
---|
1781 | { |
---|
1782 | b= mulMod2 (b, bufFactors[l], yToD); |
---|
1783 | } |
---|
1784 | } |
---|
1785 | } |
---|
1786 | p.append (b); |
---|
1787 | } |
---|
1788 | |
---|
1789 | CFListIterator j= p; |
---|
1790 | for (CFListIterator i= recResult; i.hasItem(); i++, j++) |
---|
1791 | e -= i.getItem()*j.getItem(); |
---|
1792 | |
---|
1793 | if (e.isZero()) |
---|
1794 | return recResult; |
---|
1795 | CanonicalForm coeffE; |
---|
1796 | CFList s; |
---|
1797 | result= recResult; |
---|
1798 | CanonicalForm g; |
---|
1799 | for (int i= 1; i < d; i++) |
---|
1800 | { |
---|
1801 | if (degree (e, y) > 0) |
---|
1802 | coeffE= e[i]; |
---|
1803 | else |
---|
1804 | coeffE= 0; |
---|
1805 | if (!coeffE.isZero()) |
---|
1806 | { |
---|
1807 | CFListIterator k= result; |
---|
1808 | CFListIterator l= p; |
---|
1809 | int ii= 0; |
---|
1810 | j= recResult; |
---|
1811 | for (; j.hasItem(); j++, k++, l++, ii++) |
---|
1812 | { |
---|
1813 | g= coeffE*j.getItem(); |
---|
1814 | if (degree (bufFactors[ii], y) <= 0) |
---|
1815 | g= mod (g, bufFactors[ii]); |
---|
1816 | else |
---|
1817 | g= mod (g, bufFactors[ii][0]); |
---|
1818 | k.getItem() += g*power (y, i); |
---|
1819 | e -= mulMod2 (g*power(y, i), l.getItem(), yToD); |
---|
1820 | DEBOUTLN (cerr, "mod (e, power (y, i + 1))= " << |
---|
1821 | mod (e, power (y, i + 1))); |
---|
1822 | } |
---|
1823 | } |
---|
1824 | if (e.isZero()) |
---|
1825 | break; |
---|
1826 | } |
---|
1827 | |
---|
1828 | DEBOUTLN (cerr, "mod (e, y)= " << mod (e, y)); |
---|
1829 | |
---|
1830 | #ifdef DEBUGOUTPUT |
---|
1831 | CanonicalForm test= 0; |
---|
1832 | j= p; |
---|
1833 | for (CFListIterator i= result; i.hasItem(); i++, j++) |
---|
1834 | test += mod (i.getItem()*j.getItem(), power (y, d)); |
---|
1835 | DEBOUTLN (cerr, "test= " << test); |
---|
1836 | #endif |
---|
1837 | return result; |
---|
1838 | } |
---|
1839 | } |
---|
1840 | |
---|
1841 | static inline |
---|
1842 | CFList |
---|
1843 | multiRecDiophantine (const CanonicalForm& F, const CFList& factors, |
---|
1844 | const CFList& recResult, const CFList& M, const int d) |
---|
1845 | { |
---|
1846 | Variable y= F.mvar(); |
---|
1847 | CFList result; |
---|
1848 | CFListIterator i; |
---|
1849 | CanonicalForm e= 1; |
---|
1850 | CFListIterator j= factors; |
---|
1851 | CFList p; |
---|
1852 | CFArray bufFactors= CFArray (factors.length()); |
---|
1853 | CanonicalForm yToD= power (y, d); |
---|
1854 | int k= 0; |
---|
1855 | for (CFListIterator i= factors; i.hasItem(); i++, k++) |
---|
1856 | bufFactors [k]= i.getItem(); |
---|
1857 | CanonicalForm b; |
---|
1858 | CFList buf= M; |
---|
1859 | buf.removeLast(); |
---|
1860 | buf.append (yToD); |
---|
1861 | for (k= 0; k < factors.length(); k++) //TODO compute b's faster |
---|
1862 | { |
---|
1863 | b= 1; |
---|
1864 | if (fdivides (bufFactors[k], F)) |
---|
1865 | b= F/bufFactors[k]; |
---|
1866 | else |
---|
1867 | { |
---|
1868 | for (int l= 0; l < factors.length(); l++) |
---|
1869 | { |
---|
1870 | if (l == k) |
---|
1871 | continue; |
---|
1872 | else |
---|
1873 | { |
---|
1874 | b= mulMod (b, bufFactors[l], buf); |
---|
1875 | } |
---|
1876 | } |
---|
1877 | } |
---|
1878 | p.append (b); |
---|
1879 | } |
---|
1880 | j= p; |
---|
1881 | for (CFListIterator i= recResult; i.hasItem(); i++, j++) |
---|
1882 | e -= mulMod (i.getItem(), j.getItem(), M); |
---|
1883 | |
---|
1884 | if (e.isZero()) |
---|
1885 | return recResult; |
---|
1886 | CanonicalForm coeffE; |
---|
1887 | CFList s; |
---|
1888 | result= recResult; |
---|
1889 | CanonicalForm g; |
---|
1890 | for (int i= 1; i < d; i++) |
---|
1891 | { |
---|
1892 | if (degree (e, y) > 0) |
---|
1893 | coeffE= e[i]; |
---|
1894 | else |
---|
1895 | coeffE= 0; |
---|
1896 | if (!coeffE.isZero()) |
---|
1897 | { |
---|
1898 | CFListIterator k= result; |
---|
1899 | CFListIterator l= p; |
---|
1900 | j= recResult; |
---|
1901 | int ii= 0; |
---|
1902 | CanonicalForm dummy; |
---|
1903 | for (; j.hasItem(); j++, k++, l++, ii++) |
---|
1904 | { |
---|
1905 | g= mulMod (coeffE, j.getItem(), M); |
---|
1906 | if (degree (bufFactors[ii], y) <= 0) |
---|
1907 | divrem (g, mod (bufFactors[ii], Variable (y.level() - 1)), dummy, |
---|
1908 | g, M); |
---|
1909 | else |
---|
1910 | divrem (g, bufFactors[ii][0], dummy, g, M); |
---|
1911 | k.getItem() += g*power (y, i); |
---|
1912 | e -= mulMod (g*power (y, i), l.getItem(), M); |
---|
1913 | } |
---|
1914 | } |
---|
1915 | |
---|
1916 | if (e.isZero()) |
---|
1917 | break; |
---|
1918 | } |
---|
1919 | |
---|
1920 | #ifdef DEBUGOUTPUT |
---|
1921 | CanonicalForm test= 0; |
---|
1922 | j= p; |
---|
1923 | for (CFListIterator i= result; i.hasItem(); i++, j++) |
---|
1924 | test += mod (i.getItem()*j.getItem(), power (y, d)); |
---|
1925 | DEBOUTLN (cerr, "test= " << test); |
---|
1926 | #endif |
---|
1927 | return result; |
---|
1928 | } |
---|
1929 | |
---|
1930 | static inline |
---|
1931 | void |
---|
1932 | henselStep (const CanonicalForm& F, const CFList& factors, CFArray& bufFactors, |
---|
1933 | const CFList& diophant, CFMatrix& M, CFArray& Pi, int j, |
---|
1934 | const CFList& MOD) |
---|
1935 | { |
---|
1936 | CanonicalForm E; |
---|
1937 | CanonicalForm xToJ= power (F.mvar(), j); |
---|
1938 | Variable x= F.mvar(); |
---|
1939 | // compute the error |
---|
1940 | if (j == 1) |
---|
1941 | { |
---|
1942 | E= F[j]; |
---|
1943 | #ifdef DEBUGOUTPUT |
---|
1944 | CanonicalForm test= 1; |
---|
1945 | for (int i= 0; i < factors.length(); i++) |
---|
1946 | { |
---|
1947 | if (i == 0) |
---|
1948 | test= mulMod (test, mod (bufFactors [i], xToJ), MOD); |
---|
1949 | else |
---|
1950 | test= mulMod (test, bufFactors[i], MOD); |
---|
1951 | } |
---|
1952 | CanonicalForm test2= mod (F-test, xToJ); |
---|
1953 | |
---|
1954 | test2= mod (test2, MOD); |
---|
1955 | DEBOUTLN (cerr, "test= " << test2); |
---|
1956 | #endif |
---|
1957 | } |
---|
1958 | else |
---|
1959 | { |
---|
1960 | #ifdef DEBUGOUTPUT |
---|
1961 | CanonicalForm test= 1; |
---|
1962 | for (int i= 0; i < factors.length(); i++) |
---|
1963 | { |
---|
1964 | if (i == 0) |
---|
1965 | test *= mod (bufFactors [i], power (x, j)); |
---|
1966 | else |
---|
1967 | test *= bufFactors[i]; |
---|
1968 | } |
---|
1969 | test= mod (test, power (x, j)); |
---|
1970 | test= mod (test, MOD); |
---|
1971 | CanonicalForm test2= mod (F, power (x, j - 1)) - mod (test, power (x, j-1)); |
---|
1972 | DEBOUTLN (cerr, "test= " << test2); |
---|
1973 | #endif |
---|
1974 | |
---|
1975 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
1976 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
1977 | else |
---|
1978 | E= F[j]; |
---|
1979 | } |
---|
1980 | |
---|
1981 | CFArray buf= CFArray (diophant.length()); |
---|
1982 | bufFactors[0]= mod (factors.getFirst(), power (F.mvar(), j + 1)); |
---|
1983 | int k= 0; |
---|
1984 | // actual lifting |
---|
1985 | CanonicalForm dummy, rest1; |
---|
1986 | for (CFListIterator i= diophant; i.hasItem(); i++, k++) |
---|
1987 | { |
---|
1988 | if (degree (bufFactors[k], x) > 0) |
---|
1989 | { |
---|
1990 | if (k > 0) |
---|
1991 | divrem (E, bufFactors[k] [0], dummy, rest1, MOD); |
---|
1992 | else |
---|
1993 | rest1= E; |
---|
1994 | } |
---|
1995 | else |
---|
1996 | divrem (E, bufFactors[k], dummy, rest1, MOD); |
---|
1997 | |
---|
1998 | buf[k]= mulMod (i.getItem(), rest1, MOD); |
---|
1999 | |
---|
2000 | if (degree (bufFactors[k], x) > 0) |
---|
2001 | divrem (buf[k], bufFactors[k] [0], dummy, buf[k], MOD); |
---|
2002 | else |
---|
2003 | divrem (buf[k], bufFactors[k], dummy, buf[k], MOD); |
---|
2004 | } |
---|
2005 | for (k= 1; k < factors.length(); k++) |
---|
2006 | bufFactors[k] += xToJ*buf[k]; |
---|
2007 | |
---|
2008 | // update Pi [0] |
---|
2009 | int degBuf0= degree (bufFactors[0], x); |
---|
2010 | int degBuf1= degree (bufFactors[1], x); |
---|
2011 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2012 | M (j + 1, 1)= mulMod (bufFactors[0] [j], bufFactors[1] [j], MOD); |
---|
2013 | CanonicalForm uIZeroJ; |
---|
2014 | if (j == 1) |
---|
2015 | { |
---|
2016 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2017 | uIZeroJ= mulMod ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
2018 | (bufFactors[1] [0] + buf[1]), MOD) - M(1, 1) - M(j + 1, 1); |
---|
2019 | else if (degBuf0 > 0) |
---|
2020 | uIZeroJ= mulMod (bufFactors[0] [j], bufFactors[1], MOD); |
---|
2021 | else if (degBuf1 > 0) |
---|
2022 | uIZeroJ= mulMod (bufFactors[0], buf[1], MOD); |
---|
2023 | else |
---|
2024 | uIZeroJ= 0; |
---|
2025 | Pi [0] += xToJ*uIZeroJ; |
---|
2026 | } |
---|
2027 | else |
---|
2028 | { |
---|
2029 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2030 | uIZeroJ= mulMod ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
2031 | (bufFactors[1] [0] + buf[1]), MOD) - M(1, 1) - M(j + 1, 1); |
---|
2032 | else if (degBuf0 > 0) |
---|
2033 | uIZeroJ= mulMod (bufFactors[0] [j], bufFactors[1], MOD); |
---|
2034 | else if (degBuf1 > 0) |
---|
2035 | uIZeroJ= mulMod (bufFactors[0], buf[1], MOD); |
---|
2036 | else |
---|
2037 | uIZeroJ= 0; |
---|
2038 | Pi [0] += xToJ*uIZeroJ; |
---|
2039 | } |
---|
2040 | |
---|
2041 | CFArray tmp= CFArray (factors.length() - 1); |
---|
2042 | for (k= 0; k < factors.length() - 1; k++) |
---|
2043 | tmp[k]= 0; |
---|
2044 | CFIterator one, two; |
---|
2045 | one= bufFactors [0]; |
---|
2046 | two= bufFactors [1]; |
---|
2047 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2048 | { |
---|
2049 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
2050 | { |
---|
2051 | if (k != j - k + 1) |
---|
2052 | { |
---|
2053 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
2054 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
2055 | { |
---|
2056 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
2057 | (bufFactors[1] [k] + two.coeff()), MOD) - M (k + 1, 1) - |
---|
2058 | M (j - k + 2, 1); |
---|
2059 | one++; |
---|
2060 | two++; |
---|
2061 | } |
---|
2062 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
2063 | { |
---|
2064 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
2065 | bufFactors[1] [k], MOD) - M (k + 1, 1); |
---|
2066 | one++; |
---|
2067 | } |
---|
2068 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
2069 | { |
---|
2070 | tmp[0] += mulMod (bufFactors[0] [k], (bufFactors[1] [k] + |
---|
2071 | two.coeff()), MOD) - M (k + 1, 1); |
---|
2072 | two++; |
---|
2073 | } |
---|
2074 | } |
---|
2075 | else |
---|
2076 | { |
---|
2077 | tmp[0] += M (k + 1, 1); |
---|
2078 | } |
---|
2079 | } |
---|
2080 | } |
---|
2081 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
2082 | |
---|
2083 | // update Pi [l] |
---|
2084 | int degPi, degBuf; |
---|
2085 | for (int l= 1; l < factors.length() - 1; l++) |
---|
2086 | { |
---|
2087 | degPi= degree (Pi [l - 1], x); |
---|
2088 | degBuf= degree (bufFactors[l + 1], x); |
---|
2089 | if (degPi > 0 && degBuf > 0) |
---|
2090 | M (j + 1, l + 1)= mulMod (Pi [l - 1] [j], bufFactors[l + 1] [j], MOD); |
---|
2091 | if (j == 1) |
---|
2092 | { |
---|
2093 | if (degPi > 0 && degBuf > 0) |
---|
2094 | Pi [l] += xToJ*(mulMod ((Pi [l - 1] [0] + Pi [l - 1] [j]), |
---|
2095 | (bufFactors[l + 1] [0] + buf[l + 1]), MOD) - M (j + 1, l +1)- |
---|
2096 | M (1, l + 1)); |
---|
2097 | else if (degPi > 0) |
---|
2098 | Pi [l] += xToJ*(mulMod (Pi [l - 1] [j], bufFactors[l + 1], MOD)); |
---|
2099 | else if (degBuf > 0) |
---|
2100 | Pi [l] += xToJ*(mulMod (Pi [l - 1], buf[l + 1], MOD)); |
---|
2101 | } |
---|
2102 | else |
---|
2103 | { |
---|
2104 | if (degPi > 0 && degBuf > 0) |
---|
2105 | { |
---|
2106 | uIZeroJ= mulMod (uIZeroJ, bufFactors [l + 1] [0], MOD); |
---|
2107 | uIZeroJ += mulMod (Pi [l - 1] [0], buf [l + 1], MOD); |
---|
2108 | } |
---|
2109 | else if (degPi > 0) |
---|
2110 | uIZeroJ= mulMod (uIZeroJ, bufFactors [l + 1], MOD); |
---|
2111 | else if (degBuf > 0) |
---|
2112 | { |
---|
2113 | uIZeroJ= mulMod (uIZeroJ, bufFactors [l + 1] [0], MOD); |
---|
2114 | uIZeroJ += mulMod (Pi [l - 1], buf[l + 1], MOD); |
---|
2115 | } |
---|
2116 | Pi[l] += xToJ*uIZeroJ; |
---|
2117 | } |
---|
2118 | one= bufFactors [l + 1]; |
---|
2119 | two= Pi [l - 1]; |
---|
2120 | if (two.hasTerms() && two.exp() == j + 1) |
---|
2121 | { |
---|
2122 | if (degBuf > 0 && degPi > 0) |
---|
2123 | { |
---|
2124 | tmp[l] += mulMod (two.coeff(), bufFactors[l + 1][0], MOD); |
---|
2125 | two++; |
---|
2126 | } |
---|
2127 | else if (degPi > 0) |
---|
2128 | { |
---|
2129 | tmp[l] += mulMod (two.coeff(), bufFactors[l + 1], MOD); |
---|
2130 | two++; |
---|
2131 | } |
---|
2132 | } |
---|
2133 | if (degBuf > 0 && degPi > 0) |
---|
2134 | { |
---|
2135 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
2136 | { |
---|
2137 | if (k != j - k + 1) |
---|
2138 | { |
---|
2139 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
2140 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
2141 | { |
---|
2142 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
2143 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1) - |
---|
2144 | M (j - k + 2, l + 1); |
---|
2145 | one++; |
---|
2146 | two++; |
---|
2147 | } |
---|
2148 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
2149 | { |
---|
2150 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
2151 | Pi[l - 1] [k], MOD) - M (k + 1, l + 1); |
---|
2152 | one++; |
---|
2153 | } |
---|
2154 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
2155 | { |
---|
2156 | tmp[l] += mulMod (bufFactors[l + 1] [k], |
---|
2157 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1); |
---|
2158 | two++; |
---|
2159 | } |
---|
2160 | } |
---|
2161 | else |
---|
2162 | tmp[l] += M (k + 1, l + 1); |
---|
2163 | } |
---|
2164 | } |
---|
2165 | Pi[l] += tmp[l]*xToJ*F.mvar(); |
---|
2166 | } |
---|
2167 | |
---|
2168 | return; |
---|
2169 | } |
---|
2170 | |
---|
2171 | CFList |
---|
2172 | henselLift23 (const CFList& eval, const CFList& factors, const int* l, CFList& |
---|
2173 | diophant, CFArray& Pi, CFMatrix& M) |
---|
2174 | { |
---|
2175 | CFList buf= factors; |
---|
2176 | int k= 0; |
---|
2177 | int liftBoundBivar= l[k]; |
---|
2178 | diophant= biDiophantine (eval.getFirst(), buf, liftBoundBivar); |
---|
2179 | CFList MOD; |
---|
2180 | MOD.append (power (Variable (2), liftBoundBivar)); |
---|
2181 | CFArray bufFactors= CFArray (factors.length()); |
---|
2182 | k= 0; |
---|
2183 | CFListIterator j= eval; |
---|
2184 | j++; |
---|
2185 | buf.removeFirst(); |
---|
2186 | buf.insert (LC (j.getItem(), 1)); |
---|
2187 | for (CFListIterator i= buf; i.hasItem(); i++, k++) |
---|
2188 | bufFactors[k]= i.getItem(); |
---|
2189 | Pi= CFArray (factors.length() - 1); |
---|
2190 | CFListIterator i= buf; |
---|
2191 | i++; |
---|
2192 | Variable y= j.getItem().mvar(); |
---|
2193 | Pi [0]= mulMod (i.getItem(), mod (buf.getFirst(), y), MOD); |
---|
2194 | M (1, 1)= Pi [0]; |
---|
2195 | k= 1; |
---|
2196 | if (i.hasItem()) |
---|
2197 | i++; |
---|
2198 | for (; i.hasItem(); i++, k++) |
---|
2199 | { |
---|
2200 | Pi [k]= mulMod (Pi [k - 1], i.getItem(), MOD); |
---|
2201 | M (1, k + 1)= Pi [k]; |
---|
2202 | } |
---|
2203 | |
---|
2204 | for (int d= 1; d < l[1]; d++) |
---|
2205 | henselStep (j.getItem(), buf, bufFactors, diophant, M, Pi, d, MOD); |
---|
2206 | CFList result; |
---|
2207 | for (k= 1; k < factors.length(); k++) |
---|
2208 | result.append (bufFactors[k]); |
---|
2209 | return result; |
---|
2210 | } |
---|
2211 | |
---|
2212 | void |
---|
2213 | henselLiftResume (const CanonicalForm& F, CFList& factors, int start, int end, |
---|
2214 | CFArray& Pi, const CFList& diophant, CFMatrix& M, |
---|
2215 | const CFList& MOD) |
---|
2216 | { |
---|
2217 | CFArray bufFactors= CFArray (factors.length()); |
---|
2218 | int i= 0; |
---|
2219 | CanonicalForm xToStart= power (F.mvar(), start); |
---|
2220 | for (CFListIterator k= factors; k.hasItem(); k++, i++) |
---|
2221 | { |
---|
2222 | if (i == 0) |
---|
2223 | bufFactors[i]= mod (k.getItem(), xToStart); |
---|
2224 | else |
---|
2225 | bufFactors[i]= k.getItem(); |
---|
2226 | } |
---|
2227 | for (i= start; i < end; i++) |
---|
2228 | henselStep (F, factors, bufFactors, diophant, M, Pi, i, MOD); |
---|
2229 | |
---|
2230 | CFListIterator k= factors; |
---|
2231 | for (i= 0; i < factors.length(); k++, i++) |
---|
2232 | k.getItem()= bufFactors [i]; |
---|
2233 | factors.removeFirst(); |
---|
2234 | return; |
---|
2235 | } |
---|
2236 | |
---|
2237 | CFList |
---|
2238 | henselLift (const CFList& F, const CFList& factors, const CFList& MOD, CFList& |
---|
2239 | diophant, CFArray& Pi, CFMatrix& M, const int lOld, const int |
---|
2240 | lNew) |
---|
2241 | { |
---|
2242 | diophant= multiRecDiophantine (F.getFirst(), factors, diophant, MOD, lOld); |
---|
2243 | int k= 0; |
---|
2244 | CFArray bufFactors= CFArray (factors.length()); |
---|
2245 | for (CFListIterator i= factors; i.hasItem(); i++, k++) |
---|
2246 | { |
---|
2247 | if (k == 0) |
---|
2248 | bufFactors[k]= LC (F.getLast(), 1); |
---|
2249 | else |
---|
2250 | bufFactors[k]= i.getItem(); |
---|
2251 | } |
---|
2252 | CFList buf= factors; |
---|
2253 | buf.removeFirst(); |
---|
2254 | buf.insert (LC (F.getLast(), 1)); |
---|
2255 | CFListIterator i= buf; |
---|
2256 | i++; |
---|
2257 | Variable y= F.getLast().mvar(); |
---|
2258 | Variable x= F.getFirst().mvar(); |
---|
2259 | CanonicalForm xToLOld= power (x, lOld); |
---|
2260 | Pi [0]= mod (Pi[0], xToLOld); |
---|
2261 | M (1, 1)= Pi [0]; |
---|
2262 | k= 1; |
---|
2263 | if (i.hasItem()) |
---|
2264 | i++; |
---|
2265 | for (; i.hasItem(); i++, k++) |
---|
2266 | { |
---|
2267 | Pi [k]= mod (Pi [k], xToLOld); |
---|
2268 | M (1, k + 1)= Pi [k]; |
---|
2269 | } |
---|
2270 | |
---|
2271 | for (int d= 1; d < lNew; d++) |
---|
2272 | henselStep (F.getLast(), buf, bufFactors, diophant, M, Pi, d, MOD); |
---|
2273 | CFList result; |
---|
2274 | for (k= 1; k < factors.length(); k++) |
---|
2275 | result.append (bufFactors[k]); |
---|
2276 | return result; |
---|
2277 | } |
---|
2278 | |
---|
2279 | CFList |
---|
2280 | henselLift (const CFList& eval, const CFList& factors, const int* l, const int |
---|
2281 | lLength, bool sort) |
---|
2282 | { |
---|
2283 | CFList diophant; |
---|
2284 | CFList buf= factors; |
---|
2285 | buf.insert (LC (eval.getFirst(), 1)); |
---|
2286 | if (sort) |
---|
2287 | sortList (buf, Variable (1)); |
---|
2288 | CFArray Pi; |
---|
2289 | CFMatrix M= CFMatrix (l[1], factors.length()); |
---|
2290 | CFList result= henselLift23 (eval, buf, l, diophant, Pi, M); |
---|
2291 | if (eval.length() == 2) |
---|
2292 | return result; |
---|
2293 | CFList MOD; |
---|
2294 | for (int i= 0; i < 2; i++) |
---|
2295 | MOD.append (power (Variable (i + 2), l[i])); |
---|
2296 | CFListIterator j= eval; |
---|
2297 | j++; |
---|
2298 | CFList bufEval; |
---|
2299 | bufEval.append (j.getItem()); |
---|
2300 | j++; |
---|
2301 | |
---|
2302 | for (int i= 2; i < lLength && j.hasItem(); i++, j++) |
---|
2303 | { |
---|
2304 | result.insert (LC (bufEval.getFirst(), 1)); |
---|
2305 | bufEval.append (j.getItem()); |
---|
2306 | M= CFMatrix (l[i], factors.length()); |
---|
2307 | result= henselLift (bufEval, result, MOD, diophant, Pi, M, l[i - 1], l[i]); |
---|
2308 | MOD.append (power (Variable (i + 2), l[i])); |
---|
2309 | bufEval.removeFirst(); |
---|
2310 | } |
---|
2311 | return result; |
---|
2312 | } |
---|
2313 | |
---|
2314 | void |
---|
2315 | henselStep122 (const CanonicalForm& F, const CFList& factors, |
---|
2316 | CFArray& bufFactors, const CFList& diophant, CFMatrix& M, |
---|
2317 | CFArray& Pi, int j, const CFArray& LCs) |
---|
2318 | { |
---|
2319 | Variable x= F.mvar(); |
---|
2320 | CanonicalForm xToJ= power (x, j); |
---|
2321 | |
---|
2322 | CanonicalForm E; |
---|
2323 | // compute the error |
---|
2324 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
2325 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
2326 | else |
---|
2327 | E= F[j]; |
---|
2328 | |
---|
2329 | CFArray buf= CFArray (diophant.length()); |
---|
2330 | |
---|
2331 | int k= 0; |
---|
2332 | CanonicalForm remainder; |
---|
2333 | // actual lifting |
---|
2334 | for (CFListIterator i= diophant; i.hasItem(); i++, k++) |
---|
2335 | { |
---|
2336 | if (degree (bufFactors[k], x) > 0) |
---|
2337 | remainder= modNTL (E, bufFactors[k] [0]); |
---|
2338 | else |
---|
2339 | remainder= modNTL (E, bufFactors[k]); |
---|
2340 | buf[k]= mulNTL (i.getItem(), remainder); |
---|
2341 | if (degree (bufFactors[k], x) > 0) |
---|
2342 | buf[k]= modNTL (buf[k], bufFactors[k] [0]); |
---|
2343 | else |
---|
2344 | buf[k]= modNTL (buf[k], bufFactors[k]); |
---|
2345 | } |
---|
2346 | |
---|
2347 | for (k= 0; k < factors.length(); k++) |
---|
2348 | bufFactors[k] += xToJ*buf[k]; |
---|
2349 | |
---|
2350 | // update Pi [0] |
---|
2351 | int degBuf0= degree (bufFactors[0], x); |
---|
2352 | int degBuf1= degree (bufFactors[1], x); |
---|
2353 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2354 | { |
---|
2355 | M (j + 1, 1)= mulNTL (bufFactors[0] [j], bufFactors[1] [j]); |
---|
2356 | if (j + 2 <= M.rows()) |
---|
2357 | M (j + 2, 1)= mulNTL (bufFactors[0] [j + 1], bufFactors[1] [j + 1]); |
---|
2358 | } |
---|
2359 | CanonicalForm uIZeroJ; |
---|
2360 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2361 | uIZeroJ= mulNTL(bufFactors[0][0],buf[1])+mulNTL (bufFactors[1][0], buf[0]); |
---|
2362 | else if (degBuf0 > 0) |
---|
2363 | uIZeroJ= mulNTL (buf[0], bufFactors[1]); |
---|
2364 | else if (degBuf1 > 0) |
---|
2365 | uIZeroJ= mulNTL (bufFactors[0], buf [1]); |
---|
2366 | else |
---|
2367 | uIZeroJ= 0; |
---|
2368 | Pi [0] += xToJ*uIZeroJ; |
---|
2369 | |
---|
2370 | CFArray tmp= CFArray (factors.length() - 1); |
---|
2371 | for (k= 0; k < factors.length() - 1; k++) |
---|
2372 | tmp[k]= 0; |
---|
2373 | CFIterator one, two; |
---|
2374 | one= bufFactors [0]; |
---|
2375 | two= bufFactors [1]; |
---|
2376 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2377 | { |
---|
2378 | while (one.hasTerms() && one.exp() > j) one++; |
---|
2379 | while (two.hasTerms() && two.exp() > j) two++; |
---|
2380 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
2381 | { |
---|
2382 | if (one.hasTerms() && two.hasTerms()) |
---|
2383 | { |
---|
2384 | if (k != j - k + 1) |
---|
2385 | { |
---|
2386 | if ((one.hasTerms() && one.exp() == j - k + 1) && + |
---|
2387 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
2388 | { |
---|
2389 | tmp[0] += mulNTL ((bufFactors[0][k]+one.coeff()),(bufFactors[1][k] + |
---|
2390 | two.coeff())) - M (k + 1, 1) - M (j - k + 2, 1); |
---|
2391 | one++; |
---|
2392 | two++; |
---|
2393 | } |
---|
2394 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
2395 | { |
---|
2396 | tmp[0] += mulNTL ((bufFactors[0][k]+one.coeff()), bufFactors[1] [k]) - |
---|
2397 | M (k + 1, 1); |
---|
2398 | one++; |
---|
2399 | } |
---|
2400 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
2401 | { |
---|
2402 | tmp[0] += mulNTL (bufFactors[0][k],(bufFactors[1][k] + two.coeff())) - |
---|
2403 | M (k + 1, 1); |
---|
2404 | two++; |
---|
2405 | } |
---|
2406 | } |
---|
2407 | else |
---|
2408 | tmp[0] += M (k + 1, 1); |
---|
2409 | } |
---|
2410 | } |
---|
2411 | } |
---|
2412 | |
---|
2413 | if (degBuf0 >= j + 1 && degBuf1 >= j + 1) |
---|
2414 | { |
---|
2415 | if (j + 2 <= M.rows()) |
---|
2416 | tmp [0] += mulNTL ((bufFactors [0] [j + 1]+ bufFactors [0] [0]), |
---|
2417 | (bufFactors [1] [j + 1] + bufFactors [1] [0])) |
---|
2418 | - M(1,1) - M (j + 2,1); |
---|
2419 | } |
---|
2420 | else if (degBuf0 >= j + 1) |
---|
2421 | { |
---|
2422 | if (degBuf1 > 0) |
---|
2423 | tmp[0] += mulNTL (bufFactors [0] [j+1], bufFactors [1] [0]); |
---|
2424 | else |
---|
2425 | tmp[0] += mulNTL (bufFactors [0] [j+1], bufFactors [1]); |
---|
2426 | } |
---|
2427 | else if (degBuf1 >= j + 1) |
---|
2428 | { |
---|
2429 | if (degBuf0 > 0) |
---|
2430 | tmp[0] += mulNTL (bufFactors [0] [0], bufFactors [1] [j + 1]); |
---|
2431 | else |
---|
2432 | tmp[0] += mulNTL (bufFactors [0], bufFactors [1] [j + 1]); |
---|
2433 | } |
---|
2434 | |
---|
2435 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
2436 | return; |
---|
2437 | } |
---|
2438 | |
---|
2439 | void |
---|
2440 | henselLift122 (const CanonicalForm& F, CFList& factors, int l, CFArray& Pi, |
---|
2441 | CFList& diophant, CFMatrix& M, const CFArray& LCs, bool sort) |
---|
2442 | { |
---|
2443 | if (sort) |
---|
2444 | sortList (factors, Variable (1)); |
---|
2445 | Pi= CFArray (factors.length() - 2); |
---|
2446 | CFList bufFactors2= factors; |
---|
2447 | bufFactors2.removeFirst(); |
---|
2448 | diophant.removeFirst(); |
---|
2449 | CFListIterator iter= diophant; |
---|
2450 | CanonicalForm s,t; |
---|
2451 | extgcd (bufFactors2.getFirst(), bufFactors2.getLast(), s, t); |
---|
2452 | diophant= CFList(); |
---|
2453 | diophant.append (t); |
---|
2454 | diophant.append (s); |
---|
2455 | DEBOUTLN (cerr, "diophant= " << diophant); |
---|
2456 | |
---|
2457 | CFArray bufFactors= CFArray (bufFactors2.length()); |
---|
2458 | int i= 0; |
---|
2459 | for (CFListIterator k= bufFactors2; k.hasItem(); i++, k++) |
---|
2460 | bufFactors[i]= replaceLc (k.getItem(), LCs [i]); |
---|
2461 | |
---|
2462 | Variable x= F.mvar(); |
---|
2463 | if (degree (bufFactors[0], x) > 0 && degree (bufFactors [1], x) > 0) |
---|
2464 | { |
---|
2465 | M (1, 1)= mulNTL (bufFactors [0] [0], bufFactors[1] [0]); |
---|
2466 | Pi [0]= M (1, 1) + (mulNTL (bufFactors [0] [1], bufFactors[1] [0]) + |
---|
2467 | mulNTL (bufFactors [0] [0], bufFactors [1] [1]))*x; |
---|
2468 | } |
---|
2469 | else if (degree (bufFactors[0], x) > 0) |
---|
2470 | { |
---|
2471 | M (1, 1)= mulNTL (bufFactors [0] [0], bufFactors[1]); |
---|
2472 | Pi [0]= M (1, 1) + |
---|
2473 | mulNTL (bufFactors [0] [1], bufFactors[1])*x; |
---|
2474 | } |
---|
2475 | else if (degree (bufFactors[1], x) > 0) |
---|
2476 | { |
---|
2477 | M (1, 1)= mulNTL (bufFactors [0], bufFactors[1] [0]); |
---|
2478 | Pi [0]= M (1, 1) + |
---|
2479 | mulNTL (bufFactors [0], bufFactors[1] [1])*x; |
---|
2480 | } |
---|
2481 | else |
---|
2482 | { |
---|
2483 | M (1, 1)= mulNTL (bufFactors [0], bufFactors[1]); |
---|
2484 | Pi [0]= M (1, 1); |
---|
2485 | } |
---|
2486 | |
---|
2487 | for (i= 1; i < l; i++) |
---|
2488 | henselStep122 (F, bufFactors2, bufFactors, diophant, M, Pi, i, LCs); |
---|
2489 | |
---|
2490 | factors= CFList(); |
---|
2491 | for (i= 0; i < bufFactors.size(); i++) |
---|
2492 | factors.append (bufFactors[i]); |
---|
2493 | return; |
---|
2494 | } |
---|
2495 | |
---|
2496 | /// solve \f$ E=sum_{i= 1}^{r}{\sigma_{i}prod_{j=1, j\neq i}^{r}{f_{i}}}\f$ |
---|
2497 | /// mod M, products contains \f$ prod_{j=1, j\neq i}^{r}{f_{i}}} \f$ |
---|
2498 | static inline |
---|
2499 | CFList |
---|
2500 | diophantine (const CFList& recResult, const CFList& factors, |
---|
2501 | const CFList& products, const CFList& M, const CanonicalForm& E, |
---|
2502 | bool& bad) |
---|
2503 | { |
---|
2504 | if (M.isEmpty()) |
---|
2505 | { |
---|
2506 | CFList result; |
---|
2507 | CFListIterator j= factors; |
---|
2508 | CanonicalForm buf; |
---|
2509 | for (CFListIterator i= recResult; i.hasItem(); i++, j++) |
---|
2510 | { |
---|
2511 | ASSERT (E.isUnivariate() || E.inCoeffDomain(), |
---|
2512 | "constant or univariate poly expected"); |
---|
2513 | ASSERT (i.getItem().isUnivariate() || i.getItem().inCoeffDomain(), |
---|
2514 | "constant or univariate poly expected"); |
---|
2515 | ASSERT (j.getItem().isUnivariate() || j.getItem().inCoeffDomain(), |
---|
2516 | "constant or univariate poly expected"); |
---|
2517 | buf= mulNTL (E, i.getItem()); |
---|
2518 | result.append (modNTL (buf, j.getItem())); |
---|
2519 | } |
---|
2520 | return result; |
---|
2521 | } |
---|
2522 | Variable y= M.getLast().mvar(); |
---|
2523 | CFList bufFactors= factors; |
---|
2524 | for (CFListIterator i= bufFactors; i.hasItem(); i++) |
---|
2525 | i.getItem()= mod (i.getItem(), y); |
---|
2526 | CFList bufProducts= products; |
---|
2527 | for (CFListIterator i= bufProducts; i.hasItem(); i++) |
---|
2528 | i.getItem()= mod (i.getItem(), y); |
---|
2529 | CFList buf= M; |
---|
2530 | buf.removeLast(); |
---|
2531 | CanonicalForm bufE= mod (E, y); |
---|
2532 | CFList recDiophantine= diophantine (recResult, bufFactors, bufProducts, buf, |
---|
2533 | bufE, bad); |
---|
2534 | |
---|
2535 | if (bad) |
---|
2536 | return CFList(); |
---|
2537 | |
---|
2538 | CanonicalForm e= E; |
---|
2539 | CFListIterator j= products; |
---|
2540 | for (CFListIterator i= recDiophantine; i.hasItem(); i++, j++) |
---|
2541 | e -= i.getItem()*j.getItem(); |
---|
2542 | |
---|
2543 | CFList result= recDiophantine; |
---|
2544 | int d= degree (M.getLast()); |
---|
2545 | CanonicalForm coeffE; |
---|
2546 | for (int i= 1; i < d; i++) |
---|
2547 | { |
---|
2548 | if (degree (e, y) > 0) |
---|
2549 | coeffE= e[i]; |
---|
2550 | else |
---|
2551 | coeffE= 0; |
---|
2552 | if (!coeffE.isZero()) |
---|
2553 | { |
---|
2554 | CFListIterator k= result; |
---|
2555 | recDiophantine= diophantine (recResult, bufFactors, bufProducts, buf, |
---|
2556 | coeffE, bad); |
---|
2557 | if (bad) |
---|
2558 | return CFList(); |
---|
2559 | CFListIterator l= products; |
---|
2560 | for (j= recDiophantine; j.hasItem(); j++, k++, l++) |
---|
2561 | { |
---|
2562 | k.getItem() += j.getItem()*power (y, i); |
---|
2563 | e -= j.getItem()*power (y, i)*l.getItem(); |
---|
2564 | } |
---|
2565 | } |
---|
2566 | if (e.isZero()) |
---|
2567 | break; |
---|
2568 | } |
---|
2569 | if (!e.isZero()) |
---|
2570 | { |
---|
2571 | bad= true; |
---|
2572 | return CFList(); |
---|
2573 | } |
---|
2574 | return result; |
---|
2575 | } |
---|
2576 | |
---|
2577 | static inline |
---|
2578 | void |
---|
2579 | henselStep2 (const CanonicalForm& F, const CFList& factors, CFArray& bufFactors, |
---|
2580 | const CFList& diophant, CFMatrix& M, CFArray& Pi, |
---|
2581 | const CFList& products, int j, const CFList& MOD, bool& noOneToOne) |
---|
2582 | { |
---|
2583 | CanonicalForm E; |
---|
2584 | CanonicalForm xToJ= power (F.mvar(), j); |
---|
2585 | Variable x= F.mvar(); |
---|
2586 | |
---|
2587 | // compute the error |
---|
2588 | #ifdef DEBUGOUTPUT |
---|
2589 | CanonicalForm test= 1; |
---|
2590 | for (int i= 0; i < factors.length(); i++) |
---|
2591 | { |
---|
2592 | if (i == 0) |
---|
2593 | test *= mod (bufFactors [i], power (x, j)); |
---|
2594 | else |
---|
2595 | test *= bufFactors[i]; |
---|
2596 | } |
---|
2597 | test= mod (test, power (x, j)); |
---|
2598 | test= mod (test, MOD); |
---|
2599 | CanonicalForm test2= mod (F, power (x, j - 1)) - mod (test, power (x, j-1)); |
---|
2600 | DEBOUTLN (cerr, "test= " << test2); |
---|
2601 | #endif |
---|
2602 | |
---|
2603 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
2604 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
2605 | else |
---|
2606 | E= F[j]; |
---|
2607 | |
---|
2608 | CFArray buf= CFArray (diophant.length()); |
---|
2609 | |
---|
2610 | // actual lifting |
---|
2611 | CFList diophantine2= diophantine (diophant, factors, products, MOD, E, |
---|
2612 | noOneToOne); |
---|
2613 | |
---|
2614 | if (noOneToOne) |
---|
2615 | return; |
---|
2616 | |
---|
2617 | int k= 0; |
---|
2618 | for (CFListIterator i= diophantine2; k < factors.length(); k++, i++) |
---|
2619 | { |
---|
2620 | buf[k]= i.getItem(); |
---|
2621 | bufFactors[k] += xToJ*i.getItem(); |
---|
2622 | } |
---|
2623 | |
---|
2624 | // update Pi [0] |
---|
2625 | int degBuf0= degree (bufFactors[0], x); |
---|
2626 | int degBuf1= degree (bufFactors[1], x); |
---|
2627 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2628 | { |
---|
2629 | M (j + 1, 1)= mulMod (bufFactors[0] [j], bufFactors[1] [j], MOD); |
---|
2630 | if (j + 2 <= M.rows()) |
---|
2631 | M (j + 2, 1)= mulMod (bufFactors[0] [j + 1], bufFactors[1] [j + 1], MOD); |
---|
2632 | } |
---|
2633 | CanonicalForm uIZeroJ; |
---|
2634 | |
---|
2635 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2636 | uIZeroJ= mulMod (bufFactors[0] [0], buf[1], MOD) + |
---|
2637 | mulMod (bufFactors[1] [0], buf[0], MOD); |
---|
2638 | else if (degBuf0 > 0) |
---|
2639 | uIZeroJ= mulMod (buf[0], bufFactors[1], MOD); |
---|
2640 | else if (degBuf1 > 0) |
---|
2641 | uIZeroJ= mulMod (bufFactors[0], buf[1], MOD); |
---|
2642 | else |
---|
2643 | uIZeroJ= 0; |
---|
2644 | Pi [0] += xToJ*uIZeroJ; |
---|
2645 | |
---|
2646 | CFArray tmp= CFArray (factors.length() - 1); |
---|
2647 | for (k= 0; k < factors.length() - 1; k++) |
---|
2648 | tmp[k]= 0; |
---|
2649 | CFIterator one, two; |
---|
2650 | one= bufFactors [0]; |
---|
2651 | two= bufFactors [1]; |
---|
2652 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2653 | { |
---|
2654 | while (one.hasTerms() && one.exp() > j) one++; |
---|
2655 | while (two.hasTerms() && two.exp() > j) two++; |
---|
2656 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
2657 | { |
---|
2658 | if (k != j - k + 1) |
---|
2659 | { |
---|
2660 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
2661 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
2662 | { |
---|
2663 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
2664 | (bufFactors[1] [k] + two.coeff()), MOD) - M (k + 1, 1) - |
---|
2665 | M (j - k + 2, 1); |
---|
2666 | one++; |
---|
2667 | two++; |
---|
2668 | } |
---|
2669 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
2670 | { |
---|
2671 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
2672 | bufFactors[1] [k], MOD) - M (k + 1, 1); |
---|
2673 | one++; |
---|
2674 | } |
---|
2675 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
2676 | { |
---|
2677 | tmp[0] += mulMod (bufFactors[0] [k], (bufFactors[1] [k] + |
---|
2678 | two.coeff()), MOD) - M (k + 1, 1); |
---|
2679 | two++; |
---|
2680 | } |
---|
2681 | } |
---|
2682 | else |
---|
2683 | { |
---|
2684 | tmp[0] += M (k + 1, 1); |
---|
2685 | } |
---|
2686 | } |
---|
2687 | } |
---|
2688 | |
---|
2689 | if (degBuf0 >= j + 1 && degBuf1 >= j + 1) |
---|
2690 | { |
---|
2691 | if (j + 2 <= M.rows()) |
---|
2692 | tmp [0] += mulMod ((bufFactors [0] [j + 1]+ bufFactors [0] [0]), |
---|
2693 | (bufFactors [1] [j + 1] + bufFactors [1] [0]), MOD) |
---|
2694 | - M(1,1) - M (j + 2,1); |
---|
2695 | } |
---|
2696 | else if (degBuf0 >= j + 1) |
---|
2697 | { |
---|
2698 | if (degBuf1 > 0) |
---|
2699 | tmp[0] += mulMod (bufFactors [0] [j+1], bufFactors [1] [0], MOD); |
---|
2700 | else |
---|
2701 | tmp[0] += mulMod (bufFactors [0] [j+1], bufFactors [1], MOD); |
---|
2702 | } |
---|
2703 | else if (degBuf1 >= j + 1) |
---|
2704 | { |
---|
2705 | if (degBuf0 > 0) |
---|
2706 | tmp[0] += mulMod (bufFactors [0] [0], bufFactors [1] [j + 1], MOD); |
---|
2707 | else |
---|
2708 | tmp[0] += mulMod (bufFactors [0], bufFactors [1] [j + 1], MOD); |
---|
2709 | } |
---|
2710 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
2711 | |
---|
2712 | // update Pi [l] |
---|
2713 | int degPi, degBuf; |
---|
2714 | for (int l= 1; l < factors.length() - 1; l++) |
---|
2715 | { |
---|
2716 | degPi= degree (Pi [l - 1], x); |
---|
2717 | degBuf= degree (bufFactors[l + 1], x); |
---|
2718 | if (degPi > 0 && degBuf > 0) |
---|
2719 | { |
---|
2720 | M (j + 1, l + 1)= mulMod (Pi [l - 1] [j], bufFactors[l + 1] [j], MOD); |
---|
2721 | if (j + 2 <= M.rows()) |
---|
2722 | M (j + 2, l + 1)= mulMod (Pi [l - 1] [j + 1], bufFactors[l + 1] [j + 1], |
---|
2723 | MOD); |
---|
2724 | } |
---|
2725 | |
---|
2726 | if (degPi > 0 && degBuf > 0) |
---|
2727 | uIZeroJ= mulMod (Pi[l -1] [0], buf[l + 1], MOD) + |
---|
2728 | mulMod (uIZeroJ, bufFactors[l+1] [0], MOD); |
---|
2729 | else if (degPi > 0) |
---|
2730 | uIZeroJ= mulMod (uIZeroJ, bufFactors[l + 1], MOD); |
---|
2731 | else if (degBuf > 0) |
---|
2732 | uIZeroJ= mulMod (Pi[l - 1], buf[1], MOD); |
---|
2733 | else |
---|
2734 | uIZeroJ= 0; |
---|
2735 | |
---|
2736 | Pi [l] += xToJ*uIZeroJ; |
---|
2737 | |
---|
2738 | one= bufFactors [l + 1]; |
---|
2739 | two= Pi [l - 1]; |
---|
2740 | if (degBuf > 0 && degPi > 0) |
---|
2741 | { |
---|
2742 | while (one.hasTerms() && one.exp() > j) one++; |
---|
2743 | while (two.hasTerms() && two.exp() > j) two++; |
---|
2744 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
2745 | { |
---|
2746 | if (k != j - k + 1) |
---|
2747 | { |
---|
2748 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
2749 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
2750 | { |
---|
2751 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
2752 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1) - |
---|
2753 | M (j - k + 2, l + 1); |
---|
2754 | one++; |
---|
2755 | two++; |
---|
2756 | } |
---|
2757 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
2758 | { |
---|
2759 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
2760 | Pi[l - 1] [k], MOD) - M (k + 1, l + 1); |
---|
2761 | one++; |
---|
2762 | } |
---|
2763 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
2764 | { |
---|
2765 | tmp[l] += mulMod (bufFactors[l + 1] [k], |
---|
2766 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1); |
---|
2767 | two++; |
---|
2768 | } |
---|
2769 | } |
---|
2770 | else |
---|
2771 | tmp[l] += M (k + 1, l + 1); |
---|
2772 | } |
---|
2773 | } |
---|
2774 | |
---|
2775 | if (degPi >= j + 1 && degBuf >= j + 1) |
---|
2776 | { |
---|
2777 | if (j + 2 <= M.rows()) |
---|
2778 | tmp [l] += mulMod ((Pi [l - 1] [j + 1]+ Pi [l - 1] [0]), |
---|
2779 | (bufFactors [l + 1] [j + 1] + bufFactors [l + 1] [0]) |
---|
2780 | , MOD) - M(1,l+1) - M (j + 2,l+1); |
---|
2781 | } |
---|
2782 | else if (degPi >= j + 1) |
---|
2783 | { |
---|
2784 | if (degBuf > 0) |
---|
2785 | tmp[l] += mulMod (Pi [l - 1] [j+1], bufFactors [l + 1] [0], MOD); |
---|
2786 | else |
---|
2787 | tmp[l] += mulMod (Pi [l - 1] [j+1], bufFactors [l + 1], MOD); |
---|
2788 | } |
---|
2789 | else if (degBuf >= j + 1) |
---|
2790 | { |
---|
2791 | if (degPi > 0) |
---|
2792 | tmp[l] += mulMod (Pi [l - 1] [0], bufFactors [l + 1] [j + 1], MOD); |
---|
2793 | else |
---|
2794 | tmp[l] += mulMod (Pi [l - 1], bufFactors [l + 1] [j + 1], MOD); |
---|
2795 | } |
---|
2796 | |
---|
2797 | Pi[l] += tmp[l]*xToJ*F.mvar(); |
---|
2798 | } |
---|
2799 | return; |
---|
2800 | } |
---|
2801 | |
---|
2802 | // wrt. Variable (1) |
---|
2803 | CanonicalForm replaceLC (const CanonicalForm& F, const CanonicalForm& c) |
---|
2804 | { |
---|
2805 | if (degree (F, 1) <= 0) |
---|
2806 | return c; |
---|
2807 | else |
---|
2808 | { |
---|
2809 | CanonicalForm result= swapvar (F, Variable (F.level() + 1), Variable (1)); |
---|
2810 | result += (swapvar (c, Variable (F.level() + 1), Variable (1)) |
---|
2811 | - LC (result))*power (result.mvar(), degree (result)); |
---|
2812 | return swapvar (result, Variable (F.level() + 1), Variable (1)); |
---|
2813 | } |
---|
2814 | } |
---|
2815 | |
---|
2816 | CFList |
---|
2817 | henselLift232 (const CFList& eval, const CFList& factors, int* l, CFList& |
---|
2818 | diophant, CFArray& Pi, CFMatrix& M, const CFList& LCs1, |
---|
2819 | const CFList& LCs2, bool& bad) |
---|
2820 | { |
---|
2821 | CFList buf= factors; |
---|
2822 | int k= 0; |
---|
2823 | int liftBoundBivar= l[k]; |
---|
2824 | CFList bufbuf= factors; |
---|
2825 | Variable v= Variable (2); |
---|
2826 | |
---|
2827 | CFList MOD; |
---|
2828 | MOD.append (power (Variable (2), liftBoundBivar)); |
---|
2829 | CFArray bufFactors= CFArray (factors.length()); |
---|
2830 | k= 0; |
---|
2831 | CFListIterator j= eval; |
---|
2832 | j++; |
---|
2833 | CFListIterator iter1= LCs1; |
---|
2834 | CFListIterator iter2= LCs2; |
---|
2835 | iter1++; |
---|
2836 | iter2++; |
---|
2837 | bufFactors[0]= replaceLC (buf.getFirst(), iter1.getItem()); |
---|
2838 | bufFactors[1]= replaceLC (buf.getLast(), iter2.getItem()); |
---|
2839 | |
---|
2840 | CFListIterator i= buf; |
---|
2841 | i++; |
---|
2842 | Variable y= j.getItem().mvar(); |
---|
2843 | if (y.level() != 3) |
---|
2844 | y= Variable (3); |
---|
2845 | |
---|
2846 | Pi[0]= mod (Pi[0], power (v, liftBoundBivar)); |
---|
2847 | M (1, 1)= Pi[0]; |
---|
2848 | if (degree (bufFactors[0], y) > 0 && degree (bufFactors [1], y) > 0) |
---|
2849 | Pi [0] += (mulMod (bufFactors [0] [1], bufFactors[1] [0], MOD) + |
---|
2850 | mulMod (bufFactors [0] [0], bufFactors [1] [1], MOD))*y; |
---|
2851 | else if (degree (bufFactors[0], y) > 0) |
---|
2852 | Pi [0] += mulMod (bufFactors [0] [1], bufFactors[1], MOD)*y; |
---|
2853 | else if (degree (bufFactors[1], y) > 0) |
---|
2854 | Pi [0] += mulMod (bufFactors [0], bufFactors[1] [1], MOD)*y; |
---|
2855 | |
---|
2856 | CFList products; |
---|
2857 | for (int i= 0; i < bufFactors.size(); i++) |
---|
2858 | { |
---|
2859 | if (degree (bufFactors[i], y) > 0) |
---|
2860 | products.append (eval.getFirst()/bufFactors[i] [0]); |
---|
2861 | else |
---|
2862 | products.append (eval.getFirst()/bufFactors[i]); |
---|
2863 | } |
---|
2864 | |
---|
2865 | for (int d= 1; d < l[1]; d++) |
---|
2866 | { |
---|
2867 | henselStep2 (j.getItem(), buf, bufFactors, diophant, M, Pi, products, d, MOD, bad); |
---|
2868 | if (bad) |
---|
2869 | return CFList(); |
---|
2870 | } |
---|
2871 | CFList result; |
---|
2872 | for (k= 0; k < factors.length(); k++) |
---|
2873 | result.append (bufFactors[k]); |
---|
2874 | return result; |
---|
2875 | } |
---|
2876 | |
---|
2877 | |
---|
2878 | CFList |
---|
2879 | henselLift2 (const CFList& F, const CFList& factors, const CFList& MOD, CFList& |
---|
2880 | diophant, CFArray& Pi, CFMatrix& M, const int lOld, int& |
---|
2881 | lNew, const CFList& LCs1, const CFList& LCs2, bool& bad) |
---|
2882 | { |
---|
2883 | int k= 0; |
---|
2884 | CFArray bufFactors= CFArray (factors.length()); |
---|
2885 | bufFactors[0]= replaceLC (factors.getFirst(), LCs1.getLast()); |
---|
2886 | bufFactors[1]= replaceLC (factors.getLast(), LCs2.getLast()); |
---|
2887 | CFList buf= factors; |
---|
2888 | Variable y= F.getLast().mvar(); |
---|
2889 | Variable x= F.getFirst().mvar(); |
---|
2890 | CanonicalForm xToLOld= power (x, lOld); |
---|
2891 | Pi [0]= mod (Pi[0], xToLOld); |
---|
2892 | M (1, 1)= Pi [0]; |
---|
2893 | |
---|
2894 | if (degree (bufFactors[0], y) > 0 && degree (bufFactors [1], y) > 0) |
---|
2895 | Pi [0] += (mulMod (bufFactors [0] [1], bufFactors[1] [0], MOD) + |
---|
2896 | mulMod (bufFactors [0] [0], bufFactors [1] [1], MOD))*y; |
---|
2897 | else if (degree (bufFactors[0], y) > 0) |
---|
2898 | Pi [0] += mulMod (bufFactors [0] [1], bufFactors[1], MOD)*y; |
---|
2899 | else if (degree (bufFactors[1], y) > 0) |
---|
2900 | Pi [0] += mulMod (bufFactors [0], bufFactors[1] [1], MOD)*y; |
---|
2901 | |
---|
2902 | CFList products; |
---|
2903 | for (int i= 0; i < bufFactors.size(); i++) |
---|
2904 | { |
---|
2905 | if (degree (bufFactors[i], y) > 0) |
---|
2906 | { |
---|
2907 | if (!fdivides (bufFactors[i] [0], F.getFirst())) |
---|
2908 | { |
---|
2909 | bad= true; |
---|
2910 | return CFList(); |
---|
2911 | } |
---|
2912 | products.append (F.getFirst()/bufFactors[i] [0]); |
---|
2913 | } |
---|
2914 | else |
---|
2915 | { |
---|
2916 | if (!fdivides (bufFactors[i], F.getFirst())) |
---|
2917 | { |
---|
2918 | bad= true; |
---|
2919 | return CFList(); |
---|
2920 | } |
---|
2921 | products.append (F.getFirst()/bufFactors[i]); |
---|
2922 | } |
---|
2923 | } |
---|
2924 | |
---|
2925 | for (int d= 1; d < lNew; d++) |
---|
2926 | { |
---|
2927 | henselStep2 (F.getLast(), buf, bufFactors, diophant, M, Pi, products, d, MOD, bad); |
---|
2928 | if (bad) |
---|
2929 | return CFList(); |
---|
2930 | } |
---|
2931 | |
---|
2932 | CFList result; |
---|
2933 | for (k= 0; k < factors.length(); k++) |
---|
2934 | result.append (bufFactors[k]); |
---|
2935 | return result; |
---|
2936 | } |
---|
2937 | |
---|
2938 | CFList |
---|
2939 | henselLift2 (const CFList& eval, const CFList& factors, int* l, const int |
---|
2940 | lLength, bool sort, const CFList& LCs1, const CFList& LCs2, |
---|
2941 | const CFArray& Pi, const CFList& diophant, bool& bad) |
---|
2942 | { |
---|
2943 | CFList bufDiophant= diophant; |
---|
2944 | CFList buf= factors; |
---|
2945 | if (sort) |
---|
2946 | sortList (buf, Variable (1)); |
---|
2947 | CFArray bufPi= Pi; |
---|
2948 | CFMatrix M= CFMatrix (l[1], factors.length()); |
---|
2949 | CFList result= henselLift232(eval, buf, l, bufDiophant, bufPi, M, LCs1, LCs2, |
---|
2950 | bad); |
---|
2951 | if (bad) |
---|
2952 | return CFList(); |
---|
2953 | |
---|
2954 | if (eval.length() == 2) |
---|
2955 | return result; |
---|
2956 | CFList MOD; |
---|
2957 | for (int i= 0; i < 2; i++) |
---|
2958 | MOD.append (power (Variable (i + 2), l[i])); |
---|
2959 | CFListIterator j= eval; |
---|
2960 | j++; |
---|
2961 | CFList bufEval; |
---|
2962 | bufEval.append (j.getItem()); |
---|
2963 | j++; |
---|
2964 | CFListIterator jj= LCs1; |
---|
2965 | CFListIterator jjj= LCs2; |
---|
2966 | CFList bufLCs1, bufLCs2; |
---|
2967 | jj++, jjj++; |
---|
2968 | bufLCs1.append (jj.getItem()); |
---|
2969 | bufLCs2.append (jjj.getItem()); |
---|
2970 | jj++, jjj++; |
---|
2971 | |
---|
2972 | for (int i= 2; i < lLength && j.hasItem(); i++, j++, jj++, jjj++) |
---|
2973 | { |
---|
2974 | bufEval.append (j.getItem()); |
---|
2975 | bufLCs1.append (jj.getItem()); |
---|
2976 | bufLCs2.append (jjj.getItem()); |
---|
2977 | M= CFMatrix (l[i], factors.length()); |
---|
2978 | result= henselLift2 (bufEval, result, MOD, bufDiophant, bufPi, M, l[i - 1], |
---|
2979 | l[i], bufLCs1, bufLCs2, bad); |
---|
2980 | if (bad) |
---|
2981 | return CFList(); |
---|
2982 | MOD.append (power (Variable (i + 2), l[i])); |
---|
2983 | bufEval.removeFirst(); |
---|
2984 | bufLCs1.removeFirst(); |
---|
2985 | bufLCs2.removeFirst(); |
---|
2986 | } |
---|
2987 | return result; |
---|
2988 | } |
---|
2989 | |
---|
2990 | CFList |
---|
2991 | nonMonicHenselLift23 (const CanonicalForm& F, const CFList& factors, const |
---|
2992 | CFList& LCs, CFList& diophant, CFArray& Pi, int liftBound, |
---|
2993 | int bivarLiftBound, bool& bad) |
---|
2994 | { |
---|
2995 | CFList bufFactors2= factors; |
---|
2996 | |
---|
2997 | Variable y= Variable (2); |
---|
2998 | for (CFListIterator i= bufFactors2; i.hasItem(); i++) |
---|
2999 | i.getItem()= mod (i.getItem(), y); |
---|
3000 | |
---|
3001 | CanonicalForm bufF= F; |
---|
3002 | bufF= mod (bufF, y); |
---|
3003 | bufF= mod (bufF, Variable (3)); |
---|
3004 | |
---|
3005 | diophant= diophantine (bufF, bufFactors2); |
---|
3006 | |
---|
3007 | CFMatrix M= CFMatrix (liftBound, bufFactors2.length() - 1); |
---|
3008 | |
---|
3009 | Pi= CFArray (bufFactors2.length() - 1); |
---|
3010 | |
---|
3011 | CFArray bufFactors= CFArray (bufFactors2.length()); |
---|
3012 | CFListIterator j= LCs; |
---|
3013 | int i= 0; |
---|
3014 | for (CFListIterator k= factors; k.hasItem(); j++, k++, i++) |
---|
3015 | bufFactors[i]= replaceLC (k.getItem(), j.getItem()); |
---|
3016 | |
---|
3017 | //initialise Pi |
---|
3018 | Variable v= Variable (3); |
---|
3019 | CanonicalForm yToL= power (y, bivarLiftBound); |
---|
3020 | if (degree (bufFactors[0], v) > 0 && degree (bufFactors [1], v) > 0) |
---|
3021 | { |
---|
3022 | M (1, 1)= mulMod2 (bufFactors [0] [0], bufFactors[1] [0], yToL); |
---|
3023 | Pi [0]= M (1,1) + (mulMod2 (bufFactors [0] [1], bufFactors[1] [0], yToL) + |
---|
3024 | mulMod2 (bufFactors [0] [0], bufFactors [1] [1], yToL))*v; |
---|
3025 | } |
---|
3026 | else if (degree (bufFactors[0], v) > 0) |
---|
3027 | { |
---|
3028 | M (1,1)= mulMod2 (bufFactors [0] [0], bufFactors [1], yToL); |
---|
3029 | Pi [0]= M(1,1) + mulMod2 (bufFactors [0] [1], bufFactors[1], yToL)*v; |
---|
3030 | } |
---|
3031 | else if (degree (bufFactors[1], v) > 0) |
---|
3032 | { |
---|
3033 | M (1,1)= mulMod2 (bufFactors [0], bufFactors [1] [0], yToL); |
---|
3034 | Pi [0]= M (1,1) + mulMod2 (bufFactors [0], bufFactors[1] [1], yToL)*v; |
---|
3035 | } |
---|
3036 | else |
---|
3037 | { |
---|
3038 | M (1,1)= mulMod2 (bufFactors [0], bufFactors [1], yToL); |
---|
3039 | Pi [0]= M (1,1); |
---|
3040 | } |
---|
3041 | |
---|
3042 | for (i= 1; i < Pi.size(); i++) |
---|
3043 | { |
---|
3044 | if (degree (Pi[i-1], v) > 0 && degree (bufFactors [i+1], v) > 0) |
---|
3045 | { |
---|
3046 | M (1,i+1)= mulMod2 (Pi[i-1] [0], bufFactors[i+1] [0], yToL); |
---|
3047 | Pi [i]= M (1,i+1) + (mulMod2 (Pi[i-1] [1], bufFactors[i+1] [0], yToL) + |
---|
3048 | mulMod2 (Pi[i-1] [0], bufFactors [i+1] [1], yToL))*v; |
---|
3049 | } |
---|
3050 | else if (degree (Pi[i-1], v) > 0) |
---|
3051 | { |
---|
3052 | M (1,i+1)= mulMod2 (Pi[i-1] [0], bufFactors [i+1], yToL); |
---|
3053 | Pi [i]= M(1,i+1) + mulMod2 (Pi[i-1] [1], bufFactors[i+1], yToL)*v; |
---|
3054 | } |
---|
3055 | else if (degree (bufFactors[i+1], v) > 0) |
---|
3056 | { |
---|
3057 | M (1,i+1)= mulMod2 (Pi[i-1], bufFactors [i+1] [0], yToL); |
---|
3058 | Pi [i]= M (1,i+1) + mulMod2 (Pi[i-1], bufFactors[i+1] [1], yToL)*v; |
---|
3059 | } |
---|
3060 | else |
---|
3061 | { |
---|
3062 | M (1,i+1)= mulMod2 (Pi [i-1], bufFactors [i+1], yToL); |
---|
3063 | Pi [i]= M (1,i+1); |
---|
3064 | } |
---|
3065 | } |
---|
3066 | |
---|
3067 | CFList products; |
---|
3068 | bufF= mod (F, Variable (3)); |
---|
3069 | for (CFListIterator k= factors; k.hasItem(); k++) |
---|
3070 | products.append (bufF/k.getItem()); |
---|
3071 | |
---|
3072 | CFList MOD= CFList (power (v, liftBound)); |
---|
3073 | MOD.insert (yToL); |
---|
3074 | for (int d= 1; d < liftBound; d++) |
---|
3075 | { |
---|
3076 | henselStep2 (F, factors, bufFactors, diophant, M, Pi, products, d, MOD, bad); |
---|
3077 | if (bad) |
---|
3078 | return CFList(); |
---|
3079 | } |
---|
3080 | |
---|
3081 | CFList result; |
---|
3082 | for (i= 0; i < factors.length(); i++) |
---|
3083 | result.append (bufFactors[i]); |
---|
3084 | return result; |
---|
3085 | } |
---|
3086 | |
---|
3087 | CFList |
---|
3088 | nonMonicHenselLift (const CFList& F, const CFList& factors, const CFList& LCs, |
---|
3089 | CFList& diophant, CFArray& Pi, CFMatrix& M, const int lOld, |
---|
3090 | int& lNew, const CFList& MOD, bool& noOneToOne |
---|
3091 | ) |
---|
3092 | { |
---|
3093 | |
---|
3094 | int k= 0; |
---|
3095 | CFArray bufFactors= CFArray (factors.length()); |
---|
3096 | CFListIterator j= LCs; |
---|
3097 | for (CFListIterator i= factors; i.hasItem(); i++, j++, k++) |
---|
3098 | bufFactors [k]= replaceLC (i.getItem(), j.getItem()); |
---|
3099 | |
---|
3100 | Variable y= F.getLast().mvar(); |
---|
3101 | Variable x= F.getFirst().mvar(); |
---|
3102 | CanonicalForm xToLOld= power (x, lOld); |
---|
3103 | |
---|
3104 | Pi [0]= mod (Pi[0], xToLOld); |
---|
3105 | M (1, 1)= Pi [0]; |
---|
3106 | |
---|
3107 | if (degree (bufFactors[0], y) > 0 && degree (bufFactors [1], y) > 0) |
---|
3108 | Pi [0] += (mulMod (bufFactors [0] [1], bufFactors[1] [0], MOD) + |
---|
3109 | mulMod (bufFactors [0] [0], bufFactors [1] [1], MOD))*y; |
---|
3110 | else if (degree (bufFactors[0], y) > 0) |
---|
3111 | Pi [0] += mulMod (bufFactors [0] [1], bufFactors[1], MOD)*y; |
---|
3112 | else if (degree (bufFactors[1], y) > 0) |
---|
3113 | Pi [0] += mulMod (bufFactors [0], bufFactors[1] [1], MOD)*y; |
---|
3114 | |
---|
3115 | for (int i= 1; i < Pi.size(); i++) |
---|
3116 | { |
---|
3117 | Pi [i]= mod (Pi [i], xToLOld); |
---|
3118 | M (1, i + 1)= Pi [i]; |
---|
3119 | |
---|
3120 | if (degree (Pi[i-1], y) > 0 && degree (bufFactors [i+1], y) > 0) |
---|
3121 | Pi [i] += (mulMod (Pi[i-1] [1], bufFactors[i+1] [0], MOD) + |
---|
3122 | mulMod (Pi[i-1] [0], bufFactors [i+1] [1], MOD))*y; |
---|
3123 | else if (degree (Pi[i-1], y) > 0) |
---|
3124 | Pi [i] += mulMod (Pi[i-1] [1], bufFactors[i+1], MOD)*y; |
---|
3125 | else if (degree (bufFactors[i+1], y) > 0) |
---|
3126 | Pi [i] += mulMod (Pi[i-1], bufFactors[i+1] [1], MOD)*y; |
---|
3127 | } |
---|
3128 | |
---|
3129 | CFList products; |
---|
3130 | CanonicalForm bufF= F.getFirst(); |
---|
3131 | |
---|
3132 | for (int i= 0; i < bufFactors.size(); i++) |
---|
3133 | { |
---|
3134 | if (degree (bufFactors[i], y) > 0) |
---|
3135 | { |
---|
3136 | if (!fdivides (bufFactors[i] [0], bufF)) |
---|
3137 | { |
---|
3138 | noOneToOne= true; |
---|
3139 | return factors; |
---|
3140 | } |
---|
3141 | products.append (bufF/bufFactors[i] [0]); |
---|
3142 | } |
---|
3143 | else |
---|
3144 | { |
---|
3145 | if (!fdivides (bufFactors[i], bufF)) |
---|
3146 | { |
---|
3147 | noOneToOne= true; |
---|
3148 | return factors; |
---|
3149 | } |
---|
3150 | products.append (bufF/bufFactors[i]); |
---|
3151 | } |
---|
3152 | } |
---|
3153 | |
---|
3154 | for (int d= 1; d < lNew; d++) |
---|
3155 | { |
---|
3156 | henselStep2 (F.getLast(), factors, bufFactors, diophant, M, Pi, products, d, |
---|
3157 | MOD, noOneToOne); |
---|
3158 | if (noOneToOne) |
---|
3159 | return CFList(); |
---|
3160 | } |
---|
3161 | |
---|
3162 | CFList result; |
---|
3163 | for (k= 0; k < factors.length(); k++) |
---|
3164 | result.append (bufFactors[k]); |
---|
3165 | return result; |
---|
3166 | } |
---|
3167 | |
---|
3168 | CFList |
---|
3169 | nonMonicHenselLift (const CFList& eval, const CFList& factors, |
---|
3170 | CFList* const& LCs, CFList& diophant, CFArray& Pi, |
---|
3171 | int* liftBound, int length, bool& noOneToOne |
---|
3172 | ) |
---|
3173 | { |
---|
3174 | CFList bufDiophant= diophant; |
---|
3175 | CFList buf= factors; |
---|
3176 | CFArray bufPi= Pi; |
---|
3177 | CFMatrix M= CFMatrix (liftBound[1], factors.length() - 1); |
---|
3178 | int k= 0; |
---|
3179 | |
---|
3180 | CFList result= |
---|
3181 | nonMonicHenselLift23 (eval.getFirst(), factors, LCs [0], diophant, bufPi, |
---|
3182 | liftBound[1], liftBound[0], noOneToOne); |
---|
3183 | |
---|
3184 | if (noOneToOne) |
---|
3185 | return CFList(); |
---|
3186 | |
---|
3187 | if (eval.length() == 1) |
---|
3188 | return result; |
---|
3189 | |
---|
3190 | k++; |
---|
3191 | CFList MOD; |
---|
3192 | for (int i= 0; i < 2; i++) |
---|
3193 | MOD.append (power (Variable (i + 2), liftBound[i])); |
---|
3194 | |
---|
3195 | CFListIterator j= eval; |
---|
3196 | CFList bufEval; |
---|
3197 | bufEval.append (j.getItem()); |
---|
3198 | j++; |
---|
3199 | |
---|
3200 | for (int i= 2; i <= length && j.hasItem(); i++, j++, k++) |
---|
3201 | { |
---|
3202 | bufEval.append (j.getItem()); |
---|
3203 | M= CFMatrix (liftBound[i], factors.length() - 1); |
---|
3204 | result= nonMonicHenselLift (bufEval, result, LCs [i-1], diophant, bufPi, M, |
---|
3205 | liftBound[i-1], liftBound[i], MOD, noOneToOne); |
---|
3206 | if (noOneToOne) |
---|
3207 | return result; |
---|
3208 | MOD.append (power (Variable (i + 2), liftBound[i])); |
---|
3209 | bufEval.removeFirst(); |
---|
3210 | } |
---|
3211 | |
---|
3212 | return result; |
---|
3213 | } |
---|
3214 | |
---|
3215 | #endif |
---|
3216 | /* HAVE_NTL */ |
---|
3217 | |
---|