1 | /*****************************************************************************\ |
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2 | * Computer Algebra System SINGULAR |
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3 | \*****************************************************************************/ |
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4 | /** @file facHensel.h |
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5 | * |
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6 | * This file defines functions for Hensel lifting and modular multiplication. |
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7 | * |
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8 | * ABSTRACT: function are used for bi- and multivariate factorization over |
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9 | * finite fields |
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10 | * |
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11 | * @author Martin Lee |
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12 | * |
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13 | * @internal @version \$Id$ |
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14 | * |
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15 | **/ |
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16 | /*****************************************************************************/ |
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17 | |
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18 | #ifndef FAC_HENSEL_H |
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19 | #define FAC_HENSEL_H |
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20 | |
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21 | #include <config.h> |
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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 "canonicalform.h" |
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27 | #include "cf_iter.h" |
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28 | #include "templates/ftmpl_functions.h" |
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29 | #include "algext.h" |
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30 | |
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31 | /// multiplication of univariate polys over a finite field using NTL, if we are |
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32 | /// in GF factory's default multiplication is used. |
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33 | /// |
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34 | /// @return @a mulNTL returns F*G |
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35 | CanonicalForm |
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36 | mulNTL (const CanonicalForm& F, ///< [in] a univariate poly |
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37 | const CanonicalForm& G ///< [in] a univariate poly |
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38 | ); |
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39 | |
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40 | /// mod of univariate polys over a finite field using NTL, if we are |
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41 | /// in GF factory's default mod is used. |
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42 | /// |
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43 | /// @return @a modNTL returns F mod G |
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44 | CanonicalForm |
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45 | modNTL (const CanonicalForm& F, ///< [in] a univariate poly |
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46 | const CanonicalForm& G ///< [in] a univariate poly |
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47 | ); |
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48 | |
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49 | /// division of univariate polys over a finite field using NTL, if we are |
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50 | /// in GF factory's default division is used. |
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51 | /// |
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52 | /// @return @a divNTL returns F/G |
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53 | CanonicalForm |
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54 | divNTL (const CanonicalForm& F, ///< [in] a univariate poly |
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55 | const CanonicalForm& G ///< [in] a univariate poly |
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56 | ); |
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57 | |
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58 | /*/// division with remainder of univariate polys over a finite field using NTL, |
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59 | /// if we are in GF factory's default division with remainder is used. |
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60 | void |
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61 | divremNTL (const CanonicalForm& F, ///< [in] a univariate poly |
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62 | const CanonicalForm& G, ///< [in] a univariate poly |
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63 | CanonicalForm& Q, ///< [in,out] dividend |
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64 | CanonicalForm& R ///< [in,out] remainder |
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65 | );*/ |
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66 | |
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67 | /// splits @a F into degree (F, x)/m polynomials each of degree less than @a m |
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68 | /// in @a x. |
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69 | /// |
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70 | /// @return @a split returns a list of polynomials of degree less than @a m in |
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71 | /// @a x. If degree (F, x) <= 0, F is returned. |
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72 | /// @sa divrem32(), divrem21() |
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73 | static inline |
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74 | CFList split (const CanonicalForm& F, ///< [in] some poly |
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75 | const int m, ///< [in] some int |
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76 | const Variable& x ///< [in] some Variable |
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77 | ); |
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78 | |
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79 | /// division with remainder of @a F by |
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80 | /// @a G wrt Variable (1) modulo @a M. |
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81 | /// |
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82 | /// @sa divrem(), divrem21(), divrem2() |
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83 | static inline |
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84 | void divrem32 (const CanonicalForm& F, ///< [in] poly, s.t. 3*(degree (G, 1)/2)> |
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85 | ///< degree (F, 1) |
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86 | const CanonicalForm& G, ///< [in] some poly |
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87 | CanonicalForm& Q, ///< [in,out] dividend |
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88 | CanonicalForm& R, ///< [in,out] remainder, degree (R, 1) < |
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89 | ///< degree (G, 1) |
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90 | const CFList& M ///< [in] only contains powers of |
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91 | ///< Variables of level higher than 1 |
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92 | ); |
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93 | |
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94 | /// division with remainder of @a F by |
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95 | /// @a G wrt Variable (1) modulo @a M. |
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96 | /// |
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97 | /// @sa divrem(), divrem32(), divrem2() |
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98 | static inline |
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99 | void divrem21 (const CanonicalForm& F, ///< [in] poly, s.t. 2*degree (G, 1) > |
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100 | ///< degree (F, 1) |
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101 | const CanonicalForm& G, ///< [in] some poly |
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102 | CanonicalForm& Q, ///< [in,out] dividend |
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103 | CanonicalForm& R, ///< [in,out] remainder, degree (R, 1) < |
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104 | ///< degree (G, 1) |
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105 | const CFList& M ///< [in] only contains powers of |
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106 | ///< Variables of level higher than 1 |
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107 | ); |
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108 | |
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109 | /// division with remainder of @a F by |
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110 | /// @a G wrt Variable (1) modulo @a M. |
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111 | /// |
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112 | /// @return @a Q returns the dividend, @a R returns the remainder. |
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113 | /// @sa divrem() |
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114 | void divrem2 (const CanonicalForm& F, ///< [in] bivariate, compressed polynomial |
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115 | const CanonicalForm& G, ///< [in] bivariate, compressed polynomial |
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116 | CanonicalForm& Q, ///< [in,out] dividend |
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117 | CanonicalForm& R, ///< [in,out] remainder, degree (R, 1) < |
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118 | ///< degree (G, 1) |
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119 | const CanonicalForm& M ///< [in] power of Variable (2) |
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120 | ); |
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121 | |
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122 | /// division with remainder of @a F by |
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123 | /// @a G wrt Variable (1) modulo @a MOD. |
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124 | /// |
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125 | /// @sa divrem2() |
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126 | void divrem ( |
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127 | const CanonicalForm& F, ///< [in] multivariate, compressed polynomial |
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128 | const CanonicalForm& G, ///< [in] multivariate, compressed polynomial |
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129 | CanonicalForm& Q, ///< [in,out] dividend |
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130 | CanonicalForm& R, ///< [in,out] remainder, degree (R, 1) < |
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131 | ///< degree (G, 1) |
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132 | const CFList& MOD ///< [in] only contains powers of |
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133 | ///< Variables of level higher than 1 |
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134 | ); |
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135 | |
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136 | |
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137 | /// division with remainder of @a F by |
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138 | /// @a G wrt Variable (1) modulo @a M using Newton inversion |
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139 | /// |
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140 | /// @return @a Q returns the dividend, @a R returns the remainder. |
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141 | /// @sa divrem2(), newtonDiv() |
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142 | void |
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143 | newtonDivrem (const CanonicalForm& F, ///< [in] bivariate, compressed polynomial |
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144 | const CanonicalForm& G, ///< [in] bivariate, compressed polynomial |
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145 | ///< which is monic in Variable (1) |
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146 | CanonicalForm& Q, ///< [in,out] dividend |
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147 | CanonicalForm& R, ///< [in,out] remainder, degree (R, 1) < |
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148 | ///< degree (G, 1) |
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149 | const CanonicalForm& M ///< [in] power of Variable (2) |
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150 | ); |
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151 | |
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152 | /// division of @a F by |
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153 | /// @a G wrt Variable (1) modulo @a M using Newton inversion |
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154 | /// |
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155 | /// @return @a newtonDiv returns the dividend |
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156 | /// @sa divrem2(), newtonDivrem() |
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157 | CanonicalForm |
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158 | newtonDiv (const CanonicalForm& F, ///< [in] bivariate, compressed polynomial |
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159 | const CanonicalForm& G, ///< [in] bivariate, compressed polynomial |
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160 | ///< which is monic in Variable (1) |
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161 | const CanonicalForm& M ///< [in] power of Variable (2) |
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162 | ); |
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163 | |
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164 | /// reduce @a F modulo elements in @a M. |
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165 | /// |
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166 | /// @return @a mod returns @a F modulo @a M |
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167 | CanonicalForm mod (const CanonicalForm& F, ///< [in] compressed polynomial |
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168 | const CFList& M ///< [in] list containing only |
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169 | ///< univariate polynomials |
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170 | ); |
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171 | |
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172 | /// Karatsuba style modular multiplication for bivariate polynomials. |
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173 | /// |
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174 | /// @return @a mulMod2 returns @a A * @a B mod @a M. |
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175 | CanonicalForm |
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176 | mulMod2 (const CanonicalForm& A, ///< [in] bivariate, compressed polynomial |
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177 | const CanonicalForm& B, ///< [in] bivariate, compressed polynomial |
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178 | const CanonicalForm& M ///< [in] power of Variable (2) |
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179 | ); |
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180 | |
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181 | /// Karatsuba style modular multiplication for multivariate polynomials. |
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182 | /// |
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183 | /// @return @a mulMod2 returns @a A * @a B mod @a MOD. |
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184 | CanonicalForm |
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185 | mulMod (const CanonicalForm& A, ///< [in] multivariate, compressed polynomial |
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186 | const CanonicalForm& B, ///< [in] multivariate, compressed polynomial |
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187 | const CFList& MOD ///< [in] only contains powers of |
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188 | ///< Variables of level higher than 1 |
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189 | ); |
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190 | |
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191 | /// product of all elements in @a L modulo @a M via divide-and-conquer. |
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192 | /// |
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193 | /// @return @a prodMod returns product of all elements in @a L modulo @a M. |
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194 | CanonicalForm |
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195 | prodMod (const CFList& L, ///< [in] contains only bivariate, compressed |
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196 | ///< polynomials |
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197 | const CanonicalForm& M ///< [in] power of Variable (2) |
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198 | ); |
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199 | |
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200 | /// product of all elements in @a L modulo @a M via divide-and-conquer. |
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201 | /// |
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202 | /// @return @a prodMod returns product of all elements in @a L modulo @a M. |
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203 | CanonicalForm |
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204 | prodMod (const CFList& L, ///< [in] contains multivariate, compressed |
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205 | ///< polynomials |
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206 | const CFList& M ///< [in] contains only powers of Variables |
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207 | ); |
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208 | |
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209 | /// sort a list of polynomials by their degree in @a x. |
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210 | /// |
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211 | void sortList (CFList& list, ///< [in, out] list of polys, sorted list |
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212 | const Variable& x ///< [in] some Variable |
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213 | ); |
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214 | |
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215 | /// solve the univariate diophantine equation |
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216 | /// \f$ 1\equiv \sum_{i= 1}^{r} {\delta_{i} F/g_{i}} \f$. |
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217 | /// Where \f$ F= \prod_{i= 1}^{r} {g_{i}} \f$ and \f$ F \f$ is squarefree |
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218 | /// the \f$ \delta_{i} \f$ have degree less than the degree of \f$ g_{i} \f$. |
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219 | /// |
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220 | /// @return @a diophantine returns a list of polynomials \f$ \delta_{i} \f$ as |
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221 | /// specified above |
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222 | /// @sa biDiophantine(), multiRecDiophantine() |
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223 | static inline |
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224 | CFList diophantine (const CanonicalForm& F, ///< [in] compressed, bivariate |
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225 | ///< polynomial |
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226 | const CFList& factors ///< [in] a list of factors, as |
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227 | ///< specified above, including |
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228 | ///< LC (F, Variable (1)) as first |
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229 | ///< element |
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230 | ); |
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231 | |
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232 | /// Hensel lift from univariate to bivariate. |
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233 | /// |
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234 | /// @sa henselLiftResume12(), henselLift23(), henselLiftResume(), henselLift() |
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235 | void |
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236 | henselLift12 (const CanonicalForm& F, ///< [in] compressed, bivariate poly |
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237 | CFList& factors, ///< [in, out] monic univariate factors of |
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238 | ///< F, including leading coefficient as |
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239 | ///< first element. Returns monic lifted |
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240 | ///< factors without the leading |
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241 | ///< coefficient |
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242 | int l, ///< [in] lifting precision |
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243 | CFArray& Pi, ///< [in,out] stores intermediate results |
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244 | CFList& diophant, ///< [in,out] result of diophantine() |
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245 | CFMatrix& M, ///< [in,out] stores intermediate results |
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246 | bool sort= true ///< [in] sort factors by degree in |
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247 | ///< Variable(1) |
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248 | ); |
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249 | |
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250 | /// resume Hensel lift from univariate to bivariate. Assumes factors are lifted |
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251 | /// to precision Variable (2)^start and lifts them to precision Variable (2)^end |
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252 | /// |
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253 | /// @sa henselLift12(), henselLift23(), henselLiftResume(), henselLift() |
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254 | void |
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255 | henselLiftResume12 (const CanonicalForm& F, ///< [in] compressed, bivariate poly |
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256 | CFList& factors, ///< [in,out] monic factors of F, |
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257 | ///< lifted to precision start, |
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258 | ///< including leading coefficient |
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259 | ///< as first element. Returns monic |
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260 | ///< lifted factors without the |
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261 | ///< leading coefficient |
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262 | int start, ///< [in] starting precision |
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263 | int end, ///< [in] end precision |
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264 | CFArray& Pi, ///< [in,out] stores intermediate |
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265 | ///< results |
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266 | const CFList& diophant, ///< [in] result of diophantine |
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267 | CFMatrix& M ///< [in,out] stores intermediate |
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268 | ///< results |
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269 | ); |
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270 | |
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271 | /// solves the bivariate polynomial diophantine equation |
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272 | /// \f$ 1\equiv \sum_{i= 1}^{r} {\delta_{i} F/g_{i}} \ mod\ y^{d} \f$, |
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273 | /// where \f$ F= \prod_{i= 1}^{r} {g_{i}} \ mod\ y^{d}\f$ and |
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274 | /// \f$ F \in K[x][y]\f$ is squarefree, the \f$ \delta_{i} \f$ have degree less |
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275 | /// than the degree of \f$ g_{i} \f$ in x. |
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276 | /// |
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277 | /// @return @a biDiophantine returns a list of polynomials \f$ \delta_{i} \f$ as |
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278 | /// specified above |
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279 | /// @sa diophantine(), multiRecDiophantine() |
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280 | static inline |
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281 | CFList |
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282 | biDiophantine (const CanonicalForm& F, ///< [in] compressed, bivariate poly |
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283 | const CFList& factors, ///< [in] list of monic bivariate factors |
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284 | ///< including LC (F, Variable (1)) as |
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285 | ///< first element |
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286 | const int d ///< [in] precision |
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287 | ); |
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288 | |
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289 | /// solve the multivariate polynomial diophantine equation |
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290 | /// \f$ 1\equiv \sum_{i= 1}^{r} {\delta_{i} F/g_{i}} \ mod\ <M,F.mvar()^{d}>\f$, |
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291 | /// where \f$ F= \prod_{i= 1}^{r} {g_{i}} \ mod\ <M,F.mvar()^{d}>\f$ and |
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292 | /// \f$ F \in K[x][x_1,\ldots , x_n]\f$ is squarefree, the \f$ \delta_{i} \f$ |
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293 | /// have degree less than the degree of \f$ g_{i} \f$ in x. |
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294 | /// |
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295 | /// @return @a multiDiophantine returns a list of polynomials \f$ \delta_{i} \f$ |
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296 | /// as specified above |
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297 | /// @sa diophantine(), biDiophantine() |
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298 | static inline |
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299 | CFList |
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300 | multiRecDiophantine ( |
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301 | const CanonicalForm& F, ///< [in] compressed, |
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302 | ///< multivariate polynomial |
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303 | const CFList& factors, ///< [in] list of monic factors |
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304 | ///< including LC (F, Variable (1)) as |
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305 | ///< first element |
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306 | const CFList& recResult, ///< [in] result of above equation |
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307 | ///< modulo M |
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308 | const CFList& M, ///< [in] a list of powers of Variables |
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309 | ///< of level higher than 1 |
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310 | const int d ///< [in] precision |
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311 | ); |
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312 | |
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313 | /// Hensel lifting from bivariate to trivariate. |
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314 | /// |
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315 | /// @return @a henselLift23 returns a list of polynomials lifted to precision |
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316 | /// Variable (3)^l[1] |
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317 | /// @sa henselLift12(), henselLiftResume12(), henselLiftResume(), henselLift() |
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318 | CFList |
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319 | henselLift23 (const CFList& eval, ///< [in] contains compressed, bivariate |
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320 | ///< as first element and trivariate one as |
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321 | ///< second element |
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322 | const CFList& factors, ///< [in] monic bivariate factors, |
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323 | ///< including leading coefficient |
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324 | ///< as first element. |
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325 | const int* l, ///< [in] l[0]: precision of bivariate |
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326 | ///< lifting, l[1] as above |
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327 | CFList& diophant, ///< [in,out] returns the result of |
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328 | ///< biDiophantine() |
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329 | CFArray& Pi, ///< [in,out] stores intermediate results |
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330 | CFMatrix& M ///< [in,out] stores intermediate results |
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331 | ); |
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332 | |
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333 | /// resume Hensel lifting. |
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334 | /// |
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335 | /// @sa henselLift12(), henselLiftResume12(), henselLift23(), henselLift() |
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336 | void |
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337 | henselLiftResume ( |
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338 | const CanonicalForm& F, ///< [in] compressed, multivariate poly |
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339 | CFList& factors, ///< [in,out] monic multivariate factors |
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340 | ///< lifted to precision F.mvar()^start, |
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341 | ///< including leading coefficient |
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342 | ///< as first element. Returns factors |
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343 | ///< lifted to precision F.mvar()^end |
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344 | int start, ///< [in] starting precision |
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345 | int end, ///< [in] end precision |
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346 | CFArray& Pi, ///< [in,out] stores intermediate results |
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347 | const CFList& diophant, ///< [in] result of multiRecDiophantine() |
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348 | CFMatrix& M, ///< [in, out] stores intermediate results |
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349 | const CFList& MOD ///< [in] a list of powers of Variables |
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350 | ///< of level higher than 1 |
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351 | ); |
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352 | |
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353 | /// Hensel lifting |
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354 | /// |
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355 | /// @return @a henselLift returns a list of polynomials lifted to |
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356 | /// precision F.getLast().mvar()^l_new |
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357 | /// @sa henselLift12(), henselLiftResume12(), henselLift23(), henselLiftResume() |
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358 | CFList |
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359 | henselLift (const CFList& F, ///< [in] two compressed, multivariate |
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360 | ///< polys F and G |
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361 | const CFList& factors, ///< [in] monic multivariate factors |
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362 | ///< including leading coefficient |
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363 | ///< as first element. |
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364 | const CFList& MOD, ///< [in] a list of powers of Variables |
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365 | ///< of level higher than 1 |
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366 | CFList& diophant, ///< [in,out] result of multiRecDiophantine() |
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367 | CFArray& Pi, ///< [in,out] stores intermediate results |
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368 | CFMatrix& M, ///< [in,out] stores intermediate results |
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369 | const int lOld, ///< [in] lifting precision of F.mvar() |
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370 | const int lNew ///< [in] lifting precision of G.mvar() |
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371 | ); |
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372 | |
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373 | /// Hensel lifting from bivariate to multivariate |
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374 | /// |
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375 | /// @return @a henselLift returns a list of lifted factors |
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376 | /// @sa henselLift12(), henselLiftResume12(), henselLift23(), henselLiftResume() |
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377 | CFList |
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378 | henselLift (const CFList& eval, ///< [in] a list of polynomials the last |
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379 | ///< element is a compressed multivariate |
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380 | ///< poly, last but one element equals the |
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381 | ///< last elements modulo its main variable |
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382 | ///< and so on. The first element is a |
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383 | ///< compressed bivariate poly. |
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384 | const CFList& factors, ///< [in] bivariate factors, including |
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385 | ///< leading coefficient |
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386 | const int* l, ///< [in] lifting bounds |
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387 | const int lLength, ///< [in] length of l |
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388 | bool sort= true ///< [in] sort factors by degree in |
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389 | ///< Variable(1) |
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390 | ); |
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391 | |
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392 | /// two factor Hensel lifting from univariate to bivariate, factors need not to |
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393 | /// be monic |
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394 | void |
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395 | henselLift122 (const CanonicalForm& F,///< [in] a bivariate poly |
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396 | CFList& factors, ///< [in, out] a list of univariate polys |
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397 | ///< lifted factors |
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398 | int l, ///< [in] lift bound |
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399 | CFArray& Pi, ///< [in, out] stores intermediate results |
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400 | CFList& diophant, ///< [in, out] result of diophantine |
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401 | CFMatrix& M, ///< [in, out] stores intermediate results |
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402 | const CFArray& LCs, ///< [in] leading coefficients |
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403 | bool sort ///< [in] if true factors are sorted by |
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404 | ///< their degree |
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405 | ); |
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406 | |
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407 | /// two factor Hensel lifting from bivariate to multivariate, factors need not |
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408 | /// to be monic |
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409 | /// |
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410 | /// @return @a henselLift122 returns a list of lifted factors |
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411 | CFList |
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412 | henselLift2 (const CFList& eval, ///< [in] a list of polynomials the last |
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413 | ///< element is a compressed multivariate |
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414 | ///< poly, last but one element equals the |
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415 | ///< last elements modulo its main variable |
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416 | ///< and so on. The first element is a |
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417 | ///< compressed bivariate poly. |
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418 | const CFList& factors,///< [in] bivariate factors |
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419 | int* l, ///< [in] lift bounds |
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420 | const int lLength, ///< [in] length of l |
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421 | bool sort, ///< [in] if true factors are sorted by |
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422 | ///< their degree in Variable(1) |
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423 | const CFList& LCs1, ///< [in] a list of evaluated LC of first |
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424 | ///< factor |
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425 | const CFList& LCs2, ///< [in] a list of evaluated LC of second |
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426 | ///< factor |
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427 | const CFArray& Pi, ///< [in] intermediate result |
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428 | const CFList& diophant,///< [in] result of diophantine |
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429 | bool& noOneToOne ///< [in,out] check for one to one |
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430 | ///< correspondence |
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431 | ); |
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432 | |
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433 | /// Hensel lifting of non monic factors, needs correct leading coefficients of |
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434 | /// factors and a one to one corresponds between bivariate and multivariate |
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435 | /// factors to succeed |
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436 | /// |
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437 | /// @return @a nonMonicHenselLift returns a list of lifted factors |
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438 | /// such that prod (factors) == eval.getLast() if there is a one to one |
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439 | /// correspondence |
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440 | CFList |
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441 | nonMonicHenselLift (const CFList& eval, ///< [in] a list of polys the last |
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442 | ///< element is a compressed |
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443 | ///< multivariate poly, last but one |
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444 | ///< element equals the last elements |
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445 | ///< modulo its main variable and so |
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446 | ///< on. The first element is a |
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447 | ///< compressed poly in 3 variables |
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448 | const CFList& factors, ///< [in] a list of bivariate factors |
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449 | CFList* const& LCs, ///< [in] leading coefficients, |
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450 | ///< evaluated in the same way as |
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451 | ///< eval |
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452 | CFList& diophant, ///< [in, out] solution of univariate |
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453 | ///< diophantine equation |
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454 | CFArray& Pi, ///< [in, out] buffer intermediate |
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455 | ///< results |
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456 | int* liftBound, ///< [in] lifting bounds |
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457 | int length, ///< [in] length of lifting bounds |
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458 | bool& noOneToOne ///< [in, out] check for one to one |
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459 | ///< correspondence |
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460 | ); |
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461 | #endif |
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462 | /* FAC_HENSEL_H */ |
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463 | |
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