[a2dd9b2] | 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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[c1b9927] | 5 | * |
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[a2dd9b2] | 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 | * @internal |
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| 12 | * @version \$Id$ |
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| 13 | * |
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| 14 | **/ |
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| 15 | /*****************************************************************************/ |
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| 16 | |
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[e4fe2b] | 17 | #include "config.h" |
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[a2dd9b2] | 18 | #include <stdlib.h> |
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| 19 | |
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| 20 | #include "canonicalform.h" |
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| 21 | #include "cf_iter.h" |
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| 22 | #include "cf_algorithm.h" |
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| 23 | #include "cfNewtonPolygon.h" |
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| 24 | #include "templates/ftmpl_functions.h" |
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| 25 | #include "algext.h" |
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| 26 | |
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| 27 | static |
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| 28 | void translate (int** points, int* point, int sizePoints) //make point to 0 |
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| 29 | { |
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| 30 | for (int i= 0; i < sizePoints; i++) |
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| 31 | { |
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| 32 | points[i] [0] -= point [0]; |
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| 33 | points[i] [1] -= point [1]; |
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| 34 | } |
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| 35 | } |
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| 36 | |
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| 37 | static |
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| 38 | int smallestPointIndex (int** points, int sizePoints) |
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| 39 | { |
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| 40 | int min= 0; |
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| 41 | for (int i= 1; i < sizePoints; i++) |
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| 42 | { |
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[c1b9927] | 43 | if (points[i][0] < points[min][0] || |
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[a2dd9b2] | 44 | (points[i] [0] == points[min] [0] && points[i] [1] < points[min] [1])) |
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| 45 | min= i; |
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| 46 | } |
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| 47 | return min; |
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| 48 | } |
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| 49 | |
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| 50 | static |
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| 51 | void swap (int** points, int i, int j) |
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| 52 | { |
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| 53 | int* tmp= points[i]; |
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| 54 | points[i]= points[j]; |
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| 55 | points[j]= tmp; |
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| 56 | } |
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| 57 | |
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| 58 | static |
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| 59 | bool isLess (int* point1, int* point2) |
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| 60 | { |
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| 61 | int area= point1[0]*point2[1]- point1[1]*point2[0]; |
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| 62 | if (area > 0) return true; |
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| 63 | if (area == 0) |
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| 64 | { |
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[c1b9927] | 65 | return (abs (point1[0]) + abs (point1[1]) > |
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[a2dd9b2] | 66 | abs (point2[0]) + abs (point2[1])); |
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| 67 | } |
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| 68 | return false; |
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| 69 | } |
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| 70 | |
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| 71 | static |
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| 72 | void quickSort (int lo, int hi, int** points) |
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| 73 | { |
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| 74 | int i= lo, j= hi; |
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| 75 | int* point= new int [2]; |
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| 76 | point [0]= points [(lo+hi)/2] [0]; |
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| 77 | point [1]= points [(lo+hi)/2] [1]; |
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| 78 | while (i <= j) |
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| 79 | { |
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| 80 | while (isLess (points [i], point) && i < hi) i++; |
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| 81 | while (isLess (point, points[j]) && j > lo) j--; |
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| 82 | if (i <= j) |
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| 83 | { |
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| 84 | swap (points, i, j); |
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| 85 | i++; |
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| 86 | j--; |
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| 87 | } |
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| 88 | } |
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| 89 | delete [] point; |
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| 90 | if (lo < j) quickSort (lo, j, points); |
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| 91 | if (i < hi) quickSort (i, hi, points); |
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| 92 | } |
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| 93 | |
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| 94 | static |
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| 95 | void sort (int** points, int sizePoints) |
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| 96 | { |
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| 97 | quickSort (1, sizePoints - 1, points); |
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| 98 | } |
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| 99 | |
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| 100 | // check whether p2 is convex |
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| 101 | static |
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| 102 | bool isConvex (int* point1, int* point2, int* point3) |
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| 103 | { |
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| 104 | int relArea= (point1[0] - point2[0])*(point3[1] - point2[1]) - |
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| 105 | (point1[1] - point2[1])*(point3[0] - point2[0]); |
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| 106 | if (relArea < 0) |
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| 107 | return true; |
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| 108 | if (relArea == 0) |
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| 109 | { |
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| 110 | return !(abs (point1[0] - point3[0]) + abs (point1[1] - point3[1]) >= |
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| 111 | (abs (point2[0] - point1[0]) + abs (point2[1] - point1[1]) + |
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| 112 | abs (point2[0] - point3[0]) + abs (point2[1] - point3[1]))); |
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| 113 | } |
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| 114 | return false; |
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| 115 | } |
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| 116 | |
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| 117 | static |
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| 118 | bool isConvex (int** points, int i) |
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| 119 | { |
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| 120 | return isConvex (points[i - 1], points [i], points [i + 1]); |
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| 121 | } |
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| 122 | |
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| 123 | int grahamScan (int** points, int sizePoints) |
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| 124 | { |
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| 125 | swap (points, 0, smallestPointIndex (points, sizePoints)); |
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| 126 | int * minusPoint= new int [2]; |
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| 127 | minusPoint [0]= points[0] [0]; |
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| 128 | minusPoint [1]= points[0] [1]; |
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| 129 | translate (points, minusPoint, sizePoints); |
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| 130 | sort (points, sizePoints); |
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| 131 | minusPoint[0]= - minusPoint[0]; |
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| 132 | minusPoint[1]= - minusPoint[1]; |
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| 133 | translate (points, minusPoint, sizePoints); //reverse translation |
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| 134 | delete [] minusPoint; |
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| 135 | int i= 3, k= 3; |
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| 136 | while (k < sizePoints) |
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| 137 | { |
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| 138 | swap (points, i, k); |
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| 139 | while (!isConvex (points, i - 1)) |
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| 140 | { |
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| 141 | swap (points, i - 1, i); |
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| 142 | i--; |
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| 143 | } |
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| 144 | k++; |
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| 145 | i++; |
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| 146 | } |
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[6fd83c4] | 147 | if (i + 1 <= sizePoints || i == sizePoints) |
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[a2dd9b2] | 148 | { |
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| 149 | int relArea= |
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| 150 | (points [i-2][0] - points [i-1][0])*(points [0][1] - points [i-1][1])- |
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| 151 | (points [i-2][1] - points [i-1][1])*(points [0][0] - points [i-1][0]); |
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| 152 | if (relArea == 0) |
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| 153 | { |
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| 154 | if (abs (points [i-2][0] - points [0][0]) + |
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| 155 | abs (points [i-2][1] - points [0][1]) >= |
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| 156 | abs (points [i-1][0] - points [i-2][0]) + |
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| 157 | abs (points [i-1][1] - points [i-2][1]) + |
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| 158 | abs (points [i-1][0] - points [0][0]) + |
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| 159 | abs (points [i-1][1] - points [0][1])) |
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| 160 | i--; |
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| 161 | } |
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| 162 | } |
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| 163 | return i; |
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| 164 | } |
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| 165 | |
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| 166 | //points[i] [0] is x-coordinate, points [i] [1] is y-coordinate |
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| 167 | int polygon (int** points, int sizePoints) |
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| 168 | { |
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| 169 | if (sizePoints < 3) return sizePoints; |
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| 170 | return grahamScan (points, sizePoints); |
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| 171 | } |
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| 172 | |
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| 173 | static |
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| 174 | int* getDegrees (const CanonicalForm& F, int& sizeOfOutput) |
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| 175 | { |
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[96992c] | 176 | if (F.inCoeffDomain()) |
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| 177 | { |
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| 178 | int* result= new int [1]; |
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| 179 | result [0]= 0; |
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| 180 | sizeOfOutput= 1; |
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| 181 | return result; |
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| 182 | } |
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[a2dd9b2] | 183 | sizeOfOutput= size (F); |
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| 184 | int* result= new int [sizeOfOutput]; |
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| 185 | int j= 0; |
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| 186 | for (CFIterator i= F; i.hasTerms(); i++, j++) |
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| 187 | result [j]= i.exp(); |
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| 188 | return result; |
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| 189 | } |
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| 190 | |
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| 191 | //get points in Z^2 whose convex hull is the Newton polygon |
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| 192 | int ** getPoints (const CanonicalForm& F, int& n) |
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| 193 | { |
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| 194 | n= size (F); |
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| 195 | int ** points= new int* [n]; |
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| 196 | for (int i= 0; i < n; i++) |
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| 197 | points [i]= new int [2]; |
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| 198 | |
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| 199 | int j= 0; |
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| 200 | int * buf; |
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| 201 | int bufSize; |
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| 202 | if (F.isUnivariate() && F.level() == 1) |
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| 203 | { |
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| 204 | for (CFIterator i= F; i.hasTerms(); i++, j++) |
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| 205 | { |
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| 206 | points [j] [0]= i.exp(); |
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| 207 | points [j] [1]= 0; |
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| 208 | } |
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| 209 | return points; |
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| 210 | } |
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| 211 | for (CFIterator i= F; i.hasTerms(); i++) |
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| 212 | { |
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| 213 | buf= getDegrees (i.coeff(), bufSize); |
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| 214 | for (int k= 0; k < bufSize; k++, j++) |
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| 215 | { |
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| 216 | points [j] [0]= i.exp(); |
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| 217 | points [j] [1]= buf [k]; |
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| 218 | } |
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| 219 | delete [] buf; |
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| 220 | } |
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| 221 | return points; |
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| 222 | } |
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| 223 | |
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| 224 | // assumes a bivariate poly as input |
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| 225 | int ** newtonPolygon (const CanonicalForm& F, int& sizeOfNewtonPoly) |
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| 226 | { |
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| 227 | int sizeF= size (F); |
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| 228 | int ** points= new int* [sizeF]; |
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| 229 | for (int i= 0; i < sizeF; i++) |
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| 230 | points [i]= new int [2]; |
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| 231 | int j= 0; |
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| 232 | int * buf; |
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| 233 | int bufSize; |
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| 234 | for (CFIterator i= F; i.hasTerms(); i++) |
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| 235 | { |
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| 236 | buf= getDegrees (i.coeff(), bufSize); |
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| 237 | for (int k= 0; k < bufSize; k++, j++) |
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| 238 | { |
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| 239 | points [j] [0]= i.exp(); |
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| 240 | points [j] [1]= buf [k]; |
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| 241 | } |
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| 242 | delete [] buf; |
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| 243 | } |
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| 244 | |
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| 245 | int n= polygon (points, sizeF); |
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| 246 | |
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| 247 | int ** result= new int* [n]; |
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| 248 | for (int i= 0; i < n; i++) |
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| 249 | { |
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| 250 | result [i]= new int [2]; |
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| 251 | result [i] [0]= points [i] [0]; |
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| 252 | result [i] [1]= points [i] [1]; |
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| 253 | } |
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| 254 | |
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| 255 | sizeOfNewtonPoly= n; |
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[eee11c] | 256 | for (int i= 0; i < sizeF; i++) |
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[a2dd9b2] | 257 | delete [] points[i]; |
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| 258 | delete [] points; |
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| 259 | |
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| 260 | return result; |
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| 261 | } |
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| 262 | |
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| 263 | // assumes first sizePoints entries of points form a Newton polygon |
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| 264 | bool isInPolygon (int ** points, int sizePoints, int* point) |
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| 265 | { |
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| 266 | int ** buf= new int* [sizePoints + 1]; |
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| 267 | for (int i= 0; i < sizePoints; i++) |
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| 268 | { |
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| 269 | buf [i]= new int [2]; |
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| 270 | buf [i] [0]= points [i] [0]; |
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| 271 | buf [i] [1]= points [i] [1]; |
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| 272 | } |
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| 273 | buf [sizePoints]= new int [2]; |
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| 274 | buf [sizePoints] [0]= point [0]; |
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| 275 | buf [sizePoints] [1]= point [1]; |
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| 276 | int sizeBuf= sizePoints + 1; |
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| 277 | |
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| 278 | swap (buf, 0, smallestPointIndex (buf, sizeBuf)); |
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| 279 | int * minusPoint= new int [2]; |
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| 280 | minusPoint [0]= buf[0] [0]; |
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| 281 | minusPoint [1]= buf[0] [1]; |
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| 282 | translate (buf, minusPoint, sizeBuf); |
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| 283 | sort (buf, sizeBuf); |
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| 284 | minusPoint[0]= - minusPoint[0]; |
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| 285 | minusPoint[1]= - minusPoint[1]; |
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| 286 | translate (buf, minusPoint, sizeBuf); //reverse translation |
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| 287 | delete [] minusPoint; |
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| 288 | |
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| 289 | if (buf [0] [0] == point [0] && buf [0] [1] == point [1]) |
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| 290 | { |
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| 291 | for (int i= 0; i < sizeBuf; i++) |
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| 292 | delete [] buf[i]; |
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| 293 | delete [] buf; |
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| 294 | return false; |
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| 295 | } |
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| 296 | |
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| 297 | for (int i= 1; i < sizeBuf-1; i++) |
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| 298 | { |
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| 299 | if (buf [i] [0] == point [0] && buf [i] [1] == point [1]) |
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| 300 | { |
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| 301 | bool result= !isConvex (buf, i); |
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| 302 | for (int i= 0; i < sizeBuf; i++) |
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| 303 | delete [] buf [i]; |
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| 304 | delete [] buf; |
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| 305 | return result; |
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| 306 | } |
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| 307 | } |
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[c1b9927] | 308 | |
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[a2dd9b2] | 309 | if (buf [sizeBuf - 1] [0] == point [0] && buf [sizeBuf-1] [1] == point [1]) |
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| 310 | { |
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| 311 | buf [1] [0]= point [0]; |
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| 312 | buf [1] [1]= point [1]; |
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| 313 | buf [2] [0]= buf [0] [0]; |
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| 314 | buf [2] [1]= buf [0] [1]; |
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| 315 | buf [0] [0]= buf [sizeBuf-2] [0]; |
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| 316 | buf [0] [1]= buf [sizeBuf-2] [1]; |
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| 317 | bool result= !isConvex (buf, 1); |
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| 318 | for (int i= 0; i < sizeBuf; i++) |
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| 319 | delete [] buf [i]; |
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| 320 | delete [] buf; |
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| 321 | return result; |
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| 322 | } |
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| 323 | for (int i= 0; i < sizeBuf; i++) |
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| 324 | delete [] buf [i]; |
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| 325 | delete [] buf; |
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| 326 | |
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| 327 | return false; |
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| 328 | } |
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| 329 | |
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| 330 | void lambda (int** points, int sizePoints) |
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| 331 | { |
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| 332 | for (int i= 0; i < sizePoints; i++) |
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| 333 | points [i] [1]= points [i] [1] - points [i] [0]; |
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| 334 | } |
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| 335 | |
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| 336 | void lambdaInverse (int** points, int sizePoints) |
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| 337 | { |
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| 338 | for (int i= 0; i < sizePoints; i++) |
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| 339 | points [i] [1]= points [i] [1] + points [i] [0]; |
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| 340 | } |
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| 341 | |
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| 342 | void tau (int** points, int sizePoints, int k) |
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| 343 | { |
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| 344 | for (int i= 0; i < sizePoints; i++) |
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| 345 | points [i] [1]= points [i] [1] + k; |
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| 346 | } |
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| 347 | |
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| 348 | void mu (int** points, int sizePoints) |
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| 349 | { |
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| 350 | int tmp; |
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| 351 | for (int i= 0; i < sizePoints; i++) |
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| 352 | { |
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| 353 | tmp= points [i] [0]; |
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| 354 | points [i] [0]= points [i] [1]; |
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| 355 | points [i] [1]= tmp; |
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| 356 | } |
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| 357 | } |
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| 358 | |
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| 359 | void getMaxMin (int** points, int sizePoints, int& minDiff, int& minSum, |
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| 360 | int& maxDiff, int& maxSum, int& maxX, int& maxY |
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| 361 | ) |
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| 362 | { |
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| 363 | minDiff= points [0] [1] - points [0] [0]; |
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| 364 | minSum= points [0] [1] + points [0] [0]; |
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| 365 | maxDiff= points [0] [1] - points [0] [0]; |
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| 366 | maxSum= points [0] [1] + points [0] [0]; |
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| 367 | maxX= points [0] [1]; |
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| 368 | maxY= points [0] [0]; |
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| 369 | int diff, sum; |
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| 370 | for (int i= 1; i < sizePoints; i++) |
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| 371 | { |
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| 372 | diff= points [i] [1] - points [i] [0]; |
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| 373 | sum= points [i] [1] + points [i] [0]; |
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| 374 | minDiff= tmin (minDiff, diff); |
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| 375 | minSum= tmin (minSum, sum); |
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| 376 | maxDiff= tmax (maxDiff, diff); |
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| 377 | maxSum= tmax (maxSum, sum); |
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| 378 | maxX= tmax (maxX, points [i] [1]); |
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| 379 | maxY= tmax (maxY, points [i] [0]); |
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| 380 | } |
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| 381 | } |
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| 382 | |
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[2072126] | 383 | #ifdef HAVE_NTL |
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[a2dd9b2] | 384 | void convexDense(int** points, int sizePoints, mat_ZZ& M, vec_ZZ& A) |
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| 385 | { |
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| 386 | if (sizePoints < 3) |
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| 387 | { |
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| 388 | if (sizePoints == 2) |
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| 389 | { |
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| 390 | int maxX= (points [1] [1] < points [0] [1])?points [0] [1]:points [1] [1]; |
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| 391 | int maxY= (points [1] [0] < points [0] [0])?points [0] [0]:points [1] [0]; |
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| 392 | long u,v,g; |
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| 393 | XGCD (g, u, v, maxX, maxY); |
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| 394 | M.SetDims (2,2); |
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| 395 | A.SetLength (2); |
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| 396 | if (points [0] [1] != points [0] [0] && points [1] [0] != points [1] [1]) |
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| 397 | { |
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| 398 | M (1,1)= -u; |
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| 399 | M (1,2)= v; |
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| 400 | M (2,1)= maxY/g; |
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| 401 | M (2,2)= maxX/g; |
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| 402 | A (1)= u*maxX; |
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| 403 | A (2)= -(maxY/g)*maxX; |
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| 404 | } |
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| 405 | else |
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| 406 | { |
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| 407 | M (1,1)= u; |
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| 408 | M (1,2)= v; |
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| 409 | M (2,1)= -maxY/g; |
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| 410 | M (2,2)= maxX/g; |
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| 411 | A (1)= to_ZZ (0); |
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| 412 | A (2)= to_ZZ (0); |
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| 413 | } |
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| 414 | } |
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| 415 | else if (sizePoints == 1) |
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| 416 | { |
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| 417 | ident (M, 2); |
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| 418 | A.SetLength (2); |
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| 419 | A (1)= to_ZZ (0); |
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| 420 | A (2)= to_ZZ (0); |
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| 421 | } |
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| 422 | return; |
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| 423 | } |
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| 424 | A.SetLength (2); |
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| 425 | A (1)= to_ZZ (0); |
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| 426 | A (2)= to_ZZ (0); |
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| 427 | ident (M, 2); |
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| 428 | mat_ZZ Mu; |
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| 429 | Mu.SetDims (2, 2); |
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| 430 | Mu (2,1)= to_ZZ (1); |
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| 431 | Mu (1,2)= to_ZZ (1); |
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| 432 | Mu (1,1)= to_ZZ (0); |
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| 433 | Mu (2,2)= to_ZZ (0); |
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| 434 | mat_ZZ Lambda; |
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| 435 | Lambda.SetDims (2, 2); |
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| 436 | Lambda (1,1)= to_ZZ (1); |
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| 437 | Lambda (1,2)= to_ZZ (-1); |
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| 438 | Lambda (2,2)= to_ZZ (1); |
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| 439 | Lambda (2,1)= to_ZZ (0); |
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| 440 | mat_ZZ InverseLambda; |
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| 441 | InverseLambda.SetDims (2,2); |
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| 442 | InverseLambda (1,1)= to_ZZ (1); |
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| 443 | InverseLambda (1,2)= to_ZZ (1); |
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| 444 | InverseLambda (2,2)= to_ZZ (1); |
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| 445 | InverseLambda (2,1)= to_ZZ (0); |
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| 446 | ZZ tmp; |
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| 447 | int minDiff, minSum, maxDiff, maxSum, maxX, maxY, b, d, f, h; |
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| 448 | getMaxMin (points, sizePoints, minDiff, minSum, maxDiff, maxSum, maxX, maxY); |
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| 449 | do |
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| 450 | { |
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| 451 | if (maxX < maxY) |
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| 452 | { |
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| 453 | mu (points, sizePoints); |
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| 454 | M= Mu*M; |
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| 455 | tmp= A (1); |
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| 456 | A (1)= A (2); |
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| 457 | A (2)= tmp; |
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| 458 | } |
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| 459 | getMaxMin (points, sizePoints, minDiff, minSum, maxDiff, maxSum, maxX, maxY); |
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| 460 | b= maxX - maxDiff; |
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| 461 | d= maxX + maxY - maxSum; |
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| 462 | f= maxY + minDiff; |
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| 463 | h= minSum; |
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| 464 | if (b + f > maxY) |
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| 465 | { |
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| 466 | lambda (points, sizePoints); |
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| 467 | tau (points, sizePoints, maxY - f); |
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| 468 | M= Lambda*M; |
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| 469 | A [0] += (maxY-f); |
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| 470 | maxX= maxX + maxY - b - f; |
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| 471 | } |
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| 472 | else if (d + h > maxY) |
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| 473 | { |
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| 474 | lambdaInverse (points, sizePoints); |
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| 475 | tau (points, sizePoints, -h); |
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| 476 | M= InverseLambda*M; |
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| 477 | A [0] += (-h); |
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| 478 | maxX= maxX + maxY - d - h; |
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| 479 | } |
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| 480 | else |
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| 481 | return; |
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| 482 | } while (1); |
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| 483 | } |
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| 484 | |
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| 485 | CanonicalForm |
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| 486 | compress (const CanonicalForm& F, mat_ZZ& M, vec_ZZ& A) |
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| 487 | { |
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| 488 | int n; |
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| 489 | int ** newtonPolyg= newtonPolygon (F, n); |
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| 490 | convexDense (newtonPolyg, n, M, A); |
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| 491 | CanonicalForm result= 0; |
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| 492 | ZZ expX, expY; |
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| 493 | Variable x= Variable (1); |
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| 494 | Variable y= Variable (2); |
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| 495 | |
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| 496 | ZZ minExpX, minExpY; |
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| 497 | |
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| 498 | int k= 0; |
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| 499 | Variable alpha; |
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| 500 | for (CFIterator i= F; i.hasTerms(); i++) |
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| 501 | { |
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| 502 | if (i.coeff().inCoeffDomain() && hasFirstAlgVar (i.coeff(), alpha)) |
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| 503 | { |
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| 504 | expX= i.exp()*M (1,2) + A (1); |
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| 505 | expY= i.exp()*M (2,2) + A (2); |
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| 506 | if (k == 0) |
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| 507 | { |
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| 508 | minExpY= expY; |
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| 509 | minExpX= expX; |
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| 510 | k= 1; |
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| 511 | } |
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| 512 | else |
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| 513 | { |
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| 514 | minExpY= (minExpY > expY) ? expY : minExpY; |
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| 515 | minExpX= (minExpX > expX) ? expX : minExpX; |
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| 516 | } |
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| 517 | result += i.coeff()*power (x, to_long (expX))*power (y, to_long (expY)); |
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| 518 | continue; |
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| 519 | } |
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| 520 | CFIterator j= i.coeff(); |
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| 521 | if (k == 0) |
---|
| 522 | { |
---|
| 523 | expX= j.exp()*M (1,1) + i.exp()*M (1,2) + A (1); |
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| 524 | expY= j.exp()*M (2,1) + i.exp()*M (2,2) + A (2); |
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| 525 | minExpX= expX; |
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| 526 | minExpY= expY; |
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| 527 | result += j.coeff()*power (x, to_long (expX))*power (y, to_long (expY)); |
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| 528 | j++; |
---|
| 529 | k= 1; |
---|
| 530 | } |
---|
| 531 | |
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| 532 | for (; j.hasTerms(); j++) |
---|
| 533 | { |
---|
| 534 | expX= j.exp()*M (1,1) + i.exp()*M (1,2) + A (1); |
---|
| 535 | expY= j.exp()*M (2,1) + i.exp()*M (2,2) + A (2); |
---|
| 536 | result += j.coeff()*power (x, to_long (expX))*power (y, to_long (expY)); |
---|
| 537 | minExpY= (minExpY > expY) ? expY : minExpY; |
---|
| 538 | minExpX= (minExpX > expX) ? expX : minExpX; |
---|
| 539 | } |
---|
| 540 | } |
---|
| 541 | |
---|
| 542 | if (to_long (minExpX) < 0) |
---|
| 543 | { |
---|
| 544 | result *= power (x,-to_long(minExpX)); |
---|
| 545 | result /= CanonicalForm (x, 0); |
---|
| 546 | } |
---|
| 547 | else |
---|
| 548 | result /= power (x,to_long(minExpX)); |
---|
| 549 | |
---|
| 550 | if (to_long (minExpY) < 0) |
---|
| 551 | { |
---|
| 552 | result *= power (y,-to_long(minExpY)); |
---|
| 553 | result /= CanonicalForm (y, 0); |
---|
| 554 | } |
---|
| 555 | else |
---|
| 556 | result /= power (y,to_long(minExpY)); |
---|
| 557 | |
---|
| 558 | CanonicalForm tmp= LC (result); |
---|
| 559 | if (tmp.inPolyDomain() && degree (tmp) <= 0) |
---|
| 560 | { |
---|
| 561 | int d= degree (result); |
---|
| 562 | Variable x= result.mvar(); |
---|
| 563 | result -= tmp*power (x, d); |
---|
| 564 | result += Lc (tmp)*power (x, d); |
---|
| 565 | } |
---|
| 566 | |
---|
| 567 | for (int i= 0; i < n; i++) |
---|
| 568 | delete [] newtonPolyg [i]; |
---|
| 569 | delete [] newtonPolyg; |
---|
| 570 | |
---|
| 571 | M= inv (M); |
---|
| 572 | return result; |
---|
| 573 | } |
---|
| 574 | |
---|
| 575 | CanonicalForm |
---|
| 576 | decompress (const CanonicalForm& F, const mat_ZZ& inverseM, const vec_ZZ& A) |
---|
| 577 | { |
---|
| 578 | CanonicalForm result= 0; |
---|
| 579 | ZZ expX, expY; |
---|
| 580 | Variable x= Variable (1); |
---|
| 581 | Variable y= Variable (2); |
---|
| 582 | ZZ minExpX, minExpY; |
---|
| 583 | if (F.isUnivariate() && F.level() == 1) |
---|
| 584 | { |
---|
| 585 | CFIterator i= F; |
---|
| 586 | expX= (i.exp() - A (1))*inverseM (1,1) + (-A (2))*inverseM (1,2); |
---|
| 587 | expY= (i.exp() - A (1))*inverseM (2,1) + (-A (2))*inverseM (2,2); |
---|
| 588 | minExpX= expX; |
---|
| 589 | minExpY= expY; |
---|
| 590 | result += i.coeff()*power (x, to_long (expX))*power (y, to_long (expY)); |
---|
| 591 | i++; |
---|
| 592 | for (; i.hasTerms(); i++) |
---|
| 593 | { |
---|
| 594 | expX= (i.exp() - A (1))*inverseM (1,1) + (-A (2))*inverseM (1,2); |
---|
| 595 | expY= (i.exp() - A (1))*inverseM (2,1) + (-A (2))*inverseM (2,2); |
---|
| 596 | result += i.coeff()*power (x, to_long (expX))*power (y, to_long (expY)); |
---|
| 597 | minExpY= (minExpY > expY) ? expY : minExpY; |
---|
| 598 | minExpX= (minExpX > expX) ? expX : minExpX; |
---|
| 599 | } |
---|
| 600 | |
---|
| 601 | if (to_long (minExpX) < 0) |
---|
| 602 | { |
---|
| 603 | result *= power (x,-to_long(minExpX)); |
---|
| 604 | result /= CanonicalForm (x, 0); |
---|
| 605 | } |
---|
| 606 | else |
---|
| 607 | result /= power (x,to_long(minExpX)); |
---|
| 608 | |
---|
| 609 | if (to_long (minExpY) < 0) |
---|
| 610 | { |
---|
| 611 | result *= power (y,-to_long(minExpY)); |
---|
| 612 | result /= CanonicalForm (y, 0); |
---|
| 613 | } |
---|
| 614 | else |
---|
| 615 | result /= power (y,to_long(minExpY)); |
---|
| 616 | |
---|
| 617 | return result/ Lc (result); //normalize |
---|
| 618 | } |
---|
| 619 | |
---|
| 620 | int k= 0; |
---|
| 621 | Variable alpha; |
---|
| 622 | for (CFIterator i= F; i.hasTerms(); i++) |
---|
| 623 | { |
---|
| 624 | if (i.coeff().inCoeffDomain() && hasFirstAlgVar (i.coeff(), alpha)) |
---|
| 625 | { |
---|
| 626 | expX= -A(1)*inverseM (1,1) + (i.exp() - A (2))*inverseM (1,2); |
---|
| 627 | expY= -A(1)*inverseM (2,1) + (i.exp() - A (2))*inverseM (2,2); |
---|
| 628 | if (k == 0) |
---|
| 629 | { |
---|
| 630 | minExpY= expY; |
---|
| 631 | minExpX= expX; |
---|
| 632 | k= 1; |
---|
| 633 | } |
---|
| 634 | else |
---|
| 635 | { |
---|
| 636 | minExpY= (minExpY > expY) ? expY : minExpY; |
---|
| 637 | minExpX= (minExpX > expX) ? expX : minExpX; |
---|
| 638 | } |
---|
| 639 | result += i.coeff()*power (x, to_long (expX))*power (y, to_long (expY)); |
---|
| 640 | continue; |
---|
| 641 | } |
---|
| 642 | CFIterator j= i.coeff(); |
---|
| 643 | if (k == 0) |
---|
| 644 | { |
---|
| 645 | expX= (j.exp() - A (1))*inverseM (1,1) + (i.exp() - A (2))*inverseM (1,2); |
---|
| 646 | expY= (j.exp() - A (1))*inverseM (2,1) + (i.exp() - A (2))*inverseM (2,2); |
---|
| 647 | minExpX= expX; |
---|
| 648 | minExpY= expY; |
---|
| 649 | result += j.coeff()*power (x, to_long (expX))*power (y, to_long (expY)); |
---|
| 650 | j++; |
---|
| 651 | k= 1; |
---|
| 652 | } |
---|
| 653 | |
---|
| 654 | for (; j.hasTerms(); j++) |
---|
| 655 | { |
---|
| 656 | expX= (j.exp() - A (1))*inverseM (1,1) + (i.exp() - A (2))*inverseM (1,2); |
---|
| 657 | expY= (j.exp() - A (1))*inverseM (2,1) + (i.exp() - A (2))*inverseM (2,2); |
---|
| 658 | result += j.coeff()*power (x, to_long (expX))*power (y, to_long (expY)); |
---|
| 659 | minExpY= (minExpY > expY) ? expY : minExpY; |
---|
| 660 | minExpX= (minExpX > expX) ? expX : minExpX; |
---|
| 661 | } |
---|
| 662 | } |
---|
| 663 | |
---|
| 664 | if (to_long (minExpX) < 0) |
---|
| 665 | { |
---|
| 666 | result *= power (x,-to_long(minExpX)); |
---|
| 667 | result /= CanonicalForm (x, 0); |
---|
| 668 | } |
---|
| 669 | else |
---|
| 670 | result /= power (x,to_long(minExpX)); |
---|
| 671 | |
---|
| 672 | if (to_long (minExpY) < 0) |
---|
| 673 | { |
---|
| 674 | result *= power (y,-to_long(minExpY)); |
---|
| 675 | result /= CanonicalForm (y, 0); |
---|
| 676 | } |
---|
| 677 | else |
---|
| 678 | result /= power (y,to_long(minExpY)); |
---|
| 679 | |
---|
| 680 | return result/Lc (result); //normalize |
---|
| 681 | } |
---|
[2072126] | 682 | #endif |
---|
[6fd83c4] | 683 | |
---|
| 684 | //assumes the input is a Newton polygon of a bivariate polynomial which is |
---|
| 685 | //primitive wrt. x and y, i.e. there is at least one point of the polygon lying |
---|
| 686 | //on the x-axis and one lying on the y-axis |
---|
| 687 | int* getRightSide (int** polygon, int sizeOfPolygon, int& sizeOfOutput) |
---|
| 688 | { |
---|
| 689 | int maxY= polygon [0][0]; |
---|
| 690 | int indexY= 0; |
---|
| 691 | for (int i= 1; i < sizeOfPolygon; i++) |
---|
| 692 | { |
---|
| 693 | if (maxY < polygon [i][0]) |
---|
| 694 | { |
---|
| 695 | maxY= polygon [i][0]; |
---|
| 696 | indexY= i; |
---|
| 697 | } |
---|
| 698 | else if (maxY == polygon [i][0]) |
---|
| 699 | { |
---|
| 700 | if (polygon [indexY][1] < polygon[i][1]) |
---|
| 701 | indexY= i; |
---|
| 702 | } |
---|
| 703 | if (maxY > polygon [i][0]) |
---|
| 704 | break; |
---|
| 705 | } |
---|
| 706 | |
---|
| 707 | int count= -1; |
---|
| 708 | for (int i= indexY; i < sizeOfPolygon; i++) |
---|
| 709 | { |
---|
| 710 | if (polygon[i][0] == 0) |
---|
| 711 | { |
---|
| 712 | count= i - indexY; |
---|
| 713 | break; |
---|
| 714 | } |
---|
| 715 | } |
---|
| 716 | |
---|
| 717 | int * result; |
---|
| 718 | int index= 0; |
---|
| 719 | if (count < 0) |
---|
| 720 | { |
---|
| 721 | result= new int [sizeOfPolygon - indexY]; |
---|
| 722 | sizeOfOutput= sizeOfPolygon - indexY; |
---|
| 723 | count= sizeOfPolygon - indexY - 1; |
---|
| 724 | result [0]= polygon[sizeOfPolygon - 1][0] - polygon [0] [0]; |
---|
| 725 | index= 1; |
---|
| 726 | } |
---|
| 727 | else |
---|
| 728 | { |
---|
| 729 | sizeOfOutput= count; |
---|
| 730 | result= new int [count]; |
---|
| 731 | } |
---|
| 732 | |
---|
| 733 | for (int i= indexY + count; i > indexY; i--, index++) |
---|
| 734 | result [index]= polygon [i - 1] [0] - polygon [i] [0]; |
---|
| 735 | |
---|
| 736 | return result; |
---|
| 737 | } |
---|