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Eigen-unsupported
3.3.3
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00001 // This file is part of Eigen, a lightweight C++ template library 00002 // for linear algebra. 00003 // 00004 // Copyright (C) 2011, 2013 Jitse Niesen <jitse@maths.leeds.ac.uk> 00005 // Copyright (C) 2011 Chen-Pang He <jdh8@ms63.hinet.net> 00006 // 00007 // This Source Code Form is subject to the terms of the Mozilla 00008 // Public License v. 2.0. If a copy of the MPL was not distributed 00009 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 00010 00011 #ifndef EIGEN_MATRIX_LOGARITHM 00012 #define EIGEN_MATRIX_LOGARITHM 00013 00014 namespace Eigen { 00015 00016 namespace internal { 00017 00018 template <typename Scalar> 00019 struct matrix_log_min_pade_degree 00020 { 00021 static const int value = 3; 00022 }; 00023 00024 template <typename Scalar> 00025 struct matrix_log_max_pade_degree 00026 { 00027 typedef typename NumTraits<Scalar>::Real RealScalar; 00028 static const int value = std::numeric_limits<RealScalar>::digits<= 24? 5: // single precision 00029 std::numeric_limits<RealScalar>::digits<= 53? 7: // double precision 00030 std::numeric_limits<RealScalar>::digits<= 64? 8: // extended precision 00031 std::numeric_limits<RealScalar>::digits<=106? 10: // double-double 00032 11; // quadruple precision 00033 }; 00034 00036 template <typename MatrixType> 00037 void matrix_log_compute_2x2(const MatrixType& A, MatrixType& result) 00038 { 00039 typedef typename MatrixType::Scalar Scalar; 00040 typedef typename MatrixType::RealScalar RealScalar; 00041 using std::abs; 00042 using std::ceil; 00043 using std::imag; 00044 using std::log; 00045 00046 Scalar logA00 = log(A(0,0)); 00047 Scalar logA11 = log(A(1,1)); 00048 00049 result(0,0) = logA00; 00050 result(1,0) = Scalar(0); 00051 result(1,1) = logA11; 00052 00053 Scalar y = A(1,1) - A(0,0); 00054 if (y==Scalar(0)) 00055 { 00056 result(0,1) = A(0,1) / A(0,0); 00057 } 00058 else if ((abs(A(0,0)) < RealScalar(0.5)*abs(A(1,1))) || (abs(A(0,0)) > 2*abs(A(1,1)))) 00059 { 00060 result(0,1) = A(0,1) * (logA11 - logA00) / y; 00061 } 00062 else 00063 { 00064 // computation in previous branch is inaccurate if A(1,1) \approx A(0,0) 00065 int unwindingNumber = static_cast<int>(ceil((imag(logA11 - logA00) - RealScalar(EIGEN_PI)) / RealScalar(2*EIGEN_PI))); 00066 result(0,1) = A(0,1) * (numext::log1p(y/A(0,0)) + Scalar(0,2*EIGEN_PI*unwindingNumber)) / y; 00067 } 00068 } 00069 00070 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = float) */ 00071 inline int matrix_log_get_pade_degree(float normTminusI) 00072 { 00073 const float maxNormForPade[] = { 2.5111573934555054e-1 /* degree = 3 */ , 4.0535837411880493e-1, 00074 5.3149729967117310e-1 }; 00075 const int minPadeDegree = matrix_log_min_pade_degree<float>::value; 00076 const int maxPadeDegree = matrix_log_max_pade_degree<float>::value; 00077 int degree = minPadeDegree; 00078 for (; degree <= maxPadeDegree; ++degree) 00079 if (normTminusI <= maxNormForPade[degree - minPadeDegree]) 00080 break; 00081 return degree; 00082 } 00083 00084 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = double) */ 00085 inline int matrix_log_get_pade_degree(double normTminusI) 00086 { 00087 const double maxNormForPade[] = { 1.6206284795015624e-2 /* degree = 3 */ , 5.3873532631381171e-2, 00088 1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 }; 00089 const int minPadeDegree = matrix_log_min_pade_degree<double>::value; 00090 const int maxPadeDegree = matrix_log_max_pade_degree<double>::value; 00091 int degree = minPadeDegree; 00092 for (; degree <= maxPadeDegree; ++degree) 00093 if (normTminusI <= maxNormForPade[degree - minPadeDegree]) 00094 break; 00095 return degree; 00096 } 00097 00098 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = long double) */ 00099 inline int matrix_log_get_pade_degree(long double normTminusI) 00100 { 00101 #if LDBL_MANT_DIG == 53 // double precision 00102 const long double maxNormForPade[] = { 1.6206284795015624e-2L /* degree = 3 */ , 5.3873532631381171e-2L, 00103 1.1352802267628681e-1L, 1.8662860613541288e-1L, 2.642960831111435e-1L }; 00104 #elif LDBL_MANT_DIG <= 64 // extended precision 00105 const long double maxNormForPade[] = { 5.48256690357782863103e-3L /* degree = 3 */, 2.34559162387971167321e-2L, 00106 5.84603923897347449857e-2L, 1.08486423756725170223e-1L, 1.68385767881294446649e-1L, 00107 2.32777776523703892094e-1L }; 00108 #elif LDBL_MANT_DIG <= 106 // double-double 00109 const long double maxNormForPade[] = { 8.58970550342939562202529664318890e-5L /* degree = 3 */, 00110 9.34074328446359654039446552677759e-4L, 4.26117194647672175773064114582860e-3L, 00111 1.21546224740281848743149666560464e-2L, 2.61100544998339436713088248557444e-2L, 00112 4.66170074627052749243018566390567e-2L, 7.32585144444135027565872014932387e-2L, 00113 1.05026503471351080481093652651105e-1L }; 00114 #else // quadruple precision 00115 const long double maxNormForPade[] = { 4.7419931187193005048501568167858103e-5L /* degree = 3 */, 00116 5.8853168473544560470387769480192666e-4L, 2.9216120366601315391789493628113520e-3L, 00117 8.8415758124319434347116734705174308e-3L, 1.9850836029449446668518049562565291e-2L, 00118 3.6688019729653446926585242192447447e-2L, 5.9290962294020186998954055264528393e-2L, 00119 8.6998436081634343903250580992127677e-2L, 1.1880960220216759245467951592883642e-1L }; 00120 #endif 00121 const int minPadeDegree = matrix_log_min_pade_degree<long double>::value; 00122 const int maxPadeDegree = matrix_log_max_pade_degree<long double>::value; 00123 int degree = minPadeDegree; 00124 for (; degree <= maxPadeDegree; ++degree) 00125 if (normTminusI <= maxNormForPade[degree - minPadeDegree]) 00126 break; 00127 return degree; 00128 } 00129 00130 /* \brief Compute Pade approximation to matrix logarithm */ 00131 template <typename MatrixType> 00132 void matrix_log_compute_pade(MatrixType& result, const MatrixType& T, int degree) 00133 { 00134 typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar; 00135 const int minPadeDegree = 3; 00136 const int maxPadeDegree = 11; 00137 assert(degree >= minPadeDegree && degree <= maxPadeDegree); 00138 00139 const RealScalar nodes[][maxPadeDegree] = { 00140 { 0.1127016653792583114820734600217600L, 0.5000000000000000000000000000000000L, // degree 3 00141 0.8872983346207416885179265399782400L }, 00142 { 0.0694318442029737123880267555535953L, 0.3300094782075718675986671204483777L, // degree 4 00143 0.6699905217924281324013328795516223L, 0.9305681557970262876119732444464048L }, 00144 { 0.0469100770306680036011865608503035L, 0.2307653449471584544818427896498956L, // degree 5 00145 0.5000000000000000000000000000000000L, 0.7692346550528415455181572103501044L, 00146 0.9530899229693319963988134391496965L }, 00147 { 0.0337652428984239860938492227530027L, 0.1693953067668677431693002024900473L, // degree 6 00148 0.3806904069584015456847491391596440L, 0.6193095930415984543152508608403560L, 00149 0.8306046932331322568306997975099527L, 0.9662347571015760139061507772469973L }, 00150 { 0.0254460438286207377369051579760744L, 0.1292344072003027800680676133596058L, // degree 7 00151 0.2970774243113014165466967939615193L, 0.5000000000000000000000000000000000L, 00152 0.7029225756886985834533032060384807L, 0.8707655927996972199319323866403942L, 00153 0.9745539561713792622630948420239256L }, 00154 { 0.0198550717512318841582195657152635L, 0.1016667612931866302042230317620848L, // degree 8 00155 0.2372337950418355070911304754053768L, 0.4082826787521750975302619288199080L, 00156 0.5917173212478249024697380711800920L, 0.7627662049581644929088695245946232L, 00157 0.8983332387068133697957769682379152L, 0.9801449282487681158417804342847365L }, 00158 { 0.0159198802461869550822118985481636L, 0.0819844463366821028502851059651326L, // degree 9 00159 0.1933142836497048013456489803292629L, 0.3378732882980955354807309926783317L, 00160 0.5000000000000000000000000000000000L, 0.6621267117019044645192690073216683L, 00161 0.8066857163502951986543510196707371L, 0.9180155536633178971497148940348674L, 00162 0.9840801197538130449177881014518364L }, 00163 { 0.0130467357414141399610179939577740L, 0.0674683166555077446339516557882535L, // degree 10 00164 0.1602952158504877968828363174425632L, 0.2833023029353764046003670284171079L, 00165 0.4255628305091843945575869994351400L, 0.5744371694908156054424130005648600L, 00166 0.7166976970646235953996329715828921L, 0.8397047841495122031171636825574368L, 00167 0.9325316833444922553660483442117465L, 0.9869532642585858600389820060422260L }, 00168 { 0.0108856709269715035980309994385713L, 0.0564687001159523504624211153480364L, // degree 11 00169 0.1349239972129753379532918739844233L, 0.2404519353965940920371371652706952L, 00170 0.3652284220238275138342340072995692L, 0.5000000000000000000000000000000000L, 00171 0.6347715779761724861657659927004308L, 0.7595480646034059079628628347293048L, 00172 0.8650760027870246620467081260155767L, 0.9435312998840476495375788846519636L, 00173 0.9891143290730284964019690005614287L } }; 00174 00175 const RealScalar weights[][maxPadeDegree] = { 00176 { 0.2777777777777777777777777777777778L, 0.4444444444444444444444444444444444L, // degree 3 00177 0.2777777777777777777777777777777778L }, 00178 { 0.1739274225687269286865319746109997L, 0.3260725774312730713134680253890003L, // degree 4 00179 0.3260725774312730713134680253890003L, 0.1739274225687269286865319746109997L }, 00180 { 0.1184634425280945437571320203599587L, 0.2393143352496832340206457574178191L, // degree 5 00181 0.2844444444444444444444444444444444L, 0.2393143352496832340206457574178191L, 00182 0.1184634425280945437571320203599587L }, 00183 { 0.0856622461895851725201480710863665L, 0.1803807865240693037849167569188581L, // degree 6 00184 0.2339569672863455236949351719947755L, 0.2339569672863455236949351719947755L, 00185 0.1803807865240693037849167569188581L, 0.0856622461895851725201480710863665L }, 00186 { 0.0647424830844348466353057163395410L, 0.1398526957446383339507338857118898L, // degree 7 00187 0.1909150252525594724751848877444876L, 0.2089795918367346938775510204081633L, 00188 0.1909150252525594724751848877444876L, 0.1398526957446383339507338857118898L, 00189 0.0647424830844348466353057163395410L }, 00190 { 0.0506142681451881295762656771549811L, 0.1111905172266872352721779972131204L, // degree 8 00191 0.1568533229389436436689811009933007L, 0.1813418916891809914825752246385978L, 00192 0.1813418916891809914825752246385978L, 0.1568533229389436436689811009933007L, 00193 0.1111905172266872352721779972131204L, 0.0506142681451881295762656771549811L }, 00194 { 0.0406371941807872059859460790552618L, 0.0903240803474287020292360156214564L, // degree 9 00195 0.1303053482014677311593714347093164L, 0.1561735385200014200343152032922218L, 00196 0.1651196775006298815822625346434870L, 0.1561735385200014200343152032922218L, 00197 0.1303053482014677311593714347093164L, 0.0903240803474287020292360156214564L, 00198 0.0406371941807872059859460790552618L }, 00199 { 0.0333356721543440687967844049466659L, 0.0747256745752902965728881698288487L, // degree 10 00200 0.1095431812579910219977674671140816L, 0.1346333596549981775456134607847347L, 00201 0.1477621123573764350869464973256692L, 0.1477621123573764350869464973256692L, 00202 0.1346333596549981775456134607847347L, 0.1095431812579910219977674671140816L, 00203 0.0747256745752902965728881698288487L, 0.0333356721543440687967844049466659L }, 00204 { 0.0278342835580868332413768602212743L, 0.0627901847324523123173471496119701L, // degree 11 00205 0.0931451054638671257130488207158280L, 0.1165968822959952399592618524215876L, 00206 0.1314022722551233310903444349452546L, 0.1364625433889503153572417641681711L, 00207 0.1314022722551233310903444349452546L, 0.1165968822959952399592618524215876L, 00208 0.0931451054638671257130488207158280L, 0.0627901847324523123173471496119701L, 00209 0.0278342835580868332413768602212743L } }; 00210 00211 MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows()); 00212 result.setZero(T.rows(), T.rows()); 00213 for (int k = 0; k < degree; ++k) { 00214 RealScalar weight = weights[degree-minPadeDegree][k]; 00215 RealScalar node = nodes[degree-minPadeDegree][k]; 00216 result += weight * (MatrixType::Identity(T.rows(), T.rows()) + node * TminusI) 00217 .template triangularView<Upper>().solve(TminusI); 00218 } 00219 } 00220 00223 template <typename MatrixType> 00224 void matrix_log_compute_big(const MatrixType& A, MatrixType& result) 00225 { 00226 typedef typename MatrixType::Scalar Scalar; 00227 typedef typename NumTraits<Scalar>::Real RealScalar; 00228 using std::pow; 00229 00230 int numberOfSquareRoots = 0; 00231 int numberOfExtraSquareRoots = 0; 00232 int degree; 00233 MatrixType T = A, sqrtT; 00234 00235 int maxPadeDegree = matrix_log_max_pade_degree<Scalar>::value; 00236 const RealScalar maxNormForPade = maxPadeDegree<= 5? 5.3149729967117310e-1L: // single precision 00237 maxPadeDegree<= 7? 2.6429608311114350e-1L: // double precision 00238 maxPadeDegree<= 8? 2.32777776523703892094e-1L: // extended precision 00239 maxPadeDegree<=10? 1.05026503471351080481093652651105e-1L: // double-double 00240 1.1880960220216759245467951592883642e-1L; // quadruple precision 00241 00242 while (true) { 00243 RealScalar normTminusI = (T - MatrixType::Identity(T.rows(), T.rows())).cwiseAbs().colwise().sum().maxCoeff(); 00244 if (normTminusI < maxNormForPade) { 00245 degree = matrix_log_get_pade_degree(normTminusI); 00246 int degree2 = matrix_log_get_pade_degree(normTminusI / RealScalar(2)); 00247 if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1)) 00248 break; 00249 ++numberOfExtraSquareRoots; 00250 } 00251 matrix_sqrt_triangular(T, sqrtT); 00252 T = sqrtT.template triangularView<Upper>(); 00253 ++numberOfSquareRoots; 00254 } 00255 00256 matrix_log_compute_pade(result, T, degree); 00257 result *= pow(RealScalar(2), numberOfSquareRoots); 00258 } 00259 00268 template <typename MatrixType> 00269 class MatrixLogarithmAtomic 00270 { 00271 public: 00276 MatrixType compute(const MatrixType& A); 00277 }; 00278 00279 template <typename MatrixType> 00280 MatrixType MatrixLogarithmAtomic<MatrixType>::compute(const MatrixType& A) 00281 { 00282 using std::log; 00283 MatrixType result(A.rows(), A.rows()); 00284 if (A.rows() == 1) 00285 result(0,0) = log(A(0,0)); 00286 else if (A.rows() == 2) 00287 matrix_log_compute_2x2(A, result); 00288 else 00289 matrix_log_compute_big(A, result); 00290 return result; 00291 } 00292 00293 } // end of namespace internal 00294 00307 template<typename Derived> class MatrixLogarithmReturnValue 00308 : public ReturnByValue<MatrixLogarithmReturnValue<Derived> > 00309 { 00310 public: 00311 typedef typename Derived::Scalar Scalar; 00312 typedef typename Derived::Index Index; 00313 00314 protected: 00315 typedef typename internal::ref_selector<Derived>::type DerivedNested; 00316 00317 public: 00318 00323 explicit MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { } 00324 00329 template <typename ResultType> 00330 inline void evalTo(ResultType& result) const 00331 { 00332 typedef typename internal::nested_eval<Derived, 10>::type DerivedEvalType; 00333 typedef typename internal::remove_all<DerivedEvalType>::type DerivedEvalTypeClean; 00334 typedef internal::traits<DerivedEvalTypeClean> Traits; 00335 static const int RowsAtCompileTime = Traits::RowsAtCompileTime; 00336 static const int ColsAtCompileTime = Traits::ColsAtCompileTime; 00337 typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar; 00338 typedef Matrix<ComplexScalar, Dynamic, Dynamic, 0, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType; 00339 typedef internal::MatrixLogarithmAtomic<DynMatrixType> AtomicType; 00340 AtomicType atomic; 00341 00342 internal::matrix_function_compute<DerivedEvalTypeClean>::run(m_A, atomic, result); 00343 } 00344 00345 Index rows() const { return m_A.rows(); } 00346 Index cols() const { return m_A.cols(); } 00347 00348 private: 00349 const DerivedNested m_A; 00350 }; 00351 00352 namespace internal { 00353 template<typename Derived> 00354 struct traits<MatrixLogarithmReturnValue<Derived> > 00355 { 00356 typedef typename Derived::PlainObject ReturnType; 00357 }; 00358 } 00359 00360 00361 /********** MatrixBase method **********/ 00362 00363 00364 template <typename Derived> 00365 const MatrixLogarithmReturnValue<Derived> MatrixBase<Derived>::log() const 00366 { 00367 eigen_assert(rows() == cols()); 00368 return MatrixLogarithmReturnValue<Derived>(derived()); 00369 } 00370 00371 } // end namespace Eigen 00372 00373 #endif // EIGEN_MATRIX_LOGARITHM