1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: solve.lib,v 1.26 2003-07-18 14:13:15 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 | @format |
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32 | m: precision of output in digits ( 4 <= m), if basering is not ring of |
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33 | complex numbers; |
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34 | l: precision of internal computation in decimal digits ( l >=8 ) |
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35 | only if the basering is not complex or complex with smaller precision; |
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36 | n: control of multiplicity of roots or of splitting of f into |
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37 | squarefree factors |
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38 | n < 0, no split of f (good, if all roots are simple) |
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39 | n >= 0, try to split |
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40 | n = 0, return only different roots |
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41 | n > 0, find all roots (with multiplicity) |
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42 | s: s != 0, returns ERROR if | f(root) | > 0.1^m (when computing in the |
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43 | current ring) |
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44 | ( default: m, l, n, s = 8, 30, 1, 0 ) |
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45 | @end format |
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46 | ASSUME: f is a univariate polynomial;@* |
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47 | basering has characteristic 0 and is either complex or without |
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48 | parameters. |
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49 | RETURN: list of (complex) roots of the polynomial f, depending on n. The |
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50 | result is of type |
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51 | @format |
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52 | string: if the basering is not complex, |
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53 | number: otherwise. |
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54 | @end format |
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55 | NOTE: If printlevel >0: displays comments ( default = 0 ). |
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56 | If s != 0 and if the procedure stops with ERROR, try a higher |
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57 | internal precision m. |
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58 | EXAMPLE: example laguerre_solve; shows an example |
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59 | " |
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60 | { |
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61 | if (char(basering)!=0){ERROR("characteristic of basering not 0");} |
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62 | if ((charstr(basering)[1]=="0") and (npars(basering)!=0)) |
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63 | {ERROR("basering has parameters");} |
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64 | int OLD_COMPLEX=0; |
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65 | int iv=checkv(f); // check for variable appearing in f |
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66 | if(iv==0){ERROR("Wrong polynomial!");} |
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67 | poly v=var(iv); // f univariate in v |
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68 | |
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69 | int solutionprec=8;// set the control |
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70 | int numberprec=30; |
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71 | int splitcontrol=1; |
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72 | int rootcheck=0; |
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73 | if(size(#)>0){solutionprec=#[1];if(solutionprec<4){solutionprec=4;}} |
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74 | if(size(#)>1){numberprec=#[2];if(numberprec<8){numberprec=8;}} |
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75 | if(solutionprec>numberprec){numberprec=solutionprec;} |
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76 | if(size(#)>2){splitcontrol=#[3];} |
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77 | if(size(#)>3){rootcheck=#[4];} |
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78 | int prot=printlevel-voice+2; |
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79 | int ringprec=0; |
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80 | |
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81 | poly p=divzero(f,iv); // divide out zeros as solution |
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82 | int iz=deg(f)-deg(p); // multiplicity of zero solution |
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83 | if(prot!=0) |
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84 | { |
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85 | string pout; |
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86 | string nl=newline; |
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87 | pout="//BEGIN laguerre_solve"; |
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88 | if(iz!=0){pout=pout+nl+"//zeros: divide out "+string(iz);} |
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89 | dbprint(prot,pout); |
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90 | } |
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91 | string ss,tt,oo; |
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92 | ss="";oo=ss; |
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93 | if(npars(basering)==1) |
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94 | { |
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95 | if(OLD_COMPLEX) |
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96 | { |
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97 | tt="1,"+string(par(1)); |
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98 | if(tt==charstr(basering)) |
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99 | {ss=tt;ringprec=system("getPrecDigits");} |
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100 | } |
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101 | else |
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102 | { |
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103 | tt=charstr(basering); |
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104 | if(size(tt)>7) |
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105 | { |
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106 | if(string(tt[1..7])=="complex") |
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107 | { |
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108 | ss=tt; |
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109 | ringprec=system("getPrecDigits"); |
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110 | } |
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111 | } |
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112 | } |
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113 | } |
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114 | |
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115 | list roots,simple; |
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116 | if(deg(p)==0) // only zero was root |
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117 | { |
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118 | roots=addzero(roots,ss,iz,splitcontrol); |
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119 | if(prot!=0){dbprint(prot,"//END laguerre_solve");} |
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120 | return(roots); |
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121 | } |
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122 | |
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123 | if(prot!=0)// more informations |
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124 | { |
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125 | pout="//control: complex ring with precision "+string(numberprec); |
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126 | if(size(ss)==0){pout=pout+nl+ |
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127 | "// basering not complex, hence solutiontype string"; |
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128 | if(solutionprec<numberprec){pout=pout+nl+ |
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129 | "// with precision "+string(solutionprec);}} |
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130 | if(splitcontrol<0){pout=pout+nl+ "// no spliting";} |
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131 | if(splitcontrol==0){pout=pout+nl+"// output without multiple roots";} |
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132 | if(rootcheck){pout=pout+nl+ |
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133 | "// check roots with precision "+string(solutionprec);} |
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134 | dbprint(prot,pout); |
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135 | } |
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136 | |
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137 | def rn = basering;// set the complex ground field |
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138 | if (ringprec<numberprec) |
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139 | { |
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140 | tt="ring lagc=(complex,"+string(numberprec)+","+string(numberprec)+ |
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141 | "),"+string(var(iv))+",lp;"; |
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142 | execute(tt); |
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143 | poly p=imap(rn,p); |
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144 | poly v=var(1); |
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145 | } |
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146 | int ima=0; |
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147 | if(size(ss)!=0){ima=checkim(p);} |
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148 | number prc=0.1;// set precision of the solution |
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149 | prc=prc^solutionprec; |
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150 | if(prot!=0) |
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151 | { |
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152 | if(ringprec<numberprec){pout="//working in: "+tt;} |
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153 | if((size(ss)!=0)&&(ima!=0)){pout=pout+nl+ |
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154 | "// polynomial has complex coefficients";} |
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155 | dbprint(prot,pout); |
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156 | } |
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157 | |
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158 | int i1=1; |
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159 | int i2=1; |
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160 | ideal SPLIT=p; |
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161 | if(splitcontrol>=0)// splitting |
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162 | { |
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163 | if(prot!=0){dbprint(prot,"//split in working ring:");} |
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164 | SPLIT=splitsqrfree(p,v); |
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165 | i1=size(SPLIT); |
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166 | if((i1==1)&&(charstr(rn)=="0")) |
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167 | { |
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168 | if(prot!=0){dbprint(prot,"//split exact in basering:");} |
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169 | setring rn; |
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170 | if(v>1) |
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171 | { |
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172 | ideal SQQQQ=splitsqrfree(p,v); |
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173 | setring lagc; |
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174 | SPLIT=imap(rn,SQQQQ); |
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175 | } |
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176 | else |
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177 | { |
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178 | oo="ring exa=0,"+string(var(1))+",lp;"; |
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179 | execute(oo); |
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180 | ideal SQQQQ=splitsqrfree(imap(rn,p),var(1)); |
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181 | setring lagc; |
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182 | SPLIT=imap(exa,SQQQQ); |
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183 | kill exa; |
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184 | } |
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185 | i1=size(SPLIT); |
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186 | } |
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187 | if(prot!=0) |
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188 | { |
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189 | if(i1>1) |
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190 | { |
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191 | int i3=deg(SPLIT[1]); |
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192 | pout="//results of split(the squarefree factors):"; |
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193 | if(i3>0){pout=pout+nl+ |
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194 | "// multiplicity "+string(i2)+", degree "+string(i3);} |
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195 | while(i2<i1) |
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196 | { |
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197 | i2++; |
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198 | i3=deg(SPLIT[i2]); |
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199 | if(i3>0){pout=pout+nl+ |
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200 | "// multiplicity "+string(i2)+", degree "+string(i3);} |
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201 | } |
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202 | dbprint(prot,pout); |
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203 | i2=1; |
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204 | } |
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205 | else |
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206 | { |
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207 | if(charstr(rn)=="0"){dbprint(prot,"// polynomial is squarefree");} |
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208 | else{dbprint(prot,"// split without result");} |
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209 | } |
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210 | } |
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211 | } |
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212 | |
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213 | p=SPLIT[1];// the first part |
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214 | if(deg(p)>0) |
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215 | { |
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216 | roots=laguerre(p,numberprec,1);// the ring is already complex, hence numberprec is dummy |
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217 | if((size(roots)==0)||(string(roots[1])=="0")){ERROR("laguerre: no roots found");} |
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218 | if(rootcheck){checkroots(p,v,roots,ima,prc);} |
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219 | } |
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220 | while(i2<i1) |
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221 | { |
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222 | i2++; |
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223 | p=SPLIT[i2];// the part with multiplicity i2 |
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224 | if(deg(p)>0) |
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225 | { |
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226 | simple=laguerre(p,numberprec,1); |
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227 | if((size(simple)==0)||(string(simple[1])=="0")){ERROR("laguerre: no roots found");} |
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228 | if(rootcheck){checkroots(p,v,simple,ima,prc);} |
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229 | if(splitcontrol==0)// no multiple roots |
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230 | { |
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231 | roots=roots+simple; |
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232 | } |
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233 | else// multiple roots |
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234 | { |
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235 | roots=roots+makemult(simple,i2); |
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236 | } |
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237 | } |
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238 | } |
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239 | |
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240 | if((solutionprec<numberprec)&&(size(ss)==0))// shorter output |
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241 | { |
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242 | oo="ring lout=(complex,"+string(solutionprec)+",1)," |
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243 | +string(var(1))+",lp;"; |
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244 | execute(oo); |
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245 | list roots=imap(lagc,roots); |
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246 | roots=transroots(roots); |
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247 | if(iz>0){roots=addzero(roots,ss,iz,splitcontrol);} |
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248 | if(prot!=0){dbprint(prot,"//END laguerre_solve");} |
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249 | return(roots); |
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250 | } |
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251 | if(size(ss)==0){roots=transroots(roots);}// transform to string |
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252 | else // or map in basering |
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253 | { |
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254 | if(ringprec<numberprec) |
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255 | { |
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256 | setring rn; |
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257 | list roots=imap(lagc,roots); |
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258 | } |
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259 | } |
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260 | if(iz>0){roots=addzero(roots,ss,iz,splitcontrol);} |
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261 | if(prot!=0){dbprint(prot,"//END laguerre_solve");} |
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262 | return(roots); |
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263 | } |
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264 | example |
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265 | { |
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266 | "EXAMPLE:";echo=2; |
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267 | // Find all roots of an univariate polynomial using Laguerre's method: |
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268 | ring rs1= 0,(x,y),lp; |
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269 | poly f = 15x5 + x3 + x2 - 10; |
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270 | // 10 digits precision |
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271 | laguerre_solve(f,10); |
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272 | |
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273 | // Now with complex coefficients, |
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274 | // internal precision is 30 digits (default) |
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275 | printlevel=2; |
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276 | ring rsc= (real,10,i),x,lp; |
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277 | poly f = (15.4+i*5)*x^5 + (25.0e-2+i*2)*x^3 + x2 - 10*i; |
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278 | list l = laguerre_solve(f); |
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279 | l; |
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280 | // check result, value of substituted poly should be near to zero |
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281 | // remember that l contains a list of strings |
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282 | // in the case of a different ring |
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283 | subst(f,x,l[1]); |
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284 | subst(f,x,l[2]); |
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285 | } |
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286 | ////////////////////////////////////////////////////////////////////////////// |
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287 | // subprocedures for laguerre_solve |
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288 | /* |
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289 | * if p depends only on var(i) |
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290 | * returns i |
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291 | * otherwise 0 |
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292 | */ |
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293 | static proc checkv(poly p) |
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294 | { |
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295 | int n=nvars(basering); |
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296 | int i=0; |
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297 | int v; |
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298 | |
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299 | while (n>0) |
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300 | { |
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301 | if ((p-subst(p,var(n),0))!=0) |
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302 | { |
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303 | i++; |
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304 | if (i>1){return(0);} |
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305 | v=n; |
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306 | } |
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307 | n--; |
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308 | } |
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309 | return(v); |
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310 | } |
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311 | /* |
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312 | * if p has only real coefficients |
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313 | * returns 0 |
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314 | * otherwise 1 |
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315 | */ |
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316 | static proc checkim(poly p) |
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317 | { |
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318 | poly q=p; |
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319 | |
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320 | while(q!=0) |
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321 | { |
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322 | if(impart(leadcoef(q))!=0){return(1);} |
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323 | q=q-lead(q); |
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324 | } |
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325 | return(0); |
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326 | } |
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327 | /* |
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328 | * make multiplicity m |
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329 | */ |
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330 | static proc makemult(list si,int m) |
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331 | { |
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332 | int k0=0; |
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333 | int k1=size(si); |
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334 | int k2,k3; |
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335 | number ro; |
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336 | list msi; |
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337 | |
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338 | for(k2=1;k2<=k1;k2++) |
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339 | { |
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340 | ro=si[k2]; |
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341 | for(k3=m;k3>0;k3--){k0++;msi[k0]=ro;} |
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342 | } |
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343 | return(msi); |
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344 | } |
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345 | /* |
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346 | * returns 1 for n<1 |
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347 | */ |
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348 | static proc cmp1(number n) |
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349 | { |
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350 | number r=repart(n); |
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351 | number i=impart(n); |
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352 | number c=r*r+i*i; |
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353 | if(c>1){return(1);} |
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354 | else{return(0);} |
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355 | } |
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356 | /* |
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357 | * exact division of polys f/g |
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358 | * (should be internal) |
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359 | */ |
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360 | static proc exdiv(poly f,poly g,poly v) |
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361 | { |
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362 | int d1=deg(f); |
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363 | int d2=deg(g); |
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364 | poly r0=f; |
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365 | poly rf=0; |
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366 | poly h; |
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367 | number n,m; |
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368 | |
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369 | m=leadcoef(g); |
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370 | while ((r0!=0)&&(d1>=d2)) |
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371 | { |
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372 | n=leadcoef(r0)/m; |
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373 | h=n*v^(d1-d2); |
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374 | rf=rf+h; |
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375 | r0=r0-h*g; |
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376 | d1=deg(r0); |
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377 | } |
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378 | return(cleardenom(rf)); |
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379 | } |
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380 | /* |
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381 | * p is univariant in x |
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382 | * perform a split of p into squarefree factors |
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383 | * such that the returned ideal 'split' consists of |
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384 | * the faktors, i.e. |
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385 | * p = n * product ( split[i]^i ) , n a number |
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386 | */ |
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387 | static proc splitsqrfree(poly p, poly x) |
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388 | { |
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389 | int dd=deg(p); |
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390 | if(dd==1){return(p);} |
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391 | int i=1; |
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392 | int j; |
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393 | ideal h,split; |
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394 | poly high; |
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395 | |
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396 | h=interred(ideal(p,diff(p,x))); |
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397 | if(deg(h[1])==0){return(p);} |
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398 | high=h[1]; |
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399 | split[1]=exdiv(p,high,x); |
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400 | while(1) |
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401 | { |
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402 | h=interred(ideal(split[i],high)); |
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403 | j=deg(h[1]); |
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404 | if(j==0){return(p);} |
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405 | if(deg(h[1])==deg(split[i])) |
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406 | { |
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407 | split=split,split[i]; |
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408 | split[i]=1; |
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409 | } |
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410 | else |
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411 | { |
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412 | split[i]=exdiv(split[i],h[1],x); |
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413 | split=split,h[1]; |
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414 | dd=dd-deg(split[i])*i; |
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415 | } |
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416 | j=j*(i+1); |
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417 | if(j==dd){break;} |
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418 | if(j>dd){return(p);} |
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419 | high=exdiv(high,h[1],x); |
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420 | if(deg(high)==0){return(p);}; |
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421 | i++; |
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422 | } |
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423 | return(split); |
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424 | } |
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425 | /* |
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426 | * see checkroots |
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427 | */ |
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428 | static proc nerr(number n,number m) |
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429 | { |
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430 | int r; |
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431 | number z=0; |
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432 | number nr=repart(n); |
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433 | number ni=impart(n); |
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434 | if(nr<z){nr=z-nr;} |
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435 | if(ni<z){ni=nr-ni;} |
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436 | else{ni=nr+ni;} |
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437 | if(ni<m){r=0;} |
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438 | else{r=1;} |
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439 | return(r); |
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440 | } |
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441 | /* |
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442 | * returns ERROR for nerr(p(r[i]))>=pr |
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443 | */ |
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444 | static proc checkroots(poly p,poly v,list r,int ima,number pr) |
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445 | { |
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446 | int i=0; |
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447 | int j; |
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448 | number n,m; |
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449 | ideal li; |
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450 | |
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451 | while(i<size(r)) |
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452 | { |
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453 | i++; |
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454 | n=r[i]; |
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455 | j=cmp1(n); |
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456 | if(j!=0){li[1]=v/n-1;m=1;} |
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457 | else{li[1]=v-n;m=n;} |
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458 | if((ima==0)&&(impart(n)!=0)) |
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459 | { |
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460 | i++; |
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461 | n=r[i]; |
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462 | if(j!=0){li[1]=li[1]*(v/n-1);} |
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463 | else{li[1]=li[1]*(v-n);m=m*n;} |
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464 | } |
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465 | attrib(li,"isSB",1); |
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466 | n=leadcoef(reduce(p,li));n=n/m; |
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467 | if(n!=0) |
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468 | {if(nerr(n,pr)!=0){ERROR("Unsufficient accuracy!");}} |
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469 | } |
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470 | } |
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471 | /* |
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472 | * transforms thr result to string |
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473 | */ |
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474 | static proc transroots(list r) |
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475 | { |
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476 | int i=size(r); |
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477 | while (i>0) |
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478 | { |
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479 | r[i]=string(r[i]); |
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480 | i--; |
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481 | } |
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482 | return(r); |
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483 | } |
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484 | /* |
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485 | * returns a poly without zeroroots |
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486 | */ |
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487 | static proc divzero(poly f,int iv); |
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488 | { |
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489 | poly p=f; |
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490 | poly q=p; |
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491 | poly r; |
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492 | while(p==q) |
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493 | { |
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494 | q=p/var(iv); |
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495 | r=q*var(iv); |
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496 | if(r==p){p=q;} |
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497 | } |
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498 | return(p); |
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499 | } |
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500 | /* |
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501 | * add zeros to solution |
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502 | */ |
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503 | static proc addzero(list zz,string ss,int iz,int a) |
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504 | { |
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505 | int i=1; |
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506 | int j=size(zz); |
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507 | |
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508 | if(size(ss)==0){zz[j+1]="0";} |
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509 | else{zz[j+1]=number(0);} |
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510 | if(a==0){return(zz);} |
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511 | while(i<iz) |
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512 | { |
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513 | i++; |
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514 | if(size(ss)==0){zz[j+i]="0";} |
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515 | else{zz[j+i]=number(0);} |
---|
516 | } |
---|
517 | return(zz); |
---|
518 | } |
---|
519 | /////////////////////////////////////////////////////////////////////////////// |
---|
520 | |
---|
521 | proc solve( ideal G, list # ) |
---|
522 | "USAGE: solve(G [, m, n, l] ); G = ideal, |
---|
523 | m, n, l = integers (control parameters of the method) |
---|
524 | @format |
---|
525 | m: precision of output in digits ( 4 <= m) and of the |
---|
526 | generated ring of complex numbers; |
---|
527 | n: control of multiplicity |
---|
528 | n = 0, return all different roots |
---|
529 | n != 0, find all roots (with multiplicity) |
---|
530 | l: precision of internal computation in decimal digits ( l >=8 ) |
---|
531 | only if the basering is not complex or complex with smaller |
---|
532 | precision, |
---|
533 | ( default: m, n, l = 8, 0, 30 or |
---|
534 | for (n != 0 and size(#) = 2) l = 60 ) |
---|
535 | @end format |
---|
536 | ASSUME: the ideal is 0-dimensional;@* |
---|
537 | basering has characteristic 0 and is either complex or |
---|
538 | without parameters; |
---|
539 | RETURN: list of solutions of the ideal G, depending on n; one solution is a |
---|
540 | list of complex numbers in the generated output ring (the new |
---|
541 | basering). |
---|
542 | @format |
---|
543 | The result is a list L |
---|
544 | n = 0: a list of all different solutions (L[i]), |
---|
545 | n != 0: a list of two elements, |
---|
546 | L[i][1] contains all different solutions with the same multiplicity |
---|
547 | L[i][2] the multiplicity |
---|
548 | L is ordered w.r.t. multiplicity (the smallest first). |
---|
549 | @end format |
---|
550 | NOTE: If the problem is not 0-dim. the procedure stops with ERROR, if the |
---|
551 | ideal G is not a lex. standard basis, it is generated with internal |
---|
552 | computation (Hilbert driven), if the input-ring (with char 0) has |
---|
553 | the name "<A>", the lexicographical and complex output-ring has the |
---|
554 | name "<A>C". |
---|
555 | EXAMPLE: example solve; shows an example |
---|
556 | " |
---|
557 | { |
---|
558 | // test if basering admissible |
---|
559 | if (char(basering)!=0){ERROR("characteristic of basering not 0");} |
---|
560 | if ((charstr(basering)[1]=="0") and (npars(basering)!=0)){ERROR("basering has parameters");} |
---|
561 | |
---|
562 | // some global settings and control |
---|
563 | int outprec = 8; |
---|
564 | int mu = 0; |
---|
565 | int prec = 30; |
---|
566 | if (size(#)>0){outprec = #[1];if (outprec<4){outprec = 4;}} |
---|
567 | if (size(#)>1){mu = #[2];} |
---|
568 | if (size(#)>2){prec = #[3];if (prec<8){prec = 8;}} |
---|
569 | else {if(mu!=0){prec = 60;}} |
---|
570 | if (outprec>prec){prec = outprec;} |
---|
571 | string rinC = nameof(basering)+"C"; |
---|
572 | string sord = ordstr(basering); |
---|
573 | int nv = nvars(basering); |
---|
574 | def rin = basering; |
---|
575 | intvec ovec = option(get); |
---|
576 | option(redSB); |
---|
577 | option(returnSB); |
---|
578 | int sb = attrib(G,"isSB"); |
---|
579 | int lp = 0; |
---|
580 | if (size(sord)==size("C,lp()"+string(nv))) |
---|
581 | { |
---|
582 | lp = find(sord,"lp"); |
---|
583 | } |
---|
584 | |
---|
585 | // ERROR |
---|
586 | if (sb){if (dim(G)!=0){ERROR("ideal not zero-dimensional");}} |
---|
587 | |
---|
588 | // the trivial homog case |
---|
589 | if (homog(G)) |
---|
590 | { |
---|
591 | if (sb==0) |
---|
592 | { |
---|
593 | execute("ring dphom=("+charstr(rin)+"),("+ |
---|
594 | varstr(rin)+"),dp;"); |
---|
595 | ideal G = std(imap(rin,G)); |
---|
596 | if (dim(G)!=0){ERROR("ideal not zero-dimensional");} |
---|
597 | } |
---|
598 | changering(rinC,outprec); |
---|
599 | list ret0; |
---|
600 | if (mu==0){ret0[1] = zerolist(nv);} |
---|
601 | else{ret0[1] = list(zerolist(nv),list(vdim(G)));} |
---|
602 | option(set,ovec); |
---|
603 | keepring basering; |
---|
604 | return(ret0); |
---|
605 | } |
---|
606 | |
---|
607 | // look for reduced standard basis in lex |
---|
608 | if (sb*lp==0) |
---|
609 | { |
---|
610 | if (sb==0) |
---|
611 | { |
---|
612 | execute("ring dphilb=("+charstr(rin)+"),("+ |
---|
613 | varstr(rin)+"),dp;"); |
---|
614 | ideal G = imap(rin,G); |
---|
615 | G = std(G); |
---|
616 | if (dim(G)!=0){ERROR("ideal not zero-dimensional");} |
---|
617 | } |
---|
618 | else |
---|
619 | { |
---|
620 | def dphilb = basering; |
---|
621 | } |
---|
622 | execute("ring lexhilb=("+charstr(rin)+"),("+ |
---|
623 | varstr(rin)+"),lp;"); |
---|
624 | option(redTail); |
---|
625 | ideal H = fglm(dphilb,G); |
---|
626 | kill dphilb; |
---|
627 | H = simplify(H,2); |
---|
628 | if (lp){setring rin;} |
---|
629 | else |
---|
630 | { |
---|
631 | execute("ring lplex=("+charstr(rin)+"),("+ |
---|
632 | varstr(rin)+"),lp;"); |
---|
633 | } |
---|
634 | ideal H = imap(lexhilb,H); |
---|
635 | kill lexhilb; |
---|
636 | } |
---|
637 | else{ideal H = interred(G);} |
---|
638 | |
---|
639 | // only 1 variable |
---|
640 | def hr = basering; |
---|
641 | if (nv==1) |
---|
642 | { |
---|
643 | if ((mu==0) and (charstr(basering)[1]=="0")) // special case |
---|
644 | { |
---|
645 | list L = laguerre_solve(H[1],prec,prec,mu,0); // list of strings |
---|
646 | changering(rinC,outprec); |
---|
647 | list sp; |
---|
648 | for (int ii=1; ii<=size(L); ii++ ) |
---|
649 | { |
---|
650 | execute("sp[ii]="+L[ii]); |
---|
651 | } |
---|
652 | keepring basering; |
---|
653 | return(sp); |
---|
654 | } |
---|
655 | else |
---|
656 | { |
---|
657 | execute("ring internC=(complex,"+string(prec)+ |
---|
658 | "),("+varstr(basering)+"),lp;"); |
---|
659 | |
---|
660 | ideal H = imap(hr,H); |
---|
661 | list sp = splittolist(splitsqrfree(H[1],var(1))); |
---|
662 | int jj = size(sp); |
---|
663 | while(jj>0) |
---|
664 | { |
---|
665 | sp[jj][1] = laguerre(sp[jj][1],prec,1); |
---|
666 | jj--; |
---|
667 | } |
---|
668 | setring hr; |
---|
669 | changering(rinC,outprec); |
---|
670 | list sp=imap(internC,sp); |
---|
671 | |
---|
672 | keepring basering; |
---|
673 | if(mu!=0){return(sp);} |
---|
674 | jj = size(sp); |
---|
675 | list ll=sp[jj][1]; |
---|
676 | while(jj>1) |
---|
677 | { |
---|
678 | jj--; |
---|
679 | ll = sp[jj][1]+ll; |
---|
680 | } |
---|
681 | return(ll); |
---|
682 | } |
---|
683 | } |
---|
684 | |
---|
685 | |
---|
686 | // the triangular sets (not univariate case) |
---|
687 | attrib(H,"isSB",1); |
---|
688 | if (mu==0) |
---|
689 | { |
---|
690 | list sp = triangMH(H); // faster, but destroy multiplicity |
---|
691 | } |
---|
692 | else |
---|
693 | { |
---|
694 | list sp = triangM(H); |
---|
695 | } |
---|
696 | |
---|
697 | // create the complex ring and map the result |
---|
698 | if (outprec<prec) |
---|
699 | { |
---|
700 | execute("ring internC=(complex,"+string(prec)+ |
---|
701 | "),("+varstr(hr)+"),lp;"); |
---|
702 | } |
---|
703 | else |
---|
704 | { |
---|
705 | changering(rinC,prec); |
---|
706 | } |
---|
707 | list triC = imap(hr,sp); |
---|
708 | |
---|
709 | // solve the tridiagonal systems |
---|
710 | int js = size(triC); |
---|
711 | list ret1; |
---|
712 | if (mu==0) |
---|
713 | { |
---|
714 | ret1 = trisolve(list(),triC[1],prec); |
---|
715 | while (js>1) |
---|
716 | { |
---|
717 | ret1 = trisolve(list(),triC[js],prec)+ret1; |
---|
718 | js--; |
---|
719 | } |
---|
720 | } |
---|
721 | else |
---|
722 | { |
---|
723 | ret1 = mutrisolve(list(),triC[1],prec); |
---|
724 | while (js>1) |
---|
725 | { |
---|
726 | ret1 = addlist(mutrisolve(list(),triC[js],prec),ret1,1); |
---|
727 | js--; |
---|
728 | } |
---|
729 | ret1 = finalclear(ret1); |
---|
730 | } |
---|
731 | |
---|
732 | // final computations |
---|
733 | option(set,ovec); |
---|
734 | if (outprec==prec) |
---|
735 | { |
---|
736 | keepring basering; |
---|
737 | return(ret1); |
---|
738 | } |
---|
739 | changering(rinC,outprec); |
---|
740 | keepring basering; |
---|
741 | return(imap(internC,ret1)); |
---|
742 | } |
---|
743 | example |
---|
744 | { |
---|
745 | "EXAMPLE:";echo=2; |
---|
746 | // Find all roots of a multivariate ideal using triangular sets: |
---|
747 | int d=4;// with these 3 parameters you may construct |
---|
748 | int t=3;// very hard problems for 'solve' |
---|
749 | int s=2; |
---|
750 | int i; |
---|
751 | ring A=0,(x(1..d)),dp; |
---|
752 | poly p=-1; |
---|
753 | for(i=d;i>0;i--){p=p+x(i)^s;} |
---|
754 | ideal I=x(d)^t-x(d)^s+p; |
---|
755 | for(i=d-1;i>0;i--){I=x(i)^t-x(i)^s+p,I;} |
---|
756 | I; |
---|
757 | // the mutiplicity is |
---|
758 | vdim(std(I)); |
---|
759 | list l1=solve(I,6,0); |
---|
760 | // the current ring is |
---|
761 | AC; |
---|
762 | // you must start with char. 0 |
---|
763 | setring A; |
---|
764 | list l2=solve(I,6,1); |
---|
765 | // the number of different solutions is |
---|
766 | size(l1); |
---|
767 | // this is equal to |
---|
768 | size(l2[1][1])+size(l2[2][1]); |
---|
769 | // the number of solutions with multiplicity is |
---|
770 | size(l2[1][1])*l2[1][2]+size(l2[2][1])*l2[2][2]; |
---|
771 | // the solutions with multiplicity |
---|
772 | l2[2][2]; |
---|
773 | // are |
---|
774 | l2[2][1]; |
---|
775 | } |
---|
776 | ////////////////////////////////////////////////////////////////////////////// |
---|
777 | // subprocedures for solve |
---|
778 | |
---|
779 | /* ----------------------- support ----------------------- */ |
---|
780 | /* |
---|
781 | * the complex ring with precision outprec |
---|
782 | * has the well defined name: rinC |
---|
783 | * 1. if such a ring exists with the precision outprec, |
---|
784 | * this will be the current ring |
---|
785 | * 2. otherwise such a ring will be created |
---|
786 | */ |
---|
787 | static proc changering(string rinC, int outprec) |
---|
788 | { |
---|
789 | string rinDC = "ring "+rinC+"=(complex,"+string(outprec)+ |
---|
790 | "),("+varstr(basering)+"),lp;"; |
---|
791 | string h = "int ex=defined("+rinC+");"; |
---|
792 | |
---|
793 | execute(h); |
---|
794 | if (ex) |
---|
795 | { |
---|
796 | h = "setring "+rinC+";"; |
---|
797 | execute(h); |
---|
798 | if (system("getPrecDigits")==outprec) |
---|
799 | {"// name of current ring: "+rinC;} |
---|
800 | else |
---|
801 | { |
---|
802 | execute("kill "+rinC+";"); |
---|
803 | execute(rinDC); |
---|
804 | execute("export "+rinC+";"); |
---|
805 | "// name of new current ring: "+rinC; |
---|
806 | } |
---|
807 | } |
---|
808 | else |
---|
809 | { |
---|
810 | execute(rinDC); |
---|
811 | execute("export "+rinC+";"); |
---|
812 | "// name of new current ring: "+rinC; |
---|
813 | } |
---|
814 | keepring basering; |
---|
815 | } |
---|
816 | |
---|
817 | /* |
---|
818 | * return one zero-solution |
---|
819 | */ |
---|
820 | static proc zerolist(int nv) |
---|
821 | { |
---|
822 | list ret; |
---|
823 | int i; |
---|
824 | number o=0; |
---|
825 | |
---|
826 | for (i=nv;i>0;i--){ret[i] = o;} |
---|
827 | return(ret); |
---|
828 | } |
---|
829 | |
---|
830 | /* ----------------------- check solution ----------------------- */ |
---|
831 | static proc multsol(list ff, int c) |
---|
832 | { |
---|
833 | int i,j; |
---|
834 | |
---|
835 | i = 0; |
---|
836 | j = size(ff); |
---|
837 | while (j>0) |
---|
838 | { |
---|
839 | if(c){i = i+ff[j][2]*size(ff[j][1]);} |
---|
840 | else{i = i+size(ff[j][1]);} |
---|
841 | j--; |
---|
842 | } |
---|
843 | return(i); |
---|
844 | } |
---|
845 | |
---|
846 | /* |
---|
847 | * the inputideal A => zero ? |
---|
848 | */ |
---|
849 | static proc checksol(ideal A, list lr) |
---|
850 | { |
---|
851 | int d = nvars(basering); |
---|
852 | list ro; |
---|
853 | ideal re,h; |
---|
854 | int i,j,k; |
---|
855 | |
---|
856 | for (i=size(lr);i>0;i--) |
---|
857 | { |
---|
858 | ro = lr[i]; |
---|
859 | for (j=d;j>0;j--) |
---|
860 | { |
---|
861 | re[j] = var(j)-ro[j]; |
---|
862 | } |
---|
863 | attrib(re,"isSB",1); |
---|
864 | k = size(reduce(A,re)); |
---|
865 | if (k){return(i);} |
---|
866 | } |
---|
867 | return(0); |
---|
868 | } |
---|
869 | |
---|
870 | /* |
---|
871 | * compare 2 solutions: returns 0 for equal |
---|
872 | */ |
---|
873 | static proc cmpn(list a,list b) |
---|
874 | { |
---|
875 | int ii; |
---|
876 | |
---|
877 | for(ii=size(a);ii>0;ii--){if(a[ii]!=b[ii]) break;} |
---|
878 | return(ii); |
---|
879 | } |
---|
880 | |
---|
881 | /* |
---|
882 | * delete equal solutions in the list |
---|
883 | */ |
---|
884 | static proc delequal(list r, int w) |
---|
885 | { |
---|
886 | list h; |
---|
887 | int i,j,k,c; |
---|
888 | |
---|
889 | if (w) |
---|
890 | { |
---|
891 | k = size(r); |
---|
892 | h = r[k][1]; |
---|
893 | k--; |
---|
894 | while (k>0) |
---|
895 | { |
---|
896 | h = r[k][1]+h; |
---|
897 | k--; |
---|
898 | } |
---|
899 | } |
---|
900 | else{h = r;} |
---|
901 | k=size(h); |
---|
902 | i=1; |
---|
903 | while(i<k) |
---|
904 | { |
---|
905 | j=k; |
---|
906 | while(j>i) |
---|
907 | { |
---|
908 | c=cmpn(h[i],h[j]); |
---|
909 | if(c==0) |
---|
910 | { |
---|
911 | h=delete(h,j); |
---|
912 | k--; |
---|
913 | } |
---|
914 | j--; |
---|
915 | } |
---|
916 | i++; |
---|
917 | } |
---|
918 | return(h); |
---|
919 | } |
---|
920 | |
---|
921 | /* ----------------------- substitution ----------------------- */ |
---|
922 | /* |
---|
923 | * instead of subst(T,var(v),n), much faster |
---|
924 | * need option(redSB) ! |
---|
925 | */ |
---|
926 | static proc linreduce(ideal T, int v, number n) |
---|
927 | { |
---|
928 | ideal re = var(v)-n; |
---|
929 | attrib (re,"isSB",1); |
---|
930 | return (reduce(T,re)); |
---|
931 | } |
---|
932 | |
---|
933 | /* ----------------------- triangular solution ----------------------- */ |
---|
934 | /* |
---|
935 | * solution of one tridiagonal system T |
---|
936 | * with precision prec |
---|
937 | * T[1] is univariant in var(1) |
---|
938 | * list o is empty for the first call |
---|
939 | */ |
---|
940 | static proc trisolve(list o, ideal T, int prec) |
---|
941 | { |
---|
942 | list lroots,ll; |
---|
943 | ideal S; |
---|
944 | int i,d; |
---|
945 | |
---|
946 | d = size(T); |
---|
947 | S = interred(ideal(T[1],diff(T[1],var(d)))); |
---|
948 | if (deg(S[1])) |
---|
949 | { |
---|
950 | T[1] = exdiv(T[1],S[1],var(d)); |
---|
951 | } |
---|
952 | ll = laguerre(T[1],prec,1); |
---|
953 | for (i=size(ll);i>0;i--){ll[i] = list(ll[i])+o;} |
---|
954 | if (d==1){return(ll);} |
---|
955 | for (i=size(ll);i>0;i--) |
---|
956 | { |
---|
957 | S = linreduce(ideal(T[2..d]),d,ll[i][1]); |
---|
958 | lroots = trisolve(ll[i],S,prec)+lroots; |
---|
959 | } |
---|
960 | return(lroots); |
---|
961 | } |
---|
962 | |
---|
963 | /* ------------------- triangular solution (mult) ------------------- */ |
---|
964 | /* |
---|
965 | * recompute equal solutions w.r.t. multiplicity |
---|
966 | */ |
---|
967 | static proc finalclear(list b) |
---|
968 | { |
---|
969 | list a = b; |
---|
970 | list r; |
---|
971 | int i,l,ju,j,k,ku,mu,c; |
---|
972 | |
---|
973 | // a[i] only |
---|
974 | i = 1; |
---|
975 | while (i<=size(a)) |
---|
976 | { |
---|
977 | ju = size(a[i][1]); |
---|
978 | j = 1; |
---|
979 | while (j<=ju) |
---|
980 | { |
---|
981 | mu = 1; |
---|
982 | k = j+1; |
---|
983 | while (k<=ju) |
---|
984 | { |
---|
985 | c = cmpn(a[i][1][j],a[i][1][k]); |
---|
986 | if (c==0) |
---|
987 | { |
---|
988 | a[i][1] = delete(a[i][1],k); |
---|
989 | ju--; |
---|
990 | mu++; |
---|
991 | } |
---|
992 | else{k++;} |
---|
993 | } |
---|
994 | if (mu>1) |
---|
995 | { |
---|
996 | r[1] = a[i]; |
---|
997 | r[1][1] = list(a[i][1][j]); |
---|
998 | a[i][1] = delete(a[i][1],j); |
---|
999 | a = addlist(r,a,mu); |
---|
1000 | ju--; |
---|
1001 | } |
---|
1002 | else{j++;} |
---|
1003 | } |
---|
1004 | if (ju==0){a = delete(a,i);} |
---|
1005 | else{i++;} |
---|
1006 | } |
---|
1007 | // a[i], a[l] |
---|
1008 | i = 1; |
---|
1009 | while (i<size(a)) |
---|
1010 | { |
---|
1011 | ju = size(a[i][1]); |
---|
1012 | l = i+1; |
---|
1013 | while (l<=size(a)) |
---|
1014 | { |
---|
1015 | ku = size(a[l][1]); |
---|
1016 | j = 1; |
---|
1017 | while (j<=ju) |
---|
1018 | { |
---|
1019 | mu = 0; |
---|
1020 | k = 1; |
---|
1021 | while (k<=ku) |
---|
1022 | { |
---|
1023 | c = cmpn(a[i][1][j],a[l][1][k]); |
---|
1024 | if (c==0) |
---|
1025 | { |
---|
1026 | mu = a[i][2]+a[l][2]; |
---|
1027 | r[1] = a[l]; |
---|
1028 | r[1][1] = list(a[l][1][k]); |
---|
1029 | r[1][2] = mu; |
---|
1030 | a[l][1] = delete(a[l][1],k); |
---|
1031 | a = addlist(r,a,1); |
---|
1032 | ku--; |
---|
1033 | break; |
---|
1034 | } |
---|
1035 | else{k++;} |
---|
1036 | } |
---|
1037 | if (mu) |
---|
1038 | { |
---|
1039 | a[i][1] = delete(a[i][1],j); |
---|
1040 | ju--; |
---|
1041 | } |
---|
1042 | else{j++;} |
---|
1043 | } |
---|
1044 | if (ku){l++;} |
---|
1045 | else |
---|
1046 | { |
---|
1047 | a = delete(a,l); |
---|
1048 | } |
---|
1049 | } |
---|
1050 | if (ju){i++;} |
---|
1051 | else |
---|
1052 | { |
---|
1053 | a = delete(a,i); |
---|
1054 | } |
---|
1055 | } |
---|
1056 | return(a); |
---|
1057 | } |
---|
1058 | |
---|
1059 | /* |
---|
1060 | * convert to list |
---|
1061 | */ |
---|
1062 | static proc splittolist(ideal sp) |
---|
1063 | { |
---|
1064 | int j = size(sp); |
---|
1065 | list spl = list(list(sp[j],j)); |
---|
1066 | |
---|
1067 | j--; |
---|
1068 | while (j>0) |
---|
1069 | { |
---|
1070 | if (deg(sp[j])) |
---|
1071 | { |
---|
1072 | spl = list(list(sp[j],j))+spl; |
---|
1073 | } |
---|
1074 | j--; |
---|
1075 | } |
---|
1076 | return(spl); |
---|
1077 | } |
---|
1078 | |
---|
1079 | /* |
---|
1080 | * multiply the multiplicity |
---|
1081 | */ |
---|
1082 | static proc multlist(list a, int m) |
---|
1083 | { |
---|
1084 | int i; |
---|
1085 | for (i=size(a);i>0;i--){a[i][2] = a[i][2]*m;} |
---|
1086 | return(a); |
---|
1087 | } |
---|
1088 | |
---|
1089 | /* |
---|
1090 | * a+b w.r.t. to multiplicity as ordering |
---|
1091 | * (programming like spolys) |
---|
1092 | */ |
---|
1093 | static proc addlist(list a, list b, int m) |
---|
1094 | { |
---|
1095 | int i,j,k,l,s; |
---|
1096 | list r = list(); |
---|
1097 | |
---|
1098 | if (m>1){a = multlist(a,m);} |
---|
1099 | k = size(a); |
---|
1100 | l = size(b); |
---|
1101 | i = 1; |
---|
1102 | j = 1; |
---|
1103 | while ((i<=k)&&(j<=l)) |
---|
1104 | { |
---|
1105 | s = a[i][2]-b[j][2]; |
---|
1106 | if (s>=0) |
---|
1107 | { |
---|
1108 | r = r+list(b[j]); |
---|
1109 | j++; |
---|
1110 | if (s==0) |
---|
1111 | { |
---|
1112 | s = size(r); |
---|
1113 | r[s][1] = r[s][1]+a[i][1]; |
---|
1114 | i++; |
---|
1115 | } |
---|
1116 | } |
---|
1117 | else |
---|
1118 | { |
---|
1119 | r = r+list(a[i]); |
---|
1120 | i++; |
---|
1121 | } |
---|
1122 | } |
---|
1123 | if (i>k) |
---|
1124 | { |
---|
1125 | if (j<=l){r = r+list(b[j..l]);} |
---|
1126 | } |
---|
1127 | else{r = r+list(a[i..k]);} |
---|
1128 | return(r); |
---|
1129 | } |
---|
1130 | |
---|
1131 | /* |
---|
1132 | * solution of one tridiagonal system T with multiplicity |
---|
1133 | * with precision prec |
---|
1134 | * T[1] is univariant in var(1) |
---|
1135 | * list o is empty for the first call |
---|
1136 | */ |
---|
1137 | static proc mutrisolve(list o, ideal T, int prec) |
---|
1138 | { |
---|
1139 | list lroots,ll,sp; |
---|
1140 | ideal S,h; |
---|
1141 | int i,d,m,z; |
---|
1142 | |
---|
1143 | d = size(T); |
---|
1144 | sp = splittolist(splitsqrfree(T[1],var(d))); |
---|
1145 | if (d==1){return(l_mutrisolve(sp,o,prec));} |
---|
1146 | z = size(sp); |
---|
1147 | while (z>0) |
---|
1148 | { |
---|
1149 | m = sp[z][2]; |
---|
1150 | ll = laguerre(sp[z][1],prec,1); |
---|
1151 | i = size(ll); |
---|
1152 | while(i>0) |
---|
1153 | { |
---|
1154 | h = linreduce(ideal(T[2..d]),d,ll[i]); |
---|
1155 | if (size(lroots)) |
---|
1156 | { |
---|
1157 | lroots = addlist(mutrisolve(list(ll[i])+o,h,prec),lroots,m); |
---|
1158 | } |
---|
1159 | else |
---|
1160 | { |
---|
1161 | lroots = mutrisolve(list(ll[i])+o,h,prec); |
---|
1162 | if (m>1){lroots=multlist(lroots,m);} |
---|
1163 | } |
---|
1164 | i--; |
---|
1165 | } |
---|
1166 | z--; |
---|
1167 | } |
---|
1168 | return(lroots); |
---|
1169 | } |
---|
1170 | |
---|
1171 | /* |
---|
1172 | * the last call, we are ready |
---|
1173 | */ |
---|
1174 | static proc l_mutrisolve(list sp, list o, int prec) |
---|
1175 | { |
---|
1176 | list lroots,ll; |
---|
1177 | int z,m,i; |
---|
1178 | |
---|
1179 | z = size(sp); |
---|
1180 | while (z>0) |
---|
1181 | { |
---|
1182 | m = sp[z][2]; |
---|
1183 | ll = laguerre(sp[z][1],prec,1); |
---|
1184 | for (i=size(ll);i>0;i--){ll[i] = list(ll[i])+o;} |
---|
1185 | if (size(lroots)) |
---|
1186 | { |
---|
1187 | lroots = addlist(list(list(ll,m)),lroots,1); |
---|
1188 | } |
---|
1189 | else |
---|
1190 | { |
---|
1191 | lroots = list(list(ll,m)); |
---|
1192 | } |
---|
1193 | z--; |
---|
1194 | } |
---|
1195 | return(lroots); |
---|
1196 | } |
---|
1197 | /////////////////////////////////////////////////////////////////////////////// |
---|
1198 | |
---|
1199 | proc ures_solve( ideal gls, list # ) |
---|
1200 | "USAGE: ures_solve(i [, k, p] ); i = ideal, k, p = integers |
---|
1201 | @format |
---|
1202 | k=0: use sparse resultant matrix of Gelfand, Kapranov and Zelevinsky, |
---|
1203 | k=1: use resultant matrix of Macaulay which works only for |
---|
1204 | homogeneous ideals, |
---|
1205 | p>0: defines precision of the long floats for internal computation |
---|
1206 | if the basering is not complex (in decimal digits), |
---|
1207 | (default: k=0, p=30) |
---|
1208 | @end format |
---|
1209 | ASSUME: i is a zerodimensional ideal with |
---|
1210 | nvars(basering) = ncols(i) = number of vars |
---|
1211 | actually occurring in i, |
---|
1212 | RETURN: list of all (complex) roots of the polynomial system i = 0; the |
---|
1213 | result is of type |
---|
1214 | @format |
---|
1215 | string: if the basering is not complex, |
---|
1216 | number: otherwise. |
---|
1217 | @end format |
---|
1218 | EXAMPLE: example ures_solve; shows an example |
---|
1219 | " |
---|
1220 | { |
---|
1221 | int typ=0;// defaults |
---|
1222 | int prec=30; |
---|
1223 | |
---|
1224 | if ( size(#) > 0 ) |
---|
1225 | { |
---|
1226 | typ= #[1]; |
---|
1227 | if ( typ < 0 || typ > 1 ) |
---|
1228 | { |
---|
1229 | ERROR("Valid values for second parameter k are: |
---|
1230 | 0: use sparse Resultant (default) |
---|
1231 | 1: use Macaulay Resultant"); |
---|
1232 | } |
---|
1233 | } |
---|
1234 | if ( size(#) > 1 ) |
---|
1235 | { |
---|
1236 | prec= #[2]; |
---|
1237 | if ( prec < 8 ) |
---|
1238 | { |
---|
1239 | prec = 8; |
---|
1240 | } |
---|
1241 | } |
---|
1242 | |
---|
1243 | return(uressolve(gls,typ,prec,1)); |
---|
1244 | // the last nonzero parameter gives an extra run of |
---|
1245 | // Laguerre's algorithm leading to better results |
---|
1246 | } |
---|
1247 | example |
---|
1248 | { |
---|
1249 | "EXAMPLE:";echo=2; |
---|
1250 | // compute the intersection points of two curves |
---|
1251 | ring rsq = 0,(x,y),lp; |
---|
1252 | ideal gls= x2 + y2 - 10, x2 + xy + 2y2 - 16; |
---|
1253 | ures_solve(gls,0,16); |
---|
1254 | // result is a list (x,y)-coordinates as strings |
---|
1255 | |
---|
1256 | // now with complex coefficient field, precision is 20 digits |
---|
1257 | ring rsc= (real,20,I),(x,y),lp; |
---|
1258 | ideal i = (2+3*I)*x2 + (0.35+I*45.0e-2)*y2 - 8, x2 + xy + (42.7)*y2; |
---|
1259 | list l= ures_solve(i,0,10); |
---|
1260 | // result is a list of (x,y)-coordinates of complex numbers |
---|
1261 | l; |
---|
1262 | // check the result |
---|
1263 | subst(subst(i[1],x,l[1][1]),y,l[1][2]); |
---|
1264 | } |
---|
1265 | /////////////////////////////////////////////////////////////////////////////// |
---|
1266 | |
---|
1267 | proc mp_res_mat( ideal i, list # ) |
---|
1268 | "USAGE: mp_res_mat(i [, k] ); i ideal, k integer, |
---|
1269 | @format |
---|
1270 | k=0: sparse resultant matrix of Gelfand, Kapranov and Zelevinsky, |
---|
1271 | k=1: resultant matrix of Macaulay (k=0 is default) |
---|
1272 | @end format |
---|
1273 | ASSUME: The number of elements in the input system must be the number of |
---|
1274 | variables in the basering plus one; |
---|
1275 | if k=1 then i must be homogeneous. |
---|
1276 | RETURN: module representing the multipolynomial resultant matrix |
---|
1277 | EXAMPLE: example mp_res_mat; shows an example |
---|
1278 | " |
---|
1279 | { |
---|
1280 | int typ=0; |
---|
1281 | |
---|
1282 | if ( size(#) > 0 ) |
---|
1283 | { |
---|
1284 | typ= #[1]; |
---|
1285 | if ( typ < 0 || typ > 1 ) |
---|
1286 | { |
---|
1287 | ERROR("Valid values for third parameter are: |
---|
1288 | 0: sparse resultant (default) |
---|
1289 | 1: Macaulay resultant"); |
---|
1290 | } |
---|
1291 | } |
---|
1292 | |
---|
1293 | return(mpresmat(i,typ)); |
---|
1294 | |
---|
1295 | } |
---|
1296 | example |
---|
1297 | { |
---|
1298 | "EXAMPLE:";echo=2; |
---|
1299 | // compute resultant matrix in ring with parameters (sparse resultant matrix) |
---|
1300 | ring rsq= (0,u0,u1,u2),(x1,x2),lp; |
---|
1301 | ideal i= u0+u1*x1+u2*x2,x1^2 + x2^2 - 10,x1^2 + x1*x2 + 2*x2^2 - 16; |
---|
1302 | module m = mp_res_mat(i); |
---|
1303 | print(m); |
---|
1304 | // computing sparse resultant |
---|
1305 | det(m); |
---|
1306 | |
---|
1307 | // compute resultant matrix (Macaulay resultant matrix) |
---|
1308 | ring rdq= (0,u0,u1,u2),(x0,x1,x2),lp; |
---|
1309 | ideal h= homog(imap(rsq,i),x0); |
---|
1310 | h; |
---|
1311 | |
---|
1312 | module m = mp_res_mat(h,1); |
---|
1313 | print(m); |
---|
1314 | // computing Macaulay resultant (should be the same as above!) |
---|
1315 | det(m); |
---|
1316 | |
---|
1317 | // compute numerical sparse resultant matrix |
---|
1318 | setring rsq; |
---|
1319 | ideal ir= 15+2*x1+5*x2,x1^2 + x2^2 - 10,x1^2 + x1*x2 + 2*x2^2 - 16; |
---|
1320 | module mn = mp_res_mat(ir); |
---|
1321 | print(mn); |
---|
1322 | // computing sparse resultant |
---|
1323 | det(mn); |
---|
1324 | } |
---|
1325 | /////////////////////////////////////////////////////////////////////////////// |
---|
1326 | |
---|
1327 | proc interpolate( ideal p, ideal w, int d ) |
---|
1328 | "USAGE: interpolate(p,v,d); p,v=ideals of numbers, d=integer |
---|
1329 | ASSUME: Ground field K is the field of rational numbers, p and v are lists |
---|
1330 | of elements of the ground field K with p[j] != -1,0,1, size(p) = n |
---|
1331 | (= number of vars) and size(v)=N=(d+1)^n. |
---|
1332 | RETURN: poly f, the unique polynomial f of degree n*d with prescribed values |
---|
1333 | v[i] at the points p(i)=(p[1]^(i-1),..,p[n]^(i-1)), i=1,..,N. |
---|
1334 | NOTE: mainly useful when n=1, i.e. f is satisfying f(p^(i-1)) = v[i], |
---|
1335 | i=1..d+1. |
---|
1336 | SEE ALSO: vandermonde. |
---|
1337 | EXAMPLE: example interpolate; shows an example |
---|
1338 | " |
---|
1339 | { |
---|
1340 | return(vandermonde(p,w,d)); |
---|
1341 | } |
---|
1342 | example |
---|
1343 | { |
---|
1344 | "EXAMPLE:"; echo=2; |
---|
1345 | ring r1 = 0,(x),lp; |
---|
1346 | // determine f with deg(f) = 4 and |
---|
1347 | // v = values of f at points 3^0, 3^1, 3^2, 3^3, 3^4 |
---|
1348 | ideal v=16,0,11376,1046880,85949136; |
---|
1349 | interpolate( 3, v, 4 ); |
---|
1350 | } |
---|
1351 | /////////////////////////////////////////////////////////////////////////////// |
---|
1352 | |
---|
1353 | static proc psubst( int d, int dd, int n, list resl, |
---|
1354 | ideal fi, int elem, int nv, int prec ) |
---|
1355 | { |
---|
1356 | // nv: number of ring variables (fixed value) |
---|
1357 | // elem: number of elements in ideal fi (fixed value) |
---|
1358 | // fi: input ideal (fixed value) |
---|
1359 | // rl: output list of roots |
---|
1360 | // resl: actual list of roots |
---|
1361 | // n: |
---|
1362 | // dd: actual element of fi |
---|
1363 | // d: actual variable |
---|
1364 | |
---|
1365 | int olddd=dd; |
---|
1366 | |
---|
1367 | if ( pdebug>=1 ) |
---|
1368 | {"// 0 step "+string(dd)+" of "+string(elem);} |
---|
1369 | |
---|
1370 | if ( dd <= elem ) |
---|
1371 | { |
---|
1372 | int loop = 1; |
---|
1373 | int k; |
---|
1374 | list lsr,lh; |
---|
1375 | poly ps; |
---|
1376 | int thedd; |
---|
1377 | |
---|
1378 | if ( pdebug>=1 ) |
---|
1379 | {"// 1 dd = "+string(dd);} |
---|
1380 | |
---|
1381 | thedd=0; |
---|
1382 | while ( (dd+1 <= elem) && loop ) |
---|
1383 | { |
---|
1384 | ps= fi[dd+1]; |
---|
1385 | |
---|
1386 | if ( n-1 > 0 ) |
---|
1387 | { |
---|
1388 | if ( pdebug>=2 ) |
---|
1389 | { |
---|
1390 | "// 2 ps=fi["+string(dd+1)+"]"+" size=" |
---|
1391 | +string(size(coeffs(ps,var(n-1)))) |
---|
1392 | +" leadexp(ps)="+string(leadexp(ps)); |
---|
1393 | } |
---|
1394 | if ( size(coeffs(ps,var(n-1))) == 1 ) |
---|
1395 | { |
---|
1396 | dd++; |
---|
1397 | // hier Leading-Exponent pruefen??? |
---|
1398 | // oder ist das Poly immer als letztes in der Liste?!? |
---|
1399 | // leadexp(ps) |
---|
1400 | } |
---|
1401 | else |
---|
1402 | { |
---|
1403 | loop=0; |
---|
1404 | } |
---|
1405 | } |
---|
1406 | else |
---|
1407 | { |
---|
1408 | if ( pdebug>=2 ) |
---|
1409 | { |
---|
1410 | "// 2 ps=fi["+string(dd+1)+"]"+" leadexp(ps)=" |
---|
1411 | +string(leadexp(ps)); |
---|
1412 | } |
---|
1413 | dd++; |
---|
1414 | } |
---|
1415 | } |
---|
1416 | thedd=dd; |
---|
1417 | ps= fi[thedd]; |
---|
1418 | |
---|
1419 | if ( pdebug>=1 ) |
---|
1420 | { |
---|
1421 | "// 3 fi["+string(thedd-1)+"]"+" leadexp(fi[thedd-1])=" |
---|
1422 | +string(leadexp(fi[thedd-1])); |
---|
1423 | "// 3 ps=fi["+string(thedd)+"]"+" leadexp(ps)=" |
---|
1424 | +string(leadexp(ps)); |
---|
1425 | } |
---|
1426 | |
---|
1427 | for ( k= nv; k > nv-d; k-- ) |
---|
1428 | { |
---|
1429 | if ( pdebug>=2 ) |
---|
1430 | { |
---|
1431 | "// 4 subst(fi["+string(thedd)+"]," |
---|
1432 | +string(var(k))+","+string(resl[k])+");"; |
---|
1433 | } |
---|
1434 | ps = subst(ps,var(k),resl[k]); |
---|
1435 | } |
---|
1436 | |
---|
1437 | if ( pdebug>=2 ) |
---|
1438 | { "// 5 substituted ps="+string(ps); } |
---|
1439 | |
---|
1440 | if ( ps != 0 ) |
---|
1441 | { |
---|
1442 | lsr= laguerre_solve( ps, prec, prec, 0 ); |
---|
1443 | } |
---|
1444 | else |
---|
1445 | { |
---|
1446 | if ( pdebug>=1 ) |
---|
1447 | { "// 30 ps == 0, thats not cool..."; } |
---|
1448 | lsr=@ln; // lsr=number(0); |
---|
1449 | } |
---|
1450 | |
---|
1451 | if ( pdebug>=1 ) |
---|
1452 | { "// 6 laguerre_solve found roots: lsr["+string(size(lsr))+"]"; } |
---|
1453 | |
---|
1454 | if ( size(lsr) > 1 ) |
---|
1455 | { |
---|
1456 | if ( pdebug>=1 ) |
---|
1457 | { |
---|
1458 | "// 10 checking roots found before, range " |
---|
1459 | +string(dd-olddd)+" -- "+string(dd); |
---|
1460 | "// 10 thedd = "+string(thedd); |
---|
1461 | } |
---|
1462 | |
---|
1463 | int i,j,l; |
---|
1464 | int ls=size(lsr); |
---|
1465 | int lss; |
---|
1466 | poly pss; |
---|
1467 | list nares; |
---|
1468 | int rroot; |
---|
1469 | int nares_size; |
---|
1470 | |
---|
1471 | |
---|
1472 | for ( i = 1; i <= ls; i++ ) // lsr[1..ls] |
---|
1473 | { |
---|
1474 | rroot=1; |
---|
1475 | |
---|
1476 | if ( pdebug>=2 ) |
---|
1477 | {"// 13 root lsr["+string(i)+"] = "+string(lsr[i]);} |
---|
1478 | for ( l = 0; l <= dd-olddd; l++ ) |
---|
1479 | { |
---|
1480 | if ( l+olddd != thedd ) |
---|
1481 | { |
---|
1482 | if ( pdebug>=2 ) |
---|
1483 | {"// 11 checking ideal element "+string(l+olddd);} |
---|
1484 | ps=fi[l+olddd]; |
---|
1485 | if ( pdebug>=3 ) |
---|
1486 | {"// 14 ps=fi["+string(l+olddd)+"]";} |
---|
1487 | for ( k= nv; k > nv-d; k-- ) |
---|
1488 | { |
---|
1489 | if ( pdebug>=3 ) |
---|
1490 | { |
---|
1491 | "// 11 subst(fi["+string(olddd+l)+"]," |
---|
1492 | +string(var(k))+","+string(resl[k])+");"; |
---|
1493 | } |
---|
1494 | ps = subst(ps,var(k),resl[k]); |
---|
1495 | |
---|
1496 | } |
---|
1497 | |
---|
1498 | pss=subst(ps,var(k),lsr[i]); // k=nv-d |
---|
1499 | if ( pdebug>=3 ) |
---|
1500 | { "// 15 0 == "+string(pss); } |
---|
1501 | if ( pss != 0 ) |
---|
1502 | { |
---|
1503 | if ( system("complexNearZero", |
---|
1504 | leadcoef(pss), |
---|
1505 | prec) ) |
---|
1506 | { |
---|
1507 | if ( pdebug>=2 ) |
---|
1508 | { "// 16 root "+string(i)+" is a real root"; } |
---|
1509 | } |
---|
1510 | else |
---|
1511 | { |
---|
1512 | if ( pdebug>=2 ) |
---|
1513 | { "// 17 0 == "+string(pss); } |
---|
1514 | rroot=0; |
---|
1515 | } |
---|
1516 | } |
---|
1517 | |
---|
1518 | } |
---|
1519 | } |
---|
1520 | |
---|
1521 | if ( rroot == 1 ) // add root to list ? |
---|
1522 | { |
---|
1523 | if ( size(nares) > 0 ) |
---|
1524 | { |
---|
1525 | nares=nares[1..size(nares)],lsr[i]; |
---|
1526 | } |
---|
1527 | else |
---|
1528 | { |
---|
1529 | nares=lsr[i]; |
---|
1530 | } |
---|
1531 | if ( pdebug>=2 ) |
---|
1532 | { "// 18 added root to list nares"; } |
---|
1533 | } |
---|
1534 | } |
---|
1535 | |
---|
1536 | nares_size=size(nares); |
---|
1537 | if ( nares_size == 0 ) |
---|
1538 | { |
---|
1539 | "Numerical problem: No root found..."; |
---|
1540 | "Output may be incorrect!"; |
---|
1541 | nares=@ln; |
---|
1542 | } |
---|
1543 | |
---|
1544 | if ( pdebug>=1 ) |
---|
1545 | { "// 20 found <"+string(size(nares))+"> roots"; } |
---|
1546 | |
---|
1547 | for ( i= 1; i <= nares_size; i++ ) |
---|
1548 | { |
---|
1549 | resl[nv-d]= nares[i]; |
---|
1550 | |
---|
1551 | if ( dd < elem ) |
---|
1552 | { |
---|
1553 | if ( i > 1 ) |
---|
1554 | { |
---|
1555 | rn@++; |
---|
1556 | } |
---|
1557 | psubst( d+1, dd+1, n-1, resl, fi, elem, nv, prec ); |
---|
1558 | } |
---|
1559 | else |
---|
1560 | { |
---|
1561 | if ( i > 1 ) { rn@++; } //bug found by O.Labs |
---|
1562 | if ( pdebug>=1 ) |
---|
1563 | {"// 30_1 <"+string(rn@)+"> "+string(size(resl))+" <-----";} |
---|
1564 | if ( pdebug>=2 ) |
---|
1565 | { resl; } |
---|
1566 | rlist[rn@]=resl; |
---|
1567 | } |
---|
1568 | } |
---|
1569 | } |
---|
1570 | else |
---|
1571 | { |
---|
1572 | if ( pdebug>=2 ) |
---|
1573 | { "// 21 found root to be: "+string(lsr[1]); } |
---|
1574 | resl[nv-d]= lsr[1]; |
---|
1575 | |
---|
1576 | if ( dd < elem ) |
---|
1577 | { |
---|
1578 | psubst( d+1, dd+1, n-1, resl, fi, elem, nv, prec ); |
---|
1579 | } |
---|
1580 | else |
---|
1581 | { |
---|
1582 | if ( pdebug>=1 ) |
---|
1583 | { "// 30_2 <"+string(rn@)+"> "+string(size(resl))+" <-----";} |
---|
1584 | if ( pdebug>=2 ) |
---|
1585 | { resl; } |
---|
1586 | rlist[rn@]=resl; |
---|
1587 | } |
---|
1588 | } |
---|
1589 | } |
---|
1590 | } |
---|
1591 | |
---|
1592 | /////////////////////////////////////////////////////////////////////////////// |
---|
1593 | |
---|
1594 | proc fglm_solve( ideal fi, list # ) |
---|
1595 | "USAGE: fglm_solve(i [, p] ); i ideal, p integer |
---|
1596 | ASSUME: the ground field has char 0. |
---|
1597 | RETURN: a list of numbers, the complex roots of i; |
---|
1598 | p>0: gives precision of complex numbers in decimal digits (default: |
---|
1599 | p=30). |
---|
1600 | NOTE: The procedure uses a standard basis of i to determine all complex |
---|
1601 | roots of i. |
---|
1602 | It creates a ring rC with the same number of variables but with |
---|
1603 | complex coefficients (and precision p). |
---|
1604 | EXAMPLE: example fglm_solve; shows an example |
---|
1605 | " |
---|
1606 | { |
---|
1607 | int prec=30; |
---|
1608 | |
---|
1609 | if ( size(#)>=1 && typeof(#[1])=="int") |
---|
1610 | { |
---|
1611 | prec=#[1]; |
---|
1612 | } |
---|
1613 | |
---|
1614 | lex_solve(stdfglm(fi),prec); |
---|
1615 | keepring basering; |
---|
1616 | } |
---|
1617 | example |
---|
1618 | { |
---|
1619 | "EXAMPLE:";echo=2; |
---|
1620 | ring r = 0,(x,y),lp; |
---|
1621 | // compute the intersection points of two curves |
---|
1622 | ideal s= x2 + y2 - 10, x2 + xy + 2y2 - 16; |
---|
1623 | fglm_solve(s,10); |
---|
1624 | rlist; |
---|
1625 | } |
---|
1626 | |
---|
1627 | /////////////////////////////////////////////////////////////////////////////// |
---|
1628 | |
---|
1629 | proc lex_solve( ideal fi, list # ) |
---|
1630 | "USAGE: lex_solve( i[,p] ); i=ideal, p=integer, |
---|
1631 | @format |
---|
1632 | p>0: gives precision of complex numbers in decimal digits (default: p=30). |
---|
1633 | @end format |
---|
1634 | ASSUME: i is a reduced lexicographical Groebner bases of a zero-dimensional |
---|
1635 | ideal, sorted by increasing leading terms. |
---|
1636 | RETURN: nothing |
---|
1637 | CREATE: The procedure creates a complec ring with the same variables but |
---|
1638 | with complex coefficients (and precision p). |
---|
1639 | In this ring a list rlist of numbers is created, in which the complex |
---|
1640 | roots of i are stored. |
---|
1641 | EXAMPLE: example lex_solve; shows an example |
---|
1642 | " |
---|
1643 | { |
---|
1644 | int prec=30; |
---|
1645 | |
---|
1646 | if ( size(#)>=1 && typeof(#[1])=="int") |
---|
1647 | { |
---|
1648 | prec=#[1]; |
---|
1649 | } |
---|
1650 | |
---|
1651 | if ( !defined(pdebug) ) |
---|
1652 | { |
---|
1653 | int pdebug; |
---|
1654 | pdebug=0; |
---|
1655 | export pdebug; |
---|
1656 | } |
---|
1657 | |
---|
1658 | string orings= nameof(basering); |
---|
1659 | def oring= basering; |
---|
1660 | |
---|
1661 | // change the ground field to complex numbers |
---|
1662 | string nrings= "ring "+orings+"C=(complex,"+string(prec) |
---|
1663 | +"),("+varstr(basering)+"),lp;"; |
---|
1664 | execute(nrings); |
---|
1665 | |
---|
1666 | if ( pdebug>=0 ) |
---|
1667 | { "// name of new ring: "+string(nameof(basering));} |
---|
1668 | |
---|
1669 | // map fi from old to new ring |
---|
1670 | ideal fi= imap(oring,fi); |
---|
1671 | |
---|
1672 | // list with entry 0 (number) |
---|
1673 | number nn=0; |
---|
1674 | if ( !defined(@ln) ) |
---|
1675 | { |
---|
1676 | list @ln; |
---|
1677 | export @ln; |
---|
1678 | } |
---|
1679 | @ln=nn; |
---|
1680 | |
---|
1681 | int idelem= size(fi); |
---|
1682 | int nv= nvars(basering); |
---|
1683 | int i,j,k,lis; |
---|
1684 | list resl,li; |
---|
1685 | |
---|
1686 | if ( !defined(rlist) ) |
---|
1687 | { |
---|
1688 | list rlist; |
---|
1689 | export rlist; |
---|
1690 | } |
---|
1691 | |
---|
1692 | if ( !defined(rn@) ) |
---|
1693 | { |
---|
1694 | int rn@; |
---|
1695 | export rn@; |
---|
1696 | } |
---|
1697 | rn@=0; |
---|
1698 | |
---|
1699 | li= laguerre_solve(fi[1],prec,prec,0); |
---|
1700 | lis= size(li); |
---|
1701 | |
---|
1702 | if ( pdebug>=1 ) |
---|
1703 | {"// laguerre found roots: "+string(size(li));} |
---|
1704 | |
---|
1705 | for ( j= 1; j <= lis; j++ ) |
---|
1706 | { |
---|
1707 | if ( pdebug>=1 ) |
---|
1708 | {"// root "+string(j);} |
---|
1709 | rn@++; |
---|
1710 | resl[nv]= li[j]; |
---|
1711 | psubst( 1, 2, nv-1, resl, fi, idelem, nv, prec ); |
---|
1712 | } |
---|
1713 | |
---|
1714 | if ( pdebug>=0 ) |
---|
1715 | {"// list of roots: "+nameof(rlist);} |
---|
1716 | |
---|
1717 | // keep the ring and exit |
---|
1718 | keepring basering; |
---|
1719 | } |
---|
1720 | example |
---|
1721 | { |
---|
1722 | "EXAMPLE:";echo=2; |
---|
1723 | ring r = 0,(x,y),lp; |
---|
1724 | // compute the intersection points of two curves |
---|
1725 | ideal s= x2 + y2 - 10, x2 + xy + 2y2 - 16; |
---|
1726 | lex_solve(stdfglm(s),10); |
---|
1727 | rlist; |
---|
1728 | } |
---|
1729 | |
---|
1730 | /////////////////////////////////////////////////////////////////////////////// |
---|
1731 | |
---|
1732 | proc triangLf_solve( ideal fi, list # ) |
---|
1733 | "USAGE: triangLf_solve(i [, p] ); i ideal, p integer, |
---|
1734 | p>0: gives precision of complex numbers in digits (default: p=30). |
---|
1735 | ASSUME: the ground field has char 0; i is a zero-dimensional ideal |
---|
1736 | RETURN: nothing |
---|
1737 | CREATE: The procedure creates a ring rC with the same number of variables but |
---|
1738 | with complex coefficients (and precision p).@* |
---|
1739 | In rC a list rlist of numbers is created, in which the complex |
---|
1740 | roots of i are stored.@* |
---|
1741 | The proc uses a triangular system (Lazard's Algorithm with |
---|
1742 | factorization) computed from a standard basis to determine recursively |
---|
1743 | all complex roots with Laguerre's algorithm of input ideal i. |
---|
1744 | EXAMPLE: example triangLf_solve; shows an example |
---|
1745 | " |
---|
1746 | { |
---|
1747 | int prec=30; |
---|
1748 | |
---|
1749 | if ( size(#)>=1 && typeof(#[1])=="int") |
---|
1750 | { |
---|
1751 | prec=#[1]; |
---|
1752 | } |
---|
1753 | |
---|
1754 | triang_solve(triangLfak(stdfglm(fi)),prec); |
---|
1755 | keepring basering; |
---|
1756 | } |
---|
1757 | example |
---|
1758 | { |
---|
1759 | "EXAMPLE:";echo=2; |
---|
1760 | ring r = 0,(x,y),lp; |
---|
1761 | // compute the intersection points of two curves |
---|
1762 | ideal s= x2 + y2 - 10, x2 + xy + 2y2 - 16; |
---|
1763 | triangLf_solve(s,10); |
---|
1764 | rlist; |
---|
1765 | } |
---|
1766 | |
---|
1767 | /////////////////////////////////////////////////////////////////////////////// |
---|
1768 | |
---|
1769 | proc triangM_solve( ideal fi, list # ) |
---|
1770 | "USAGE: triangM_solve(i [, p ] ); i=ideal, p=integer, |
---|
1771 | p>0: gives precision of complex numbers in digits (default: p=30). |
---|
1772 | ASSUME: the ground field has char 0;@* |
---|
1773 | i zero-dimensional ideal |
---|
1774 | RETURN: nothing |
---|
1775 | CREATE: The procedure creates a ring rC with the same number of variables but |
---|
1776 | with complex coefficients (and precision p).@* |
---|
1777 | In rC a list rlist of numbers is created, in which the complex |
---|
1778 | roots of i are stored.@* |
---|
1779 | The proc uses a triangular system (Moellers Algorithm) computed from a |
---|
1780 | standard basis to determine recursively all complex roots with |
---|
1781 | Laguerre's algorithm of input ideal i. |
---|
1782 | EXAMPLE: example triangM_solve; shows an example |
---|
1783 | " |
---|
1784 | { |
---|
1785 | int prec=30; |
---|
1786 | |
---|
1787 | if ( size(#)>=1 && typeof(#[1])=="int") |
---|
1788 | { |
---|
1789 | prec=#[1]; |
---|
1790 | } |
---|
1791 | |
---|
1792 | triang_solve(triangM(stdfglm(fi)),prec); |
---|
1793 | keepring basering; |
---|
1794 | } |
---|
1795 | example |
---|
1796 | { |
---|
1797 | "EXAMPLE:";echo=2; |
---|
1798 | ring r = 0,(x,y),lp; |
---|
1799 | // compute the intersection points of two curves |
---|
1800 | ideal s= x2 + y2 - 10, x2 + xy + 2y2 - 16; |
---|
1801 | triangM_solve(s,10); |
---|
1802 | rlist; |
---|
1803 | } |
---|
1804 | |
---|
1805 | /////////////////////////////////////////////////////////////////////////////// |
---|
1806 | |
---|
1807 | proc triangL_solve( ideal fi, list # ) |
---|
1808 | "USAGE: triangL_solve(i [, p] ); i=ideal, p=integer,@* |
---|
1809 | p>0: gives precision of complex numbers in digits (default: p=30). |
---|
1810 | ASSUME: the ground field has char 0; i is a zero-dimensional ideal. |
---|
1811 | RETURN: nothing |
---|
1812 | CREATE: The procedure creates a ring rC with the same number of variables but |
---|
1813 | with complex coefficients (and precision p).@* |
---|
1814 | In rC a list rlist of numbers is created, in which the complex |
---|
1815 | roots of i are stored.@* |
---|
1816 | The proc uses a triangular system (Lazard's Algorithm) computed from |
---|
1817 | a standard basis to determine recursively all complex roots with |
---|
1818 | Laguerre's algorithm of input ideal i. |
---|
1819 | EXAMPLE: example triangL_solve; shows an example |
---|
1820 | " |
---|
1821 | { |
---|
1822 | int prec=30; |
---|
1823 | |
---|
1824 | if ( size(#)>=1 && typeof(#[1])=="int") |
---|
1825 | { |
---|
1826 | prec=#[1]; |
---|
1827 | } |
---|
1828 | |
---|
1829 | triang_solve(triangL(stdfglm(fi)),prec); |
---|
1830 | keepring basering; |
---|
1831 | } |
---|
1832 | example |
---|
1833 | { |
---|
1834 | "EXAMPLE:";echo=2; |
---|
1835 | ring r = 0,(x,y),lp; |
---|
1836 | // compute the intersection points of two curves |
---|
1837 | ideal s= x2 + y2 - 10, x2 + xy + 2y2 - 16; |
---|
1838 | triangL_solve(s,10); |
---|
1839 | rlist; |
---|
1840 | } |
---|
1841 | |
---|
1842 | |
---|
1843 | /////////////////////////////////////////////////////////////////////////////// |
---|
1844 | |
---|
1845 | proc triang_solve( list lfi, int prec, list # ) |
---|
1846 | "USAGE: triang_solve(l,p [, d] ); l=list, p,d=integers,@* |
---|
1847 | l a list of finitely many triangular systems, such that the union of |
---|
1848 | their varieties equals the variety of the initial ideal.@* |
---|
1849 | p>0: gives precision of complex numbers in digits,@* |
---|
1850 | d>0: gives precision (1<d<p) for near-zero-determination,@* |
---|
1851 | (default: d=1/2*p). |
---|
1852 | ASSUME: the ground field has char 0;@* |
---|
1853 | l was computed using Algorithm of Lazard or Algorithm of Moeller |
---|
1854 | (see triang.lib). |
---|
1855 | RETURN: nothing |
---|
1856 | CREATE: The procedure creates a ring rC with the same number of variables but |
---|
1857 | with complex coefficients (and precision p).@* |
---|
1858 | In rC a list rlist of numbers is created, in which the complex |
---|
1859 | roots of i are stored.@* |
---|
1860 | EXAMPLE: example triang_solve; shows an example |
---|
1861 | " |
---|
1862 | { |
---|
1863 | if ( !defined(pdebug) ) |
---|
1864 | { |
---|
1865 | int pdebug; |
---|
1866 | export pdebug; |
---|
1867 | } |
---|
1868 | pdebug=0; |
---|
1869 | |
---|
1870 | string orings= nameof(basering); |
---|
1871 | def oring= basering; |
---|
1872 | |
---|
1873 | // change the ground field to complex numbers |
---|
1874 | string nrings= "ring "+orings+"C=(real,"+string(prec) |
---|
1875 | +",I),("+varstr(basering)+"),lp;"; |
---|
1876 | execute(nrings); |
---|
1877 | |
---|
1878 | if ( pdebug>=0 ) |
---|
1879 | { "// name of new ring: "+string(nameof(basering));} |
---|
1880 | |
---|
1881 | // list with entry 0 (number) |
---|
1882 | number nn=0; |
---|
1883 | if ( !defined(@ln) ) |
---|
1884 | { |
---|
1885 | list @ln; |
---|
1886 | export @ln; |
---|
1887 | } |
---|
1888 | @ln=nn; |
---|
1889 | |
---|
1890 | // set number of digits for zero-comparison of roots |
---|
1891 | if ( !defined(myCompDigits) ) |
---|
1892 | { |
---|
1893 | int myCompDigits; |
---|
1894 | export myCompDigits; |
---|
1895 | } |
---|
1896 | if ( size(#)>=1 && typeof(#[1])=="int" ) |
---|
1897 | { |
---|
1898 | myCompDigits=#[1]; |
---|
1899 | } |
---|
1900 | else |
---|
1901 | { |
---|
1902 | myCompDigits=(system("getPrecDigits")); |
---|
1903 | } |
---|
1904 | |
---|
1905 | if ( pdebug>=1 ) |
---|
1906 | {"// myCompDigits="+string(myCompDigits);} |
---|
1907 | |
---|
1908 | int idelem; |
---|
1909 | int nv= nvars(basering); |
---|
1910 | int i,j,k,lis; |
---|
1911 | list resu,li; |
---|
1912 | |
---|
1913 | if ( !defined(rlist) ) |
---|
1914 | { |
---|
1915 | list rlist; |
---|
1916 | export rlist; |
---|
1917 | } |
---|
1918 | |
---|
1919 | if ( !defined(rn@) ) |
---|
1920 | { |
---|
1921 | int rn@; |
---|
1922 | export rn@; |
---|
1923 | } |
---|
1924 | rn@=0; |
---|
1925 | |
---|
1926 | // map the list |
---|
1927 | list lfi= imap(oring,lfi); |
---|
1928 | |
---|
1929 | int slfi= size(lfi); |
---|
1930 | ideal fi; |
---|
1931 | |
---|
1932 | for ( i= 1; i <= slfi; i++ ) |
---|
1933 | { |
---|
1934 | // map fi from old to new ring |
---|
1935 | fi= lfi[i]; //imap(oring,lfi[i]); |
---|
1936 | |
---|
1937 | idelem= size(fi); |
---|
1938 | |
---|
1939 | // solve fi[1] |
---|
1940 | li= laguerre_solve(fi[1],myCompDigits,myCompDigits,0); |
---|
1941 | lis= size(li); |
---|
1942 | |
---|
1943 | if ( pdebug>=1 ) |
---|
1944 | {"// laguerre found roots: "+string(size(li));} |
---|
1945 | |
---|
1946 | for ( j= 1; j <= lis; j++ ) |
---|
1947 | { |
---|
1948 | if ( pdebug>=1 ) |
---|
1949 | {"// root "+string(j);} |
---|
1950 | rn@++; |
---|
1951 | resu[nv]= li[j]; |
---|
1952 | psubst( 1, 2, nv-1, resu, fi, idelem, nv, myCompDigits ); |
---|
1953 | } |
---|
1954 | } |
---|
1955 | |
---|
1956 | if ( pdebug>=0 ) |
---|
1957 | {"// list of roots: "+nameof(rlist);} |
---|
1958 | // keep the ring and exit |
---|
1959 | keepring basering; |
---|
1960 | } |
---|
1961 | example |
---|
1962 | { |
---|
1963 | "EXAMPLE:";echo=2; |
---|
1964 | ring r = 0,(x,y),lp; |
---|
1965 | // compute the intersection points of two curves |
---|
1966 | ideal s= x2 + y2 - 10, x2 + xy + 2y2 - 16; |
---|
1967 | triang_solve(triangLfak(stdfglm(s)),10); |
---|
1968 | rlist; |
---|
1969 | } |
---|
1970 | |
---|
1971 | /////////////////////////////////////////////////////////////////////////////// |
---|
1972 | |
---|
1973 | proc simplexOut(list l) |
---|
1974 | "USAGE: simplexOut(l); l list |
---|
1975 | ASSUME: l is the output of simplex. |
---|
1976 | RETURN: nothing. The procedure prints the computed solution of simplex |
---|
1977 | (as strings) in a nice format. |
---|
1978 | SEE ALSO: simplex |
---|
1979 | EXAMPLE: example simplexOut; shows an example |
---|
1980 | " |
---|
1981 | { |
---|
1982 | int i,j; |
---|
1983 | matrix m= l[1]; |
---|
1984 | intvec iposv= l[3]; |
---|
1985 | int icase= l[2]; |
---|
1986 | |
---|
1987 | int cols= ncols(m); |
---|
1988 | int rows= nrows(m); |
---|
1989 | |
---|
1990 | int N= l[6]; |
---|
1991 | |
---|
1992 | if ( 1 == icase ) // objective function is unbound |
---|
1993 | { |
---|
1994 | "objective function is unbound"; |
---|
1995 | return(); |
---|
1996 | } |
---|
1997 | if ( -1 == icase ) // no solution satisfies the given constraints |
---|
1998 | { |
---|
1999 | "no solution satisfies the given constraints"; |
---|
2000 | return(); |
---|
2001 | } |
---|
2002 | if ( -2 == icase ) // other error |
---|
2003 | { |
---|
2004 | "an error occurred during simplex computation!"; |
---|
2005 | return(); |
---|
2006 | } |
---|
2007 | |
---|
2008 | for ( i = 1; i <= rows; i++ ) |
---|
2009 | { |
---|
2010 | if (i == 1) |
---|
2011 | { |
---|
2012 | "z = "+string(m[1][1]); |
---|
2013 | } |
---|
2014 | else |
---|
2015 | { |
---|
2016 | if ( iposv[i-1] <= N ) |
---|
2017 | { |
---|
2018 | "x"+string(iposv[i-1])+" = "+string(m[i,1]); |
---|
2019 | } |
---|
2020 | // else |
---|
2021 | // { |
---|
2022 | // "Y"; iposv[i-1]-N+1; |
---|
2023 | // } |
---|
2024 | } |
---|
2025 | } |
---|
2026 | } |
---|
2027 | example |
---|
2028 | { |
---|
2029 | "EXAMPLE:";echo=2; |
---|
2030 | ring r = (real,10),(x),lp; |
---|
2031 | |
---|
2032 | // consider the max. problem: |
---|
2033 | // |
---|
2034 | // maximize x(1) + x(2) + 3*x(3) - 0.5*x(4) |
---|
2035 | // |
---|
2036 | // with constraints: x(1) + 2*x(3) <= 740 |
---|
2037 | // 2*x(2) - 7*x(4) <= 0 |
---|
2038 | // x(2) - x(3) + 2*x(4) >= 0.5 |
---|
2039 | // x(1) + x(2) + x(3) + x(4) = 9 |
---|
2040 | // |
---|
2041 | matrix sm[5][5]= 0, 1, 1, 3,-0.5, |
---|
2042 | 740,-1, 0,-2, 0, |
---|
2043 | 0, 0,-2, 0, 7, |
---|
2044 | 0.5, 0,-1, 1,-2, |
---|
2045 | 9,-1,-1,-1,-1; |
---|
2046 | |
---|
2047 | int n = 4; // number of constraints |
---|
2048 | int m = 4; // number of variables |
---|
2049 | int m1= 2; // number of <= constraints |
---|
2050 | int m2= 1; // number of >= constraints |
---|
2051 | int m3= 1; // number of == constraints |
---|
2052 | |
---|
2053 | list sol=simplex(sm, n, m, m1, m2, m3); |
---|
2054 | simplexOut(sol); |
---|
2055 | } |
---|
2056 | |
---|
2057 | |
---|
2058 | // local Variables: *** |
---|
2059 | // c-set-style: bsd *** |
---|
2060 | // End: *** |
---|