1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: solve.lib,v 1.37 2009-01-06 14:15:21 Singular Exp $"; |
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3 | category="Symbolic-numerical solving"; |
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4 | info=" |
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5 | LIBRARY: solve.lib Complex Solving of Polynomial Systems |
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6 | AUTHOR: Moritz Wenk, email: wenk@mathematik.uni-kl.de |
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7 | Wilfred Pohl, email: pohl@mathematik.uni-kl.de |
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8 | |
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9 | PROCEDURES: |
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10 | laguerre_solve(p,[..]); find all roots of univariate polynomial p |
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11 | solve(i,[..]); all roots of 0-dim. ideal i using triangular sets |
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12 | ures_solve(i,[..]); find all roots of 0-dimensional ideal i with resultants |
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13 | mp_res_mat(i,[..]); multipolynomial resultant matrix of ideal i |
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14 | interpolate(p,v,d); interpolate poly from evaluation points i and results j |
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15 | fglm_solve(i,[..]); find roots of 0-dim. ideal using FGLM and lex_solve |
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16 | lex_solve(i,p,[..]); find roots of reduced lexicographic standard basis |
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17 | simplexOut(l); prints solution of simplex in nice format |
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18 | triangLf_solve(l,[..]); find roots using triangular sys. (factorizing Lazard) |
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19 | triangM_solve(l,[..]); find roots of given triangular system (Moeller) |
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20 | triangL_solve(l,[..]); find roots using triangular system (Lazard) |
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21 | triang_solve(l,p,[..]); find roots of given triangular system |
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22 | "; |
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23 | |
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24 | LIB "triang.lib"; // needed for triang_solve |
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25 | |
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26 | /////////////////////////////////////////////////////////////////////////////// |
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27 | |
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28 | proc laguerre_solve( poly f, list # ) |
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29 | "USAGE: laguerre_solve(f [, m, l, n, s] ); f = polynomial,@* |
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30 | m, l, n, s = integers (control parameters of the method) |
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31 | m: precision of output in digits ( 4 <= m), if basering is not ring of |
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32 | complex numbers; |
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33 | l: precision of internal computation in decimal digits ( l >=8 ) |
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34 | only if the basering is not complex or complex with smaller precision; |
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35 | n: control of multiplicity of roots or of splitting of f into |
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36 | squarefree factors |
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37 | n < 0, no split of f (good, if all roots are simple) |
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38 | n >= 0, try to split |
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39 | n = 0, return only different roots |
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40 | n > 0, find all roots (with multiplicity) |
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41 | s: s != 0, returns ERROR if | f(root) | > 0.1^m (when computing in the |
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42 | current ring) |
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43 | ( default: m, l, n, s = 8, 30, 1, 0 ) |
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44 | ASSUME: f is a univariate polynomial;@* |
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45 | basering has characteristic 0 and is either complex or without |
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46 | parameters. |
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47 | RETURN: list of (complex) roots of the polynomial f, depending on n. The |
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48 | entries of the result are of type@* |
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49 | string: if the basering is not complex,@* |
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50 | number: otherwise. |
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51 | NOTE: If printlevel >0: displays comments ( default = 0 ). |
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52 | If s != 0 and if the procedure stops with ERROR, try a higher |
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53 | internal precision m. |
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54 | EXAMPLE: example laguerre_solve; shows an example |
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55 | " |
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56 | { |
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57 | if (char(basering)!=0){ERROR("characteristic of basering not 0");} |
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58 | if ((charstr(basering)[1]=="0") and (npars(basering)!=0)) |
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59 | {ERROR("basering has parameters");} |
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60 | int OLD_COMPLEX=0; |
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61 | int iv=checkv(f); // check for variable appearing in f |
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62 | if(iv==0){ERROR("Wrong polynomial!");} |
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63 | poly v=var(iv); // f univariate in v |
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64 | |
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65 | int solutionprec=8;// set the control |
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66 | int numberprec=30; |
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67 | int splitcontrol=1; |
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68 | int rootcheck=0; |
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69 | if(size(#)>0){solutionprec=#[1];if(solutionprec<4){solutionprec=4;}} |
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70 | if(size(#)>1){numberprec=#[2];if(numberprec<8){numberprec=8;}} |
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71 | if(solutionprec>numberprec){numberprec=solutionprec;} |
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72 | if(size(#)>2){splitcontrol=#[3];} |
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73 | if(size(#)>3){rootcheck=#[4];} |
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74 | int prot=printlevel-voice+2; |
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75 | int ringprec=0; |
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76 | |
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77 | poly p=divzero(f,iv); // divide out zeros as solution |
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78 | int iz=deg(f)-deg(p); // multiplicity of zero solution |
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79 | if(prot!=0) |
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80 | { |
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81 | string pout; |
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82 | string nl=newline; |
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83 | pout="//BEGIN laguerre_solve"; |
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84 | if(iz!=0){pout=pout+nl+"//zeros: divide out "+string(iz);} |
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85 | dbprint(prot,pout); |
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86 | } |
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87 | string ss,tt,oo; |
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88 | ss="";oo=ss; |
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89 | if(npars(basering)==1) |
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90 | { |
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91 | if(OLD_COMPLEX) |
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92 | { |
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93 | tt="1,"+string(par(1)); |
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94 | if(tt==charstr(basering)) |
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95 | {ss=tt;ringprec=system("getPrecDigits");} |
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96 | } |
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97 | else |
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98 | { |
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99 | tt=charstr(basering); |
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100 | if(size(tt)>7) |
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101 | { |
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102 | if(string(tt[1..7])=="complex") |
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103 | { |
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104 | ss=tt; |
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105 | ringprec=system("getPrecDigits"); |
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106 | } |
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107 | } |
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108 | } |
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109 | } |
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110 | |
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111 | list roots,simple; |
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112 | if(deg(p)==0) // only zero was root |
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113 | { |
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114 | roots=addzero(roots,ss,iz,splitcontrol); |
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115 | if(prot!=0){dbprint(prot,"//END laguerre_solve");} |
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116 | return(roots); |
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117 | } |
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118 | |
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119 | if(prot!=0)// more informations |
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120 | { |
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121 | pout="//control: complex ring with precision "+string(numberprec); |
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122 | if(size(ss)==0){pout=pout+nl+ |
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123 | "// basering not complex, hence solutiontype string"; |
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124 | if(solutionprec<numberprec){pout=pout+nl+ |
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125 | "// with precision "+string(solutionprec);}} |
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126 | if(splitcontrol<0){pout=pout+nl+ "// no spliting";} |
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127 | if(splitcontrol==0){pout=pout+nl+"// output without multiple roots";} |
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128 | if(rootcheck){pout=pout+nl+ |
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129 | "// check roots with precision "+string(solutionprec);} |
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130 | dbprint(prot,pout); |
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131 | } |
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132 | |
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133 | def rn = basering;// set the complex ground field |
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134 | if (ringprec<numberprec) |
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135 | { |
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136 | tt="ring lagc=(complex,"+string(numberprec)+","+string(numberprec)+ |
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137 | "),"+string(var(iv))+",lp;"; |
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138 | execute(tt); |
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139 | poly p=imap(rn,p); |
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140 | poly v=var(1); |
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141 | } |
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142 | int ima=0; |
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143 | if(size(ss)!=0){ima=checkim(p);} |
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144 | number prc=0.1;// set precision of the solution |
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145 | prc=prc^solutionprec; |
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146 | if(prot!=0) |
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147 | { |
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148 | if(ringprec<numberprec){pout="//working in: "+tt;} |
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149 | if((size(ss)!=0)&&(ima!=0)){pout=pout+nl+ |
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150 | "// polynomial has complex coefficients";} |
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151 | dbprint(prot,pout); |
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152 | } |
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153 | |
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154 | int i1=1; |
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155 | int i2=1; |
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156 | ideal SPLIT=p; |
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157 | if(splitcontrol>=0)// splitting |
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158 | { |
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159 | if(prot!=0){dbprint(prot,"//split in working ring:");} |
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160 | SPLIT=splitsqrfree(p,v); |
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161 | i1=size(SPLIT); |
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162 | if((i1==1)&&(charstr(rn)=="0")) |
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163 | { |
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164 | if(prot!=0){dbprint(prot,"//split exact in basering:");} |
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165 | setring rn; |
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166 | if(v>1) |
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167 | { |
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168 | ideal SQQQQ=splitsqrfree(p,v); |
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169 | setring lagc; |
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170 | SPLIT=imap(rn,SQQQQ); |
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171 | } |
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172 | else |
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173 | { |
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174 | oo="ring exa=0,"+string(var(1))+",lp;"; |
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175 | execute(oo); |
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176 | ideal SQQQQ=splitsqrfree(imap(rn,p),var(1)); |
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177 | setring lagc; |
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178 | SPLIT=imap(exa,SQQQQ); |
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179 | kill exa; |
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180 | } |
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181 | i1=size(SPLIT); |
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182 | } |
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183 | if(prot!=0) |
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184 | { |
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185 | if(i1>1) |
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186 | { |
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187 | int i3=deg(SPLIT[1]); |
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188 | pout="//results of split(the squarefree factors):"; |
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189 | if(i3>0){pout=pout+nl+ |
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190 | "// multiplicity "+string(i2)+", degree "+string(i3);} |
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191 | while(i2<i1) |
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192 | { |
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193 | i2++; |
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194 | i3=deg(SPLIT[i2]); |
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195 | if(i3>0){pout=pout+nl+ |
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196 | "// multiplicity "+string(i2)+", degree "+string(i3);} |
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197 | } |
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198 | dbprint(prot,pout); |
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199 | i2=1; |
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200 | } |
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201 | else |
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202 | { |
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203 | if(charstr(rn)=="0"){dbprint(prot,"// polynomial is squarefree");} |
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204 | else{dbprint(prot,"// split without result");} |
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205 | } |
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206 | } |
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207 | } |
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208 | |
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209 | p=SPLIT[1];// the first part |
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210 | if(deg(p)>0) |
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211 | { |
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212 | roots=laguerre(p,numberprec,1);// the ring is already complex, hence numberprec is dummy |
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213 | if((size(roots)==0)||(string(roots[1])=="0")){ERROR("laguerre: no roots found");} |
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214 | if(rootcheck){checkroots(p,v,roots,ima,prc);} |
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215 | } |
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216 | while(i2<i1) |
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217 | { |
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218 | i2++; |
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219 | p=SPLIT[i2];// the part with multiplicity i2 |
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220 | if(deg(p)>0) |
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221 | { |
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222 | simple=laguerre(p,numberprec,1); |
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223 | if((size(simple)==0)||(string(simple[1])=="0")){ERROR("laguerre: no roots found");} |
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224 | if(rootcheck){checkroots(p,v,simple,ima,prc);} |
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225 | if(splitcontrol==0)// no multiple roots |
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226 | { |
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227 | roots=roots+simple; |
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228 | } |
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229 | else// multiple roots |
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230 | { |
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231 | roots=roots+makemult(simple,i2); |
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232 | } |
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233 | } |
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234 | } |
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235 | |
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236 | if((solutionprec<numberprec)&&(size(ss)==0))// shorter output |
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237 | { |
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238 | oo="ring lout=(complex,"+string(solutionprec)+",1)," |
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239 | +string(var(1))+",lp;"; |
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240 | execute(oo); |
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241 | list roots=imap(lagc,roots); |
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242 | roots=transroots(roots); |
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243 | if(iz>0){roots=addzero(roots,ss,iz,splitcontrol);} |
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244 | if(prot!=0){dbprint(prot,"//END laguerre_solve");} |
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245 | return(roots); |
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246 | } |
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247 | if(size(ss)==0){roots=transroots(roots);}// transform to string |
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248 | else // or map in basering |
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249 | { |
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250 | if(ringprec<numberprec) |
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251 | { |
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252 | setring rn; |
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253 | list roots=imap(lagc,roots); |
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254 | } |
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255 | } |
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256 | if(iz>0){roots=addzero(roots,ss,iz,splitcontrol);} |
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257 | if(prot!=0){dbprint(prot,"//END laguerre_solve");} |
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258 | return(roots); |
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259 | } |
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260 | example |
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261 | { |
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262 | "EXAMPLE:";echo=2; |
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263 | // Find all roots of an univariate polynomial using Laguerre's method: |
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264 | ring rs1= 0,(x,y),lp; |
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265 | poly f = 15x5 + x3 + x2 - 10; |
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266 | // 10 digits precision |
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267 | laguerre_solve(f,10); |
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268 | |
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269 | // Now with complex coefficients, |
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270 | // internal precision is 30 digits (default) |
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271 | printlevel=2; |
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272 | ring rsc= (real,10,i),x,lp; |
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273 | poly f = (15.4+i*5)*x^5 + (25.0e-2+i*2)*x^3 + x2 - 10*i; |
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274 | list l = laguerre_solve(f); |
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275 | l; |
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276 | // check result, value of substituted poly should be near to zero |
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277 | // remember that l contains a list of strings |
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278 | // in the case of a different ring |
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279 | subst(f,x,l[1]); |
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280 | subst(f,x,l[2]); |
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281 | } |
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282 | ////////////////////////////////////////////////////////////////////////////// |
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283 | // subprocedures for laguerre_solve |
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284 | /* |
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285 | * if p depends only on var(i) |
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286 | * returns i |
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287 | * otherwise 0 |
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288 | */ |
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289 | static proc checkv(poly p) |
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290 | { |
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291 | int n=nvars(basering); |
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292 | int i=0; |
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293 | int v; |
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294 | |
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295 | while (n>0) |
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296 | { |
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297 | if ((p-subst(p,var(n),0))!=0) |
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298 | { |
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299 | i++; |
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300 | if (i>1){return(0);} |
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301 | v=n; |
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302 | } |
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303 | n--; |
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304 | } |
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305 | return(v); |
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306 | } |
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307 | /* |
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308 | * if p has only real coefficients |
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309 | * returns 0 |
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310 | * otherwise 1 |
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311 | */ |
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312 | static proc checkim(poly p) |
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313 | { |
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314 | poly q=p; |
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315 | |
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316 | while(q!=0) |
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317 | { |
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318 | if(impart(leadcoef(q))!=0){return(1);} |
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319 | q=q-lead(q); |
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320 | } |
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321 | return(0); |
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322 | } |
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323 | /* |
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324 | * make multiplicity m |
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325 | */ |
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326 | static proc makemult(list si,int m) |
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327 | { |
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328 | int k0=0; |
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329 | int k1=size(si); |
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330 | int k2,k3; |
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331 | number ro; |
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332 | list msi; |
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333 | |
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334 | for(k2=1;k2<=k1;k2++) |
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335 | { |
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336 | ro=si[k2]; |
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337 | for(k3=m;k3>0;k3--){k0++;msi[k0]=ro;} |
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338 | } |
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339 | return(msi); |
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340 | } |
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341 | /* |
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342 | * returns 1 for n<1 |
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343 | */ |
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344 | static proc cmp1(number n) |
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345 | { |
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346 | number r=repart(n); |
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347 | number i=impart(n); |
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348 | number c=r*r+i*i; |
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349 | if(c>1){return(1);} |
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350 | else{return(0);} |
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351 | } |
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352 | /* |
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353 | * exact division of polys f/g |
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354 | * (should be internal) |
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355 | */ |
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356 | static proc exdiv(poly f,poly g,poly v) |
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357 | { |
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358 | int d1=deg(f); |
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359 | int d2=deg(g); |
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360 | poly r0=f; |
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361 | poly rf=0; |
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362 | poly h; |
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363 | number n,m; |
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364 | |
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365 | m=leadcoef(g); |
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366 | while ((r0!=0)&&(d1>=d2)) |
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367 | { |
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368 | n=leadcoef(r0)/m; |
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369 | h=n*v^(d1-d2); |
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370 | rf=rf+h; |
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371 | r0=r0-h*g; |
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372 | d1=deg(r0); |
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373 | } |
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374 | return(cleardenom(rf)); |
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375 | } |
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376 | /* |
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377 | * p is univariant in x |
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378 | * perform a split of p into squarefree factors |
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379 | * such that the returned ideal 'split' consists of |
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380 | * the faktors, i.e. |
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381 | * p = n * product ( split[i]^i ) , n a number |
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382 | */ |
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383 | static proc splitsqrfree(poly p, poly x) |
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384 | { |
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385 | int dd=deg(p); |
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386 | if(dd==1){return(p);} |
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387 | int i=1; |
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388 | int j; |
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389 | ideal h,split; |
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390 | poly high; |
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391 | |
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392 | h=interred(ideal(p,diff(p,x))); |
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393 | if(deg(h[1])==0){return(p);} |
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394 | high=h[1]; |
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395 | split[1]=exdiv(p,high,x); |
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396 | while(1) |
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397 | { |
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398 | h=interred(ideal(split[i],high)); |
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399 | j=deg(h[1]); |
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400 | if(j==0){return(p);} |
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401 | if(deg(h[1])==deg(split[i])) |
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402 | { |
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403 | split=split,split[i]; |
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404 | split[i]=1; |
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405 | } |
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406 | else |
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407 | { |
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408 | split[i]=exdiv(split[i],h[1],x); |
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409 | split=split,h[1]; |
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410 | dd=dd-deg(split[i])*i; |
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411 | } |
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412 | j=j*(i+1); |
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413 | if(j==dd){break;} |
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414 | if(j>dd){return(p);} |
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415 | high=exdiv(high,h[1],x); |
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416 | if(deg(high)==0){return(p);}; |
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417 | i++; |
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418 | } |
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419 | return(split); |
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420 | } |
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421 | /* |
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422 | * see checkroots |
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423 | */ |
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424 | static proc nerr(number n,number m) |
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425 | { |
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426 | int r; |
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427 | number z=0; |
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428 | number nr=repart(n); |
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429 | number ni=impart(n); |
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430 | if(nr<z){nr=z-nr;} |
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431 | if(ni<z){ni=nr-ni;} |
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432 | else{ni=nr+ni;} |
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433 | if(ni<m){r=0;} |
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434 | else{r=1;} |
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435 | return(r); |
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436 | } |
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437 | /* |
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438 | * returns ERROR for nerr(p(r[i]))>=pr |
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439 | */ |
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440 | static proc checkroots(poly p,poly v,list r,int ima,number pr) |
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441 | { |
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442 | int i=0; |
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443 | int j; |
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444 | number n,m; |
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445 | ideal li; |
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446 | |
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447 | while(i<size(r)) |
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448 | { |
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449 | i++; |
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450 | n=r[i]; |
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451 | j=cmp1(n); |
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452 | if(j!=0){li[1]=v/n-1;m=1;} |
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453 | else{li[1]=v-n;m=n;} |
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454 | if((ima==0)&&(impart(n)!=0)) |
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455 | { |
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456 | i++; |
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457 | n=r[i]; |
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458 | if(j!=0){li[1]=li[1]*(v/n-1);} |
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459 | else{li[1]=li[1]*(v-n);m=m*n;} |
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460 | } |
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461 | attrib(li,"isSB",1); |
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462 | n=leadcoef(reduce(p,li));n=n/m; |
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463 | if(n!=0) |
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464 | {if(nerr(n,pr)!=0){ERROR("Unsufficient accuracy!");}} |
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465 | } |
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466 | } |
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467 | /* |
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468 | * transforms thr result to string |
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469 | */ |
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470 | static proc transroots(list r) |
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471 | { |
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472 | int i=size(r); |
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473 | while (i>0) |
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474 | { |
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475 | r[i]=string(r[i]); |
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476 | i--; |
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477 | } |
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478 | return(r); |
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479 | } |
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480 | /* |
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481 | * returns a poly without zeroroots |
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482 | */ |
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483 | static proc divzero(poly f,int iv) |
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484 | { |
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485 | poly p=f; |
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486 | poly q=p; |
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487 | poly r; |
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488 | while(p==q) |
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489 | { |
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490 | q=p/var(iv); |
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491 | r=q*var(iv); |
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492 | if(r==p){p=q;} |
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493 | } |
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494 | return(p); |
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495 | } |
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496 | /* |
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497 | * add zeros to solution |
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498 | */ |
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499 | static proc addzero(list zz,string ss,int iz,int a) |
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500 | { |
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501 | int i=1; |
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502 | int j=size(zz); |
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503 | |
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504 | if(size(ss)==0){zz[j+1]="0";} |
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505 | else{zz[j+1]=number(0);} |
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506 | if(a==0){return(zz);} |
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507 | while(i<iz) |
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508 | { |
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509 | i++; |
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510 | if(size(ss)==0){zz[j+i]="0";} |
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511 | else{zz[j+i]=number(0);} |
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512 | } |
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513 | return(zz); |
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514 | } |
---|
515 | /////////////////////////////////////////////////////////////////////////////// |
---|
516 | |
---|
517 | proc solve( ideal G, list # ) |
---|
518 | "USAGE: solve(G [, m, n [, l]] [,\"oldring\"] [,\"nodisplay\"] ); G = ideal, |
---|
519 | m, n, l = integers (control parameters of the method), outR ring |
---|
520 | m: precision of output in digits ( 4 <= m) and of the generated ring |
---|
521 | of complex numbers; |
---|
522 | n: control of multiplicity |
---|
523 | n = 0, return all different roots |
---|
524 | n != 0, find all roots (with multiplicity) |
---|
525 | l: precision of internal computation in decimal digits ( l >=8 ) |
---|
526 | only if the basering is not complex or complex with smaller |
---|
527 | precision, @* |
---|
528 | [default: (m,n,l) = (8,0,30), or if only (m,n) are set explicitly |
---|
529 | with n!=0, then (m,n,l) = (m,n,60) ] |
---|
530 | ASSUME: the ideal is 0-dimensional;@* |
---|
531 | basering has characteristic 0 and is either complex or without |
---|
532 | parameters; |
---|
533 | RETURN: (1) If called without the additional parameter @code{\"oldring\"}: @* |
---|
534 | ring @code{R} with the same number of variables but with complex |
---|
535 | coefficients (and precision m). @code{R} comes with a list |
---|
536 | @code{SOL} of numbers, in which complex roots of G are stored: @* |
---|
537 | * If n = 0, @code{SOL} is the list of all different solutions, each |
---|
538 | of them being represented by a list of numbers. @* |
---|
539 | * If n != 0, @code{SOL} is a list of two list: SOL[i][1] is the list |
---|
540 | of all different solutions with the multiplicity SOL[i][2].@* |
---|
541 | SOL is ordered w.r.t. multiplicity (the smallest first). @* |
---|
542 | (2) If called with the additional parameter @code{\"oldring\"}, the |
---|
543 | procedure looks for an appropriate ring (at top level) in which |
---|
544 | the solutions can be stored (interactive). @* |
---|
545 | The user may then select an appropriate ring and choose a name for |
---|
546 | the output list in this ring. The list is exported directly to the |
---|
547 | selected ring and the return value is a string \"result exported to\" |
---|
548 | + name of the selected ring. |
---|
549 | NOTE: If the problem is not 0-dim. the procedure stops with ERROR. If the |
---|
550 | ideal G is not a lexicographic Groebner basis, the lexicographic |
---|
551 | Groebner basis is computed internally (Hilbert driven). @* |
---|
552 | The computed solutions are displayed, unless @code{solve} is called |
---|
553 | with the additional parameter @code{\"nodisplay\"}. |
---|
554 | EXAMPLE: example solve; shows an example |
---|
555 | " |
---|
556 | { |
---|
557 | // test if basering admissible |
---|
558 | if (char(basering)!=0){ERROR("characteristic of basering not 0");} |
---|
559 | if ((charstr(basering)[1]=="0") and (npars(basering)!=0)) |
---|
560 | { ERROR("basering has parameters"); } |
---|
561 | |
---|
562 | // some global settings and control |
---|
563 | int oldr, nodisp, ii, jj; |
---|
564 | list LL; |
---|
565 | int outprec = 8; |
---|
566 | int mu = 0; |
---|
567 | int prec = 30; |
---|
568 | // check additional parameters... |
---|
569 | if (size(#)>0) |
---|
570 | { |
---|
571 | int sofar=1; |
---|
572 | if (typeof(#[1])=="int") |
---|
573 | { |
---|
574 | outprec = #[1]; |
---|
575 | if (outprec<4){outprec = 4;} |
---|
576 | if (size(#)>1) |
---|
577 | { |
---|
578 | if (typeof(#[2])=="int") |
---|
579 | { |
---|
580 | mu = #[2]; |
---|
581 | if (size(#)>2) |
---|
582 | { |
---|
583 | if (typeof(#[3])=="int") |
---|
584 | { |
---|
585 | prec = #[3]; |
---|
586 | if (prec<8){prec = 8;} |
---|
587 | } |
---|
588 | else |
---|
589 | { |
---|
590 | if(mu!=0){prec = 60;} |
---|
591 | if (#[3]=="oldring"){ oldr=1; } |
---|
592 | if (#[3]=="nodisplay"){ nodisp=1; } |
---|
593 | } |
---|
594 | sofar=3; |
---|
595 | } |
---|
596 | } |
---|
597 | else |
---|
598 | { |
---|
599 | if (#[2]=="oldring"){ oldr=1; } |
---|
600 | if (#[2]=="nodisplay"){ nodisp=1; } |
---|
601 | } |
---|
602 | sofar=2; |
---|
603 | } |
---|
604 | } |
---|
605 | else |
---|
606 | { |
---|
607 | if (#[1]=="oldring"){ oldr=1; } |
---|
608 | if (#[1]=="nodisplay"){ nodisp=1; } |
---|
609 | } |
---|
610 | for (ii=sofar+1;ii<=size(#);ii++) |
---|
611 | { // check for additional strings |
---|
612 | if (typeof(#[ii])=="string") |
---|
613 | { |
---|
614 | if (#[ii]=="oldring"){ oldr=1; } |
---|
615 | if (#[ii]=="nodisplay"){ nodisp=1; } |
---|
616 | } |
---|
617 | } |
---|
618 | } |
---|
619 | if (outprec>prec){prec = outprec;} |
---|
620 | // if interaktive version is chosen -- choice of basering (Top::`outR`) |
---|
621 | // and name for list of solutions (outL): |
---|
622 | if (oldr==1) |
---|
623 | { |
---|
624 | list Out; |
---|
625 | LL=names(Top); |
---|
626 | for (ii=1;ii<=size(LL);ii++) |
---|
627 | { |
---|
628 | if (typeof(`LL[ii]`)=="ring") |
---|
629 | { |
---|
630 | if (find(charstr(`LL[ii]`),"complex,"+string(outprec))) |
---|
631 | { |
---|
632 | jj++; |
---|
633 | Out[jj]=LL[ii]; |
---|
634 | } |
---|
635 | } |
---|
636 | } |
---|
637 | if (size(Out)>0) |
---|
638 | { |
---|
639 | print("// *** You may select between the following rings for storing "+ |
---|
640 | "the list of"); |
---|
641 | print("// *** complex solutions:"); |
---|
642 | Out; |
---|
643 | print("// *** Enter the number of the chosen ring"); |
---|
644 | print("// *** (0: none of them => new ring created and returned)"); |
---|
645 | string chosen; |
---|
646 | while (chosen=="") { chosen=read(""); } |
---|
647 | execute("def tchosen = "+chosen); |
---|
648 | if (typeof(tchosen)=="int") |
---|
649 | { |
---|
650 | if ((tchosen>0) and (tchosen<=size(Out))) |
---|
651 | { |
---|
652 | string outR = Out[tchosen]; |
---|
653 | print("// *** You have chosen the ring "+ outR +". In this ring" |
---|
654 | +" the following objects"); |
---|
655 | print("//*** are defined:"); |
---|
656 | listvar(Top::`outR`); |
---|
657 | print("// *** Enter a name for the list of solutions (different "+ |
---|
658 | "from existing names):"); |
---|
659 | string outL; |
---|
660 | while (outL==""){ outL=read(""); } |
---|
661 | } |
---|
662 | } |
---|
663 | } |
---|
664 | else |
---|
665 | { |
---|
666 | print("No appropriate ring for storing the list of solutions found " + |
---|
667 | "=> new ring created and returned"); |
---|
668 | } |
---|
669 | if (not(defined(outR))) { oldr=0; } |
---|
670 | } |
---|
671 | |
---|
672 | // string rinC = nameof(basering)+"C"; |
---|
673 | string sord = ordstr(basering); |
---|
674 | int nv = nvars(basering); |
---|
675 | def rin = basering; |
---|
676 | intvec ovec = option(get); |
---|
677 | option(redSB); |
---|
678 | option(returnSB); |
---|
679 | int sb = attrib(G,"isSB"); |
---|
680 | int lp = 0; |
---|
681 | if (size(sord)==size("C,lp()"+string(nv))) |
---|
682 | { |
---|
683 | lp = find(sord,"lp"); |
---|
684 | } |
---|
685 | |
---|
686 | // ERROR |
---|
687 | if (sb){if (dim(G)!=0){ERROR("ideal not zero-dimensional");}} |
---|
688 | |
---|
689 | // the trivial homogeneous case (unique solution: (0,...0)) |
---|
690 | if (homog(G)) |
---|
691 | { |
---|
692 | if (sb==0) |
---|
693 | { |
---|
694 | execute("ring dphom=("+charstr(rin)+"),("+ |
---|
695 | varstr(rin)+"),dp;"); |
---|
696 | ideal G = std(imap(rin,G)); |
---|
697 | if (dim(G)!=0){ERROR("ideal not zero-dimensional");} |
---|
698 | int vdG=vdim(G); |
---|
699 | } |
---|
700 | if (oldr!=1) |
---|
701 | { |
---|
702 | execute("ring rinC =(complex,"+string(outprec)+ |
---|
703 | "),("+varstr(basering)+"),lp;"); |
---|
704 | list SOL; |
---|
705 | if (mu==0){SOL[1] = zerolist(nv);} |
---|
706 | else{SOL[1] = list(zerolist(nv),list(vdG));} |
---|
707 | export SOL; |
---|
708 | if (nodisp==0) { print(SOL); } |
---|
709 | option(set,ovec); |
---|
710 | dbprint( printlevel-voice+3," |
---|
711 | // 'solve' created a ring, in which a list SOL of numbers (the complex solutions) |
---|
712 | // is stored. |
---|
713 | // To access the list of complex solutions, type (if the name R was assigned |
---|
714 | // to the return value): |
---|
715 | setring R; SOL; "); |
---|
716 | return(rinC); |
---|
717 | } |
---|
718 | else |
---|
719 | { |
---|
720 | setring (Top::`outR`); |
---|
721 | list SOL; |
---|
722 | if (mu==0){SOL[1] = zerolist(nv);} |
---|
723 | else{SOL[1] = list(zerolist(nv),list(vdG));} |
---|
724 | execute("def "+outL + "=SOL;"); |
---|
725 | execute("export "+outL+";"); |
---|
726 | if (nodisp==0) { print(SOL); } |
---|
727 | option(set,ovec); |
---|
728 | kill SOL; |
---|
729 | return("result exported to "+outR+" as list "+outL); |
---|
730 | } |
---|
731 | } |
---|
732 | |
---|
733 | // look for reduced standard basis in lex |
---|
734 | if (sb*lp==0) |
---|
735 | { |
---|
736 | if (sb==0) |
---|
737 | { |
---|
738 | execute("ring dphilb=("+charstr(rin)+"),("+ varstr(rin)+"),dp;"); |
---|
739 | ideal G = imap(rin,G); |
---|
740 | G = std(G); |
---|
741 | if (dim(G)!=0){ERROR("ideal not zero-dimensional");} |
---|
742 | } |
---|
743 | else |
---|
744 | { |
---|
745 | def dphilb = basering; |
---|
746 | G=interred(G); |
---|
747 | attrib(G,"isSB",1); |
---|
748 | } |
---|
749 | execute("ring lexhilb=("+charstr(rin)+"),("+ varstr(rin)+"),lp;"); |
---|
750 | option(redTail); |
---|
751 | ideal H = fglm(dphilb,G); |
---|
752 | kill dphilb; |
---|
753 | H = simplify(H,2); |
---|
754 | if (lp){setring rin;} |
---|
755 | else |
---|
756 | { |
---|
757 | execute("ring lplex=("+charstr(rin)+"),("+varstr(rin)+"),lp;"); |
---|
758 | } |
---|
759 | ideal H = imap(lexhilb,H); |
---|
760 | kill lexhilb; |
---|
761 | } |
---|
762 | else{ideal H = interred(G);} |
---|
763 | |
---|
764 | // only 1 variable |
---|
765 | def hr = basering; |
---|
766 | if (nv==1) |
---|
767 | { |
---|
768 | if ((mu==0) and (charstr(basering)[1]=="0")) |
---|
769 | { // special case |
---|
770 | list L = laguerre_solve(H[1],prec,prec,mu,0); // list of strings |
---|
771 | if (oldr!=1) |
---|
772 | { |
---|
773 | execute("ring rinC =(complex,"+string(outprec)+"),("+varstr(basering)+"),lp;"); |
---|
774 | list SOL; |
---|
775 | for (ii=1; ii<=size(L); ii++ ) { execute("SOL[ii]=number("+L[ii]+");"); } |
---|
776 | export SOL; |
---|
777 | if (nodisp==0) { print(SOL); } |
---|
778 | option(set,ovec); |
---|
779 | dbprint( printlevel-voice+3," |
---|
780 | // 'solve' created a ring, in which a list SOL of numbers (the complex solutions) |
---|
781 | // is stored. |
---|
782 | // To access the list of complex solutions, type (if the name R was assigned |
---|
783 | // to the return value): |
---|
784 | setring R; SOL; "); |
---|
785 | return(rinC); |
---|
786 | } |
---|
787 | else |
---|
788 | { |
---|
789 | setring (Top::`outR`); |
---|
790 | list SOL; |
---|
791 | for (ii=1; ii<=size(L); ii++ ) { execute("SOL[ii]="+L[ii]+";"); } |
---|
792 | execute("def "+outL + "=SOL;"); |
---|
793 | execute("export "+outL+";"); |
---|
794 | if (nodisp==0) { print(SOL); } |
---|
795 | option(set,ovec); |
---|
796 | kill SOL; |
---|
797 | return("result exported to "+outR+" as list "+outL); |
---|
798 | } |
---|
799 | } |
---|
800 | else |
---|
801 | { |
---|
802 | execute("ring internC=(complex,"+string(prec)+"),("+varstr(basering)+"),lp;"); |
---|
803 | ideal H = imap(hr,H); |
---|
804 | list sp = splittolist(splitsqrfree(H[1],var(1))); |
---|
805 | jj = size(sp); |
---|
806 | while(jj>0) |
---|
807 | { |
---|
808 | sp[jj][1] = laguerre(sp[jj][1],prec,1); |
---|
809 | jj--; |
---|
810 | } |
---|
811 | setring hr; |
---|
812 | if (oldr!=1) |
---|
813 | { |
---|
814 | execute("ring rinC =(complex,"+string(outprec)+"),("+varstr(basering)+"),lp;"); |
---|
815 | list SOL; |
---|
816 | list sp=imap(internC,sp); |
---|
817 | if(mu!=0){ SOL=sp; } |
---|
818 | else |
---|
819 | { |
---|
820 | jj = size(sp); |
---|
821 | SOL=sp[jj][1]; |
---|
822 | while(jj>1) |
---|
823 | { |
---|
824 | jj--; |
---|
825 | SOL = sp[jj][1]+SOL; |
---|
826 | } |
---|
827 | } |
---|
828 | export SOL; |
---|
829 | if (nodisp==0) { print(SOL); } |
---|
830 | option(set,ovec); |
---|
831 | dbprint( printlevel-voice+3," |
---|
832 | // 'solve' created a ring, in which a list SOL of numbers (the complex solutions) |
---|
833 | // is stored. |
---|
834 | // To access the list of complex solutions, type (if the name R was assigned |
---|
835 | // to the return value): |
---|
836 | setring R; SOL; "); |
---|
837 | return(rinC); |
---|
838 | } |
---|
839 | else |
---|
840 | { |
---|
841 | setring (Top::`outR`); |
---|
842 | list SOL; |
---|
843 | list sp=imap(internC,sp); |
---|
844 | if(mu!=0){ SOL=sp; } |
---|
845 | else |
---|
846 | { |
---|
847 | jj = size(sp); |
---|
848 | SOL=sp[jj][1]; |
---|
849 | while(jj>1) |
---|
850 | { |
---|
851 | jj--; |
---|
852 | SOL = sp[jj][1]+SOL; |
---|
853 | } |
---|
854 | } |
---|
855 | kill sp; |
---|
856 | execute("def "+outL + "=SOL;"); |
---|
857 | execute("export "+outL+";"); |
---|
858 | if (nodisp==0) { print(SOL); } |
---|
859 | option(set,ovec); |
---|
860 | kill SOL; |
---|
861 | return("result exported to "+outR+" as list "+outL); |
---|
862 | } |
---|
863 | } |
---|
864 | } |
---|
865 | |
---|
866 | // the triangular sets (not univariate case) |
---|
867 | attrib(H,"isSB",1); |
---|
868 | if (mu==0) |
---|
869 | { |
---|
870 | list sp = triangMH(H); // faster, but destroy multiplicity |
---|
871 | } |
---|
872 | else |
---|
873 | { |
---|
874 | list sp = triangM(H); |
---|
875 | } |
---|
876 | |
---|
877 | // create the complex ring and map the result |
---|
878 | if (outprec<prec) |
---|
879 | { |
---|
880 | execute("ring internC=(complex,"+string(prec)+"),("+varstr(hr)+"),lp;"); |
---|
881 | } |
---|
882 | else |
---|
883 | { |
---|
884 | execute("ring rinC =(complex,"+string(outprec)+"),("+varstr(basering)+"),lp;"); |
---|
885 | } |
---|
886 | list triC = imap(hr,sp); |
---|
887 | |
---|
888 | // solve the tridiagonal systems |
---|
889 | int js = size(triC); |
---|
890 | list ret1; |
---|
891 | if (mu==0) |
---|
892 | { |
---|
893 | ret1 = trisolve(list(),triC[1],prec); |
---|
894 | while (js>1) |
---|
895 | { |
---|
896 | ret1 = trisolve(list(),triC[js],prec)+ret1; |
---|
897 | js--; |
---|
898 | } |
---|
899 | } |
---|
900 | else |
---|
901 | { |
---|
902 | ret1 = mutrisolve(list(),triC[1],prec); |
---|
903 | while (js>1) |
---|
904 | { |
---|
905 | ret1 = addlist(mutrisolve(list(),triC[js],prec),ret1,1); |
---|
906 | js--; |
---|
907 | } |
---|
908 | ret1 = finalclear(ret1); |
---|
909 | } |
---|
910 | |
---|
911 | // final computations |
---|
912 | option(set,ovec); |
---|
913 | if (outprec==prec) |
---|
914 | { // we are in ring rinC |
---|
915 | if (oldr!=1) |
---|
916 | { |
---|
917 | list SOL=ret1; |
---|
918 | export SOL; |
---|
919 | if (nodisp==0) { print(SOL); } |
---|
920 | dbprint( printlevel-voice+3," |
---|
921 | // 'solve' created a ring, in which a list SOL of numbers (the complex solutions) |
---|
922 | // is stored. |
---|
923 | // To access the list of complex solutions, type (if the name R was assigned |
---|
924 | // to the return value): |
---|
925 | setring R; SOL; "); |
---|
926 | return(rinC); |
---|
927 | } |
---|
928 | else |
---|
929 | { |
---|
930 | setring (Top::`outR`); |
---|
931 | list SOL=imap(rinC,ret1); |
---|
932 | execute("def "+outL + "=SOL;"); |
---|
933 | execute("export "+outL+";"); |
---|
934 | if (nodisp==0) { print(SOL); } |
---|
935 | kill SOL; |
---|
936 | return("result exported to "+outR+" as list "+outL); |
---|
937 | } |
---|
938 | } |
---|
939 | else |
---|
940 | { |
---|
941 | if (oldr!=1) |
---|
942 | { |
---|
943 | execute("ring rinC =(complex,"+string(outprec)+"),("+varstr(basering)+"),lp;"); |
---|
944 | list SOL=imap(internC,ret1); |
---|
945 | export SOL; |
---|
946 | if (nodisp==0) { print(SOL); } |
---|
947 | dbprint( printlevel-voice+3," |
---|
948 | // 'solve' created a ring, in which a list SOL of numbers (the complex solutions) |
---|
949 | // is stored. |
---|
950 | // To access the list of complex solutions, type (if the name R was assigned |
---|
951 | // to the return value): |
---|
952 | setring R; SOL; "); |
---|
953 | return(rinC); |
---|
954 | } |
---|
955 | else |
---|
956 | { |
---|
957 | setring (Top::`outR`); |
---|
958 | list SOL=imap(internC,ret1); |
---|
959 | execute("def "+outL + "=SOL;"); |
---|
960 | execute("export "+outL+";"); |
---|
961 | if (nodisp==0) { print(SOL); } |
---|
962 | kill SOL; |
---|
963 | return("result exported to "+outR+" as list "+outL); |
---|
964 | } |
---|
965 | } |
---|
966 | } |
---|
967 | example |
---|
968 | { |
---|
969 | "EXAMPLE:";echo=2; |
---|
970 | // Find all roots of a multivariate ideal using triangular sets: |
---|
971 | int d,t,s = 4,3,2 ; |
---|
972 | int i; |
---|
973 | ring A=0,x(1..d),dp; |
---|
974 | poly p=-1; |
---|
975 | for (i=d; i>0; i--) { p=p+x(i)^s; } |
---|
976 | ideal I = x(d)^t-x(d)^s+p; |
---|
977 | for (i=d-1; i>0; i--) { I=x(i)^t-x(i)^s+p,I; } |
---|
978 | I; |
---|
979 | // the multiplicity is |
---|
980 | vdim(std(I)); |
---|
981 | def AC=solve(I,6,0,"nodisplay"); // solutions should not be displayed |
---|
982 | // list of solutions is stored in AC as the list SOL (default name) |
---|
983 | setring AC; |
---|
984 | size(SOL); // number of different solutions |
---|
985 | SOL[5]; // the 5th solution |
---|
986 | // you must start with char. 0 |
---|
987 | setring A; |
---|
988 | def AC1=solve(I,6,1,"nodisplay"); |
---|
989 | setring AC1; |
---|
990 | size(SOL); // number of different multiplicities |
---|
991 | SOL[1][1][1]; // a solution with |
---|
992 | SOL[1][2]; // multiplicity 1 |
---|
993 | SOL[2][1][1]; // a solution with |
---|
994 | SOL[2][2]; // multiplicity 12 |
---|
995 | // the number of different solutions is equal to |
---|
996 | size(SOL[1][1])+size(SOL[2][1]); |
---|
997 | // the number of complex solutions (counted with multiplicities) is |
---|
998 | size(SOL[1][1])*SOL[1][2]+size(SOL[2][1])*SOL[2][2]; |
---|
999 | } |
---|
1000 | ////////////////////////////////////////////////////////////////////////////// |
---|
1001 | // subprocedures for solve |
---|
1002 | |
---|
1003 | |
---|
1004 | /* |
---|
1005 | * return one zero-solution |
---|
1006 | */ |
---|
1007 | static proc zerolist(int nv) |
---|
1008 | { |
---|
1009 | list ret; |
---|
1010 | int i; |
---|
1011 | number o=0; |
---|
1012 | |
---|
1013 | for (i=nv;i>0;i--){ret[i] = o;} |
---|
1014 | return(ret); |
---|
1015 | } |
---|
1016 | |
---|
1017 | /* ----------------------- check solution ----------------------- */ |
---|
1018 | static proc multsol(list ff, int c) |
---|
1019 | { |
---|
1020 | int i,j; |
---|
1021 | |
---|
1022 | i = 0; |
---|
1023 | j = size(ff); |
---|
1024 | while (j>0) |
---|
1025 | { |
---|
1026 | if(c){i = i+ff[j][2]*size(ff[j][1]);} |
---|
1027 | else{i = i+size(ff[j][1]);} |
---|
1028 | j--; |
---|
1029 | } |
---|
1030 | return(i); |
---|
1031 | } |
---|
1032 | |
---|
1033 | /* |
---|
1034 | * the inputideal A => zero ? |
---|
1035 | */ |
---|
1036 | static proc checksol(ideal A, list lr) |
---|
1037 | { |
---|
1038 | int d = nvars(basering); |
---|
1039 | list ro; |
---|
1040 | ideal re,h; |
---|
1041 | int i,j,k; |
---|
1042 | |
---|
1043 | for (i=size(lr);i>0;i--) |
---|
1044 | { |
---|
1045 | ro = lr[i]; |
---|
1046 | for (j=d;j>0;j--) |
---|
1047 | { |
---|
1048 | re[j] = var(j)-ro[j]; |
---|
1049 | } |
---|
1050 | attrib(re,"isSB",1); |
---|
1051 | k = size(reduce(A,re,1)); |
---|
1052 | if (k){return(i);} |
---|
1053 | } |
---|
1054 | return(0); |
---|
1055 | } |
---|
1056 | |
---|
1057 | /* |
---|
1058 | * compare 2 solutions: returns 0 for equal |
---|
1059 | */ |
---|
1060 | static proc cmpn(list a,list b) |
---|
1061 | { |
---|
1062 | int ii; |
---|
1063 | |
---|
1064 | for(ii=size(a);ii>0;ii--){if(a[ii]!=b[ii]) break;} |
---|
1065 | return(ii); |
---|
1066 | } |
---|
1067 | |
---|
1068 | /* |
---|
1069 | * delete equal solutions in the list |
---|
1070 | */ |
---|
1071 | static proc delequal(list r, int w) |
---|
1072 | { |
---|
1073 | list h; |
---|
1074 | int i,j,k,c; |
---|
1075 | |
---|
1076 | if (w) |
---|
1077 | { |
---|
1078 | k = size(r); |
---|
1079 | h = r[k][1]; |
---|
1080 | k--; |
---|
1081 | while (k>0) |
---|
1082 | { |
---|
1083 | h = r[k][1]+h; |
---|
1084 | k--; |
---|
1085 | } |
---|
1086 | } |
---|
1087 | else{h = r;} |
---|
1088 | k=size(h); |
---|
1089 | i=1; |
---|
1090 | while(i<k) |
---|
1091 | { |
---|
1092 | j=k; |
---|
1093 | while(j>i) |
---|
1094 | { |
---|
1095 | c=cmpn(h[i],h[j]); |
---|
1096 | if(c==0) |
---|
1097 | { |
---|
1098 | h=delete(h,j); |
---|
1099 | k--; |
---|
1100 | } |
---|
1101 | j--; |
---|
1102 | } |
---|
1103 | i++; |
---|
1104 | } |
---|
1105 | return(h); |
---|
1106 | } |
---|
1107 | |
---|
1108 | /* ----------------------- substitution ----------------------- */ |
---|
1109 | /* |
---|
1110 | * instead of subst(T,var(v),n), much faster |
---|
1111 | * need option(redSB) ! |
---|
1112 | */ |
---|
1113 | static proc linreduce(ideal T, int v, number n) |
---|
1114 | { |
---|
1115 | ideal re = var(v)-n; |
---|
1116 | attrib (re,"isSB",1); |
---|
1117 | return (reduce(T,re)); |
---|
1118 | } |
---|
1119 | |
---|
1120 | /* ----------------------- triangular solution ----------------------- */ |
---|
1121 | /* |
---|
1122 | * solution of one tridiagonal system T |
---|
1123 | * with precision prec |
---|
1124 | * T[1] is univariant in var(1) |
---|
1125 | * list o is empty for the first call |
---|
1126 | */ |
---|
1127 | static proc trisolve(list o, ideal T, int prec) |
---|
1128 | { |
---|
1129 | list lroots,ll; |
---|
1130 | ideal S; |
---|
1131 | int i,d; |
---|
1132 | |
---|
1133 | d = size(T); |
---|
1134 | S = interred(ideal(T[1],diff(T[1],var(d)))); |
---|
1135 | if (deg(S[1])) |
---|
1136 | { |
---|
1137 | T[1] = exdiv(T[1],S[1],var(d)); |
---|
1138 | } |
---|
1139 | ll = laguerre(T[1],prec,1); |
---|
1140 | for (i=size(ll);i>0;i--){ll[i] = list(ll[i])+o;} |
---|
1141 | if (d==1){return(ll);} |
---|
1142 | for (i=size(ll);i>0;i--) |
---|
1143 | { |
---|
1144 | S = linreduce(ideal(T[2..d]),d,ll[i][1]); |
---|
1145 | lroots = trisolve(ll[i],S,prec)+lroots; |
---|
1146 | } |
---|
1147 | return(lroots); |
---|
1148 | } |
---|
1149 | |
---|
1150 | /* ------------------- triangular solution (mult) ------------------- */ |
---|
1151 | /* |
---|
1152 | * recompute equal solutions w.r.t. multiplicity |
---|
1153 | */ |
---|
1154 | static proc finalclear(list b) |
---|
1155 | { |
---|
1156 | list a = b; |
---|
1157 | list r; |
---|
1158 | int i,l,ju,j,k,ku,mu,c; |
---|
1159 | |
---|
1160 | // a[i] only |
---|
1161 | i = 1; |
---|
1162 | while (i<=size(a)) |
---|
1163 | { |
---|
1164 | ju = size(a[i][1]); |
---|
1165 | j = 1; |
---|
1166 | while (j<=ju) |
---|
1167 | { |
---|
1168 | mu = 1; |
---|
1169 | k = j+1; |
---|
1170 | while (k<=ju) |
---|
1171 | { |
---|
1172 | c = cmpn(a[i][1][j],a[i][1][k]); |
---|
1173 | if (c==0) |
---|
1174 | { |
---|
1175 | a[i][1] = delete(a[i][1],k); |
---|
1176 | ju--; |
---|
1177 | mu++; |
---|
1178 | } |
---|
1179 | else{k++;} |
---|
1180 | } |
---|
1181 | if (mu>1) |
---|
1182 | { |
---|
1183 | r[1] = a[i]; |
---|
1184 | r[1][1] = list(a[i][1][j]); |
---|
1185 | a[i][1] = delete(a[i][1],j); |
---|
1186 | a = addlist(r,a,mu); |
---|
1187 | ju--; |
---|
1188 | } |
---|
1189 | else{j++;} |
---|
1190 | } |
---|
1191 | if (ju==0){a = delete(a,i);} |
---|
1192 | else{i++;} |
---|
1193 | } |
---|
1194 | // a[i], a[l] |
---|
1195 | i = 1; |
---|
1196 | while (i<size(a)) |
---|
1197 | { |
---|
1198 | ju = size(a[i][1]); |
---|
1199 | l = i+1; |
---|
1200 | while (l<=size(a)) |
---|
1201 | { |
---|
1202 | ku = size(a[l][1]); |
---|
1203 | j = 1; |
---|
1204 | while (j<=ju) |
---|
1205 | { |
---|
1206 | mu = 0; |
---|
1207 | k = 1; |
---|
1208 | while (k<=ku) |
---|
1209 | { |
---|
1210 | c = cmpn(a[i][1][j],a[l][1][k]); |
---|
1211 | if (c==0) |
---|
1212 | { |
---|
1213 | mu = a[i][2]+a[l][2]; |
---|
1214 | r[1] = a[l]; |
---|
1215 | r[1][1] = list(a[l][1][k]); |
---|
1216 | r[1][2] = mu; |
---|
1217 | a[l][1] = delete(a[l][1],k); |
---|
1218 | a = addlist(r,a,1); |
---|
1219 | ku--; |
---|
1220 | break; |
---|
1221 | } |
---|
1222 | else{k++;} |
---|
1223 | } |
---|
1224 | if (mu) |
---|
1225 | { |
---|
1226 | a[i][1] = delete(a[i][1],j); |
---|
1227 | ju--; |
---|
1228 | } |
---|
1229 | else{j++;} |
---|
1230 | } |
---|
1231 | if (ku){l++;} |
---|
1232 | else |
---|
1233 | { |
---|
1234 | a = delete(a,l); |
---|
1235 | } |
---|
1236 | } |
---|
1237 | if (ju){i++;} |
---|
1238 | else |
---|
1239 | { |
---|
1240 | a = delete(a,i); |
---|
1241 | } |
---|
1242 | } |
---|
1243 | return(a); |
---|
1244 | } |
---|
1245 | |
---|
1246 | /* |
---|
1247 | * convert to list |
---|
1248 | */ |
---|
1249 | static proc splittolist(ideal sp) |
---|
1250 | { |
---|
1251 | int j = size(sp); |
---|
1252 | list spl = list(list(sp[j],j)); |
---|
1253 | |
---|
1254 | j--; |
---|
1255 | while (j>0) |
---|
1256 | { |
---|
1257 | if (deg(sp[j])) |
---|
1258 | { |
---|
1259 | spl = list(list(sp[j],j))+spl; |
---|
1260 | } |
---|
1261 | j--; |
---|
1262 | } |
---|
1263 | return(spl); |
---|
1264 | } |
---|
1265 | |
---|
1266 | /* |
---|
1267 | * multiply the multiplicity |
---|
1268 | */ |
---|
1269 | static proc multlist(list a, int m) |
---|
1270 | { |
---|
1271 | int i; |
---|
1272 | for (i=size(a);i>0;i--){a[i][2] = a[i][2]*m;} |
---|
1273 | return(a); |
---|
1274 | } |
---|
1275 | |
---|
1276 | /* |
---|
1277 | * a+b w.r.t. to multiplicity as ordering |
---|
1278 | * (programming like spolys) |
---|
1279 | */ |
---|
1280 | static proc addlist(list a, list b, int m) |
---|
1281 | { |
---|
1282 | int i,j,k,l,s; |
---|
1283 | list r = list(); |
---|
1284 | |
---|
1285 | if (m>1){a = multlist(a,m);} |
---|
1286 | k = size(a); |
---|
1287 | l = size(b); |
---|
1288 | i = 1; |
---|
1289 | j = 1; |
---|
1290 | while ((i<=k)&&(j<=l)) |
---|
1291 | { |
---|
1292 | s = a[i][2]-b[j][2]; |
---|
1293 | if (s>=0) |
---|
1294 | { |
---|
1295 | r = r+list(b[j]); |
---|
1296 | j++; |
---|
1297 | if (s==0) |
---|
1298 | { |
---|
1299 | s = size(r); |
---|
1300 | r[s][1] = r[s][1]+a[i][1]; |
---|
1301 | i++; |
---|
1302 | } |
---|
1303 | } |
---|
1304 | else |
---|
1305 | { |
---|
1306 | r = r+list(a[i]); |
---|
1307 | i++; |
---|
1308 | } |
---|
1309 | } |
---|
1310 | if (i>k) |
---|
1311 | { |
---|
1312 | if (j<=l){r = r+list(b[j..l]);} |
---|
1313 | } |
---|
1314 | else{r = r+list(a[i..k]);} |
---|
1315 | return(r); |
---|
1316 | } |
---|
1317 | |
---|
1318 | /* |
---|
1319 | * solution of one tridiagonal system T with multiplicity |
---|
1320 | * with precision prec |
---|
1321 | * T[1] is univariant in var(1) |
---|
1322 | * list o is empty for the first call |
---|
1323 | */ |
---|
1324 | static proc mutrisolve(list o, ideal T, int prec) |
---|
1325 | { |
---|
1326 | list lroots,ll,sp; |
---|
1327 | ideal S,h; |
---|
1328 | int i,d,m,z; |
---|
1329 | |
---|
1330 | d = size(T); |
---|
1331 | sp = splittolist(splitsqrfree(T[1],var(d))); |
---|
1332 | if (d==1){return(l_mutrisolve(sp,o,prec));} |
---|
1333 | z = size(sp); |
---|
1334 | while (z>0) |
---|
1335 | { |
---|
1336 | m = sp[z][2]; |
---|
1337 | ll = laguerre(sp[z][1],prec,1); |
---|
1338 | i = size(ll); |
---|
1339 | while(i>0) |
---|
1340 | { |
---|
1341 | h = linreduce(ideal(T[2..d]),d,ll[i]); |
---|
1342 | if (size(lroots)) |
---|
1343 | { |
---|
1344 | lroots = addlist(mutrisolve(list(ll[i])+o,h,prec),lroots,m); |
---|
1345 | } |
---|
1346 | else |
---|
1347 | { |
---|
1348 | lroots = mutrisolve(list(ll[i])+o,h,prec); |
---|
1349 | if (m>1){lroots=multlist(lroots,m);} |
---|
1350 | } |
---|
1351 | i--; |
---|
1352 | } |
---|
1353 | z--; |
---|
1354 | } |
---|
1355 | return(lroots); |
---|
1356 | } |
---|
1357 | |
---|
1358 | /* |
---|
1359 | * the last call, we are ready |
---|
1360 | */ |
---|
1361 | static proc l_mutrisolve(list sp, list o, int prec) |
---|
1362 | { |
---|
1363 | list lroots,ll; |
---|
1364 | int z,m,i; |
---|
1365 | |
---|
1366 | z = size(sp); |
---|
1367 | while (z>0) |
---|
1368 | { |
---|
1369 | m = sp[z][2]; |
---|
1370 | ll = laguerre(sp[z][1],prec,1); |
---|
1371 | for (i=size(ll);i>0;i--){ll[i] = list(ll[i])+o;} |
---|
1372 | if (size(lroots)) |
---|
1373 | { |
---|
1374 | lroots = addlist(list(list(ll,m)),lroots,1); |
---|
1375 | } |
---|
1376 | else |
---|
1377 | { |
---|
1378 | lroots = list(list(ll,m)); |
---|
1379 | } |
---|
1380 | z--; |
---|
1381 | } |
---|
1382 | return(lroots); |
---|
1383 | } |
---|
1384 | /////////////////////////////////////////////////////////////////////////////// |
---|
1385 | |
---|
1386 | proc ures_solve( ideal gls, list # ) |
---|
1387 | "USAGE: ures_solve(i [, k, p] ); i = ideal, k, p = integers |
---|
1388 | k=0: use sparse resultant matrix of Gelfand, Kapranov and Zelevinsky, @* |
---|
1389 | k=1: use resultant matrix of Macaulay which works only for |
---|
1390 | homogeneous ideals,@* |
---|
1391 | p>0: defines precision of the long floats for internal computation |
---|
1392 | if the basering is not complex (in decimal digits), |
---|
1393 | (default: k=0, p=30) |
---|
1394 | ASSUME: i is a zerodimensional ideal given by a quadratic system, that is,@* |
---|
1395 | nvars(basering) = ncols(i) = number of vars actually occurring in i, |
---|
1396 | RETURN: If the ground field is the field of complex numbers: list of numbers |
---|
1397 | (the complex roots of the polynomial system i=0). @* |
---|
1398 | Otherwise: ring @code{R} with the same number of variables but with |
---|
1399 | complex coefficients (and precision p). @code{R} comes with a list |
---|
1400 | @code{SOL} of numbers, in which complex roots of the polynomial |
---|
1401 | system i are stored: @* |
---|
1402 | EXAMPLE: example ures_solve; shows an example |
---|
1403 | " |
---|
1404 | { |
---|
1405 | int typ=0;// defaults |
---|
1406 | int prec=30; |
---|
1407 | |
---|
1408 | if ( size(#) > 0 ) |
---|
1409 | { |
---|
1410 | typ= #[1]; |
---|
1411 | if ( typ < 0 || typ > 1 ) |
---|
1412 | { |
---|
1413 | ERROR("Valid values for second parameter k are: |
---|
1414 | 0: use sparse Resultant (default) |
---|
1415 | 1: use Macaulay Resultant"); |
---|
1416 | } |
---|
1417 | } |
---|
1418 | if ( size(#) > 1 ) |
---|
1419 | { |
---|
1420 | prec= #[2]; |
---|
1421 | if ( prec < 8 ) |
---|
1422 | { |
---|
1423 | prec = 8; |
---|
1424 | } |
---|
1425 | } |
---|
1426 | |
---|
1427 | list LL=uressolve(gls,typ,prec,1); |
---|
1428 | int sizeLL=size(LL); |
---|
1429 | if (sizeLL==0) |
---|
1430 | { |
---|
1431 | dbprint(printlevel-voice+3,"No solution found!"); |
---|
1432 | return(list()); |
---|
1433 | } |
---|
1434 | if (typeof(LL[1][1])=="string") |
---|
1435 | { |
---|
1436 | int ii,jj; |
---|
1437 | int nv=size(LL[1]); |
---|
1438 | execute("ring rinC =(complex,"+string(prec)+",I),(" |
---|
1439 | +varstr(basering)+"),lp;"); |
---|
1440 | list SOL,SOLnew; |
---|
1441 | for (ii=1; ii<=sizeLL; ii++) |
---|
1442 | { |
---|
1443 | SOLnew=list(); |
---|
1444 | for (jj=1; jj<=nv; jj++) |
---|
1445 | { |
---|
1446 | execute("SOLnew["+string(jj)+"]="+LL[ii][jj]+";"); |
---|
1447 | } |
---|
1448 | SOL[ii]=SOLnew; |
---|
1449 | } |
---|
1450 | kill SOLnew; |
---|
1451 | export SOL; |
---|
1452 | dbprint( printlevel-voice+3," |
---|
1453 | // 'ures_solve' created a ring, in which a list SOL of numbers (the complex |
---|
1454 | // solutions) is stored. |
---|
1455 | // To access the list of complex solutions, type (if the name R was assigned |
---|
1456 | // to the return value): |
---|
1457 | setring R; SOL; "); |
---|
1458 | return(rinC); |
---|
1459 | } |
---|
1460 | else |
---|
1461 | { |
---|
1462 | return(LL); |
---|
1463 | } |
---|
1464 | } |
---|
1465 | example |
---|
1466 | { |
---|
1467 | "EXAMPLE:";echo=2; |
---|
1468 | // compute the intersection points of two curves |
---|
1469 | ring rsq = 0,(x,y),lp; |
---|
1470 | ideal gls= x2 + y2 - 10, x2 + xy + 2y2 - 16; |
---|
1471 | def R=ures_solve(gls,0,16); |
---|
1472 | setring R; SOL; |
---|
1473 | } |
---|
1474 | /////////////////////////////////////////////////////////////////////////////// |
---|
1475 | |
---|
1476 | proc mp_res_mat( ideal i, list # ) |
---|
1477 | "USAGE: mp_res_mat(i [, k] ); i ideal, k integer, |
---|
1478 | k=0: sparse resultant matrix of Gelfand, Kapranov and Zelevinsky, |
---|
1479 | k=1: resultant matrix of Macaulay (k=0 is default) |
---|
1480 | ASSUME: The number of elements in the input system must be the number of |
---|
1481 | variables in the basering plus one; |
---|
1482 | if k=1 then i must be homogeneous. |
---|
1483 | RETURN: module representing the multipolynomial resultant matrix |
---|
1484 | EXAMPLE: example mp_res_mat; shows an example |
---|
1485 | " |
---|
1486 | { |
---|
1487 | int typ=0; |
---|
1488 | |
---|
1489 | if ( size(#) > 0 ) |
---|
1490 | { |
---|
1491 | typ= #[1]; |
---|
1492 | if ( typ < 0 || typ > 1 ) |
---|
1493 | { |
---|
1494 | ERROR("Valid values for third parameter are: |
---|
1495 | 0: sparse resultant (default) |
---|
1496 | 1: Macaulay resultant"); |
---|
1497 | } |
---|
1498 | } |
---|
1499 | return(mpresmat(i,typ)); |
---|
1500 | } |
---|
1501 | example |
---|
1502 | { |
---|
1503 | "EXAMPLE:";echo=2; |
---|
1504 | // compute resultant matrix in ring with parameters (sparse resultant matrix) |
---|
1505 | ring rsq= (0,u0,u1,u2),(x1,x2),lp; |
---|
1506 | ideal i= u0+u1*x1+u2*x2,x1^2 + x2^2 - 10,x1^2 + x1*x2 + 2*x2^2 - 16; |
---|
1507 | module m = mp_res_mat(i); |
---|
1508 | print(m); |
---|
1509 | // computing sparse resultant |
---|
1510 | det(m); |
---|
1511 | |
---|
1512 | // compute resultant matrix (Macaulay resultant matrix) |
---|
1513 | ring rdq= (0,u0,u1,u2),(x0,x1,x2),lp; |
---|
1514 | ideal h= homog(imap(rsq,i),x0); |
---|
1515 | h; |
---|
1516 | |
---|
1517 | module m = mp_res_mat(h,1); |
---|
1518 | print(m); |
---|
1519 | // computing Macaulay resultant (should be the same as above!) |
---|
1520 | det(m); |
---|
1521 | |
---|
1522 | // compute numerical sparse resultant matrix |
---|
1523 | setring rsq; |
---|
1524 | ideal ir= 15+2*x1+5*x2,x1^2 + x2^2 - 10,x1^2 + x1*x2 + 2*x2^2 - 16; |
---|
1525 | module mn = mp_res_mat(ir); |
---|
1526 | print(mn); |
---|
1527 | // computing sparse resultant |
---|
1528 | det(mn); |
---|
1529 | } |
---|
1530 | /////////////////////////////////////////////////////////////////////////////// |
---|
1531 | |
---|
1532 | proc interpolate( ideal p, ideal w, int d ) |
---|
1533 | "USAGE: interpolate(p,v,d); p,v=ideals of numbers, d=integer |
---|
1534 | ASSUME: Ground field K is the field of rational numbers, p and v are lists |
---|
1535 | of elements of the ground field K with p[j] != -1,0,1, size(p) = n |
---|
1536 | (= number of vars) and size(v)=N=(d+1)^n. |
---|
1537 | RETURN: poly f, the unique polynomial f of degree n*d with prescribed values |
---|
1538 | v[i] at the points p(i)=(p[1]^(i-1),..,p[n]^(i-1)), i=1,..,N. |
---|
1539 | NOTE: mainly useful when n=1, i.e. f is satisfying f(p^(i-1)) = v[i], |
---|
1540 | i=1..d+1. |
---|
1541 | SEE ALSO: vandermonde. |
---|
1542 | EXAMPLE: example interpolate; shows an example |
---|
1543 | " |
---|
1544 | { |
---|
1545 | return(vandermonde(p,w,d)); |
---|
1546 | } |
---|
1547 | example |
---|
1548 | { |
---|
1549 | "EXAMPLE:"; echo=2; |
---|
1550 | ring r1 = 0,(x),lp; |
---|
1551 | // determine f with deg(f) = 4 and |
---|
1552 | // v = values of f at points 3^0, 3^1, 3^2, 3^3, 3^4 |
---|
1553 | ideal v=16,0,11376,1046880,85949136; |
---|
1554 | interpolate( 3, v, 4 ); |
---|
1555 | } |
---|
1556 | |
---|
1557 | /////////////////////////////////////////////////////////////////////////////// |
---|
1558 | // changed for Singular 3 |
---|
1559 | // Return value is now a list: (rlist, rn@) |
---|
1560 | static proc psubst( int d, int dd, int n, list resl, |
---|
1561 | ideal fi, int elem, int nv, int prec, int rn@, list rlist) |
---|
1562 | { |
---|
1563 | // nv: number of ring variables (fixed value) |
---|
1564 | // elem: number of elements in ideal fi (fixed value) |
---|
1565 | // fi: input ideal (fixed value) |
---|
1566 | // rl: output list of roots |
---|
1567 | // resl: actual list of roots |
---|
1568 | // n: |
---|
1569 | // dd: actual element of fi |
---|
1570 | // d: actual variable |
---|
1571 | |
---|
1572 | list LL; |
---|
1573 | int pdebug; |
---|
1574 | int olddd=dd; |
---|
1575 | |
---|
1576 | dbprint(printlevel-voice+2, "// 0 step "+string(dd)+" of "+string(elem) ); |
---|
1577 | |
---|
1578 | if ( dd <= elem ) |
---|
1579 | { |
---|
1580 | int loop = 1; |
---|
1581 | int k; |
---|
1582 | list lsr,lh; |
---|
1583 | poly ps; |
---|
1584 | int thedd; |
---|
1585 | |
---|
1586 | dbprint( printlevel-voice+1,"// 1 dd = "+string(dd) ); |
---|
1587 | |
---|
1588 | thedd=0; |
---|
1589 | while ( (dd+1 <= elem) && loop ) |
---|
1590 | { |
---|
1591 | ps= fi[dd+1]; |
---|
1592 | |
---|
1593 | if ( n-1 > 0 ) |
---|
1594 | { |
---|
1595 | dbprint( printlevel-voice, |
---|
1596 | "// 2 ps=fi["+string(dd+1)+"]"+" size=" |
---|
1597 | +string(size(coeffs(ps,var(n-1)))) |
---|
1598 | +" leadexp(ps)="+string(leadexp(ps)) ); |
---|
1599 | |
---|
1600 | if ( size(coeffs(ps,var(n-1))) == 1 ) |
---|
1601 | { |
---|
1602 | dd++; |
---|
1603 | // hier Leading-Exponent pruefen??? |
---|
1604 | // oder ist das Poly immer als letztes in der Liste?!? |
---|
1605 | // leadexp(ps) |
---|
1606 | } |
---|
1607 | else |
---|
1608 | { |
---|
1609 | loop=0; |
---|
1610 | } |
---|
1611 | } |
---|
1612 | else |
---|
1613 | { |
---|
1614 | dbprint( printlevel-voice, |
---|
1615 | "// 2 ps=fi["+string(dd+1)+"]"+" leadexp(ps)=" |
---|
1616 | +string(leadexp(ps)) ); |
---|
1617 | dd++; |
---|
1618 | } |
---|
1619 | } |
---|
1620 | thedd=dd; |
---|
1621 | ps= fi[thedd]; |
---|
1622 | |
---|
1623 | dbprint( printlevel-voice+1, |
---|
1624 | "// 3 fi["+string(thedd-1)+"]"+" leadexp(fi[thedd-1])=" |
---|
1625 | +string(leadexp(fi[thedd-1])) ); |
---|
1626 | dbprint( printlevel-voice+1, |
---|
1627 | "// 3 ps=fi["+string(thedd)+"]"+" leadexp(ps)=" |
---|
1628 | +string(leadexp(ps)) ); |
---|
1629 | |
---|
1630 | for ( k= nv; k > nv-d; k-- ) |
---|
1631 | { |
---|
1632 | dbprint( printlevel-voice, |
---|
1633 | "// 4 subst(fi["+string(thedd)+"]," |
---|
1634 | +string(var(k))+","+string(resl[k])+");" ); |
---|
1635 | ps = subst(ps,var(k),resl[k]); |
---|
1636 | } |
---|
1637 | |
---|
1638 | dbprint( printlevel-voice, "// 5 substituted ps="+string(ps) ); |
---|
1639 | |
---|
1640 | if ( ps != 0 ) |
---|
1641 | { |
---|
1642 | lsr= laguerre_solve( ps, prec, prec, 0 ); |
---|
1643 | } |
---|
1644 | else |
---|
1645 | { |
---|
1646 | dbprint( printlevel-voice+1,"// 30 ps == 0, thats not cool..."); |
---|
1647 | lsr=list(number(0)); |
---|
1648 | } |
---|
1649 | |
---|
1650 | dbprint( printlevel-voice+1, |
---|
1651 | "// 6 laguerre_solve found roots: lsr["+string(size(lsr))+"]" ); |
---|
1652 | |
---|
1653 | if ( size(lsr) > 1 ) |
---|
1654 | { |
---|
1655 | dbprint( printlevel-voice+1, |
---|
1656 | "// 10 checking roots found before, range " |
---|
1657 | +string(dd-olddd)+" -- "+string(dd) ); |
---|
1658 | dbprint( printlevel-voice+1, |
---|
1659 | "// 10 thedd = "+string(thedd) ); |
---|
1660 | |
---|
1661 | int i,j,l; |
---|
1662 | int ls=size(lsr); |
---|
1663 | int lss; |
---|
1664 | poly pss; |
---|
1665 | list nares; |
---|
1666 | int rroot; |
---|
1667 | int nares_size; |
---|
1668 | |
---|
1669 | |
---|
1670 | for ( i = 1; i <= ls; i++ ) // lsr[1..ls] |
---|
1671 | { |
---|
1672 | rroot=1; |
---|
1673 | |
---|
1674 | if ( pdebug>=2 ) |
---|
1675 | {"// 13 root lsr["+string(i)+"] = "+string(lsr[i]);} |
---|
1676 | for ( l = 0; l <= dd-olddd; l++ ) |
---|
1677 | { |
---|
1678 | if ( l+olddd != thedd ) |
---|
1679 | { |
---|
1680 | if ( pdebug>=2 ) |
---|
1681 | {"// 11 checking ideal element "+string(l+olddd);} |
---|
1682 | ps=fi[l+olddd]; |
---|
1683 | if ( pdebug>=3 ) |
---|
1684 | {"// 14 ps=fi["+string(l+olddd)+"]";} |
---|
1685 | for ( k= nv; k > nv-d; k-- ) |
---|
1686 | { |
---|
1687 | if ( pdebug>=3 ) |
---|
1688 | { |
---|
1689 | "// 11 subst(fi["+string(olddd+l)+"]," |
---|
1690 | +string(var(k))+","+string(resl[k])+");"; |
---|
1691 | } |
---|
1692 | ps = subst(ps,var(k),resl[k]); |
---|
1693 | |
---|
1694 | } |
---|
1695 | |
---|
1696 | pss=subst(ps,var(k),lsr[i]); // k=nv-d |
---|
1697 | if ( pdebug>=3 ) |
---|
1698 | { "// 15 0 == "+string(pss); } |
---|
1699 | if ( pss != 0 ) |
---|
1700 | { |
---|
1701 | if ( system("complexNearZero", |
---|
1702 | leadcoef(pss), |
---|
1703 | prec) ) |
---|
1704 | { |
---|
1705 | if ( pdebug>=2 ) |
---|
1706 | { "// 16 root "+string(i)+" is a real root"; } |
---|
1707 | } |
---|
1708 | else |
---|
1709 | { |
---|
1710 | if ( pdebug>=2 ) |
---|
1711 | { "// 17 0 == "+string(pss); } |
---|
1712 | rroot=0; |
---|
1713 | } |
---|
1714 | } |
---|
1715 | |
---|
1716 | } |
---|
1717 | } |
---|
1718 | |
---|
1719 | if ( rroot == 1 ) // add root to list ? |
---|
1720 | { |
---|
1721 | if ( size(nares) > 0 ) |
---|
1722 | { |
---|
1723 | nares=nares[1..size(nares)],lsr[i]; |
---|
1724 | } |
---|
1725 | else |
---|
1726 | { |
---|
1727 | nares=lsr[i]; |
---|
1728 | } |
---|
1729 | if ( pdebug>=2 ) |
---|
1730 | { "// 18 added root to list nares"; } |
---|
1731 | } |
---|
1732 | } |
---|
1733 | |
---|
1734 | nares_size=size(nares); |
---|
1735 | if ( nares_size == 0 ) |
---|
1736 | { |
---|
1737 | "Numerical problem: No root found..."; |
---|
1738 | "Output may be incorrect!"; |
---|
1739 | nares=list(number(0)); |
---|
1740 | } |
---|
1741 | |
---|
1742 | if ( pdebug>=1 ) |
---|
1743 | { "// 20 found <"+string(size(nares))+"> roots"; } |
---|
1744 | |
---|
1745 | for ( i= 1; i <= nares_size; i++ ) |
---|
1746 | { |
---|
1747 | resl[nv-d]= nares[i]; |
---|
1748 | |
---|
1749 | if ( dd < elem ) |
---|
1750 | { |
---|
1751 | if ( i > 1 ) |
---|
1752 | { |
---|
1753 | rn@++; |
---|
1754 | } |
---|
1755 | LL = psubst( d+1, dd+1, n-1, resl, fi, elem, nv, prec, |
---|
1756 | rn@, rlist ); |
---|
1757 | rlist = LL[1]; |
---|
1758 | rn@ = LL[2]; |
---|
1759 | } |
---|
1760 | else |
---|
1761 | { |
---|
1762 | if ( i > 1 ) { rn@++; } //bug found by O.Labs |
---|
1763 | if ( pdebug>=1 ) |
---|
1764 | {"// 30_1 <"+string(rn@)+"> "+string(size(resl))+" <-----";} |
---|
1765 | if ( pdebug>=2 ){ resl; } |
---|
1766 | rlist[rn@]=resl; |
---|
1767 | } |
---|
1768 | } |
---|
1769 | } |
---|
1770 | else |
---|
1771 | { |
---|
1772 | if ( pdebug>=2 ) |
---|
1773 | { "// 21 found root to be: "+string(lsr[1]); } |
---|
1774 | resl[nv-d]= lsr[1]; |
---|
1775 | |
---|
1776 | if ( dd < elem ) |
---|
1777 | { |
---|
1778 | LL= psubst( d+1, dd+1, n-1, resl, fi, elem, nv, prec, |
---|
1779 | rn@, rlist ); |
---|
1780 | rlist = LL[1]; |
---|
1781 | rn@ = LL[2]; |
---|
1782 | } |
---|
1783 | else |
---|
1784 | { |
---|
1785 | if ( pdebug>=1 ) |
---|
1786 | { "// 30_2 <"+string(rn@)+"> "+string(size(resl))+" <-----";} |
---|
1787 | if ( pdebug>=2 ) |
---|
1788 | { resl; } |
---|
1789 | rlist[rn@]=resl; |
---|
1790 | } |
---|
1791 | } |
---|
1792 | } |
---|
1793 | return(list(rlist,rn@)); |
---|
1794 | } |
---|
1795 | |
---|
1796 | /////////////////////////////////////////////////////////////////////////////// |
---|
1797 | |
---|
1798 | proc fglm_solve( ideal fi, list # ) |
---|
1799 | "USAGE: fglm_solve(i [, p] ); i ideal, p integer |
---|
1800 | ASSUME: the ground field has char 0. |
---|
1801 | RETURN: ring @code{R} with the same number of variables but with complex |
---|
1802 | coefficients (and precision p). @code{R} comes with a list |
---|
1803 | @code{rlist} of numbers, in which the complex roots of i are stored.@* |
---|
1804 | p>0: gives precision of complex numbers in decimal digits [default: |
---|
1805 | p=30]. |
---|
1806 | NOTE: The procedure uses a standard basis of i to determine all complex |
---|
1807 | roots of i. |
---|
1808 | EXAMPLE: example fglm_solve; shows an example |
---|
1809 | " |
---|
1810 | { |
---|
1811 | int prec=30; |
---|
1812 | |
---|
1813 | if ( size(#)>=1 && typeof(#[1])=="int") |
---|
1814 | { |
---|
1815 | prec=#[1]; |
---|
1816 | } |
---|
1817 | |
---|
1818 | def R = lex_solve(stdfglm(fi),prec); |
---|
1819 | dbprint( printlevel-voice+3," |
---|
1820 | // 'fglm_solve' created a ring, in which a list rlist of numbers (the |
---|
1821 | // complex solutions) is stored. |
---|
1822 | // To access the list of complex solutions, type (if the name R was assigned |
---|
1823 | // to the return value): |
---|
1824 | setring R; rlist; "); |
---|
1825 | return(R); |
---|
1826 | } |
---|
1827 | example |
---|
1828 | { |
---|
1829 | "EXAMPLE:";echo=2; |
---|
1830 | ring r = 0,(x,y),lp; |
---|
1831 | // compute the intersection points of two curves |
---|
1832 | ideal s = x2 + y2 - 10, x2 + xy + 2y2 - 16; |
---|
1833 | def R = fglm_solve(s,10); |
---|
1834 | setring R; rlist; |
---|
1835 | } |
---|
1836 | |
---|
1837 | /////////////////////////////////////////////////////////////////////////////// |
---|
1838 | |
---|
1839 | proc lex_solve( ideal fi, list # ) |
---|
1840 | "USAGE: lex_solve( i[,p] ); i=ideal, p=integer, |
---|
1841 | p>0: gives precision of complex numbers in decimal digits (default: p=30). |
---|
1842 | ASSUME: i is a reduced lexicographical Groebner bases of a zero-dimensional |
---|
1843 | ideal, sorted by increasing leading terms. |
---|
1844 | RETURN: ring @code{R} with the same number of variables but with complex |
---|
1845 | coefficients (and precision p). @code{R} comes with a list |
---|
1846 | @code{rlist} of numbers, in which the complex roots of i are stored. |
---|
1847 | EXAMPLE: example lex_solve; shows an example |
---|
1848 | " |
---|
1849 | { |
---|
1850 | int prec=30; |
---|
1851 | list LL; |
---|
1852 | |
---|
1853 | if ( size(#)>=1 && typeof(#[1])=="int") |
---|
1854 | { |
---|
1855 | prec=#[1]; |
---|
1856 | } |
---|
1857 | |
---|
1858 | if ( !defined(pdebug) ) { int pdebug; } |
---|
1859 | def oring= basering; |
---|
1860 | |
---|
1861 | // change the ground field to complex numbers |
---|
1862 | string nrings= "ring RC =(complex,"+string(prec) |
---|
1863 | +"),("+varstr(basering)+"),lp;"; |
---|
1864 | execute(nrings); |
---|
1865 | |
---|
1866 | // map fi from old to new ring |
---|
1867 | ideal fi= imap(oring,fi); |
---|
1868 | |
---|
1869 | int idelem= size(fi); |
---|
1870 | int nv= nvars(basering); |
---|
1871 | int i,j,k,lis; |
---|
1872 | list resl,li; |
---|
1873 | |
---|
1874 | if ( !defined(rlist) ) |
---|
1875 | { |
---|
1876 | list rlist; |
---|
1877 | export rlist; |
---|
1878 | } |
---|
1879 | |
---|
1880 | li= laguerre_solve(fi[1],prec,prec,0); |
---|
1881 | lis= size(li); |
---|
1882 | |
---|
1883 | dbprint(printlevel-voice+2,"// laguerre found roots: "+string(size(li))); |
---|
1884 | int rn@; |
---|
1885 | |
---|
1886 | for ( j= 1; j <= lis; j++ ) |
---|
1887 | { |
---|
1888 | dbprint(printlevel-voice+1,"// root "+string(j) ); |
---|
1889 | rn@++; |
---|
1890 | resl[nv]= li[j]; |
---|
1891 | LL = psubst( 1, 2, nv-1, resl, fi, idelem, nv, prec, rn@, rlist ); |
---|
1892 | rlist=LL[1]; |
---|
1893 | rn@=LL[2]; |
---|
1894 | } |
---|
1895 | |
---|
1896 | dbprint( printlevel-voice+3," |
---|
1897 | // 'lex_solve' created a ring, in which a list rlist of numbers (the |
---|
1898 | // complex solutions) is stored. |
---|
1899 | // To access the list of complex solutions, type (if the name R was assigned |
---|
1900 | // to the return value): |
---|
1901 | setring R; rlist; "); |
---|
1902 | |
---|
1903 | return(RC); |
---|
1904 | } |
---|
1905 | example |
---|
1906 | { |
---|
1907 | "EXAMPLE:";echo=2; |
---|
1908 | ring r = 0,(x,y),lp; |
---|
1909 | // compute the intersection points of two curves |
---|
1910 | ideal s = x2 + y2 - 10, x2 + xy + 2y2 - 16; |
---|
1911 | def R = lex_solve(stdfglm(s),10); |
---|
1912 | setring R; rlist; |
---|
1913 | } |
---|
1914 | |
---|
1915 | /////////////////////////////////////////////////////////////////////////////// |
---|
1916 | |
---|
1917 | proc triangLf_solve( ideal fi, list # ) |
---|
1918 | "USAGE: triangLf_solve(i [, p] ); i ideal, p integer, |
---|
1919 | p>0: gives precision of complex numbers in digits (default: p=30). |
---|
1920 | ASSUME: the ground field has char 0; i is a zero-dimensional ideal |
---|
1921 | RETURN: ring @code{R} with the same number of variables but with complex |
---|
1922 | coefficients (and precision p). @code{R} comes with a list |
---|
1923 | @code{rlist} of numbers, in which the complex roots of i are stored. |
---|
1924 | NOTE: The procedure uses a triangular system (Lazard's Algorithm with |
---|
1925 | factorization) computed from a standard basis to determine |
---|
1926 | recursively all complex roots of the input ideal i with Laguerre's |
---|
1927 | algorithm. |
---|
1928 | EXAMPLE: example triangLf_solve; shows an example |
---|
1929 | " |
---|
1930 | { |
---|
1931 | int prec=30; |
---|
1932 | |
---|
1933 | if ( size(#)>=1 && typeof(#[1])=="int") |
---|
1934 | { |
---|
1935 | prec=#[1]; |
---|
1936 | } |
---|
1937 | |
---|
1938 | def R=triang_solve(triangLfak(stdfglm(fi)),prec); |
---|
1939 | dbprint( printlevel-voice+3," |
---|
1940 | // 'triangLf_solve' created a ring, in which a list rlist of numbers (the |
---|
1941 | // complex solutions) is stored. |
---|
1942 | // To access the list of complex solutions, type (if the name R was assigned |
---|
1943 | // to the return value): |
---|
1944 | setring R; rlist; "); |
---|
1945 | return(R); |
---|
1946 | } |
---|
1947 | example |
---|
1948 | { |
---|
1949 | "EXAMPLE:";echo=2; |
---|
1950 | ring r = 0,(x,y),lp; |
---|
1951 | // compute the intersection points of two curves |
---|
1952 | ideal s = x2 + y2 - 10, x2 + xy + 2y2 - 16; |
---|
1953 | def R = triangLf_solve(s,10); |
---|
1954 | setring R; rlist; |
---|
1955 | } |
---|
1956 | |
---|
1957 | /////////////////////////////////////////////////////////////////////////////// |
---|
1958 | |
---|
1959 | proc triangM_solve( ideal fi, list # ) |
---|
1960 | "USAGE: triangM_solve(i [, p ] ); i=ideal, p=integer, |
---|
1961 | p>0: gives precision of complex numbers in digits (default: p=30). |
---|
1962 | ASSUME: the ground field has char 0;@* |
---|
1963 | i zero-dimensional ideal |
---|
1964 | RETURN: ring @code{R} with the same number of variables but with complex |
---|
1965 | coefficients (and precision p). @code{R} comes with a list |
---|
1966 | @code{rlist} of numbers, in which the complex roots of i are stored. |
---|
1967 | NOTE: The procedure uses a triangular system (Moellers Algorithm) computed |
---|
1968 | from a standard basis of input ideal i to determine recursively all |
---|
1969 | complex roots with Laguerre's algorithm. |
---|
1970 | EXAMPLE: example triangM_solve; shows an example |
---|
1971 | " |
---|
1972 | { |
---|
1973 | int prec=30; |
---|
1974 | |
---|
1975 | if ( size(#)>=1 && typeof(#[1])=="int") |
---|
1976 | { |
---|
1977 | prec=#[1]; |
---|
1978 | } |
---|
1979 | |
---|
1980 | def R = triang_solve(triangM(stdfglm(fi)),prec); |
---|
1981 | dbprint( printlevel-voice+3," |
---|
1982 | // 'triangM_solve' created a ring, in which a list rlist of numbers (the |
---|
1983 | // complex solutions) is stored. |
---|
1984 | // To access the list of complex solutions, type (if the name R was assigned |
---|
1985 | // to the return value): |
---|
1986 | setring R; rlist; "); |
---|
1987 | return(R); |
---|
1988 | } |
---|
1989 | example |
---|
1990 | { |
---|
1991 | "EXAMPLE:";echo=2; |
---|
1992 | ring r = 0,(x,y),lp; |
---|
1993 | // compute the intersection points of two curves |
---|
1994 | ideal s = x2 + y2 - 10, x2 + xy + 2y2 - 16; |
---|
1995 | def R = triangM_solve(s,10); |
---|
1996 | setring R; rlist; |
---|
1997 | } |
---|
1998 | |
---|
1999 | /////////////////////////////////////////////////////////////////////////////// |
---|
2000 | |
---|
2001 | proc triangL_solve( ideal fi, list # ) |
---|
2002 | "USAGE: triangL_solve(i [, p] ); i=ideal, p=integer,@* |
---|
2003 | p>0: gives precision of complex numbers in digits (default: p=30). |
---|
2004 | ASSUME: the ground field has char 0; i is a zero-dimensional ideal. |
---|
2005 | RETURN: ring @code{R} with the same number of variables, but with complex |
---|
2006 | coefficients (and precision p). @code{R} comes with a list |
---|
2007 | @code{rlist} of numbers, in which the complex roots of i are stored. |
---|
2008 | NOTE: The procedure uses a triangular system (Lazard's Algorithm) computed |
---|
2009 | from a standard basis of input ideal i to determine recursively all |
---|
2010 | complex roots with Laguerre's algorithm. |
---|
2011 | EXAMPLE: example triangL_solve; shows an example |
---|
2012 | " |
---|
2013 | { |
---|
2014 | int prec=30; |
---|
2015 | |
---|
2016 | if ( size(#)>=1 && typeof(#[1])=="int") |
---|
2017 | { |
---|
2018 | prec=#[1]; |
---|
2019 | } |
---|
2020 | |
---|
2021 | def R=triang_solve(triangL(stdfglm(fi)),prec); |
---|
2022 | dbprint( printlevel-voice+3," |
---|
2023 | // 'triangL_solve' created a ring, in which a list rlist of numbers (the |
---|
2024 | // complex solutions) is stored. |
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2025 | // To access the list of complex solutions, type (if the name R was assigned |
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2026 | // to the return value): |
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2027 | setring R; rlist; "); |
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2028 | return(R); |
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2029 | } |
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2030 | example |
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2031 | { |
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2032 | "EXAMPLE:";echo=2; |
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2033 | ring r = 0,(x,y),lp; |
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2034 | // compute the intersection points of two curves |
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2035 | ideal s = x2 + y2 - 10, x2 + xy + 2y2 - 16; |
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2036 | def R = triangL_solve(s,10); |
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2037 | setring R; rlist; |
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2038 | } |
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2039 | |
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2040 | |
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2041 | /////////////////////////////////////////////////////////////////////////////// |
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2042 | |
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2043 | proc triang_solve( list lfi, int prec, list # ) |
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2044 | "USAGE: triang_solve(l,p [,d] ); l=list, p,d=integers@* |
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2045 | l a list of finitely many triangular systems, such that the union of |
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2046 | their varieties equals the variety of the initial ideal.@* |
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2047 | p>0: gives precision of complex numbers in digits,@* |
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2048 | d>0: gives precision (1<d<p) for near-zero-determination,@* |
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2049 | (default: d=1/2*p). |
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2050 | ASSUME: the ground field has char 0;@* |
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2051 | l was computed using the algorithm of Lazard or the algorithm of |
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2052 | Moeller (see triang.lib). |
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2053 | RETURN: ring @code{R} with the same number of variables, but with complex |
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2054 | coefficients (and precision p). @code{R} comes with a list |
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2055 | @code{rlist} of numbers, in which the complex roots of l are stored.@* |
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2056 | EXAMPLE: example triang_solve; shows an example |
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2057 | " |
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2058 | { |
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2059 | def oring= basering; |
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2060 | list LL; |
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2061 | |
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2062 | // change the ground field to complex numbers |
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2063 | string nrings= "ring RC =(complex,"+string(prec) |
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2064 | +",I),("+varstr(basering)+"),lp;"; |
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2065 | execute(nrings); |
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2066 | |
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2067 | // list with entry 0 (number) |
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2068 | number nn=0; |
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2069 | |
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2070 | // set number of digits for zero-comparison of roots |
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2071 | if ( !defined(myCompDigits) ) |
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2072 | { |
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2073 | int myCompDigits; |
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2074 | } |
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2075 | if ( size(#)>=1 && typeof(#[1])=="int" ) |
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2076 | { |
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2077 | myCompDigits=#[1]; |
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2078 | } |
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2079 | else |
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2080 | { |
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2081 | myCompDigits=(system("getPrecDigits")); |
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2082 | } |
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2083 | |
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2084 | dbprint( printlevel-voice+2,"// myCompDigits="+string(myCompDigits) ); |
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2085 | |
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2086 | int idelem; |
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2087 | int nv= nvars(basering); |
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2088 | int i,j,lis; |
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2089 | list resu,li; |
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2090 | |
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2091 | if ( !defined(rlist) ) |
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2092 | { |
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2093 | list rlist; |
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2094 | export rlist; |
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2095 | } |
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2096 | |
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2097 | int rn@=0; |
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2098 | |
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2099 | // map the list |
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2100 | list lfi= imap(oring,lfi); |
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2101 | int slfi= size(lfi); |
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2102 | |
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2103 | ideal fi; |
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2104 | for ( i= 1; i <= slfi; i++ ) |
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2105 | { |
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2106 | // map fi from old to new ring |
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2107 | fi= lfi[i]; //imap(oring,lfi[i]); |
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2108 | |
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2109 | idelem= size(fi); |
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2110 | |
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2111 | // solve fi[1] |
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2112 | li= laguerre_solve(fi[1],myCompDigits,myCompDigits,0); |
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2113 | lis= size(li); |
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2114 | |
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2115 | dbprint( printlevel-voice+2,"// laguerre found roots: "+string(lis) ); |
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2116 | |
---|
2117 | for ( j= 1; j <= lis; j++ ) |
---|
2118 | { |
---|
2119 | dbprint( printlevel-voice+2,"// root "+string(j) ); |
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2120 | rn@++; |
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2121 | resu[nv]= li[j]; |
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2122 | LL = psubst( 1, 2, nv-1, resu, fi, idelem, nv, myCompDigits, |
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2123 | rn@, rlist ); |
---|
2124 | rlist = LL[1]; |
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2125 | rn@ = LL[2]; |
---|
2126 | } |
---|
2127 | } |
---|
2128 | |
---|
2129 | dbprint( printlevel-voice+3," |
---|
2130 | // 'triang_solve' created a ring, in which a list rlist of numbers (the |
---|
2131 | // complex solutions) is stored. |
---|
2132 | // To access the list of complex solutions, type (if the name R was assigned |
---|
2133 | // to the return value): |
---|
2134 | setring R; rlist; "); |
---|
2135 | |
---|
2136 | return(RC); |
---|
2137 | } |
---|
2138 | example |
---|
2139 | { |
---|
2140 | "EXAMPLE:";echo=2; |
---|
2141 | ring r = 0,(x,y),lp; |
---|
2142 | // compute the intersection points of two curves |
---|
2143 | ideal s= x2 + y2 - 10, x2 + xy + 2y2 - 16; |
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2144 | def R=triang_solve(triangLfak(stdfglm(s)),10); |
---|
2145 | setring R; rlist; |
---|
2146 | } |
---|
2147 | |
---|
2148 | /////////////////////////////////////////////////////////////////////////////// |
---|
2149 | |
---|
2150 | proc simplexOut(list l) |
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2151 | "USAGE: simplexOut(l); l list |
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2152 | ASSUME: l is the output of simplex. |
---|
2153 | RETURN: Nothing. The procedure prints the computed solution of simplex |
---|
2154 | (as strings) in a nice format. |
---|
2155 | SEE ALSO: simplex |
---|
2156 | EXAMPLE: example simplexOut; shows an example |
---|
2157 | " |
---|
2158 | { |
---|
2159 | int i,j; |
---|
2160 | matrix m= l[1]; |
---|
2161 | intvec iposv= l[3]; |
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2162 | int icase= l[2]; |
---|
2163 | |
---|
2164 | int cols= ncols(m); |
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2165 | int rows= nrows(m); |
---|
2166 | |
---|
2167 | int N= l[6]; |
---|
2168 | |
---|
2169 | if ( 1 == icase ) // objective function is unbound |
---|
2170 | { |
---|
2171 | "objective function is unbound"; |
---|
2172 | return(); |
---|
2173 | } |
---|
2174 | if ( -1 == icase ) // no solution satisfies the given constraints |
---|
2175 | { |
---|
2176 | "no solution satisfies the given constraints"; |
---|
2177 | return(); |
---|
2178 | } |
---|
2179 | if ( -2 == icase ) // other error |
---|
2180 | { |
---|
2181 | "an error occurred during simplex computation!"; |
---|
2182 | return(); |
---|
2183 | } |
---|
2184 | |
---|
2185 | for ( i = 1; i <= rows; i++ ) |
---|
2186 | { |
---|
2187 | if (i == 1) |
---|
2188 | { |
---|
2189 | "z = "+string(m[1][1]); |
---|
2190 | } |
---|
2191 | else |
---|
2192 | { |
---|
2193 | if ( iposv[i-1] <= N ) |
---|
2194 | { |
---|
2195 | "x"+string(iposv[i-1])+" = "+string(m[i,1]); |
---|
2196 | } |
---|
2197 | // else |
---|
2198 | // { |
---|
2199 | // "Y"; iposv[i-1]-N+1; |
---|
2200 | // } |
---|
2201 | } |
---|
2202 | } |
---|
2203 | } |
---|
2204 | example |
---|
2205 | { |
---|
2206 | "EXAMPLE:";echo=2; |
---|
2207 | ring r = (real,10),(x),lp; |
---|
2208 | |
---|
2209 | // consider the max. problem: |
---|
2210 | // |
---|
2211 | // maximize x(1) + x(2) + 3*x(3) - 0.5*x(4) |
---|
2212 | // |
---|
2213 | // with constraints: x(1) + 2*x(3) <= 740 |
---|
2214 | // 2*x(2) - 7*x(4) <= 0 |
---|
2215 | // x(2) - x(3) + 2*x(4) >= 0.5 |
---|
2216 | // x(1) + x(2) + x(3) + x(4) = 9 |
---|
2217 | // |
---|
2218 | matrix sm[5][5]= 0, 1, 1, 3,-0.5, |
---|
2219 | 740,-1, 0,-2, 0, |
---|
2220 | 0, 0,-2, 0, 7, |
---|
2221 | 0.5, 0,-1, 1,-2, |
---|
2222 | 9,-1,-1,-1,-1; |
---|
2223 | |
---|
2224 | int n = 4; // number of constraints |
---|
2225 | int m = 4; // number of variables |
---|
2226 | int m1= 2; // number of <= constraints |
---|
2227 | int m2= 1; // number of >= constraints |
---|
2228 | int m3= 1; // number of == constraints |
---|
2229 | |
---|
2230 | list sol=simplex(sm, n, m, m1, m2, m3); |
---|
2231 | simplexOut(sol); |
---|
2232 | } |
---|
2233 | |
---|
2234 | |
---|
2235 | // local Variables: *** |
---|
2236 | // c-set-style: bsd *** |
---|
2237 | // End: *** |
---|