1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. 3 // 4 // Copyright (C) 2011, 2013 Jitse Niesen <jitse (at) maths.leeds.ac.uk> 5 // Copyright (C) 2011 Chen-Pang He <jdh8 (at) ms63.hinet.net> 6 // 7 // This Source Code Form is subject to the terms of the Mozilla 8 // Public License v. 2.0. If a copy of the MPL was not distributed 9 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 10 11 #ifndef EIGEN_MATRIX_LOGARITHM 12 #define EIGEN_MATRIX_LOGARITHM 13 14 namespace Eigen { 15 16 namespace internal { 17 18 template <typename Scalar> 19 struct matrix_log_min_pade_degree 20 { 21 static const int value = 3; 22 }; 23 24 template <typename Scalar> 25 struct matrix_log_max_pade_degree 26 { 27 typedef typename NumTraits<Scalar>::Real RealScalar; 28 static const int value = std::numeric_limits<RealScalar>::digits<= 24? 5: // single precision 29 std::numeric_limits<RealScalar>::digits<= 53? 7: // double precision 30 std::numeric_limits<RealScalar>::digits<= 64? 8: // extended precision 31 std::numeric_limits<RealScalar>::digits<=106? 10: // double-double 32 11; // quadruple precision 33 }; 34 35 /** \brief Compute logarithm of 2x2 triangular matrix. */ 36 template <typename MatrixType> 37 void matrix_log_compute_2x2(const MatrixType& A, MatrixType& result) 38 { 39 typedef typename MatrixType::Scalar Scalar; 40 typedef typename MatrixType::RealScalar RealScalar; 41 using std::abs; 42 using std::ceil; 43 using std::imag; 44 using std::log; 45 46 Scalar logA00 = log(A(0,0)); 47 Scalar logA11 = log(A(1,1)); 48 49 result(0,0) = logA00; 50 result(1,0) = Scalar(0); 51 result(1,1) = logA11; 52 53 Scalar y = A(1,1) - A(0,0); 54 if (y==Scalar(0)) 55 { 56 result(0,1) = A(0,1) / A(0,0); 57 } 58 else if ((abs(A(0,0)) < RealScalar(0.5)*abs(A(1,1))) || (abs(A(0,0)) > 2*abs(A(1,1)))) 59 { 60 result(0,1) = A(0,1) * (logA11 - logA00) / y; 61 } 62 else 63 { 64 // computation in previous branch is inaccurate if A(1,1) \approx A(0,0) 65 int unwindingNumber = static_cast<int>(ceil((imag(logA11 - logA00) - RealScalar(EIGEN_PI)) / RealScalar(2*EIGEN_PI))); 66 result(0,1) = A(0,1) * (numext::log1p(y/A(0,0)) + Scalar(0,2*EIGEN_PI*unwindingNumber)) / y; 67 } 68 } 69 70 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = float) */ 71 inline int matrix_log_get_pade_degree(float normTminusI) 72 { 73 const float maxNormForPade[] = { 2.5111573934555054e-1 /* degree = 3 */ , 4.0535837411880493e-1, 74 5.3149729967117310e-1 }; 75 const int minPadeDegree = matrix_log_min_pade_degree<float>::value; 76 const int maxPadeDegree = matrix_log_max_pade_degree<float>::value; 77 int degree = minPadeDegree; 78 for (; degree <= maxPadeDegree; ++degree) 79 if (normTminusI <= maxNormForPade[degree - minPadeDegree]) 80 break; 81 return degree; 82 } 83 84 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = double) */ 85 inline int matrix_log_get_pade_degree(double normTminusI) 86 { 87 const double maxNormForPade[] = { 1.6206284795015624e-2 /* degree = 3 */ , 5.3873532631381171e-2, 88 1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 }; 89 const int minPadeDegree = matrix_log_min_pade_degree<double>::value; 90 const int maxPadeDegree = matrix_log_max_pade_degree<double>::value; 91 int degree = minPadeDegree; 92 for (; degree <= maxPadeDegree; ++degree) 93 if (normTminusI <= maxNormForPade[degree - minPadeDegree]) 94 break; 95 return degree; 96 } 97 98 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = long double) */ 99 inline int matrix_log_get_pade_degree(long double normTminusI) 100 { 101 #if LDBL_MANT_DIG == 53 // double precision 102 const long double maxNormForPade[] = { 1.6206284795015624e-2L /* degree = 3 */ , 5.3873532631381171e-2L, 103 1.1352802267628681e-1L, 1.8662860613541288e-1L, 2.642960831111435e-1L }; 104 #elif LDBL_MANT_DIG <= 64 // extended precision 105 const long double maxNormForPade[] = { 5.48256690357782863103e-3L /* degree = 3 */, 2.34559162387971167321e-2L, 106 5.84603923897347449857e-2L, 1.08486423756725170223e-1L, 1.68385767881294446649e-1L, 107 2.32777776523703892094e-1L }; 108 #elif LDBL_MANT_DIG <= 106 // double-double 109 const long double maxNormForPade[] = { 8.58970550342939562202529664318890e-5L /* degree = 3 */, 110 9.34074328446359654039446552677759e-4L, 4.26117194647672175773064114582860e-3L, 111 1.21546224740281848743149666560464e-2L, 2.61100544998339436713088248557444e-2L, 112 4.66170074627052749243018566390567e-2L, 7.32585144444135027565872014932387e-2L, 113 1.05026503471351080481093652651105e-1L }; 114 #else // quadruple precision 115 const long double maxNormForPade[] = { 4.7419931187193005048501568167858103e-5L /* degree = 3 */, 116 5.8853168473544560470387769480192666e-4L, 2.9216120366601315391789493628113520e-3L, 117 8.8415758124319434347116734705174308e-3L, 1.9850836029449446668518049562565291e-2L, 118 3.6688019729653446926585242192447447e-2L, 5.9290962294020186998954055264528393e-2L, 119 8.6998436081634343903250580992127677e-2L, 1.1880960220216759245467951592883642e-1L }; 120 #endif 121 const int minPadeDegree = matrix_log_min_pade_degree<long double>::value; 122 const int maxPadeDegree = matrix_log_max_pade_degree<long double>::value; 123 int degree = minPadeDegree; 124 for (; degree <= maxPadeDegree; ++degree) 125 if (normTminusI <= maxNormForPade[degree - minPadeDegree]) 126 break; 127 return degree; 128 } 129 130 /* \brief Compute Pade approximation to matrix logarithm */ 131 template <typename MatrixType> 132 void matrix_log_compute_pade(MatrixType& result, const MatrixType& T, int degree) 133 { 134 typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar; 135 const int minPadeDegree = 3; 136 const int maxPadeDegree = 11; 137 assert(degree >= minPadeDegree && degree <= maxPadeDegree); 138 139 const RealScalar nodes[][maxPadeDegree] = { 140 { 0.1127016653792583114820734600217600L, 0.5000000000000000000000000000000000L, // degree 3 141 0.8872983346207416885179265399782400L }, 142 { 0.0694318442029737123880267555535953L, 0.3300094782075718675986671204483777L, // degree 4 143 0.6699905217924281324013328795516223L, 0.9305681557970262876119732444464048L }, 144 { 0.0469100770306680036011865608503035L, 0.2307653449471584544818427896498956L, // degree 5 145 0.5000000000000000000000000000000000L, 0.7692346550528415455181572103501044L, 146 0.9530899229693319963988134391496965L }, 147 { 0.0337652428984239860938492227530027L, 0.1693953067668677431693002024900473L, // degree 6 148 0.3806904069584015456847491391596440L, 0.6193095930415984543152508608403560L, 149 0.8306046932331322568306997975099527L, 0.9662347571015760139061507772469973L }, 150 { 0.0254460438286207377369051579760744L, 0.1292344072003027800680676133596058L, // degree 7 151 0.2970774243113014165466967939615193L, 0.5000000000000000000000000000000000L, 152 0.7029225756886985834533032060384807L, 0.8707655927996972199319323866403942L, 153 0.9745539561713792622630948420239256L }, 154 { 0.0198550717512318841582195657152635L, 0.1016667612931866302042230317620848L, // degree 8 155 0.2372337950418355070911304754053768L, 0.4082826787521750975302619288199080L, 156 0.5917173212478249024697380711800920L, 0.7627662049581644929088695245946232L, 157 0.8983332387068133697957769682379152L, 0.9801449282487681158417804342847365L }, 158 { 0.0159198802461869550822118985481636L, 0.0819844463366821028502851059651326L, // degree 9 159 0.1933142836497048013456489803292629L, 0.3378732882980955354807309926783317L, 160 0.5000000000000000000000000000000000L, 0.6621267117019044645192690073216683L, 161 0.8066857163502951986543510196707371L, 0.9180155536633178971497148940348674L, 162 0.9840801197538130449177881014518364L }, 163 { 0.0130467357414141399610179939577740L, 0.0674683166555077446339516557882535L, // degree 10 164 0.1602952158504877968828363174425632L, 0.2833023029353764046003670284171079L, 165 0.4255628305091843945575869994351400L, 0.5744371694908156054424130005648600L, 166 0.7166976970646235953996329715828921L, 0.8397047841495122031171636825574368L, 167 0.9325316833444922553660483442117465L, 0.9869532642585858600389820060422260L }, 168 { 0.0108856709269715035980309994385713L, 0.0564687001159523504624211153480364L, // degree 11 169 0.1349239972129753379532918739844233L, 0.2404519353965940920371371652706952L, 170 0.3652284220238275138342340072995692L, 0.5000000000000000000000000000000000L, 171 0.6347715779761724861657659927004308L, 0.7595480646034059079628628347293048L, 172 0.8650760027870246620467081260155767L, 0.9435312998840476495375788846519636L, 173 0.9891143290730284964019690005614287L } }; 174 175 const RealScalar weights[][maxPadeDegree] = { 176 { 0.2777777777777777777777777777777778L, 0.4444444444444444444444444444444444L, // degree 3 177 0.2777777777777777777777777777777778L }, 178 { 0.1739274225687269286865319746109997L, 0.3260725774312730713134680253890003L, // degree 4 179 0.3260725774312730713134680253890003L, 0.1739274225687269286865319746109997L }, 180 { 0.1184634425280945437571320203599587L, 0.2393143352496832340206457574178191L, // degree 5 181 0.2844444444444444444444444444444444L, 0.2393143352496832340206457574178191L, 182 0.1184634425280945437571320203599587L }, 183 { 0.0856622461895851725201480710863665L, 0.1803807865240693037849167569188581L, // degree 6 184 0.2339569672863455236949351719947755L, 0.2339569672863455236949351719947755L, 185 0.1803807865240693037849167569188581L, 0.0856622461895851725201480710863665L }, 186 { 0.0647424830844348466353057163395410L, 0.1398526957446383339507338857118898L, // degree 7 187 0.1909150252525594724751848877444876L, 0.2089795918367346938775510204081633L, 188 0.1909150252525594724751848877444876L, 0.1398526957446383339507338857118898L, 189 0.0647424830844348466353057163395410L }, 190 { 0.0506142681451881295762656771549811L, 0.1111905172266872352721779972131204L, // degree 8 191 0.1568533229389436436689811009933007L, 0.1813418916891809914825752246385978L, 192 0.1813418916891809914825752246385978L, 0.1568533229389436436689811009933007L, 193 0.1111905172266872352721779972131204L, 0.0506142681451881295762656771549811L }, 194 { 0.0406371941807872059859460790552618L, 0.0903240803474287020292360156214564L, // degree 9 195 0.1303053482014677311593714347093164L, 0.1561735385200014200343152032922218L, 196 0.1651196775006298815822625346434870L, 0.1561735385200014200343152032922218L, 197 0.1303053482014677311593714347093164L, 0.0903240803474287020292360156214564L, 198 0.0406371941807872059859460790552618L }, 199 { 0.0333356721543440687967844049466659L, 0.0747256745752902965728881698288487L, // degree 10 200 0.1095431812579910219977674671140816L, 0.1346333596549981775456134607847347L, 201 0.1477621123573764350869464973256692L, 0.1477621123573764350869464973256692L, 202 0.1346333596549981775456134607847347L, 0.1095431812579910219977674671140816L, 203 0.0747256745752902965728881698288487L, 0.0333356721543440687967844049466659L }, 204 { 0.0278342835580868332413768602212743L, 0.0627901847324523123173471496119701L, // degree 11 205 0.0931451054638671257130488207158280L, 0.1165968822959952399592618524215876L, 206 0.1314022722551233310903444349452546L, 0.1364625433889503153572417641681711L, 207 0.1314022722551233310903444349452546L, 0.1165968822959952399592618524215876L, 208 0.0931451054638671257130488207158280L, 0.0627901847324523123173471496119701L, 209 0.0278342835580868332413768602212743L } }; 210 211 MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows()); 212 result.setZero(T.rows(), T.rows()); 213 for (int k = 0; k < degree; ++k) { 214 RealScalar weight = weights[degree-minPadeDegree][k]; 215 RealScalar node = nodes[degree-minPadeDegree][k]; 216 result += weight * (MatrixType::Identity(T.rows(), T.rows()) + node * TminusI) 217 .template triangularView<Upper>().solve(TminusI); 218 } 219 } 220 221 /** \brief Compute logarithm of triangular matrices with size > 2. 222 * \details This uses a inverse scale-and-square algorithm. */ 223 template <typename MatrixType> 224 void matrix_log_compute_big(const MatrixType& A, MatrixType& result) 225 { 226 typedef typename MatrixType::Scalar Scalar; 227 typedef typename NumTraits<Scalar>::Real RealScalar; 228 using std::pow; 229 230 int numberOfSquareRoots = 0; 231 int numberOfExtraSquareRoots = 0; 232 int degree; 233 MatrixType T = A, sqrtT; 234 235 int maxPadeDegree = matrix_log_max_pade_degree<Scalar>::value; 236 const RealScalar maxNormForPade = maxPadeDegree<= 5? 5.3149729967117310e-1L: // single precision 237 maxPadeDegree<= 7? 2.6429608311114350e-1L: // double precision 238 maxPadeDegree<= 8? 2.32777776523703892094e-1L: // extended precision 239 maxPadeDegree<=10? 1.05026503471351080481093652651105e-1L: // double-double 240 1.1880960220216759245467951592883642e-1L; // quadruple precision 241 242 while (true) { 243 RealScalar normTminusI = (T - MatrixType::Identity(T.rows(), T.rows())).cwiseAbs().colwise().sum().maxCoeff(); 244 if (normTminusI < maxNormForPade) { 245 degree = matrix_log_get_pade_degree(normTminusI); 246 int degree2 = matrix_log_get_pade_degree(normTminusI / RealScalar(2)); 247 if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1)) 248 break; 249 ++numberOfExtraSquareRoots; 250 } 251 matrix_sqrt_triangular(T, sqrtT); 252 T = sqrtT.template triangularView<Upper>(); 253 ++numberOfSquareRoots; 254 } 255 256 matrix_log_compute_pade(result, T, degree); 257 result *= pow(RealScalar(2), numberOfSquareRoots); 258 } 259 260 /** \ingroup MatrixFunctions_Module 261 * \class MatrixLogarithmAtomic 262 * \brief Helper class for computing matrix logarithm of atomic matrices. 263 * 264 * Here, an atomic matrix is a triangular matrix whose diagonal entries are close to each other. 265 * 266 * \sa class MatrixFunctionAtomic, MatrixBase::log() 267 */ 268 template <typename MatrixType> 269 class MatrixLogarithmAtomic 270 { 271 public: 272 /** \brief Compute matrix logarithm of atomic matrix 273 * \param[in] A argument of matrix logarithm, should be upper triangular and atomic 274 * \returns The logarithm of \p A. 275 */ 276 MatrixType compute(const MatrixType& A); 277 }; 278 279 template <typename MatrixType> 280 MatrixType MatrixLogarithmAtomic<MatrixType>::compute(const MatrixType& A) 281 { 282 using std::log; 283 MatrixType result(A.rows(), A.rows()); 284 if (A.rows() == 1) 285 result(0,0) = log(A(0,0)); 286 else if (A.rows() == 2) 287 matrix_log_compute_2x2(A, result); 288 else 289 matrix_log_compute_big(A, result); 290 return result; 291 } 292 293 } // end of namespace internal 294 295 /** \ingroup MatrixFunctions_Module 296 * 297 * \brief Proxy for the matrix logarithm of some matrix (expression). 298 * 299 * \tparam Derived Type of the argument to the matrix function. 300 * 301 * This class holds the argument to the matrix function until it is 302 * assigned or evaluated for some other reason (so the argument 303 * should not be changed in the meantime). It is the return type of 304 * MatrixBase::log() and most of the time this is the only way it 305 * is used. 306 */ 307 template<typename Derived> class MatrixLogarithmReturnValue 308 : public ReturnByValue<MatrixLogarithmReturnValue<Derived> > 309 { 310 public: 311 typedef typename Derived::Scalar Scalar; 312 typedef typename Derived::Index Index; 313 314 protected: 315 typedef typename internal::ref_selector<Derived>::type DerivedNested; 316 317 public: 318 319 /** \brief Constructor. 320 * 321 * \param[in] A %Matrix (expression) forming the argument of the matrix logarithm. 322 */ 323 explicit MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { } 324 325 /** \brief Compute the matrix logarithm. 326 * 327 * \param[out] result Logarithm of \p A, where \A is as specified in the constructor. 328 */ 329 template <typename ResultType> 330 inline void evalTo(ResultType& result) const 331 { 332 typedef typename internal::nested_eval<Derived, 10>::type DerivedEvalType; 333 typedef typename internal::remove_all<DerivedEvalType>::type DerivedEvalTypeClean; 334 typedef internal::traits<DerivedEvalTypeClean> Traits; 335 static const int RowsAtCompileTime = Traits::RowsAtCompileTime; 336 static const int ColsAtCompileTime = Traits::ColsAtCompileTime; 337 typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar; 338 typedef Matrix<ComplexScalar, Dynamic, Dynamic, 0, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType; 339 typedef internal::MatrixLogarithmAtomic<DynMatrixType> AtomicType; 340 AtomicType atomic; 341 342 internal::matrix_function_compute<typename DerivedEvalTypeClean::PlainObject>::run(m_A, atomic, result); 343 } 344 345 Index rows() const { return m_A.rows(); } 346 Index cols() const { return m_A.cols(); } 347 348 private: 349 const DerivedNested m_A; 350 }; 351 352 namespace internal { 353 template<typename Derived> 354 struct traits<MatrixLogarithmReturnValue<Derived> > 355 { 356 typedef typename Derived::PlainObject ReturnType; 357 }; 358 } 359 360 361 /********** MatrixBase method **********/ 362 363 364 template <typename Derived> 365 const MatrixLogarithmReturnValue<Derived> MatrixBase<Derived>::log() const 366 { 367 eigen_assert(rows() == cols()); 368 return MatrixLogarithmReturnValue<Derived>(derived()); 369 } 370 371 } // end namespace Eigen 372 373 #endif // EIGEN_MATRIX_LOGARITHM 374
