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 | |
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29 | #ifdef HAVE_NTL |
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30 | #include <NTL/lzz_pEX.h> |
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31 | #include "NTLconvert.h" |
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32 | |
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33 | static inline |
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34 | CanonicalForm |
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35 | mulNTL (const CanonicalForm& F, const CanonicalForm& G) |
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36 | { |
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37 | if (F.inCoeffDomain() || G.inCoeffDomain()) |
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38 | return F*G; |
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39 | ASSERT (F.isUnivariate() && G.isUnivariate(), "expected univariate polys"); |
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40 | ASSERT (F.level() == G.level(), "expected polys of same level"); |
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41 | if (CFFactory::gettype() == GaloisFieldDomain) |
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42 | return F*G; |
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43 | zz_p::init (getCharacteristic()); |
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44 | Variable alpha; |
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45 | CanonicalForm result; |
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46 | if (hasFirstAlgVar (F, alpha) || hasFirstAlgVar (G, alpha)) |
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47 | { |
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48 | zz_pX NTLMipo= convertFacCF2NTLzzpX (getMipo (alpha)); |
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49 | zz_pE::init (NTLMipo); |
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50 | zz_pEX NTLF= convertFacCF2NTLzz_pEX (F, NTLMipo); |
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51 | zz_pEX NTLG= convertFacCF2NTLzz_pEX (G, NTLMipo); |
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52 | mul (NTLF, NTLF, NTLG); |
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53 | result= convertNTLzz_pEX2CF(NTLF, F.mvar(), alpha); |
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54 | } |
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55 | else |
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56 | { |
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57 | zz_pX NTLF= convertFacCF2NTLzzpX (F); |
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58 | zz_pX NTLG= convertFacCF2NTLzzpX (G); |
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59 | mul (NTLF, NTLF, NTLG); |
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60 | result= convertNTLzzpX2CF(NTLF, F.mvar()); |
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61 | } |
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62 | return result; |
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63 | } |
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64 | |
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65 | static inline |
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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 | static inline |
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103 | CanonicalForm |
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104 | divNTL (const CanonicalForm& F, const CanonicalForm& G) |
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105 | { |
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106 | if (F.inCoeffDomain() && G.isUnivariate()) |
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107 | return F; |
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108 | else if (F.inCoeffDomain() && G.inCoeffDomain()) |
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109 | return div (F, G); |
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110 | else if (F.isUnivariate() && G.inCoeffDomain()) |
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111 | return div (F,G); |
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112 | |
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113 | ASSERT (F.isUnivariate() && G.isUnivariate(), "expected univariate polys"); |
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114 | ASSERT (F.level() == G.level(), "expected polys of same level"); |
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115 | if (CFFactory::gettype() == GaloisFieldDomain) |
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116 | return div (F, G); |
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117 | zz_p::init (getCharacteristic()); |
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118 | Variable alpha; |
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119 | CanonicalForm result; |
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120 | if (hasFirstAlgVar (F, alpha) || hasFirstAlgVar (G, alpha)) |
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121 | { |
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122 | zz_pX NTLMipo= convertFacCF2NTLzzpX(getMipo (alpha)); |
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123 | zz_pE::init (NTLMipo); |
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124 | zz_pEX NTLF= convertFacCF2NTLzz_pEX (F, NTLMipo); |
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125 | zz_pEX NTLG= convertFacCF2NTLzz_pEX (G, NTLMipo); |
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126 | div (NTLF, NTLF, NTLG); |
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127 | result= convertNTLzz_pEX2CF(NTLF, F.mvar(), alpha); |
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128 | } |
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129 | else |
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130 | { |
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131 | zz_pX NTLF= convertFacCF2NTLzzpX (F); |
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132 | zz_pX NTLG= convertFacCF2NTLzzpX (G); |
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133 | div (NTLF, NTLF, NTLG); |
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134 | result= convertNTLzzpX2CF(NTLF, F.mvar()); |
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135 | } |
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136 | return result; |
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137 | } |
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138 | |
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139 | /*static inline |
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140 | void |
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141 | divremNTL (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
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142 | CanonicalForm& R) |
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143 | { |
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144 | if (F.inCoeffDomain() && G.isUnivariate()) |
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145 | { |
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146 | R= F; |
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147 | Q= 0; |
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148 | } |
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149 | else if (F.inCoeffDomain() && G.inCoeffDomain()) |
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150 | { |
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151 | divrem (F, G, Q, R); |
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152 | return; |
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153 | } |
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154 | else if (F.isUnivariate() && G.inCoeffDomain()) |
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155 | { |
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156 | divrem (F, G, Q, R); |
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157 | return; |
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158 | } |
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159 | |
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160 | ASSERT (F.isUnivariate() && G.isUnivariate(), "expected univariate polys"); |
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161 | ASSERT (F.level() == G.level(), "expected polys of same level"); |
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162 | if (CFFactory::gettype() == GaloisFieldDomain) |
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163 | { |
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164 | divrem (F, G, Q, R); |
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165 | return; |
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166 | } |
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167 | zz_p::init (getCharacteristic()); |
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168 | Variable alpha; |
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169 | CanonicalForm result; |
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170 | if (hasFirstAlgVar (F, alpha) || hasFirstAlgVar (G, alpha)) |
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171 | { |
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172 | zz_pX NTLMipo= convertFacCF2NTLzzpX(getMipo (alpha)); |
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173 | zz_pE::init (NTLMipo); |
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174 | zz_pEX NTLF= convertFacCF2NTLzz_pEX (F, NTLMipo); |
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175 | zz_pEX NTLG= convertFacCF2NTLzz_pEX (G, NTLMipo); |
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176 | zz_pEX NTLQ; |
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177 | zz_pEX NTLR; |
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178 | DivRem (NTLQ, NTLR, NTLF, NTLG); |
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179 | Q= convertNTLzz_pEX2CF(NTLQ, F.mvar(), alpha); |
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180 | R= convertNTLzz_pEX2CF(NTLR, F.mvar(), alpha); |
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181 | return; |
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182 | } |
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183 | else |
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184 | { |
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185 | zz_pX NTLF= convertFacCF2NTLzzpX (F); |
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186 | zz_pX NTLG= convertFacCF2NTLzzpX (G); |
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187 | zz_pX NTLQ; |
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188 | zz_pX NTLR; |
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189 | DivRem (NTLQ, NTLR, NTLF, NTLG); |
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190 | Q= convertNTLzzpX2CF(NTLQ, F.mvar()); |
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191 | R= convertNTLzzpX2CF(NTLR, F.mvar()); |
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192 | return; |
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193 | } |
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194 | }*/ |
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195 | |
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196 | CanonicalForm mod (const CanonicalForm& F, const CFList& M) |
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197 | { |
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198 | CanonicalForm A= F; |
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199 | for (CFListIterator i= M; i.hasItem(); i++) |
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200 | A= mod (A, i.getItem()); |
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201 | return A; |
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202 | } |
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203 | |
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204 | CanonicalForm mulMod2 (const CanonicalForm& A, const CanonicalForm& B, |
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205 | const CanonicalForm& M) |
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206 | { |
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207 | if (A.isZero() || B.isZero()) |
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208 | return 0; |
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209 | |
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210 | ASSERT (M.isUnivariate(), "M must be univariate"); |
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211 | |
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212 | CanonicalForm F= mod (A, M); |
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213 | CanonicalForm G= mod (B, M); |
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214 | if (F.inCoeffDomain() || G.inCoeffDomain()) |
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215 | return F*G; |
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216 | Variable y= M.mvar(); |
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217 | int degF= degree (F, y); |
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218 | int degG= degree (G, y); |
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219 | |
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220 | if ((degF < 1 && degG < 1) && (F.isUnivariate() && G.isUnivariate()) && |
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221 | (F.level() == G.level())) |
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222 | { |
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223 | CanonicalForm result= mulNTL (F, G); |
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224 | return mod (result, M); |
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225 | } |
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226 | else if (degF <= 1 && degG <= 1) |
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227 | { |
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228 | CanonicalForm result= F*G; |
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229 | return mod (result, M); |
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230 | } |
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231 | |
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232 | int m= (int) ceil (degree (M)/2.0); |
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233 | if (degF >= m || degG >= m) |
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234 | { |
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235 | CanonicalForm MLo= power (y, m); |
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236 | CanonicalForm MHi= power (y, degree (M) - m); |
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237 | CanonicalForm F0= mod (F, MLo); |
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238 | CanonicalForm F1= div (F, MLo); |
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239 | CanonicalForm G0= mod (G, MLo); |
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240 | CanonicalForm G1= div (G, MLo); |
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241 | CanonicalForm F0G1= mulMod2 (F0, G1, MHi); |
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242 | CanonicalForm F1G0= mulMod2 (F1, G0, MHi); |
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243 | CanonicalForm F0G0= mulMod2 (F0, G0, M); |
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244 | return F0G0 + MLo*(F0G1 + F1G0); |
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245 | } |
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246 | else |
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247 | { |
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248 | m= (int) ceil (tmax (degF, degG)/2.0); |
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249 | CanonicalForm yToM= power (y, m); |
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250 | CanonicalForm F0= mod (F, yToM); |
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251 | CanonicalForm F1= div (F, yToM); |
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252 | CanonicalForm G0= mod (G, yToM); |
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253 | CanonicalForm G1= div (G, yToM); |
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254 | CanonicalForm H00= mulMod2 (F0, G0, M); |
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255 | CanonicalForm H11= mulMod2 (F1, G1, M); |
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256 | CanonicalForm H01= mulMod2 (F0 + F1, G0 + G1, M); |
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257 | return H11*yToM*yToM + (H01 - H11 - H00)*yToM + H00; |
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258 | } |
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259 | DEBOUTLN (cerr, "fatal end in mulMod2"); |
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260 | } |
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261 | |
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262 | CanonicalForm mulMod (const CanonicalForm& A, const CanonicalForm& B, |
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263 | const CFList& MOD) |
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264 | { |
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265 | if (A.isZero() || B.isZero()) |
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266 | return 0; |
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267 | |
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268 | if (MOD.length() == 1) |
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269 | return mulMod2 (A, B, MOD.getLast()); |
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270 | |
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271 | CanonicalForm M= MOD.getLast(); |
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272 | CanonicalForm F= mod (A, M); |
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273 | CanonicalForm G= mod (B, M); |
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274 | if (F.inCoeffDomain() || G.inCoeffDomain()) |
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275 | return F*G; |
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276 | Variable y= M.mvar(); |
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277 | int degF= degree (F, y); |
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278 | int degG= degree (G, y); |
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279 | |
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280 | if ((degF <= 1 && F.level() <= M.level()) && |
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281 | (degG <= 1 && G.level() <= M.level())) |
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282 | { |
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283 | CFList buf= MOD; |
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284 | buf.removeLast(); |
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285 | if (degF == 1 && degG == 1) |
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286 | { |
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287 | CanonicalForm F0= mod (F, y); |
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288 | CanonicalForm F1= div (F, y); |
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289 | CanonicalForm G0= mod (G, y); |
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290 | CanonicalForm G1= div (G, y); |
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291 | if (degree (M) > 2) |
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292 | { |
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293 | CanonicalForm H00= mulMod (F0, G0, buf); |
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294 | CanonicalForm H11= mulMod (F1, G1, buf); |
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295 | CanonicalForm H01= mulMod (F0 + F1, G0 + G1, buf); |
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296 | return H11*y*y + (H01 - H00 - H11)*y + H00; |
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297 | } |
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298 | else //here degree (M) == 2 |
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299 | { |
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300 | buf.append (y); |
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301 | CanonicalForm F0G1= mulMod (F0, G1, buf); |
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302 | CanonicalForm F1G0= mulMod (F1, G0, buf); |
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303 | CanonicalForm F0G0= mulMod (F0, G0, MOD); |
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304 | CanonicalForm result= F0G0 + y*(F0G1 + F1G0); |
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305 | return result; |
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306 | } |
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307 | } |
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308 | else if (degF == 1 && degG == 0) |
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309 | return mulMod (div (F, y), G, buf)*y + mulMod (mod (F, y), G, buf); |
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310 | else if (degF == 0 && degG == 1) |
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311 | return mulMod (div (G, y), F, buf)*y + mulMod (mod (G, y), F, buf); |
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312 | else |
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313 | return mulMod (F, G, buf); |
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314 | } |
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315 | int m= (int) ceil (degree (M)/2.0); |
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316 | if (degF >= m || degG >= m) |
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317 | { |
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318 | CanonicalForm MLo= power (y, m); |
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319 | CanonicalForm MHi= power (y, degree (M) - m); |
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320 | CanonicalForm F0= mod (F, MLo); |
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321 | CanonicalForm F1= div (F, MLo); |
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322 | CanonicalForm G0= mod (G, MLo); |
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323 | CanonicalForm G1= div (G, MLo); |
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324 | CFList buf= MOD; |
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325 | buf.removeLast(); |
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326 | buf.append (MHi); |
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327 | CanonicalForm F0G1= mulMod (F0, G1, buf); |
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328 | CanonicalForm F1G0= mulMod (F1, G0, buf); |
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329 | CanonicalForm F0G0= mulMod (F0, G0, MOD); |
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330 | return F0G0 + MLo*(F0G1 + F1G0); |
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331 | } |
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332 | else |
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333 | { |
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334 | m= (int) ceil (tmax (degF, degG)/2.0); |
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335 | CanonicalForm yToM= power (y, m); |
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336 | CanonicalForm F0= mod (F, yToM); |
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337 | CanonicalForm F1= div (F, yToM); |
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338 | CanonicalForm G0= mod (G, yToM); |
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339 | CanonicalForm G1= div (G, yToM); |
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340 | CanonicalForm H00= mulMod (F0, G0, MOD); |
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341 | CanonicalForm H11= mulMod (F1, G1, MOD); |
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342 | CanonicalForm H01= mulMod (F0 + F1, G0 + G1, MOD); |
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343 | return H11*yToM*yToM + (H01 - H11 - H00)*yToM + H00; |
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344 | } |
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345 | DEBOUTLN (cerr, "fatal end in mulMod"); |
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346 | } |
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347 | |
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348 | CanonicalForm prodMod (const CFList& L, const CanonicalForm& M) |
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349 | { |
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350 | if (L.isEmpty()) |
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351 | return 1; |
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352 | int l= L.length(); |
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353 | if (l == 1) |
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354 | return mod (L.getFirst(), M); |
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355 | else if (l == 2) { |
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356 | CanonicalForm result= mulMod2 (L.getFirst(), L.getLast(), M); |
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357 | return result; |
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358 | } |
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359 | else |
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360 | { |
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361 | l /= 2; |
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362 | CFList tmp1, tmp2; |
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363 | CFListIterator i= L; |
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364 | CanonicalForm buf1, buf2; |
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365 | for (int j= 1; j <= l; j++, i++) |
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366 | tmp1.append (i.getItem()); |
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367 | tmp2= Difference (L, tmp1); |
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368 | buf1= prodMod (tmp1, M); |
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369 | buf2= prodMod (tmp2, M); |
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370 | CanonicalForm result= mulMod2 (buf1, buf2, M); |
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371 | return result; |
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372 | } |
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373 | } |
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374 | |
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375 | CanonicalForm prodMod (const CFList& L, const CFList& M) |
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376 | { |
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377 | if (L.isEmpty()) |
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378 | return 1; |
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379 | else if (L.length() == 1) |
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380 | return L.getFirst(); |
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381 | else if (L.length() == 2) |
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382 | return mulMod (L.getFirst(), L.getLast(), M); |
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383 | else |
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384 | { |
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385 | int l= L.length()/2; |
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386 | CFListIterator i= L; |
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387 | CFList tmp1, tmp2; |
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388 | CanonicalForm buf1, buf2; |
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389 | for (int j= 1; j <= l; j++, i++) |
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390 | tmp1.append (i.getItem()); |
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391 | tmp2= Difference (L, tmp1); |
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392 | buf1= prodMod (tmp1, M); |
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393 | buf2= prodMod (tmp2, M); |
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394 | return mulMod (buf1, buf2, M); |
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395 | } |
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396 | } |
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397 | |
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398 | static inline |
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399 | CFList split (const CanonicalForm& F, const int m, const Variable& x) |
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400 | { |
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401 | CanonicalForm A= F; |
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402 | CanonicalForm buf= 0; |
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403 | bool swap= false; |
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404 | if (degree (A, x) <= 0) |
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405 | return CFList(A); |
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406 | else if (x.level() != A.level()) |
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407 | { |
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408 | swap= true; |
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409 | A= swapvar (A, x, A.mvar()); |
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410 | } |
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411 | |
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412 | int j= (int) floor ((double) degree (A)/ m); |
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413 | CFList result; |
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414 | CFIterator i= A; |
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415 | for (; j >= 0; j--) |
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416 | { |
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417 | while (i.hasTerms() && i.exp() - j*m >= 0) |
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418 | { |
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419 | if (swap) |
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420 | buf += i.coeff()*power (A.mvar(), i.exp() - j*m); |
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421 | else |
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422 | buf += i.coeff()*power (x, i.exp() - j*m); |
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423 | i++; |
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424 | } |
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425 | if (swap) |
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426 | result.append (swapvar (buf, x, F.mvar())); |
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427 | else |
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428 | result.append (buf); |
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429 | buf= 0; |
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430 | } |
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431 | return result; |
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432 | } |
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433 | |
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434 | static inline |
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435 | void divrem32 (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
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436 | CanonicalForm& R, const CFList& M); |
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437 | |
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438 | static inline |
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439 | void divrem21 (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
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440 | CanonicalForm& R, const CFList& M) |
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441 | { |
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442 | CanonicalForm A= mod (F, M); |
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443 | CanonicalForm B= mod (G, M); |
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444 | Variable x= Variable (1); |
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445 | int degB= degree (B, x); |
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446 | int degA= degree (A, x); |
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447 | if (degA < degB) |
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448 | { |
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449 | Q= 0; |
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450 | R= A; |
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451 | return; |
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452 | } |
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453 | ASSERT (2*degB > degA, "expected degree (F, 1) < 2*degree (G, 1)"); |
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454 | if (degB < 1) |
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455 | { |
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456 | divrem (A, B, Q, R); |
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457 | Q= mod (Q, M); |
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458 | R= mod (R, M); |
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459 | return; |
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460 | } |
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461 | |
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462 | int m= (int) ceil ((double) (degB + 1)/2.0) + 1; |
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463 | CFList splitA= split (A, m, x); |
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464 | CFList splitB= split (B, m, x); |
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465 | if (splitA.length() == 3) |
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466 | splitA.insert (0); |
---|
467 | if (splitA.length() == 2) |
---|
468 | { |
---|
469 | splitA.insert (0); |
---|
470 | splitA.insert (0); |
---|
471 | } |
---|
472 | if (splitA.length() == 1) |
---|
473 | { |
---|
474 | splitA.insert (0); |
---|
475 | splitA.insert (0); |
---|
476 | splitA.insert (0); |
---|
477 | } |
---|
478 | |
---|
479 | CanonicalForm xToM= power (x, m); |
---|
480 | |
---|
481 | CFListIterator i= splitA; |
---|
482 | CanonicalForm H= i.getItem(); |
---|
483 | i++; |
---|
484 | H *= xToM; |
---|
485 | H += i.getItem(); |
---|
486 | i++; |
---|
487 | H *= xToM; |
---|
488 | H += i.getItem(); |
---|
489 | i++; |
---|
490 | |
---|
491 | divrem32 (H, B, Q, R, M); |
---|
492 | |
---|
493 | CFList splitR= split (R, m, x); |
---|
494 | if (splitR.length() == 1) |
---|
495 | splitR.insert (0); |
---|
496 | |
---|
497 | H= splitR.getFirst(); |
---|
498 | H *= xToM; |
---|
499 | H += splitR.getLast(); |
---|
500 | H *= xToM; |
---|
501 | H += i.getItem(); |
---|
502 | |
---|
503 | CanonicalForm bufQ; |
---|
504 | divrem32 (H, B, bufQ, R, M); |
---|
505 | |
---|
506 | Q *= xToM; |
---|
507 | Q += bufQ; |
---|
508 | return; |
---|
509 | } |
---|
510 | |
---|
511 | static inline |
---|
512 | void divrem32 (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
---|
513 | CanonicalForm& R, const CFList& M) |
---|
514 | { |
---|
515 | CanonicalForm A= mod (F, M); |
---|
516 | CanonicalForm B= mod (G, M); |
---|
517 | Variable x= Variable (1); |
---|
518 | int degB= degree (B, x); |
---|
519 | int degA= degree (A, x); |
---|
520 | if (degA < degB) |
---|
521 | { |
---|
522 | Q= 0; |
---|
523 | R= A; |
---|
524 | return; |
---|
525 | } |
---|
526 | ASSERT (3*(degB/2) > degA, "expected degree (F, 1) < 3*(degree (G, 1)/2)"); |
---|
527 | if (degB < 1) |
---|
528 | { |
---|
529 | divrem (A, B, Q, R); |
---|
530 | Q= mod (Q, M); |
---|
531 | R= mod (R, M); |
---|
532 | return; |
---|
533 | } |
---|
534 | int m= (int) ceil ((double) (degB + 1)/ 2.0); |
---|
535 | |
---|
536 | CFList splitA= split (A, m, x); |
---|
537 | CFList splitB= split (B, m, x); |
---|
538 | |
---|
539 | if (splitA.length() == 2) |
---|
540 | { |
---|
541 | splitA.insert (0); |
---|
542 | } |
---|
543 | if (splitA.length() == 1) |
---|
544 | { |
---|
545 | splitA.insert (0); |
---|
546 | splitA.insert (0); |
---|
547 | } |
---|
548 | CanonicalForm xToM= power (x, m); |
---|
549 | |
---|
550 | CanonicalForm H; |
---|
551 | CFListIterator i= splitA; |
---|
552 | i++; |
---|
553 | |
---|
554 | if (degree (splitA.getFirst(), x) < degree (splitB.getFirst(), x)) |
---|
555 | { |
---|
556 | H= splitA.getFirst()*xToM + i.getItem(); |
---|
557 | divrem21 (H, splitB.getFirst(), Q, R, M); |
---|
558 | } |
---|
559 | else |
---|
560 | { |
---|
561 | R= splitA.getFirst()*xToM + i.getItem() + splitB.getFirst() - |
---|
562 | splitB.getFirst()*xToM; |
---|
563 | Q= xToM - 1; |
---|
564 | } |
---|
565 | |
---|
566 | H= mulMod (Q, splitB.getLast(), M); |
---|
567 | |
---|
568 | R= R*xToM + splitA.getLast() - H; |
---|
569 | |
---|
570 | while (degree (R, x) >= degB) |
---|
571 | { |
---|
572 | xToM= power (x, degree (R, x) - degB); |
---|
573 | Q += LC (R, x)*xToM; |
---|
574 | R -= mulMod (LC (R, x), B, M)*xToM; |
---|
575 | Q= mod (Q, M); |
---|
576 | R= mod (R, M); |
---|
577 | } |
---|
578 | |
---|
579 | return; |
---|
580 | } |
---|
581 | |
---|
582 | void divrem2 (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
---|
583 | CanonicalForm& R, const CanonicalForm& M) |
---|
584 | { |
---|
585 | CanonicalForm A= mod (F, M); |
---|
586 | CanonicalForm B= mod (G, M); |
---|
587 | Variable x= Variable (1); |
---|
588 | int degB= degree (B, x); |
---|
589 | if (degB > degree (A, x)) |
---|
590 | { |
---|
591 | Q= 0; |
---|
592 | R= A; |
---|
593 | return; |
---|
594 | } |
---|
595 | |
---|
596 | CFList splitA= split (A, degB, x); |
---|
597 | |
---|
598 | CanonicalForm xToDegB= power (x, degB); |
---|
599 | CanonicalForm H, bufQ; |
---|
600 | Q= 0; |
---|
601 | CFListIterator i= splitA; |
---|
602 | H= i.getItem()*xToDegB; |
---|
603 | i++; |
---|
604 | H += i.getItem(); |
---|
605 | CFList buf; |
---|
606 | while (i.hasItem()) |
---|
607 | { |
---|
608 | buf= CFList (M); |
---|
609 | divrem21 (H, B, bufQ, R, buf); |
---|
610 | i++; |
---|
611 | if (i.hasItem()) |
---|
612 | H= R*xToDegB + i.getItem(); |
---|
613 | Q *= xToDegB; |
---|
614 | Q += bufQ; |
---|
615 | } |
---|
616 | return; |
---|
617 | } |
---|
618 | |
---|
619 | void divrem (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
---|
620 | CanonicalForm& R, const CFList& MOD) |
---|
621 | { |
---|
622 | CanonicalForm A= mod (F, MOD); |
---|
623 | CanonicalForm B= mod (G, MOD); |
---|
624 | Variable x= Variable (1); |
---|
625 | int degB= degree (B, x); |
---|
626 | if (degB > degree (A, x)) |
---|
627 | { |
---|
628 | Q= 0; |
---|
629 | R= A; |
---|
630 | return; |
---|
631 | } |
---|
632 | |
---|
633 | if (degB == 0) |
---|
634 | { |
---|
635 | divrem (A, B, Q, R); |
---|
636 | Q= mod (Q, MOD); |
---|
637 | R= mod (R, MOD); |
---|
638 | return; |
---|
639 | } |
---|
640 | CFList splitA= split (A, degB, x); |
---|
641 | |
---|
642 | CanonicalForm xToDegB= power (x, degB); |
---|
643 | CanonicalForm H, bufQ; |
---|
644 | Q= 0; |
---|
645 | CFListIterator i= splitA; |
---|
646 | H= i.getItem()*xToDegB; |
---|
647 | i++; |
---|
648 | H += i.getItem(); |
---|
649 | while (i.hasItem()) |
---|
650 | { |
---|
651 | divrem21 (H, B, bufQ, R, MOD); |
---|
652 | i++; |
---|
653 | if (i.hasItem()) |
---|
654 | H= R*xToDegB + i.getItem(); |
---|
655 | Q *= xToDegB; |
---|
656 | Q += bufQ; |
---|
657 | } |
---|
658 | return; |
---|
659 | } |
---|
660 | |
---|
661 | void sortList (CFList& list, const Variable& x) |
---|
662 | { |
---|
663 | int l= 1; |
---|
664 | int k= 1; |
---|
665 | CanonicalForm buf; |
---|
666 | CFListIterator m; |
---|
667 | for (CFListIterator i= list; l <= list.length(); i++, l++) |
---|
668 | { |
---|
669 | for (CFListIterator j= list; k <= list.length() - l; k++) |
---|
670 | { |
---|
671 | m= j; |
---|
672 | m++; |
---|
673 | if (degree (j.getItem(), x) > degree (m.getItem(), x)) |
---|
674 | { |
---|
675 | buf= m.getItem(); |
---|
676 | m.getItem()= j.getItem(); |
---|
677 | j.getItem()= buf; |
---|
678 | j++; |
---|
679 | j.getItem()= m.getItem(); |
---|
680 | } |
---|
681 | else |
---|
682 | j++; |
---|
683 | } |
---|
684 | k= 1; |
---|
685 | } |
---|
686 | } |
---|
687 | |
---|
688 | static inline |
---|
689 | CFList diophantine (const CanonicalForm& F, const CFList& factors) |
---|
690 | { |
---|
691 | CanonicalForm buf1, buf2, buf3, S, T; |
---|
692 | CFListIterator i= factors; |
---|
693 | CFList result; |
---|
694 | if (i.hasItem()) |
---|
695 | i++; |
---|
696 | buf1= F/factors.getFirst(); |
---|
697 | buf2= divNTL (F, i.getItem()); |
---|
698 | buf3= extgcd (buf1, buf2, S, T); |
---|
699 | result.append (S); |
---|
700 | result.append (T); |
---|
701 | if (i.hasItem()) |
---|
702 | i++; |
---|
703 | for (; i.hasItem(); i++) |
---|
704 | { |
---|
705 | buf1= divNTL (F, i.getItem()); |
---|
706 | buf3= extgcd (buf3, buf1, S, T); |
---|
707 | CFListIterator k= factors; |
---|
708 | for (CFListIterator j= result; j.hasItem(); j++, k++) |
---|
709 | { |
---|
710 | j.getItem()= mulNTL (j.getItem(), S); |
---|
711 | j.getItem()= modNTL (j.getItem(), k.getItem()); |
---|
712 | } |
---|
713 | result.append (T); |
---|
714 | } |
---|
715 | return result; |
---|
716 | } |
---|
717 | |
---|
718 | void |
---|
719 | henselStep12 (const CanonicalForm& F, const CFList& factors, |
---|
720 | CFArray& bufFactors, const CFList& diophant, CFMatrix& M, |
---|
721 | CFArray& Pi, int j) |
---|
722 | { |
---|
723 | CanonicalForm E; |
---|
724 | CanonicalForm xToJ= power (F.mvar(), j); |
---|
725 | Variable x= F.mvar(); |
---|
726 | // compute the error |
---|
727 | if (j == 1) |
---|
728 | E= F[j]; |
---|
729 | else |
---|
730 | { |
---|
731 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
732 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
733 | else |
---|
734 | E= F[j]; |
---|
735 | } |
---|
736 | |
---|
737 | CFArray buf= CFArray (diophant.length()); |
---|
738 | bufFactors[0]= mod (factors.getFirst(), power (F.mvar(), j + 1)); |
---|
739 | int k= 0; |
---|
740 | CanonicalForm remainder; |
---|
741 | // actual lifting |
---|
742 | for (CFListIterator i= diophant; i.hasItem(); i++, k++) |
---|
743 | { |
---|
744 | if (degree (bufFactors[k], x) > 0) |
---|
745 | { |
---|
746 | if (k > 0) |
---|
747 | remainder= modNTL (E, bufFactors[k] [0]); |
---|
748 | else |
---|
749 | remainder= E; |
---|
750 | } |
---|
751 | else |
---|
752 | remainder= modNTL (E, bufFactors[k]); |
---|
753 | |
---|
754 | buf[k]= mulNTL (i.getItem(), remainder); |
---|
755 | if (degree (bufFactors[k], x) > 0) |
---|
756 | buf[k]= modNTL (buf[k], bufFactors[k] [0]); |
---|
757 | else |
---|
758 | buf[k]= modNTL (buf[k], bufFactors[k]); |
---|
759 | } |
---|
760 | for (k= 1; k < factors.length(); k++) |
---|
761 | bufFactors[k] += xToJ*buf[k]; |
---|
762 | |
---|
763 | // update Pi [0] |
---|
764 | int degBuf0= degree (bufFactors[0], x); |
---|
765 | int degBuf1= degree (bufFactors[1], x); |
---|
766 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
767 | M (j + 1, 1)= mulNTL (bufFactors[0] [j], bufFactors[1] [j]); |
---|
768 | CanonicalForm uIZeroJ; |
---|
769 | if (j == 1) |
---|
770 | { |
---|
771 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
772 | uIZeroJ= mulNTL ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
773 | (bufFactors[1] [0] + buf[1])) - M(1, 1) - M(j + 1, 1); |
---|
774 | else if (degBuf0 > 0) |
---|
775 | uIZeroJ= mulNTL (bufFactors[0] [j], bufFactors[1]); |
---|
776 | else if (degBuf1 > 0) |
---|
777 | uIZeroJ= mulNTL (bufFactors[0], buf[1]); |
---|
778 | else |
---|
779 | uIZeroJ= 0; |
---|
780 | Pi [0] += xToJ*uIZeroJ; |
---|
781 | } |
---|
782 | else |
---|
783 | { |
---|
784 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
785 | uIZeroJ= mulNTL ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
786 | (bufFactors[1] [0] + buf[1])) - M(1, 1) - M(j + 1, 1); |
---|
787 | else if (degBuf0 > 0) |
---|
788 | uIZeroJ= mulNTL (bufFactors[0] [j], bufFactors[1]); |
---|
789 | else if (degBuf1 > 0) |
---|
790 | uIZeroJ= mulNTL (bufFactors[0], buf[1]); |
---|
791 | else |
---|
792 | uIZeroJ= 0; |
---|
793 | Pi [0] += xToJ*uIZeroJ; |
---|
794 | } |
---|
795 | CFArray tmp= CFArray (factors.length() - 1); |
---|
796 | for (k= 0; k < factors.length() - 1; k++) |
---|
797 | tmp[k]= 0; |
---|
798 | CFIterator one, two; |
---|
799 | one= bufFactors [0]; |
---|
800 | two= bufFactors [1]; |
---|
801 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
802 | { |
---|
803 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
804 | { |
---|
805 | if (k != j - k + 1) |
---|
806 | { |
---|
807 | if (one.exp() == j - k + 1 && two.exp() == j - k + 1) |
---|
808 | { |
---|
809 | tmp[0] += mulNTL ((bufFactors[0] [k] + one.coeff()), (bufFactors[1] [k] + |
---|
810 | two.coeff())) - M (k + 1, 1) - M (j - k + 2, 1); |
---|
811 | one++; |
---|
812 | two++; |
---|
813 | } |
---|
814 | else if (one.exp() == j - k + 1) |
---|
815 | { |
---|
816 | tmp[0] += mulNTL ((bufFactors[0] [k] + one.coeff()), bufFactors[1] [k]) - |
---|
817 | M (k + 1, 1); |
---|
818 | one++; |
---|
819 | } |
---|
820 | else if (two.exp() == j - k + 1) |
---|
821 | { |
---|
822 | tmp[0] += mulNTL (bufFactors[0] [k], (bufFactors[1] [k] + two.coeff())) - |
---|
823 | M (k + 1, 1); |
---|
824 | two++; |
---|
825 | } |
---|
826 | } |
---|
827 | else |
---|
828 | { |
---|
829 | tmp[0] += M (k + 1, 1); |
---|
830 | } |
---|
831 | } |
---|
832 | } |
---|
833 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
834 | |
---|
835 | // update Pi [l] |
---|
836 | int degPi, degBuf; |
---|
837 | for (int l= 1; l < factors.length() - 1; l++) |
---|
838 | { |
---|
839 | degPi= degree (Pi [l - 1], x); |
---|
840 | degBuf= degree (bufFactors[l + 1], x); |
---|
841 | if (degPi > 0 && degBuf > 0) |
---|
842 | M (j + 1, l + 1)= mulNTL (Pi [l - 1] [j], bufFactors[l + 1] [j]); |
---|
843 | if (j == 1) |
---|
844 | { |
---|
845 | if (degPi > 0 && degBuf > 0) |
---|
846 | Pi [l] += xToJ*(mulNTL (Pi [l - 1] [0] + Pi [l - 1] [j], |
---|
847 | bufFactors[l + 1] [0] + buf[l + 1]) - M (j + 1, l +1) - |
---|
848 | M (1, l + 1)); |
---|
849 | else if (degPi > 0) |
---|
850 | Pi [l] += xToJ*(mulNTL (Pi [l - 1] [j], bufFactors[l + 1])); |
---|
851 | else if (degBuf > 0) |
---|
852 | Pi [l] += xToJ*(mulNTL (Pi [l - 1], buf[l + 1])); |
---|
853 | } |
---|
854 | else |
---|
855 | { |
---|
856 | if (degPi > 0 && degBuf > 0) |
---|
857 | { |
---|
858 | uIZeroJ= mulNTL (uIZeroJ, bufFactors [l + 1] [0]); |
---|
859 | uIZeroJ += mulNTL (Pi [l - 1] [0], buf [l + 1]); |
---|
860 | } |
---|
861 | else if (degPi > 0) |
---|
862 | uIZeroJ= mulNTL (uIZeroJ, bufFactors [l + 1]); |
---|
863 | else if (degBuf > 0) |
---|
864 | { |
---|
865 | uIZeroJ= mulNTL (uIZeroJ, bufFactors [l + 1] [0]); |
---|
866 | uIZeroJ += mulNTL (Pi [l - 1], buf[l + 1]); |
---|
867 | } |
---|
868 | Pi[l] += xToJ*uIZeroJ; |
---|
869 | } |
---|
870 | one= bufFactors [l + 1]; |
---|
871 | two= Pi [l - 1]; |
---|
872 | if (two.exp() == j + 1) |
---|
873 | { |
---|
874 | if (degBuf > 0 && degPi > 0) |
---|
875 | { |
---|
876 | tmp[l] += mulNTL (two.coeff(), bufFactors[l + 1][0]); |
---|
877 | two++; |
---|
878 | } |
---|
879 | else if (degPi > 0) |
---|
880 | { |
---|
881 | tmp[l] += mulNTL (two.coeff(), bufFactors[l + 1]); |
---|
882 | two++; |
---|
883 | } |
---|
884 | } |
---|
885 | if (degBuf > 0 && degPi > 0) |
---|
886 | { |
---|
887 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
888 | { |
---|
889 | if (k != j - k + 1) |
---|
890 | { |
---|
891 | if (one.exp() == j - k + 1 && two.exp() == j - k + 1) |
---|
892 | { |
---|
893 | tmp[l] += mulNTL ((bufFactors[l + 1] [k] + one.coeff()), (Pi[l - 1] [k] + |
---|
894 | two.coeff())) - M (k + 1, l + 1) - M (j - k + 2, l + 1); |
---|
895 | one++; |
---|
896 | two++; |
---|
897 | } |
---|
898 | else if (one.exp() == j - k + 1) |
---|
899 | { |
---|
900 | tmp[l] += mulNTL ((bufFactors[l + 1] [k] + one.coeff()), Pi[l - 1] [k]) - |
---|
901 | M (k + 1, l + 1); |
---|
902 | one++; |
---|
903 | } |
---|
904 | else if (two.exp() == j - k + 1) |
---|
905 | { |
---|
906 | tmp[l] += mulNTL (bufFactors[l + 1] [k], (Pi[l - 1] [k] + two.coeff())) - |
---|
907 | M (k + 1, l + 1); |
---|
908 | two++; |
---|
909 | } |
---|
910 | } |
---|
911 | else |
---|
912 | tmp[l] += M (k + 1, l + 1); |
---|
913 | } |
---|
914 | } |
---|
915 | Pi[l] += tmp[l]*xToJ*F.mvar(); |
---|
916 | } |
---|
917 | return; |
---|
918 | } |
---|
919 | |
---|
920 | void |
---|
921 | henselLift12 (const CanonicalForm& F, CFList& factors, int l, CFArray& Pi, |
---|
922 | CFList& diophant, CFMatrix& M) |
---|
923 | { |
---|
924 | sortList (factors, Variable (1)); |
---|
925 | Pi= CFArray (factors.length() - 1); |
---|
926 | CFListIterator j= factors; |
---|
927 | diophant= diophantine (F[0], factors); |
---|
928 | DEBOUTLN (cerr, "diophant= " << diophant); |
---|
929 | j++; |
---|
930 | Pi [0]= mulNTL (j.getItem(), mod (factors.getFirst(), F.mvar())); |
---|
931 | M (1, 1)= Pi [0]; |
---|
932 | int i= 1; |
---|
933 | if (j.hasItem()) |
---|
934 | j++; |
---|
935 | for (j; j.hasItem(); j++, i++) |
---|
936 | { |
---|
937 | Pi [i]= mulNTL (Pi [i - 1], j.getItem()); |
---|
938 | M (1, i + 1)= Pi [i]; |
---|
939 | } |
---|
940 | CFArray bufFactors= CFArray (factors.length()); |
---|
941 | i= 0; |
---|
942 | for (CFListIterator k= factors; k.hasItem(); i++, k++) |
---|
943 | { |
---|
944 | if (i == 0) |
---|
945 | bufFactors[i]= mod (k.getItem(), F.mvar()); |
---|
946 | else |
---|
947 | bufFactors[i]= k.getItem(); |
---|
948 | } |
---|
949 | for (i= 1; i < l; i++) |
---|
950 | henselStep12 (F, factors, bufFactors, diophant, M, Pi, i); |
---|
951 | |
---|
952 | CFListIterator k= factors; |
---|
953 | for (i= 0; i < factors.length (); i++, k++) |
---|
954 | k.getItem()= bufFactors[i]; |
---|
955 | factors.removeFirst(); |
---|
956 | return; |
---|
957 | } |
---|
958 | |
---|
959 | void |
---|
960 | henselLiftResume12 (const CanonicalForm& F, CFList& factors, int start, int |
---|
961 | end, CFArray& Pi, const CFList& diophant, CFMatrix& M) |
---|
962 | { |
---|
963 | CFArray bufFactors= CFArray (factors.length()); |
---|
964 | int i= 0; |
---|
965 | CanonicalForm xToStart= power (F.mvar(), start); |
---|
966 | for (CFListIterator k= factors; k.hasItem(); k++, i++) |
---|
967 | { |
---|
968 | if (i == 0) |
---|
969 | bufFactors[i]= mod (k.getItem(), xToStart); |
---|
970 | else |
---|
971 | bufFactors[i]= k.getItem(); |
---|
972 | } |
---|
973 | for (i= start; i < end; i++) |
---|
974 | henselStep12 (F, factors, bufFactors, diophant, M, Pi, i); |
---|
975 | |
---|
976 | CFListIterator k= factors; |
---|
977 | for (i= 0; i < factors.length(); k++, i++) |
---|
978 | k.getItem()= bufFactors [i]; |
---|
979 | factors.removeFirst(); |
---|
980 | return; |
---|
981 | } |
---|
982 | |
---|
983 | static inline |
---|
984 | CFList |
---|
985 | biDiophantine (const CanonicalForm& F, const CFList& factors, const int d) |
---|
986 | { |
---|
987 | Variable y= F.mvar(); |
---|
988 | CFList result; |
---|
989 | if (y.level() == 1) |
---|
990 | { |
---|
991 | result= diophantine (F, factors); |
---|
992 | return result; |
---|
993 | } |
---|
994 | else |
---|
995 | { |
---|
996 | CFList buf= factors; |
---|
997 | for (CFListIterator i= buf; i.hasItem(); i++) |
---|
998 | i.getItem()= mod (i.getItem(), y); |
---|
999 | CanonicalForm A= mod (F, y); |
---|
1000 | int bufD= 1; |
---|
1001 | CFList recResult= biDiophantine (A, buf, bufD); |
---|
1002 | CanonicalForm e= 1; |
---|
1003 | CFList p; |
---|
1004 | CFArray bufFactors= CFArray (factors.length()); |
---|
1005 | CanonicalForm yToD= power (y, d); |
---|
1006 | int k= 0; |
---|
1007 | for (CFListIterator i= factors; i.hasItem(); i++, k++) |
---|
1008 | { |
---|
1009 | bufFactors [k]= i.getItem(); |
---|
1010 | } |
---|
1011 | CanonicalForm b; |
---|
1012 | for (k= 0; k < factors.length(); k++) //TODO compute b's faster |
---|
1013 | { |
---|
1014 | b= 1; |
---|
1015 | if (fdivides (bufFactors[k], F)) |
---|
1016 | b= F/bufFactors[k]; |
---|
1017 | else |
---|
1018 | { |
---|
1019 | for (int l= 0; l < factors.length(); l++) |
---|
1020 | { |
---|
1021 | if (l == k) |
---|
1022 | continue; |
---|
1023 | else |
---|
1024 | { |
---|
1025 | b= mulMod2 (b, bufFactors[l], yToD); |
---|
1026 | } |
---|
1027 | } |
---|
1028 | } |
---|
1029 | p.append (b); |
---|
1030 | } |
---|
1031 | |
---|
1032 | CFListIterator j= p; |
---|
1033 | for (CFListIterator i= recResult; i.hasItem(); i++, j++) |
---|
1034 | e -= i.getItem()*j.getItem(); |
---|
1035 | |
---|
1036 | if (e.isZero()) |
---|
1037 | return recResult; |
---|
1038 | CanonicalForm coeffE; |
---|
1039 | CFList s; |
---|
1040 | result= recResult; |
---|
1041 | CanonicalForm g; |
---|
1042 | for (int i= 1; i < d; i++) |
---|
1043 | { |
---|
1044 | if (degree (e, y) > 0) |
---|
1045 | coeffE= e[i]; |
---|
1046 | else |
---|
1047 | coeffE= 0; |
---|
1048 | if (!coeffE.isZero()) |
---|
1049 | { |
---|
1050 | CFListIterator k= result; |
---|
1051 | CFListIterator l= p; |
---|
1052 | int ii= 0; |
---|
1053 | j= recResult; |
---|
1054 | for (; j.hasItem(); j++, k++, l++, ii++) |
---|
1055 | { |
---|
1056 | g= coeffE*j.getItem(); |
---|
1057 | if (degree (bufFactors[ii], y) <= 0) |
---|
1058 | g= mod (g, bufFactors[ii]); |
---|
1059 | else |
---|
1060 | g= mod (g, bufFactors[ii][0]); |
---|
1061 | k.getItem() += g*power (y, i); |
---|
1062 | e -= mulMod2 (g*power(y, i), l.getItem(), yToD); |
---|
1063 | DEBOUTLN (cerr, "mod (e, power (y, i + 1))= " << |
---|
1064 | mod (e, power (y, i + 1))); |
---|
1065 | } |
---|
1066 | } |
---|
1067 | if (e.isZero()) |
---|
1068 | break; |
---|
1069 | } |
---|
1070 | |
---|
1071 | DEBOUTLN (cerr, "mod (e, y)= " << mod (e, y)); |
---|
1072 | |
---|
1073 | #ifdef DEBUGOUTPUT |
---|
1074 | CanonicalForm test= 0; |
---|
1075 | j= p; |
---|
1076 | for (CFListIterator i= result; i.hasItem(); i++, j++) |
---|
1077 | test += mod (i.getItem()*j.getItem(), power (y, d)); |
---|
1078 | DEBOUTLN (cerr, "test= " << test); |
---|
1079 | #endif |
---|
1080 | return result; |
---|
1081 | } |
---|
1082 | } |
---|
1083 | |
---|
1084 | static inline |
---|
1085 | CFList |
---|
1086 | multiRecDiophantine (const CanonicalForm& F, const CFList& factors, |
---|
1087 | const CFList& recResult, const CFList& M, const int d) |
---|
1088 | { |
---|
1089 | Variable y= F.mvar(); |
---|
1090 | CFList result; |
---|
1091 | CFListIterator i; |
---|
1092 | CanonicalForm e= 1; |
---|
1093 | CFListIterator j= factors; |
---|
1094 | CFList p; |
---|
1095 | CFArray bufFactors= CFArray (factors.length()); |
---|
1096 | CanonicalForm yToD= power (y, d); |
---|
1097 | int k= 0; |
---|
1098 | for (CFListIterator i= factors; i.hasItem(); i++, k++) |
---|
1099 | bufFactors [k]= i.getItem(); |
---|
1100 | CanonicalForm b; |
---|
1101 | CFList buf= M; |
---|
1102 | buf.removeLast(); |
---|
1103 | buf.append (yToD); |
---|
1104 | for (k= 0; k < factors.length(); k++) //TODO compute b's faster |
---|
1105 | { |
---|
1106 | b= 1; |
---|
1107 | if (fdivides (bufFactors[k], F)) |
---|
1108 | b= F/bufFactors[k]; |
---|
1109 | else |
---|
1110 | { |
---|
1111 | for (int l= 0; l < factors.length(); l++) |
---|
1112 | { |
---|
1113 | if (l == k) |
---|
1114 | continue; |
---|
1115 | else |
---|
1116 | { |
---|
1117 | b= mulMod (b, bufFactors[l], buf); |
---|
1118 | } |
---|
1119 | } |
---|
1120 | } |
---|
1121 | p.append (b); |
---|
1122 | } |
---|
1123 | j= p; |
---|
1124 | for (CFListIterator i= recResult; i.hasItem(); i++, j++) |
---|
1125 | e -= mulMod (i.getItem(), j.getItem(), M); |
---|
1126 | |
---|
1127 | if (e.isZero()) |
---|
1128 | return recResult; |
---|
1129 | CanonicalForm coeffE; |
---|
1130 | CFList s; |
---|
1131 | result= recResult; |
---|
1132 | CanonicalForm g; |
---|
1133 | for (int i= 1; i < d; i++) |
---|
1134 | { |
---|
1135 | if (degree (e, y) > 0) |
---|
1136 | coeffE= e[i]; |
---|
1137 | else |
---|
1138 | coeffE= 0; |
---|
1139 | if (!coeffE.isZero()) |
---|
1140 | { |
---|
1141 | CFListIterator k= result; |
---|
1142 | CFListIterator l= p; |
---|
1143 | j= recResult; |
---|
1144 | int ii= 0; |
---|
1145 | CanonicalForm dummy; |
---|
1146 | for (; j.hasItem(); j++, k++, l++, ii++) |
---|
1147 | { |
---|
1148 | g= mulMod (coeffE, j.getItem(), M); |
---|
1149 | if (degree (bufFactors[ii], y) <= 0) |
---|
1150 | divrem (g, mod (bufFactors[ii], Variable (y.level() - 1)), dummy, |
---|
1151 | g, M); |
---|
1152 | else |
---|
1153 | divrem (g, bufFactors[ii][0], dummy, g, M); |
---|
1154 | k.getItem() += g*power (y, i); |
---|
1155 | e -= mulMod (g*power (y, i), l.getItem(), M); |
---|
1156 | } |
---|
1157 | } |
---|
1158 | |
---|
1159 | if (e.isZero()) |
---|
1160 | break; |
---|
1161 | } |
---|
1162 | |
---|
1163 | #ifdef DEBUGOUTPUT |
---|
1164 | CanonicalForm test= 0; |
---|
1165 | j= p; |
---|
1166 | for (CFListIterator i= result; i.hasItem(); i++, j++) |
---|
1167 | test += mod (i.getItem()*j.getItem(), power (y, d)); |
---|
1168 | DEBOUTLN (cerr, "test= " << test); |
---|
1169 | #endif |
---|
1170 | return result; |
---|
1171 | } |
---|
1172 | |
---|
1173 | static inline |
---|
1174 | void |
---|
1175 | henselStep (const CanonicalForm& F, const CFList& factors, CFArray& bufFactors, |
---|
1176 | const CFList& diophant, CFMatrix& M, CFArray& Pi, int j, |
---|
1177 | const CFList& MOD) |
---|
1178 | { |
---|
1179 | CanonicalForm E; |
---|
1180 | CanonicalForm xToJ= power (F.mvar(), j); |
---|
1181 | Variable x= F.mvar(); |
---|
1182 | // compute the error |
---|
1183 | if (j == 1) |
---|
1184 | { |
---|
1185 | E= F[j]; |
---|
1186 | #ifdef DEBUGOUTPUT |
---|
1187 | CanonicalForm test= 1; |
---|
1188 | for (int i= 0; i < factors.length(); i++) |
---|
1189 | { |
---|
1190 | if (i == 0) |
---|
1191 | test= mulMod (test, mod (bufFactors [i], xToJ), MOD); |
---|
1192 | else |
---|
1193 | test= mulMod (test, bufFactors[i], MOD); |
---|
1194 | } |
---|
1195 | CanonicalForm test2= mod (F-test, xToJ); |
---|
1196 | |
---|
1197 | test2= mod (test2, MOD); |
---|
1198 | DEBOUTLN (cerr, "test= " << test2); |
---|
1199 | #endif |
---|
1200 | } |
---|
1201 | else |
---|
1202 | { |
---|
1203 | #ifdef DEBUGOUTPUT |
---|
1204 | CanonicalForm test= 1; |
---|
1205 | for (int i= 0; i < factors.length(); i++) |
---|
1206 | { |
---|
1207 | if (i == 0) |
---|
1208 | test *= mod (bufFactors [i], power (x, j)); |
---|
1209 | else |
---|
1210 | test *= bufFactors[i]; |
---|
1211 | } |
---|
1212 | test= mod (test, power (x, j)); |
---|
1213 | test= mod (test, MOD); |
---|
1214 | CanonicalForm test2= mod (F, power (x, j - 1)) - mod (test, power (x, j-1)); |
---|
1215 | DEBOUTLN (cerr, "test= " << test2); |
---|
1216 | #endif |
---|
1217 | |
---|
1218 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
1219 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
1220 | else |
---|
1221 | E= F[j]; |
---|
1222 | } |
---|
1223 | |
---|
1224 | CFArray buf= CFArray (diophant.length()); |
---|
1225 | bufFactors[0]= mod (factors.getFirst(), power (F.mvar(), j + 1)); |
---|
1226 | int k= 0; |
---|
1227 | // actual lifting |
---|
1228 | CanonicalForm dummy, rest1; |
---|
1229 | for (CFListIterator i= diophant; i.hasItem(); i++, k++) |
---|
1230 | { |
---|
1231 | if (degree (bufFactors[k], x) > 0) |
---|
1232 | { |
---|
1233 | if (k > 0) |
---|
1234 | divrem (E, bufFactors[k] [0], dummy, rest1, MOD); |
---|
1235 | else |
---|
1236 | rest1= E; |
---|
1237 | } |
---|
1238 | else |
---|
1239 | divrem (E, bufFactors[k], dummy, rest1, MOD); |
---|
1240 | |
---|
1241 | buf[k]= mulMod (i.getItem(), rest1, MOD); |
---|
1242 | |
---|
1243 | if (degree (bufFactors[k], x) > 0) |
---|
1244 | divrem (buf[k], bufFactors[k] [0], dummy, buf[k], MOD); |
---|
1245 | else |
---|
1246 | divrem (buf[k], bufFactors[k], dummy, buf[k], MOD); |
---|
1247 | } |
---|
1248 | for (k= 1; k < factors.length(); k++) |
---|
1249 | bufFactors[k] += xToJ*buf[k]; |
---|
1250 | |
---|
1251 | // update Pi [0] |
---|
1252 | int degBuf0= degree (bufFactors[0], x); |
---|
1253 | int degBuf1= degree (bufFactors[1], x); |
---|
1254 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1255 | M (j + 1, 1)= mulMod (bufFactors[0] [j], bufFactors[1] [j], MOD); |
---|
1256 | CanonicalForm uIZeroJ; |
---|
1257 | if (j == 1) |
---|
1258 | { |
---|
1259 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1260 | uIZeroJ= mulMod ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
1261 | (bufFactors[1] [0] + buf[1]), MOD) - M(1, 1) - M(j + 1, 1); |
---|
1262 | else if (degBuf0 > 0) |
---|
1263 | uIZeroJ= mulMod (bufFactors[0] [j], bufFactors[1], MOD); |
---|
1264 | else if (degBuf1 > 0) |
---|
1265 | uIZeroJ= mulMod (bufFactors[0], buf[1], MOD); |
---|
1266 | else |
---|
1267 | uIZeroJ= 0; |
---|
1268 | Pi [0] += xToJ*uIZeroJ; |
---|
1269 | } |
---|
1270 | else |
---|
1271 | { |
---|
1272 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1273 | uIZeroJ= mulMod ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
1274 | (bufFactors[1] [0] + buf[1]), MOD) - M(1, 1) - M(j + 1, 1); |
---|
1275 | else if (degBuf0 > 0) |
---|
1276 | uIZeroJ= mulMod (bufFactors[0] [j], bufFactors[1], MOD); |
---|
1277 | else if (degBuf1 > 0) |
---|
1278 | uIZeroJ= mulMod (bufFactors[0], buf[1], MOD); |
---|
1279 | else |
---|
1280 | uIZeroJ= 0; |
---|
1281 | Pi [0] += xToJ*uIZeroJ; |
---|
1282 | } |
---|
1283 | |
---|
1284 | CFArray tmp= CFArray (factors.length() - 1); |
---|
1285 | for (k= 0; k < factors.length() - 1; k++) |
---|
1286 | tmp[k]= 0; |
---|
1287 | CFIterator one, two; |
---|
1288 | one= bufFactors [0]; |
---|
1289 | two= bufFactors [1]; |
---|
1290 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1291 | { |
---|
1292 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
1293 | { |
---|
1294 | if (k != j - k + 1) |
---|
1295 | { |
---|
1296 | if (one.exp() == j - k + 1 && two.exp() == j - k + 1) |
---|
1297 | { |
---|
1298 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
1299 | (bufFactors[1] [k] + two.coeff()), MOD) - M (k + 1, 1) - |
---|
1300 | M (j - k + 2, 1); |
---|
1301 | one++; |
---|
1302 | two++; |
---|
1303 | } |
---|
1304 | else if (one.exp() == j - k + 1) |
---|
1305 | { |
---|
1306 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
1307 | bufFactors[1] [k], MOD) - M (k + 1, 1); |
---|
1308 | one++; |
---|
1309 | } |
---|
1310 | else if (two.exp() == j - k + 1) |
---|
1311 | { |
---|
1312 | tmp[0] += mulMod (bufFactors[0] [k], (bufFactors[1] [k] + |
---|
1313 | two.coeff()), MOD) - M (k + 1, 1); |
---|
1314 | two++; |
---|
1315 | } |
---|
1316 | } |
---|
1317 | else |
---|
1318 | { |
---|
1319 | tmp[0] += M (k + 1, 1); |
---|
1320 | } |
---|
1321 | } |
---|
1322 | } |
---|
1323 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
1324 | |
---|
1325 | // update Pi [l] |
---|
1326 | int degPi, degBuf; |
---|
1327 | for (int l= 1; l < factors.length() - 1; l++) |
---|
1328 | { |
---|
1329 | degPi= degree (Pi [l - 1], x); |
---|
1330 | degBuf= degree (bufFactors[l + 1], x); |
---|
1331 | if (degPi > 0 && degBuf > 0) |
---|
1332 | M (j + 1, l + 1)= mulMod (Pi [l - 1] [j], bufFactors[l + 1] [j], MOD); |
---|
1333 | if (j == 1) |
---|
1334 | { |
---|
1335 | if (degPi > 0 && degBuf > 0) |
---|
1336 | Pi [l] += xToJ*(mulMod ((Pi [l - 1] [0] + Pi [l - 1] [j]), |
---|
1337 | (bufFactors[l + 1] [0] + buf[l + 1]), MOD) - M (j + 1, l +1)- |
---|
1338 | M (1, l + 1)); |
---|
1339 | else if (degPi > 0) |
---|
1340 | Pi [l] += xToJ*(mulMod (Pi [l - 1] [j], bufFactors[l + 1], MOD)); |
---|
1341 | else if (degBuf > 0) |
---|
1342 | Pi [l] += xToJ*(mulMod (Pi [l - 1], buf[l + 1], MOD)); |
---|
1343 | } |
---|
1344 | else |
---|
1345 | { |
---|
1346 | if (degPi > 0 && degBuf > 0) |
---|
1347 | { |
---|
1348 | uIZeroJ= mulMod (uIZeroJ, bufFactors [l + 1] [0], MOD); |
---|
1349 | uIZeroJ += mulMod (Pi [l - 1] [0], buf [l + 1], MOD); |
---|
1350 | } |
---|
1351 | else if (degPi > 0) |
---|
1352 | uIZeroJ= mulMod (uIZeroJ, bufFactors [l + 1], MOD); |
---|
1353 | else if (degBuf > 0) |
---|
1354 | { |
---|
1355 | uIZeroJ= mulMod (uIZeroJ, bufFactors [l + 1] [0], MOD); |
---|
1356 | uIZeroJ += mulMod (Pi [l - 1], buf[l + 1], MOD); |
---|
1357 | } |
---|
1358 | Pi[l] += xToJ*uIZeroJ; |
---|
1359 | } |
---|
1360 | one= bufFactors [l + 1]; |
---|
1361 | two= Pi [l - 1]; |
---|
1362 | if (two.exp() == j + 1) |
---|
1363 | { |
---|
1364 | if (degBuf > 0 && degPi > 0) |
---|
1365 | { |
---|
1366 | tmp[l] += mulMod (two.coeff(), bufFactors[l + 1][0], MOD); |
---|
1367 | two++; |
---|
1368 | } |
---|
1369 | else if (degPi > 0) |
---|
1370 | { |
---|
1371 | tmp[l] += mulMod (two.coeff(), bufFactors[l + 1], MOD); |
---|
1372 | two++; |
---|
1373 | } |
---|
1374 | } |
---|
1375 | if (degBuf > 0 && degPi > 0) |
---|
1376 | { |
---|
1377 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
1378 | { |
---|
1379 | if (k != j - k + 1) |
---|
1380 | { |
---|
1381 | if (one.exp() == j - k + 1 && two.exp() == j - k + 1) |
---|
1382 | { |
---|
1383 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
1384 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1) - |
---|
1385 | M (j - k + 2, l + 1); |
---|
1386 | one++; |
---|
1387 | two++; |
---|
1388 | } |
---|
1389 | else if (one.exp() == j - k + 1) |
---|
1390 | { |
---|
1391 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
1392 | Pi[l - 1] [k], MOD) - M (k + 1, l + 1); |
---|
1393 | one++; |
---|
1394 | } |
---|
1395 | else if (two.exp() == j - k + 1) |
---|
1396 | { |
---|
1397 | tmp[l] += mulMod (bufFactors[l + 1] [k], |
---|
1398 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1); |
---|
1399 | two++; |
---|
1400 | } |
---|
1401 | } |
---|
1402 | else |
---|
1403 | tmp[l] += M (k + 1, l + 1); |
---|
1404 | } |
---|
1405 | } |
---|
1406 | Pi[l] += tmp[l]*xToJ*F.mvar(); |
---|
1407 | } |
---|
1408 | |
---|
1409 | return; |
---|
1410 | } |
---|
1411 | |
---|
1412 | CFList |
---|
1413 | henselLift23 (const CFList& eval, const CFList& factors, const int* l, CFList& |
---|
1414 | diophant, CFArray& Pi, CFMatrix& M) |
---|
1415 | { |
---|
1416 | CFList buf= factors; |
---|
1417 | int k= 0; |
---|
1418 | int liftBound; |
---|
1419 | int liftBoundBivar= l[k]; |
---|
1420 | diophant= biDiophantine (eval.getFirst(), buf, liftBoundBivar); |
---|
1421 | CFList MOD; |
---|
1422 | MOD.append (power (Variable (2), liftBoundBivar)); |
---|
1423 | CFArray bufFactors= CFArray (factors.length()); |
---|
1424 | k= 0; |
---|
1425 | CFListIterator j= eval; |
---|
1426 | j++; |
---|
1427 | buf.removeFirst(); |
---|
1428 | buf.insert (LC (j.getItem(), 1)); |
---|
1429 | for (CFListIterator i= buf; i.hasItem(); i++, k++) |
---|
1430 | bufFactors[k]= i.getItem(); |
---|
1431 | Pi= CFArray (factors.length() - 1); |
---|
1432 | CFListIterator i= buf; |
---|
1433 | i++; |
---|
1434 | Variable y= j.getItem().mvar(); |
---|
1435 | Pi [0]= mulMod (i.getItem(), mod (buf.getFirst(), y), MOD); |
---|
1436 | M (1, 1)= Pi [0]; |
---|
1437 | k= 1; |
---|
1438 | if (i.hasItem()) |
---|
1439 | i++; |
---|
1440 | for (i; i.hasItem(); i++, k++) |
---|
1441 | { |
---|
1442 | Pi [k]= mulMod (Pi [k - 1], i.getItem(), MOD); |
---|
1443 | M (1, k + 1)= Pi [k]; |
---|
1444 | } |
---|
1445 | |
---|
1446 | for (int d= 1; d < l[1]; d++) |
---|
1447 | henselStep (j.getItem(), buf, bufFactors, diophant, M, Pi, d, MOD); |
---|
1448 | CFList result; |
---|
1449 | for (k= 1; k < factors.length(); k++) |
---|
1450 | result.append (bufFactors[k]); |
---|
1451 | return result; |
---|
1452 | } |
---|
1453 | |
---|
1454 | void |
---|
1455 | henselLiftResume (const CanonicalForm& F, CFList& factors, int start, int end, |
---|
1456 | CFArray& Pi, const CFList& diophant, CFMatrix& M, |
---|
1457 | const CFList& MOD) |
---|
1458 | { |
---|
1459 | CFArray bufFactors= CFArray (factors.length()); |
---|
1460 | int i= 0; |
---|
1461 | CanonicalForm xToStart= power (F.mvar(), start); |
---|
1462 | for (CFListIterator k= factors; k.hasItem(); k++, i++) |
---|
1463 | { |
---|
1464 | if (i == 0) |
---|
1465 | bufFactors[i]= mod (k.getItem(), xToStart); |
---|
1466 | else |
---|
1467 | bufFactors[i]= k.getItem(); |
---|
1468 | } |
---|
1469 | for (i= start; i < end; i++) |
---|
1470 | henselStep (F, factors, bufFactors, diophant, M, Pi, i, MOD); |
---|
1471 | |
---|
1472 | CFListIterator k= factors; |
---|
1473 | for (i= 0; i < factors.length(); k++, i++) |
---|
1474 | k.getItem()= bufFactors [i]; |
---|
1475 | factors.removeFirst(); |
---|
1476 | return; |
---|
1477 | } |
---|
1478 | |
---|
1479 | CFList |
---|
1480 | henselLift (const CFList& F, const CFList& factors, const CFList& MOD, CFList& |
---|
1481 | diophant, CFArray& Pi, CFMatrix& M, const int lOld, const int |
---|
1482 | lNew) |
---|
1483 | { |
---|
1484 | diophant= multiRecDiophantine (F.getFirst(), factors, diophant, MOD, lOld); |
---|
1485 | int k= 0; |
---|
1486 | CFArray bufFactors= CFArray (factors.length()); |
---|
1487 | for (CFListIterator i= factors; i.hasItem(); i++, k++) |
---|
1488 | { |
---|
1489 | if (k == 0) |
---|
1490 | bufFactors[k]= LC (F.getLast(), 1); |
---|
1491 | else |
---|
1492 | bufFactors[k]= i.getItem(); |
---|
1493 | } |
---|
1494 | CFList buf= factors; |
---|
1495 | buf.removeFirst(); |
---|
1496 | buf.insert (LC (F.getLast(), 1)); |
---|
1497 | CFListIterator i= buf; |
---|
1498 | i++; |
---|
1499 | Variable y= F.getLast().mvar(); |
---|
1500 | Variable x= F.getFirst().mvar(); |
---|
1501 | CanonicalForm xToLOld= power (x, lOld); |
---|
1502 | Pi [0]= mod (Pi[0], xToLOld); |
---|
1503 | M (1, 1)= Pi [0]; |
---|
1504 | k= 1; |
---|
1505 | if (i.hasItem()) |
---|
1506 | i++; |
---|
1507 | for (i; i.hasItem(); i++, k++) |
---|
1508 | { |
---|
1509 | Pi [k]= mod (Pi [k], xToLOld); |
---|
1510 | M (1, k + 1)= Pi [k]; |
---|
1511 | } |
---|
1512 | |
---|
1513 | for (int d= 1; d < lNew; d++) |
---|
1514 | henselStep (F.getLast(), buf, bufFactors, diophant, M, Pi, d, MOD); |
---|
1515 | CFList result; |
---|
1516 | for (k= 1; k < factors.length(); k++) |
---|
1517 | result.append (bufFactors[k]); |
---|
1518 | return result; |
---|
1519 | } |
---|
1520 | |
---|
1521 | CFList |
---|
1522 | henselLift (const CFList& eval, const CFList& factors, const int* l, const int |
---|
1523 | lLength) |
---|
1524 | { |
---|
1525 | CFList diophant; |
---|
1526 | CFList buf= factors; |
---|
1527 | buf.insert (LC (eval.getFirst(), 1)); |
---|
1528 | sortList (buf, Variable (1)); |
---|
1529 | CFArray Pi; |
---|
1530 | CFMatrix M= CFMatrix (l[1], factors.length()); |
---|
1531 | CFList result= henselLift23 (eval, buf, l, diophant, Pi, M); |
---|
1532 | if (eval.length() == 2) |
---|
1533 | return result; |
---|
1534 | CFList MOD; |
---|
1535 | for (int i= 0; i < 2; i++) |
---|
1536 | MOD.append (power (Variable (i + 2), l[i])); |
---|
1537 | CFListIterator j= eval; |
---|
1538 | j++; |
---|
1539 | CFList bufEval; |
---|
1540 | bufEval.append (j.getItem()); |
---|
1541 | j++; |
---|
1542 | |
---|
1543 | for (int i= 2; i <= lLength && j.hasItem(); i++, j++) |
---|
1544 | { |
---|
1545 | result.insert (LC (bufEval.getFirst(), 1)); |
---|
1546 | bufEval.append (j.getItem()); |
---|
1547 | M= CFMatrix (l[i], factors.length()); |
---|
1548 | result= henselLift (bufEval, result, MOD, diophant, Pi, M, l[i - 1], l[i]); |
---|
1549 | MOD.append (power (Variable (i + 2), l[i])); |
---|
1550 | bufEval.removeFirst(); |
---|
1551 | } |
---|
1552 | return result; |
---|
1553 | } |
---|
1554 | |
---|
1555 | #endif |
---|
1556 | /* HAVE_NTL */ |
---|
1557 | |
---|