1 | typedef int perm[100]; |
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2 | static void mpReplace(int j, int n, int &sign, int *perm); |
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3 | static int mpNextperm(perm * z, int max); |
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4 | static poly mp_Leibnitz(matrix a, const ring); |
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5 | static poly minuscopy (poly p, const ring); |
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6 | static poly p_Insert(poly p1, poly p2, const ring); |
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7 | |
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8 | static void mp_PartClean(matrix, int, int, const ring); |
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9 | static void mp_FinalClean(matrix, const ring); |
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10 | static int mp_PrepareRow (matrix, int, int, const ring); |
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11 | static int mp_PreparePiv (matrix, int, int, const ring); |
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12 | static int mp_PivBar(matrix, int, int, const ring); |
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13 | static int mp_PivRow(matrix, int, int, const ring); |
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14 | static float mp_PolyWeight(poly, const ring); |
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15 | static void mp_SwapRow(matrix, int, int, int, const ring); |
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16 | static void mp_SwapCol(matrix, int, int, int, const ring); |
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17 | static void mp_ElimBar(matrix, matrix, poly, int, int, const ring); |
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18 | |
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19 | /*2 |
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20 | * prepare one step of 'Bareiss' algorithm |
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21 | * for application in minor |
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22 | */ |
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23 | static int mp_PrepareRow (matrix a, int lr, int lc, const ring R) |
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24 | { |
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25 | int r; |
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26 | |
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27 | r = mp_PivBar(a,lr,lc, R); |
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28 | if(r==0) return 0; |
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29 | if(r<lr) mp_SwapRow(a, r, lr, lc, R); |
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30 | return 1; |
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31 | } |
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32 | |
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33 | /*2 |
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34 | * prepare one step of 'Bareiss' algorithm |
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35 | * for application in minor |
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36 | */ |
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37 | static int mp_PreparePiv (matrix a, int lr, int lc, const ring R) |
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38 | { |
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39 | int c; |
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40 | |
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41 | c = mp_PivRow(a, lr, lc, R); |
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42 | if(c==0) return 0; |
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43 | if(c<lc) mp_SwapCol(a, c, lr, lc, R); |
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44 | return 1; |
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45 | } |
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46 | |
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47 | /* |
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48 | * find best row |
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49 | */ |
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50 | static int mp_PivBar(matrix a, int lr, int lc, const ring R) |
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51 | { |
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52 | float f1, f2; |
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53 | poly *q1; |
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54 | int i,j,io; |
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55 | |
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56 | io = -1; |
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57 | f1 = 1.0e30; |
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58 | for (i=lr-1;i>=0;i--) |
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59 | { |
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60 | q1 = &(a->m)[i*a->ncols]; |
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61 | f2 = 0.0; |
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62 | for (j=lc-1;j>=0;j--) |
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63 | { |
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64 | if (q1[j]!=NULL) |
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65 | f2 += mp_PolyWeight(q1[j], R); |
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66 | } |
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67 | if ((f2!=0.0) && (f2<f1)) |
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68 | { |
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69 | f1 = f2; |
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70 | io = i; |
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71 | } |
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72 | } |
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73 | if (io<0) return 0; |
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74 | else return io+1; |
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75 | } |
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76 | |
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77 | /* |
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78 | * find pivot in the last row |
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79 | */ |
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80 | static int mp_PivRow(matrix a, int lr, int lc, const ring R) |
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81 | { |
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82 | float f1, f2; |
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83 | poly *q1; |
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84 | int j,jo; |
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85 | |
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86 | jo = -1; |
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87 | f1 = 1.0e30; |
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88 | q1 = &(a->m)[(lr-1)*a->ncols]; |
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89 | for (j=lc-1;j>=0;j--) |
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90 | { |
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91 | if (q1[j]!=NULL) |
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92 | { |
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93 | f2 = mp_PolyWeight(q1[j], R); |
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94 | if (f2<f1) |
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95 | { |
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96 | f1 = f2; |
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97 | jo = j; |
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98 | } |
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99 | } |
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100 | } |
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101 | if (jo<0) return 0; |
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102 | else return jo+1; |
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103 | } |
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104 | |
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105 | /* |
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106 | * weigth of a polynomial, for pivot strategy |
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107 | */ |
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108 | static float mp_PolyWeight(poly p, const ring R) |
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109 | { |
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110 | int i; |
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111 | float res; |
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112 | |
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113 | if (pNext(p) == NULL) |
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114 | { |
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115 | res = (float)n_Size(p_GetCoeff(p, R), R); |
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116 | for (i=rVar(R);i>0;i--) |
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117 | { |
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118 | if(p_GetExp(p,i, R)!=0) |
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119 | { |
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120 | res += 2.0; |
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121 | break; |
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122 | } |
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123 | } |
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124 | } |
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125 | else |
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126 | { |
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127 | res = 0.0; |
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128 | do |
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129 | { |
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130 | res += (float)n_Size(p_GetCoeff(p, R), R) + 2.0; |
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131 | pIter(p); |
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132 | } |
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133 | while (p); |
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134 | } |
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135 | return res; |
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136 | } |
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137 | |
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138 | static void mpSwapRow(matrix a, int pos, int lr, int lc) |
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139 | { |
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140 | poly sw; |
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141 | int j; |
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142 | poly* a2 = a->m; |
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143 | poly* a1 = &a2[a->ncols*(pos-1)]; |
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144 | |
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145 | a2 = &a2[a->ncols*(lr-1)]; |
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146 | for (j=lc-1; j>=0; j--) |
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147 | { |
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148 | sw = a1[j]; |
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149 | a1[j] = a2[j]; |
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150 | a2[j] = sw; |
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151 | } |
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152 | } |
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153 | |
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154 | static void mpSwapCol(matrix a, int pos, int lr, int lc) |
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155 | { |
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156 | poly sw; |
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157 | int j; |
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158 | poly* a2 = a->m; |
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159 | poly* a1 = &a2[pos-1]; |
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160 | |
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161 | a2 = &a2[lc-1]; |
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162 | for (j=a->ncols*(lr-1); j>=0; j-=a->ncols) |
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163 | { |
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164 | sw = a1[j]; |
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165 | a1[j] = a2[j]; |
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166 | a2[j] = sw; |
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167 | } |
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168 | } |
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169 | |
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170 | |
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171 | /* |
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172 | /// entries of a are minors and go to result (only if not in R) |
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173 | void mp_MinorToResult(ideal result, int &elems, matrix a, int r, int c, |
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174 | ideal R, const ring R) |
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175 | { |
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176 | poly *q1; |
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177 | int e=IDELEMS(result); |
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178 | int i,j; |
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179 | |
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180 | if (R != NULL) |
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181 | { |
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182 | for (i=r-1;i>=0;i--) |
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183 | { |
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184 | q1 = &(a->m)[i*a->ncols]; |
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185 | for (j=c-1;j>=0;j--) |
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186 | { |
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187 | if (q1[j]!=NULL) q1[j] = kNF(R,currQuotient,q1[j]); |
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188 | } |
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189 | } |
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190 | } |
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191 | for (i=r-1;i>=0;i--) |
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192 | { |
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193 | q1 = &(a->m)[i*a->ncols]; |
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194 | for (j=c-1;j>=0;j--) |
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195 | { |
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196 | if (q1[j]!=NULL) |
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197 | { |
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198 | if (elems>=e) |
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199 | { |
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200 | if(e<SIZE_OF_SYSTEM_PAGE) |
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201 | { |
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202 | pEnlargeSet(&(result->m),e,e); |
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203 | e += e; |
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204 | } |
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205 | else |
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206 | { |
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207 | pEnlargeSet(&(result->m),e,SIZE_OF_SYSTEM_PAGE); |
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208 | e += SIZE_OF_SYSTEM_PAGE; |
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209 | } |
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210 | IDELEMS(result) =e; |
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211 | } |
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212 | result->m[elems] = q1[j]; |
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213 | q1[j] = NULL; |
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214 | elems++; |
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215 | } |
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216 | } |
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217 | } |
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218 | } |
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219 | |
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220 | /// produces recursively the ideal of all arxar-minors of a |
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221 | void mp_RecMin(int ar,ideal result,int &elems,matrix a,int lr,int lc, |
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222 | poly barDiv, ideal R, const ring R) |
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223 | { |
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224 | int k; |
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225 | int kr=lr-1,kc=lc-1; |
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226 | matrix nextLevel=mpNew(kr,kc); |
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227 | |
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228 | loop |
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229 | { |
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230 | // --- look for an optimal row and bring it to last position ------------ |
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231 | if(mpPrepareRow(a,lr,lc)==0) break; |
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232 | // --- now take all pivots from the last row ------------ |
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233 | k = lc; |
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234 | loop |
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235 | { |
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236 | if(mpPreparePiv(a,lr,k)==0) break; |
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237 | mpElimBar(a,nextLevel,barDiv,lr,k); |
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238 | k--; |
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239 | if (ar>1) |
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240 | { |
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241 | mpRecMin(ar-1,result,elems,nextLevel,kr,k,a->m[kr*a->ncols+k],R); |
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242 | mpPartClean(nextLevel,kr,k); |
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243 | } |
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244 | else mpMinorToResult(result,elems,nextLevel,kr,k,R); |
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245 | if (ar>k-1) break; |
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246 | } |
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247 | if (ar>=kr) break; |
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248 | // --- now we have to take out the last row...------------ |
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249 | lr = kr; |
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250 | kr--; |
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251 | } |
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252 | mpFinalClean(nextLevel); |
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253 | } |
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254 | */ |
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255 | |
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256 | |
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257 | |
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258 | /* |
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259 | * C++ classes for Bareiss algorithm |
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260 | */ |
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261 | class row_col_weight |
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262 | { |
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263 | private: |
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264 | int ym, yn; |
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265 | public: |
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266 | float *wrow, *wcol; |
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267 | row_col_weight() : ym(0) {} |
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268 | row_col_weight(int, int); |
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269 | ~row_col_weight(); |
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270 | }; |
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271 | |
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272 | /*2 |
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273 | * a submatrix M of a matrix X[m,n]: |
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274 | * 0 <= i < s_m <= a_m |
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275 | * 0 <= j < s_n <= a_n |
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276 | * M = ( Xarray[qrow[i],qcol[j]] ) |
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277 | * if a_m = a_n and s_m = s_n |
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278 | * det(X) = sign*div^(s_m-1)*det(M) |
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279 | * resticted pivot for elimination |
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280 | * 0 <= j < piv_s |
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281 | */ |
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282 | class mp_permmatrix |
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283 | { |
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284 | private: |
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285 | int a_m, a_n, s_m, s_n, sign, piv_s; |
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286 | int *qrow, *qcol; |
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287 | poly *Xarray; |
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288 | ring R; |
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289 | |
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290 | void mpInitMat(); |
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291 | poly * mpRowAdr(int); |
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292 | poly * mpColAdr(int); |
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293 | void mpRowWeight(float *); |
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294 | void mpColWeight(float *); |
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295 | void mpRowSwap(int, int); |
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296 | void mpColSwap(int, int); |
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297 | public: |
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298 | mp_permmatrix() : a_m(0), R(NULL) {} |
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299 | mp_permmatrix(matrix, const ring); |
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300 | mp_permmatrix(mp_permmatrix *); |
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301 | ~mp_permmatrix(); |
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302 | int mpGetRow(); |
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303 | int mpGetCol(); |
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304 | int mpGetRdim(); |
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305 | int mpGetCdim(); |
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306 | int mpGetSign(); |
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307 | void mpSetSearch(int s); |
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308 | void mpSaveArray(); |
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309 | poly mpGetElem(int, int); |
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310 | void mpSetElem(poly, int, int); |
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311 | void mpDelElem(int, int); |
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312 | void mpElimBareiss(poly); |
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313 | int mpPivotBareiss(row_col_weight *); |
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314 | int mpPivotRow(row_col_weight *, int); |
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315 | void mpToIntvec(intvec *); |
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316 | void mpRowReorder(); |
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317 | void mpColReorder(); |
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318 | }; |
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319 | |
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320 | |
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321 | |
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322 | /*2 |
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323 | *returns the determinant of the matrix m; |
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324 | *uses Bareiss algorithm |
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325 | */ |
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326 | poly mp_DetBareiss (matrix a, const ring R) |
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327 | { |
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328 | int s; |
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329 | poly div, res; |
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330 | if (MATROWS(a) != MATCOLS(a)) |
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331 | { |
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332 | Werror("det of %d x %d matrix",MATROWS(a),MATCOLS(a)); |
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333 | return NULL; |
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334 | } |
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335 | matrix c = mp_Copy(a, R); |
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336 | mp_permmatrix *Bareiss = new mp_permmatrix(c, R); |
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337 | row_col_weight w(Bareiss->mpGetRdim(), Bareiss->mpGetCdim()); |
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338 | |
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339 | /* Bareiss */ |
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340 | div = NULL; |
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341 | while(Bareiss->mpPivotBareiss(&w)) |
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342 | { |
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343 | Bareiss->mpElimBareiss(div); |
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344 | div = Bareiss->mpGetElem(Bareiss->mpGetRdim(), Bareiss->mpGetCdim()); |
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345 | } |
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346 | Bareiss->mpRowReorder(); |
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347 | Bareiss->mpColReorder(); |
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348 | Bareiss->mpSaveArray(); |
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349 | s = Bareiss->mpGetSign(); |
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350 | delete Bareiss; |
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351 | |
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352 | /* result */ |
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353 | res = MATELEM(c,1,1); |
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354 | MATELEM(c,1,1) = NULL; |
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355 | id_Delete((ideal *)&c, R); |
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356 | if (s < 0) |
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357 | res = p_Neg(res, R); |
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358 | return res; |
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359 | } |
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360 | |
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361 | |
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362 | |
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363 | /*2 |
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364 | *returns the determinant of the matrix m; |
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365 | *uses Newtons formulea for symmetric functions |
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366 | */ |
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367 | poly mp_Det (matrix m, const ring R) |
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368 | { |
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369 | int i,j,k,n; |
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370 | poly p,q; |
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371 | matrix a, s; |
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372 | matrix ma[100]; |
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373 | number c=NULL, d=NULL, ONE=NULL; |
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374 | |
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375 | n = MATROWS(m); |
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376 | if (n != MATCOLS(m)) |
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377 | { |
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378 | Werror("det of %d x %d matrix",n,MATCOLS(m)); |
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379 | return NULL; |
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380 | } |
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381 | k=rChar(R); |
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382 | if ((k > 0) && (k <= n)) |
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383 | return mp_Leibnitz(m, R); |
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384 | ONE = n_Init(1, R); |
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385 | ma[1]=mp_Copy(m, R); |
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386 | k = (n+1) / 2; |
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387 | s = mpNew(1, n); |
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388 | MATELEM(s,1,1) = mp_Trace(m, R); |
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389 | for (i=2; i<=k; i++) |
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390 | { |
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391 | //ma[i] = mpNew(n,n); |
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392 | ma[i]=mp_Mult(ma[i-1], ma[1], R); |
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393 | MATELEM(s,1,i) = mp_Trace(ma[i], R); |
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394 | p_Test(MATELEM(s,1,i), R); |
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395 | } |
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396 | for (i=k+1; i<=n; i++) |
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397 | { |
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398 | MATELEM(s,1,i) = TraceOfProd(ma[i / 2], ma[(i+1) / 2], n, R); |
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399 | p_Test(MATELEM(s,1,i), R); |
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400 | } |
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401 | for (i=1; i<=k; i++) |
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402 | id_Delete((ideal *)&(ma[i]), R); |
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403 | /* the array s contains the traces of the powers of the matrix m, |
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404 | * these are the power sums of the eigenvalues of m */ |
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405 | a = mpNew(1,n); |
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406 | MATELEM(a,1,1) = minuscopy(MATELEM(s,1,1), R); |
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407 | for (i=2; i<=n; i++) |
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408 | { |
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409 | p = p_Copy(MATELEM(s,1,i), R); |
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410 | for (j=i-1; j>=1; j--) |
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411 | { |
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412 | q = pp_Mult_qq(MATELEM(s,1,j), MATELEM(a,1,i-j), R); |
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413 | p_Test(q, R); |
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414 | p = p_Add_q(p,q, R); |
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415 | } |
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416 | // c= -1/i |
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417 | d = n_Init(-(int)i, R); |
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418 | c = n_Div(ONE, d, R); |
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419 | n_Delete(&d, R); |
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420 | |
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421 | p_Mult_nn(p, c, R); |
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422 | p_Test(p, R); |
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423 | MATELEM(a,1,i) = p; |
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424 | n_Delete(&c, R); |
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425 | } |
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426 | /* the array a contains the elementary symmetric functions of the |
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427 | * eigenvalues of m */ |
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428 | for (i=1; i<=n-1; i++) |
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429 | { |
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430 | //p_Delete(&(MATELEM(a,1,i)), R); |
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431 | p_Delete(&(MATELEM(s,1,i)), R); |
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432 | } |
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433 | p_Delete(&(MATELEM(s,1,n)), R); |
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434 | /* up to a sign, the determinant is the n-th elementary symmetric function */ |
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435 | if ((n/2)*2 < n) |
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436 | { |
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437 | d = n_Init(-1, R); |
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438 | p_Mult_nn(MATELEM(a,1,n), d, R); |
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439 | n_Delete(&d, R); |
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440 | } |
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441 | n_Delete(&ONE, R); |
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442 | id_Delete((ideal *)&s, R); |
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443 | poly result=MATELEM(a,1,n); |
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444 | MATELEM(a,1,n)=NULL; |
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445 | id_Delete((ideal *)&a, R); |
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446 | return result; |
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447 | } |
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448 | |
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449 | /*2 |
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450 | * compute all ar-minors of the matrix a |
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451 | */ |
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452 | matrix mp_Wedge(matrix a, int ar, const ring R) |
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453 | { |
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454 | int i,j,k,l; |
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455 | int *rowchoise,*colchoise; |
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456 | BOOLEAN rowch,colch; |
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457 | matrix result; |
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458 | matrix tmp; |
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459 | poly p; |
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460 | |
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461 | i = binom(a->nrows,ar); |
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462 | j = binom(a->ncols,ar); |
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463 | |
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464 | rowchoise=(int *)omAlloc(ar*sizeof(int)); |
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465 | colchoise=(int *)omAlloc(ar*sizeof(int)); |
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466 | result =mpNew(i,j); |
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467 | tmp=mpNew(ar,ar); |
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468 | l = 1; /* k,l:the index in result*/ |
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469 | idInitChoise(ar,1,a->nrows,&rowch,rowchoise); |
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470 | while (!rowch) |
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471 | { |
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472 | k=1; |
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473 | idInitChoise(ar,1,a->ncols,&colch,colchoise); |
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474 | while (!colch) |
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475 | { |
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476 | for (i=1; i<=ar; i++) |
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477 | { |
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478 | for (j=1; j<=ar; j++) |
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479 | { |
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480 | MATELEM(tmp,i,j) = MATELEM(a,rowchoise[i-1],colchoise[j-1]); |
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481 | } |
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482 | } |
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483 | p = mp_DetBareiss(tmp, R); |
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484 | if ((k+l) & 1) p=p_Neg(p, R); |
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485 | MATELEM(result,l,k) = p; |
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486 | k++; |
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487 | idGetNextChoise(ar,a->ncols,&colch,colchoise); |
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488 | } |
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489 | idGetNextChoise(ar,a->nrows,&rowch,rowchoise); |
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490 | l++; |
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491 | } |
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492 | |
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493 | /*delete the matrix tmp*/ |
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494 | for (i=1; i<=ar; i++) |
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495 | { |
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496 | for (j=1; j<=ar; j++) MATELEM(tmp,i,j) = NULL; |
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497 | } |
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498 | id_Delete((ideal *) &tmp, R); |
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499 | return (result); |
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500 | } |
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501 | |
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502 | ///*2 |
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503 | //*homogenize all elements of matrix (not the matrix itself) |
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504 | //*/ |
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505 | //matrix mpHomogen(matrix a, int v) |
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506 | //{ |
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507 | // int i,j; |
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508 | // poly p; |
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509 | // |
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510 | // for (i=1;i<=MATROWS(a);i++) |
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511 | // { |
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512 | // for (j=1;j<=MATCOLS(a);j++) |
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513 | // { |
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514 | // p=pHomogen(MATELEM(a,i,j),v); |
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515 | // p_Delete(&(MATELEM(a,i,j)), ?); |
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516 | // MATELEM(a,i,j)=p; |
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517 | // } |
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518 | // } |
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519 | // return a; |
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520 | //} |
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521 | |
---|
522 | |
---|
523 | |
---|
524 | /* --------------- internal stuff ------------------- */ |
---|
525 | |
---|
526 | row_col_weight::row_col_weight(int i, int j) |
---|
527 | { |
---|
528 | ym = i; |
---|
529 | yn = j; |
---|
530 | wrow = (float *)omAlloc(i*sizeof(float)); |
---|
531 | wcol = (float *)omAlloc(j*sizeof(float)); |
---|
532 | } |
---|
533 | |
---|
534 | row_col_weight::~row_col_weight() |
---|
535 | { |
---|
536 | if (ym!=0) |
---|
537 | { |
---|
538 | omFreeSize((ADDRESS)wcol, yn*sizeof(float)); |
---|
539 | omFreeSize((ADDRESS)wrow, ym*sizeof(float)); |
---|
540 | } |
---|
541 | } |
---|
542 | |
---|
543 | mp_permmatrix::mp_permmatrix(matrix A, const ring r) : sign(1), R(r) |
---|
544 | { |
---|
545 | a_m = A->nrows; |
---|
546 | a_n = A->ncols; |
---|
547 | this->mpInitMat(); |
---|
548 | Xarray = A->m; |
---|
549 | } |
---|
550 | |
---|
551 | mp_permmatrix::mp_permmatrix(mp_permmatrix *M) |
---|
552 | { |
---|
553 | poly p, *athis, *aM; |
---|
554 | int i, j; |
---|
555 | |
---|
556 | a_m = M->s_m; |
---|
557 | a_n = M->s_n; |
---|
558 | sign = M->sign; |
---|
559 | R = M->R; |
---|
560 | |
---|
561 | this->mpInitMat(); |
---|
562 | Xarray = (poly *)omAlloc0(a_m*a_n*sizeof(poly)); |
---|
563 | for (i=a_m-1; i>=0; i--) |
---|
564 | { |
---|
565 | athis = this->mpRowAdr(i); |
---|
566 | aM = M->mpRowAdr(i); |
---|
567 | for (j=a_n-1; j>=0; j--) |
---|
568 | { |
---|
569 | p = aM[M->qcol[j]]; |
---|
570 | if (p) |
---|
571 | { |
---|
572 | athis[j] = p_Copy(p, R); |
---|
573 | } |
---|
574 | } |
---|
575 | } |
---|
576 | } |
---|
577 | |
---|
578 | mp_permmatrix::~mp_permmatrix() |
---|
579 | { |
---|
580 | int k; |
---|
581 | |
---|
582 | if (a_m != 0) |
---|
583 | { |
---|
584 | omFreeSize((ADDRESS)qrow,a_m*sizeof(int)); |
---|
585 | omFreeSize((ADDRESS)qcol,a_n*sizeof(int)); |
---|
586 | if (Xarray != NULL) |
---|
587 | { |
---|
588 | for (k=a_m*a_n-1; k>=0; k--) |
---|
589 | p_Delete(&Xarray[k], R); |
---|
590 | omFreeSize((ADDRESS)Xarray,a_m*a_n*sizeof(poly)); |
---|
591 | } |
---|
592 | } |
---|
593 | } |
---|
594 | |
---|
595 | int mp_permmatrix::mpGetRdim() { return s_m; } |
---|
596 | |
---|
597 | int mp_permmatrix::mpGetCdim() { return s_n; } |
---|
598 | |
---|
599 | int mp_permmatrix::mpGetSign() { return sign; } |
---|
600 | |
---|
601 | void mp_permmatrix::mpSetSearch(int s) { piv_s = s; } |
---|
602 | |
---|
603 | void mp_permmatrix::mpSaveArray() { Xarray = NULL; } |
---|
604 | |
---|
605 | poly mp_permmatrix::mpGetElem(int r, int c) |
---|
606 | { |
---|
607 | return Xarray[a_n*qrow[r]+qcol[c]]; |
---|
608 | } |
---|
609 | |
---|
610 | void mp_permmatrix::mpSetElem(poly p, int r, int c) |
---|
611 | { |
---|
612 | Xarray[a_n*qrow[r]+qcol[c]] = p; |
---|
613 | } |
---|
614 | |
---|
615 | void mp_permmatrix::mpDelElem(int r, int c) |
---|
616 | { |
---|
617 | p_Delete(&Xarray[a_n*qrow[r]+qcol[c]], R); |
---|
618 | } |
---|
619 | |
---|
620 | /* |
---|
621 | * the Bareiss-type elimination with division by div (div != NULL) |
---|
622 | */ |
---|
623 | void mp_permmatrix::mpElimBareiss(poly div) |
---|
624 | { |
---|
625 | poly piv, elim, q1, q2, *ap, *a; |
---|
626 | int i, j, jj; |
---|
627 | |
---|
628 | ap = this->mpRowAdr(s_m); |
---|
629 | piv = ap[qcol[s_n]]; |
---|
630 | for(i=s_m-1; i>=0; i--) |
---|
631 | { |
---|
632 | a = this->mpRowAdr(i); |
---|
633 | elim = a[qcol[s_n]]; |
---|
634 | if (elim != NULL) |
---|
635 | { |
---|
636 | elim = p_Neg(elim, R); |
---|
637 | for (j=s_n-1; j>=0; j--) |
---|
638 | { |
---|
639 | q2 = NULL; |
---|
640 | jj = qcol[j]; |
---|
641 | if (ap[jj] != NULL) |
---|
642 | { |
---|
643 | q2 = SM_MULT(ap[jj], elim, div, R); |
---|
644 | if (a[jj] != NULL) |
---|
645 | { |
---|
646 | q1 = SM_MULT(a[jj], piv, div, R); |
---|
647 | p_Delete(&a[jj], R); |
---|
648 | q2 = p_Add_q(q2, q1, R); |
---|
649 | } |
---|
650 | } |
---|
651 | else if (a[jj] != NULL) |
---|
652 | { |
---|
653 | q2 = SM_MULT(a[jj], piv, div, R); |
---|
654 | } |
---|
655 | if ((q2!=NULL) && div) |
---|
656 | SM_DIV(q2, div, R); |
---|
657 | a[jj] = q2; |
---|
658 | } |
---|
659 | p_Delete(&a[qcol[s_n]], R); |
---|
660 | } |
---|
661 | else |
---|
662 | { |
---|
663 | for (j=s_n-1; j>=0; j--) |
---|
664 | { |
---|
665 | jj = qcol[j]; |
---|
666 | if (a[jj] != NULL) |
---|
667 | { |
---|
668 | q2 = SM_MULT(a[jj], piv, div, R); |
---|
669 | p_Delete(&a[jj], R); |
---|
670 | if (div) |
---|
671 | SM_DIV(q2, div, R); |
---|
672 | a[jj] = q2; |
---|
673 | } |
---|
674 | } |
---|
675 | } |
---|
676 | } |
---|
677 | } |
---|
678 | |
---|
679 | /*2 |
---|
680 | * pivot strategy for Bareiss algorithm |
---|
681 | */ |
---|
682 | int mp_permmatrix::mpPivotBareiss(row_col_weight *C) |
---|
683 | { |
---|
684 | poly p, *a; |
---|
685 | int i, j, iopt, jopt; |
---|
686 | float sum, f1, f2, fo, r, ro, lp; |
---|
687 | float *dr = C->wrow, *dc = C->wcol; |
---|
688 | |
---|
689 | fo = 1.0e20; |
---|
690 | ro = 0.0; |
---|
691 | iopt = jopt = -1; |
---|
692 | |
---|
693 | s_n--; |
---|
694 | s_m--; |
---|
695 | if (s_m == 0) |
---|
696 | return 0; |
---|
697 | if (s_n == 0) |
---|
698 | { |
---|
699 | for(i=s_m; i>=0; i--) |
---|
700 | { |
---|
701 | p = this->mpRowAdr(i)[qcol[0]]; |
---|
702 | if (p) |
---|
703 | { |
---|
704 | f1 = mp_PolyWeight(p, R); |
---|
705 | if (f1 < fo) |
---|
706 | { |
---|
707 | fo = f1; |
---|
708 | if (iopt >= 0) |
---|
709 | p_Delete(&(this->mpRowAdr(iopt)[qcol[0]]), R); |
---|
710 | iopt = i; |
---|
711 | } |
---|
712 | else |
---|
713 | p_Delete(&(this->mpRowAdr(i)[qcol[0]]), R); |
---|
714 | } |
---|
715 | } |
---|
716 | if (iopt >= 0) |
---|
717 | mpReplace(iopt, s_m, sign, qrow); |
---|
718 | return 0; |
---|
719 | } |
---|
720 | this->mpRowWeight(dr); |
---|
721 | this->mpColWeight(dc); |
---|
722 | sum = 0.0; |
---|
723 | for(i=s_m; i>=0; i--) |
---|
724 | sum += dr[i]; |
---|
725 | for(i=s_m; i>=0; i--) |
---|
726 | { |
---|
727 | r = dr[i]; |
---|
728 | a = this->mpRowAdr(i); |
---|
729 | for(j=s_n; j>=0; j--) |
---|
730 | { |
---|
731 | p = a[qcol[j]]; |
---|
732 | if (p) |
---|
733 | { |
---|
734 | lp = mp_PolyWeight(p, R); |
---|
735 | ro = r - lp; |
---|
736 | f1 = ro * (dc[j]-lp); |
---|
737 | if (f1 != 0.0) |
---|
738 | { |
---|
739 | f2 = lp * (sum - ro - dc[j]); |
---|
740 | f2 += f1; |
---|
741 | } |
---|
742 | else |
---|
743 | f2 = lp-r-dc[j]; |
---|
744 | if (f2 < fo) |
---|
745 | { |
---|
746 | fo = f2; |
---|
747 | iopt = i; |
---|
748 | jopt = j; |
---|
749 | } |
---|
750 | } |
---|
751 | } |
---|
752 | } |
---|
753 | if (iopt < 0) |
---|
754 | return 0; |
---|
755 | mpReplace(iopt, s_m, sign, qrow); |
---|
756 | mpReplace(jopt, s_n, sign, qcol); |
---|
757 | return 1; |
---|
758 | } |
---|
759 | |
---|
760 | /*2 |
---|
761 | * pivot strategy for Bareiss algorithm with defined row |
---|
762 | */ |
---|
763 | int mp_permmatrix::mpPivotRow(row_col_weight *C, int row) |
---|
764 | { |
---|
765 | poly p, *a; |
---|
766 | int j, iopt, jopt; |
---|
767 | float sum, f1, f2, fo, r, ro, lp; |
---|
768 | float *dr = C->wrow, *dc = C->wcol; |
---|
769 | |
---|
770 | fo = 1.0e20; |
---|
771 | ro = 0.0; |
---|
772 | iopt = jopt = -1; |
---|
773 | |
---|
774 | s_n--; |
---|
775 | s_m--; |
---|
776 | if (s_m == 0) |
---|
777 | return 0; |
---|
778 | if (s_n == 0) |
---|
779 | { |
---|
780 | p = this->mpRowAdr(row)[qcol[0]]; |
---|
781 | if (p) |
---|
782 | { |
---|
783 | f1 = mp_PolyWeight(p, R); |
---|
784 | if (f1 < fo) |
---|
785 | { |
---|
786 | fo = f1; |
---|
787 | if (iopt >= 0) |
---|
788 | p_Delete(&(this->mpRowAdr(iopt)[qcol[0]]), R); |
---|
789 | iopt = row; |
---|
790 | } |
---|
791 | else |
---|
792 | p_Delete(&(this->mpRowAdr(row)[qcol[0]]), R); |
---|
793 | } |
---|
794 | if (iopt >= 0) |
---|
795 | mpReplace(iopt, s_m, sign, qrow); |
---|
796 | return 0; |
---|
797 | } |
---|
798 | this->mpRowWeight(dr); |
---|
799 | this->mpColWeight(dc); |
---|
800 | sum = 0.0; |
---|
801 | for(j=s_m; j>=0; j--) |
---|
802 | sum += dr[j]; |
---|
803 | r = dr[row]; |
---|
804 | a = this->mpRowAdr(row); |
---|
805 | for(j=s_n; j>=0; j--) |
---|
806 | { |
---|
807 | p = a[qcol[j]]; |
---|
808 | if (p) |
---|
809 | { |
---|
810 | lp = mp_PolyWeight(p, R); |
---|
811 | ro = r - lp; |
---|
812 | f1 = ro * (dc[j]-lp); |
---|
813 | if (f1 != 0.0) |
---|
814 | { |
---|
815 | f2 = lp * (sum - ro - dc[j]); |
---|
816 | f2 += f1; |
---|
817 | } |
---|
818 | else |
---|
819 | f2 = lp-r-dc[j]; |
---|
820 | if (f2 < fo) |
---|
821 | { |
---|
822 | fo = f2; |
---|
823 | iopt = row; |
---|
824 | jopt = j; |
---|
825 | } |
---|
826 | } |
---|
827 | } |
---|
828 | if (iopt < 0) |
---|
829 | return 0; |
---|
830 | mpReplace(iopt, s_m, sign, qrow); |
---|
831 | mpReplace(jopt, s_n, sign, qcol); |
---|
832 | return 1; |
---|
833 | } |
---|
834 | |
---|
835 | void mp_permmatrix::mpToIntvec(intvec *v) |
---|
836 | { |
---|
837 | int i; |
---|
838 | |
---|
839 | for (i=v->rows()-1; i>=0; i--) |
---|
840 | (*v)[i] = qcol[i]+1; |
---|
841 | } |
---|
842 | |
---|
843 | void mp_permmatrix::mpRowReorder() |
---|
844 | { |
---|
845 | int k, i, i1, i2; |
---|
846 | |
---|
847 | if (a_m > a_n) |
---|
848 | k = a_m - a_n; |
---|
849 | else |
---|
850 | k = 0; |
---|
851 | for (i=a_m-1; i>=k; i--) |
---|
852 | { |
---|
853 | i1 = qrow[i]; |
---|
854 | if (i1 != i) |
---|
855 | { |
---|
856 | this->mpRowSwap(i1, i); |
---|
857 | i2 = 0; |
---|
858 | while (qrow[i2] != i) i2++; |
---|
859 | qrow[i2] = i1; |
---|
860 | } |
---|
861 | } |
---|
862 | } |
---|
863 | |
---|
864 | void mp_permmatrix::mpColReorder() |
---|
865 | { |
---|
866 | int k, j, j1, j2; |
---|
867 | |
---|
868 | if (a_n > a_m) |
---|
869 | k = a_n - a_m; |
---|
870 | else |
---|
871 | k = 0; |
---|
872 | for (j=a_n-1; j>=k; j--) |
---|
873 | { |
---|
874 | j1 = qcol[j]; |
---|
875 | if (j1 != j) |
---|
876 | { |
---|
877 | this->mpColSwap(j1, j); |
---|
878 | j2 = 0; |
---|
879 | while (qcol[j2] != j) j2++; |
---|
880 | qcol[j2] = j1; |
---|
881 | } |
---|
882 | } |
---|
883 | } |
---|
884 | |
---|
885 | // private |
---|
886 | void mp_permmatrix::mpInitMat() |
---|
887 | { |
---|
888 | int k; |
---|
889 | |
---|
890 | s_m = a_m; |
---|
891 | s_n = a_n; |
---|
892 | piv_s = 0; |
---|
893 | qrow = (int *)omAlloc(a_m*sizeof(int)); |
---|
894 | qcol = (int *)omAlloc(a_n*sizeof(int)); |
---|
895 | for (k=a_m-1; k>=0; k--) qrow[k] = k; |
---|
896 | for (k=a_n-1; k>=0; k--) qcol[k] = k; |
---|
897 | } |
---|
898 | |
---|
899 | poly * mp_permmatrix::mpRowAdr(int r) |
---|
900 | { |
---|
901 | return &(Xarray[a_n*qrow[r]]); |
---|
902 | } |
---|
903 | |
---|
904 | poly * mp_permmatrix::mpColAdr(int c) |
---|
905 | { |
---|
906 | return &(Xarray[qcol[c]]); |
---|
907 | } |
---|
908 | |
---|
909 | void mp_permmatrix::mpRowWeight(float *wrow) |
---|
910 | { |
---|
911 | poly p, *a; |
---|
912 | int i, j; |
---|
913 | float count; |
---|
914 | |
---|
915 | for (i=s_m; i>=0; i--) |
---|
916 | { |
---|
917 | a = this->mpRowAdr(i); |
---|
918 | count = 0.0; |
---|
919 | for(j=s_n; j>=0; j--) |
---|
920 | { |
---|
921 | p = a[qcol[j]]; |
---|
922 | if (p) |
---|
923 | count += mp_PolyWeight(p, R); |
---|
924 | } |
---|
925 | wrow[i] = count; |
---|
926 | } |
---|
927 | } |
---|
928 | |
---|
929 | void mp_permmatrix::mpColWeight(float *wcol) |
---|
930 | { |
---|
931 | poly p, *a; |
---|
932 | int i, j; |
---|
933 | float count; |
---|
934 | |
---|
935 | for (j=s_n; j>=0; j--) |
---|
936 | { |
---|
937 | a = this->mpColAdr(j); |
---|
938 | count = 0.0; |
---|
939 | for(i=s_m; i>=0; i--) |
---|
940 | { |
---|
941 | p = a[a_n*qrow[i]]; |
---|
942 | if (p) |
---|
943 | count += mp_PolyWeight(p, R); |
---|
944 | } |
---|
945 | wcol[j] = count; |
---|
946 | } |
---|
947 | } |
---|
948 | |
---|
949 | void mp_permmatrix::mpRowSwap(int i1, int i2) |
---|
950 | { |
---|
951 | poly p, *a1, *a2; |
---|
952 | int j; |
---|
953 | |
---|
954 | a1 = &(Xarray[a_n*i1]); |
---|
955 | a2 = &(Xarray[a_n*i2]); |
---|
956 | for (j=a_n-1; j>= 0; j--) |
---|
957 | { |
---|
958 | p = a1[j]; |
---|
959 | a1[j] = a2[j]; |
---|
960 | a2[j] = p; |
---|
961 | } |
---|
962 | } |
---|
963 | |
---|
964 | void mp_permmatrix::mpColSwap(int j1, int j2) |
---|
965 | { |
---|
966 | poly p, *a1, *a2; |
---|
967 | int i, k = a_n*a_m; |
---|
968 | |
---|
969 | a1 = &(Xarray[j1]); |
---|
970 | a2 = &(Xarray[j2]); |
---|
971 | for (i=0; i< k; i+=a_n) |
---|
972 | { |
---|
973 | p = a1[i]; |
---|
974 | a1[i] = a2[i]; |
---|
975 | a2[i] = p; |
---|
976 | } |
---|
977 | } |
---|
978 | |
---|
979 | int mp_permmatrix::mpGetRow() |
---|
980 | { |
---|
981 | return qrow[s_m]; |
---|
982 | } |
---|
983 | |
---|
984 | int mp_permmatrix::mpGetCol() |
---|
985 | { |
---|
986 | return qcol[s_n]; |
---|
987 | } |
---|
988 | |
---|
989 | /// perform replacement for pivot strategy in Bareiss algorithm |
---|
990 | /// change sign of determinant |
---|
991 | static void mpReplace(int j, int n, int &sign, int *perm) |
---|
992 | { |
---|
993 | int k; |
---|
994 | |
---|
995 | if (j != n) |
---|
996 | { |
---|
997 | k = perm[n]; |
---|
998 | perm[n] = perm[j]; |
---|
999 | perm[j] = k; |
---|
1000 | sign = -sign; |
---|
1001 | } |
---|
1002 | } |
---|
1003 | |
---|
1004 | static int mpNextperm(perm * z, int max) |
---|
1005 | { |
---|
1006 | int s, i, k, t; |
---|
1007 | s = max; |
---|
1008 | do |
---|
1009 | { |
---|
1010 | s--; |
---|
1011 | } |
---|
1012 | while ((s > 0) && ((*z)[s] >= (*z)[s+1])); |
---|
1013 | if (s==0) |
---|
1014 | return 0; |
---|
1015 | do |
---|
1016 | { |
---|
1017 | (*z)[s]++; |
---|
1018 | k = 0; |
---|
1019 | do |
---|
1020 | { |
---|
1021 | k++; |
---|
1022 | } |
---|
1023 | while (((*z)[k] != (*z)[s]) && (k!=s)); |
---|
1024 | } |
---|
1025 | while (k < s); |
---|
1026 | for (i=s+1; i <= max; i++) |
---|
1027 | { |
---|
1028 | (*z)[i]=0; |
---|
1029 | do |
---|
1030 | { |
---|
1031 | (*z)[i]++; |
---|
1032 | k=0; |
---|
1033 | do |
---|
1034 | { |
---|
1035 | k++; |
---|
1036 | } |
---|
1037 | while (((*z)[k] != (*z)[i]) && (k != i)); |
---|
1038 | } |
---|
1039 | while (k < i); |
---|
1040 | } |
---|
1041 | s = max+1; |
---|
1042 | do |
---|
1043 | { |
---|
1044 | s--; |
---|
1045 | } |
---|
1046 | while ((s > 0) && ((*z)[s] > (*z)[s+1])); |
---|
1047 | t = 1; |
---|
1048 | for (i=1; i<max; i++) |
---|
1049 | for (k=i+1; k<=max; k++) |
---|
1050 | if ((*z)[k] < (*z)[i]) |
---|
1051 | t = -t; |
---|
1052 | (*z)[0] = t; |
---|
1053 | return s; |
---|
1054 | } |
---|
1055 | |
---|
1056 | static poly mp_Leibnitz(matrix a, const ring R) |
---|
1057 | { |
---|
1058 | int i, e, n; |
---|
1059 | poly p, d; |
---|
1060 | perm z; |
---|
1061 | |
---|
1062 | n = MATROWS(a); |
---|
1063 | memset(&z,0,(n+2)*sizeof(int)); |
---|
1064 | p = p_One(R); |
---|
1065 | for (i=1; i <= n; i++) |
---|
1066 | p = p_Mult_q(p, p_Copy(MATELEM(a, i, i), R), R); |
---|
1067 | d = p; |
---|
1068 | for (i=1; i<= n; i++) |
---|
1069 | z[i] = i; |
---|
1070 | z[0]=1; |
---|
1071 | e = 1; |
---|
1072 | if (n!=1) |
---|
1073 | { |
---|
1074 | while (e) |
---|
1075 | { |
---|
1076 | e = mpNextperm((perm *)&z, n); |
---|
1077 | p = p_One(R); |
---|
1078 | for (i = 1; i <= n; i++) |
---|
1079 | p = p_Mult_q(p, p_Copy(MATELEM(a, i, z[i]), R), R); |
---|
1080 | if (z[0] > 0) |
---|
1081 | d = p_Add_q(d, p, R); |
---|
1082 | else |
---|
1083 | d = p_Sub(d, p, R); |
---|
1084 | } |
---|
1085 | } |
---|
1086 | return d; |
---|
1087 | } |
---|
1088 | |
---|
1089 | static void mp_ElimBar(matrix a0, matrix re, poly div, int lr, int lc, const ring R) |
---|
1090 | { |
---|
1091 | int r=lr-1, c=lc-1; |
---|
1092 | poly *b = a0->m, *x = re->m; |
---|
1093 | poly piv, elim, q1, q2, *ap, *a, *q; |
---|
1094 | int i, j; |
---|
1095 | |
---|
1096 | ap = &b[r*a0->ncols]; |
---|
1097 | piv = ap[c]; |
---|
1098 | for(j=c-1; j>=0; j--) |
---|
1099 | if (ap[j] != NULL) ap[j] = p_Neg(ap[j], R); |
---|
1100 | for(i=r-1; i>=0; i--) |
---|
1101 | { |
---|
1102 | a = &b[i*a0->ncols]; |
---|
1103 | q = &x[i*re->ncols]; |
---|
1104 | if (a[c] != NULL) |
---|
1105 | { |
---|
1106 | elim = a[c]; |
---|
1107 | for (j=c-1; j>=0; j--) |
---|
1108 | { |
---|
1109 | q1 = NULL; |
---|
1110 | if (a[j] != NULL) |
---|
1111 | { |
---|
1112 | q1 = SM_MULT(a[j], piv, div, R); |
---|
1113 | if (ap[j] != NULL) |
---|
1114 | { |
---|
1115 | q2 = SM_MULT(ap[j], elim, div, R); |
---|
1116 | q1 = p_Add_q(q1,q2, R); |
---|
1117 | } |
---|
1118 | } |
---|
1119 | else if (ap[j] != NULL) |
---|
1120 | q1 = SM_MULT(ap[j], elim, div, R); |
---|
1121 | if (q1 != NULL) |
---|
1122 | { |
---|
1123 | if (div) |
---|
1124 | SM_DIV(q1, div, R); |
---|
1125 | q[j] = q1; |
---|
1126 | } |
---|
1127 | } |
---|
1128 | } |
---|
1129 | else |
---|
1130 | { |
---|
1131 | for (j=c-1; j>=0; j--) |
---|
1132 | { |
---|
1133 | if (a[j] != NULL) |
---|
1134 | { |
---|
1135 | q1 = SM_MULT(a[j], piv, div, R); |
---|
1136 | if (div) |
---|
1137 | SM_DIV(q1, div, R); |
---|
1138 | q[j] = q1; |
---|
1139 | } |
---|
1140 | } |
---|
1141 | } |
---|
1142 | } |
---|
1143 | } |
---|
1144 | |
---|