1 | /*****************************************************************************\ |
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2 | * Computer Algebra System SINGULAR |
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3 | \*****************************************************************************/ |
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4 | /** @file cfNewtonPolygon.cc |
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5 | * |
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6 | * This file provides functions to compute the Newton polygon of a bivariate |
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7 | * polynomial |
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8 | * |
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9 | * @author Martin Lee |
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10 | * |
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11 | **/ |
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12 | /*****************************************************************************/ |
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13 | |
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14 | #include "config.h" |
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15 | #include <stdlib.h> |
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16 | |
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17 | #include "canonicalform.h" |
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18 | #include "cf_iter.h" |
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19 | #include "cf_algorithm.h" |
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20 | #include "cf_primes.h" |
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21 | #include "cf_reval.h" |
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22 | #include "cfNewtonPolygon.h" |
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23 | #include "templates/ftmpl_functions.h" |
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24 | #include "algext.h" |
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25 | |
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26 | static |
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27 | void translate (int** points, int* point, int sizePoints) //make point to 0 |
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28 | { |
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29 | for (int i= 0; i < sizePoints; i++) |
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30 | { |
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31 | points[i] [0] -= point [0]; |
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32 | points[i] [1] -= point [1]; |
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33 | } |
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34 | } |
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35 | |
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36 | static |
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37 | int smallestPointIndex (int** points, int sizePoints) |
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38 | { |
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39 | int min= 0; |
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40 | for (int i= 1; i < sizePoints; i++) |
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41 | { |
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42 | if (points[i][0] < points[min][0] || |
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43 | (points[i] [0] == points[min] [0] && points[i] [1] < points[min] [1])) |
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44 | min= i; |
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45 | } |
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46 | return min; |
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47 | } |
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48 | |
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49 | static |
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50 | void swap (int** points, int i, int j) |
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51 | { |
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52 | int* tmp= points[i]; |
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53 | points[i]= points[j]; |
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54 | points[j]= tmp; |
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55 | } |
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56 | |
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57 | static |
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58 | bool isLess (int* point1, int* point2) |
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59 | { |
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60 | int area= point1[0]*point2[1]- point1[1]*point2[0]; |
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61 | if (area > 0) return true; |
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62 | if (area == 0) |
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63 | { |
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64 | return (abs (point1[0]) + abs (point1[1]) > |
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65 | abs (point2[0]) + abs (point2[1])); |
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66 | } |
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67 | return false; |
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68 | } |
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69 | |
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70 | static |
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71 | void quickSort (int lo, int hi, int** points) |
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72 | { |
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73 | int i= lo, j= hi; |
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74 | int* point= new int [2]; |
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75 | point [0]= points [(lo+hi)/2] [0]; |
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76 | point [1]= points [(lo+hi)/2] [1]; |
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77 | while (i <= j) |
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78 | { |
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79 | while (isLess (points [i], point) && i < hi) i++; |
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80 | while (isLess (point, points[j]) && j > lo) j--; |
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81 | if (i <= j) |
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82 | { |
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83 | swap (points, i, j); |
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84 | i++; |
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85 | j--; |
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86 | } |
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87 | } |
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88 | delete [] point; |
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89 | if (lo < j) quickSort (lo, j, points); |
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90 | if (i < hi) quickSort (i, hi, points); |
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91 | } |
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92 | |
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93 | static |
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94 | void sort (int** points, int sizePoints) |
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95 | { |
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96 | quickSort (1, sizePoints - 1, points); |
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97 | } |
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98 | |
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99 | // check whether p2 is convex |
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100 | static |
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101 | bool isConvex (int* point1, int* point2, int* point3) |
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102 | { |
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103 | int relArea= (point1[0] - point2[0])*(point3[1] - point2[1]) - |
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104 | (point1[1] - point2[1])*(point3[0] - point2[0]); |
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105 | if (relArea < 0) |
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106 | return true; |
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107 | if (relArea == 0) |
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108 | { |
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109 | return !(abs (point1[0] - point3[0]) + abs (point1[1] - point3[1]) >= |
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110 | (abs (point2[0] - point1[0]) + abs (point2[1] - point1[1]) + |
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111 | abs (point2[0] - point3[0]) + abs (point2[1] - point3[1]))); |
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112 | } |
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113 | return false; |
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114 | } |
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115 | |
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116 | static |
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117 | bool isConvex (int** points, int i) |
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118 | { |
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119 | return isConvex (points[i - 1], points [i], points [i + 1]); |
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120 | } |
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121 | |
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122 | int grahamScan (int** points, int sizePoints) |
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123 | { |
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124 | swap (points, 0, smallestPointIndex (points, sizePoints)); |
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125 | int * minusPoint= new int [2]; |
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126 | minusPoint [0]= points[0] [0]; |
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127 | minusPoint [1]= points[0] [1]; |
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128 | translate (points, minusPoint, sizePoints); |
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129 | sort (points, sizePoints); |
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130 | minusPoint[0]= - minusPoint[0]; |
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131 | minusPoint[1]= - minusPoint[1]; |
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132 | translate (points, minusPoint, sizePoints); //reverse translation |
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133 | delete [] minusPoint; |
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134 | int i= 3, k= 3; |
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135 | while (k < sizePoints) |
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136 | { |
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137 | swap (points, i, k); |
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138 | while (!isConvex (points, i - 1)) |
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139 | { |
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140 | swap (points, i - 1, i); |
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141 | i--; |
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142 | } |
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143 | k++; |
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144 | i++; |
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145 | } |
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146 | if (i + 1 <= sizePoints || i == sizePoints) |
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147 | { |
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148 | int relArea= |
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149 | (points [i-2][0] - points [i-1][0])*(points [0][1] - points [i-1][1])- |
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150 | (points [i-2][1] - points [i-1][1])*(points [0][0] - points [i-1][0]); |
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151 | if (relArea == 0) |
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152 | { |
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153 | if (abs (points [i-2][0] - points [0][0]) + |
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154 | abs (points [i-2][1] - points [0][1]) >= |
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155 | abs (points [i-1][0] - points [i-2][0]) + |
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156 | abs (points [i-1][1] - points [i-2][1]) + |
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157 | abs (points [i-1][0] - points [0][0]) + |
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158 | abs (points [i-1][1] - points [0][1])) |
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159 | i--; |
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160 | } |
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161 | } |
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162 | return i; |
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163 | } |
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164 | |
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165 | //points[i] [0] is x-coordinate, points [i] [1] is y-coordinate |
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166 | int polygon (int** points, int sizePoints) |
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167 | { |
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168 | if (sizePoints < 3) return sizePoints; |
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169 | return grahamScan (points, sizePoints); |
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170 | } |
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171 | |
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172 | static |
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173 | int* getDegrees (const CanonicalForm& F, int& sizeOfOutput) |
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174 | { |
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175 | if (F.inCoeffDomain()) |
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176 | { |
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177 | int* result= new int [1]; |
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178 | result [0]= 0; |
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179 | sizeOfOutput= 1; |
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180 | return result; |
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181 | } |
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182 | sizeOfOutput= size (F); |
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183 | int* result= new int [sizeOfOutput]; |
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184 | int j= 0; |
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185 | for (CFIterator i= F; i.hasTerms(); i++, j++) |
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186 | result [j]= i.exp(); |
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187 | return result; |
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188 | } |
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189 | |
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190 | //get points in Z^2 whose convex hull is the Newton polygon |
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191 | int ** getPoints (const CanonicalForm& F, int& n) |
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192 | { |
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193 | n= size (F); |
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194 | int ** points= new int* [n]; |
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195 | for (int i= 0; i < n; i++) |
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196 | points [i]= new int [2]; |
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197 | |
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198 | int j= 0; |
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199 | int * buf; |
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200 | int bufSize; |
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201 | if (F.isUnivariate() && F.level() == 1) |
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202 | { |
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203 | for (CFIterator i= F; i.hasTerms(); i++, j++) |
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204 | { |
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205 | points [j] [0]= i.exp(); |
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206 | points [j] [1]= 0; |
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207 | } |
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208 | return points; |
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209 | } |
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210 | for (CFIterator i= F; i.hasTerms(); i++) |
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211 | { |
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212 | buf= getDegrees (i.coeff(), bufSize); |
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213 | for (int k= 0; k < bufSize; k++, j++) |
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214 | { |
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215 | points [j] [0]= i.exp(); |
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216 | points [j] [1]= buf [k]; |
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217 | } |
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218 | delete [] buf; |
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219 | } |
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220 | return points; |
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221 | } |
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222 | |
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223 | int ** |
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224 | merge (int ** points1, int sizePoints1, int ** points2, int sizePoints2, |
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225 | int& sizeResult) |
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226 | { |
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227 | int i, j; |
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228 | sizeResult= sizePoints1+sizePoints2; |
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229 | for (i= 0; i < sizePoints1; i++) |
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230 | { |
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231 | for (j= 0; j < sizePoints2; j++) |
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232 | { |
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233 | if (points1[i][0] != points2[j][0]) |
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234 | continue; |
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235 | else |
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236 | { |
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237 | if (points1[i][1] != points2[j][1]) |
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238 | continue; |
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239 | else |
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240 | { |
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241 | points2[j][0]= -1; |
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242 | points2[j][1]= -1; |
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243 | sizeResult--; |
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244 | } |
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245 | } |
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246 | } |
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247 | } |
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248 | if (sizeResult == 0) |
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249 | return points1; |
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250 | |
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251 | int ** result= new int *[sizeResult]; |
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252 | for (i= 0; i < sizeResult; i++) |
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253 | result [i]= new int [2]; |
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254 | |
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255 | int k= 0; |
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256 | for (i= 0; i < sizePoints1; i++, k++) |
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257 | { |
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258 | result[k][0]= points1[i][0]; |
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259 | result[k][1]= points1[i][1]; |
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260 | } |
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261 | for (i= 0; i < sizePoints2; i++) |
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262 | { |
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263 | if (points2[i][0] < 0) |
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264 | continue; |
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265 | else |
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266 | { |
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267 | result[k][0]= points2[i][0]; |
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268 | result[k][1]= points2[i][1]; |
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269 | k++; |
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270 | } |
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271 | } |
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272 | return result; |
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273 | } |
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274 | |
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275 | // assumes a bivariate poly as input |
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276 | int ** newtonPolygon (const CanonicalForm& F, int& sizeOfNewtonPoly) |
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277 | { |
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278 | int sizeF= size (F); |
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279 | int ** points= new int* [sizeF]; |
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280 | for (int i= 0; i < sizeF; i++) |
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281 | points [i]= new int [2]; |
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282 | int j= 0; |
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283 | int * buf; |
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284 | int bufSize; |
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285 | for (CFIterator i= F; i.hasTerms(); i++) |
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286 | { |
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287 | buf= getDegrees (i.coeff(), bufSize); |
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288 | for (int k= 0; k < bufSize; k++, j++) |
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289 | { |
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290 | points [j] [0]= i.exp(); |
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291 | points [j] [1]= buf [k]; |
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292 | } |
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293 | delete [] buf; |
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294 | } |
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295 | |
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296 | int n= polygon (points, sizeF); |
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297 | |
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298 | int ** result= new int* [n]; |
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299 | for (int i= 0; i < n; i++) |
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300 | { |
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301 | result [i]= new int [2]; |
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302 | result [i] [0]= points [i] [0]; |
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303 | result [i] [1]= points [i] [1]; |
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304 | } |
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305 | |
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306 | sizeOfNewtonPoly= n; |
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307 | for (int i= 0; i < sizeF; i++) |
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308 | delete [] points[i]; |
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309 | delete [] points; |
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310 | |
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311 | return result; |
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312 | } |
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313 | |
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314 | // assumes a bivariate polys as input |
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315 | int ** newtonPolygon (const CanonicalForm& F, const CanonicalForm& G, |
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316 | int& sizeOfNewtonPoly) |
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317 | { |
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318 | int sizeF= size (F); |
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319 | int ** pointsF= new int* [sizeF]; |
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320 | for (int i= 0; i < sizeF; i++) |
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321 | pointsF [i]= new int [2]; |
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322 | int j= 0; |
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323 | int * buf; |
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324 | int bufSize; |
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325 | for (CFIterator i= F; i.hasTerms(); i++) |
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326 | { |
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327 | buf= getDegrees (i.coeff(), bufSize); |
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328 | for (int k= 0; k < bufSize; k++, j++) |
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329 | { |
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330 | pointsF [j] [0]= i.exp(); |
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331 | pointsF [j] [1]= buf [k]; |
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332 | } |
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333 | delete [] buf; |
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334 | } |
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335 | |
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336 | int sizeG= size (G); |
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337 | int ** pointsG= new int* [sizeG]; |
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338 | for (int i= 0; i < sizeG; i++) |
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339 | pointsG [i]= new int [2]; |
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340 | j= 0; |
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341 | for (CFIterator i= G; i.hasTerms(); i++) |
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342 | { |
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343 | buf= getDegrees (i.coeff(), bufSize); |
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344 | for (int k= 0; k < bufSize; k++, j++) |
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345 | { |
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346 | pointsG [j] [0]= i.exp(); |
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347 | pointsG [j] [1]= buf [k]; |
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348 | } |
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349 | delete [] buf; |
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350 | } |
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351 | |
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352 | int sizePoints; |
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353 | int ** points= merge (pointsF, sizeF, pointsG, sizeG, sizePoints); |
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354 | |
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355 | int n= polygon (points, sizePoints); |
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356 | |
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357 | int ** result= new int* [n]; |
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358 | for (int i= 0; i < n; i++) |
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359 | { |
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360 | result [i]= new int [2]; |
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361 | result [i] [0]= points [i] [0]; |
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362 | result [i] [1]= points [i] [1]; |
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363 | } |
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364 | |
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365 | sizeOfNewtonPoly= n; |
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366 | for (int i= 0; i < sizeF; i++) |
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367 | delete [] pointsF[i]; |
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368 | delete [] pointsF; |
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369 | for (int i= 0; i < sizeG; i++) |
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370 | delete [] pointsG[i]; |
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371 | delete [] pointsG; |
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372 | |
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373 | return result; |
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374 | } |
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375 | |
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376 | // assumes first sizePoints entries of points form a Newton polygon |
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377 | bool isInPolygon (int ** points, int sizePoints, int* point) |
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378 | { |
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379 | int ** buf= new int* [sizePoints + 1]; |
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380 | for (int i= 0; i < sizePoints; i++) |
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381 | { |
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382 | buf [i]= new int [2]; |
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383 | buf [i] [0]= points [i] [0]; |
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384 | buf [i] [1]= points [i] [1]; |
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385 | } |
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386 | buf [sizePoints]= new int [2]; |
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387 | buf [sizePoints] [0]= point [0]; |
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388 | buf [sizePoints] [1]= point [1]; |
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389 | int sizeBuf= sizePoints + 1; |
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390 | |
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391 | swap (buf, 0, smallestPointIndex (buf, sizeBuf)); |
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392 | int * minusPoint= new int [2]; |
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393 | minusPoint [0]= buf[0] [0]; |
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394 | minusPoint [1]= buf[0] [1]; |
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395 | translate (buf, minusPoint, sizeBuf); |
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396 | sort (buf, sizeBuf); |
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397 | minusPoint[0]= - minusPoint[0]; |
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398 | minusPoint[1]= - minusPoint[1]; |
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399 | translate (buf, minusPoint, sizeBuf); //reverse translation |
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400 | delete [] minusPoint; |
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401 | |
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402 | if (buf [0] [0] == point [0] && buf [0] [1] == point [1]) |
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403 | { |
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404 | for (int i= 0; i < sizeBuf; i++) |
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405 | delete [] buf[i]; |
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406 | delete [] buf; |
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407 | return false; |
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408 | } |
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409 | |
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410 | for (int i= 1; i < sizeBuf-1; i++) |
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411 | { |
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412 | if (buf [i] [0] == point [0] && buf [i] [1] == point [1]) |
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413 | { |
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414 | bool result= !isConvex (buf, i); |
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415 | for (int i= 0; i < sizeBuf; i++) |
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416 | delete [] buf [i]; |
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417 | delete [] buf; |
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418 | return result; |
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419 | } |
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420 | } |
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421 | |
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422 | if (buf [sizeBuf - 1] [0] == point [0] && buf [sizeBuf-1] [1] == point [1]) |
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423 | { |
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424 | buf [1] [0]= point [0]; |
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425 | buf [1] [1]= point [1]; |
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426 | buf [2] [0]= buf [0] [0]; |
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427 | buf [2] [1]= buf [0] [1]; |
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428 | buf [0] [0]= buf [sizeBuf-2] [0]; |
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429 | buf [0] [1]= buf [sizeBuf-2] [1]; |
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430 | bool result= !isConvex (buf, 1); |
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431 | for (int i= 0; i < sizeBuf; i++) |
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432 | delete [] buf [i]; |
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433 | delete [] buf; |
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434 | return result; |
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435 | } |
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436 | for (int i= 0; i < sizeBuf; i++) |
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437 | delete [] buf [i]; |
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438 | delete [] buf; |
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439 | |
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440 | return false; |
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441 | } |
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442 | |
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443 | void lambda (int** points, int sizePoints) |
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444 | { |
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445 | for (int i= 0; i < sizePoints; i++) |
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446 | points [i] [1]= points [i] [1] - points [i] [0]; |
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447 | } |
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448 | |
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449 | void lambdaInverse (int** points, int sizePoints) |
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450 | { |
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451 | for (int i= 0; i < sizePoints; i++) |
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452 | points [i] [1]= points [i] [1] + points [i] [0]; |
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453 | } |
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454 | |
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455 | void tau (int** points, int sizePoints, int k) |
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456 | { |
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457 | for (int i= 0; i < sizePoints; i++) |
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458 | points [i] [1]= points [i] [1] + k; |
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459 | } |
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460 | |
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461 | void mu (int** points, int sizePoints) |
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462 | { |
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463 | int tmp; |
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464 | for (int i= 0; i < sizePoints; i++) |
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465 | { |
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466 | tmp= points [i] [0]; |
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467 | points [i] [0]= points [i] [1]; |
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468 | points [i] [1]= tmp; |
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469 | } |
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470 | } |
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471 | |
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472 | void getMaxMin (int** points, int sizePoints, int& minDiff, int& minSum, |
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473 | int& maxDiff, int& maxSum, int& maxX, int& maxY |
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474 | ) |
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475 | { |
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476 | minDiff= points [0] [1] - points [0] [0]; |
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477 | minSum= points [0] [1] + points [0] [0]; |
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478 | maxDiff= points [0] [1] - points [0] [0]; |
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479 | maxSum= points [0] [1] + points [0] [0]; |
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480 | maxX= points [0] [1]; |
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481 | maxY= points [0] [0]; |
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482 | int diff, sum; |
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483 | for (int i= 1; i < sizePoints; i++) |
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484 | { |
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485 | diff= points [i] [1] - points [i] [0]; |
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486 | sum= points [i] [1] + points [i] [0]; |
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487 | minDiff= tmin (minDiff, diff); |
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488 | minSum= tmin (minSum, sum); |
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489 | maxDiff= tmax (maxDiff, diff); |
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490 | maxSum= tmax (maxSum, sum); |
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491 | maxX= tmax (maxX, points [i] [1]); |
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492 | maxY= tmax (maxY, points [i] [0]); |
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493 | } |
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494 | } |
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495 | |
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496 | #ifdef HAVE_NTL |
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497 | void convexDense(int** points, int sizePoints, mat_ZZ& M, vec_ZZ& A) |
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498 | { |
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499 | if (sizePoints < 3) |
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500 | { |
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501 | if (sizePoints == 2) |
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502 | { |
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503 | int maxX= (points [1] [1] < points [0] [1])?points [0] [1]:points [1] [1]; |
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504 | int maxY= (points [1] [0] < points [0] [0])?points [0] [0]:points [1] [0]; |
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505 | long u,v,g; |
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506 | XGCD (g, u, v, maxX, maxY); |
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507 | M.SetDims (2,2); |
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508 | A.SetLength (2); |
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509 | if (points [0] [1] != points [0] [0] && points [1] [0] != points [1] [1]) |
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510 | { |
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511 | M (1,1)= -u; |
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512 | M (1,2)= v; |
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513 | M (2,1)= maxY/g; |
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514 | M (2,2)= maxX/g; |
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515 | A (1)= u*maxX; |
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516 | A (2)= -(maxY/g)*maxX; |
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517 | } |
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518 | else |
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519 | { |
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520 | M (1,1)= u; |
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521 | M (1,2)= v; |
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522 | M (2,1)= -maxY/g; |
---|
523 | M (2,2)= maxX/g; |
---|
524 | A (1)= to_ZZ (0); |
---|
525 | A (2)= to_ZZ (0); |
---|
526 | } |
---|
527 | } |
---|
528 | else if (sizePoints == 1) |
---|
529 | { |
---|
530 | ident (M, 2); |
---|
531 | A.SetLength (2); |
---|
532 | A (1)= to_ZZ (0); |
---|
533 | A (2)= to_ZZ (0); |
---|
534 | } |
---|
535 | return; |
---|
536 | } |
---|
537 | A.SetLength (2); |
---|
538 | A (1)= to_ZZ (0); |
---|
539 | A (2)= to_ZZ (0); |
---|
540 | ident (M, 2); |
---|
541 | mat_ZZ Mu; |
---|
542 | Mu.SetDims (2, 2); |
---|
543 | Mu (2,1)= to_ZZ (1); |
---|
544 | Mu (1,2)= to_ZZ (1); |
---|
545 | Mu (1,1)= to_ZZ (0); |
---|
546 | Mu (2,2)= to_ZZ (0); |
---|
547 | mat_ZZ Lambda; |
---|
548 | Lambda.SetDims (2, 2); |
---|
549 | Lambda (1,1)= to_ZZ (1); |
---|
550 | Lambda (1,2)= to_ZZ (-1); |
---|
551 | Lambda (2,2)= to_ZZ (1); |
---|
552 | Lambda (2,1)= to_ZZ (0); |
---|
553 | mat_ZZ InverseLambda; |
---|
554 | InverseLambda.SetDims (2,2); |
---|
555 | InverseLambda (1,1)= to_ZZ (1); |
---|
556 | InverseLambda (1,2)= to_ZZ (1); |
---|
557 | InverseLambda (2,2)= to_ZZ (1); |
---|
558 | InverseLambda (2,1)= to_ZZ (0); |
---|
559 | ZZ tmp; |
---|
560 | int minDiff, minSum, maxDiff, maxSum, maxX, maxY, b, d, f, h; |
---|
561 | getMaxMin (points, sizePoints, minDiff, minSum, maxDiff, maxSum, maxX, maxY); |
---|
562 | do |
---|
563 | { |
---|
564 | if (maxX < maxY) |
---|
565 | { |
---|
566 | mu (points, sizePoints); |
---|
567 | M= Mu*M; |
---|
568 | tmp= A (1); |
---|
569 | A (1)= A (2); |
---|
570 | A (2)= tmp; |
---|
571 | } |
---|
572 | getMaxMin (points, sizePoints, minDiff, minSum, maxDiff, maxSum, maxX, maxY); |
---|
573 | b= maxX - maxDiff; |
---|
574 | d= maxX + maxY - maxSum; |
---|
575 | f= maxY + minDiff; |
---|
576 | h= minSum; |
---|
577 | if (b + f > maxY) |
---|
578 | { |
---|
579 | lambda (points, sizePoints); |
---|
580 | tau (points, sizePoints, maxY - f); |
---|
581 | M= Lambda*M; |
---|
582 | A [0] += (maxY-f); |
---|
583 | maxX= maxX + maxY - b - f; |
---|
584 | } |
---|
585 | else if (d + h > maxY) |
---|
586 | { |
---|
587 | lambdaInverse (points, sizePoints); |
---|
588 | tau (points, sizePoints, -h); |
---|
589 | M= InverseLambda*M; |
---|
590 | A [0] += (-h); |
---|
591 | maxX= maxX + maxY - d - h; |
---|
592 | } |
---|
593 | else |
---|
594 | return; |
---|
595 | } while (1); |
---|
596 | } |
---|
597 | |
---|
598 | CanonicalForm |
---|
599 | compress (const CanonicalForm& F, mat_ZZ& M, vec_ZZ& A, bool computeMA) |
---|
600 | { |
---|
601 | int n; |
---|
602 | int ** newtonPolyg= NULL; |
---|
603 | if (computeMA) |
---|
604 | { |
---|
605 | newtonPolyg= newtonPolygon (F, n); |
---|
606 | convexDense (newtonPolyg, n, M, A); |
---|
607 | } |
---|
608 | CanonicalForm result= 0; |
---|
609 | ZZ expX, expY; |
---|
610 | Variable x= Variable (1); |
---|
611 | Variable y= Variable (2); |
---|
612 | |
---|
613 | ZZ minExpX, minExpY; |
---|
614 | |
---|
615 | int k= 0; |
---|
616 | Variable alpha; |
---|
617 | for (CFIterator i= F; i.hasTerms(); i++) |
---|
618 | { |
---|
619 | if (i.coeff().inCoeffDomain() && hasFirstAlgVar (i.coeff(), alpha)) |
---|
620 | { |
---|
621 | expX= i.exp()*M (1,2) + A (1); |
---|
622 | expY= i.exp()*M (2,2) + A (2); |
---|
623 | if (k == 0) |
---|
624 | { |
---|
625 | minExpY= expY; |
---|
626 | minExpX= expX; |
---|
627 | k= 1; |
---|
628 | } |
---|
629 | else |
---|
630 | { |
---|
631 | minExpY= (minExpY > expY) ? expY : minExpY; |
---|
632 | minExpX= (minExpX > expX) ? expX : minExpX; |
---|
633 | } |
---|
634 | result += i.coeff()*power (x, to_long (expX))*power (y, to_long (expY)); |
---|
635 | continue; |
---|
636 | } |
---|
637 | CFIterator j= i.coeff(); |
---|
638 | if (k == 0) |
---|
639 | { |
---|
640 | expX= j.exp()*M (1,1) + i.exp()*M (1,2) + A (1); |
---|
641 | expY= j.exp()*M (2,1) + i.exp()*M (2,2) + A (2); |
---|
642 | minExpX= expX; |
---|
643 | minExpY= expY; |
---|
644 | result += j.coeff()*power (x, to_long (expX))*power (y, to_long (expY)); |
---|
645 | j++; |
---|
646 | k= 1; |
---|
647 | } |
---|
648 | |
---|
649 | for (; j.hasTerms(); j++) |
---|
650 | { |
---|
651 | expX= j.exp()*M (1,1) + i.exp()*M (1,2) + A (1); |
---|
652 | expY= j.exp()*M (2,1) + i.exp()*M (2,2) + A (2); |
---|
653 | result += j.coeff()*power (x, to_long (expX))*power (y, to_long (expY)); |
---|
654 | minExpY= (minExpY > expY) ? expY : minExpY; |
---|
655 | minExpX= (minExpX > expX) ? expX : minExpX; |
---|
656 | } |
---|
657 | } |
---|
658 | |
---|
659 | if (to_long (minExpX) < 0) |
---|
660 | { |
---|
661 | result *= power (x,-to_long(minExpX)); |
---|
662 | result /= CanonicalForm (x, 0); |
---|
663 | } |
---|
664 | else |
---|
665 | result /= power (x,to_long(minExpX)); |
---|
666 | |
---|
667 | if (to_long (minExpY) < 0) |
---|
668 | { |
---|
669 | result *= power (y,-to_long(minExpY)); |
---|
670 | result /= CanonicalForm (y, 0); |
---|
671 | } |
---|
672 | else |
---|
673 | result /= power (y,to_long(minExpY)); |
---|
674 | |
---|
675 | CanonicalForm tmp= LC (result); |
---|
676 | if (tmp.inPolyDomain() && degree (tmp) <= 0) |
---|
677 | { |
---|
678 | int d= degree (result); |
---|
679 | Variable x= result.mvar(); |
---|
680 | result -= tmp*power (x, d); |
---|
681 | result += Lc (tmp)*power (x, d); |
---|
682 | } |
---|
683 | |
---|
684 | if (computeMA) |
---|
685 | { |
---|
686 | for (int i= 0; i < n; i++) |
---|
687 | delete [] newtonPolyg [i]; |
---|
688 | delete [] newtonPolyg; |
---|
689 | M= inv (M); |
---|
690 | } |
---|
691 | |
---|
692 | return result; |
---|
693 | } |
---|
694 | |
---|
695 | CanonicalForm |
---|
696 | decompress (const CanonicalForm& F, const mat_ZZ& inverseM, const vec_ZZ& A) |
---|
697 | { |
---|
698 | CanonicalForm result= 0; |
---|
699 | ZZ expX, expY; |
---|
700 | Variable x= Variable (1); |
---|
701 | Variable y= Variable (2); |
---|
702 | ZZ minExpX, minExpY; |
---|
703 | if (F.isUnivariate() && F.level() == 1) |
---|
704 | { |
---|
705 | CFIterator i= F; |
---|
706 | expX= (i.exp() - A (1))*inverseM (1,1) + (-A (2))*inverseM (1,2); |
---|
707 | expY= (i.exp() - A (1))*inverseM (2,1) + (-A (2))*inverseM (2,2); |
---|
708 | minExpX= expX; |
---|
709 | minExpY= expY; |
---|
710 | result += i.coeff()*power (x, to_long (expX))*power (y, to_long (expY)); |
---|
711 | i++; |
---|
712 | for (; i.hasTerms(); i++) |
---|
713 | { |
---|
714 | expX= (i.exp() - A (1))*inverseM (1,1) + (-A (2))*inverseM (1,2); |
---|
715 | expY= (i.exp() - A (1))*inverseM (2,1) + (-A (2))*inverseM (2,2); |
---|
716 | result += i.coeff()*power (x, to_long (expX))*power (y, to_long (expY)); |
---|
717 | minExpY= (minExpY > expY) ? expY : minExpY; |
---|
718 | minExpX= (minExpX > expX) ? expX : minExpX; |
---|
719 | } |
---|
720 | |
---|
721 | if (to_long (minExpX) < 0) |
---|
722 | { |
---|
723 | result *= power (x,-to_long(minExpX)); |
---|
724 | result /= CanonicalForm (x, 0); |
---|
725 | } |
---|
726 | else |
---|
727 | result /= power (x,to_long(minExpX)); |
---|
728 | |
---|
729 | if (to_long (minExpY) < 0) |
---|
730 | { |
---|
731 | result *= power (y,-to_long(minExpY)); |
---|
732 | result /= CanonicalForm (y, 0); |
---|
733 | } |
---|
734 | else |
---|
735 | result /= power (y,to_long(minExpY)); |
---|
736 | |
---|
737 | return result/ Lc (result); //normalize |
---|
738 | } |
---|
739 | |
---|
740 | int k= 0; |
---|
741 | Variable alpha; |
---|
742 | for (CFIterator i= F; i.hasTerms(); i++) |
---|
743 | { |
---|
744 | if (i.coeff().inCoeffDomain() && hasFirstAlgVar (i.coeff(), alpha)) |
---|
745 | { |
---|
746 | expX= -A(1)*inverseM (1,1) + (i.exp() - A (2))*inverseM (1,2); |
---|
747 | expY= -A(1)*inverseM (2,1) + (i.exp() - A (2))*inverseM (2,2); |
---|
748 | if (k == 0) |
---|
749 | { |
---|
750 | minExpY= expY; |
---|
751 | minExpX= expX; |
---|
752 | k= 1; |
---|
753 | } |
---|
754 | else |
---|
755 | { |
---|
756 | minExpY= (minExpY > expY) ? expY : minExpY; |
---|
757 | minExpX= (minExpX > expX) ? expX : minExpX; |
---|
758 | } |
---|
759 | result += i.coeff()*power (x, to_long (expX))*power (y, to_long (expY)); |
---|
760 | continue; |
---|
761 | } |
---|
762 | CFIterator j= i.coeff(); |
---|
763 | if (k == 0) |
---|
764 | { |
---|
765 | expX= (j.exp() - A (1))*inverseM (1,1) + (i.exp() - A (2))*inverseM (1,2); |
---|
766 | expY= (j.exp() - A (1))*inverseM (2,1) + (i.exp() - A (2))*inverseM (2,2); |
---|
767 | minExpX= expX; |
---|
768 | minExpY= expY; |
---|
769 | result += j.coeff()*power (x, to_long (expX))*power (y, to_long (expY)); |
---|
770 | j++; |
---|
771 | k= 1; |
---|
772 | } |
---|
773 | |
---|
774 | for (; j.hasTerms(); j++) |
---|
775 | { |
---|
776 | expX= (j.exp() - A (1))*inverseM (1,1) + (i.exp() - A (2))*inverseM (1,2); |
---|
777 | expY= (j.exp() - A (1))*inverseM (2,1) + (i.exp() - A (2))*inverseM (2,2); |
---|
778 | result += j.coeff()*power (x, to_long (expX))*power (y, to_long (expY)); |
---|
779 | minExpY= (minExpY > expY) ? expY : minExpY; |
---|
780 | minExpX= (minExpX > expX) ? expX : minExpX; |
---|
781 | } |
---|
782 | } |
---|
783 | |
---|
784 | if (to_long (minExpX) < 0) |
---|
785 | { |
---|
786 | result *= power (x,-to_long(minExpX)); |
---|
787 | result /= CanonicalForm (x, 0); |
---|
788 | } |
---|
789 | else |
---|
790 | result /= power (x,to_long(minExpX)); |
---|
791 | |
---|
792 | if (to_long (minExpY) < 0) |
---|
793 | { |
---|
794 | result *= power (y,-to_long(minExpY)); |
---|
795 | result /= CanonicalForm (y, 0); |
---|
796 | } |
---|
797 | else |
---|
798 | result /= power (y,to_long(minExpY)); |
---|
799 | |
---|
800 | return result/Lc (result); //normalize |
---|
801 | } |
---|
802 | #endif |
---|
803 | |
---|
804 | //assumes the input is a Newton polygon of a bivariate polynomial which is |
---|
805 | //primitive wrt. x and y, i.e. there is at least one point of the polygon lying |
---|
806 | //on the x-axis and one lying on the y-axis |
---|
807 | int* getRightSide (int** polygon, int sizeOfPolygon, int& sizeOfOutput) |
---|
808 | { |
---|
809 | int maxY= polygon [0][0]; |
---|
810 | int indexY= 0; |
---|
811 | for (int i= 1; i < sizeOfPolygon; i++) |
---|
812 | { |
---|
813 | if (maxY < polygon [i][0]) |
---|
814 | { |
---|
815 | maxY= polygon [i][0]; |
---|
816 | indexY= i; |
---|
817 | } |
---|
818 | else if (maxY == polygon [i][0]) |
---|
819 | { |
---|
820 | if (polygon [indexY][1] < polygon[i][1]) |
---|
821 | indexY= i; |
---|
822 | } |
---|
823 | if (maxY > polygon [i][0]) |
---|
824 | break; |
---|
825 | } |
---|
826 | |
---|
827 | int count= -1; |
---|
828 | for (int i= indexY; i < sizeOfPolygon; i++) |
---|
829 | { |
---|
830 | if (polygon[i][0] == 0) |
---|
831 | { |
---|
832 | count= i - indexY; |
---|
833 | break; |
---|
834 | } |
---|
835 | } |
---|
836 | |
---|
837 | int * result; |
---|
838 | int index= 0; |
---|
839 | if (count < 0) |
---|
840 | { |
---|
841 | result= new int [sizeOfPolygon - indexY]; |
---|
842 | sizeOfOutput= sizeOfPolygon - indexY; |
---|
843 | count= sizeOfPolygon - indexY - 1; |
---|
844 | result [0]= polygon[sizeOfPolygon - 1][0] - polygon [0] [0]; |
---|
845 | index= 1; |
---|
846 | } |
---|
847 | else |
---|
848 | { |
---|
849 | sizeOfOutput= count; |
---|
850 | result= new int [count]; |
---|
851 | } |
---|
852 | |
---|
853 | for (int i= indexY + count; i > indexY; i--, index++) |
---|
854 | result [index]= polygon [i - 1] [0] - polygon [i] [0]; |
---|
855 | |
---|
856 | return result; |
---|
857 | } |
---|
858 | |
---|
859 | bool irreducibilityTest (const CanonicalForm& F) |
---|
860 | { |
---|
861 | ASSERT (getNumVars (F) == 2, "expected bivariate polynomial"); |
---|
862 | ASSERT (getCharacteristic() == 0, "expected polynomial over integers or rationals"); |
---|
863 | |
---|
864 | int sizeOfNewtonPolygon; |
---|
865 | int ** newtonPolyg= newtonPolygon (F, sizeOfNewtonPolygon); |
---|
866 | if (sizeOfNewtonPolygon == 3) |
---|
867 | { |
---|
868 | bool check1= |
---|
869 | (newtonPolyg[0][0]==0 || newtonPolyg[1][0]==0 || newtonPolyg[2][0]==0); |
---|
870 | if (check1) |
---|
871 | { |
---|
872 | bool check2= |
---|
873 | (newtonPolyg[0][1]==0 || newtonPolyg[1][1]==0 || newtonPolyg[2][0]==0); |
---|
874 | if (check2) |
---|
875 | { |
---|
876 | bool isRat= isOn (SW_RATIONAL); |
---|
877 | if (isRat) |
---|
878 | Off (SW_RATIONAL); |
---|
879 | CanonicalForm tmp= gcd (newtonPolyg[0][0],newtonPolyg[0][1]); // maybe it's better to use plain intgcd |
---|
880 | tmp= gcd (tmp, newtonPolyg[1][0]); |
---|
881 | tmp= gcd (tmp, newtonPolyg[1][1]); |
---|
882 | tmp= gcd (tmp, newtonPolyg[2][0]); |
---|
883 | tmp= gcd (tmp, newtonPolyg[2][1]); |
---|
884 | if (isRat) |
---|
885 | On (SW_RATIONAL); |
---|
886 | for (int i= 0; i < sizeOfNewtonPolygon; i++) |
---|
887 | delete [] newtonPolyg [i]; |
---|
888 | delete [] newtonPolyg; |
---|
889 | return (tmp==1); |
---|
890 | } |
---|
891 | } |
---|
892 | } |
---|
893 | for (int i= 0; i < sizeOfNewtonPolygon; i++) |
---|
894 | delete [] newtonPolyg [i]; |
---|
895 | delete [] newtonPolyg; |
---|
896 | return false; |
---|
897 | } |
---|
898 | |
---|
899 | bool absIrredTest (const CanonicalForm& F) |
---|
900 | { |
---|
901 | ASSERT (getNumVars (F) == 2, "expected bivariate polynomial"); |
---|
902 | ASSERT (factorize (F).length() <= 2, " expected irreducible polynomial"); |
---|
903 | |
---|
904 | int sizeOfNewtonPolygon; |
---|
905 | int ** newtonPolyg= newtonPolygon (F, sizeOfNewtonPolygon); |
---|
906 | bool isRat= isOn (SW_RATIONAL); |
---|
907 | if (isRat) |
---|
908 | Off (SW_RATIONAL); |
---|
909 | int p=getCharacteristic(); |
---|
910 | int d=1; |
---|
911 | char bufGFName='Z'; |
---|
912 | bool GF= (CFFactory::gettype()==GaloisFieldDomain); |
---|
913 | if (GF) |
---|
914 | { |
---|
915 | d= getGFDegree(); |
---|
916 | bufGFName=gf_name; |
---|
917 | } |
---|
918 | |
---|
919 | setCharacteristic(0); |
---|
920 | |
---|
921 | CanonicalForm g= gcd (newtonPolyg[0][0], newtonPolyg[0][1]); //maybe it's better to use plain intgcd |
---|
922 | |
---|
923 | int i= 1; |
---|
924 | while (!g.isOne() && i < sizeOfNewtonPolygon) |
---|
925 | { |
---|
926 | g= gcd (g, newtonPolyg[i][0]); |
---|
927 | g= gcd (g, newtonPolyg[i][1]); |
---|
928 | i++; |
---|
929 | } |
---|
930 | |
---|
931 | bool result= g.isOne(); |
---|
932 | |
---|
933 | if (GF) |
---|
934 | setCharacteristic (p, d, bufGFName); |
---|
935 | else |
---|
936 | setCharacteristic(p); |
---|
937 | |
---|
938 | if (isRat) |
---|
939 | On (SW_RATIONAL); |
---|
940 | |
---|
941 | for (int i= 0; i < sizeOfNewtonPolygon; i++) |
---|
942 | delete [] newtonPolyg[i]; |
---|
943 | |
---|
944 | delete [] newtonPolyg; |
---|
945 | |
---|
946 | return result; |
---|
947 | } |
---|
948 | |
---|
949 | bool modularIrredTest (const CanonicalForm& F) |
---|
950 | { |
---|
951 | ASSERT (getNumVars (F) == 2, "expected bivariate polynomial"); |
---|
952 | ASSERT (factorize (F).length() <= 2, " expected irreducible polynomial"); |
---|
953 | |
---|
954 | bool isRat= isOn (SW_RATIONAL); |
---|
955 | if (isRat) |
---|
956 | Off (SW_RATIONAL); |
---|
957 | |
---|
958 | if (isRat) |
---|
959 | ASSERT (bCommonDen (F).isOne(), "poly over Z expected"); |
---|
960 | |
---|
961 | CanonicalForm Fp, N= maxNorm (F); |
---|
962 | int tdeg= totaldegree (F); |
---|
963 | |
---|
964 | int i= 0; |
---|
965 | //TODO: maybe it's better to choose the characteristic as large as possible |
---|
966 | // as factorization over large finite field will be faster |
---|
967 | //TODO: handle those cases where our factory primes are not enough |
---|
968 | //TODO: factorize coefficients corresponding to the vertices of the Newton |
---|
969 | // polygon and only try the obtained factors |
---|
970 | if (N < cf_getSmallPrime (cf_getNumSmallPrimes()-1)) |
---|
971 | { |
---|
972 | while (i < cf_getNumSmallPrimes() && N > cf_getSmallPrime(i)) |
---|
973 | { |
---|
974 | setCharacteristic (cf_getSmallPrime (i)); |
---|
975 | Fp= F.mapinto(); |
---|
976 | i++; |
---|
977 | if (totaldegree (Fp) == tdeg) |
---|
978 | { |
---|
979 | if (absIrredTest (Fp)) |
---|
980 | { |
---|
981 | CFFList factors= factorize (Fp); |
---|
982 | if (factors.length() == 2 && factors.getLast().exp() == 1) |
---|
983 | { |
---|
984 | if (isRat) |
---|
985 | On (SW_RATIONAL); |
---|
986 | setCharacteristic (0); |
---|
987 | return true; |
---|
988 | } |
---|
989 | } |
---|
990 | } |
---|
991 | setCharacteristic (0); |
---|
992 | } |
---|
993 | } |
---|
994 | else |
---|
995 | { |
---|
996 | while (i < cf_getNumPrimes() && N > cf_getPrime (i)) |
---|
997 | { |
---|
998 | setCharacteristic (cf_getPrime (i)); |
---|
999 | Fp= F.mapinto(); |
---|
1000 | i++; |
---|
1001 | if (totaldegree (Fp) == tdeg) |
---|
1002 | { |
---|
1003 | if (absIrredTest (Fp)) |
---|
1004 | { |
---|
1005 | CFFList factors= factorize (Fp); |
---|
1006 | if (factors.length() == 2 && factors.getLast().exp() == 1) |
---|
1007 | { |
---|
1008 | if (isRat) |
---|
1009 | On (SW_RATIONAL); |
---|
1010 | setCharacteristic (0); |
---|
1011 | return true; |
---|
1012 | } |
---|
1013 | } |
---|
1014 | } |
---|
1015 | setCharacteristic (0); |
---|
1016 | } |
---|
1017 | } |
---|
1018 | |
---|
1019 | if (isRat) |
---|
1020 | On (SW_RATIONAL); |
---|
1021 | |
---|
1022 | return false; |
---|
1023 | } |
---|
1024 | |
---|
1025 | bool |
---|
1026 | modularIrredTestWithShift (const CanonicalForm& F) |
---|
1027 | { |
---|
1028 | ASSERT (getNumVars (F) == 2, "expected bivariate polynomial"); |
---|
1029 | ASSERT (factorize (F).length() <= 2, " expected irreducible polynomial"); |
---|
1030 | |
---|
1031 | bool isRat= isOn (SW_RATIONAL); |
---|
1032 | if (isRat) |
---|
1033 | Off (SW_RATIONAL); |
---|
1034 | |
---|
1035 | if (isRat) |
---|
1036 | ASSERT (bCommonDen (F).isOne(), "poly over Z expected"); |
---|
1037 | |
---|
1038 | Variable x= Variable (1); |
---|
1039 | Variable y= Variable (2); |
---|
1040 | CanonicalForm Fp; |
---|
1041 | int tdeg= totaldegree (F); |
---|
1042 | |
---|
1043 | REvaluation E; |
---|
1044 | |
---|
1045 | setCharacteristic (2); |
---|
1046 | Fp= F.mapinto(); |
---|
1047 | |
---|
1048 | E= REvaluation (1,2, FFRandom()); |
---|
1049 | |
---|
1050 | E.nextpoint(); |
---|
1051 | |
---|
1052 | Fp= Fp (x+E[1], x); |
---|
1053 | Fp= Fp (y+E[2], y); |
---|
1054 | |
---|
1055 | if (tdeg == totaldegree (Fp)) |
---|
1056 | { |
---|
1057 | if (absIrredTest (Fp)) |
---|
1058 | { |
---|
1059 | CFFList factors= factorize (Fp); |
---|
1060 | if (factors.length() == 2 && factors.getLast().exp() == 1) |
---|
1061 | { |
---|
1062 | if (isRat) |
---|
1063 | On (SW_RATIONAL); |
---|
1064 | setCharacteristic (0); |
---|
1065 | return true; |
---|
1066 | } |
---|
1067 | } |
---|
1068 | } |
---|
1069 | |
---|
1070 | E.nextpoint(); |
---|
1071 | |
---|
1072 | Fp= Fp (x+E[1], x); |
---|
1073 | Fp= Fp (y+E[2], y); |
---|
1074 | |
---|
1075 | if (tdeg == totaldegree (Fp)) |
---|
1076 | { |
---|
1077 | if (absIrredTest (Fp)) |
---|
1078 | { |
---|
1079 | CFFList factors= factorize (Fp); |
---|
1080 | if (factors.length() == 2 && factors.getLast().exp() == 1) |
---|
1081 | { |
---|
1082 | if (isRat) |
---|
1083 | On (SW_RATIONAL); |
---|
1084 | setCharacteristic (0); |
---|
1085 | return true; |
---|
1086 | } |
---|
1087 | } |
---|
1088 | } |
---|
1089 | |
---|
1090 | int i= 0; |
---|
1091 | while (cf_getSmallPrime (i) < 102) |
---|
1092 | { |
---|
1093 | setCharacteristic (cf_getSmallPrime (i)); |
---|
1094 | i++; |
---|
1095 | E= REvaluation (1, 2, FFRandom()); |
---|
1096 | |
---|
1097 | for (int j= 0; j < 3; j++) |
---|
1098 | { |
---|
1099 | Fp= F.mapinto(); |
---|
1100 | E.nextpoint(); |
---|
1101 | Fp= Fp (x+E[1], x); |
---|
1102 | Fp= Fp (y+E[2], y); |
---|
1103 | |
---|
1104 | if (tdeg == totaldegree (Fp)) |
---|
1105 | { |
---|
1106 | if (absIrredTest (Fp)) |
---|
1107 | { |
---|
1108 | CFFList factors= factorize (Fp); |
---|
1109 | if (factors.length() == 2 && factors.getLast().exp() == 1) |
---|
1110 | { |
---|
1111 | if (isRat) |
---|
1112 | On (SW_RATIONAL); |
---|
1113 | setCharacteristic (0); |
---|
1114 | return true; |
---|
1115 | } |
---|
1116 | } |
---|
1117 | } |
---|
1118 | } |
---|
1119 | } |
---|
1120 | |
---|
1121 | setCharacteristic (0); |
---|
1122 | if (isRat) |
---|
1123 | On (SW_RATIONAL); |
---|
1124 | |
---|
1125 | return false; |
---|
1126 | } |
---|
1127 | |
---|
1128 | |
---|