1 | ////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: standard.lib,v 1.70 2005-05-10 17:56:16 Singular Exp $"; |
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3 | category="Miscellaneous"; |
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4 | info=" |
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5 | LIBRARY: standard.lib Procedures which are always loaded at Start-up |
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6 | |
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7 | PROCEDURES: |
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8 | stdfglm(ideal[,ord]) standard basis of ideal via fglm [and ordering ord] |
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9 | stdhilb(ideal[,h]) standard basis of ideal using the Hilbert function |
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10 | groebner(ideal/module) standard basis using a heuristically chosen method |
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11 | quot(any,any[,n]) quotient using heuristically chosen method |
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12 | res(ideal/module,[i]) free resolution of ideal or module |
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13 | sprintf(fmt,...) returns fomatted string |
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14 | fprintf(link,fmt,..) writes formatted string to link |
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15 | printf(fmt,...) displays formatted string |
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16 | "; |
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17 | |
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18 | ////////////////////////////////////////////////////////////////////////////// |
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19 | |
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20 | proc stdfglm (ideal i, list #) |
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21 | "SYNTAX: @code{stdfglm (} ideal_expression @code{)} @* |
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22 | @code{stdfglm (} ideal_expression@code{,} string_expression @code{)} |
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23 | TYPE: ideal |
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24 | PURPOSE: computes the standard basis of the ideal in the basering |
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25 | via @code{fglm} (from the ordering given as the second argument |
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26 | to the ordering of the basering).@* |
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27 | If no second argument is given, \"dp\" is used. |
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28 | SEE ALSO: fglm, groebner, std, stdhilb |
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29 | KEYWORDS: fglm |
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30 | EXAMPLE: example stdfglm; shows an example" |
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31 | { |
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32 | string os; |
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33 | def dr= basering; |
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34 | if( (size(#)==0) or (typeof(#[1]) != "string") ) |
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35 | { |
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36 | os = "dp(" + string( nvars(dr) ) + ")"; |
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37 | if ( (find( ordstr(dr), os ) != 0) and (find( ordstr(dr), "a") == 0) ) |
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38 | { |
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39 | os= "Dp"; |
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40 | } |
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41 | else |
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42 | { |
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43 | os= "dp"; |
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44 | } |
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45 | } |
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46 | else { os = #[1]; } |
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47 | execute("ring sr=("+charstr(dr)+"),("+varstr(dr)+"),"+os+";"); |
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48 | ideal i= fetch(dr,i); |
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49 | intvec opt= option(get); |
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50 | option(redSB); |
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51 | i=std(i); |
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52 | option(set,opt); |
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53 | setring dr; |
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54 | return (fglm(sr,i)); |
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55 | } |
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56 | example |
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57 | { "EXAMPLE:"; echo = 2; |
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58 | ring r=0,(x,y,z),lp; |
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59 | ideal i=y3+x2,x2y+x2,x3-x2,z4-x2-y; |
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60 | ideal i1=stdfglm(i); //uses fglm from "dp" to "lp" |
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61 | i1; |
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62 | ideal i2=stdfglm(i,"Dp"); //uses fglm from "Dp" to "lp" |
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63 | i2; |
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64 | } |
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65 | ///////////////////////////////////////////////////////////////////////////// |
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66 | |
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67 | proc stdhilb(ideal i,list #) |
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68 | "SYNTAX: @code{stdhilb (} ideal_expression @code{)} @* |
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69 | @code{stdhilb (} ideal_expression@code{,} intvec_expression @code{)} |
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70 | TYPE: ideal |
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71 | PURPOSE: computes the standard basis of the homogeneous ideal in the basering, |
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72 | via a Hilbert driven standard basis computation.@* |
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73 | An optional second argument will be used as 1st Hilbert function. |
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74 | ASSUME: The optional second argument is the first Hilbert series as computed |
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75 | by @code{hilb}. |
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76 | SEE ALSO: stdfglm, std, groebner |
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77 | KEYWORDS: Hilbert function |
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78 | EXAMPLE: example stdhilb; shows an example" |
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79 | { |
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80 | def R=basering; |
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81 | |
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82 | if((homog(i)==1)||(ordstr(basering)[1]=="d")) |
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83 | { |
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84 | if ((size(#)!=0)&&(homog(i)==1)) |
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85 | { |
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86 | return(std(i,#[1])); |
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87 | } |
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88 | return(std(i)); |
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89 | } |
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90 | |
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91 | execute("ring S = ("+charstr(R)+"),("+varstr(R)+",@t),dp;"); |
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92 | ideal i=homog(imap(R,i),@t); |
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93 | intvec v=hilb(std(i),1); |
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94 | execute("ring T = ("+charstr(R)+"),("+varstr(R)+",@t),("+ordstr(R)+");"); |
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95 | ideal i=fetch(S,i); |
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96 | ideal a=std(i,v); |
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97 | setring R; |
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98 | map phi=T,maxideal(1),1; |
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99 | ideal a=phi(a); |
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100 | |
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101 | int k,j; |
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102 | poly m; |
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103 | int c=ncols(i); |
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104 | |
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105 | for(j=1;j<c;j++) |
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106 | { |
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107 | if(deg(a[j])==0) |
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108 | { |
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109 | a=ideal(1); |
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110 | attrib(a,"isSB",1); |
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111 | return(a); |
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112 | } |
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113 | if(deg(a[j])>0) |
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114 | { |
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115 | m=lead(a[j]); |
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116 | for(k=j+1;k<=c;k++) |
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117 | { |
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118 | if(size(lead(a[k])/m)>0) |
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119 | { |
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120 | a[k]=0; |
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121 | } |
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122 | } |
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123 | } |
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124 | } |
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125 | a=simplify(a,2); |
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126 | attrib(a,"isSB",1); |
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127 | return(a); |
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128 | } |
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129 | example |
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130 | { "EXAMPLE:"; echo = 2; |
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131 | ring r=0,(x,y,z),dp; |
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132 | ideal i=y3+x2,x2y+x2,x3-x2,z4-x2-y; |
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133 | ideal i1=stdhilb(i); i1; |
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134 | // the latter computation is equivalent to: |
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135 | intvec v=hilb(i,1); |
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136 | ideal i2=stdhilb(i,v); i2; |
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137 | } |
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138 | ////////////////////////////////////////////////////////////////////////// |
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139 | |
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140 | proc groebner(def i, list #) |
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141 | "SYNTAX: @code{groebner (} ideal_expression @code{)} @* |
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142 | @code{groebner (} module_expression @code{)} @* |
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143 | @code{groebner (} ideal_expression@code{,} int_expression @code{)} @* |
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144 | @code{groebner (} module_expression@code{,} int_expression @code{)} |
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145 | TYPE: type of the first argument |
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146 | PURPOSE: computes the standard basis of the first argument @code{I} |
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147 | (ideal or module), by a heuristically chosen method: if the |
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148 | ordering of the current ring is a local ordering, or if it is a |
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149 | non-block ordering and the current ring has no parameters, then |
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150 | @code{std(I)} is returned. Otherwise, @code{I} is mapped into a |
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151 | ring with no parameters and ordering dp, where its Hilbert series |
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152 | is computed. This is followed by a Hilbert-series based std |
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153 | computation in the original ring. |
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154 | NOTE: If a 2nd argument @code{wait} is given, then the computation proceeds |
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155 | at most @code{wait} seconds. That is, if no result could be computed in |
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156 | @code{wait} seconds, then the computation is interrupted, 0 is returned, |
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157 | a warning message is displayed, and the global variable |
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158 | @code{groebner_error} is defined. |
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159 | SEE ALSO: stdhilb, stdfglm, std |
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160 | KEYWORDS: time limit on computations; MP, groebner basis computations |
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161 | EXAMPLE: example groebner; shows an example" |
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162 | { |
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163 | def P=basering; |
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164 | |
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165 | // we have two arguments -- try to use MPfork links |
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166 | if (size(#) > 0) |
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167 | { |
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168 | if (system("with", "MP")) |
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169 | { |
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170 | if (typeof(#[1]) == "int") |
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171 | { |
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172 | int wait = #[1]; |
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173 | int j = 10; |
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174 | |
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175 | string bs = nameof(basering); |
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176 | link l_fork = "MPtcp:fork"; |
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177 | open(l_fork); |
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178 | write(l_fork, quote(system("pid"))); |
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179 | int pid = read(l_fork); |
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180 | write(l_fork, quote(groebner(eval(i)))); |
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181 | |
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182 | // sleep in small intervalls for appr. one second |
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183 | if (wait > 0) |
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184 | { |
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185 | while(j < 1000000) |
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186 | { |
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187 | if (status(l_fork, "read", "ready", j)) {break;} |
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188 | j = j + j; |
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189 | } |
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190 | } |
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191 | |
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192 | // sleep in intervalls of one second from now on |
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193 | j = 1; |
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194 | while (j < wait) |
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195 | { |
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196 | if (status(l_fork, "read", "ready", 1000000)) {break;} |
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197 | j = j + 1; |
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198 | } |
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199 | |
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200 | if (status(l_fork, "read", "ready")) |
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201 | { |
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202 | def result = read(l_fork); |
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203 | if (bs != nameof(basering)) |
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204 | { |
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205 | def PP = basering; |
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206 | setring P; |
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207 | def result = imap(PP, result); |
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208 | kill PP; |
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209 | } |
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210 | if (defined(groebner_error)) |
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211 | { |
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212 | kill groebner_error; |
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213 | } |
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214 | kill l_fork; |
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215 | } |
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216 | else |
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217 | { |
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218 | ideal result; |
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219 | if (! defined(groebner_error)) |
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220 | { |
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221 | int groebner_error = 1; |
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222 | export groebner_error; |
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223 | } |
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224 | "// ** groebner did not finish"; |
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225 | j = system("sh", "kill " + string(pid)); |
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226 | } |
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227 | return (result); |
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228 | } |
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229 | else |
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230 | { |
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231 | "// ** groebner needs int as 2nd arg"; |
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232 | } |
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233 | } |
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234 | else |
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235 | { |
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236 | "// ** groebner with two args is not supported in this configuration"; |
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237 | } |
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238 | } |
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239 | |
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240 | // we are still here -- do the actual computation |
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241 | string ordstr_P = ordstr(P); |
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242 | if ((find(ordstr_P,"s") > 0) |
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243 | ||(find(ordstr_P,"M") > 0) |
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244 | ||(find(ordstr_P,"w") > 0) |
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245 | ||(find(ordstr_P,"W") > 0)) |
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246 | { |
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247 | //spaeter den lokalen fall ueber lp oder aehnlich behandeln |
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248 | return(std(i)); |
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249 | } |
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250 | |
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251 | int IsSimple_P; |
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252 | if (system("nblocks") <= 2) |
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253 | { |
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254 | if (find(ordstr_P, "M") <= 0) |
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255 | { |
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256 | IsSimple_P = 1; |
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257 | } |
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258 | } |
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259 | int npars_P = npars(P); |
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260 | |
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261 | // return std if no parameters and (dp or wp) |
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262 | if ((npars_P <= 1) && IsSimple_P) |
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263 | { |
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264 | if (find(ordstr_P, "d") > 0) |
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265 | { |
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266 | return (std(i)); |
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267 | } |
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268 | if (find(ordstr_P,"w") > 0) |
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269 | { |
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270 | return (std(i)); |
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271 | } |
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272 | } |
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273 | |
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274 | // reset options |
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275 | intvec opt=option(get); |
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276 | int p_opt; |
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277 | string s_opt = option(); |
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278 | option(none); |
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279 | // turn on option(prot) and/or option(mem), if previously set |
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280 | if (find(s_opt, "prot")) |
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281 | { |
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282 | option(prot); |
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283 | p_opt = 1; |
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284 | } |
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285 | if (find(s_opt, "mem")) |
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286 | { option(mem); } |
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287 | if (find(s_opt, "intStrategy")) |
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288 | { option(intStrategy); } |
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289 | |
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290 | // construct ring in which first std computation is done |
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291 | string varstr_P = varstr(P); |
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292 | string parstr_P = parstr(P); |
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293 | int is_homog = (homog(i) && (npars_P <= 1)); |
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294 | int add_vars = 0; |
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295 | string ri = "ring Phelp ="; |
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296 | |
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297 | if (npars_P > 0) |
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298 | { |
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299 | for(int k=ncols(i); k>0; k--) { i[k]=cleardenom(i[k]); } |
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300 | } |
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301 | // more than one parameters are converted to ring variables |
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302 | if (npars_P > 1) |
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303 | { |
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304 | ri = ri + string(char(P)) + ",(" + varstr_P + "," + parstr_P; |
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305 | add_vars = npars_P; |
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306 | } |
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307 | else |
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308 | { |
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309 | ri = ri + "(" + charstr(P) + "),(" + varstr_P; |
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310 | } |
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311 | |
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312 | // a homogenizing variable is added, if necessary |
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313 | if (! is_homog) |
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314 | { |
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315 | ri = ri + ",@t"; |
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316 | add_vars = add_vars + 1; |
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317 | } |
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318 | // ordering is set to (dp, C) |
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319 | ri = ri + "),(dp,C);"; |
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320 | |
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321 | // change the ring |
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322 | execute(ri); |
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323 | |
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324 | // get ideal from previous ring |
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325 | if (is_homog) |
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326 | { |
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327 | ideal qh = imap(P, i); |
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328 | } |
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329 | else |
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330 | { |
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331 | // and homogenize |
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332 | ideal qh=homog(imap(P,i),@t); |
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333 | } |
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334 | |
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335 | // compute std and hilbert series |
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336 | if (p_opt) |
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337 | { |
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338 | "std in " + ri[13, size(ri) - 13]; |
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339 | } |
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340 | intvec hi=hilb(std(qh),1); |
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341 | |
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342 | if (add_vars == 0) |
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343 | { |
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344 | // no additional variables were introduced |
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345 | setring P; // can immediately change to original ring |
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346 | // simply compute std with hilbert series in original ring |
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347 | if (p_opt) |
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348 | { |
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349 | "std with hilb in basering"; |
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350 | } |
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351 | i = std(i, hi); |
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352 | } |
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353 | else |
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354 | { |
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355 | // additional variables were introduced |
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356 | // need another intermediate ring |
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357 | ri = "ring Phelp1 = (" + charstr(Phelp) |
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358 | + "),(" + varstr(Phelp) + "),(" + ordstr_P; |
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359 | |
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360 | // for lp wit at most one parameter, we do not need a block ordering |
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361 | if ( ! (IsSimple_P && (add_vars <2) && find(ordstr_P, "l"))) |
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362 | { |
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363 | // need block ordering |
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364 | ri = ri + ", dp(" + string(add_vars) + ")"; |
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365 | } |
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366 | ri = ri + ");"; |
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367 | |
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368 | // change to intermediate ring |
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369 | execute(ri); |
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370 | ideal qh = imap(Phelp, qh); |
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371 | kill Phelp; |
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372 | if (p_opt) |
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373 | { |
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374 | "std with hilb in " + ri[14,size(ri)-14]; |
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375 | } |
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376 | // compute std with Hilbert series |
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377 | qh = std(qh, hi); |
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378 | // subst 1 for homogenizing var |
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379 | if (!is_homog) |
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380 | { |
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381 | if (p_opt) |
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382 | { |
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383 | "dehomogenization"; |
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384 | } |
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385 | qh = subst(qh, @t, 1); |
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386 | } |
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387 | |
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388 | // go back to original ring |
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389 | setring P; |
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390 | // get ideal, delete zeros and clean SB |
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391 | if (p_opt) |
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392 | { |
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393 | "imap to original ring"; |
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394 | } |
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395 | i = imap(Phelp1,qh); |
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396 | if (p_opt) |
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397 | { |
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398 | "simplification"; |
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399 | } |
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400 | i = simplify(i, 34); |
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401 | kill Phelp1; |
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402 | } |
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403 | |
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404 | // clean-up time |
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405 | option(set, opt); |
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406 | if (find(s_opt, "redSB") > 0) |
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407 | { |
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408 | if (p_opt) |
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409 | { |
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410 | "interreduction"; |
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411 | } |
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412 | i=interred(i); |
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413 | } |
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414 | attrib(i, "isSB", 1); |
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415 | return (i); |
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416 | } |
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417 | example |
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418 | { "EXAMPLE: "; echo=2; |
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419 | ring r=0,(a,b,c,d),lp; |
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420 | option(prot); |
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421 | ideal i=a+b+c+d,ab+ad+bc+cd,abc+abd+acd+bcd,abcd-1; // cyclic 4 |
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422 | groebner(i); |
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423 | ring rp=(0,a,b),(c,d), lp; |
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424 | ideal i=imap(r,i); |
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425 | ideal j=groebner(i); |
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426 | option(noprot); |
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427 | j; simplify(j,1); std(i); |
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428 | if (system("with","MP")) {groebner(i,0);} |
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429 | defined(groebner_error); |
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430 | } |
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431 | ////////////////////////////////////////////////////////////////////////// |
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432 | |
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433 | proc res(list #) |
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434 | "@c we do texinfo here: |
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435 | @cindex resolution, computation of |
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436 | @table @code |
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437 | @item @strong{Syntax:} |
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438 | @code{res (} ideal_expression@code{,} int_expression @code{[,} any_expression @code{])} |
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439 | @*@code{res (} module_expression@code{,} int_expression @code{[,} any_expression @code{])} |
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440 | @item @strong{Type:} |
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441 | resolution |
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442 | @item @strong{Purpose:} |
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443 | computes a (possibly minimal) free resolution of an ideal or module using |
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444 | a heuristically chosen method. |
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445 | @* The second (int) argument (say, @code{k}) specifies the length of |
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446 | the resolution. If it is not positive then @code{k} is assumed to be the |
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447 | number of variables of the basering. |
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448 | @* If a third argument is given, the returned resolution is minimized. |
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449 | |
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450 | Depending on the input, the returned resolution is computed using the |
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451 | following methods: |
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452 | @table @asis |
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453 | @item @strong{quotient rings:} |
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454 | @code{nres} (classical method using syzygies) , see @ref{nres}. |
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455 | |
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456 | @item @strong{homogeneous ideals and k=0:} |
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457 | @code{lres} (La'Scala's method), see @ref{lres}. |
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458 | |
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459 | @item @strong{not minimized resolution and (homogeneous input with k not 0, or local rings):} |
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460 | @code{sres} (Schreyer's method), see @ref{sres}. |
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461 | |
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462 | @item @strong{all other inputs:} |
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463 | @code{mres} (classical method), see @ref{mres}. |
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464 | @end table |
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465 | @item @strong{Note:} |
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466 | Accessing single elements of a resolution may require that some partial |
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467 | computations have to be finished and may therefore take some time. |
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468 | @end table |
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469 | @c ref |
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470 | See also |
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471 | @ref{betti}; |
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472 | @ref{ideal}; |
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473 | @ref{minres}; |
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474 | @ref{module}; |
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475 | @ref{mres}; |
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476 | @ref{nres}; |
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477 | @ref{lres}; |
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478 | @ref{hres}; |
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479 | @ref{sres}. |
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480 | @ref{resolution} |
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481 | @c ref |
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482 | " |
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483 | { |
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484 | def P=basering; |
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485 | if (size(#) < 2) |
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486 | { |
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487 | ERROR("res: need at least two arguments: ideal/module, int"); |
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488 | } |
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489 | |
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490 | def m=#[1]; //the ideal or module |
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491 | int i=#[2]; //the length of the resolution |
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492 | if (i< 0) { i=0;} |
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493 | |
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494 | string varstr_P = varstr(P); |
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495 | |
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496 | int p_opt; |
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497 | string s_opt = option(); |
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498 | // set p_opt, if option(prot) is set |
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499 | if (find(s_opt, "prot")) |
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500 | { |
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501 | p_opt = 1; |
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502 | } |
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503 | |
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504 | if(size(ideal(basering)) > 0) |
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505 | { |
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506 | // the quick hack for qrings - seems to fit most needs |
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507 | // (lres is not implemented for qrings, sres is not so efficient) |
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508 | if (p_opt) { "using nres";} |
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509 | return(nres(m,i)); |
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510 | } |
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511 | |
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512 | if(homog(m)==1) |
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513 | { |
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514 | resolution re; |
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515 | if (((i==0) or (i>=nvars(basering))) && typeof(m) != "module") |
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516 | { |
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517 | //LaScala for the homogeneous case and i == 0 |
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518 | if (p_opt) { "using lres";} |
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519 | re=lres(m,i); |
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520 | if(size(#)>2) |
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521 | { |
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522 | re=minres(re); |
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523 | } |
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524 | } |
---|
525 | else |
---|
526 | { |
---|
527 | if(size(#)>2) |
---|
528 | { |
---|
529 | if (p_opt) { "using mres";} |
---|
530 | re=mres(m,i); |
---|
531 | } |
---|
532 | else |
---|
533 | { |
---|
534 | if (p_opt) { "using sres";} |
---|
535 | re=sres(std(m),i); |
---|
536 | } |
---|
537 | } |
---|
538 | return(re); |
---|
539 | } |
---|
540 | |
---|
541 | //mres for the global non homogeneous case |
---|
542 | if(find(ordstr(P),"s")==0) |
---|
543 | { |
---|
544 | string ri= "ring Phelp =" |
---|
545 | +string(char(P))+",("+varstr_P+"),(dp,C);"; |
---|
546 | execute(ri); |
---|
547 | def m=imap(P,m); |
---|
548 | if (p_opt) { "using mres in another ring";} |
---|
549 | list re=mres(m,i); |
---|
550 | setring P; |
---|
551 | resolution result=imap(Phelp,re); |
---|
552 | if (size(#) > 2) {result = minres(result);} |
---|
553 | return(result); |
---|
554 | } |
---|
555 | |
---|
556 | //sres for the local case and not minimal resolution |
---|
557 | if(size(#)<=2) |
---|
558 | { |
---|
559 | string ri= "ring Phelp =" |
---|
560 | +string(char(P))+",("+varstr_P+"),(ls,c);"; |
---|
561 | execute(ri); |
---|
562 | def m=imap(P,m); |
---|
563 | m=std(m); |
---|
564 | if (p_opt) { "using sres in another ring";} |
---|
565 | list re=sres(m,i); |
---|
566 | setring P; |
---|
567 | resolution result=imap(Phelp,re); |
---|
568 | return(result); |
---|
569 | } |
---|
570 | |
---|
571 | //mres for the local case and minimal resolution |
---|
572 | string ri= "ring Phelp =" |
---|
573 | +string(char(P))+",("+varstr_P+"),(ls,C);"; |
---|
574 | execute(ri); |
---|
575 | def m=imap(P,m); |
---|
576 | if (p_opt) { "using mres in another ring";} |
---|
577 | list re=mres(m,i); |
---|
578 | setring P; |
---|
579 | resolution result=imap(Phelp,re); |
---|
580 | result = minres(result); |
---|
581 | return(result); |
---|
582 | } |
---|
583 | example |
---|
584 | {"EXAMPLE:"; echo = 2; |
---|
585 | ring r=0,(x,y,z),dp; |
---|
586 | ideal i=xz,yz,x3-y3; |
---|
587 | def l=res(i,0); // homogeneous ideal: uses lres |
---|
588 | l; |
---|
589 | print(betti(l), "betti"); // input to betti may be of type resolution |
---|
590 | l[2]; // element access may take some time |
---|
591 | i=i,x+1; |
---|
592 | l=res(i,0); // inhomogeneous ideal: uses mres |
---|
593 | l; |
---|
594 | ring rs=0,(x,y,z),ds; |
---|
595 | ideal i=imap(r,i); |
---|
596 | def l=res(i,0); // local ring not minimized: uses sres |
---|
597 | l; |
---|
598 | res(i,0,0); // local ring and minimized: uses mres |
---|
599 | } |
---|
600 | ///////////////////////////////////////////////////////////////////////// |
---|
601 | |
---|
602 | proc quot (m1,m2,list #) |
---|
603 | "SYNTAX: @code{quot (} module_expression@code{,} module_expression @code{)} |
---|
604 | @*@code{quot (} module_expression@code{,} module_expression@code{,} |
---|
605 | int_expression @code{)} |
---|
606 | @*@code{quot (} ideal_expression@code{,} ideal_expression @code{)} |
---|
607 | @*@code{quot (} ideal_expression@code{,} ideal_expression@code{,} |
---|
608 | int_expression @code{)} |
---|
609 | TYPE: ideal |
---|
610 | SYNTAX: @code{quot (} module_expression@code{,} ideal_expression @code{)} |
---|
611 | TYPE: module |
---|
612 | PURPOSE: computes the quotient of the 1st and the 2nd argument. |
---|
613 | If a 3rd argument 'n' is given the n-th method is used |
---|
614 | (n=1...5). |
---|
615 | SEE ALSO: quotient |
---|
616 | EXAMPLE: example quot; shows an example" |
---|
617 | { |
---|
618 | if (((typeof(m1)!="ideal") and (typeof(m1)!="module")) |
---|
619 | or ((typeof(m2)!="ideal") and (typeof(m2)!="module"))) |
---|
620 | { |
---|
621 | "USAGE: quot(m1, m2[, n]); m1, m2 two submodules of k^s,"; |
---|
622 | " n (optional) integer (1<= n <=5)"; |
---|
623 | "RETURN: the quotient of m1 and m2"; |
---|
624 | "EXAMPLE: example quot; shows an example"; |
---|
625 | return(); |
---|
626 | } |
---|
627 | if (typeof(m1)!=typeof(m2)) |
---|
628 | { |
---|
629 | return(quotient(m1,m2)); |
---|
630 | } |
---|
631 | if (size(#)>0) |
---|
632 | { |
---|
633 | if (typeof(#[1])=="int" ) |
---|
634 | { |
---|
635 | return(quot1(m1,m2,#[1])); |
---|
636 | } |
---|
637 | } |
---|
638 | else |
---|
639 | { |
---|
640 | return(quot1(m1,m2,2)); |
---|
641 | } |
---|
642 | } |
---|
643 | example |
---|
644 | { "EXAMPLE:"; echo = 2; |
---|
645 | ring r=181,(x,y,z),(c,ls); |
---|
646 | ideal id1=maxideal(4); |
---|
647 | ideal id2=x2+xyz,y2-z3y,z3+y5xz; |
---|
648 | option(prot); |
---|
649 | ideal id3=quotient(id1,id2); |
---|
650 | id3; |
---|
651 | ideal id4=quot(id1,id2,1); |
---|
652 | id4; |
---|
653 | ideal id5=quot(id1,id2,2); |
---|
654 | id5; |
---|
655 | } |
---|
656 | |
---|
657 | static proc quot1 (module m1, module m2,int n) |
---|
658 | "USAGE: quot1(m1, m2, n); m1, m2 two submodules of k^s, |
---|
659 | n integer (1<= n <=5) |
---|
660 | RETURN: the quotient of m1 and m2 |
---|
661 | EXAMPLE: example quot1; shows an example" |
---|
662 | { |
---|
663 | if (n==1) |
---|
664 | { |
---|
665 | return(quotient1(m1,m2)); |
---|
666 | } |
---|
667 | else |
---|
668 | { |
---|
669 | if (n==2) |
---|
670 | { |
---|
671 | return(quotient2(m1,m2)); |
---|
672 | } |
---|
673 | else |
---|
674 | { |
---|
675 | if (n==3) |
---|
676 | { |
---|
677 | return(quotient3(m1,m2)); |
---|
678 | } |
---|
679 | else |
---|
680 | { |
---|
681 | if (n==4) |
---|
682 | { |
---|
683 | return(quotient4(m1,m2)); |
---|
684 | } |
---|
685 | else |
---|
686 | { |
---|
687 | if (n==5) |
---|
688 | { |
---|
689 | return(quotient5(m1,m2)); |
---|
690 | } |
---|
691 | else |
---|
692 | { |
---|
693 | return(quotient(m1,m2)); |
---|
694 | } |
---|
695 | } |
---|
696 | } |
---|
697 | } |
---|
698 | } |
---|
699 | } |
---|
700 | example |
---|
701 | { "EXAMPLE:"; echo = 2; |
---|
702 | ring r=181,(x,y,z),(c,ls); |
---|
703 | ideal id1=maxideal(4); |
---|
704 | ideal id2=x2+xyz,y2-z3y,z3+y5xz; |
---|
705 | option(prot); |
---|
706 | ideal id6=quotient(id1,id2); |
---|
707 | id6; |
---|
708 | ideal id7=quot1(id1,id2,1); |
---|
709 | id7; |
---|
710 | ideal id8=quot1(id1,id2,2); |
---|
711 | id8; |
---|
712 | } |
---|
713 | |
---|
714 | static proc quotient0(module a,module b) |
---|
715 | { |
---|
716 | module mm=b+a; |
---|
717 | resolution rs=lres(mm,0); |
---|
718 | list I=list(rs); |
---|
719 | matrix M=I[2]; |
---|
720 | matrix A[1][nrows(M)]=M[1..nrows(M),1]; |
---|
721 | ideal i=A; |
---|
722 | return (i); |
---|
723 | } |
---|
724 | proc quotient1(module a,module b) //17sec |
---|
725 | "USAGE: quotient1(m1, m2); m1, m2 two submodules of k^s, |
---|
726 | RETURN: the quotient of m1 and m2" |
---|
727 | { |
---|
728 | int i; |
---|
729 | a=std(a); |
---|
730 | module dummy; |
---|
731 | module B=NF(b,a)+dummy; |
---|
732 | ideal re=quotient(a,module(B[1])); |
---|
733 | for(i=2;i<=ncols(B);i++) |
---|
734 | { |
---|
735 | re=intersect1(re,quotient(a,module(B[i]))); |
---|
736 | } |
---|
737 | return(re); |
---|
738 | } |
---|
739 | proc quotient2(module a,module b) //13sec |
---|
740 | "USAGE: quotient2(m1, m2); m1, m2 two submodules of k^s, |
---|
741 | RETURN: the quotient of m1 and m2" |
---|
742 | { |
---|
743 | a=std(a); |
---|
744 | module dummy; |
---|
745 | module bb=NF(b,a)+dummy; |
---|
746 | int i=ncols(bb); |
---|
747 | ideal re=quotient(a,module(bb[i])); |
---|
748 | bb[i]=0; |
---|
749 | module temp; |
---|
750 | module temp1; |
---|
751 | module bbb; |
---|
752 | int mx; |
---|
753 | i=i-1; |
---|
754 | while (1) |
---|
755 | { |
---|
756 | if (i==0) break; |
---|
757 | temp = a+bb*re; |
---|
758 | temp1 = lead(interred(temp)); |
---|
759 | mx=ncols(a); |
---|
760 | if (ncols(temp1)>ncols(a)) |
---|
761 | { |
---|
762 | mx=ncols(temp1); |
---|
763 | } |
---|
764 | temp1 = matrix(temp1,1,mx)-matrix(lead(a),1,mx); |
---|
765 | temp1 = dummy+temp1; |
---|
766 | if (deg(temp1[1])<0) break; |
---|
767 | re=intersect1(re,quotient(a,module(bb[i]))); |
---|
768 | bb[i]=0; |
---|
769 | i = i-1; |
---|
770 | } |
---|
771 | return(re); |
---|
772 | } |
---|
773 | proc quotient3(module a,module b) //89sec |
---|
774 | "USAGE: quotient3(m1, m2); m1, m2 two submodules of k^s, |
---|
775 | only for global rings |
---|
776 | RETURN: the quotient of m1 and m2" |
---|
777 | { |
---|
778 | string s="ring @newr=("+charstr(basering)+ |
---|
779 | "),("+varstr(basering)+",@t,@w),dp;"; |
---|
780 | def @newP=basering; |
---|
781 | execute(s); |
---|
782 | module b=imap(@newP,b); |
---|
783 | module a=imap(@newP,a); |
---|
784 | int i; |
---|
785 | int j=ncols(b); |
---|
786 | vector @b; |
---|
787 | for(i=1;i<=j;i++) |
---|
788 | { |
---|
789 | @b=@b+@t^(i-1)*@w^(j-i+1)*b[i]; |
---|
790 | } |
---|
791 | ideal re=quotient(a,module(@b)); |
---|
792 | setring @newP; |
---|
793 | ideal re=imap(@newr,re); |
---|
794 | return(re); |
---|
795 | } |
---|
796 | proc quotient5(module a,module b) //89sec |
---|
797 | "USAGE: quotient5(m1, m2); m1, m2 two submodules of k^s, |
---|
798 | only for global rings |
---|
799 | RETURN: the quotient of m1 and m2" |
---|
800 | { |
---|
801 | string s="ring @newr=("+charstr(basering)+ |
---|
802 | "),("+varstr(basering)+",@t),dp;"; |
---|
803 | def @newP=basering; |
---|
804 | execute(s); |
---|
805 | module b=imap(@newP,b); |
---|
806 | module a=imap(@newP,a); |
---|
807 | int i; |
---|
808 | int j=ncols(b); |
---|
809 | vector @b; |
---|
810 | for(i=1;i<=j;i++) |
---|
811 | { |
---|
812 | @b=@b+@t^(i-1)*b[i]; |
---|
813 | } |
---|
814 | @b=homog(@b,@w); |
---|
815 | ideal re=quotient(a,module(@b)); |
---|
816 | setring @newP; |
---|
817 | ideal re=imap(@newr,re); |
---|
818 | return(re); |
---|
819 | } |
---|
820 | proc quotient4(module a,module b) //95sec |
---|
821 | "USAGE: quotient4(m1, m2); m1, m2 two submodules of k^s, |
---|
822 | only for global rings |
---|
823 | RETURN: the quotient of m1 and m2" |
---|
824 | { |
---|
825 | string s="ring @newr=("+charstr(basering)+ |
---|
826 | "),("+varstr(basering)+",@t),dp;"; |
---|
827 | def @newP=basering; |
---|
828 | execute(s); |
---|
829 | module b=imap(@newP,b); |
---|
830 | module a=imap(@newP,a); |
---|
831 | int i; |
---|
832 | vector @b=b[1]; |
---|
833 | for(i=2;i<=ncols(b);i++) |
---|
834 | { |
---|
835 | @b=@b+@t^(i-1)*b[i]; |
---|
836 | } |
---|
837 | matrix sy=modulo(@b,a); |
---|
838 | ideal re=sy; |
---|
839 | setring @newP; |
---|
840 | ideal re=imap(@newr,re); |
---|
841 | return(re); |
---|
842 | } |
---|
843 | static proc intersect1(ideal i,ideal j) |
---|
844 | { |
---|
845 | def R=basering; |
---|
846 | execute("ring gnir = ("+charstr(basering)+"), |
---|
847 | ("+varstr(basering)+",@t),(C,dp);"); |
---|
848 | ideal i=var(nvars(basering))*imap(R,i)+(var(nvars(basering))-1)*imap(R,j); |
---|
849 | ideal j=eliminate(i,var(nvars(basering))); |
---|
850 | setring R; |
---|
851 | map phi=gnir,maxideal(1); |
---|
852 | return(phi(j)); |
---|
853 | } |
---|
854 | |
---|
855 | ////////////////////////////////////////////////////////////////// |
---|
856 | /// |
---|
857 | /// sprintf, fprintf printf |
---|
858 | /// |
---|
859 | proc sprintf(string fmt, list #) |
---|
860 | "SYNTAX: @code{sprintf (} string_expression @code{[,} any_expressions |
---|
861 | @code{] )} |
---|
862 | RETURN: string |
---|
863 | PURPOSE: @code{sprintf(fmt,...);} performs output formatting. The first |
---|
864 | argument is a format control string. Additional arguments may be |
---|
865 | required, depending on the content of the control string. A series |
---|
866 | of output characters is generated as directed by the control string; |
---|
867 | these characters are returned as a string. @* |
---|
868 | The control string @code{fmt} is simply text to be copied, |
---|
869 | except that the string may contain conversion specifications.@* |
---|
870 | Do @code{help print;} for a listing of valid conversion |
---|
871 | specifications. As an addition to the conversions of @code{print}, |
---|
872 | the @code{%n} and @code{%2} conversion specification does not |
---|
873 | consume an additional argument, but simply generates a newline |
---|
874 | character. |
---|
875 | NOTE: If one of the additional arguments is a list, then it should be |
---|
876 | enclosed once more into a @code{list()} command, since passing a list |
---|
877 | as an argument flattens the list by one level. |
---|
878 | SEE ALSO: fprintf, printf, print, string |
---|
879 | EXAMPLE : example sprintf; shows an example |
---|
880 | " |
---|
881 | { |
---|
882 | int sfmt = size(fmt); |
---|
883 | if (sfmt <= 1) |
---|
884 | { |
---|
885 | return (fmt); |
---|
886 | } |
---|
887 | int next, l, nnext; |
---|
888 | string ret; |
---|
889 | list formats = "%l", "%s", "%2l", "%2s", "%t", "%;", "%p", "%b", "%n", "%2"; |
---|
890 | while (1) |
---|
891 | { |
---|
892 | if (size(#) <= 0) |
---|
893 | { |
---|
894 | return (ret + fmt); |
---|
895 | } |
---|
896 | nnext = 0; |
---|
897 | while (nnext < sfmt) |
---|
898 | { |
---|
899 | nnext = find(fmt, "%", nnext + 1); |
---|
900 | if (nnext == 0) |
---|
901 | { |
---|
902 | next = 0; |
---|
903 | break; |
---|
904 | } |
---|
905 | l = 1; |
---|
906 | while (l <= size(formats)) |
---|
907 | { |
---|
908 | next = find(fmt, formats[l], nnext); |
---|
909 | if (next == nnext) break; |
---|
910 | l++; |
---|
911 | } |
---|
912 | if (next == nnext) break; |
---|
913 | } |
---|
914 | if (next == 0) |
---|
915 | { |
---|
916 | return (ret + fmt); |
---|
917 | } |
---|
918 | if (formats[l] != "%2" && formats[l] != "%n") |
---|
919 | { |
---|
920 | ret = ret + fmt[1, next - 1] + print(#[1], formats[l]); |
---|
921 | # = delete(#, 1); |
---|
922 | } |
---|
923 | else |
---|
924 | { |
---|
925 | ret = ret + fmt[1, next - 1] + print("", "%2s"); |
---|
926 | } |
---|
927 | if (size(fmt) <= (next + size(formats[l]) - 1)) |
---|
928 | { |
---|
929 | return (ret); |
---|
930 | } |
---|
931 | fmt = fmt[next + size(formats[l]), size(fmt)-next-size(formats[l]) + 1]; |
---|
932 | } |
---|
933 | } |
---|
934 | example |
---|
935 | { "EXAMPLE:"; echo=2; |
---|
936 | ring r=0,(x,y,z),dp; |
---|
937 | module m=[1,y],[0,x+z]; |
---|
938 | intmat M=betti(mres(m,0)); |
---|
939 | list l = r, m, M; |
---|
940 | string s = sprintf("s:%s,%n l:%l", 1, 2); s; |
---|
941 | s = sprintf("s:%n%s", l); s; |
---|
942 | s = sprintf("s:%2%s", list(l)); s; |
---|
943 | s = sprintf("2l:%n%2l", list(l)); s; |
---|
944 | s = sprintf("%p", list(l)); s; |
---|
945 | s = sprintf("%;", list(l)); s; |
---|
946 | s = sprintf("%b", M); s; |
---|
947 | } |
---|
948 | |
---|
949 | proc printf(string fmt, list #) |
---|
950 | "SYNTAX: @code{printf (} string_expression @code{[,} any_expressions@code{] )} |
---|
951 | RETURN: none |
---|
952 | PURPOSE: @code{printf(fmt,...);} performs output formatting. The first |
---|
953 | argument is a format control string. Additional arguments may be |
---|
954 | required, depending on the content of the control string. A series |
---|
955 | of output characters is generated as directed by the control string; |
---|
956 | these characters are displayed (i.e., printed to standard out). @* |
---|
957 | The control string @code{fmt} is simply text to be copied, except |
---|
958 | that the string may contain conversion specifications. @* |
---|
959 | Do @code{help print;} for a listing of valid conversion |
---|
960 | specifications. As an addition to the conversions of @code{print}, |
---|
961 | the @code{%n} and @code{%2} conversion specification does not |
---|
962 | consume an additional argument, but simply generates a newline |
---|
963 | character. |
---|
964 | NOTE: If one of the additional arguments is a list, then it should be |
---|
965 | enclosed once more into a @code{list()} command, since passing a |
---|
966 | list as an argument flattens the list by one level. |
---|
967 | SEE ALSO: sprintf, fprintf, print, string |
---|
968 | EXAMPLE : example printf; shows an example |
---|
969 | " |
---|
970 | { |
---|
971 | write("", sprintf(fmt, #)); |
---|
972 | } |
---|
973 | example |
---|
974 | { "EXAMPLE:"; echo=2; |
---|
975 | ring r=0,(x,y,z),dp; |
---|
976 | module m=[1,y],[0,x+z]; |
---|
977 | intmat M=betti(mres(m,0)); |
---|
978 | list l=r,m,M; |
---|
979 | printf("s:%s,l:%l",1,2); |
---|
980 | printf("s:%s",l); |
---|
981 | printf("s:%s",list(l)); |
---|
982 | printf("2l:%2l",list(l)); |
---|
983 | printf("%p",list(l)); |
---|
984 | printf("%;",list(l)); |
---|
985 | printf("%b",M); |
---|
986 | } |
---|
987 | |
---|
988 | |
---|
989 | proc fprintf(link l, string fmt, list #) |
---|
990 | "SYNTAX: @code{fprintf (} link_expression@code{,} string_expression @code{[,} |
---|
991 | any_expressions@code{] )} |
---|
992 | RETURN: none |
---|
993 | PURPOSE: @code{fprintf(l,fmt,...);} performs output formatting. |
---|
994 | The second argument is a format control string. Additional |
---|
995 | arguments may be required, depending on the content of the |
---|
996 | control string. A series of output characters is generated as |
---|
997 | directed by the control string; these characters are |
---|
998 | written to the link l. |
---|
999 | The control string @code{fmt} is simply text to be copied, except |
---|
1000 | that the string may contain conversion specifications.@* |
---|
1001 | Do @code{help print;} for a listing of valid conversion |
---|
1002 | specifications. As an addition to the conversions of @code{print}, |
---|
1003 | the @code{%n} and @code{%2} conversion specification does not |
---|
1004 | consume an additional argument, but simply generates a newline |
---|
1005 | character. |
---|
1006 | NOTE: If one of the additional arguments is a list, then it should be |
---|
1007 | enclosed once more into a @code{list()} command, since passing |
---|
1008 | a list as an argument flattens the list by one level. |
---|
1009 | SEE ALSO: sprintf, printf, print, string |
---|
1010 | EXAMPLE : example fprintf; shows an example |
---|
1011 | " |
---|
1012 | { |
---|
1013 | write(l, sprintf(fmt, #)); |
---|
1014 | } |
---|
1015 | example |
---|
1016 | { "EXAMPLE:"; echo=2; |
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1017 | ring r=0,(x,y,z),dp; |
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1018 | module m=[1,y],[0,x+z]; |
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1019 | intmat M=betti(mres(m,0)); |
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1020 | list l=r,m,M; |
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1021 | link li=""; // link to stdout |
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1022 | fprintf(li,"s:%s,l:%l",1,2); |
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1023 | fprintf(li,"s:%s",l); |
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1024 | fprintf(li,"s:%s",list(l)); |
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1025 | fprintf(li,"2l:%2l",list(l)); |
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1026 | fprintf(li,"%p",list(l)); |
---|
1027 | fprintf(li,"%;",list(l)); |
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1028 | fprintf(li,"%b",M); |
---|
1029 | } |
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1030 | |
---|
1031 | ////////////////////////////////////////////////////////////////////////// |
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1032 | |
---|
1033 | /* |
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1034 | proc minres(list #) |
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1035 | { |
---|
1036 | if (size(#) == 2) |
---|
1037 | { |
---|
1038 | if (typeof(#[1]) == "ideal" || typeof(#[1]) == "module") |
---|
1039 | { |
---|
1040 | if (typeof(#[2] == "int")) |
---|
1041 | { |
---|
1042 | return (res(#[1],#[2],1)); |
---|
1043 | } |
---|
1044 | } |
---|
1045 | } |
---|
1046 | |
---|
1047 | if (typeof(#[1]) == "resolution") |
---|
1048 | { |
---|
1049 | return minimizeres(#[1]); |
---|
1050 | } |
---|
1051 | else |
---|
1052 | { |
---|
1053 | return minimizeres(#); |
---|
1054 | } |
---|
1055 | |
---|
1056 | } |
---|
1057 | */ |
---|