1 | /*****************************************************************************\ |
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2 | * Computer Algebra System SINGULAR |
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3 | \*****************************************************************************/ |
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4 | /** @file lineareAlgebra.h |
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5 | * |
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6 | * This file provides basic linear algebra functionality. |
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7 | * |
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8 | * ABSTRACT: This file provides basic algorithms from linear algebra over |
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9 | * any SINGULAR-supported field. |
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10 | * For the time being, the procedures defined in this file expect matrices |
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11 | * containing objects of the SINGULAR type 'number'. This means that, when |
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12 | * 'currentRing' represents some polynomial ring K[x_1, x_2, ..., x_n], then |
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13 | * the entries of the matrices are 'numbers' representing elements of K (and |
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14 | * NOT 'polys' in K[x_1, x_2, ..., x_n]). |
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15 | * This restriction may become obselete in the future. |
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16 | * |
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17 | * @author Frank Seelisch |
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18 | * |
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19 | * @internal @version \$Id$ |
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20 | * |
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21 | **/ |
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22 | /*****************************************************************************/ |
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23 | |
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24 | #ifndef LINEAR_ALGEBRA_H |
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25 | #define LINEAR_ALGEBRA_H |
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26 | |
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27 | // include basic SINGULAR structures |
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28 | #include <kernel/structs.h> |
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29 | |
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30 | /** |
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31 | * LU-decomposition of a given (m x n)-matrix. |
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32 | * |
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33 | * Given an (m x n) matrix A, the method computes its LU-decomposition, |
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34 | * that is, it computes matrices P, L, and U such that<br> |
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35 | * - P * A = L * U,<br> |
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36 | * - P is an (m x m) permutation matrix, i.e., its row/columns form the |
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37 | * standard basis of K^m,<br> |
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38 | * - L is an (m x m) matrix in lower triangular form with all diagonal |
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39 | * entries equal to 1,<br> |
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40 | * - U is an (m x n) matrix in upper row echelon form.<br> |
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41 | * From these conditions, it follows immediately that also A = P * L * U, |
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42 | * since P is self-inverse.<br> |
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43 | * |
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44 | * The method will modify all argument matrices except aMat, so that |
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45 | * they will finally contain the matrices P, L, and U as specified |
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46 | * above. |
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47 | **/ |
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48 | void luDecomp( |
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49 | const matrix aMat, /**< [in] the initial matrix A, */ |
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50 | matrix &pMat, /**< [out] the row permutation matrix P */ |
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51 | matrix &lMat, /**< [out] the lower triangular matrix L */ |
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52 | matrix &uMat /**< [out] the upper row echelon matrix U */ |
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53 | ); |
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54 | |
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55 | /** |
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56 | * Returns a pivot score for any non-zero matrix entry. |
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57 | * |
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58 | * The smaller the score the better will n serve as a pivot element |
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59 | * in subsequent Gauss elimination steps. |
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60 | * |
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61 | * @return the pivot score of n |
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62 | **/ |
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63 | int pivotScore( |
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64 | number n /**< [in] a non-zero matrix entry */ |
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65 | ); |
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66 | |
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67 | /** |
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68 | * Finds the best pivot element in some part of a given matrix. |
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69 | * |
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70 | * Given any matrix A with valid row indices r1..r2 and valid column |
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71 | * indices c1..c2, this method finds the best pivot element for |
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72 | * subsequent Gauss elimination steps in A[r1..r2, c1..c2]. "Best" |
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73 | * here means best with respect to the implementation of the method |
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74 | * 'pivotScore(number n)'.<br> |
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75 | * In the case that all elements in A[r1..r2, c1..c2] are zero, the |
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76 | * method returns false, otherwise true. |
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77 | * |
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78 | * @return false if all relevant matrix entries are zero, true otherwise |
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79 | * @sa pivotScore(number n) |
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80 | **/ |
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81 | bool pivot( |
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82 | const matrix aMat, /**< [in] any matrix with number entries */ |
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83 | const int r1, /**< [in] lower row index */ |
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84 | const int r2, /**< [in] upper row index */ |
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85 | const int c1, /**< [in] lower column index */ |
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86 | const int c2, /**< [in] upper column index */ |
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87 | int* bestR, /**< [out] address of row index of best |
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88 | pivot element */ |
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89 | int* bestC /**< [out] address of column index of |
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90 | best pivot element */ |
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91 | ); |
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92 | |
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93 | /** |
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94 | * Computes the inverse of a given (n x n)-matrix. |
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95 | * |
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96 | * This method expects an (n x n)-matrix, that is, it must have as many |
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97 | * rows as columns. Inversion of the first argument is attempted via the |
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98 | * LU-decomposition. There are two cases:<br> |
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99 | * 1) The matrix is not invertible. Then the method returns false, and |
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100 | * &iMat remains unchanged.<br> |
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101 | * 2) The matrix is invertible. Then the method returns true, and the |
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102 | * content of iMat is the inverse of aMat. |
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103 | * |
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104 | * @return true iff aMat is invertible, false otherwise |
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105 | * @sa luInverseFromLUDecomp(const matrix pMat, const matrix lMat, |
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106 | * const matrix uMat, matrix &iMat) |
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107 | **/ |
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108 | bool luInverse( |
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109 | const matrix aMat, /**< [in] matrix to be inverted */ |
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110 | matrix &iMat /**< [out] inverse of aMat if |
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111 | invertible */ |
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112 | ); |
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113 | |
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114 | /** |
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115 | * Computes the inverse of a given (n x n)-matrix in upper right |
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116 | * triangular form. |
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117 | * |
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118 | * This method expects a quadratic matrix, that is, it must have as |
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119 | * many rows as columns. Moreover, uMat[i, j] = 0, at least for all |
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120 | * i > j, that is, u is in upper right triangular form.<br> |
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121 | * If the argument diagonalIsOne is true, then we know additionally, |
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122 | * that uMat[i, i] = 1, for all i. In this case uMat is invertible. |
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123 | * Contrariwise, if diagonalIsOne is false, we do not know anything |
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124 | * about the diagonal entries. (Note that they may still all be |
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125 | * 1.)<br> |
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126 | * In general, there are two cases:<br> |
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127 | * 1) The matrix is not invertible. Then the method returns false, |
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128 | * and &iMat remains unchanged.<br> |
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129 | * 2) The matrix is invertible. Then the method returns true, and |
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130 | * the content of iMat is the inverse of uMat. |
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131 | * |
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132 | * @return true iff uMat is invertible, false otherwise |
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133 | **/ |
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134 | bool upperRightTriangleInverse( |
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135 | const matrix uMat, /**< [in] (n x n)-matrix in upper right |
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136 | triangular form */ |
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137 | matrix &iMat, /**< [out] inverse of uMat if invertible */ |
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138 | bool diagonalIsOne /**< [in] if true, then all diagonal |
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139 | entries of uMat are 1 */ |
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140 | ); |
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141 | |
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142 | /** |
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143 | * Computes the inverse of a given (n x n)-matrix in lower left |
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144 | * triangular form. |
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145 | * |
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146 | * This method expects an (n x n)-matrix, that is, it must have as |
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147 | * many rows as columns. Moreover, lMat[i,j] = 0, at least for all |
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148 | * j > i, that ism lMat is in lower left triangular form.<br> |
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149 | * If the argument diagonalIsOne is true, then we know additionally, |
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150 | * that lMat[i, i] = 1, for all i. In this case lMat is invertible. |
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151 | * Contrariwise, if diagonalIsOne is false, we do not know anything |
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152 | * about the diagonal entries. (Note that they may still all be |
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153 | * 1.)<br> |
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154 | * In general, there are two cases:<br> |
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155 | * 1) The matrix is not invertible. Then the method returns false, |
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156 | * and &iMat remains unchanged.<br> |
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157 | * 2) The matrix is invertible. Then the method returns true, and |
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158 | * the content of iMat is the inverse of lMat. |
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159 | * |
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160 | * @return true iff lMat is invertible, false otherwise |
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161 | **/ |
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162 | bool lowerLeftTriangleInverse( |
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163 | const matrix lMat, /**< [in] (n x n)-matrix in lower left |
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164 | triangular form */ |
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165 | matrix &iMat, /**< [out] inverse of lMat if invertible */ |
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166 | bool diagonalIsOne /**< [in] if true, then all diagonal |
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167 | entries of lMat are 1 */ |
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168 | ); |
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169 | |
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170 | /** |
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171 | * Computes the inverse of an (n x n)-matrix which is given by its LU- |
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172 | * decomposition. |
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173 | * |
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174 | * With A denoting the matrix to be inverted, the method expects the |
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175 | * LU-decomposition of A, that is, pMat * A = lMat * uMat, where |
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176 | * the argument matrices have the appropriate proteries; see method |
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177 | * 'luDecomp(const matrix aMat, matrix &pMat, matrix &lMat, |
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178 | * matrix &uMat)'.<br> |
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179 | * Furthermore, uMat is expected to be an (n x n)-matrix. Then A^(-1) |
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180 | * is computed according to A^(-1) = uMat^(-1) * lMat^(-1) * pMat, |
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181 | * since pMat is self-inverse. This will work if and only if uMat is |
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182 | * invertible, because lMat and pMat are in any case invertible.<br> |
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183 | * In general, there are two cases:<br> |
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184 | * 1) uMat and hence A is not invertible. Then the method returns |
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185 | * false, and &iMat remains unchanged.<br> |
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186 | * 2) uMat and hence A is invertible. Then the method returns true, |
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187 | * and the content of iMat is the inverse of A. |
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188 | * |
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189 | * @return true if A is invertible, false otherwise |
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190 | * @sa luInverse(const matrix aMat, matrix &iMat) |
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191 | **/ |
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192 | bool luInverseFromLUDecomp( |
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193 | const matrix pMat, /**< [in] permutation matrix of an LU- |
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194 | decomposition */ |
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195 | const matrix lMat, /**< [in] lower left matrix of an LU- |
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196 | decomposition */ |
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197 | const matrix uMat, /**< [in] upper right matrix of an LU- |
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198 | decomposition */ |
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199 | matrix &iMat /**< [out] inverse of A if invertible */ |
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200 | ); |
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201 | |
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202 | /** |
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203 | * Computes the rank of a given (m x n)-matrix. |
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204 | * |
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205 | * The matrix may already be given in row echelon form, e.g., when |
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206 | * the user has before called luDecomp and passes the upper triangular |
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207 | * matrix U to luRank. In this case, the second argument can be set to |
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208 | * true in order to pass this piece of information to the method. |
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209 | * Otherwise, luDecomp will be called first to compute the matrix U. |
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210 | * The rank is then read off the matrix U. |
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211 | * |
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212 | * @return the rank as an int |
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213 | * @sa luDecomp(const matrix aMat, matrix &pMat, matrix &lMat, matrix &uMat) |
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214 | **/ |
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215 | int luRank( |
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216 | const matrix aMat, /**< [in] input matrix */ |
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217 | const bool isRowEchelon /**< [in] if true then aMat is assumed to be |
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218 | already in row echelon form */ |
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219 | ); |
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220 | |
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221 | /** |
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222 | * Solves the linear system A * x = b, where A is an (m x n)-matrix |
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223 | * which is given by its LU-decomposition. |
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224 | * |
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225 | * The method expects the LU-decomposition of A, that is, |
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226 | * pMat * A = lMat * uMat, where the argument matrices have the |
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227 | * appropriate proteries; see method |
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228 | * 'luDecomp(const matrix aMat, matrix &pMat, matrix &lMat, |
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229 | * matrix &uMat)'.<br> |
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230 | * Instead of trying to invert A and return x = A^(-1)*b, this |
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231 | * method |
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232 | * 1) computes b' = pMat * b, |
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233 | * 2) solves the simple system L * y = b', and then |
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234 | * 3) solves the simple system U * x = y. |
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235 | * Note that steps 1) and 2) will always work, as L is in any case |
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236 | * invertible. Moreover, y is uniquely determined. Step 3) will only |
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237 | * work if and only if y is in the column span of U. In that case, |
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238 | * there may be infinitely many solutions. |
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239 | * Thus, there are three cases:<br> |
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240 | * 1) There is no solution. Then the method returns false, and &xVec |
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241 | * as well as &H remain unchanged.<br> |
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242 | * 2) There is a unique solution. Then the method returns true and |
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243 | * H is the 1x1 matrix with zero entry.<br> |
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244 | * 3) There are infinitely many solutions. Then the method returns |
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245 | * true and some solution of the given original linear system. |
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246 | * Furthermore, the columns of H span the solution space of the |
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247 | * homogeneous linear system. The dimension of the solution |
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248 | * space is then the number of columns of H. |
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249 | * |
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250 | * @return true if there is at least one solution, false otherwise |
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251 | **/ |
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252 | bool luSolveViaLUDecomp( |
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253 | const matrix pMat, /**< [in] permutation matrix of an LU- |
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254 | decomposition */ |
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255 | const matrix lMat, /**< [in] lower left matrix of an LU- |
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256 | decomposition */ |
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257 | const matrix uMat, /**< [in] upper right matrix of an LU- |
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258 | decomposition */ |
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259 | const matrix bVec, /**< [in] right-hand side of the linear |
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260 | system to be solved */ |
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261 | matrix &xVec, /**< [out] solution of A*x = b */ |
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262 | matrix &H /**< [out] matrix with columns spanning |
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263 | homogeneous solution space */ |
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264 | ); |
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265 | |
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266 | /** |
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267 | * Solves the linear system A * x = b, where A is an (m x n)-matrix |
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268 | * which is given by its LDU-decomposition. |
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269 | * |
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270 | * The method expects the LDU-decomposition of A, that is, |
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271 | * pMat * A = lMat * dMat^(-1) * uMat, where the argument matrices have |
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272 | * the appropriate proteries; see method |
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273 | * 'lduDecomp(const matrix aMat, matrix &pMat, matrix &lMat, |
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274 | * matrix &dMat, matrix &uMat, poly &l, poly &u, poly &lTimesU)'.<br> |
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275 | * Instead of trying to invert A and return x = A^(-1)*b, this |
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276 | * method |
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277 | * 1) computes b' = l * pMat * b, |
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278 | * 2) solves the simple system L * y = b', |
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279 | * 3) computes y' = u * dMat * y, |
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280 | * 4) solves the simple system U * y'' = y', |
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281 | * 5) computes x = 1/(lTimesU) * y''. |
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282 | * Note that steps 1), 2) and 3) will always work, as L is in any case |
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283 | * invertible. Moreover, y and thus y' are uniquely determined. |
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284 | * Step 4) will only work if and only if y' is in the column span of U. |
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285 | * In that case, there may be infinitely many solutions. |
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286 | * In contrast to luSolveViaLUDecomp, this algorithm guarantees that |
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287 | * all divisions which have to be performed in steps 2) and 4) will |
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288 | * work without remainder. This is due to properties of the given LDU- |
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289 | * decomposition. Only the final step 5) performs a division of a vector |
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290 | * by a member of the ground field. (Suppose the ground field is Q or |
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291 | * some rational function field. Then, if A contains only integers or |
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292 | * polynomials, respectively, then steps 1) - 4) will keep all data |
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293 | * integer or polynomial, respectively. This may speed up computations |
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294 | * considerably.) |
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295 | * For the outcome, there are three cases:<br> |
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296 | * 1) There is no solution. Then the method returns false, and &xVec |
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297 | * as well as &H remain unchanged.<br> |
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298 | * 2) There is a unique solution. Then the method returns true and |
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299 | * H is the 1x1 matrix with zero entry.<br> |
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300 | * 3) There are infinitely many solutions. Then the method returns |
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301 | * true and some solution of the given original linear system. |
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302 | * Furthermore, the columns of H span the solution space of the |
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303 | * homogeneous linear system. The dimension of the solution |
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304 | * space is then the number of columns of H. |
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305 | * |
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306 | * @return true if there is at least one solution, false otherwise |
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307 | **/ |
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308 | bool luSolveViaLDUDecomp( |
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309 | const matrix pMat, /**< [in] permutation matrix of an LDU- |
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310 | decomposition */ |
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311 | const matrix lMat, /**< [in] lower left matrix of an LDU- |
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312 | decomposition */ |
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313 | const matrix dMat, /**< [in] diagonal matrix of an LDU- |
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314 | decomposition */ |
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315 | const matrix uMat, /**< [in] upper right matrix of an LDU- |
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316 | decomposition */ |
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317 | const poly l, /**< [in] pivot product l of an LDU decomp. */ |
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318 | const poly u, /**< [in] pivot product u of an LDU decomp. */ |
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319 | const poly lTimesU, /**< [in] product of l and u */ |
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320 | const matrix bVec, /**< [in] right-hand side of the linear |
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321 | system to be solved */ |
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322 | matrix &xVec, /**< [out] solution of A*x = b */ |
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323 | matrix &H /**< [out] matrix with columns spanning |
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324 | homogeneous solution space */ |
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325 | ); |
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326 | |
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327 | /** |
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328 | * Creates a new matrix which is the (nxn) unit matrix, and returns true |
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329 | * in case of success. |
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330 | * |
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331 | * The method will be successful whenever n >= 1. In this case and only then |
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332 | * true will be returned and the new (nxn) unit matrix will reside inside |
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333 | * the second argument. |
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334 | * |
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335 | * @return true iff the (nxn) unit matrix could be build |
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336 | **/ |
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337 | bool unitMatrix( |
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338 | const int n, /**< [in] size of the matrix */ |
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339 | matrix &unitMat /**< [out] the new (nxn) unit matrix */ |
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340 | ); |
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341 | |
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342 | /** |
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343 | * Creates a new matrix which is a submatrix of the first argument, and |
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344 | * returns true in case of success. |
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345 | * |
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346 | * The method will be successful whenever rowIndex1 <= rowIndex2 and |
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347 | * colIndex1 <= colIndex2. In this case and only then true will be |
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348 | * returned and the last argument will afterwards contain a copy of the |
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349 | * respective submatrix of the first argument. |
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350 | * Note that this method may also be used to copy an entire matrix. |
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351 | * |
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352 | * @return true iff the submatrix could be build |
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353 | **/ |
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354 | bool subMatrix( |
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355 | const matrix aMat, /**< [in] the input matrix */ |
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356 | const int rowIndex1, /**< [in] lower row index */ |
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357 | const int rowIndex2, /**< [in] higher row index */ |
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358 | const int colIndex1, /**< [in] lower column index */ |
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359 | const int colIndex2, /**< [in] higher column index */ |
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360 | matrix &subMat /**< [out] the new submatrix */ |
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361 | ); |
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362 | |
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363 | /** |
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364 | * Swaps two rows of a given matrix and thereby modifies it. |
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365 | * |
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366 | * The method expects two valid, distinct row indices, i.e. in |
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367 | * [1..n], where n is the number of rows in aMat. |
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368 | **/ |
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369 | void swapRows( |
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370 | int row1, /**< [in] index of first row to swap */ |
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371 | int row2, /**< [in] index of second row to swap */ |
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372 | matrix& aMat /**< [in/out] matrix subject to swapping */ |
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373 | ); |
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374 | |
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375 | /** |
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376 | * Swaps two columns of a given matrix and thereby modifies it. |
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377 | * |
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378 | * The method expects two valid, distinct column indices, i.e. in |
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379 | * [1..n], where n is the number of columns in aMat. |
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380 | **/ |
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381 | void swapColumns( |
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382 | int column1, /**< [in] index of first column to swap */ |
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383 | int column2, /**< [in] index of second column to swap */ |
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384 | matrix& aMat /**< [in/out] matrix subject to swapping */ |
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385 | ); |
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386 | |
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387 | /** |
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388 | * Creates a new matrix which contains the first argument in the top left |
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389 | * corner, and the second in the bottom right. |
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390 | * |
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391 | * All other entries of the resulting matrix which will be created in the |
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392 | * third argument, are zero. |
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393 | **/ |
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394 | void matrixBlock( |
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395 | const matrix aMat, /**< [in] the top left input matrix */ |
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396 | const matrix bMat, /**< [in] the bottom right input matrix */ |
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397 | matrix &block /**< [out] the new block matrix */ |
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398 | ); |
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399 | |
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400 | /** |
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401 | * Computes the characteristic polynomial from a quadratic (2x2) matrix |
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402 | * and returns true in case of success. |
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403 | * |
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404 | * The method will be successful whenever the input matrix is a (2x2) matrix. |
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405 | * In this case, the resulting polynomial will be a univariate polynomial in |
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406 | * the first ring variable of degree 2 and it will reside in the second |
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407 | * argument. |
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408 | * The method assumes that the matrix entries are all numbers, i.e., elements |
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409 | * from the ground field/ring. |
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410 | * |
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411 | * @return true iff the input matrix was (2x2) |
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412 | **/ |
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413 | bool charPoly( |
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414 | const matrix aMat, /**< [in] the input matrix */ |
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415 | poly &charPoly /**< [out] the characteristic polynomial */ |
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416 | ); |
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417 | |
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418 | /** |
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419 | * Computes the square root of a non-negative real number and returns |
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420 | * it as a new number. |
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421 | * |
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422 | * The method assumes that the current ground field is the complex |
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423 | * numbers, and that the argument has imaginary part zero. |
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424 | * If the real part is negative, then false is returned, otherwise true. |
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425 | * The square root will be computed in the last argument by Heron's |
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426 | * iteration formula with the first argument as the starting value. The |
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427 | * iteration will stop as soon as two successive values (in the resulting |
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428 | * sequence) differ by no more than the given tolerance, which is assumed |
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429 | * to be a positive real number. |
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430 | * |
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431 | * @return the square root of the non-negative argument number |
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432 | **/ |
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433 | bool realSqrt( |
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434 | const number n, /**< [in] the input number */ |
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435 | const number tolerance, /**< [in] accuracy of iteration */ |
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436 | number &root /**< [out] the root of n */ |
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437 | ); |
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438 | |
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439 | /** |
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440 | * Computes the Hessenberg form of a given square matrix. |
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441 | * |
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442 | * The method assumes that all matrix entries are numbers coming from some |
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443 | * subfield of the reals.. |
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444 | * Afterwards, the following conditions will hold: |
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445 | * 1) hessenbergMat = pMat * aMat * pMat^{-1}, |
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446 | * 2) hessenbergMat is in Hessenberg form. |
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447 | * During the algorithm, pMat will be constructed as the product of self- |
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448 | * inverse matrices. |
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449 | * The algorithm relies on computing square roots of floating point numbers. |
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450 | * These will be combuted by Heron's iteration formula, with iteration |
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451 | * stopping when two successive approximations of the square root differ by |
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452 | * no more than the given tolerance, which is assumed to be a positve real |
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453 | * number. |
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454 | **/ |
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455 | void hessenberg( |
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456 | const matrix aMat, /**< [in] the square input matrix */ |
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457 | matrix &pMat, /**< [out] the transformation matrix */ |
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458 | matrix &hessenbergMat, /**< [out] the Hessenberg form of aMat */ |
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459 | const number tolerance /**< [in] accuracy */ |
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460 | ); |
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461 | |
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462 | /** |
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463 | * Returns all solutions of a quadratic polynomial relation with real |
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464 | * coefficients. |
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465 | * |
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466 | * The method assumes that the polynomial is univariate in the first |
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467 | * ring variable, and that the current ground field is the complex numbers. |
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468 | * The polynomial has degree <= 2. Thus, there may be |
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469 | * a) infinitely many zeros, when p == 0; then -1 is returned; |
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470 | * b) no zero, when p == constant <> 0; then 0 is returned; |
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471 | * c) one zero, when p is linear; then 1 is returned; |
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472 | * d) one double zero; then 2 is returned; |
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473 | * e) two distinct zeros; then 3 is returned; |
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474 | * In the case e), s1 and s2 will contain the two distinct solutions. |
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475 | * In cases c) and d) s1 will contain the single/double solution. |
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476 | * |
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477 | * @return the number of distinct zeros |
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478 | **/ |
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479 | int quadraticSolve( |
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480 | const poly p, /**< [in] the polynomial */ |
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481 | number &s1, /**< [out] first zero, if any */ |
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482 | number &s2, /**< [out] second zero, if any */ |
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483 | const number tolerance /**< [in] accuracy */ |
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484 | ); |
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485 | |
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486 | /** |
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487 | * Computes all eigenvalues of a given real quadratic matrix with |
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488 | * multiplicites. |
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489 | * |
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490 | * The method assumes that the current ground field is the complex numbers. |
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491 | * Computations are based on the QR double shift algorithm involving |
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492 | * Hessenberg form and householder transformations. |
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493 | * If the algorithm works, then it returns a list with two entries which |
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494 | * are again lists of the same size: |
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495 | * _[1][i] is the i-th mutually distinct eigenvalue that was found, |
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496 | * _[2][i] is the (int) multiplicity of _[1][i]. |
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497 | * If the algorithm does not work (due to an ill-posed matrix), a list with |
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498 | * the single entry (int)0 is returned. |
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499 | * 'tol1' is used for detection of deflation in the actual qr double shift |
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500 | * algorithm. |
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501 | * 'tol2' is used for ending Heron's iteration whenever square roots |
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502 | * are being computed. |
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503 | * 'tol3' is used to distinguish between distinct eigenvalues: When |
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504 | * the Euclidean distance between two computed eigenvalues is less then |
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505 | * tol3, then they will be regarded equal, resulting in a higher |
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506 | * multiplicity of the corresponding eigenvalue. |
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507 | * |
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508 | * @return a list with one entry (int)0, or two entries which are again lists |
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509 | **/ |
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510 | lists qrDoubleShift( |
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511 | const matrix A, /**< [in] the quadratic matrix */ |
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512 | const number tol1, /**< [in] tolerance for deflation */ |
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513 | const number tol2, /**< [in] tolerance for square roots */ |
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514 | const number tol3 /**< [in] tolerance for distinguishing |
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515 | eigenvalues */ |
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516 | ); |
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517 | |
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518 | /** |
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519 | * Computes a factorization of a polynomial h(x, y) in K[[x]][y] up to a |
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520 | * certain degree in x, whenever a factorization of h(0, y) is given. |
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521 | * |
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522 | * The algorithm is based on Hensel's lemma: Let h(x, y) denote a monic |
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523 | * polynomial in y of degree m + n with coefficients in K[[x]]. Suppose there |
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524 | * are two monic factors f_0(y) (of degree n) and g_0(y) of degree (m) such |
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525 | * that h(0, y) = f_0(y) * g_0(y) and <f_0, g_0> = K[y]. Fix an integer d >= 0. |
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526 | * Then there are monic polynomials in y with coefficients in K[[x]], namely |
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527 | * f(x, y) of degree n and g(x, y) of degree m such that |
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528 | * h(x, y) = f(x, y) * g(x, y) modulo <x^(d+1)> (*). |
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529 | * |
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530 | * This implementation expects h, f0, g0, and d as described and computes the |
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531 | * factors f and g. Effectively, h will be given as an element of K[x, y] since |
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532 | * all terms of h with x-degree larger than d can be ignored due to (*). |
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533 | * The method expects the ground ring to contain at least two variables; then |
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534 | * x is the first ring variable, specified by the input index xIndex, and y the |
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535 | * second one, specified by yIndex. |
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536 | * |
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537 | * This code was placed here since the algorithm works by successively solving |
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538 | * d linear equation systems. It is hence an application of other methods |
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539 | * defined in this h-file and its corresponding cc-file. |
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540 | * |
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541 | **/ |
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542 | void henselFactors( |
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543 | const int xIndex, /**< [in] index of x in {1, ..., nvars(basering)} */ |
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544 | const int yIndex, /**< [in] index of y in {1, ..., nvars(basering)} */ |
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545 | const poly h, /**< [in] the polynomial h(x, y) as above */ |
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546 | const poly f0, /**< [in] the first univariate factor of h(0, y) */ |
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547 | const poly g0, /**< [in] the second univariate factor of h(0, y) */ |
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548 | const int d, /**< [in] the degree bound, d >= 0 */ |
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549 | poly &f, /**< [out] the first factor of h(x, y) */ |
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550 | poly &g /**< [out] the second factor of h(x, y) */ |
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551 | ); |
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552 | |
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553 | /** |
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554 | * LU-decomposition of a given (m x n)-matrix with performing only those |
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555 | * divisions that yield zero remainders. |
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556 | * |
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557 | * Given an (m x n) matrix A, the method computes its LDU-decomposition, |
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558 | * that is, it computes matrices P, L, D, and U such that<br> |
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559 | * - P * A = L * D^(-1) * U,<br> |
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560 | * - P is an (m x m) permutation matrix, i.e., its row/columns form the |
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561 | * standard basis of K^m,<br> |
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562 | * - L is an (m x m) matrix in lower triangular form of full rank,<br> |
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563 | * - D is an (m x m) diagonal matrix with full rank, and<br> |
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564 | * - U is an (m x n) matrix in upper row echelon form.<br> |
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565 | * From these conditions, it follows immediately that also |
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566 | * A = P * L * D^(-1) * U, since P is self-inverse.<br> |
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567 | * |
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568 | * In contrast to luDecomp, this method only performs those divisions which |
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569 | * yield zero remainders. Hence, when e.g. computing over a rational function |
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570 | * field and starting with polynomial entries only (or over Q and starting |
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571 | * with integer entries only), then any newly computed matrix entry will again |
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572 | * be polynomial (or integer). |
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573 | * |
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574 | * The method will modify all argument matrices except aMat, so that |
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575 | * they will finally contain the matrices P, L, D, and U as specified |
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576 | * above. Moreover, in order to further speed up subsequent calls of |
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577 | * luSolveViaLDUDecomp, the two denominators and their product will also be |
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578 | * returned. |
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579 | **/ |
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580 | void lduDecomp( |
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581 | const matrix aMat, /**< [in] the initial matrix A, */ |
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582 | matrix &pMat, /**< [out] the row permutation matrix P */ |
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583 | matrix &lMat, /**< [out] the lower triangular matrix L */ |
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584 | matrix &dMat, /**< [out] the diagonal matrix D */ |
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585 | matrix &uMat, /**< [out] the upper row echelon matrix U */ |
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586 | poly &l, /**< [out] the product of pivots of L */ |
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587 | poly &u, /**< [out] the product of pivots of U */ |
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588 | poly &lTimesU /**< [out] the product of l and u */ |
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589 | ); |
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590 | |
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591 | #endif |
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592 | /* LINEAR_ALGEBRA_H */ |
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