1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="version modwalk.lib 4.0.0.0 Jun_2013 "; // $Id$ |
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3 | category = "Commutative Algebra"; |
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4 | info=" |
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5 | LIBRARY: modwalk.lib Groebner basis convertion |
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6 | |
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7 | AUTHORS: S. Oberfranz oberfran@mathematik.uni-kl.de |
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8 | |
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9 | OVERVIEW: |
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10 | |
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11 | A library for converting Groebner bases of an ideal in the polynomial |
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12 | ring over the rational numbers using modular methods. The procedures are |
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13 | inspired by the following paper: |
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14 | Elizabeth A. Arnold: Modular algorithms for computing Groebner bases. |
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15 | Journal of Symbolic Computation 35, 403-419 (2003). |
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16 | |
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17 | PROCEDURES: |
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18 | modWalk(I); standard basis conversion of I using modular methods (chinese remainder) |
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19 | "; |
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20 | |
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21 | LIB "poly.lib"; |
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22 | LIB "ring.lib"; |
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23 | LIB "parallel.lib"; |
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24 | LIB "rwalk.lib"; |
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25 | LIB "grwalk.lib"; |
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26 | LIB "modstd.lib"; |
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27 | |
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28 | |
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29 | //////////////////////////////////////////////////////////////////////////////// |
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30 | |
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31 | proc modpWalk(def II, int p, int variant, list #) |
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32 | "USAGE: modpWalk(I,p,#); I ideal, p integer, variant integer |
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33 | ASSUME: If size(#) > 0, then |
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34 | #[1] is an intvec describing the current weight vector |
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35 | #[2] is an intvec describing the target weight vector |
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36 | RETURN: ideal - a standard basis of I mod p, integer - p |
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37 | NOTE: The procedure computes a standard basis of the ideal I modulo p and |
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38 | fetches the result to the basering. |
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39 | EXAMPLE: example modpWalk; shows an example |
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40 | " |
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41 | { |
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42 | option(redSB); |
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43 | int k,nvar@r; |
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44 | def R0 = basering; |
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45 | string ordstr_R0 = ordstr(R0); |
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46 | list rl = ringlist(R0); |
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47 | int sizerl = size(rl); |
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48 | int neg = 1 - attrib(R0,"global"); |
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49 | if(typeof(II) == "ideal") |
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50 | { |
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51 | ideal I = II; |
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52 | int radius = 2; |
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53 | int pert_deg = 2; |
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54 | } |
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55 | if(typeof(II) == "list" && typeof(II[1]) == "ideal") |
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56 | { |
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57 | ideal I = II[1]; |
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58 | if(size(II) == 2) |
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59 | { |
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60 | int radius = II[2]; |
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61 | int pert_deg = 2; |
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62 | } |
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63 | if(size(II) == 3) |
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64 | { |
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65 | int radius = II[2]; |
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66 | int pert_deg = II[3]; |
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67 | } |
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68 | } |
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69 | rl[1] = p; |
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70 | int h = homog(I); |
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71 | def @r = ring(rl); |
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72 | setring @r; |
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73 | ideal i = fetch(R0,I); |
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74 | string order; |
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75 | if(system("nblocks") <= 2) |
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76 | { |
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77 | if(find(ordstr_R0, "M") + find(ordstr_R0, "lp") + find(ordstr_R0, "rp") <= 0) |
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78 | { |
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79 | order = "simple"; |
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80 | } |
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81 | } |
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82 | |
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83 | //------------------------- make i homogeneous ----------------------------- |
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84 | |
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85 | if(!mixedTest() && !h) |
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86 | { |
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87 | if(!((find(ordstr_R0, "M") > 0) || (find(ordstr_R0, "a") > 0) || neg)) |
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88 | { |
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89 | if(!((order == "simple") || (sizerl > 4))) |
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90 | { |
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91 | list rl@r = ringlist(@r); |
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92 | nvar@r = nvars(@r); |
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93 | intvec w; |
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94 | for(k = 1; k <= nvar@r; k++) |
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95 | { |
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96 | w[k] = deg(var(k)); |
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97 | } |
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98 | w[nvar@r + 1] = 1; |
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99 | rl@r[2][nvar@r + 1] = "homvar"; |
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100 | rl@r[3][2][2] = w; |
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101 | def HomR = ring(rl@r); |
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102 | setring HomR; |
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103 | ideal i = imap(@r, i); |
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104 | i = homog(i, homvar); |
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105 | } |
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106 | } |
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107 | } |
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108 | |
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109 | //------------------------- compute a standard basis mod p ----------------------------- |
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110 | |
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111 | if(variant == 1) |
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112 | { |
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113 | if(size(#)>0) |
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114 | { |
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115 | i = rwalk(i,radius,pert_deg,#); |
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116 | // rwalk(i,radius,pert_deg,#); std(i); |
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117 | } |
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118 | else |
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119 | { |
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120 | i = rwalk(i,radius,pert_deg); |
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121 | } |
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122 | } |
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123 | if(variant == 2) |
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124 | { |
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125 | if(size(#) == 2) |
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126 | { |
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127 | i = gwalk(i,#); |
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128 | } |
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129 | else |
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130 | { |
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131 | i = gwalk(i); |
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132 | } |
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133 | } |
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134 | if(variant == 3) |
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135 | { |
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136 | if(size(#) == 2) |
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137 | { |
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138 | i = frandwalk(i,radius,#); |
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139 | } |
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140 | else |
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141 | { |
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142 | i = frandwalk(i,radius); |
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143 | } |
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144 | } |
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145 | if(variant == 4) |
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146 | { |
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147 | if(size(#) == 2) |
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148 | { |
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149 | trwalk(i,radius,pert_deg,#); |
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150 | } |
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151 | else |
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152 | { |
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153 | trwalk(i,radius,pert_deg); |
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154 | } |
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155 | } |
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156 | if(!mixedTest() && !h) |
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157 | { |
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158 | if(!((find(ordstr_R0, "M") > 0) || (find(ordstr_R0, "a") > 0) || neg)) |
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159 | { |
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160 | if(!((order == "simple") || (sizerl > 4))) |
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161 | { |
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162 | i = subst(i, homvar, 1); |
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163 | i = simplify(i, 34); |
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164 | setring @r; |
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165 | i = imap(HomR, i); |
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166 | i = interred(i); |
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167 | kill HomR; |
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168 | } |
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169 | } |
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170 | } |
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171 | setring R0; |
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172 | return(list(fetch(@r,i),p)); |
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173 | } |
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174 | example |
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175 | { |
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176 | "EXAMPLE:"; echo = 2; |
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177 | option(redSB); |
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178 | |
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179 | int p = 181; |
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180 | intvec a = 2,1,3,4; |
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181 | intvec b = 1,9,1,1; |
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182 | ring ra = 0,(w,x,y,z),(a(a),lp); |
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183 | ideal I = std(cyclic(4)); |
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184 | ring rb = 0,(w,x,y,z),(a(b),lp); |
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185 | ideal I = imap(ra,I); |
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186 | modpWalk(I,p,1,a,b); |
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187 | std(I); |
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188 | /* ~ |
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189 | list P = modpWalk(I,p,2,a,b); |
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190 | P; |
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191 | list P1 = modpWalk(I,p,1,a,b); |
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192 | P1; |
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193 | ring R =181,(w,x,y,z),(a(b),lp); |
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194 | std(cyclic(4)); |
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195 | ~ |
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196 | a = 1,3,5,7,9; |
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197 | b = 5,2,7,2,6; |
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198 | ring raa = 0,(v,w,x,y,z),(a(a),lp); |
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199 | ideal I = std(cyclic(5)); |
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200 | ring rbb =0,(v,w,x,y,z),(a(b),lp); |
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201 | ideal I = imap(raa,I); |
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202 | int q = 32003; |
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203 | list Q = modpWalk(I,q,1,a,b); |
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204 | ideal J = rwalk(I,2,3,a,b); |
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205 | J; |
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206 | std(I); |
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207 | Q; |
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208 | ring r = 32003,(v,w,x,y,z),(a(b),lp); |
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209 | std(cyclic(5));*/ |
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210 | } |
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211 | |
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212 | //////////////////////////////////////////////////////////////////////////////// |
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213 | |
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214 | proc modWalk(def II, int variant, list #) |
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215 | "USAGE: modWalk(II); II ideal or list(ideal,int) |
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216 | ASSUME: If variant = 1 the random walk algorithm with radius II[2] is applied |
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217 | to II[1] if II = list(ideal, int). It is applied to II with radius 2 |
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218 | if II is an ideal. If variant = 2, the Groebner walk algorithm is |
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219 | applied to II[1] or to II, respectively. |
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220 | If size(#) > 0, then # contains either 1, 2 or 4 integers such that |
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221 | @* - #[1] is the number of available processors for the computation, |
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222 | @* - #[2] is an optional parameter for the exactness of the computation, |
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223 | if #[2] = 1, the procedure computes a standard basis for sure, |
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224 | @* - #[3] is the number of primes until the first lifting, |
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225 | @* - #[4] is the constant number of primes between two liftings until |
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226 | the computation stops. |
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227 | RETURN: a standard basis of I if no warning appears. |
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228 | NOTE: The procedure converts a standard basis of I (over the rational |
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229 | numbers) from the ordering \"a(v),lp\", "dp\" or \"Dp\" to the ordering |
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230 | \"(a(w),lp\" or \"a(1,0,...,0),lp\" by using modular methods. |
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231 | By default the procedure computes a standard basis of I for sure, but |
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232 | if the optional parameter #[2] = 0, it computes a standard basis of I |
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233 | with high probability. |
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234 | EXAMPLE: example modWalk; shows an example |
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235 | " |
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236 | { |
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237 | int TT = timer; |
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238 | int RT = rtimer; |
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239 | int i,j,pTest,sizeTest,weighted,n1; |
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240 | bigint N; |
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241 | |
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242 | def R0 = basering; |
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243 | list rl = ringlist(R0); |
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244 | if((npars(R0) > 0) || (rl[1] > 0)) |
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245 | { |
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246 | ERROR("Characteristic of basering should be zero, basering should have no parameters."); |
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247 | } |
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248 | |
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249 | if(typeof(II) == "ideal") |
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250 | { |
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251 | ideal I = II; |
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252 | kill II; |
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253 | list II; |
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254 | II[1] = I; |
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255 | II[2] = 2; |
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256 | II[3] = 2; |
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257 | } |
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258 | else |
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259 | { |
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260 | if(typeof(II) == "list" && typeof(II[1]) == "ideal") |
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261 | { |
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262 | ideal I = II[1]; |
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263 | if(size(II) == 1) |
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264 | { |
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265 | II[2] = 2; |
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266 | II[3] = 2; |
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267 | } |
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268 | if(size(II) == 2) |
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269 | { |
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270 | II[3] = 2; |
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271 | } |
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272 | |
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273 | } |
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274 | else |
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275 | { |
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276 | ERROR("Unexpected type of input."); |
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277 | } |
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278 | } |
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279 | |
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280 | //-------------------- Initialize optional parameters ------------------------ |
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281 | n1 = system("cpu"); |
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282 | if(size(#) == 0) |
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283 | { |
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284 | int exactness = 1; |
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285 | int n2 = 10; |
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286 | int n3 = 10; |
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287 | } |
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288 | else |
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289 | { |
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290 | if(size(#) == 1) |
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291 | { |
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292 | if(typeof(#[1]) == "int") |
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293 | { |
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294 | if(#[1] < n1) |
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295 | { |
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296 | n1 = #[1]; |
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297 | } |
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298 | int exactness = 1; |
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299 | if(n1 >= 10) |
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300 | { |
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301 | int n2 = n1 + 1; |
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302 | int n3 = n1; |
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303 | } |
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304 | else |
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305 | { |
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306 | int n2 = 10; |
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307 | int n3 = 10; |
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308 | } |
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309 | } |
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310 | else |
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311 | { |
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312 | ERROR("Unexpected type of input."); |
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313 | } |
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314 | } |
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315 | if(size(#) == 2) |
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316 | { |
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317 | if(typeof(#[1]) == "int" && typeof(#[2]) == "int") |
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318 | { |
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319 | if(#[1] < n1) |
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320 | { |
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321 | n1 = #[1]; |
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322 | } |
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323 | int exactness = #[2]; |
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324 | if(n1 >= 10) |
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325 | { |
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326 | int n2 = n1 + 1; |
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327 | int n3 = n1; |
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328 | } |
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329 | else |
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330 | { |
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331 | int n2 = 10; |
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332 | int n3 = 10; |
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333 | } |
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334 | } |
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335 | else |
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336 | { |
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337 | if(typeof(#[1]) == "intvec" && typeof(#[2]) == "intvec") |
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338 | { |
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339 | intvec curr_weight = #[1]; |
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340 | intvec target_weight = #[2]; |
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341 | weighted = 1; |
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342 | int n2 = 10; |
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343 | int n3 = 10; |
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344 | int exactness = 1; |
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345 | } |
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346 | else |
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347 | { |
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348 | ERROR("Unexpected type of input."); |
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349 | } |
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350 | } |
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351 | } |
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352 | if(size(#) == 3) |
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353 | { |
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354 | if(typeof(#[1]) == "intvec" && typeof(#[2]) == "intvec" && typeof(#[3]) == "int") |
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355 | { |
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356 | intvec curr_weight = #[1]; |
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357 | intvec target_weight = #[2]; |
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358 | weighted = 1; |
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359 | n1 = #[3]; |
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360 | int n2 = 10; |
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361 | int n3 = 10; |
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362 | int exactness = 1; |
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363 | } |
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364 | else |
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365 | { |
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366 | ERROR("Unexpected type of input."); |
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367 | } |
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368 | } |
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369 | if(size(#) == 4) |
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370 | { |
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371 | if(typeof(#[1]) == "intvec" && typeof(#[2]) == "intvec" && typeof(#[3]) == "int" && typeof(#[4]) == "int") |
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372 | { |
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373 | intvec curr_weight = #[1]; |
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374 | intvec target_weight = #[2]; |
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375 | weighted = 1; |
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376 | if(#[1] < n1) |
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377 | { |
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378 | n1 = #[3]; |
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379 | } |
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380 | int exactness = #[4]; |
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381 | if(n1 >= 10) |
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382 | { |
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383 | int n2 = n1 + 1; |
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384 | int n3 = n1; |
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385 | } |
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386 | else |
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387 | { |
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388 | int n2 = 10; |
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389 | int n3 = 10; |
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390 | } |
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391 | } |
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392 | else |
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393 | { |
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394 | if(typeof(#[1]) == "int" && typeof(#[2]) == "int" && typeof(#[3]) == "int" && typeof(#[4]) == "int") |
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395 | { |
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396 | if(#[1] < n1) |
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397 | { |
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398 | n1 = #[1]; |
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399 | } |
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400 | int exactness = #[2]; |
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401 | if(n1 >= #[3]) |
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402 | { |
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403 | int n2 = n1 + 1; |
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404 | } |
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405 | else |
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406 | { |
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407 | int n2 = #[3]; |
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408 | } |
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409 | if(n1 >= #[4]) |
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410 | { |
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411 | int n3 = n1; |
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412 | } |
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413 | else |
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414 | { |
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415 | int n3 = #[4]; |
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416 | } |
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417 | } |
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418 | else |
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419 | { |
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420 | ERROR("Unexpected type of input."); |
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421 | } |
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422 | } |
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423 | } |
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424 | if(size(#) == 6) |
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425 | { |
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426 | if(typeof(#[1]) == "intvec" && typeof(#[2]) == "intvec" && typeof(#[3]) == "int" && typeof(#[4]) == "int" && typeof(#[5]) == "int" && typeof(#[6]) == "int") |
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427 | { |
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428 | intvec curr_weight = #[1]; |
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429 | intvec target_weight = #[2]; |
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430 | weighted = 1; |
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431 | if(#[3] < n1) |
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432 | { |
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433 | n1 = #[3]; |
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434 | } |
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435 | int exactness = #[4]; |
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436 | if(n1 >= #[5]) |
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437 | { |
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438 | int n2 = n1 + 1; |
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439 | } |
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440 | else |
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441 | { |
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442 | int n2 = #[5]; |
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443 | } |
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444 | if(n1 >= #[6]) |
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445 | { |
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446 | int n3 = n1; |
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447 | } |
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448 | else |
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449 | { |
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450 | int n3 = #[6]; |
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451 | } |
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452 | } |
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453 | else |
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454 | { |
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455 | ERROR("Expected list(intvec,intvec,int,int,int,int) as optional parameter list."); |
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456 | } |
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457 | } |
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458 | if(size(#) == 1 || size(#) == 5 || size(#) > 6) |
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459 | { |
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460 | ERROR("Expected 0,2,3,4 or 5 optional arguments."); |
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461 | } |
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462 | } |
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463 | if(printlevel >= 10) |
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464 | { |
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465 | "n1 = "+string(n1)+", n2 = "+string(n2)+", n3 = "+string(n3)+", exactness = "+string(exactness); |
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466 | } |
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467 | |
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468 | //------------------------- Save current options ----------------------------- |
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469 | intvec opt = option(get); |
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470 | option(redSB); |
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471 | |
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472 | //-------------------- Initialize the list of primes ------------------------- |
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473 | int tt = timer; |
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474 | int rt = rtimer; |
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475 | int en = 2134567879; |
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476 | int an = 1000000000; |
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477 | intvec L = primeList(I,n2); |
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478 | if(n2 > 4) |
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479 | { |
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480 | // L[5] = prime(random(an,en)); |
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481 | } |
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482 | if(printlevel >= 10) |
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483 | { |
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484 | "CPU-time for primeList: "+string(timer-tt)+" seconds."; |
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485 | "Real-time for primeList: "+string(rtimer-rt)+" seconds."; |
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486 | } |
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487 | int h = homog(I); |
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488 | list P,T1,T2,LL,Arguments,PP; |
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489 | ideal J,K,H; |
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490 | |
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491 | //------------------- parallelized Groebner Walk in positive characteristic -------------------- |
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492 | |
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493 | if(weighted) |
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494 | { |
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495 | for(i=1; i<=size(L); i++) |
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496 | { |
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497 | Arguments[i] = list(II,L[i],variant,list(curr_weight,target_weight)); |
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498 | } |
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499 | } |
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500 | else |
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501 | { |
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502 | for(i=1; i<=size(L); i++) |
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503 | { |
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504 | Arguments[i] = list(II,L[i],variant); |
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505 | } |
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506 | } |
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507 | P = parallelWaitAll("modpWalk",Arguments); |
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508 | for(i=1; i<=size(P); i++) |
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509 | { |
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510 | T1[i] = P[i][1]; |
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511 | T2[i] = bigint(P[i][2]); |
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512 | } |
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513 | |
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514 | while(1) |
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515 | { |
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516 | LL = deleteUnluckyPrimes(T1,T2,h); |
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517 | T1 = LL[1]; |
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518 | T2 = LL[2]; |
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519 | //------------------- Now all leading ideals are the same -------------------- |
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520 | //------------------- Lift results to basering via farey --------------------- |
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521 | |
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522 | tt = timer; rt = rtimer; |
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523 | N = T2[1]; |
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524 | for(i=2; i<=size(T2); i++) |
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525 | { |
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526 | N = N*T2[i]; |
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527 | } |
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528 | H = chinrem(T1,T2); |
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529 | //J = parallelFarey(H,N,n1); |
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530 | J=farey(H,N); |
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531 | if(printlevel >= 10) |
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532 | { |
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533 | "CPU-time for lifting-process is "+string(timer - tt)+" seconds."; |
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534 | "Real-time for lifting-process is "+string(rtimer - rt)+" seconds."; |
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535 | } |
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536 | |
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537 | //---------------- Test if we already have a standard basis of I -------------- |
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538 | |
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539 | tt = timer; rt = rtimer; |
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540 | //pTest = pTestSB(I,J,L,variant); |
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541 | pTest = primeTestSB(I,J,L,variant); |
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542 | if(printlevel >= 10) |
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543 | { |
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544 | "CPU-time for pTest is "+string(timer - tt)+" seconds."; |
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545 | "Real-time for pTest is "+string(rtimer - rt)+" seconds."; |
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546 | } |
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547 | if(pTest) |
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548 | { |
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549 | if(printlevel >= 10) |
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550 | { |
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551 | "CPU-time for computation without final tests is "+string(timer - TT)+" seconds."; |
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552 | "Real-time for computation without final tests is "+string(rtimer - RT)+" seconds."; |
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553 | } |
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554 | attrib(J,"isSB",1); |
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555 | if(exactness == 0) |
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556 | { |
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557 | option(set, opt); |
---|
558 | return(J); |
---|
559 | } |
---|
560 | else |
---|
561 | { |
---|
562 | tt = timer; |
---|
563 | rt = rtimer; |
---|
564 | sizeTest = 1 - isIdealIncluded(I,J,n1); |
---|
565 | if(printlevel >= 10) |
---|
566 | { |
---|
567 | "CPU-time for checking if I subset <G> is "+string(timer - tt)+" seconds."; |
---|
568 | "Real-time for checking if I subset <G> is "+string(rtimer - rt)+" seconds."; |
---|
569 | } |
---|
570 | if(sizeTest == 0) |
---|
571 | { |
---|
572 | tt = timer; |
---|
573 | rt = rtimer; |
---|
574 | K = std(J); |
---|
575 | if(printlevel >= 10) |
---|
576 | { |
---|
577 | "CPU-time for last std-computation is "+string(timer - tt)+" seconds."; |
---|
578 | "Real-time for last std-computation is "+string(rtimer - rt)+" seconds."; |
---|
579 | } |
---|
580 | if(size(reduce(K,J)) == 0) |
---|
581 | { |
---|
582 | option(set, opt); |
---|
583 | return(J); |
---|
584 | } |
---|
585 | } |
---|
586 | } |
---|
587 | } |
---|
588 | //-------------- We do not already have a standard basis of I, therefore do the main computation for more primes -------------- |
---|
589 | |
---|
590 | T1 = H; |
---|
591 | T2 = N; |
---|
592 | j = size(L)+1; |
---|
593 | tt = timer; rt = rtimer; |
---|
594 | L = primeList(I,n3,L,n1); |
---|
595 | L; |
---|
596 | if(printlevel >= 10) |
---|
597 | { |
---|
598 | "CPU-time for primeList: "+string(timer-tt)+" seconds."; |
---|
599 | "Real-time for primeList: "+string(rtimer-rt)+" seconds."; |
---|
600 | } |
---|
601 | Arguments = list(); |
---|
602 | PP = list(); |
---|
603 | if(weighted) |
---|
604 | { |
---|
605 | for(i=j; i<=size(L); i++) |
---|
606 | { |
---|
607 | //Arguments[i-j+1] = list(II,L[i],variant,list(curr_weight,target_weight)); |
---|
608 | Arguments[size(Arguments)+1] = list(II,L[i],variant,list(curr_weight,target_weight)); |
---|
609 | } |
---|
610 | } |
---|
611 | else |
---|
612 | { |
---|
613 | for(i=j; i<=size(L); i++) |
---|
614 | { |
---|
615 | //Arguments[i-j+1] = list(II,L[i],variant); |
---|
616 | Arguments[size(Arguments)+1] = list(II,L[i],variant); |
---|
617 | } |
---|
618 | } |
---|
619 | PP = parallelWaitAll("modpWalk",Arguments); |
---|
620 | if(printlevel >= 10) |
---|
621 | { |
---|
622 | "parallel modpWalk"; |
---|
623 | // ~ |
---|
624 | } |
---|
625 | for(i=1; i<=size(PP); i++) |
---|
626 | { |
---|
627 | //P[size(P) + 1] = PP[i]; |
---|
628 | T1[size(T1) + 1] = PP[i][1]; |
---|
629 | T2[size(T2) + 1] = bigint(PP[i][2]); |
---|
630 | } |
---|
631 | } |
---|
632 | if(printlevel >= 10) |
---|
633 | { |
---|
634 | "CPU-time for computation with final tests is "+string(timer - TT)+" seconds."; |
---|
635 | "Real-time for computation with final tests is "+string(rtimer - RT)+" seconds."; |
---|
636 | } |
---|
637 | } |
---|
638 | |
---|
639 | example |
---|
640 | { |
---|
641 | "EXAMPLE:"; |
---|
642 | echo = 2; |
---|
643 | ring R=0,(x,y,z),lp; |
---|
644 | ideal I=-x+y2z-z,xz+1,x2+y2-1; |
---|
645 | // I is a standard basis in dp |
---|
646 | modWalk(I,1); |
---|
647 | modWalk(I,2,2,0); |
---|
648 | modWalk(I,3,system("cpu"),0); |
---|
649 | std(I); |
---|
650 | |
---|
651 | ring r0=0,x(1..6),dp; |
---|
652 | ideal i0=std(cyclic(6)); |
---|
653 | ring r=0,x(1..6),lp; |
---|
654 | ideal i=fetch(r0,i0); |
---|
655 | modWalk(i,1,system("cpu"),0); |
---|
656 | modWalk(i,3); |
---|
657 | } |
---|
658 | |
---|
659 | //////////////////////////////////////////////////////////////////////////////// |
---|
660 | proc isIdealIncluded(ideal I, ideal J, int n1) |
---|
661 | "USAGE: isIdealIncluded(I,J,int n1); I ideal, J ideal, n1 integer |
---|
662 | " |
---|
663 | { |
---|
664 | if(n1 > 1) |
---|
665 | { |
---|
666 | int k; |
---|
667 | list args,results; |
---|
668 | for(k=1; k<=size(I); k++) |
---|
669 | { |
---|
670 | args[k] = list(ideal(I[k]),J,1); |
---|
671 | } |
---|
672 | results = parallelWaitAll("reduce",args); |
---|
673 | for(k=1; k<=size(results); k++) |
---|
674 | { |
---|
675 | if(results[k] == 0) |
---|
676 | { |
---|
677 | return(1); |
---|
678 | } |
---|
679 | } |
---|
680 | return(0); |
---|
681 | } |
---|
682 | else |
---|
683 | { |
---|
684 | if(reduce(I,J,1) == 0) |
---|
685 | { |
---|
686 | return(1); |
---|
687 | } |
---|
688 | else |
---|
689 | { |
---|
690 | return(0); |
---|
691 | } |
---|
692 | } |
---|
693 | } |
---|
694 | |
---|
695 | //////////////////////////////////////////////////////////////////////////////// |
---|
696 | proc parallelChinrem(list T1, list T2, int n1) |
---|
697 | "USAGE: parallelChinrem(T1,T2); T1 list of ideals, T2 list of primes, n1 integer" |
---|
698 | { |
---|
699 | int i,j,k; |
---|
700 | |
---|
701 | ideal H,J; |
---|
702 | |
---|
703 | list arguments_chinrem,results_chinrem; |
---|
704 | for(i=1; i<=size(T1); i++) |
---|
705 | { |
---|
706 | J = ideal(T1[i]); |
---|
707 | attrib(J,"isSB",1); |
---|
708 | arguments_chinrem[size(arguments_chinrem)+1] = list(list(J),T2); |
---|
709 | } |
---|
710 | results_chinrem = parallelWaitAll("chinrem",arguments_chinrem); |
---|
711 | for(j=1; j <= size(results_chinrem); j++) |
---|
712 | { |
---|
713 | J = results_chinrem[j]; |
---|
714 | attrib(J,"isSB",1); |
---|
715 | if(isIdealIncluded(J,H,n1) == 0) |
---|
716 | { |
---|
717 | if(H == 0) |
---|
718 | { |
---|
719 | H = J; |
---|
720 | } |
---|
721 | else |
---|
722 | { |
---|
723 | H = H,J; |
---|
724 | } |
---|
725 | } |
---|
726 | } |
---|
727 | return(H); |
---|
728 | } |
---|
729 | |
---|
730 | //////////////////////////////////////////////////////////////////////////////// |
---|
731 | proc parallelFarey(ideal H, bigint N, int n1) |
---|
732 | "USAGE: parallelFarey(H,N,n1); H ideal, N bigint, n1 integer |
---|
733 | " |
---|
734 | { |
---|
735 | int i,j; |
---|
736 | int ii = 1; |
---|
737 | list arguments_farey,results_farey; |
---|
738 | for(i=1; i<=size(H); i++) |
---|
739 | { |
---|
740 | for(j=1; j<=size(H[i]); j++) |
---|
741 | { |
---|
742 | arguments_farey[size(arguments_farey)+1] = list(H[i][j],N); |
---|
743 | } |
---|
744 | } |
---|
745 | results_farey = parallelWaitAll("farey",arguments_farey); |
---|
746 | ideal J,K; |
---|
747 | poly f_farey; |
---|
748 | while(ii<=size(results_farey)) |
---|
749 | { |
---|
750 | for(i=1; i<=size(H); i++) |
---|
751 | { |
---|
752 | f_farey = 0; |
---|
753 | for(j=1; j<=size(H[i]); j++) |
---|
754 | { |
---|
755 | f_farey = f_farey + results_farey[ii][1]; |
---|
756 | ii++; |
---|
757 | } |
---|
758 | K = ideal(f_farey); |
---|
759 | attrib(K,"isSB",1); |
---|
760 | attrib(J,"isSB",1); |
---|
761 | if(isIdealIncluded(K,J,n1) == 0) |
---|
762 | { |
---|
763 | if(J == 0) |
---|
764 | { |
---|
765 | J = K; |
---|
766 | } |
---|
767 | else |
---|
768 | { |
---|
769 | J = J,K; |
---|
770 | } |
---|
771 | } |
---|
772 | } |
---|
773 | } |
---|
774 | return(J); |
---|
775 | } |
---|
776 | |
---|
777 | proc primeTestSB(def II, ideal J, list L, int variant, list #) |
---|
778 | "USAGE: primeTestSB(I,J,L,variant,#); I,J ideals, L intvec of primes, variant int |
---|
779 | RETURN: 1 (resp. 0) if for a randomly chosen prime p that is not in L |
---|
780 | J mod p is (resp. is not) a standard basis of I mod p |
---|
781 | EXAMPLE: example primeTestSB; shows an example |
---|
782 | " |
---|
783 | { |
---|
784 | if(typeof(II) == "ideal") |
---|
785 | { |
---|
786 | ideal I = II; |
---|
787 | int radius = 2; |
---|
788 | } |
---|
789 | if(typeof(II) == "list") |
---|
790 | { |
---|
791 | ideal I = II[1]; |
---|
792 | int radius = II[2]; |
---|
793 | } |
---|
794 | |
---|
795 | int i,j,k,p; |
---|
796 | def R = basering; |
---|
797 | list r = ringlist(R); |
---|
798 | |
---|
799 | while(!j) |
---|
800 | { |
---|
801 | j = 1; |
---|
802 | p = prime(random(1000000000,2134567879)); |
---|
803 | for(i = 1; i <= size(L); i++) |
---|
804 | { |
---|
805 | if(p == L[i]) |
---|
806 | { |
---|
807 | j = 0; |
---|
808 | break; |
---|
809 | } |
---|
810 | } |
---|
811 | if(j) |
---|
812 | { |
---|
813 | for(i = 1; i <= ncols(I); i++) |
---|
814 | { |
---|
815 | for(k = 2; k <= size(I[i]); k++) |
---|
816 | { |
---|
817 | if((denominator(leadcoef(I[i][k])) mod p) == 0) |
---|
818 | { |
---|
819 | j = 0; |
---|
820 | break; |
---|
821 | } |
---|
822 | } |
---|
823 | if(!j) |
---|
824 | { |
---|
825 | break; |
---|
826 | } |
---|
827 | } |
---|
828 | } |
---|
829 | if(j) |
---|
830 | { |
---|
831 | if(!primeTest(I,p)) |
---|
832 | { |
---|
833 | j = 0; |
---|
834 | } |
---|
835 | } |
---|
836 | } |
---|
837 | r[1] = p; |
---|
838 | def @R = ring(r); |
---|
839 | setring @R; |
---|
840 | ideal I = imap(R,I); |
---|
841 | ideal J = imap(R,J); |
---|
842 | attrib(J,"isSB",1); |
---|
843 | |
---|
844 | int t = timer; |
---|
845 | j = 1; |
---|
846 | if(isIncluded(I,J) == 0) |
---|
847 | { |
---|
848 | j = 0; |
---|
849 | } |
---|
850 | if(printlevel >= 11) |
---|
851 | { |
---|
852 | "isIncluded(I,J) takes "+string(timer - t)+" seconds"; |
---|
853 | "j = "+string(j); |
---|
854 | } |
---|
855 | t = timer; |
---|
856 | if(j) |
---|
857 | { |
---|
858 | if(size(#) > 0) |
---|
859 | { |
---|
860 | ideal K = modpWalk(I,p,variant,#)[1]; |
---|
861 | } |
---|
862 | else |
---|
863 | { |
---|
864 | ideal K = modpWalk(I,p,variant)[1]; |
---|
865 | } |
---|
866 | t = timer; |
---|
867 | if(isIncluded(J,K) == 0) |
---|
868 | { |
---|
869 | j = 0; |
---|
870 | } |
---|
871 | if(printlevel >= 11) |
---|
872 | { |
---|
873 | "isIncluded(K,J) takes "+string(timer - t)+" seconds"; |
---|
874 | "j = "+string(j); |
---|
875 | } |
---|
876 | } |
---|
877 | setring R; |
---|
878 | |
---|
879 | return(j); |
---|
880 | } |
---|
881 | example |
---|
882 | { "EXAMPLE:"; echo = 2; |
---|
883 | intvec L = 2,3,5; |
---|
884 | ring r = 0,(x,y,z),lp; |
---|
885 | ideal I = x+1,x+y+1; |
---|
886 | ideal J = x+1,y; |
---|
887 | primeTestSB(I,I,L,1); |
---|
888 | primeTestSB(I,J,L,1); |
---|
889 | } |
---|