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) 2009, 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
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5 | // Copyright (C) 2011 Chen-Pang He <jdh8@ms63.hinet.net>
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6 | //
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7 | // This Source Code Form is subject to the terms of the Mozilla
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8 | // Public License v. 2.0. If a copy of the MPL was not distributed
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9 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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10 |
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11 | #ifndef EIGEN_MATRIX_EXPONENTIAL
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12 | #define EIGEN_MATRIX_EXPONENTIAL
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13 |
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14 | #include "StemFunction.h"
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15 |
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16 | namespace Eigen {
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17 |
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18 | /** \ingroup MatrixFunctions_Module
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19 | * \brief Class for computing the matrix exponential.
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20 | * \tparam MatrixType type of the argument of the exponential,
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21 | * expected to be an instantiation of the Matrix class template.
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22 | */
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23 | template <typename MatrixType>
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24 | class MatrixExponential {
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25 |
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26 | public:
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27 |
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28 | /** \brief Constructor.
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29 | *
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30 | * The class stores a reference to \p M, so it should not be
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31 | * changed (or destroyed) before compute() is called.
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32 | *
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33 | * \param[in] M matrix whose exponential is to be computed.
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34 | */
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35 | MatrixExponential(const MatrixType &M);
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36 |
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37 | /** \brief Computes the matrix exponential.
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38 | *
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39 | * \param[out] result the matrix exponential of \p M in the constructor.
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40 | */
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41 | template <typename ResultType>
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42 | void compute(ResultType &result);
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43 |
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44 | private:
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45 |
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46 | // Prevent copying
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47 | MatrixExponential(const MatrixExponential&);
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48 | MatrixExponential& operator=(const MatrixExponential&);
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49 |
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50 | /** \brief Compute the (3,3)-Padé approximant to the exponential.
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51 | *
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52 | * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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53 | * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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54 | *
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55 | * \param[in] A Argument of matrix exponential
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56 | */
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57 | void pade3(const MatrixType &A);
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58 |
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59 | /** \brief Compute the (5,5)-Padé approximant to the exponential.
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60 | *
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61 | * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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62 | * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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63 | *
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64 | * \param[in] A Argument of matrix exponential
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65 | */
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66 | void pade5(const MatrixType &A);
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67 |
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68 | /** \brief Compute the (7,7)-Padé approximant to the exponential.
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69 | *
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70 | * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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71 | * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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72 | *
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73 | * \param[in] A Argument of matrix exponential
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74 | */
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75 | void pade7(const MatrixType &A);
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76 |
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77 | /** \brief Compute the (9,9)-Padé approximant to the exponential.
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78 | *
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79 | * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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80 | * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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81 | *
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82 | * \param[in] A Argument of matrix exponential
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83 | */
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84 | void pade9(const MatrixType &A);
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85 |
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86 | /** \brief Compute the (13,13)-Padé approximant to the exponential.
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87 | *
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88 | * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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89 | * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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90 | *
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91 | * \param[in] A Argument of matrix exponential
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92 | */
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93 | void pade13(const MatrixType &A);
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94 |
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95 | /** \brief Compute the (17,17)-Padé approximant to the exponential.
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96 | *
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97 | * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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98 | * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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99 | *
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100 | * This function activates only if your long double is double-double or quadruple.
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101 | *
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102 | * \param[in] A Argument of matrix exponential
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103 | */
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104 | void pade17(const MatrixType &A);
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105 |
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106 | /** \brief Compute Padé approximant to the exponential.
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107 | *
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108 | * Computes \c m_U, \c m_V and \c m_squarings such that
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109 | * \f$ (V+U)(V-U)^{-1} \f$ is a Padé of
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110 | * \f$ \exp(2^{-\mbox{squarings}}M) \f$ around \f$ M = 0 \f$. The
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111 | * degree of the Padé approximant and the value of
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112 | * squarings are chosen such that the approximation error is no
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113 | * more than the round-off error.
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114 | *
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115 | * The argument of this function should correspond with the (real
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116 | * part of) the entries of \c m_M. It is used to select the
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117 | * correct implementation using overloading.
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118 | */
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119 | void computeUV(double);
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120 |
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121 | /** \brief Compute Padé approximant to the exponential.
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122 | *
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123 | * \sa computeUV(double);
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124 | */
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125 | void computeUV(float);
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126 |
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127 | /** \brief Compute Padé approximant to the exponential.
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128 | *
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129 | * \sa computeUV(double);
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130 | */
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131 | void computeUV(long double);
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132 |
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133 | typedef typename internal::traits<MatrixType>::Scalar Scalar;
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134 | typedef typename NumTraits<Scalar>::Real RealScalar;
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135 | typedef typename std::complex<RealScalar> ComplexScalar;
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136 |
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137 | /** \brief Reference to matrix whose exponential is to be computed. */
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138 | typename internal::nested<MatrixType>::type m_M;
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139 |
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140 | /** \brief Odd-degree terms in numerator of Padé approximant. */
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141 | MatrixType m_U;
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142 |
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143 | /** \brief Even-degree terms in numerator of Padé approximant. */
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144 | MatrixType m_V;
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145 |
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146 | /** \brief Used for temporary storage. */
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147 | MatrixType m_tmp1;
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148 |
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149 | /** \brief Used for temporary storage. */
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150 | MatrixType m_tmp2;
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151 |
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152 | /** \brief Identity matrix of the same size as \c m_M. */
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153 | MatrixType m_Id;
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154 |
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155 | /** \brief Number of squarings required in the last step. */
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156 | int m_squarings;
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157 |
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158 | /** \brief L1 norm of m_M. */
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159 | RealScalar m_l1norm;
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160 | };
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161 |
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162 | template <typename MatrixType>
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163 | MatrixExponential<MatrixType>::MatrixExponential(const MatrixType &M) :
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164 | m_M(M),
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165 | m_U(M.rows(),M.cols()),
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166 | m_V(M.rows(),M.cols()),
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167 | m_tmp1(M.rows(),M.cols()),
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168 | m_tmp2(M.rows(),M.cols()),
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169 | m_Id(MatrixType::Identity(M.rows(), M.cols())),
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170 | m_squarings(0),
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171 | m_l1norm(M.cwiseAbs().colwise().sum().maxCoeff())
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172 | {
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173 | /* empty body */
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174 | }
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175 |
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176 | template <typename MatrixType>
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177 | template <typename ResultType>
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178 | void MatrixExponential<MatrixType>::compute(ResultType &result)
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179 | {
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180 | #if LDBL_MANT_DIG > 112 // rarely happens
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181 | if(sizeof(RealScalar) > 14) {
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182 | result = m_M.matrixFunction(StdStemFunctions<ComplexScalar>::exp);
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183 | return;
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184 | }
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185 | #endif
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186 | computeUV(RealScalar());
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187 | m_tmp1 = m_U + m_V; // numerator of Pade approximant
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188 | m_tmp2 = -m_U + m_V; // denominator of Pade approximant
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189 | result = m_tmp2.partialPivLu().solve(m_tmp1);
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190 | for (int i=0; i<m_squarings; i++)
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191 | result *= result; // undo scaling by repeated squaring
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192 | }
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193 |
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194 | template <typename MatrixType>
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195 | EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade3(const MatrixType &A)
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196 | {
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197 | const RealScalar b[] = {120., 60., 12., 1.};
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198 | m_tmp1.noalias() = A * A;
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199 | m_tmp2 = b[3]*m_tmp1 + b[1]*m_Id;
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200 | m_U.noalias() = A * m_tmp2;
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201 | m_V = b[2]*m_tmp1 + b[0]*m_Id;
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202 | }
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203 |
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204 | template <typename MatrixType>
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205 | EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade5(const MatrixType &A)
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206 | {
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207 | const RealScalar b[] = {30240., 15120., 3360., 420., 30., 1.};
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208 | MatrixType A2 = A * A;
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209 | m_tmp1.noalias() = A2 * A2;
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210 | m_tmp2 = b[5]*m_tmp1 + b[3]*A2 + b[1]*m_Id;
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211 | m_U.noalias() = A * m_tmp2;
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212 | m_V = b[4]*m_tmp1 + b[2]*A2 + b[0]*m_Id;
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213 | }
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214 |
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215 | template <typename MatrixType>
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216 | EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade7(const MatrixType &A)
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217 | {
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218 | const RealScalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.};
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219 | MatrixType A2 = A * A;
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220 | MatrixType A4 = A2 * A2;
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221 | m_tmp1.noalias() = A4 * A2;
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222 | m_tmp2 = b[7]*m_tmp1 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
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223 | m_U.noalias() = A * m_tmp2;
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224 | m_V = b[6]*m_tmp1 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
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225 | }
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226 |
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227 | template <typename MatrixType>
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228 | EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade9(const MatrixType &A)
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229 | {
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230 | const RealScalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240.,
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231 | 2162160., 110880., 3960., 90., 1.};
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232 | MatrixType A2 = A * A;
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233 | MatrixType A4 = A2 * A2;
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234 | MatrixType A6 = A4 * A2;
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235 | m_tmp1.noalias() = A6 * A2;
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236 | m_tmp2 = b[9]*m_tmp1 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
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237 | m_U.noalias() = A * m_tmp2;
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238 | m_V = b[8]*m_tmp1 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
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239 | }
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240 |
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241 | template <typename MatrixType>
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242 | EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade13(const MatrixType &A)
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243 | {
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244 | const RealScalar b[] = {64764752532480000., 32382376266240000., 7771770303897600.,
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245 | 1187353796428800., 129060195264000., 10559470521600., 670442572800.,
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246 | 33522128640., 1323241920., 40840800., 960960., 16380., 182., 1.};
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247 | MatrixType A2 = A * A;
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248 | MatrixType A4 = A2 * A2;
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249 | m_tmp1.noalias() = A4 * A2;
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250 | m_V = b[13]*m_tmp1 + b[11]*A4 + b[9]*A2; // used for temporary storage
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251 | m_tmp2.noalias() = m_tmp1 * m_V;
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252 | m_tmp2 += b[7]*m_tmp1 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
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253 | m_U.noalias() = A * m_tmp2;
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254 | m_tmp2 = b[12]*m_tmp1 + b[10]*A4 + b[8]*A2;
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255 | m_V.noalias() = m_tmp1 * m_tmp2;
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256 | m_V += b[6]*m_tmp1 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
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257 | }
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258 |
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259 | #if LDBL_MANT_DIG > 64
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260 | template <typename MatrixType>
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261 | EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade17(const MatrixType &A)
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262 | {
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263 | const RealScalar b[] = {830034394580628357120000.L, 415017197290314178560000.L,
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264 | 100610229646136770560000.L, 15720348382208870400000.L,
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265 | 1774878043152614400000.L, 153822763739893248000.L, 10608466464820224000.L,
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266 | 595373117923584000.L, 27563570274240000.L, 1060137318240000.L,
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267 | 33924394183680.L, 899510451840.L, 19554575040.L, 341863200.L, 4651200.L,
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268 | 46512.L, 306.L, 1.L};
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269 | MatrixType A2 = A * A;
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270 | MatrixType A4 = A2 * A2;
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271 | MatrixType A6 = A4 * A2;
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272 | m_tmp1.noalias() = A4 * A4;
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273 | m_V = b[17]*m_tmp1 + b[15]*A6 + b[13]*A4 + b[11]*A2; // used for temporary storage
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274 | m_tmp2.noalias() = m_tmp1 * m_V;
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275 | m_tmp2 += b[9]*m_tmp1 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
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276 | m_U.noalias() = A * m_tmp2;
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277 | m_tmp2 = b[16]*m_tmp1 + b[14]*A6 + b[12]*A4 + b[10]*A2;
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278 | m_V.noalias() = m_tmp1 * m_tmp2;
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279 | m_V += b[8]*m_tmp1 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
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280 | }
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281 | #endif
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282 |
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283 | template <typename MatrixType>
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284 | void MatrixExponential<MatrixType>::computeUV(float)
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285 | {
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286 | using std::frexp;
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287 | using std::pow;
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288 | if (m_l1norm < 4.258730016922831e-001) {
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289 | pade3(m_M);
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290 | } else if (m_l1norm < 1.880152677804762e+000) {
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291 | pade5(m_M);
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292 | } else {
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293 | const float maxnorm = 3.925724783138660f;
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294 | frexp(m_l1norm / maxnorm, &m_squarings);
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295 | if (m_squarings < 0) m_squarings = 0;
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296 | MatrixType A = m_M / Scalar(pow(2, m_squarings));
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297 | pade7(A);
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298 | }
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299 | }
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300 |
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301 | template <typename MatrixType>
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302 | void MatrixExponential<MatrixType>::computeUV(double)
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303 | {
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304 | using std::frexp;
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305 | using std::pow;
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306 | if (m_l1norm < 1.495585217958292e-002) {
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307 | pade3(m_M);
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308 | } else if (m_l1norm < 2.539398330063230e-001) {
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309 | pade5(m_M);
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310 | } else if (m_l1norm < 9.504178996162932e-001) {
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311 | pade7(m_M);
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312 | } else if (m_l1norm < 2.097847961257068e+000) {
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313 | pade9(m_M);
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314 | } else {
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315 | const double maxnorm = 5.371920351148152;
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316 | frexp(m_l1norm / maxnorm, &m_squarings);
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317 | if (m_squarings < 0) m_squarings = 0;
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318 | MatrixType A = m_M / Scalar(pow(2, m_squarings));
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319 | pade13(A);
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320 | }
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321 | }
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322 |
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323 | template <typename MatrixType>
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324 | void MatrixExponential<MatrixType>::computeUV(long double)
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325 | {
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326 | using std::frexp;
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327 | using std::pow;
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328 | #if LDBL_MANT_DIG == 53 // double precision
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329 | computeUV(double());
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330 | #elif LDBL_MANT_DIG <= 64 // extended precision
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331 | if (m_l1norm < 4.1968497232266989671e-003L) {
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332 | pade3(m_M);
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333 | } else if (m_l1norm < 1.1848116734693823091e-001L) {
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334 | pade5(m_M);
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335 | } else if (m_l1norm < 5.5170388480686700274e-001L) {
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336 | pade7(m_M);
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337 | } else if (m_l1norm < 1.3759868875587845383e+000L) {
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338 | pade9(m_M);
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339 | } else {
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340 | const long double maxnorm = 4.0246098906697353063L;
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341 | frexp(m_l1norm / maxnorm, &m_squarings);
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342 | if (m_squarings < 0) m_squarings = 0;
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343 | MatrixType A = m_M / Scalar(pow(2, m_squarings));
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344 | pade13(A);
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345 | }
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346 | #elif LDBL_MANT_DIG <= 106 // double-double
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347 | if (m_l1norm < 3.2787892205607026992947488108213e-005L) {
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348 | pade3(m_M);
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349 | } else if (m_l1norm < 6.4467025060072760084130906076332e-003L) {
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350 | pade5(m_M);
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351 | } else if (m_l1norm < 6.8988028496595374751374122881143e-002L) {
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352 | pade7(m_M);
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353 | } else if (m_l1norm < 2.7339737518502231741495857201670e-001L) {
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354 | pade9(m_M);
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355 | } else if (m_l1norm < 1.3203382096514474905666448850278e+000L) {
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356 | pade13(m_M);
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357 | } else {
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358 | const long double maxnorm = 3.2579440895405400856599663723517L;
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359 | frexp(m_l1norm / maxnorm, &m_squarings);
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360 | if (m_squarings < 0) m_squarings = 0;
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361 | MatrixType A = m_M / pow(Scalar(2), m_squarings);
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362 | pade17(A);
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363 | }
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364 | #elif LDBL_MANT_DIG <= 112 // quadruple precison
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365 | if (m_l1norm < 1.639394610288918690547467954466970e-005L) {
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366 | pade3(m_M);
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367 | } else if (m_l1norm < 4.253237712165275566025884344433009e-003L) {
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368 | pade5(m_M);
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369 | } else if (m_l1norm < 5.125804063165764409885122032933142e-002L) {
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370 | pade7(m_M);
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371 | } else if (m_l1norm < 2.170000765161155195453205651889853e-001L) {
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372 | pade9(m_M);
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373 | } else if (m_l1norm < 1.125358383453143065081397882891878e+000L) {
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374 | pade13(m_M);
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375 | } else {
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376 | const long double maxnorm = 2.884233277829519311757165057717815L;
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377 | frexp(m_l1norm / maxnorm, &m_squarings);
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378 | if (m_squarings < 0) m_squarings = 0;
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379 | MatrixType A = m_M / Scalar(pow(2, m_squarings));
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380 | pade17(A);
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381 | }
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382 | #else
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383 | // this case should be handled in compute()
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384 | eigen_assert(false && "Bug in MatrixExponential");
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385 | #endif // LDBL_MANT_DIG
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386 | }
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387 |
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388 | /** \ingroup MatrixFunctions_Module
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389 | *
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390 | * \brief Proxy for the matrix exponential of some matrix (expression).
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391 | *
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392 | * \tparam Derived Type of the argument to the matrix exponential.
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393 | *
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394 | * This class holds the argument to the matrix exponential until it
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395 | * is assigned or evaluated for some other reason (so the argument
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396 | * should not be changed in the meantime). It is the return type of
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397 | * MatrixBase::exp() and most of the time this is the only way it is
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398 | * used.
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399 | */
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400 | template<typename Derived> struct MatrixExponentialReturnValue
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401 | : public ReturnByValue<MatrixExponentialReturnValue<Derived> >
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402 | {
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403 | typedef typename Derived::Index Index;
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404 | public:
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405 | /** \brief Constructor.
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406 | *
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407 | * \param[in] src %Matrix (expression) forming the argument of the
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408 | * matrix exponential.
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409 | */
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410 | MatrixExponentialReturnValue(const Derived& src) : m_src(src) { }
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411 |
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412 | /** \brief Compute the matrix exponential.
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413 | *
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414 | * \param[out] result the matrix exponential of \p src in the
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415 | * constructor.
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416 | */
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417 | template <typename ResultType>
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418 | inline void evalTo(ResultType& result) const
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419 | {
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420 | const typename Derived::PlainObject srcEvaluated = m_src.eval();
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421 | MatrixExponential<typename Derived::PlainObject> me(srcEvaluated);
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422 | me.compute(result);
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423 | }
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424 |
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425 | Index rows() const { return m_src.rows(); }
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426 | Index cols() const { return m_src.cols(); }
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427 |
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428 | protected:
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429 | const Derived& m_src;
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430 | private:
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431 | MatrixExponentialReturnValue& operator=(const MatrixExponentialReturnValue&);
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432 | };
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433 |
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434 | namespace internal {
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435 | template<typename Derived>
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436 | struct traits<MatrixExponentialReturnValue<Derived> >
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437 | {
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438 | typedef typename Derived::PlainObject ReturnType;
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439 | };
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440 | }
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441 |
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442 | template <typename Derived>
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443 | const MatrixExponentialReturnValue<Derived> MatrixBase<Derived>::exp() const
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444 | {
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445 | eigen_assert(rows() == cols());
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446 | return MatrixExponentialReturnValue<Derived>(derived());
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447 | }
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448 |
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449 | } // end namespace Eigen
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450 |
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451 | #endif // EIGEN_MATRIX_EXPONENTIAL
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