src/MatrixFunctions/MatrixPower.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2012, 2013 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_POWER
#define EIGEN_MATRIX_POWER

namespace Eigen {

template<typename MatrixType> class MatrixPower;

template<typename MatrixType>
class MatrixPowerRetval : public ReturnByValue< MatrixPowerRetval<MatrixType> >
{
  public:
    typedef typename MatrixType::RealScalar RealScalar;
    typedef typename MatrixType::Index Index;

    MatrixPowerRetval(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p)
    { }

    template<typename ResultType>
    inline void evalTo(ResultType& res) const
    { m_pow.compute(res, m_p); }

    Index rows() const { return m_pow.rows(); }
    Index cols() const { return m_pow.cols(); }

  private:
    MatrixPower<MatrixType>& m_pow;
    const RealScalar m_p;
    MatrixPowerRetval& operator=(const MatrixPowerRetval&);
};

template<typename MatrixType>
class MatrixPowerAtomic
{
  private:
    enum {
      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
    };
    typedef typename MatrixType::Scalar Scalar;
    typedef typename MatrixType::RealScalar RealScalar;
    typedef std::complex<RealScalar> ComplexScalar;
    typedef typename MatrixType::Index Index;
    typedef Array<Scalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> ArrayType;

    const MatrixType& m_A;
    RealScalar m_p;

    void computePade(int degree, const MatrixType& IminusT, MatrixType& res) const;
    void compute2x2(MatrixType& res, RealScalar p) const;
    void computeBig(MatrixType& res) const;
    static int getPadeDegree(float normIminusT);
    static int getPadeDegree(double normIminusT);
    static int getPadeDegree(long double normIminusT);
    static ComplexScalar computeSuperDiag(const ComplexScalar&, const ComplexScalar&, RealScalar p);
    static RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p);

  public:
    MatrixPowerAtomic(const MatrixType& T, RealScalar p);
    void compute(MatrixType& res) const;
};

template<typename MatrixType>
MatrixPowerAtomic<MatrixType>::MatrixPowerAtomic(const MatrixType& T, RealScalar p) :
  m_A(T), m_p(p)
{ eigen_assert(T.rows() == T.cols()); }

template<typename MatrixType>
void MatrixPowerAtomic<MatrixType>::compute(MatrixType& res) const
{
  res.resizeLike(m_A);
  switch (m_A.rows()) {
    case 0:
      break;
    case 1:
      res(0,0) = std::pow(m_A(0,0), m_p);
      break;
    case 2:
      compute2x2(res, m_p);
      break;
    default:
      computeBig(res);
  }
}

template<typename MatrixType>
void MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, MatrixType& res) const
{
  int i = degree<<1;
  res = (m_p-degree) / ((i-1)<<1) * IminusT;
  for (--i; i; --i) {
    res = (MatrixType::Identity(IminusT.rows(), IminusT.cols()) + res).template triangularView<Upper>()
	.solve((i==1 ? -m_p : i&1 ? (-m_p-(i>>1))/(i<<1) : (m_p-(i>>1))/((i-1)<<1)) * IminusT).eval();
  }
  res += MatrixType::Identity(IminusT.rows(), IminusT.cols());
}

// This function assumes that res has the correct size (see bug 614)
template<typename MatrixType>
void MatrixPowerAtomic<MatrixType>::compute2x2(MatrixType& res, RealScalar p) const
{
  using std::abs;
  using std::pow;

  res.coeffRef(0,0) = pow(m_A.coeff(0,0), p);

  for (Index i=1; i < m_A.cols(); ++i) {
    res.coeffRef(i,i) = pow(m_A.coeff(i,i), p);
    if (m_A.coeff(i-1,i-1) == m_A.coeff(i,i))
      res.coeffRef(i-1,i) = p * pow(m_A.coeff(i,i), p-1);
    else if (2*abs(m_A.coeff(i-1,i-1)) < abs(m_A.coeff(i,i)) || 2*abs(m_A.coeff(i,i)) < abs(m_A.coeff(i-1,i-1)))
      res.coeffRef(i-1,i) = (res.coeff(i,i)-res.coeff(i-1,i-1)) / (m_A.coeff(i,i)-m_A.coeff(i-1,i-1));
    else
      res.coeffRef(i-1,i) = computeSuperDiag(m_A.coeff(i,i), m_A.coeff(i-1,i-1), p);
    res.coeffRef(i-1,i) *= m_A.coeff(i-1,i);
  }
}

template<typename MatrixType>
void MatrixPowerAtomic<MatrixType>::computeBig(MatrixType& res) const
{
  const int digits = std::numeric_limits<RealScalar>::digits;
  const RealScalar maxNormForPade = digits <=  24? 4.3386528e-1f:                           // sigle precision
				    digits <=  53? 2.789358995219730e-1:                    // double precision
				    digits <=  64? 2.4471944416607995472e-1L:               // extended precision
				    digits <= 106? 1.1016843812851143391275867258512e-1L:   // double-double
						   9.134603732914548552537150753385375e-2L; // quadruple precision
  MatrixType IminusT, sqrtT, T = m_A.template triangularView<Upper>();
  RealScalar normIminusT;
  int degree, degree2, numberOfSquareRoots = 0;
  bool hasExtraSquareRoot = false;

  /* FIXME
   * For singular T, norm(I - T) >= 1 but maxNormForPade < 1, leads to infinite
   * loop.  We should move 0 eigenvalues to bottom right corner.  We need not
   * worry about tiny values (e.g. 1e-300) because they will reach 1 if
   * repetitively sqrt'ed.
   *
   * If the 0 eigenvalues are semisimple, they can form a 0 matrix at the
   * bottom right corner.
   *
   * [ T  A ]^p   [ T^p  (T^-1 T^p A) ]
   * [      ]   = [                   ]
   * [ 0  0 ]     [  0         0      ]
   */
  for (Index i=0; i < m_A.cols(); ++i)
    eigen_assert(m_A(i,i) != RealScalar(0));

  while (true) {
    IminusT = MatrixType::Identity(m_A.rows(), m_A.cols()) - T;
    normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff();
    if (normIminusT < maxNormForPade) {
      degree = getPadeDegree(normIminusT);
      degree2 = getPadeDegree(normIminusT/2);
      if (degree - degree2 <= 1 || hasExtraSquareRoot)
	break;
      hasExtraSquareRoot = true;
    }
    MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
    T = sqrtT.template triangularView<Upper>();
    ++numberOfSquareRoots;
  }
  computePade(degree, IminusT, res);

  for (; numberOfSquareRoots; --numberOfSquareRoots) {
    compute2x2(res, std::ldexp(m_p, -numberOfSquareRoots));
    res = res.template triangularView<Upper>() * res;
  }
  compute2x2(res, m_p);
}

template<typename MatrixType>
inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(float normIminusT)
{
  const float maxNormForPade[] = { 2.8064004e-1f /* degree = 3 */ , 4.3386528e-1f };
  int degree = 3;
  for (; degree <= 4; ++degree)
    if (normIminusT <= maxNormForPade[degree - 3])
      break;
  return degree;
}

template<typename MatrixType>
inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(double normIminusT)
{
  const double maxNormForPade[] = { 1.884160592658218e-2 /* degree = 3 */ , 6.038881904059573e-2, 1.239917516308172e-1,
      1.999045567181744e-1, 2.789358995219730e-1 };
  int degree = 3;
  for (; degree <= 7; ++degree)
    if (normIminusT <= maxNormForPade[degree - 3])
      break;
  return degree;
}

template<typename MatrixType>
inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(long double normIminusT)
{
#if   LDBL_MANT_DIG == 53
  const int maxPadeDegree = 7;
  const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L,
      1.999045567181744e-1L, 2.789358995219730e-1L };
#elif LDBL_MANT_DIG <= 64
  const int maxPadeDegree = 8;
  const double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L,
      6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L };
#elif LDBL_MANT_DIG <= 106
  const int maxPadeDegree = 10;
  const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ ,
      1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L,
      2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L,
      1.1016843812851143391275867258512e-1L };
#else
  const int maxPadeDegree = 10;
  const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ ,
      6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L,
      9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L,
      3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L,
      9.134603732914548552537150753385375e-2L };
#endif
  int degree = 3;
  for (; degree <= maxPadeDegree; ++degree)
    if (normIminusT <= maxNormForPade[degree - 3])
      break;
  return degree;
}

template<typename MatrixType>
inline typename MatrixPowerAtomic<MatrixType>::ComplexScalar
MatrixPowerAtomic<MatrixType>::computeSuperDiag(const ComplexScalar& curr, const ComplexScalar& prev, RealScalar p)
{
  ComplexScalar logCurr = std::log(curr);
  ComplexScalar logPrev = std::log(prev);
  int unwindingNumber = std::ceil((numext::imag(logCurr - logPrev) - M_PI) / (2*M_PI));
  ComplexScalar w = numext::atanh2(curr - prev, curr + prev) + ComplexScalar(0, M_PI*unwindingNumber);
  return RealScalar(2) * std::exp(RealScalar(0.5) * p * (logCurr + logPrev)) * std::sinh(p * w) / (curr - prev);
}

template<typename MatrixType>
inline typename MatrixPowerAtomic<MatrixType>::RealScalar
MatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev, RealScalar p)
{
  RealScalar w = numext::atanh2(curr - prev, curr + prev);
  return 2 * std::exp(p * (std::log(curr) + std::log(prev)) / 2) * std::sinh(p * w) / (curr - prev);
}

/**
 * \ingroup MatrixFunctions_Module
 *
 * \brief Class for computing matrix powers.
 *
 * \tparam MatrixType  type of the base, expected to be an instantiation
 * of the Matrix class template.
 *
 * This class is capable of computing real/complex matrices raised to
 * an arbitrary real power. Meanwhile, it saves the result of Schur
 * decomposition if an non-integral power has even been calculated.
 * Therefore, if you want to compute multiple (>= 2) matrix powers
 * for the same matrix, using the class directly is more efficient than
 * calling MatrixBase::pow().
 *
 * Example:
 * \include MatrixPower_optimal.cpp
 * Output: \verbinclude MatrixPower_optimal.out
 */
template<typename MatrixType>
class MatrixPower
{
  private:
    enum {
      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };
    typedef typename MatrixType::Scalar Scalar;
    typedef typename MatrixType::RealScalar RealScalar;
    typedef typename MatrixType::Index Index;

  public:
    /**
     * \brief Constructor.
     *
     * \param[in] A  the base of the matrix power.
     *
     * The class stores a reference to A, so it should not be changed
     * (or destroyed) before evaluation.
     */
    explicit MatrixPower(const MatrixType& A) : m_A(A), m_conditionNumber(0)
    { eigen_assert(A.rows() == A.cols()); }

    /**
     * \brief Returns the matrix power.
     *
     * \param[in] p  exponent, a real scalar.
     * \return The expression \f$ A^p \f$, where A is specified in the
     * constructor.
     */
    const MatrixPowerRetval<MatrixType> operator()(RealScalar p)
    { return MatrixPowerRetval<MatrixType>(*this, p); }

    /**
     * \brief Compute the matrix power.
     *
     * \param[in]  p    exponent, a real scalar.
     * \param[out] res  \f$ A^p \f$ where A is specified in the
     * constructor.
     */
    template<typename ResultType>
    void compute(ResultType& res, RealScalar p);

    Index rows() const { return m_A.rows(); }
    Index cols() const { return m_A.cols(); }

  private:
    typedef std::complex<RealScalar> ComplexScalar;
    typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, MatrixType::Options,
              MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrix;

    typename MatrixType::Nested m_A;
    MatrixType m_tmp;
    ComplexMatrix m_T, m_U, m_fT;
    RealScalar m_conditionNumber;

    RealScalar modfAndInit(RealScalar, RealScalar*);

    template<typename ResultType>
    void computeIntPower(ResultType&, RealScalar);

    template<typename ResultType>
    void computeFracPower(ResultType&, RealScalar);

    template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
    static void revertSchur(
        Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
        const ComplexMatrix& T,
        const ComplexMatrix& U);

    template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
    static void revertSchur(
        Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
        const ComplexMatrix& T,
        const ComplexMatrix& U);
};

template<typename MatrixType>
template<typename ResultType>
void MatrixPower<MatrixType>::compute(ResultType& res, RealScalar p)
{
  switch (cols()) {
    case 0:
      break;
    case 1:
      res(0,0) = std::pow(m_A.coeff(0,0), p);
      break;
    default:
      RealScalar intpart, x = modfAndInit(p, &intpart);
      computeIntPower(res, intpart);
      computeFracPower(res, x);
  }
}

template<typename MatrixType>
typename MatrixPower<MatrixType>::RealScalar
MatrixPower<MatrixType>::modfAndInit(RealScalar x, RealScalar* intpart)
{
  typedef Array<RealScalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> RealArray;

  *intpart = std::floor(x);
  RealScalar res = x - *intpart;

  if (!m_conditionNumber && res) {
    const ComplexSchur<MatrixType> schurOfA(m_A);
    m_T = schurOfA.matrixT();
    m_U = schurOfA.matrixU();

    const RealArray absTdiag = m_T.diagonal().array().abs();
    m_conditionNumber = absTdiag.maxCoeff() / absTdiag.minCoeff();
  }

  if (res>RealScalar(0.5) && res>(1-res)*std::pow(m_conditionNumber, res)) {
    --res;
    ++*intpart;
  }
  return res;
}

template<typename MatrixType>
template<typename ResultType>
void MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p)
{
  RealScalar pp = std::abs(p);

  if (p<0)  m_tmp = m_A.inverse();
  else      m_tmp = m_A;

  res = MatrixType::Identity(rows(), cols());
  while (pp >= 1) {
    if (std::fmod(pp, 2) >= 1)
      res = m_tmp * res;
    m_tmp *= m_tmp;
    pp /= 2;
  }
}

template<typename MatrixType>
template<typename ResultType>
void MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p)
{
  if (p) {
    eigen_assert(m_conditionNumber);
    MatrixPowerAtomic<ComplexMatrix>(m_T, p).compute(m_fT);
    revertSchur(m_tmp, m_fT, m_U);
    res = m_tmp * res;
  }
}

template<typename MatrixType>
template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
inline void MatrixPower<MatrixType>::revertSchur(
    Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
    const ComplexMatrix& T,
    const ComplexMatrix& U)
{ res.noalias() = U * (T.template triangularView<Upper>() * U.adjoint()); }

template<typename MatrixType>
template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
inline void MatrixPower<MatrixType>::revertSchur(
    Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
    const ComplexMatrix& T,
    const ComplexMatrix& U)
{ res.noalias() = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); }

/**
 * \ingroup MatrixFunctions_Module
 *
 * \brief Proxy for the matrix power of some matrix (expression).
 *
 * \tparam Derived  type of the base, a matrix (expression).
 *
 * This class holds the arguments to the matrix power 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::pow() and related functions and most of the
 * time this is the only way it is used.
 */
template<typename Derived>
class MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Derived> >
{
  public:
    typedef typename Derived::PlainObject PlainObject;
    typedef typename Derived::RealScalar RealScalar;
    typedef typename Derived::Index Index;

    /**
     * \brief Constructor.
     *
     * \param[in] A  %Matrix (expression), the base of the matrix power.
     * \param[in] p  scalar, the exponent of the matrix power.
     */
    MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p)
    { }

    /**
     * \brief Compute the matrix power.
     *
     * \param[out] result  \f$ A^p \f$ where \p A and \p p are as in the
     * constructor.
     */
    template<typename ResultType>
    inline void evalTo(ResultType& res) const
    { MatrixPower<PlainObject>(m_A.eval()).compute(res, m_p); }

    Index rows() const { return m_A.rows(); }
    Index cols() const { return m_A.cols(); }

  private:
    const Derived& m_A;
    const RealScalar m_p;
    MatrixPowerReturnValue& operator=(const MatrixPowerReturnValue&);
};

namespace internal {

template<typename MatrixPowerType>
struct traits< MatrixPowerRetval<MatrixPowerType> >
{ typedef typename MatrixPowerType::PlainObject ReturnType; };

template<typename Derived>
struct traits< MatrixPowerReturnValue<Derived> >
{ typedef typename Derived::PlainObject ReturnType; };

}

template<typename Derived>
const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const
{ return MatrixPowerReturnValue<Derived>(derived(), p); }

} // namespace Eigen

#endif // EIGEN_MATRIX_POWER