| // This file is part of Eigen, a lightweight C++ template library |
| // for linear algebra. |
| // |
| // Copyright (C) 2011, 2013 Jitse Niesen <jitse@maths.leeds.ac.uk> |
| // Copyright (C) 2011 Chen-Pang He <jdh8@ms63.hinet.net> |
| // |
| // This Source Code Form is subject to the terms of the Mozilla |
| // Public License v. 2.0. If a copy of the MPL was not distributed |
| // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. |
| |
| #ifndef EIGEN_MATRIX_LOGARITHM |
| #define EIGEN_MATRIX_LOGARITHM |
| |
| // IWYU pragma: private |
| #include "./InternalHeaderCheck.h" |
| |
| namespace Eigen { |
| |
| namespace internal { |
| |
| template <typename Scalar> |
| struct matrix_log_min_pade_degree { |
| static const int value = 3; |
| }; |
| |
| template <typename Scalar> |
| struct matrix_log_max_pade_degree { |
| typedef typename NumTraits<Scalar>::Real RealScalar; |
| static const int value = std::numeric_limits<RealScalar>::digits <= 24 ? 5 : // single precision |
| std::numeric_limits<RealScalar>::digits <= 53 ? 7 |
| : // double precision |
| std::numeric_limits<RealScalar>::digits <= 64 ? 8 |
| : // extended precision |
| std::numeric_limits<RealScalar>::digits <= 106 ? 10 |
| : // double-double |
| 11; // quadruple precision |
| }; |
| |
| /** \brief Compute logarithm of 2x2 triangular matrix. */ |
| template <typename MatrixType> |
| void matrix_log_compute_2x2(const MatrixType& A, MatrixType& result) { |
| typedef typename MatrixType::Scalar Scalar; |
| typedef typename MatrixType::RealScalar RealScalar; |
| using std::abs; |
| using std::ceil; |
| using std::imag; |
| using std::log; |
| |
| Scalar logA00 = log(A(0, 0)); |
| Scalar logA11 = log(A(1, 1)); |
| |
| result(0, 0) = logA00; |
| result(1, 0) = Scalar(0); |
| result(1, 1) = logA11; |
| |
| Scalar y = A(1, 1) - A(0, 0); |
| if (y == Scalar(0)) { |
| result(0, 1) = A(0, 1) / A(0, 0); |
| } else if ((abs(A(0, 0)) < RealScalar(0.5) * abs(A(1, 1))) || (abs(A(0, 0)) > 2 * abs(A(1, 1)))) { |
| result(0, 1) = A(0, 1) * (logA11 - logA00) / y; |
| } else { |
| // computation in previous branch is inaccurate if A(1,1) \approx A(0,0) |
| RealScalar unwindingNumber = ceil((imag(logA11 - logA00) - RealScalar(EIGEN_PI)) / RealScalar(2 * EIGEN_PI)); |
| result(0, 1) = A(0, 1) * (numext::log1p(y / A(0, 0)) + Scalar(0, RealScalar(2 * EIGEN_PI) * unwindingNumber)) / y; |
| } |
| } |
| |
| /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = float) */ |
| inline int matrix_log_get_pade_degree(float normTminusI) { |
| const float maxNormForPade[] = {2.5111573934555054e-1 /* degree = 3 */, 4.0535837411880493e-1, 5.3149729967117310e-1}; |
| const int minPadeDegree = matrix_log_min_pade_degree<float>::value; |
| const int maxPadeDegree = matrix_log_max_pade_degree<float>::value; |
| int degree = minPadeDegree; |
| for (; degree <= maxPadeDegree; ++degree) |
| if (normTminusI <= maxNormForPade[degree - minPadeDegree]) break; |
| return degree; |
| } |
| |
| /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = double) */ |
| inline int matrix_log_get_pade_degree(double normTminusI) { |
| const double maxNormForPade[] = {1.6206284795015624e-2 /* degree = 3 */, 5.3873532631381171e-2, 1.1352802267628681e-1, |
| 1.8662860613541288e-1, 2.642960831111435e-1}; |
| const int minPadeDegree = matrix_log_min_pade_degree<double>::value; |
| const int maxPadeDegree = matrix_log_max_pade_degree<double>::value; |
| int degree = minPadeDegree; |
| for (; degree <= maxPadeDegree; ++degree) |
| if (normTminusI <= maxNormForPade[degree - minPadeDegree]) break; |
| return degree; |
| } |
| |
| /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = long double) */ |
| inline int matrix_log_get_pade_degree(long double normTminusI) { |
| #if LDBL_MANT_DIG == 53 // double precision |
| const long double maxNormForPade[] = {1.6206284795015624e-2L /* degree = 3 */, 5.3873532631381171e-2L, |
| 1.1352802267628681e-1L, 1.8662860613541288e-1L, 2.642960831111435e-1L}; |
| #elif LDBL_MANT_DIG <= 64 // extended precision |
| const long double maxNormForPade[] = {5.48256690357782863103e-3L /* degree = 3 */, |
| 2.34559162387971167321e-2L, |
| 5.84603923897347449857e-2L, |
| 1.08486423756725170223e-1L, |
| 1.68385767881294446649e-1L, |
| 2.32777776523703892094e-1L}; |
| #elif LDBL_MANT_DIG <= 106 // double-double |
| const long double maxNormForPade[] = {8.58970550342939562202529664318890e-5L /* degree = 3 */, |
| 9.34074328446359654039446552677759e-4L, |
| 4.26117194647672175773064114582860e-3L, |
| 1.21546224740281848743149666560464e-2L, |
| 2.61100544998339436713088248557444e-2L, |
| 4.66170074627052749243018566390567e-2L, |
| 7.32585144444135027565872014932387e-2L, |
| 1.05026503471351080481093652651105e-1L}; |
| #else // quadruple precision |
| const long double maxNormForPade[] = {4.7419931187193005048501568167858103e-5L /* degree = 3 */, |
| 5.8853168473544560470387769480192666e-4L, |
| 2.9216120366601315391789493628113520e-3L, |
| 8.8415758124319434347116734705174308e-3L, |
| 1.9850836029449446668518049562565291e-2L, |
| 3.6688019729653446926585242192447447e-2L, |
| 5.9290962294020186998954055264528393e-2L, |
| 8.6998436081634343903250580992127677e-2L, |
| 1.1880960220216759245467951592883642e-1L}; |
| #endif |
| const int minPadeDegree = matrix_log_min_pade_degree<long double>::value; |
| const int maxPadeDegree = matrix_log_max_pade_degree<long double>::value; |
| int degree = minPadeDegree; |
| for (; degree <= maxPadeDegree; ++degree) |
| if (normTminusI <= maxNormForPade[degree - minPadeDegree]) break; |
| return degree; |
| } |
| |
| /* \brief Compute Pade approximation to matrix logarithm */ |
| template <typename MatrixType> |
| void matrix_log_compute_pade(MatrixType& result, const MatrixType& T, int degree) { |
| typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar; |
| const int minPadeDegree = 3; |
| const int maxPadeDegree = 11; |
| eigen_assert(degree >= minPadeDegree && degree <= maxPadeDegree); |
| // FIXME this creates float-conversion-warnings if these are enabled. |
| // Either manually convert each value, or disable the warning locally |
| const RealScalar nodes[][maxPadeDegree] = { |
| {0.1127016653792583114820734600217600L, 0.5000000000000000000000000000000000L, // degree 3 |
| 0.8872983346207416885179265399782400L}, |
| {0.0694318442029737123880267555535953L, 0.3300094782075718675986671204483777L, // degree 4 |
| 0.6699905217924281324013328795516223L, 0.9305681557970262876119732444464048L}, |
| {0.0469100770306680036011865608503035L, 0.2307653449471584544818427896498956L, // degree 5 |
| 0.5000000000000000000000000000000000L, 0.7692346550528415455181572103501044L, |
| 0.9530899229693319963988134391496965L}, |
| {0.0337652428984239860938492227530027L, 0.1693953067668677431693002024900473L, // degree 6 |
| 0.3806904069584015456847491391596440L, 0.6193095930415984543152508608403560L, |
| 0.8306046932331322568306997975099527L, 0.9662347571015760139061507772469973L}, |
| {0.0254460438286207377369051579760744L, 0.1292344072003027800680676133596058L, // degree 7 |
| 0.2970774243113014165466967939615193L, 0.5000000000000000000000000000000000L, |
| 0.7029225756886985834533032060384807L, 0.8707655927996972199319323866403942L, |
| 0.9745539561713792622630948420239256L}, |
| {0.0198550717512318841582195657152635L, 0.1016667612931866302042230317620848L, // degree 8 |
| 0.2372337950418355070911304754053768L, 0.4082826787521750975302619288199080L, |
| 0.5917173212478249024697380711800920L, 0.7627662049581644929088695245946232L, |
| 0.8983332387068133697957769682379152L, 0.9801449282487681158417804342847365L}, |
| {0.0159198802461869550822118985481636L, 0.0819844463366821028502851059651326L, // degree 9 |
| 0.1933142836497048013456489803292629L, 0.3378732882980955354807309926783317L, |
| 0.5000000000000000000000000000000000L, 0.6621267117019044645192690073216683L, |
| 0.8066857163502951986543510196707371L, 0.9180155536633178971497148940348674L, |
| 0.9840801197538130449177881014518364L}, |
| {0.0130467357414141399610179939577740L, 0.0674683166555077446339516557882535L, // degree 10 |
| 0.1602952158504877968828363174425632L, 0.2833023029353764046003670284171079L, |
| 0.4255628305091843945575869994351400L, 0.5744371694908156054424130005648600L, |
| 0.7166976970646235953996329715828921L, 0.8397047841495122031171636825574368L, |
| 0.9325316833444922553660483442117465L, 0.9869532642585858600389820060422260L}, |
| {0.0108856709269715035980309994385713L, 0.0564687001159523504624211153480364L, // degree 11 |
| 0.1349239972129753379532918739844233L, 0.2404519353965940920371371652706952L, |
| 0.3652284220238275138342340072995692L, 0.5000000000000000000000000000000000L, |
| 0.6347715779761724861657659927004308L, 0.7595480646034059079628628347293048L, |
| 0.8650760027870246620467081260155767L, 0.9435312998840476495375788846519636L, |
| 0.9891143290730284964019690005614287L}}; |
| |
| const RealScalar weights[][maxPadeDegree] = { |
| {0.2777777777777777777777777777777778L, 0.4444444444444444444444444444444444L, // degree 3 |
| 0.2777777777777777777777777777777778L}, |
| {0.1739274225687269286865319746109997L, 0.3260725774312730713134680253890003L, // degree 4 |
| 0.3260725774312730713134680253890003L, 0.1739274225687269286865319746109997L}, |
| {0.1184634425280945437571320203599587L, 0.2393143352496832340206457574178191L, // degree 5 |
| 0.2844444444444444444444444444444444L, 0.2393143352496832340206457574178191L, |
| 0.1184634425280945437571320203599587L}, |
| {0.0856622461895851725201480710863665L, 0.1803807865240693037849167569188581L, // degree 6 |
| 0.2339569672863455236949351719947755L, 0.2339569672863455236949351719947755L, |
| 0.1803807865240693037849167569188581L, 0.0856622461895851725201480710863665L}, |
| {0.0647424830844348466353057163395410L, 0.1398526957446383339507338857118898L, // degree 7 |
| 0.1909150252525594724751848877444876L, 0.2089795918367346938775510204081633L, |
| 0.1909150252525594724751848877444876L, 0.1398526957446383339507338857118898L, |
| 0.0647424830844348466353057163395410L}, |
| {0.0506142681451881295762656771549811L, 0.1111905172266872352721779972131204L, // degree 8 |
| 0.1568533229389436436689811009933007L, 0.1813418916891809914825752246385978L, |
| 0.1813418916891809914825752246385978L, 0.1568533229389436436689811009933007L, |
| 0.1111905172266872352721779972131204L, 0.0506142681451881295762656771549811L}, |
| {0.0406371941807872059859460790552618L, 0.0903240803474287020292360156214564L, // degree 9 |
| 0.1303053482014677311593714347093164L, 0.1561735385200014200343152032922218L, |
| 0.1651196775006298815822625346434870L, 0.1561735385200014200343152032922218L, |
| 0.1303053482014677311593714347093164L, 0.0903240803474287020292360156214564L, |
| 0.0406371941807872059859460790552618L}, |
| {0.0333356721543440687967844049466659L, 0.0747256745752902965728881698288487L, // degree 10 |
| 0.1095431812579910219977674671140816L, 0.1346333596549981775456134607847347L, |
| 0.1477621123573764350869464973256692L, 0.1477621123573764350869464973256692L, |
| 0.1346333596549981775456134607847347L, 0.1095431812579910219977674671140816L, |
| 0.0747256745752902965728881698288487L, 0.0333356721543440687967844049466659L}, |
| {0.0278342835580868332413768602212743L, 0.0627901847324523123173471496119701L, // degree 11 |
| 0.0931451054638671257130488207158280L, 0.1165968822959952399592618524215876L, |
| 0.1314022722551233310903444349452546L, 0.1364625433889503153572417641681711L, |
| 0.1314022722551233310903444349452546L, 0.1165968822959952399592618524215876L, |
| 0.0931451054638671257130488207158280L, 0.0627901847324523123173471496119701L, |
| 0.0278342835580868332413768602212743L}}; |
| |
| MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows()); |
| result.setZero(T.rows(), T.rows()); |
| for (int k = 0; k < degree; ++k) { |
| RealScalar weight = weights[degree - minPadeDegree][k]; |
| RealScalar node = nodes[degree - minPadeDegree][k]; |
| result += |
| weight * |
| (MatrixType::Identity(T.rows(), T.rows()) + node * TminusI).template triangularView<Upper>().solve(TminusI); |
| } |
| } |
| |
| /** \brief Compute logarithm of triangular matrices with size > 2. |
| * \details This uses a inverse scale-and-square algorithm. */ |
| template <typename MatrixType> |
| void matrix_log_compute_big(const MatrixType& A, MatrixType& result) { |
| typedef typename MatrixType::Scalar Scalar; |
| typedef typename NumTraits<Scalar>::Real RealScalar; |
| using std::pow; |
| |
| int numberOfSquareRoots = 0; |
| int numberOfExtraSquareRoots = 0; |
| int degree; |
| MatrixType T = A, sqrtT; |
| |
| const int maxPadeDegree = matrix_log_max_pade_degree<Scalar>::value; |
| const RealScalar maxNormForPade = RealScalar(maxPadeDegree <= 5 ? 5.3149729967117310e-1L : // single precision |
| maxPadeDegree <= 7 ? 2.6429608311114350e-1L |
| : // double precision |
| maxPadeDegree <= 8 ? 2.32777776523703892094e-1L |
| : // extended precision |
| maxPadeDegree <= 10 ? 1.05026503471351080481093652651105e-1L |
| : // double-double |
| 1.1880960220216759245467951592883642e-1L); // quadruple precision |
| |
| while (true) { |
| RealScalar normTminusI = (T - MatrixType::Identity(T.rows(), T.rows())).cwiseAbs().colwise().sum().maxCoeff(); |
| if (normTminusI < maxNormForPade) { |
| degree = matrix_log_get_pade_degree(normTminusI); |
| int degree2 = matrix_log_get_pade_degree(normTminusI / RealScalar(2)); |
| if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1)) break; |
| ++numberOfExtraSquareRoots; |
| } |
| matrix_sqrt_triangular(T, sqrtT); |
| T = sqrtT.template triangularView<Upper>(); |
| ++numberOfSquareRoots; |
| } |
| |
| matrix_log_compute_pade(result, T, degree); |
| result *= pow(RealScalar(2), RealScalar(numberOfSquareRoots)); // TODO replace by bitshift if possible |
| } |
| |
| /** \ingroup MatrixFunctions_Module |
| * \class MatrixLogarithmAtomic |
| * \brief Helper class for computing matrix logarithm of atomic matrices. |
| * |
| * Here, an atomic matrix is a triangular matrix whose diagonal entries are close to each other. |
| * |
| * \sa class MatrixFunctionAtomic, MatrixBase::log() |
| */ |
| template <typename MatrixType> |
| class MatrixLogarithmAtomic { |
| public: |
| /** \brief Compute matrix logarithm of atomic matrix |
| * \param[in] A argument of matrix logarithm, should be upper triangular and atomic |
| * \returns The logarithm of \p A. |
| */ |
| MatrixType compute(const MatrixType& A); |
| }; |
| |
| template <typename MatrixType> |
| MatrixType MatrixLogarithmAtomic<MatrixType>::compute(const MatrixType& A) { |
| using std::log; |
| MatrixType result(A.rows(), A.rows()); |
| if (A.rows() == 1) |
| result(0, 0) = log(A(0, 0)); |
| else if (A.rows() == 2) |
| matrix_log_compute_2x2(A, result); |
| else |
| matrix_log_compute_big(A, result); |
| return result; |
| } |
| |
| } // end of namespace internal |
| |
| /** \ingroup MatrixFunctions_Module |
| * |
| * \brief Proxy for the matrix logarithm of some matrix (expression). |
| * |
| * \tparam Derived Type of the argument to the matrix function. |
| * |
| * This class holds the argument to the matrix function until it is |
| * assigned or evaluated for some other reason (so the argument |
| * should not be changed in the meantime). It is the return type of |
| * MatrixBase::log() and most of the time this is the only way it |
| * is used. |
| */ |
| template <typename Derived> |
| class MatrixLogarithmReturnValue : public ReturnByValue<MatrixLogarithmReturnValue<Derived> > { |
| public: |
| typedef typename Derived::Scalar Scalar; |
| typedef typename Derived::Index Index; |
| |
| protected: |
| typedef typename internal::ref_selector<Derived>::type DerivedNested; |
| |
| public: |
| /** \brief Constructor. |
| * |
| * \param[in] A %Matrix (expression) forming the argument of the matrix logarithm. |
| */ |
| explicit MatrixLogarithmReturnValue(const Derived& A) : m_A(A) {} |
| |
| /** \brief Compute the matrix logarithm. |
| * |
| * \param[out] result Logarithm of \c A, where \c A is as specified in the constructor. |
| */ |
| template <typename ResultType> |
| inline void evalTo(ResultType& result) const { |
| typedef typename internal::nested_eval<Derived, 10>::type DerivedEvalType; |
| typedef internal::remove_all_t<DerivedEvalType> DerivedEvalTypeClean; |
| typedef internal::traits<DerivedEvalTypeClean> Traits; |
| typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar; |
| typedef Matrix<ComplexScalar, Dynamic, Dynamic, 0, Traits::RowsAtCompileTime, Traits::ColsAtCompileTime> |
| DynMatrixType; |
| typedef internal::MatrixLogarithmAtomic<DynMatrixType> AtomicType; |
| AtomicType atomic; |
| |
| internal::matrix_function_compute<typename DerivedEvalTypeClean::PlainObject>::run(m_A, atomic, result); |
| } |
| |
| Index rows() const { return m_A.rows(); } |
| Index cols() const { return m_A.cols(); } |
| |
| private: |
| const DerivedNested m_A; |
| }; |
| |
| namespace internal { |
| template <typename Derived> |
| struct traits<MatrixLogarithmReturnValue<Derived> > { |
| typedef typename Derived::PlainObject ReturnType; |
| }; |
| } // namespace internal |
| |
| /********** MatrixBase method **********/ |
| |
| template <typename Derived> |
| const MatrixLogarithmReturnValue<Derived> MatrixBase<Derived>::log() const { |
| eigen_assert(rows() == cols()); |
| return MatrixLogarithmReturnValue<Derived>(derived()); |
| } |
| |
| } // end namespace Eigen |
| |
| #endif // EIGEN_MATRIX_LOGARITHM |
