unsupported/Eigen/src/MatrixFunctions/MatrixLogarithm.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2011 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

#ifndef M_PI
#define M_PI 3.141592653589793238462643383279503L
#endif

namespace Eigen { 

/** \ingroup MatrixFunctions_Module
  * \class MatrixLogarithmAtomic
  * \brief Helper class for computing matrix logarithm of atomic matrices.
  *
  * \internal
  * 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:

  typedef typename MatrixType::Scalar Scalar;
  // typedef typename MatrixType::Index Index;
  typedef typename NumTraits<Scalar>::Real RealScalar;
  // typedef typename internal::stem_function<Scalar>::type StemFunction;
  // typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;

  /** \brief Constructor. */
  MatrixLogarithmAtomic() { }

  /** \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);

private:

  void compute2x2(const MatrixType& A, MatrixType& result);
  void computeBig(const MatrixType& A, MatrixType& result);
  int getPadeDegree(float normTminusI);
  int getPadeDegree(double normTminusI);
  int getPadeDegree(long double normTminusI);
  void computePade(MatrixType& result, const MatrixType& T, int degree);
  void computePade3(MatrixType& result, const MatrixType& T);
  void computePade4(MatrixType& result, const MatrixType& T);
  void computePade5(MatrixType& result, const MatrixType& T);
  void computePade6(MatrixType& result, const MatrixType& T);
  void computePade7(MatrixType& result, const MatrixType& T);
  void computePade8(MatrixType& result, const MatrixType& T);
  void computePade9(MatrixType& result, const MatrixType& T);
  void computePade10(MatrixType& result, const MatrixType& T);
  void computePade11(MatrixType& result, const MatrixType& T);

  static const int minPadeDegree = 3;
  static const int maxPadeDegree = 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

  // Prevent copying
  MatrixLogarithmAtomic(const MatrixLogarithmAtomic&);
  MatrixLogarithmAtomic& operator=(const MatrixLogarithmAtomic&);
};

/** \brief Compute logarithm of triangular matrix with clustered eigenvalues. */
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)
    compute2x2(A, result);
  else
    computeBig(A, result);
  return result;
}

/** \brief Compute logarithm of 2x2 triangular matrix. */
template <typename MatrixType>
void MatrixLogarithmAtomic<MatrixType>::compute2x2(const MatrixType& A, MatrixType& result)
{
  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;

  if (A(0,0) == A(1,1)) {
    result(0,1) = A(0,1) / A(0,0);
  } else if ((abs(A(0,0)) < 0.5*abs(A(1,1))) || (abs(A(0,0)) > 2*abs(A(1,1)))) {
    result(0,1) = A(0,1) * (logA11 - logA00) / (A(1,1) - A(0,0));
  } else {
    // computation in previous branch is inaccurate if A(1,1) \approx A(0,0)
    int unwindingNumber = static_cast<int>(ceil((imag(logA11 - logA00) - M_PI) / (2*M_PI)));
    Scalar y = A(1,1) - A(0,0), x = A(1,1) + A(0,0);
    result(0,1) = A(0,1) * (Scalar(2) * numext::atanh2(y,x) + Scalar(0,2*M_PI*unwindingNumber)) / y;
  }
}

/** \brief Compute logarithm of triangular matrices with size > 2. 
  * \details This uses a inverse scale-and-square algorithm. */
template <typename MatrixType>
void MatrixLogarithmAtomic<MatrixType>::computeBig(const MatrixType& A, MatrixType& result)
{
  using std::pow;
  int numberOfSquareRoots = 0;
  int numberOfExtraSquareRoots = 0;
  int degree;
  MatrixType T = A, sqrtT;
  const RealScalar maxNormForPade = maxPadeDegree<= 5? 5.3149729967117310e-1:                     // single precision
                                    maxPadeDegree<= 7? 2.6429608311114350e-1:                     // 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 = getPadeDegree(normTminusI);
      int degree2 = getPadeDegree(normTminusI / RealScalar(2));
      if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1)) 
        break;
      ++numberOfExtraSquareRoots;
    }
    MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
    T = sqrtT.template triangularView<Upper>();
    ++numberOfSquareRoots;
  }

  computePade(result, T, degree);
  result *= pow(RealScalar(2), numberOfSquareRoots);
}

/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = float) */
template <typename MatrixType>
int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(float normTminusI)
{
  const float maxNormForPade[] = { 2.5111573934555054e-1 /* degree = 3 */ , 4.0535837411880493e-1,
            5.3149729967117310e-1 };
  int degree = 3;
  for (; degree <= maxPadeDegree; ++degree) 
    if (normTminusI <= maxNormForPade[degree - minPadeDegree])
      break;
  return degree;
}

/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = double) */
template <typename MatrixType>
int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(double normTminusI)
{
  const double maxNormForPade[] = { 1.6206284795015624e-2 /* degree = 3 */ , 5.3873532631381171e-2,
            1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 };
  int degree = 3;
  for (; degree <= maxPadeDegree; ++degree)
    if (normTminusI <= maxNormForPade[degree - minPadeDegree])
      break;
  return degree;
}

/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = long double) */
template <typename MatrixType>
int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(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
  int degree = 3;
  for (; degree <= maxPadeDegree; ++degree)
    if (normTminusI <= maxNormForPade[degree - minPadeDegree])
      break;
  return degree;
}

/* \brief Compute Pade approximation to matrix logarithm */
template <typename MatrixType>
void MatrixLogarithmAtomic<MatrixType>::computePade(MatrixType& result, const MatrixType& T, int degree)
{
  switch (degree) {
    case 3:  computePade3(result, T);  break;
    case 4:  computePade4(result, T);  break;
    case 5:  computePade5(result, T);  break;
    case 6:  computePade6(result, T);  break;
    case 7:  computePade7(result, T);  break;
    case 8:  computePade8(result, T);  break;
    case 9:  computePade9(result, T);  break;
    case 10: computePade10(result, T); break;
    case 11: computePade11(result, T); break;
    default: assert(false); // should never happen
  }
} 

template <typename MatrixType>
void MatrixLogarithmAtomic<MatrixType>::computePade3(MatrixType& result, const MatrixType& T)
{
  const int degree = 3;
  const RealScalar nodes[]   = { 0.1127016653792583114820734600217600L, 0.5000000000000000000000000000000000L,
            0.8872983346207416885179265399782400L };
  const RealScalar weights[] = { 0.2777777777777777777777777777777778L, 0.4444444444444444444444444444444444L,
            0.2777777777777777777777777777777778L };
  eigen_assert(degree <= maxPadeDegree);
  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
  result.setZero(T.rows(), T.rows());
  for (int k = 0; k < degree; ++k)
    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
                           .template triangularView<Upper>().solve(TminusI);
}

template <typename MatrixType>
void MatrixLogarithmAtomic<MatrixType>::computePade4(MatrixType& result, const MatrixType& T)
{
  const int degree = 4;
  const RealScalar nodes[]   = { 0.0694318442029737123880267555535953L, 0.3300094782075718675986671204483777L,
            0.6699905217924281324013328795516223L, 0.9305681557970262876119732444464048L };
  const RealScalar weights[] = { 0.1739274225687269286865319746109997L, 0.3260725774312730713134680253890003L,
            0.3260725774312730713134680253890003L, 0.1739274225687269286865319746109997L };
  eigen_assert(degree <= maxPadeDegree);
  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
  result.setZero(T.rows(), T.rows());
  for (int k = 0; k < degree; ++k)
    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
                           .template triangularView<Upper>().solve(TminusI);
}

template <typename MatrixType>
void MatrixLogarithmAtomic<MatrixType>::computePade5(MatrixType& result, const MatrixType& T)
{
  const int degree = 5;
  const RealScalar nodes[]   = { 0.0469100770306680036011865608503035L, 0.2307653449471584544818427896498956L,
            0.5000000000000000000000000000000000L, 0.7692346550528415455181572103501044L,
            0.9530899229693319963988134391496965L };
  const RealScalar weights[] = { 0.1184634425280945437571320203599587L, 0.2393143352496832340206457574178191L,
            0.2844444444444444444444444444444444L, 0.2393143352496832340206457574178191L,
            0.1184634425280945437571320203599587L };
  eigen_assert(degree <= maxPadeDegree);
  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
  result.setZero(T.rows(), T.rows());
  for (int k = 0; k < degree; ++k)
    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
                           .template triangularView<Upper>().solve(TminusI);
}

template <typename MatrixType>
void MatrixLogarithmAtomic<MatrixType>::computePade6(MatrixType& result, const MatrixType& T)
{
  const int degree = 6;
  const RealScalar nodes[]   = { 0.0337652428984239860938492227530027L, 0.1693953067668677431693002024900473L,
            0.3806904069584015456847491391596440L, 0.6193095930415984543152508608403560L,
            0.8306046932331322568306997975099527L, 0.9662347571015760139061507772469973L };
  const RealScalar weights[] = { 0.0856622461895851725201480710863665L, 0.1803807865240693037849167569188581L,
            0.2339569672863455236949351719947755L, 0.2339569672863455236949351719947755L,
            0.1803807865240693037849167569188581L, 0.0856622461895851725201480710863665L };
  eigen_assert(degree <= maxPadeDegree);
  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
  result.setZero(T.rows(), T.rows());
  for (int k = 0; k < degree; ++k)
    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
                           .template triangularView<Upper>().solve(TminusI);
}

template <typename MatrixType>
void MatrixLogarithmAtomic<MatrixType>::computePade7(MatrixType& result, const MatrixType& T)
{
  const int degree = 7;
  const RealScalar nodes[]   = { 0.0254460438286207377369051579760744L, 0.1292344072003027800680676133596058L,
            0.2970774243113014165466967939615193L, 0.5000000000000000000000000000000000L,
            0.7029225756886985834533032060384807L, 0.8707655927996972199319323866403942L,
            0.9745539561713792622630948420239256L };
  const RealScalar weights[] = { 0.0647424830844348466353057163395410L, 0.1398526957446383339507338857118898L,
            0.1909150252525594724751848877444876L, 0.2089795918367346938775510204081633L,
            0.1909150252525594724751848877444876L, 0.1398526957446383339507338857118898L,
            0.0647424830844348466353057163395410L };
  eigen_assert(degree <= maxPadeDegree);
  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
  result.setZero(T.rows(), T.rows());
  for (int k = 0; k < degree; ++k)
    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
                           .template triangularView<Upper>().solve(TminusI);
}

template <typename MatrixType>
void MatrixLogarithmAtomic<MatrixType>::computePade8(MatrixType& result, const MatrixType& T)
{
  const int degree = 8;
  const RealScalar nodes[]   = { 0.0198550717512318841582195657152635L, 0.1016667612931866302042230317620848L,
            0.2372337950418355070911304754053768L, 0.4082826787521750975302619288199080L,
            0.5917173212478249024697380711800920L, 0.7627662049581644929088695245946232L,
            0.8983332387068133697957769682379152L, 0.9801449282487681158417804342847365L };
  const RealScalar weights[] = { 0.0506142681451881295762656771549811L, 0.1111905172266872352721779972131204L,
            0.1568533229389436436689811009933007L, 0.1813418916891809914825752246385978L,
            0.1813418916891809914825752246385978L, 0.1568533229389436436689811009933007L,
            0.1111905172266872352721779972131204L, 0.0506142681451881295762656771549811L };
  eigen_assert(degree <= maxPadeDegree);
  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
  result.setZero(T.rows(), T.rows());
  for (int k = 0; k < degree; ++k)
    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
                           .template triangularView<Upper>().solve(TminusI);
}

template <typename MatrixType>
void MatrixLogarithmAtomic<MatrixType>::computePade9(MatrixType& result, const MatrixType& T)
{
  const int degree = 9;
  const RealScalar nodes[]   = { 0.0159198802461869550822118985481636L, 0.0819844463366821028502851059651326L,
            0.1933142836497048013456489803292629L, 0.3378732882980955354807309926783317L,
            0.5000000000000000000000000000000000L, 0.6621267117019044645192690073216683L,
            0.8066857163502951986543510196707371L, 0.9180155536633178971497148940348674L,
            0.9840801197538130449177881014518364L };
  const RealScalar weights[] = { 0.0406371941807872059859460790552618L, 0.0903240803474287020292360156214564L,
            0.1303053482014677311593714347093164L, 0.1561735385200014200343152032922218L,
            0.1651196775006298815822625346434870L, 0.1561735385200014200343152032922218L,
            0.1303053482014677311593714347093164L, 0.0903240803474287020292360156214564L,
            0.0406371941807872059859460790552618L };
  eigen_assert(degree <= maxPadeDegree);
  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
  result.setZero(T.rows(), T.rows());
  for (int k = 0; k < degree; ++k)
    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
                           .template triangularView<Upper>().solve(TminusI);
}

template <typename MatrixType>
void MatrixLogarithmAtomic<MatrixType>::computePade10(MatrixType& result, const MatrixType& T)
{
  const int degree = 10;
  const RealScalar nodes[]   = { 0.0130467357414141399610179939577740L, 0.0674683166555077446339516557882535L,
            0.1602952158504877968828363174425632L, 0.2833023029353764046003670284171079L,
            0.4255628305091843945575869994351400L, 0.5744371694908156054424130005648600L,
            0.7166976970646235953996329715828921L, 0.8397047841495122031171636825574368L,
            0.9325316833444922553660483442117465L, 0.9869532642585858600389820060422260L };
  const RealScalar weights[] = { 0.0333356721543440687967844049466659L, 0.0747256745752902965728881698288487L,
            0.1095431812579910219977674671140816L, 0.1346333596549981775456134607847347L,
            0.1477621123573764350869464973256692L, 0.1477621123573764350869464973256692L,
            0.1346333596549981775456134607847347L, 0.1095431812579910219977674671140816L,
            0.0747256745752902965728881698288487L, 0.0333356721543440687967844049466659L };
  eigen_assert(degree <= maxPadeDegree);
  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
  result.setZero(T.rows(), T.rows());
  for (int k = 0; k < degree; ++k)
    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
                           .template triangularView<Upper>().solve(TminusI);
}

template <typename MatrixType>
void MatrixLogarithmAtomic<MatrixType>::computePade11(MatrixType& result, const MatrixType& T)
{
  const int degree = 11;
  const RealScalar nodes[]   = { 0.0108856709269715035980309994385713L, 0.0564687001159523504624211153480364L,
            0.1349239972129753379532918739844233L, 0.2404519353965940920371371652706952L,
            0.3652284220238275138342340072995692L, 0.5000000000000000000000000000000000L,
            0.6347715779761724861657659927004308L, 0.7595480646034059079628628347293048L,
            0.8650760027870246620467081260155767L, 0.9435312998840476495375788846519636L,
            0.9891143290730284964019690005614287L };
  const RealScalar weights[] = { 0.0278342835580868332413768602212743L, 0.0627901847324523123173471496119701L,
            0.0931451054638671257130488207158280L, 0.1165968822959952399592618524215876L,
            0.1314022722551233310903444349452546L, 0.1364625433889503153572417641681711L,
            0.1314022722551233310903444349452546L, 0.1165968822959952399592618524215876L,
            0.0931451054638671257130488207158280L, 0.0627901847324523123173471496119701L,
            0.0278342835580868332413768602212743L };
  eigen_assert(degree <= maxPadeDegree);
  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
  result.setZero(T.rows(), T.rows());
  for (int k = 0; k < degree; ++k)
    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
                           .template triangularView<Upper>().solve(TminusI);
}

/** \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;

  /** \brief Constructor.
    *
    * \param[in]  A  %Matrix (expression) forming the argument of the matrix logarithm.
    */
  MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { }
  
  /** \brief Compute the matrix logarithm.
    *
    * \param[out]  result  Logarithm of \p A, where \A is as specified in the constructor.
    */
  template <typename ResultType>
  inline void evalTo(ResultType& result) const
  {
    typedef typename Derived::PlainObject PlainObject;
    typedef internal::traits<PlainObject> Traits;
    static const int RowsAtCompileTime = Traits::RowsAtCompileTime;
    static const int ColsAtCompileTime = Traits::ColsAtCompileTime;
    static const int Options = PlainObject::Options;
    typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
    typedef Matrix<ComplexScalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType;
    typedef MatrixLogarithmAtomic<DynMatrixType> AtomicType;
    AtomicType atomic;
    
    const PlainObject Aevaluated = m_A.eval();
    MatrixFunction<PlainObject, AtomicType> mf(Aevaluated, atomic);
    mf.compute(result);
  }

  Index rows() const { return m_A.rows(); }
  Index cols() const { return m_A.cols(); }
  
private:
  typename internal::nested<Derived>::type m_A;
  
  MatrixLogarithmReturnValue& operator=(const MatrixLogarithmReturnValue&);
};

namespace internal {
  template<typename Derived>
  struct traits<MatrixLogarithmReturnValue<Derived> >
  {
    typedef typename Derived::PlainObject ReturnType;
  };
}


/********** 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