1 | /* emacs edit mode for this file is -*- C++ -*- */ |
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2 | |
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3 | /** |
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4 | * @file cf_linsys.cc |
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5 | * |
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6 | * solve linear systems and compute determinants of matrices |
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7 | **/ |
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8 | |
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9 | #include "config.h" |
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10 | |
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11 | |
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12 | #include "cf_assert.h" |
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13 | #include "debug.h" |
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14 | #include "timing.h" |
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15 | |
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16 | #include "cf_defs.h" |
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17 | #include "cf_primes.h" |
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18 | #include "canonicalform.h" |
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19 | #include "cf_iter.h" |
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20 | #include "cf_algorithm.h" |
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21 | #include "ffops.h" |
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22 | #include "cf_primes.h" |
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23 | |
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24 | |
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25 | TIMING_DEFINE_PRINT(det_mapping) |
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26 | TIMING_DEFINE_PRINT(det_determinant) |
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27 | TIMING_DEFINE_PRINT(det_chinese) |
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28 | TIMING_DEFINE_PRINT(det_bound) |
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29 | TIMING_DEFINE_PRINT(det_numprimes) |
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30 | |
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31 | |
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32 | static bool solve ( int **extmat, int nrows, int ncols ); |
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33 | int determinant ( int **extmat, int n ); |
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34 | |
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35 | static CanonicalForm bound ( const CFMatrix & M ); |
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36 | CanonicalForm detbound ( const CFMatrix & M, int rows ); |
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37 | |
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38 | bool |
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39 | matrix_in_Z( const CFMatrix & M, int rows ) |
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40 | { |
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41 | int i, j; |
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42 | for ( i = 1; i <= rows; i++ ) |
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43 | for ( j = 1; j <= rows; j++ ) |
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44 | if ( ! M(i,j).inZ() ) |
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45 | return false; |
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46 | return true; |
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47 | } |
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48 | |
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49 | bool |
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50 | matrix_in_Z( const CFMatrix & M ) |
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51 | { |
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52 | int i, j, rows = M.rows(), cols = M.columns(); |
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53 | for ( i = 1; i <= rows; i++ ) |
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54 | for ( j = 1; j <= cols; j++ ) |
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55 | if ( ! M(i,j).inZ() ) |
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56 | return false; |
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57 | return true; |
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58 | } |
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59 | |
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60 | bool |
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61 | betterpivot ( const CanonicalForm & oldpivot, const CanonicalForm & newpivot ) |
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62 | { |
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63 | if ( newpivot.isZero() ) |
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64 | return false; |
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65 | else if ( oldpivot.isZero() ) |
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66 | return true; |
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67 | else if ( level( oldpivot ) > level( newpivot ) ) |
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68 | return true; |
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69 | else if ( level( oldpivot ) < level( newpivot ) ) |
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70 | return false; |
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71 | else |
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72 | return ( newpivot.lc() < oldpivot.lc() ); |
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73 | } |
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74 | |
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75 | VAR bool fuzzy_result; |
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76 | |
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77 | bool |
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78 | linearSystemSolve( CFMatrix & M ) |
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79 | { |
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80 | typedef int* int_ptr; |
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81 | |
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82 | if ( ! matrix_in_Z( M ) ) { |
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83 | int nrows = M.rows(), ncols = M.columns(); |
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84 | int i, j, k; |
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85 | CanonicalForm rowpivot, pivotrecip; |
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86 | // triangularization |
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87 | for ( i = 1; i <= nrows; i++ ) { |
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88 | //find "pivot" |
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89 | for (j = i; j <= nrows; j++ ) |
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90 | if ( M(j,i) != 0 ) break; |
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91 | if ( j > nrows ) return false; |
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92 | if ( j != i ) |
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93 | M.swapRow( i, j ); |
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94 | pivotrecip = 1 / M(i,i); |
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95 | for ( j = 1; j <= ncols; j++ ) |
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96 | M(i,j) *= pivotrecip; |
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97 | for ( j = i+1; j <= nrows; j++ ) { |
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98 | rowpivot = M(j,i); |
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99 | if ( rowpivot == 0 ) continue; |
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100 | for ( k = i; k <= ncols; k++ ) |
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101 | M(j,k) -= M(i,k) * rowpivot; |
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102 | } |
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103 | } |
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104 | // matrix is now upper triangular with 1s down the diagonal |
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105 | // back-substitute |
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106 | for ( i = nrows-1; i > 0; i-- ) { |
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107 | for ( j = nrows+1; j <= ncols; j++ ) { |
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108 | for ( k = i+1; k <= nrows; k++ ) |
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109 | M(i,j) -= M(k,j) * M(i,k); |
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110 | } |
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111 | } |
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112 | return true; |
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113 | } |
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114 | else { |
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115 | int rows = M.rows(), cols = M.columns(); |
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116 | CFMatrix MM( rows, cols ); |
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117 | int ** mm = new int_ptr[rows]; |
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118 | CanonicalForm Q, Qhalf, mnew, qnew, B; |
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119 | int i, j, p, pno; |
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120 | bool ok; |
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121 | |
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122 | // initialize room to hold the result and the result mod p |
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123 | for ( i = 0; i < rows; i++ ) { |
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124 | mm[i] = new int[cols]; |
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125 | } |
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126 | |
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127 | // calculate the bound for the result |
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128 | B = bound( M ); |
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129 | DEBOUTLN( cerr, "bound = " << B ); |
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130 | |
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131 | // find a first solution mod p |
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132 | pno = 0; |
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133 | do { |
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134 | DEBOUTSL( cerr ); |
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135 | DEBOUT( cerr, "trying prime(" << pno << ") = " ); |
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136 | p = cf_getBigPrime( pno ); |
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137 | DEBOUT( cerr, p ); |
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138 | DEBOUTENDL( cerr ); |
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139 | setCharacteristic( p ); |
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140 | // map matrix into char p |
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141 | for ( i = 1; i <= rows; i++ ) |
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142 | for ( j = 1; j <= cols; j++ ) |
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143 | mm[i-1][j-1] = mapinto( M(i,j) ).intval(); |
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144 | // solve mod p |
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145 | ok = solve( mm, rows, cols ); |
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146 | pno++; |
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147 | } while ( ! ok ); |
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148 | |
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149 | // initialize the result matrix with first solution |
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150 | setCharacteristic( 0 ); |
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151 | for ( i = 1; i <= rows; i++ ) |
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152 | for ( j = rows+1; j <= cols; j++ ) |
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153 | MM(i,j) = mm[i-1][j-1]; |
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154 | |
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155 | // Q so far |
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156 | Q = p; |
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157 | while ( Q < B && pno < cf_getNumBigPrimes() ) { |
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158 | do { |
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159 | DEBOUTSL( cerr ); |
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160 | DEBOUT( cerr, "trying prime(" << pno << ") = " ); |
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161 | p = cf_getBigPrime( pno ); |
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162 | DEBOUT( cerr, p ); |
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163 | DEBOUTENDL( cerr ); |
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164 | setCharacteristic( p ); |
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165 | for ( i = 1; i <= rows; i++ ) |
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166 | for ( j = 1; j <= cols; j++ ) |
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167 | mm[i-1][j-1] = mapinto( M(i,j) ).intval(); |
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168 | // solve mod p |
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169 | ok = solve( mm, rows, cols ); |
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170 | pno++; |
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171 | } while ( ! ok ); |
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172 | // found a solution mod p |
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173 | // now chinese remainder it to a solution mod Q*p |
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174 | setCharacteristic( 0 ); |
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175 | for ( i = 1; i <= rows; i++ ) |
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176 | for ( j = rows+1; j <= cols; j++ ) |
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177 | { |
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178 | chineseRemainder( MM[i][j], Q, CanonicalForm(mm[i-1][j-1]), CanonicalForm(p), mnew, qnew ); |
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179 | MM(i, j) = mnew; |
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180 | } |
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181 | Q = qnew; |
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182 | } |
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183 | if ( pno == cf_getNumBigPrimes() ) |
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184 | fuzzy_result = true; |
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185 | else |
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186 | fuzzy_result = false; |
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187 | // store the result in M |
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188 | Qhalf = Q / 2; |
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189 | for ( i = 1; i <= rows; i++ ) { |
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190 | for ( j = rows+1; j <= cols; j++ ) |
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191 | if ( MM(i,j) > Qhalf ) |
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192 | M(i,j) = MM(i,j) - Q; |
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193 | else |
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194 | M(i,j) = MM(i,j); |
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195 | delete [] mm[i-1]; |
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196 | } |
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197 | delete [] mm; |
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198 | return ! fuzzy_result; |
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199 | } |
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200 | } |
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201 | |
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202 | static bool |
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203 | fill_int_mat( const CFMatrix & M, int ** m, int rows ) |
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204 | { |
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205 | int i, j; |
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206 | bool ok = true; |
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207 | for ( i = 0; i < rows && ok; i++ ) |
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208 | for ( j = 0; j < rows && ok; j++ ) |
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209 | { |
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210 | if ( M(i+1,j+1).isZero() ) |
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211 | m[i][j] = 0; |
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212 | else |
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213 | { |
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214 | m[i][j] = mapinto( M(i+1,j+1) ).intval(); |
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215 | // ok = m[i][j] != 0; |
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216 | } |
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217 | } |
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218 | return ok; |
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219 | } |
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220 | |
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221 | CanonicalForm |
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222 | determinant( const CFMatrix & M, int rows ) |
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223 | { |
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224 | typedef int* int_ptr; |
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225 | |
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226 | ASSERT( rows <= M.rows() && rows <= M.columns() && rows > 0, "undefined determinant" ); |
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227 | if ( rows == 1 ) |
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228 | return M(1,1); |
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229 | else if ( rows == 2 ) |
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230 | return M(1,1)*M(2,2)-M(2,1)*M(1,2); |
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231 | else if ( matrix_in_Z( M, rows ) ) |
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232 | { |
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233 | int ** mm = new int_ptr[rows]; |
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234 | CanonicalForm x, q, Qhalf, B; |
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235 | int n, i, intdet, p, pno; |
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236 | for ( i = 0; i < rows; i++ ) |
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237 | { |
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238 | mm[i] = new int[rows]; |
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239 | } |
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240 | pno = 0; n = 0; |
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241 | TIMING_START(det_bound); |
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242 | B = detbound( M, rows ); |
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243 | TIMING_END(det_bound); |
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244 | q = 1; |
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245 | TIMING_START(det_numprimes); |
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246 | while ( B > q && n < cf_getNumBigPrimes() ) |
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247 | { |
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248 | q *= cf_getBigPrime( n ); |
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249 | n++; |
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250 | } |
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251 | TIMING_END(det_numprimes); |
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252 | |
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253 | CFArray X(1,n), Q(1,n); |
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254 | |
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255 | while ( pno < n ) |
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256 | { |
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257 | p = cf_getBigPrime( pno ); |
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258 | setCharacteristic( p ); |
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259 | // map matrix into char p |
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260 | TIMING_START(det_mapping); |
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261 | fill_int_mat( M, mm, rows ); |
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262 | TIMING_END(det_mapping); |
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263 | pno++; |
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264 | DEBOUT( cerr, "." ); |
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265 | TIMING_START(det_determinant); |
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266 | intdet = determinant( mm, rows ); |
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267 | TIMING_END(det_determinant); |
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268 | setCharacteristic( 0 ); |
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269 | X[pno] = intdet; |
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270 | Q[pno] = p; |
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271 | } |
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272 | TIMING_START(det_chinese); |
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273 | chineseRemainder( X, Q, x, q ); |
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274 | TIMING_END(det_chinese); |
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275 | Qhalf = q / 2; |
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276 | if ( x > Qhalf ) |
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277 | x = x - q; |
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278 | for ( i = 0; i < rows; i++ ) |
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279 | delete [] mm[i]; |
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280 | delete [] mm; |
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281 | return x; |
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282 | } |
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283 | else |
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284 | { |
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285 | CFMatrix m( M ); |
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286 | CanonicalForm divisor = 1, pivot, mji; |
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287 | int i, j, k, sign = 1; |
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288 | for ( i = 1; i <= rows; i++ ) { |
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289 | pivot = m(i,i); k = i; |
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290 | for ( j = i+1; j <= rows; j++ ) { |
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291 | if ( betterpivot( pivot, m(j,i) ) ) { |
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292 | pivot = m(j,i); |
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293 | k = j; |
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294 | } |
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295 | } |
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296 | if ( pivot.isZero() ) |
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297 | return 0; |
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298 | if ( i != k ) |
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299 | { |
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300 | m.swapRow( i, k ); |
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301 | sign = -sign; |
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302 | } |
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303 | for ( j = i+1; j <= rows; j++ ) |
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304 | { |
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305 | if ( ! m(j,i).isZero() ) |
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306 | { |
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307 | divisor *= pivot; |
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308 | mji = m(j,i); |
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309 | m(j,i) = 0; |
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310 | for ( k = i+1; k <= rows; k++ ) |
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311 | m(j,k) = m(j,k) * pivot - m(i,k)*mji; |
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312 | } |
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313 | } |
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314 | } |
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315 | pivot = sign; |
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316 | for ( i = 1; i <= rows; i++ ) |
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317 | pivot *= m(i,i); |
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318 | return pivot / divisor; |
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319 | } |
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320 | } |
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321 | |
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322 | CanonicalForm |
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323 | determinant2( const CFMatrix & M, int rows ) |
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324 | { |
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325 | typedef int* int_ptr; |
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326 | |
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327 | ASSERT( rows <= M.rows() && rows <= M.columns() && rows > 0, "undefined determinant" ); |
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328 | if ( rows == 1 ) |
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329 | return M(1,1); |
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330 | else if ( rows == 2 ) |
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331 | return M(1,1)*M(2,2)-M(2,1)*M(1,2); |
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332 | else if ( matrix_in_Z( M, rows ) ) { |
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333 | int ** mm = new int_ptr[rows]; |
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334 | CanonicalForm QQ, Q, Qhalf, mnew, q, qnew, B; |
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335 | CanonicalForm det, detnew, qdet; |
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336 | int i, p, pcount, pno, intdet; |
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337 | bool ok; |
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338 | |
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339 | // initialize room to hold the result and the result mod p |
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340 | for ( i = 0; i < rows; i++ ) { |
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341 | mm[i] = new int[rows]; |
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342 | } |
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343 | |
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344 | // calculate the bound for the result |
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345 | B = detbound( M, rows ); |
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346 | |
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347 | // find a first solution mod p |
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348 | pno = 0; |
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349 | do { |
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350 | p = cf_getBigPrime( pno ); |
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351 | setCharacteristic( p ); |
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352 | // map matrix into char p |
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353 | ok = fill_int_mat( M, mm, rows ); |
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354 | pno++; |
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355 | } while ( ! ok && pno < cf_getNumPrimes() ); |
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356 | // initialize the result matrix with first solution |
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357 | // solve mod p |
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358 | DEBOUT( cerr, "." ); |
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359 | intdet = determinant( mm, rows ); |
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360 | setCharacteristic( 0 ); |
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361 | det = intdet; |
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362 | // Q so far |
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363 | Q = p; |
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364 | QQ = p; |
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365 | while ( Q < B && cf_getNumPrimes() > pno ) { |
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366 | // find a first solution mod p |
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367 | do { |
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368 | p = cf_getBigPrime( pno ); |
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369 | setCharacteristic( p ); |
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370 | // map matrix into char p |
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371 | ok = fill_int_mat( M, mm, rows ); |
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372 | pno++; |
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373 | } while ( ! ok && pno < cf_getNumPrimes() ); |
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374 | // initialize the result matrix with first solution |
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375 | // solve mod p |
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376 | DEBOUT( cerr, "." ); |
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377 | intdet = determinant( mm, rows ); |
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378 | setCharacteristic( 0 ); |
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379 | qdet = intdet; |
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380 | // Q so far |
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381 | q = p; |
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382 | QQ *= p; |
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383 | pcount = 0; |
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384 | while ( QQ < B && cf_getNumPrimes() > pno && pcount < 500 ) { |
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385 | do { |
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386 | p = cf_getBigPrime( pno ); |
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387 | setCharacteristic( p ); |
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388 | ok = true; |
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389 | // map matrix into char p |
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390 | ok = fill_int_mat( M, mm, rows ); |
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391 | pno++; |
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392 | } while ( ! ok && cf_getNumPrimes() > pno ); |
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393 | // solve mod p |
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394 | DEBOUT( cerr, "." ); |
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395 | intdet = determinant( mm, rows ); |
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396 | // found a solution mod p |
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397 | // now chinese remainder it to a solution mod Q*p |
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398 | setCharacteristic( 0 ); |
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399 | chineseRemainder( qdet, q, intdet, p, detnew, qnew ); |
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400 | qdet = detnew; |
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401 | q = qnew; |
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402 | QQ *= p; |
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403 | pcount++; |
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404 | } |
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405 | DEBOUT( cerr, "*" ); |
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406 | chineseRemainder( det, Q, qdet, q, detnew, qnew ); |
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407 | Q = qnew; |
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408 | QQ = Q; |
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409 | det = detnew; |
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410 | } |
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411 | if ( ! ok ) |
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412 | fuzzy_result = true; |
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413 | else |
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414 | fuzzy_result = false; |
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415 | // store the result in M |
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416 | Qhalf = Q / 2; |
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417 | if ( det > Qhalf ) |
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418 | det = det - Q; |
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419 | for ( i = 0; i < rows; i++ ) |
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420 | delete [] mm[i]; |
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421 | delete [] mm; |
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422 | return det; |
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423 | } |
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424 | else { |
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425 | CFMatrix m( M ); |
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426 | CanonicalForm divisor = 1, pivot, mji; |
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427 | int i, j, k, sign = 1; |
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428 | for ( i = 1; i <= rows; i++ ) { |
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429 | pivot = m(i,i); k = i; |
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430 | for ( j = i+1; j <= rows; j++ ) { |
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431 | if ( betterpivot( pivot, m(j,i) ) ) { |
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432 | pivot = m(j,i); |
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433 | k = j; |
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434 | } |
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435 | } |
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436 | if ( pivot.isZero() ) |
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437 | return 0; |
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438 | if ( i != k ) { |
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439 | m.swapRow( i, k ); |
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440 | sign = -sign; |
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441 | } |
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442 | for ( j = i+1; j <= rows; j++ ) { |
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443 | if ( ! m(j,i).isZero() ) { |
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444 | divisor *= pivot; |
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445 | mji = m(j,i); |
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446 | m(j,i) = 0; |
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447 | for ( k = i+1; k <= rows; k++ ) |
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448 | m(j,k) = m(j,k) * pivot - m(i,k)*mji; |
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449 | } |
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450 | } |
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451 | } |
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452 | pivot = sign; |
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453 | for ( i = 1; i <= rows; i++ ) |
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454 | pivot *= m(i,i); |
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455 | return pivot / divisor; |
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456 | } |
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457 | } |
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458 | |
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459 | static CanonicalForm |
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460 | bound ( const CFMatrix & M ) |
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461 | { |
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462 | DEBINCLEVEL( cerr, "bound" ); |
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463 | int rows = M.rows(), cols = M.columns(); |
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464 | CanonicalForm sum = 0; |
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465 | int i, j; |
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466 | for ( i = 1; i <= rows; i++ ) |
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467 | for ( j = 1; j <= rows; j++ ) |
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468 | sum += M(i,j) * M(i,j); |
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469 | DEBOUTLN( cerr, "bound(matrix)^2 = " << sum ); |
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470 | CanonicalForm vmax = 0, vsum; |
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471 | for ( j = rows+1; j <= cols; j++ ) { |
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472 | vsum = 0; |
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473 | for ( i = 1; i <= rows; i++ ) |
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474 | vsum += M(i,j) * M(i,j); |
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475 | if ( vsum > vmax ) vmax = vsum; |
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476 | } |
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477 | DEBOUTLN( cerr, "bound(lhs)^2 = " << vmax ); |
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478 | sum += vmax; |
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479 | DEBOUTLN( cerr, "bound(overall)^2 = " << sum ); |
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480 | DEBDECLEVEL( cerr, "bound" ); |
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481 | return sqrt( sum ) + 1; |
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482 | } |
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483 | |
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484 | |
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485 | CanonicalForm |
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486 | detbound ( const CFMatrix & M, int rows ) |
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487 | { |
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488 | CanonicalForm sum = 0, prod = 2; |
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489 | int i, j; |
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490 | for ( i = 1; i <= rows; i++ ) { |
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491 | sum = 0; |
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492 | for ( j = 1; j <= rows; j++ ) |
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493 | sum += M(i,j) * M(i,j); |
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494 | prod *= 1 + sqrt(sum); |
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495 | } |
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496 | return prod; |
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497 | } |
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498 | |
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499 | |
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500 | // solve returns false if computation failed |
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501 | // extmat is overwritten: output is Id mat followed by solution(s) |
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502 | |
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503 | bool |
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504 | solve ( int **extmat, int nrows, int ncols ) |
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505 | { |
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506 | DEBINCLEVEL( cerr, "solve" ); |
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507 | int i, j, k; |
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508 | int rowpivot, pivotrecip; // all FF |
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509 | int * rowi; // FF |
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510 | int * rowj; // FF |
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511 | int * swap; // FF |
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512 | // triangularization |
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513 | for ( i = 0; i < nrows; i++ ) { |
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514 | //find "pivot" |
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515 | for (j = i; j < nrows; j++ ) |
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516 | if ( extmat[j][i] != 0 ) break; |
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517 | if ( j == nrows ) { |
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518 | DEBOUTLN( cerr, "solve failed" ); |
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519 | DEBDECLEVEL( cerr, "solve" ); |
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520 | return false; |
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521 | } |
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522 | if ( j != i ) { |
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523 | swap = extmat[i]; extmat[i] = extmat[j]; extmat[j] = swap; |
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524 | } |
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525 | pivotrecip = ff_inv( extmat[i][i] ); |
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526 | rowi = extmat[i]; |
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527 | for ( j = 0; j < ncols; j++ ) |
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528 | rowi[j] = ff_mul( pivotrecip, rowi[j] ); |
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529 | for ( j = i+1; j < nrows; j++ ) { |
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530 | rowj = extmat[j]; |
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531 | rowpivot = rowj[i]; |
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532 | if ( rowpivot == 0 ) continue; |
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533 | for ( k = i; k < ncols; k++ ) |
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534 | rowj[k] = ff_sub( rowj[k], ff_mul( rowpivot, rowi[k] ) ); |
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535 | } |
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536 | } |
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537 | // matrix is now upper triangular with 1s down the diagonal |
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538 | // back-substitute |
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539 | for ( i = nrows-1; i >= 0; i-- ) { |
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540 | rowi = extmat[i]; |
---|
541 | for ( j = 0; j < i; j++ ) { |
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542 | rowj = extmat[j]; |
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543 | rowpivot = rowj[i]; |
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544 | if ( rowpivot == 0 ) continue; |
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545 | for ( k = i; k < ncols; k++ ) |
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546 | rowj[k] = ff_sub( rowj[k], ff_mul( rowpivot, rowi[k] ) ); |
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547 | // for (k=nrows; k<ncols; k++) rowj[k] = ff_sub(rowj[k], ff_mul(rowpivot, rowi[k])); |
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548 | } |
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549 | } |
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550 | DEBOUTLN( cerr, "solve succeeded" ); |
---|
551 | DEBDECLEVEL( cerr, "solve" ); |
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552 | return true; |
---|
553 | } |
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554 | |
---|
555 | int |
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556 | determinant ( int **extmat, int n ) |
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557 | { |
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558 | int i, j, k; |
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559 | int divisor, multiplier, rowii, rowji; // all FF |
---|
560 | int * rowi; // FF |
---|
561 | int * rowj; // FF |
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562 | int * swap; // FF |
---|
563 | // triangularization |
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564 | multiplier = 1; |
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565 | divisor = 1; |
---|
566 | |
---|
567 | for ( i = 0; i < n; i++ ) { |
---|
568 | //find "pivot" |
---|
569 | for (j = i; j < n; j++ ) |
---|
570 | if ( extmat[j][i] != 0 ) break; |
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571 | if ( j == n ) return 0; |
---|
572 | if ( j != i ) { |
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573 | multiplier = ff_neg( multiplier ); |
---|
574 | swap = extmat[i]; extmat[i] = extmat[j]; extmat[j] = swap; |
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575 | } |
---|
576 | rowi = extmat[i]; |
---|
577 | rowii = rowi[i]; |
---|
578 | for ( j = i+1; j < n; j++ ) { |
---|
579 | rowj = extmat[j]; |
---|
580 | rowji = rowj[i]; |
---|
581 | if ( rowji == 0 ) continue; |
---|
582 | divisor = ff_mul( divisor, rowii ); |
---|
583 | for ( k = i; k < n; k++ ) |
---|
584 | rowj[k] = ff_sub( ff_mul( rowj[k], rowii ), ff_mul( rowi[k], rowji ) ); |
---|
585 | } |
---|
586 | } |
---|
587 | multiplier = ff_mul( multiplier, ff_inv( divisor ) ); |
---|
588 | for ( i = 0; i < n; i++ ) |
---|
589 | multiplier = ff_mul( multiplier, extmat[i][i] ); |
---|
590 | return multiplier; |
---|
591 | } |
---|
592 | |
---|
593 | void |
---|
594 | solveVandermondeT ( const CFArray & a, const CFArray & w, CFArray & x, const Variable & z ) |
---|
595 | { |
---|
596 | CanonicalForm Q = 1, q, p; |
---|
597 | CFIterator j; |
---|
598 | int i, n = a.size(); |
---|
599 | |
---|
600 | for ( i = 1; i <= n; i++ ) |
---|
601 | Q *= ( z - a[i] ); |
---|
602 | for ( i = 1; i <= n; i++ ) { |
---|
603 | q = Q / ( z - a[i] ); |
---|
604 | p = q / q( a[i], z ); |
---|
605 | x[i] = 0; |
---|
606 | for ( j = p; j.hasTerms(); j++ ) |
---|
607 | x[i] += w[j.exp()+1] * j.coeff(); |
---|
608 | } |
---|
609 | } |
---|