1 | // This file is part of Eigen, a lightweight C++ template library
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2 | // for linear algebra.
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3 | //
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4 | // Copyright (C) 2012, 2013 Chen-Pang He <jdh8@ms63.hinet.net>
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5 | //
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6 | // This Source Code Form is subject to the terms of the Mozilla
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7 | // Public License v. 2.0. If a copy of the MPL was not distributed
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8 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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9 |
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10 | #ifndef EIGEN_MATRIX_POWER
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11 | #define EIGEN_MATRIX_POWER
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12 |
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13 | namespace Eigen {
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14 |
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15 | template<typename MatrixType> class MatrixPower;
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16 |
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17 | template<typename MatrixType>
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18 | class MatrixPowerRetval : public ReturnByValue< MatrixPowerRetval<MatrixType> >
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19 | {
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20 | public:
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21 | typedef typename MatrixType::RealScalar RealScalar;
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22 | typedef typename MatrixType::Index Index;
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23 |
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24 | MatrixPowerRetval(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p)
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25 | { }
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26 |
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27 | template<typename ResultType>
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28 | inline void evalTo(ResultType& res) const
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29 | { m_pow.compute(res, m_p); }
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30 |
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31 | Index rows() const { return m_pow.rows(); }
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32 | Index cols() const { return m_pow.cols(); }
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33 |
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34 | private:
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35 | MatrixPower<MatrixType>& m_pow;
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36 | const RealScalar m_p;
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37 | MatrixPowerRetval& operator=(const MatrixPowerRetval&);
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38 | };
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39 |
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40 | template<typename MatrixType>
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41 | class MatrixPowerAtomic
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42 | {
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43 | private:
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44 | enum {
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45 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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46 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
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47 | };
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48 | typedef typename MatrixType::Scalar Scalar;
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49 | typedef typename MatrixType::RealScalar RealScalar;
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50 | typedef std::complex<RealScalar> ComplexScalar;
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51 | typedef typename MatrixType::Index Index;
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52 | typedef Array<Scalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> ArrayType;
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53 |
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54 | const MatrixType& m_A;
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55 | RealScalar m_p;
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56 |
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57 | void computePade(int degree, const MatrixType& IminusT, MatrixType& res) const;
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58 | void compute2x2(MatrixType& res, RealScalar p) const;
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59 | void computeBig(MatrixType& res) const;
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60 | static int getPadeDegree(float normIminusT);
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61 | static int getPadeDegree(double normIminusT);
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62 | static int getPadeDegree(long double normIminusT);
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63 | static ComplexScalar computeSuperDiag(const ComplexScalar&, const ComplexScalar&, RealScalar p);
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64 | static RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p);
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65 |
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66 | public:
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67 | MatrixPowerAtomic(const MatrixType& T, RealScalar p);
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68 | void compute(MatrixType& res) const;
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69 | };
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70 |
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71 | template<typename MatrixType>
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72 | MatrixPowerAtomic<MatrixType>::MatrixPowerAtomic(const MatrixType& T, RealScalar p) :
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73 | m_A(T), m_p(p)
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74 | { eigen_assert(T.rows() == T.cols()); }
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75 |
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76 | template<typename MatrixType>
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77 | void MatrixPowerAtomic<MatrixType>::compute(MatrixType& res) const
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78 | {
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79 | res.resizeLike(m_A);
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80 | switch (m_A.rows()) {
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81 | case 0:
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82 | break;
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83 | case 1:
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84 | res(0,0) = std::pow(m_A(0,0), m_p);
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85 | break;
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86 | case 2:
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87 | compute2x2(res, m_p);
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88 | break;
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89 | default:
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90 | computeBig(res);
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91 | }
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92 | }
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93 |
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94 | template<typename MatrixType>
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95 | void MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, MatrixType& res) const
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96 | {
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97 | int i = degree<<1;
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98 | res = (m_p-degree) / ((i-1)<<1) * IminusT;
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99 | for (--i; i; --i) {
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100 | res = (MatrixType::Identity(IminusT.rows(), IminusT.cols()) + res).template triangularView<Upper>()
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101 | .solve((i==1 ? -m_p : i&1 ? (-m_p-(i>>1))/(i<<1) : (m_p-(i>>1))/((i-1)<<1)) * IminusT).eval();
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102 | }
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103 | res += MatrixType::Identity(IminusT.rows(), IminusT.cols());
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104 | }
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105 |
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106 | // This function assumes that res has the correct size (see bug 614)
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107 | template<typename MatrixType>
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108 | void MatrixPowerAtomic<MatrixType>::compute2x2(MatrixType& res, RealScalar p) const
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109 | {
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110 | using std::abs;
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111 | using std::pow;
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112 |
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113 | res.coeffRef(0,0) = pow(m_A.coeff(0,0), p);
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114 |
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115 | for (Index i=1; i < m_A.cols(); ++i) {
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116 | res.coeffRef(i,i) = pow(m_A.coeff(i,i), p);
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117 | if (m_A.coeff(i-1,i-1) == m_A.coeff(i,i))
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118 | res.coeffRef(i-1,i) = p * pow(m_A.coeff(i,i), p-1);
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119 | else if (2*abs(m_A.coeff(i-1,i-1)) < abs(m_A.coeff(i,i)) || 2*abs(m_A.coeff(i,i)) < abs(m_A.coeff(i-1,i-1)))
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120 | res.coeffRef(i-1,i) = (res.coeff(i,i)-res.coeff(i-1,i-1)) / (m_A.coeff(i,i)-m_A.coeff(i-1,i-1));
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121 | else
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122 | res.coeffRef(i-1,i) = computeSuperDiag(m_A.coeff(i,i), m_A.coeff(i-1,i-1), p);
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123 | res.coeffRef(i-1,i) *= m_A.coeff(i-1,i);
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124 | }
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125 | }
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126 |
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127 | template<typename MatrixType>
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128 | void MatrixPowerAtomic<MatrixType>::computeBig(MatrixType& res) const
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129 | {
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130 | const int digits = std::numeric_limits<RealScalar>::digits;
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131 | const RealScalar maxNormForPade = digits <= 24? 4.3386528e-1f: // sigle precision
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132 | digits <= 53? 2.789358995219730e-1: // double precision
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133 | digits <= 64? 2.4471944416607995472e-1L: // extended precision
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134 | digits <= 106? 1.1016843812851143391275867258512e-1L: // double-double
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135 | 9.134603732914548552537150753385375e-2L; // quadruple precision
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136 | MatrixType IminusT, sqrtT, T = m_A.template triangularView<Upper>();
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137 | RealScalar normIminusT;
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138 | int degree, degree2, numberOfSquareRoots = 0;
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139 | bool hasExtraSquareRoot = false;
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140 |
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141 | /* FIXME
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142 | * For singular T, norm(I - T) >= 1 but maxNormForPade < 1, leads to infinite
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143 | * loop. We should move 0 eigenvalues to bottom right corner. We need not
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144 | * worry about tiny values (e.g. 1e-300) because they will reach 1 if
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145 | * repetitively sqrt'ed.
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146 | *
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147 | * If the 0 eigenvalues are semisimple, they can form a 0 matrix at the
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148 | * bottom right corner.
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149 | *
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150 | * [ T A ]^p [ T^p (T^-1 T^p A) ]
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151 | * [ ] = [ ]
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152 | * [ 0 0 ] [ 0 0 ]
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153 | */
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154 | for (Index i=0; i < m_A.cols(); ++i)
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155 | eigen_assert(m_A(i,i) != RealScalar(0));
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156 |
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157 | while (true) {
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158 | IminusT = MatrixType::Identity(m_A.rows(), m_A.cols()) - T;
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159 | normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff();
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160 | if (normIminusT < maxNormForPade) {
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161 | degree = getPadeDegree(normIminusT);
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162 | degree2 = getPadeDegree(normIminusT/2);
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163 | if (degree - degree2 <= 1 || hasExtraSquareRoot)
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164 | break;
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165 | hasExtraSquareRoot = true;
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166 | }
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167 | MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
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168 | T = sqrtT.template triangularView<Upper>();
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169 | ++numberOfSquareRoots;
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170 | }
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171 | computePade(degree, IminusT, res);
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172 |
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173 | for (; numberOfSquareRoots; --numberOfSquareRoots) {
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174 | compute2x2(res, std::ldexp(m_p, -numberOfSquareRoots));
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175 | res = res.template triangularView<Upper>() * res;
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176 | }
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177 | compute2x2(res, m_p);
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178 | }
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179 |
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180 | template<typename MatrixType>
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181 | inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(float normIminusT)
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182 | {
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183 | const float maxNormForPade[] = { 2.8064004e-1f /* degree = 3 */ , 4.3386528e-1f };
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184 | int degree = 3;
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185 | for (; degree <= 4; ++degree)
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186 | if (normIminusT <= maxNormForPade[degree - 3])
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187 | break;
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188 | return degree;
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189 | }
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190 |
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191 | template<typename MatrixType>
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192 | inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(double normIminusT)
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193 | {
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194 | const double maxNormForPade[] = { 1.884160592658218e-2 /* degree = 3 */ , 6.038881904059573e-2, 1.239917516308172e-1,
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195 | 1.999045567181744e-1, 2.789358995219730e-1 };
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196 | int degree = 3;
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197 | for (; degree <= 7; ++degree)
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198 | if (normIminusT <= maxNormForPade[degree - 3])
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199 | break;
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200 | return degree;
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201 | }
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202 |
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203 | template<typename MatrixType>
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204 | inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(long double normIminusT)
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205 | {
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206 | #if LDBL_MANT_DIG == 53
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207 | const int maxPadeDegree = 7;
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208 | const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L,
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209 | 1.999045567181744e-1L, 2.789358995219730e-1L };
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210 | #elif LDBL_MANT_DIG <= 64
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211 | const int maxPadeDegree = 8;
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212 | const double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L,
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213 | 6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L };
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214 | #elif LDBL_MANT_DIG <= 106
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215 | const int maxPadeDegree = 10;
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216 | const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ ,
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217 | 1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L,
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218 | 2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L,
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219 | 1.1016843812851143391275867258512e-1L };
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220 | #else
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221 | const int maxPadeDegree = 10;
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222 | const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ ,
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223 | 6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L,
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224 | 9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L,
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225 | 3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L,
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226 | 9.134603732914548552537150753385375e-2L };
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227 | #endif
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228 | int degree = 3;
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229 | for (; degree <= maxPadeDegree; ++degree)
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230 | if (normIminusT <= maxNormForPade[degree - 3])
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231 | break;
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232 | return degree;
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233 | }
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234 |
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235 | template<typename MatrixType>
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236 | inline typename MatrixPowerAtomic<MatrixType>::ComplexScalar
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237 | MatrixPowerAtomic<MatrixType>::computeSuperDiag(const ComplexScalar& curr, const ComplexScalar& prev, RealScalar p)
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238 | {
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239 | ComplexScalar logCurr = std::log(curr);
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240 | ComplexScalar logPrev = std::log(prev);
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241 | int unwindingNumber = std::ceil((numext::imag(logCurr - logPrev) - M_PI) / (2*M_PI));
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242 | ComplexScalar w = numext::atanh2(curr - prev, curr + prev) + ComplexScalar(0, M_PI*unwindingNumber);
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243 | return RealScalar(2) * std::exp(RealScalar(0.5) * p * (logCurr + logPrev)) * std::sinh(p * w) / (curr - prev);
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244 | }
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245 |
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246 | template<typename MatrixType>
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247 | inline typename MatrixPowerAtomic<MatrixType>::RealScalar
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248 | MatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev, RealScalar p)
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249 | {
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250 | RealScalar w = numext::atanh2(curr - prev, curr + prev);
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251 | return 2 * std::exp(p * (std::log(curr) + std::log(prev)) / 2) * std::sinh(p * w) / (curr - prev);
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252 | }
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253 |
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254 | /**
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255 | * \ingroup MatrixFunctions_Module
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256 | *
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257 | * \brief Class for computing matrix powers.
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258 | *
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259 | * \tparam MatrixType type of the base, expected to be an instantiation
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260 | * of the Matrix class template.
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261 | *
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262 | * This class is capable of computing real/complex matrices raised to
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263 | * an arbitrary real power. Meanwhile, it saves the result of Schur
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264 | * decomposition if an non-integral power has even been calculated.
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265 | * Therefore, if you want to compute multiple (>= 2) matrix powers
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266 | * for the same matrix, using the class directly is more efficient than
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267 | * calling MatrixBase::pow().
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268 | *
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269 | * Example:
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270 | * \include MatrixPower_optimal.cpp
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271 | * Output: \verbinclude MatrixPower_optimal.out
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272 | */
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273 | template<typename MatrixType>
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274 | class MatrixPower
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275 | {
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276 | private:
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277 | enum {
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278 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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279 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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280 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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281 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
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282 | };
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283 | typedef typename MatrixType::Scalar Scalar;
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284 | typedef typename MatrixType::RealScalar RealScalar;
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285 | typedef typename MatrixType::Index Index;
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286 |
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287 | public:
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288 | /**
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289 | * \brief Constructor.
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290 | *
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291 | * \param[in] A the base of the matrix power.
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292 | *
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293 | * The class stores a reference to A, so it should not be changed
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294 | * (or destroyed) before evaluation.
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295 | */
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296 | explicit MatrixPower(const MatrixType& A) : m_A(A), m_conditionNumber(0)
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297 | { eigen_assert(A.rows() == A.cols()); }
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298 |
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299 | /**
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300 | * \brief Returns the matrix power.
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301 | *
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302 | * \param[in] p exponent, a real scalar.
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303 | * \return The expression \f$ A^p \f$, where A is specified in the
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304 | * constructor.
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305 | */
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306 | const MatrixPowerRetval<MatrixType> operator()(RealScalar p)
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307 | { return MatrixPowerRetval<MatrixType>(*this, p); }
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308 |
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309 | /**
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310 | * \brief Compute the matrix power.
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311 | *
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312 | * \param[in] p exponent, a real scalar.
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313 | * \param[out] res \f$ A^p \f$ where A is specified in the
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314 | * constructor.
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315 | */
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316 | template<typename ResultType>
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317 | void compute(ResultType& res, RealScalar p);
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318 |
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319 | Index rows() const { return m_A.rows(); }
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320 | Index cols() const { return m_A.cols(); }
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321 |
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322 | private:
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323 | typedef std::complex<RealScalar> ComplexScalar;
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324 | typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, MatrixType::Options,
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325 | MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrix;
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326 |
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327 | typename MatrixType::Nested m_A;
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328 | MatrixType m_tmp;
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329 | ComplexMatrix m_T, m_U, m_fT;
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330 | RealScalar m_conditionNumber;
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331 |
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332 | RealScalar modfAndInit(RealScalar, RealScalar*);
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333 |
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334 | template<typename ResultType>
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335 | void computeIntPower(ResultType&, RealScalar);
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336 |
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337 | template<typename ResultType>
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338 | void computeFracPower(ResultType&, RealScalar);
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339 |
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340 | template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
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341 | static void revertSchur(
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342 | Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
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343 | const ComplexMatrix& T,
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344 | const ComplexMatrix& U);
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345 |
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346 | template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
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347 | static void revertSchur(
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348 | Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
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349 | const ComplexMatrix& T,
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350 | const ComplexMatrix& U);
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351 | };
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352 |
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353 | template<typename MatrixType>
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354 | template<typename ResultType>
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355 | void MatrixPower<MatrixType>::compute(ResultType& res, RealScalar p)
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356 | {
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357 | switch (cols()) {
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358 | case 0:
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359 | break;
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360 | case 1:
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361 | res(0,0) = std::pow(m_A.coeff(0,0), p);
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362 | break;
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363 | default:
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364 | RealScalar intpart, x = modfAndInit(p, &intpart);
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365 | computeIntPower(res, intpart);
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366 | computeFracPower(res, x);
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367 | }
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368 | }
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369 |
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370 | template<typename MatrixType>
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371 | typename MatrixPower<MatrixType>::RealScalar
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372 | MatrixPower<MatrixType>::modfAndInit(RealScalar x, RealScalar* intpart)
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373 | {
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374 | typedef Array<RealScalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> RealArray;
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375 |
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376 | *intpart = std::floor(x);
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377 | RealScalar res = x - *intpart;
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378 |
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379 | if (!m_conditionNumber && res) {
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380 | const ComplexSchur<MatrixType> schurOfA(m_A);
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381 | m_T = schurOfA.matrixT();
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382 | m_U = schurOfA.matrixU();
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383 |
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384 | const RealArray absTdiag = m_T.diagonal().array().abs();
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385 | m_conditionNumber = absTdiag.maxCoeff() / absTdiag.minCoeff();
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386 | }
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387 |
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388 | if (res>RealScalar(0.5) && res>(1-res)*std::pow(m_conditionNumber, res)) {
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389 | --res;
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390 | ++*intpart;
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391 | }
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392 | return res;
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393 | }
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394 |
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395 | template<typename MatrixType>
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396 | template<typename ResultType>
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397 | void MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p)
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398 | {
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399 | RealScalar pp = std::abs(p);
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400 |
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401 | if (p<0) m_tmp = m_A.inverse();
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402 | else m_tmp = m_A;
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403 |
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404 | res = MatrixType::Identity(rows(), cols());
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405 | while (pp >= 1) {
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406 | if (std::fmod(pp, 2) >= 1)
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407 | res = m_tmp * res;
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408 | m_tmp *= m_tmp;
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409 | pp /= 2;
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410 | }
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411 | }
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412 |
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413 | template<typename MatrixType>
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414 | template<typename ResultType>
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415 | void MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p)
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416 | {
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417 | if (p) {
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418 | eigen_assert(m_conditionNumber);
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419 | MatrixPowerAtomic<ComplexMatrix>(m_T, p).compute(m_fT);
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420 | revertSchur(m_tmp, m_fT, m_U);
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421 | res = m_tmp * res;
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422 | }
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423 | }
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424 |
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425 | template<typename MatrixType>
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426 | template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
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427 | inline void MatrixPower<MatrixType>::revertSchur(
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428 | Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
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429 | const ComplexMatrix& T,
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430 | const ComplexMatrix& U)
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431 | { res.noalias() = U * (T.template triangularView<Upper>() * U.adjoint()); }
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432 |
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433 | template<typename MatrixType>
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434 | template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
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435 | inline void MatrixPower<MatrixType>::revertSchur(
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436 | Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
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437 | const ComplexMatrix& T,
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438 | const ComplexMatrix& U)
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439 | { res.noalias() = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); }
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440 |
|
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441 | /**
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442 | * \ingroup MatrixFunctions_Module
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443 | *
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444 | * \brief Proxy for the matrix power of some matrix (expression).
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445 | *
|
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446 | * \tparam Derived type of the base, a matrix (expression).
|
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447 | *
|
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448 | * This class holds the arguments to the matrix power until it is
|
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449 | * assigned or evaluated for some other reason (so the argument
|
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450 | * should not be changed in the meantime). It is the return type of
|
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451 | * MatrixBase::pow() and related functions and most of the
|
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452 | * time this is the only way it is used.
|
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453 | */
|
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454 | template<typename Derived>
|
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455 | class MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Derived> >
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456 | {
|
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457 | public:
|
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458 | typedef typename Derived::PlainObject PlainObject;
|
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459 | typedef typename Derived::RealScalar RealScalar;
|
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460 | typedef typename Derived::Index Index;
|
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461 |
|
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462 | /**
|
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463 | * \brief Constructor.
|
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464 | *
|
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465 | * \param[in] A %Matrix (expression), the base of the matrix power.
|
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466 | * \param[in] p scalar, the exponent of the matrix power.
|
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467 | */
|
---|
468 | MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p)
|
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469 | { }
|
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470 |
|
---|
471 | /**
|
---|
472 | * \brief Compute the matrix power.
|
---|
473 | *
|
---|
474 | * \param[out] result \f$ A^p \f$ where \p A and \p p are as in the
|
---|
475 | * constructor.
|
---|
476 | */
|
---|
477 | template<typename ResultType>
|
---|
478 | inline void evalTo(ResultType& res) const
|
---|
479 | { MatrixPower<PlainObject>(m_A.eval()).compute(res, m_p); }
|
---|
480 |
|
---|
481 | Index rows() const { return m_A.rows(); }
|
---|
482 | Index cols() const { return m_A.cols(); }
|
---|
483 |
|
---|
484 | private:
|
---|
485 | const Derived& m_A;
|
---|
486 | const RealScalar m_p;
|
---|
487 | MatrixPowerReturnValue& operator=(const MatrixPowerReturnValue&);
|
---|
488 | };
|
---|
489 |
|
---|
490 | namespace internal {
|
---|
491 |
|
---|
492 | template<typename MatrixPowerType>
|
---|
493 | struct traits< MatrixPowerRetval<MatrixPowerType> >
|
---|
494 | { typedef typename MatrixPowerType::PlainObject ReturnType; };
|
---|
495 |
|
---|
496 | template<typename Derived>
|
---|
497 | struct traits< MatrixPowerReturnValue<Derived> >
|
---|
498 | { typedef typename Derived::PlainObject ReturnType; };
|
---|
499 |
|
---|
500 | }
|
---|
501 |
|
---|
502 | template<typename Derived>
|
---|
503 | const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const
|
---|
504 | { return MatrixPowerReturnValue<Derived>(derived(), p); }
|
---|
505 |
|
---|
506 | } // namespace Eigen
|
---|
507 |
|
---|
508 | #endif // EIGEN_MATRIX_POWER
|
---|