1 | /* emacs edit mode for this file is -*- C++ -*- */ |
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2 | /* $Id: cf_ops.cc,v 1.8 1997-09-29 16:07:28 schmidt Exp $ */ |
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3 | |
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4 | //{{{ docu |
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5 | // |
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6 | // cf_ops.cc - simple structural algorithms. |
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7 | // |
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8 | // A 'structural' algorithm is an algorithm which gives |
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9 | // structural information on polynomials in contrast to a |
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10 | // 'mathematical' algorithm which calculates some mathematical |
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11 | // function. |
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12 | // |
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13 | // Compare these functions with the functions in cf_algorithm.cc, |
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14 | // which are mathematical algorithms. |
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15 | // |
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16 | // Used by: allmost everywhere |
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17 | // |
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18 | // Header file: canonicalform.h |
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19 | // |
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20 | //}}} |
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21 | |
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22 | #include <config.h> |
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23 | |
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24 | #include "assert.h" |
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25 | |
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26 | #include "canonicalform.h" |
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27 | #include "variable.h" |
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28 | #include "cf_iter.h" |
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29 | |
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30 | //{{{ static Variable sv_x1, sv_x2; |
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31 | //{{{ docu |
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32 | // |
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33 | // sv_x1, sv_x2 - variables to swap by swapvar() and replacevar. |
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34 | // |
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35 | // These variables are initialized by swapvar() such that sv_x1 < |
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36 | // sv_x2. They are used by swapvar_between() and swapvar_rec() |
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37 | // to swap variables efficiently. |
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38 | // Furthermore, sv_x1 and sv_x2 are used by replacevar() and |
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39 | // replacevar_between(). |
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40 | // |
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41 | //}}} |
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42 | static Variable sv_x1, sv_x2; |
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43 | //}}} |
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44 | |
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45 | //{{{ static void swapvar_between ( const CanonicalForm & f, CanonicalForm & result, const CanonicalForm & term, int expx2 ) |
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46 | //{{{ docu |
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47 | // |
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48 | // swapvar_between() - replace occurences of sv_x1 in f with sv_x2. |
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49 | // |
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50 | // If Psi denotes the map which maps sv_x1 to sv_x2, this |
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51 | // function returns |
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52 | // |
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53 | // result + Psi(f) * term * sv_x1^expx2 |
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54 | // |
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55 | // Used by: swapvar() |
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56 | // |
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57 | //}}} |
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58 | static void |
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59 | swapvar_between ( const CanonicalForm & f, CanonicalForm & result, const CanonicalForm & term, int expx2 ) |
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60 | { |
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61 | if ( f.inCoeffDomain() || f.mvar() < sv_x1 ) |
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62 | // in this case, we do not have to replace anything |
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63 | result += term * power( sv_x1, expx2 ) * f; |
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64 | else if ( f.mvar() == sv_x1 ) |
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65 | // this is where the real work is done: this iterator |
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66 | // replaces sv_x1 with sv_x2 |
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67 | for ( CFIterator i = f; i.hasTerms(); i++ ) |
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68 | result += power( sv_x2, i.exp() ) * term * power( sv_x1, expx2 ) * i.coeff(); |
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69 | else |
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70 | // f's level is larger than sv_x1: descend down |
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71 | for ( CFIterator i = f; i.hasTerms(); i++ ) |
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72 | swapvar_between( i.coeff(), result, term * power( f.mvar(), i.exp() ), expx2 ); |
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73 | } |
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74 | static CanonicalForm |
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75 | swapvar_between1 ( const CanonicalForm & f ) |
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76 | { |
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77 | if ( f.inCoeffDomain() || f.mvar() < sv_x1 ) |
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78 | // in this case, we do not have to replace anything |
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79 | return f; |
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80 | else if ( f.mvar() == sv_x1 ) { |
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81 | // this is where the real work is done: this iterator |
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82 | // replaces sv_x1 with sv_x2 |
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83 | CanonicalForm result; |
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84 | for ( CFIterator i = f; i.hasTerms(); i++ ) |
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85 | result += power( sv_x2, i.exp() ) * i.coeff(); |
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86 | return result; |
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87 | } |
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88 | else { |
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89 | // f's level is larger than sv_x1: descend down |
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90 | CanonicalForm result; |
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91 | for ( CFIterator i = f; i.hasTerms(); i++ ) |
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92 | result += swapvar_between1( i.coeff() ) * power( f.mvar(), i.exp() ); |
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93 | return result; |
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94 | } |
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95 | } |
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96 | //}}} |
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97 | |
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98 | //{{{ static void swapvar_rec ( const CanonicalForm & f, CanonicalForm & result, const CanonicalForm & term ) |
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99 | //{{{ docu |
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100 | // |
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101 | // swapvar_between() - swap occurences of sv_x1 and sv_x2 in f. |
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102 | // |
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103 | // If Psi denotes the map which swaps sv_x1 and sv_x2, this |
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104 | // function returns |
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105 | // |
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106 | // result + Psi(f) * term |
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107 | // |
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108 | // Used by: swapvar() |
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109 | // |
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110 | //}}} |
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111 | static void |
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112 | swapvar_rec ( const CanonicalForm & f, CanonicalForm & result, const CanonicalForm & term ) |
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113 | { |
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114 | if ( f.inCoeffDomain() || f.mvar() < sv_x1 ) |
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115 | // in this case, we do not have to swap anything |
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116 | result += term * f; |
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117 | else if ( f.mvar() == sv_x2 ) |
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118 | // this is where the real work is done: this iterator |
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119 | // replaces sv_x1 with sv_x2 in the coefficients of f and |
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120 | // remembers the exponents of sv_x2 in the last argument |
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121 | // of the call to swapvar_between() |
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122 | for ( CFIterator i = f; i.hasTerms(); i++ ) |
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123 | swapvar_between( i.coeff(), result, term, i.exp() ); |
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124 | else if ( f.mvar() < sv_x2 ) |
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125 | // sv_x2 does not occur in f, but sv_x1 does. Replace it. |
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126 | swapvar_between( f, result, term, 0 ); |
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127 | else |
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128 | // f's level is larger than sv_x2: descend down |
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129 | for ( CFIterator i = f; i.hasTerms(); i++ ) |
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130 | swapvar_rec( i.coeff(), result, term * power( f.mvar(), i.exp() ) ); |
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131 | } |
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132 | static CanonicalForm |
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133 | swapvar_rec1 ( const CanonicalForm & f ) |
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134 | { |
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135 | if ( f.inCoeffDomain() || f.mvar() < sv_x1 ) |
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136 | return f; |
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137 | else if ( f.mvar() == sv_x2 ) { |
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138 | CanonicalForm result; |
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139 | for ( CFIterator i = f; i.hasTerms(); i++ ) |
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140 | result += swapvar_between1( i.coeff() ) * power( sv_x1, i.exp() ); |
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141 | return result; |
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142 | } |
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143 | else if ( f.mvar() < sv_x2 ) |
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144 | return swapvar_between1( f ); |
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145 | else { |
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146 | CanonicalForm result; |
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147 | for ( CFIterator i = f; i.hasTerms(); i++ ) |
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148 | result += swapvar_rec1( i.coeff() ) * power( f.mvar(), i.exp() ); |
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149 | return result; |
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150 | } |
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151 | } |
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152 | //}}} |
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153 | |
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154 | //{{{ CanonicalForm swapvar ( const CanonicalForm & f, const Variable & x1, const Variable & x2 ) |
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155 | //{{{ docu |
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156 | // |
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157 | // swapvar() - swap variables x1 and x2 in f. |
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158 | // |
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159 | // Returns the image of f under the map which maps x1 to x2 and |
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160 | // x2 to x1. This is done quite efficiently because it is used |
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161 | // really often. x1 and x2 should be polynomial variables. |
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162 | // |
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163 | //}}} |
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164 | CanonicalForm |
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165 | swapvar ( const CanonicalForm & f, const Variable & x1, const Variable & x2 ) |
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166 | { |
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167 | ASSERT( x1.level() > 0 && x2.level() > 0, "cannot swap algebraic Variables" ); |
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168 | if ( f.inCoeffDomain() || x1 == x2 || ( x1 > f.mvar() && x2 > f.mvar() ) ) |
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169 | return f; |
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170 | else { |
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171 | CanonicalForm result = 0; |
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172 | if ( x1 > x2 ) { |
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173 | sv_x1 = x2; sv_x2 = x1; |
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174 | } |
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175 | else { |
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176 | sv_x1 = x1; sv_x2 = x2; |
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177 | } |
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178 | if ( f.mvar() < sv_x2 ) |
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179 | // we only have to replace sv_x1 by sv_x2 |
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180 | swapvar_between( f, result, 1, 0 ); |
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181 | else |
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182 | // we really have to swap variables |
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183 | swapvar_rec( f, result, 1 ); |
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184 | return result; |
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185 | } |
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186 | } |
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187 | CanonicalForm |
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188 | swapvar1 ( const CanonicalForm & f, const Variable & x1, const Variable & x2 ) |
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189 | { |
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190 | ASSERT( x1.level() > 0 && x2.level() > 0, "cannot swap algebraic variables" ); |
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191 | if ( f.inCoeffDomain() || x1 == x2 || ( x1 > f.mvar() && x2 > f.mvar() ) ) |
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192 | return f; |
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193 | else { |
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194 | CanonicalForm result = 0; |
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195 | if ( x1 > x2 ) { |
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196 | sv_x1 = x2; sv_x2 = x1; |
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197 | } |
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198 | else { |
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199 | sv_x1 = x1; sv_x2 = x2; |
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200 | } |
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201 | if ( f.mvar() < sv_x2 ) |
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202 | // we only have to replace sv_x1 by sv_x2 |
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203 | return swapvar_between1( f ); |
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204 | else |
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205 | // we really have to swap variables |
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206 | return swapvar_rec1( f ); |
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207 | } |
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208 | } |
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209 | //}}} |
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210 | |
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211 | //{{{ static CanonicalForm replacevar_between ( const CanonicalForm & f ) |
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212 | //{{{ docu |
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213 | // |
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214 | // replacevar_between() - replace occurences of sv_x1 in f with sv_x2. |
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215 | // |
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216 | // This is allmost the same as swapvar_between() except that |
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217 | // sv_x1 may be an algebraic variable, so we have to test on |
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218 | // 'f.inBaseDomain()' instead of 'f.inCoeffDomain()' in the |
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219 | // beginning. |
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220 | // |
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221 | // Used by: replacevar() |
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222 | // |
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223 | //}}} |
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224 | static CanonicalForm |
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225 | replacevar_between ( const CanonicalForm & f ) |
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226 | { |
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227 | if ( f.inBaseDomain() ) |
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228 | return f; |
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229 | |
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230 | Variable x = f.mvar(); |
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231 | |
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232 | if ( x < sv_x1 ) |
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233 | // in this case, we do not have to replace anything |
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234 | return f; |
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235 | else if ( x == sv_x1 ) { |
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236 | // this is where the real work is done: this iterator |
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237 | // replaces sv_x1 with sv_x2 |
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238 | CanonicalForm result; |
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239 | for ( CFIterator i = f; i.hasTerms(); i++ ) |
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240 | result += power( sv_x2, i.exp() ) * i.coeff(); |
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241 | return result; |
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242 | } |
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243 | else { |
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244 | // f's level is larger than sv_x1: descend down |
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245 | CanonicalForm result; |
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246 | for ( CFIterator i = f; i.hasTerms(); i++ ) |
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247 | result += replacevar_between( i.coeff() ) * power( x, i.exp() ); |
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248 | return result; |
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249 | } |
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250 | } |
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251 | //}}} |
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252 | |
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253 | //{{{ CanonicalForm replacevar ( const CanonicalForm & f, const Variable & x1, const Variable & x2 ) |
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254 | //{{{ docu |
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255 | // |
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256 | // replacevar() - replace all occurences of x1 in f by x2. |
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257 | // |
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258 | // In contrast to swapvar(), x1 may be an algebraic variable, but |
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259 | // x2 must be a polynomial variable. |
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260 | // |
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261 | //}}} |
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262 | CanonicalForm |
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263 | replacevar ( const CanonicalForm & f, const Variable & x1, const Variable & x2 ) |
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264 | { |
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265 | ASSERT( x2.level() > 0, "cannot replace with algebraic variable" ); |
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266 | if ( f.inBaseDomain() || x1 == x2 || ( x1 > f.mvar() ) ) |
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267 | return f; |
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268 | else { |
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269 | sv_x1 = x1; |
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270 | sv_x2 = x2; |
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271 | return replacevar_between( f ); |
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272 | } |
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273 | } |
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274 | //}}} |
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275 | |
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276 | //{{{ static void fillVarsRec ( const CanonicalForm & f, int * vars ) |
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277 | //{{{ docu |
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278 | // |
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279 | // fillVarsRec - fill array describing occurences of variables in f. |
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280 | // |
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281 | // Only polynomial variables are looked up. The information is |
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282 | // stored in the arrary vars. vars should be large enough to |
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283 | // hold all information, i.e. larger than the level of f. |
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284 | // |
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285 | // Used by getVars() and getNumVars(). |
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286 | // |
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287 | //}}} |
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288 | static void |
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289 | fillVarsRec ( const CanonicalForm & f, int * vars ) |
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290 | { |
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291 | int n; |
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292 | if ( (n = f.level()) > 0 ) { |
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293 | vars[n] = 1; |
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294 | CFIterator i; |
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295 | for ( i = f; i.hasTerms(); ++i ) |
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296 | fillVarsRec( i.coeff(), vars ); |
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297 | } |
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298 | } |
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299 | //}}} |
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300 | |
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301 | //{{{ int getNumVars ( const CanonicalForm & f ) |
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302 | //{{{ docu |
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303 | // |
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304 | // getNumVars() - get number of polynomial variables in f. |
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305 | // |
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306 | //}}} |
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307 | int |
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308 | getNumVars ( const CanonicalForm & f ) |
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309 | { |
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310 | int n; |
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311 | if ( f.inCoeffDomain() ) |
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312 | return 0; |
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313 | else if ( (n = f.level()) == 1 ) |
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314 | return 1; |
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315 | else { |
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316 | int * vars = new int[ n+1 ]; |
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317 | int i; |
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318 | for ( i = 0; i < n; i++ ) vars[i] = 0; |
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319 | |
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320 | // look for variables |
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321 | for ( CFIterator I = f; I.hasTerms(); ++I ) |
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322 | fillVarsRec( I.coeff(), vars ); |
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323 | |
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324 | // count them |
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325 | int m = 0; |
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326 | for ( i = 1; i < n; i++ ) |
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327 | if ( vars[i] != 0 ) m++; |
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328 | |
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329 | delete [] vars; |
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330 | // do not forget to count our own variable |
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331 | return m+1; |
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332 | } |
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333 | } |
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334 | //}}} |
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335 | |
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336 | //{{{ CanonicalForm getVars ( const CanonicalForm & f ) |
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337 | //{{{ docu |
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338 | // |
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339 | // getVars() - get polynomial variables of f. |
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340 | // |
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341 | // Return the product of all of them, 1 if there are not any. |
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342 | // |
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343 | //}}} |
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344 | CanonicalForm |
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345 | getVars ( const CanonicalForm & f ) |
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346 | { |
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347 | int n; |
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348 | if ( f.inCoeffDomain() ) |
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349 | return 1; |
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350 | else if ( (n = f.level()) == 1 ) |
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351 | return Variable( 1 ); |
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352 | else { |
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353 | int * vars = new int[ n+1 ]; |
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354 | int i; |
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355 | for ( i = 0; i <= n; i++ ) vars[i] = 0; |
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356 | |
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357 | // look for variables |
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358 | for ( CFIterator I = f; I.hasTerms(); ++I ) |
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359 | fillVarsRec( I.coeff(), vars ); |
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360 | |
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361 | // multiply them all |
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362 | CanonicalForm result = 1; |
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363 | for ( i = n; i > 0; i-- ) |
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364 | if ( vars[i] != 0 ) result *= Variable( i ); |
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365 | |
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366 | delete [] vars; |
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367 | // do not forget our own variable |
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368 | return f.mvar() * result; |
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369 | } |
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370 | } |
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371 | //}}} |
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372 | |
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373 | //{{{ CanonicalForm apply ( const CanonicalForm & f, void (*mf)( CanonicalForm &, int & ) ) |
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374 | //{{{ docu |
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375 | // |
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376 | // apply() - apply mf to terms of f. |
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377 | // |
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378 | // Calls mf( f[i], i ) for each term f[i]*x^i of f and builds a |
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379 | // new term from the result. If f is in a coefficient domain, |
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380 | // mf( f, i ) should result in an i == 0, since otherwise it is |
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381 | // not clear which variable to use for the resulting term. |
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382 | // |
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383 | // An example: |
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384 | // |
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385 | // void |
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386 | // diff( CanonicalForm & f, int & i ) |
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387 | // { |
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388 | // f = f * i; |
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389 | // if ( i > 0 ) i--; |
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390 | // } |
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391 | // |
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392 | // Then apply( f, diff ) is differentation of f with respect to the |
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393 | // main variable of f. |
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394 | // |
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395 | //}}} |
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396 | CanonicalForm |
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397 | apply ( const CanonicalForm & f, void (*mf)( CanonicalForm &, int & ) ) |
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398 | { |
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399 | if ( f.inCoeffDomain() ) { |
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400 | int exp = 0; |
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401 | CanonicalForm result = f; |
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402 | mf( result, exp ); |
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403 | ASSERT( exp == 0, "illegal result, do not know what variable to use" ); |
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404 | return result; |
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405 | } |
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406 | else { |
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407 | CanonicalForm result, coeff; |
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408 | CFIterator i; |
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409 | int exp; |
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410 | Variable x = f.mvar(); |
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411 | for ( i = f; i.hasTerms(); i++ ) { |
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412 | coeff = i.coeff(); |
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413 | exp = i.exp(); |
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414 | mf( coeff, exp ); |
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415 | if ( ! coeff.isZero() ) |
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416 | result += power( x, exp ) * coeff; |
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417 | } |
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418 | return result; |
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419 | } |
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420 | } |
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421 | //}}} |
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422 | |
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423 | //{{{ CanonicalForm mapdomain ( const CanonicalForm & f, CanonicalForm (*mf)( const CanonicalForm & ) ) |
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424 | //{{{ docu |
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425 | // |
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426 | // mapdomain() - map all coefficients of f through mf. |
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427 | // |
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428 | // Recursively descends down through f to the coefficients which |
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429 | // are in a coefficient domain mapping each such coefficient |
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430 | // through mf and returns the result. |
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431 | // |
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432 | //}}} |
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433 | CanonicalForm |
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434 | mapdomain ( const CanonicalForm & f, CanonicalForm (*mf)( const CanonicalForm & ) ) |
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435 | { |
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436 | if ( f.inCoeffDomain() ) |
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437 | return mf( f ); |
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438 | else { |
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439 | CanonicalForm result = 0; |
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440 | CFIterator i; |
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441 | Variable x = f.mvar(); |
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442 | for ( i = f; i.hasTerms(); i++ ) |
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443 | result += power( x, i.exp() ) * mapdomain( i.coeff(), mf ); |
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444 | return result; |
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445 | } |
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446 | } |
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447 | //}}} |
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448 | |
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449 | //{{{ static void degreesRec ( const CanonicalForm & f, int * degs ) |
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450 | //{{{ docu |
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451 | // |
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452 | // degreesRec() - recursively get degrees of f. |
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453 | // |
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454 | // Used by degrees(). |
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455 | // |
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456 | //}}} |
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457 | static void |
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458 | degreesRec ( const CanonicalForm & f, int * degs ) |
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459 | { |
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460 | if ( ! f.inCoeffDomain() ) { |
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461 | int level = f.level(); |
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462 | int deg = f.degree(); |
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463 | // calculate the maximum degree of all coefficients which |
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464 | // are in the same level |
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465 | if ( degs[level] < deg ) |
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466 | degs[level] = f.degree(); |
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467 | for ( CFIterator i = f; i.hasTerms(); i++ ) |
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468 | degreesRec( i.coeff(), degs ); |
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469 | } |
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470 | } |
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471 | //}}} |
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472 | |
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473 | //{{{ int * degrees ( const CanonicalForm & f, int * degs ) |
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474 | //{{{ docu |
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475 | // |
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476 | // degress() - return the degrees of all polynomial variables in f. |
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477 | // |
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478 | // Returns 0 if f is in a coefficient domain, the degrees of f in |
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479 | // all its polynomial variables in an array of int otherwise: |
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480 | // |
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481 | // degrees( f, 0 )[i] = degree( f, Variable(i) ) |
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482 | // |
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483 | // If degs is not the zero pointer the degrees are stored in this |
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484 | // array. In this case degs should be larger than the level of |
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485 | // f. If degs is the zero pointer, an array of sufficient size |
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486 | // is allocated automatically. |
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487 | // |
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488 | //}}} |
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489 | int * |
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490 | degrees ( const CanonicalForm & f, int * degs ) |
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491 | { |
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492 | if ( f.inCoeffDomain() ) |
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493 | return 0; |
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494 | else { |
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495 | int level = f.level(); |
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496 | if ( degs == 0 ) |
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497 | degs = new int[level+1]; |
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498 | for ( int i = 0; i <= level; i++ ) |
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499 | degs[i] = 0; |
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500 | degreesRec( f, degs ); |
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501 | return degs; |
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502 | } |
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503 | } |
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504 | //}}} |
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505 | |
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506 | //{{{ int totaldegree ( const CanonicalForm & f ) |
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507 | //{{{ docu |
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508 | // |
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509 | // totaldegree() - return the total degree of f. |
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510 | // |
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511 | // If f is zero, return -1. If f is in a coefficient domain, |
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512 | // return 0. Otherwise return the total degree of f in all |
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513 | // polynomial variables. |
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514 | // |
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515 | //}}} |
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516 | int |
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517 | totaldegree ( const CanonicalForm & f ) |
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518 | { |
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519 | if ( f.isZero() ) |
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520 | return -1; |
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521 | else if ( f.inCoeffDomain() ) |
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522 | return 0; |
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523 | else { |
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524 | CFIterator i; |
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525 | int cdeg = 0, dummy; |
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526 | // calculate maximum over all coefficients of f, taking |
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527 | // in account our own exponent |
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528 | for ( i = f; i.hasTerms(); i++ ) |
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529 | if ( (dummy = totaldegree( i.coeff() ) + i.exp()) > cdeg ) |
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530 | cdeg = dummy; |
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531 | return cdeg; |
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532 | } |
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533 | } |
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534 | //}}} |
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535 | |
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536 | //{{{ int totaldegree ( const CanonicalForm & f, const Variable & v1, const Variable & v2 ) |
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537 | //{{{ docu |
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538 | // |
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539 | // totaldegree() - return the total degree of f as a polynomial |
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540 | // in the polynomial variables between v1 and v2 (inclusively). |
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541 | // |
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542 | // If f is zero, return -1. If f is in a coefficient domain, |
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543 | // return 0. Also, return 0 if v1 > v2. Otherwise, take f to be |
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544 | // a polynomial in the polynomial variables between v1 and v2 and |
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545 | // return its total degree. |
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546 | // |
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547 | //}}} |
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548 | int |
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549 | totaldegree ( const CanonicalForm & f, const Variable & v1, const Variable & v2 ) |
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550 | { |
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551 | if ( f.isZero() ) |
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552 | return -1; |
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553 | else if ( v1 > v2 ) |
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554 | return 0; |
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555 | else if ( f.inCoeffDomain() ) |
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556 | return 0; |
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557 | else if ( f.mvar() < v1 ) |
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558 | return 0; |
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559 | else if ( f.mvar() == v1 ) |
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560 | return f.degree(); |
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561 | else if ( f.mvar() > v2 ) { |
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562 | // v2's level is larger than f's level, descend down |
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563 | CFIterator i; |
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564 | int cdeg = 0, dummy; |
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565 | // calculate maximum over all coefficients of f |
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566 | for ( i = f; i.hasTerms(); i++ ) |
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567 | if ( (dummy = totaldegree( i.coeff(), v1, v2 )) > cdeg ) |
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568 | cdeg = dummy; |
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569 | return cdeg; |
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570 | } |
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571 | else { |
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572 | // v1 < f.mvar() <= v2 |
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573 | CFIterator i; |
---|
574 | int cdeg = 0, dummy; |
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575 | // calculate maximum over all coefficients of f, taking |
---|
576 | // in account our own exponent |
---|
577 | for ( i = f; i.hasTerms(); i++ ) |
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578 | if ( (dummy = totaldegree( i.coeff(), v1, v2 ) + i.exp()) > cdeg ) |
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579 | cdeg = dummy; |
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580 | return cdeg; |
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581 | } |
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582 | } |
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583 | //}}} |
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584 | |
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585 | //{{{ int size ( const CanonicalForm & f, const Variable & v ) |
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586 | //{{{ docu |
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587 | // |
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588 | // size() - count number of monomials of f with level higher |
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589 | // or equal than level of v. |
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590 | // |
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591 | // Returns one if f is in an base domain. |
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592 | // |
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593 | //}}} |
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594 | int |
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595 | size ( const CanonicalForm & f, const Variable & v ) |
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596 | { |
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597 | if ( f.inBaseDomain() ) |
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598 | return 1; |
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599 | |
---|
600 | int result = 0; |
---|
601 | CFIterator i; |
---|
602 | if ( f.mvar() < v ) |
---|
603 | // polynomials with level < v1 are counted as coefficients |
---|
604 | return 1; |
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605 | else { |
---|
606 | // polynomials with level > v2 are not counted al all |
---|
607 | for ( i = f; i.hasTerms(); i++ ) |
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608 | result += size( i.coeff(), v ); |
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609 | return result; |
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610 | } |
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611 | } |
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612 | //}}} |
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613 | |
---|
614 | //{{{ int size ( const CanonicalForm & f ) |
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615 | //{{{ docu |
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616 | // |
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617 | // size() - return number of monomials in f which are in an |
---|
618 | // coefficient domain. |
---|
619 | // |
---|
620 | // Returns one if f is in an coefficient domain. |
---|
621 | // |
---|
622 | //}}} |
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623 | int |
---|
624 | size ( const CanonicalForm & f ) |
---|
625 | { |
---|
626 | if ( f.inCoeffDomain() ) |
---|
627 | return 1; |
---|
628 | else { |
---|
629 | int result = 0; |
---|
630 | CFIterator i; |
---|
631 | for ( i = f; i.hasTerms(); i++ ) |
---|
632 | result += size( i.coeff() ); |
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633 | return result; |
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634 | } |
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635 | } |
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636 | //}}} |
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