matrix power: MatrixBase::pow(RealScalar) and MatrixBase::pow(T) where T is integral type

Eigen/src/Core/MatrixBase.h

@@ -454,6 +454,8 @@ template<typename Derived> class MatrixBase
     const MatrixFunctionReturnValue<Derived> sin() const;
     const MatrixSquareRootReturnValue<Derived> sqrt() const;
     const MatrixLogarithmReturnValue<Derived> log() const;
+    template <typename ExponentType>
+    const MatrixPowerReturnValue<Derived, ExponentType> pow(const ExponentType& p) const;
 
 #ifdef EIGEN2_SUPPORT
     template<typename ProductDerived, typename Lhs, typename Rhs>

Eigen/src/Core/util/ForwardDeclarations.h

@@ -271,6 +271,7 @@ template<typename Derived> struct MatrixExponentialReturnValue;
 template<typename Derived> class MatrixFunctionReturnValue;
 template<typename Derived> class MatrixSquareRootReturnValue;
 template<typename Derived> class MatrixLogarithmReturnValue;
+template<typename Derived, typename ExponentType> class MatrixPowerReturnValue;
 
 namespace internal {
 template <typename Scalar>

unsupported/Eigen/MatrixFunctions

@@ -57,7 +57,7 @@
 #include "src/MatrixFunctions/MatrixFunction.h"
 #include "src/MatrixFunctions/MatrixSquareRoot.h"
 #include "src/MatrixFunctions/MatrixLogarithm.h"
-
+#include "src/MatrixFunctions/MatrixPower.h"
 
 
 /** 

unsupported/Eigen/src/MatrixFunctions/MatrixPower.h

@@ -0,0 +1,953 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2012 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
+
+#ifndef M_PI
+#define M_PI 3.141592653589793238462643383279503L
+#endif
+
+namespace Eigen {
+
+/**
+ * \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.
+ * \tparam RealScalar   type of the exponent, a real scalar.
+ * \tparam PlainObject  type of the multiplier.
+ * \tparam IsInteger    used internally to select correct specialization.
+ */
+template <typename MatrixType, typename RealScalar, typename PlainObject = MatrixType,
+	  int IsInteger = NumTraits<RealScalar>::IsInteger>
+class MatrixPower
+{
+  public:
+    /**
+     * \brief Constructor.
+     *
+     * \param[in] A  the base of the matrix power.
+     * \param[in] p  the exponent of the matrix power.
+     * \param[in] b  the multiplier.
+     */
+    MatrixPower(const MatrixType& A, const RealScalar& p, const PlainObject& b) :
+      m_A(A),
+      m_p(p),
+      m_b(b),
+      m_dimA(A.cols()),
+      m_dimb(b.cols())
+    { /* empty body */ }
+
+    /**
+     * \brief Compute the matrix power.
+     *
+     * \param[out] result  \f$ A^p b \f$, as specified in the constructor.
+     */
+    template <typename ResultType> void compute(ResultType& result);
+ 
+  private:
+    typedef internal::traits<MatrixType> Traits;
+    static const int Rows = Traits::RowsAtCompileTime;
+    static const int Cols = Traits::ColsAtCompileTime;
+    static const int Options = Traits::Options;
+    static const int MaxRows = Traits::MaxRowsAtCompileTime;
+    static const int MaxCols = Traits::MaxColsAtCompileTime;
+
+    typedef typename MatrixType::Scalar Scalar;
+    typedef std::complex<RealScalar> ComplexScalar;
+    typedef typename MatrixType::Index Index;
+    typedef Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols> ComplexMatrix;
+    typedef Array<ComplexScalar, Rows, 1, ColMajor, MaxRows> ComplexArray;
+
+    /**
+     * \brief Compute the matrix power.
+     *
+     * If \c b is \em fatter than \c A, it computes \f$ A^{p_{\textrm int}}
+     * \f$ first, and then multiplies it with \c b. Otherwise,
+     * #computeChainProduct optimizes the expression.
+     *
+     * \sa computeChainProduct(ResultType&);
+     */
+    template <typename ResultType>
+    void computeIntPower(ResultType& result);
+
+    /**
+     * \brief Convert integral power of the matrix into chain product.
+     *
+     * This optimizes the matrix expression. It automatically chooses binary
+     * powering or matrix chain multiplication or solving the linear system
+     * repetitively, according to which algorithm costs less.
+     */
+    template <typename ResultType>
+    void computeChainProduct(ResultType&);
+
+    /** \brief Compute the cost of binary powering. */
+    int computeCost(RealScalar);
+
+    /** \brief Solve the linear system repetitively. */
+    template <typename ResultType>
+    void partialPivLuSolve(RealScalar, ResultType&);
+
+    /** \brief Compute Schur decomposition of #m_A. */
+    void computeSchurDecomposition();
+
+    /**
+     * \brief Split #m_p into integral part and fractional part.
+     *
+     * This method stores the integral part \f$ p_{\textrm int} \f$ into
+     * #m_pint and the fractional part \f$ p_{\textrm frac} \f$ into
+     * #m_pfrac, where #m_pfrac is in the interval \f$ (-1,1) \f$. To
+     * choose between the possibilities below, it considers the computation
+     * of \f$ A^{p_1} \f$ and \f$ A^{p_2} \f$ and determines which of these
+     * computations is the better conditioned.
+     */
+    void getFractionalExponent();
+
+    /** \brief Compute atanh (inverse hyperbolic tangent). */
+    ComplexScalar atanh(const ComplexScalar& x);
+
+    /** \brief Compute power of 2x2 triangular matrix. */
+    void compute2x2(const RealScalar& p);
+
+    /**
+     * \brief Compute power of triangular matrices with size > 2. 
+     * \details This uses a Schur-Pad&eacute; algorithm.
+     */
+    void computeBig();
+
+    /** \brief Get suitable degree for Pade approximation. (specialized for \c RealScalar = \c double) */
+    inline int getPadeDegree(double);
+/* TODO
+ *  inline int getPadeDegree(float);
+ *
+ *  inline int getPadeDegree(long double);
+ */
+    /** \brief Compute Pad&eacute; approximation to matrix fractional power. */
+    void computePade(int degree, const ComplexMatrix& IminusT);
+
+    /** \brief Get a certain coefficient of the Pad&eacute; approximation. */
+    inline RealScalar coeff(int degree);
+
+    /**
+     * \brief Store the fractional power into #m_tmp.
+     *
+     * This intended for complex matrices.
+     */
+    void computeTmp(ComplexScalar);
+
+    /**
+     * \brief Store the fractional power into #m_tmp.
+     *
+     * This is intended for real matrices. It takes the real part of
+     * \f$ U T^{p_{\textrm frac}} U^H \f$.
+     *
+     * \sa computeTmp(ComplexScalar);
+     */
+    void computeTmp(RealScalar);
+
+    const MatrixType& m_A;   ///< \brief Reference to the matrix base.
+    const RealScalar& m_p;   ///< \brief Reference to the real exponent.
+    const PlainObject& m_b;  ///< \brief Reference to the multiplier.
+    const Index m_dimA;      ///< \brief The dimension of #m_A, equivalent to %m_A.cols().
+    const Index m_dimb;      ///< \brief The dimension of #m_b, equivalent to %m_b.cols().
+    MatrixType m_tmp;        ///< \brief Used for temporary storage.
+    RealScalar m_pint;       ///< \brief Integer part of #m_p.
+    RealScalar m_pfrac;      ///< \brief Fractional part of #m_p.
+    ComplexMatrix m_T;       ///< \brief Triangular part of Schur decomposition.
+    ComplexMatrix m_U;       ///< \brief Unitary part of Schur decomposition.
+    ComplexMatrix m_fT;      ///< \brief #m_T to the power of #m_pfrac.
+    ComplexArray m_logTdiag; ///< \brief Logarithm of the main diagonal of #m_T.
+};
+
+/**
+ * \internal \ingroup MatrixFunctions_Module
+ * \brief Partial specialization for integral exponents.
+ */
+template <typename MatrixType, typename IntExponent, typename PlainObject>
+class MatrixPower<MatrixType, IntExponent, PlainObject, 1>
+{
+  public:
+    /**
+     * \brief Constructor.
+     *
+     * \param[in] A  the base of the matrix power.
+     * \param[in] p  the exponent of the matrix power.
+     * \param[in] b  the multiplier.
+     */
+    MatrixPower(const MatrixType& A, const IntExponent& p, const PlainObject& b) :
+      m_A(A),
+      m_p(p),
+      m_b(b),
+      m_dimA(A.cols()),
+      m_dimb(b.cols())
+    { /* empty body */ }
+
+    /**
+     * \brief Compute the matrix power.
+     *
+     * If \c b is \em fatter than \c A, it computes \f$ A^p \f$ first, and
+     * then multiplies it with \c b. Otherwise, #computeChainProduct
+     * optimizes the expression.
+     *
+     * \param[out] result  \f$ A^p b \f$, as specified in the constructor.
+     *
+     * \sa computeChainProduct(ResultType&);
+     */
+    template <typename ResultType> void compute(ResultType& result);
+
+  private:
+    typedef typename MatrixType::Index Index;
+
+    const MatrixType& m_A;  ///< \brief Reference to the matrix base.
+    const IntExponent& m_p; ///< \brief Reference to the real exponent.
+    const PlainObject& m_b; ///< \brief Reference to the multiplier.
+    const Index m_dimA;     ///< \brief The dimension of #m_A, equivalent to %m_A.cols().
+    const Index m_dimb;     ///< \brief The dimension of #m_b, equivalent to %m_b.cols().
+    MatrixType m_tmp;       ///< \brief Used for temporary storage.
+
+    /**
+     * \brief Convert matrix power into chain product.
+     *
+     * This optimizes the matrix expression. It automatically chooses binary
+     * powering or matrix chain multiplication or solving the linear system
+     * repetitively, according to which algorithm costs less.
+     */
+    template <typename ResultType> void computeChainProduct(ResultType& result);
+
+    /** \brief Compute the cost of binary powering. */
+    int computeCost(const IntExponent& p);
+
+    /** \brief Solve the linear system repetitively. */
+    template <typename ResultType>
+    void partialPivLuSolve(IntExponent p, ResultType& result);
+};
+
+/**
+ * \internal \ingroup MatrixFunctions_Module
+ * \brief Partial specialization for complex matrices raised to complex exponents.
+ */
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+class MatrixPower<MatrixType, std::complex<RealScalar>, PlainObject, IsInteger>
+{
+  private:
+    typedef internal::traits<MatrixType> Traits;
+    static const int Rows = Traits::RowsAtCompileTime;
+    static const int Cols = Traits::ColsAtCompileTime;
+    static const int Options = Traits::Options;
+    static const int MaxRows = Traits::MaxRowsAtCompileTime;
+    static const int MaxCols = Traits::MaxColsAtCompileTime;
+
+    typedef typename MatrixType::Scalar Scalar;
+    typedef typename MatrixType::Index Index;
+    typedef Array<Scalar, Rows, 1, ColMajor, MaxRows> ArrayType;
+
+  public:
+    /**
+     * \brief Constructor.
+     *
+     * \param[in] A  the base of the matrix power.
+     * \param[in] p  the exponent of the matrix power.
+     * \param[in] b  the multiplier.
+     */
+    MatrixPower(const MatrixType& A, const Scalar& p, const PlainObject& b) :
+      m_A(A),
+      m_p(p),
+      m_b(b),
+      m_dimA(A.cols()),
+      m_dimb(b.cols())
+    { EIGEN_STATIC_ASSERT(false, COMPLEX_POWER_OF_A_MATRIX_IS_UNDER_CONSTRUCTION) }
+
+    /**
+     * \brief Compute the matrix power.
+     *
+     * \param[out] result  \f$ A^p b \f$, as specified in the constructor.
+     */
+    template <typename ResultType> void compute(ResultType& result);
+ 
+  private:
+    /** \brief Compute Schur decomposition of #m_A. */
+    void computeSchurDecomposition();
+
+    /** \brief Compute atanh (inverse hyperbolic tangent). */
+    Scalar atanh(const Scalar& x);
+
+    /** \brief Compute power of 2x2 triangular matrix. */
+    void compute2x2(const Scalar& p);
+
+    /**
+     * \brief Compute power of triangular matrices with size > 2. 
+     * \details This uses a Schur-Pad&eacute; algorithm.
+     */
+    void computeBig();
+
+    /** \brief Get suitable degree for Pade approximation. (specialized for \c RealScalar = \c double) */
+    inline int getPadeDegree(double);
+/* TODO
+ *  inline int getPadeDegree(float);
+ *
+ *  inline int getPadeDegree(long double);
+ */
+    /** \brief Compute Pad&eacute; approximation to matrix fractional power. */
+    void computePade(int degree, const MatrixType& IminusT);
+
+    /** \brief Get a certain coefficient of the Pad&eacute; approximation. */
+    inline Scalar coeff(int degree);
+
+    const MatrixType& m_A;  ///< \brief Reference to the matrix base.
+    const Scalar& m_p;      ///< \brief Reference to the real exponent.
+    const PlainObject& m_b; ///< \brief Reference to the multiplier.
+    const Index m_dimA;     ///< \brief The dimension of #m_A, equivalent to %m_A.cols().
+    const Index m_dimb;     ///< \brief The dimension of #m_b, equivalent to %m_b.cols().
+    MatrixType m_tmp;       ///< \brief Used for temporary storage.
+    MatrixType m_T;         ///< \brief Triangular part of Schur decomposition.
+    MatrixType m_U;         ///< \brief Unitary part of Schur decomposition.
+    MatrixType m_fT;        ///< \brief #m_T to the power of #m_pfrac.
+    ArrayType m_logTdiag;   ///< \brief Logarithm of the main diagonal of #m_T.
+};
+
+/******* Specialized for real exponents *******/
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+template <typename ResultType>
+void MatrixPower<MatrixType,RealScalar,PlainObject,IsInteger>::compute(ResultType& result)
+{
+  using std::floor;
+  using std::pow;
+
+  m_pint = floor(m_p);
+  m_pfrac = m_p - m_pint;
+
+  if (m_pfrac == RealScalar(0))
+    computeIntPower(result);
+  else if (m_dimA == 1)
+    result = pow(m_A(0,0), m_p) * m_b;
+  else {
+    computeSchurDecomposition();
+    getFractionalExponent();
+    computeIntPower(result);
+    if (m_dimA == 2)
+      compute2x2(m_pfrac);
+    else
+      computeBig();
+    computeTmp(Scalar());
+    result *= m_tmp;
+  }
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+template <typename ResultType>
+void MatrixPower<MatrixType,RealScalar,PlainObject,IsInteger>::computeIntPower(ResultType& result)
+{
+  if (m_dimb > m_dimA) {
+    MatrixType tmp = MatrixType::Identity(m_A.rows(), m_A.cols());
+    computeChainProduct(tmp);
+    result = tmp * m_b;
+  } else {
+    result = m_b;
+    computeChainProduct(result);
+  }
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+template <typename ResultType>
+void MatrixPower<MatrixType,RealScalar,PlainObject,IsInteger>::computeChainProduct(ResultType& result)
+{
+  using std::frexp;
+  using std::ldexp;
+
+  const bool pIsNegative = m_pint < RealScalar(0);
+  RealScalar p = pIsNegative? -m_pint: m_pint;
+  int cost = computeCost(p);
+
+  if (pIsNegative) {
+    if (p * m_dimb <= cost * m_dimA) {
+      partialPivLuSolve(p, result);
+      return;
+    } else {
+      m_tmp = m_A.inverse();
+    }
+  } else {
+    m_tmp = m_A;
+  }
+  while (p * m_dimb > cost * m_dimA) {
+    if (fmod(p, RealScalar(2)) >= RealScalar(1)) {
+      result = m_tmp * result;
+      cost--;
+    }
+    m_tmp *= m_tmp;
+    cost--;
+    p = ldexp(p, -1);
+  }
+  for (; p >= RealScalar(1); p--)
+    result = m_tmp * result;
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+int MatrixPower<MatrixType,RealScalar,PlainObject,IsInteger>::computeCost(RealScalar p)
+{
+  using std::frexp;
+  using std::ldexp;
+  int cost, tmp;
+  frexp(p, &cost);
+  while (frexp(p, &tmp), tmp > 0) {
+    p -= ldexp(RealScalar(0.5), tmp);
+    cost++;
+  }
+  return cost;
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+template <typename ResultType>
+void MatrixPower<MatrixType,RealScalar,PlainObject,IsInteger>::partialPivLuSolve(RealScalar p, ResultType& result)
+{
+  const PartialPivLU<MatrixType> Asolver(m_A);
+  for (; p >= RealScalar(1); p--)
+    result = Asolver.solve(result);
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+void MatrixPower<MatrixType,RealScalar,PlainObject,IsInteger>::computeSchurDecomposition()
+{
+  const ComplexSchur<MatrixType> schurOfA(m_A);
+  m_T = schurOfA.matrixT();
+  m_U = schurOfA.matrixU();
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+void MatrixPower<MatrixType,RealScalar,PlainObject,IsInteger>::getFractionalExponent()
+{
+  using std::pow;
+
+  typedef Array<RealScalar, Rows, 1, ColMajor, MaxRows> RealArray;
+  const ComplexArray Tdiag = m_T.diagonal();
+  RealScalar maxAbsEival, minAbsEival, *begin, *end;
+  RealArray absTdiag;
+
+  m_logTdiag = Tdiag.log();
+  absTdiag = Tdiag.abs();
+  maxAbsEival = minAbsEival = absTdiag[0];
+  begin = absTdiag.data();
+  end = begin + m_dimA;
+
+  // This avoids traversing the array twice.
+  for (RealScalar *ptr = begin + 1; ptr < end; ptr++) {
+    if (*ptr > maxAbsEival)
+      maxAbsEival = *ptr;
+    else if (*ptr < minAbsEival)
+      minAbsEival = *ptr;
+  }
+  if (m_pfrac > RealScalar(0.5) &&  // This is just a shortcut.
+      m_pfrac > (RealScalar(1) - m_pfrac) * pow(maxAbsEival/minAbsEival, m_pfrac)) {
+    m_pfrac--;
+    m_pint++;
+  }
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+std::complex<RealScalar> MatrixPower<MatrixType,RealScalar,PlainObject,IsInteger>::atanh(const ComplexScalar& x)
+{
+  using std::abs;
+  using std::log;
+  using std::sqrt;
+
+  if (abs(x) > sqrt(NumTraits<RealScalar>::epsilon()))
+    return RealScalar(0.5) * log((RealScalar(1) + x) / (RealScalar(1) - x));
+  else
+    return x + x*x*x / RealScalar(3);
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+void MatrixPower<MatrixType,RealScalar,PlainObject,IsInteger>::compute2x2(const RealScalar& p)
+{
+  using std::abs;
+  using std::ceil;
+  using std::exp;
+  using std::imag;
+  using std::ldexp;
+  using std::log;
+  using std::pow;
+  using std::sinh;
+
+  int i, j, unwindingNumber;
+  ComplexScalar w;
+
+  m_fT(0,0) = pow(m_T(0,0), p);
+
+  for (j = 1; j < m_dimA; j++) {
+    i = j - 1;
+    m_fT(j,j) = pow(m_T(j,j), p);
+
+    if (m_T(i,i) == m_T(j,j))
+      m_fT(i,j) = p * pow(m_T(i,j), p - RealScalar(1));
+    else if (abs(m_T(i,i)) < ldexp(abs(m_T(j,j)), -1) || abs(m_T(j,j)) < ldexp(abs(m_T(i,i)), -1))
+      m_fT(i,j) = m_T(i,j) * (m_fT(j,j) - m_fT(i,i)) / (m_T(j,j) - m_T(i,i));
+    else {
+      // computation in previous branch is inaccurate if abs(m_T(j,j)) \approx abs(m_T(i,i))
+      unwindingNumber = static_cast<int>(ceil((imag(m_logTdiag[j] - m_logTdiag[i]) - M_PI) / (2 * M_PI)));
+      w = atanh((m_T(j,j) - m_T(i,i)) / (m_T(j,j) + m_T(i,i))) + ComplexScalar(0, M_PI * unwindingNumber);
+      m_fT(i,j) = m_T(i,j) * RealScalar(2) * exp(RealScalar(0.5) * p * (m_logTdiag[j] + m_logTdiag[i])) *
+	  sinh(p * w) / (m_T(j,j) - m_T(i,i));
+    }
+  }
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+void MatrixPower<MatrixType,RealScalar,PlainObject,IsInteger>::computeBig()
+{
+  using std::ldexp;
+
+  const RealScalar maxNormForPade = 2.787629930862099e-1;
+  int degree, degree2, numberOfSquareRoots = 0, numberOfExtraSquareRoots = 0;
+  ComplexMatrix IminusT, sqrtT, T = m_T;
+  RealScalar normIminusT;
+
+  while (true) {
+    IminusT = ComplexMatrix::Identity(m_A.rows(), m_A.cols()) - T;
+    normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff();
+    if (normIminusT < maxNormForPade) {
+      degree = getPadeDegree(normIminusT);
+      degree2 = getPadeDegree(normIminusT * RealScalar(0.5));
+      if (degree - degree2 <= 1 || numberOfExtraSquareRoots)
+	break;
+      numberOfExtraSquareRoots++;
+    }
+    MatrixSquareRootTriangular<ComplexMatrix>(T).compute(sqrtT);
+    T = sqrtT;
+    numberOfSquareRoots++;
+  }
+  computePade(degree, IminusT);
+
+  for (; numberOfSquareRoots; numberOfSquareRoots--) {
+    compute2x2(ldexp(m_pfrac, -numberOfSquareRoots));
+    m_fT *= m_fT;
+  }
+  compute2x2(m_pfrac);
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+inline int MatrixPower<MatrixType,RealScalar,PlainObject,IsInteger>::getPadeDegree(double normIminusT)
+{
+  const double maxNormForPade[] = { 1.882832775783885e-2 /* degree = 3 */ , 6.036100693089764e-2,
+      1.239372725584911e-1, 1.998030690604271e-1, 2.787629930862099e-1 };
+  for (int degree = 3; degree <= 7; degree++)
+    if (normIminusT <= maxNormForPade[degree - 3])
+      return degree;
+  assert(false); // this line should never be reached
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+void MatrixPower<MatrixType,RealScalar,PlainObject,IsInteger>::computePade(int degree, const ComplexMatrix& IminusT)
+{
+  degree <<= 1;
+  m_fT = coeff(degree) * IminusT;
+
+  for (int i = degree - 1; i; i--) {
+    m_fT = (ComplexMatrix::Identity(m_A.rows(), m_A.cols()) + m_fT).template triangularView<Upper>()
+	.solve(coeff(i) * IminusT).eval();
+  }
+  m_fT += ComplexMatrix::Identity(m_A.rows(), m_A.cols());
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+inline RealScalar MatrixPower<MatrixType,RealScalar,PlainObject,IsInteger>::coeff(int i)
+{
+  if (i == 1)
+    return -m_pfrac;
+  else if (i & 1)
+    return (-m_pfrac - RealScalar(i)) / RealScalar((i<<2) + 2);
+  else
+    return (m_pfrac - RealScalar(i)) / RealScalar((i<<2) - 2);
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+void MatrixPower<MatrixType,RealScalar,PlainObject,IsInteger>::computeTmp(RealScalar)
+{ m_tmp = (m_U * m_fT * m_U.adjoint()).real(); }
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+void MatrixPower<MatrixType,RealScalar,PlainObject,IsInteger>::computeTmp(ComplexScalar)
+{ m_tmp = (m_U * m_fT * m_U.adjoint()).eval(); }
+
+/******* Specialized for integral exponents *******/
+
+template <typename MatrixType, typename IntExponent, typename PlainObject>
+template <typename ResultType>
+void MatrixPower<MatrixType,IntExponent,PlainObject,1>::compute(ResultType& result)
+{
+  if (m_dimb > m_dimA) {
+    MatrixType tmp = MatrixType::Identity(m_dimA, m_dimA);
+    computeChainProduct(tmp);
+    result = tmp * m_b;
+  } else {
+    result = m_b;
+    computeChainProduct(result);
+  }
+}
+
+template <typename MatrixType, typename IntExponent, typename PlainObject>
+int MatrixPower<MatrixType,IntExponent,PlainObject,1>::computeCost(const IntExponent& p)
+{
+  int cost = 0;
+  IntExponent tmp = p;
+  for (tmp = p >> 1; tmp; tmp >>= 1)
+    cost++;
+  for (tmp = IntExponent(1); tmp <= p; tmp <<= 1)
+    if (tmp & p) cost++;
+  return cost;
+}
+
+template <typename MatrixType, typename IntExponent, typename PlainObject>
+template <typename ResultType>
+void MatrixPower<MatrixType,IntExponent,PlainObject,1>::partialPivLuSolve(IntExponent p, ResultType& result)
+{
+  const PartialPivLU<MatrixType> Asolver(m_A);
+  for(; p; p--)
+    result = Asolver.solve(result);
+}
+
+template <typename MatrixType, typename IntExponent, typename PlainObject>
+template <typename ResultType>
+void MatrixPower<MatrixType,IntExponent,PlainObject,1>::computeChainProduct(ResultType& result)
+{
+  const bool pIsNegative = m_p < IntExponent(0);
+  IntExponent p = pIsNegative? -m_p: m_p;
+  int cost = computeCost(p);
+
+  if (pIsNegative) {
+    if (p * m_dimb <= cost * m_dimA) {
+      partialPivLuSolve(p, result);
+      return;
+    } else { m_tmp = m_A.inverse(); }
+  } else { m_tmp = m_A; }
+ 
+  while (p * m_dimb > cost * m_dimA) {
+    if (p & 1) {
+      result = m_tmp * result;
+      cost--;
+    }
+    m_tmp *= m_tmp;
+    cost--;
+    p >>= 1;
+  }
+
+  for (; p; p--)
+    result = m_tmp * result;
+}
+
+/******* Specialized for complex exponents *******/
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+template <typename ResultType>
+void MatrixPower<MatrixType,std::complex<RealScalar>,PlainObject,IsInteger>::compute(ResultType& result)
+{
+  using std::floor;
+  using std::pow;
+
+  if (m_dimA == 1)
+    result = pow(m_A(0,0), m_p) * m_b;
+  else {
+    computeSchurDecomposition();
+    if (m_dimA == 2)
+      compute2x2(m_p);
+    else
+      computeBig();
+    result = m_U * m_fT * m_U.adjoint();
+  }
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+void MatrixPower<MatrixType,std::complex<RealScalar>,PlainObject,IsInteger>::computeSchurDecomposition()
+{
+  const ComplexSchur<MatrixType> schurOfA(m_A);
+  m_T = schurOfA.matrixT();
+  m_U = schurOfA.matrixU();
+  m_logTdiag = m_T.diagonal().array().log();
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+typename MatrixType::Scalar MatrixPower<MatrixType,std::complex<RealScalar>,PlainObject,IsInteger>::atanh(const Scalar& x)
+{
+  using std::abs;
+  using std::log;
+  using std::sqrt;
+
+  if (abs(x) > sqrt(NumTraits<RealScalar>::epsilon()))
+    return RealScalar(0.5) * log((RealScalar(1) + x) / (RealScalar(1) - x));
+  else
+    return x + x*x*x / RealScalar(3);
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+void MatrixPower<MatrixType,std::complex<RealScalar>,PlainObject,IsInteger>::compute2x2(const Scalar& p)
+{
+  using std::abs;
+  using std::ceil;
+  using std::exp;
+  using std::imag;
+  using std::ldexp;
+  using std::log;
+  using std::pow;
+  using std::sinh;
+
+  int i, j, unwindingNumber;
+  Scalar w;
+
+  m_fT(0,0) = pow(m_T(0,0), p);
+
+  for (j = 1; j < m_dimA; j++) {
+    i = j - 1;
+    m_fT(j,j) = pow(m_T(j,j), p);
+
+    if (m_T(i,i) == m_T(j,j))
+      m_fT(i,j) = p * pow(m_T(i,j), p - RealScalar(1));
+    else if (abs(m_T(i,i)) < ldexp(abs(m_T(j,j)), -1) || abs(m_T(j,j)) < ldexp(abs(m_T(i,i)), -1))
+      m_fT(i,j) = m_T(i,j) * (m_fT(j,j) - m_fT(i,i)) / (m_T(j,j) - m_T(i,i));
+    else {
+      // computation in previous branch is inaccurate if abs(m_T(j,j)) \approx abs(m_T(i,i))
+      unwindingNumber = static_cast<int>(ceil((imag(m_logTdiag[j] - m_logTdiag[i]) - M_PI) / (2 * M_PI)));
+      w = atanh((m_T(j,j) - m_T(i,i)) / (m_T(j,j) + m_T(i,i))) + Scalar(0, M_PI * unwindingNumber);
+      m_fT(i,j) = m_T(i,j) * RealScalar(2) * exp(RealScalar(0.5) * p * (m_logTdiag[j] + m_logTdiag[i])) *
+	  sinh(p * w) / (m_T(j,j) - m_T(i,i));
+    }
+  }
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+void MatrixPower<MatrixType,std::complex<RealScalar>,PlainObject,IsInteger>::computeBig()
+{
+  using std::abs;
+  using std::ceil;
+  using std::frexp;
+  using std::ldexp;
+
+  const RealScalar maxNormForPade = 2.787629930862099e-1;
+  int degree, degree2, numberOfSquareRoots, numberOfExtraSquareRoots = 0;
+  MatrixType IminusT, sqrtT, T = m_T;
+  RealScalar normIminusT;
+  Scalar p;
+/*
+  frexp(abs(m_p), &numberOfSquareRoots);
+  if (numberOfSquareRoots > 0)
+    p = m_p * ldexp(RealScalar(1), -numberOfSquareRoots);
+  else {
+    p = m_p;
+    numberOfSquareRoots = 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 * RealScalar(0.5));
+      if (degree - degree2 <= 1 || numberOfExtraSquareRoots)
+	break;
+      numberOfExtraSquareRoots++;
+    }
+    MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
+    T = sqrtT;
+    numberOfSquareRoots++;
+  }
+  computePade(degree, IminusT);
+
+  for (; numberOfSquareRoots; numberOfSquareRoots--) {
+    compute2x2(p * ldexp(RealScalar(1), -numberOfSquareRoots));
+    m_fT *= m_fT;
+  }
+  compute2x2(p);
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+inline int MatrixPower<MatrixType,std::complex<RealScalar>,PlainObject,IsInteger>::getPadeDegree(double normIminusT)
+{
+  const double maxNormForPade[] = { 1.882832775783885e-2 /* degree = 3 */ , 6.036100693089764e-2,
+      1.239372725584911e-1, 1.998030690604271e-1, 2.787629930862099e-1 };
+  for (int degree = 3; degree <= 7; degree++)
+    if (normIminusT <= maxNormForPade[degree - 3])
+      return degree;
+  assert(false); // this line should never be reached
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+void MatrixPower<MatrixType,std::complex<RealScalar>,PlainObject,IsInteger>::computePade(int degree, const MatrixType& IminusT)
+{
+  degree <<= 1;
+  m_fT = coeff(degree) * IminusT;
+
+  for (int i = degree - 1; i; i--) {
+    m_fT = (MatrixType::Identity(m_A.rows(), m_A.cols()) + m_fT).template triangularView<Upper>()
+	.solve(coeff(i) * IminusT).eval();
+  }
+  m_fT += MatrixType::Identity(m_A.rows(), m_A.cols());
+}
+
+template <typename MatrixType, typename RealScalar, typename PlainObject, int IsInteger>
+inline typename MatrixType::Scalar MatrixPower<MatrixType,std::complex<RealScalar>,PlainObject,IsInteger>::coeff(int i)
+{
+  if (i == 1)
+    return -m_p;
+  else if (i & 1)
+    return (-m_p - RealScalar(i)) / RealScalar((i<<2) + 2);
+  else
+    return (m_p - RealScalar(i)) / RealScalar((i<<2) - 2);
+}
+
+/**
+ * \ingroup MatrixFunctions_Module
+ *
+ * \brief Proxy for the matrix power multiplied by another matrix
+ * (expression).
+ *
+ * \tparam MatrixType    type of the base, a matrix (expression).
+ * \tparam ExponentType  type of the exponent, a scalar.
+ * \tparam Derived       type of the multiplier, a matrix (expression).
+ *
+ * This class holds the arguments to the matrix expression 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
+ * MatrixPowerReturnValue::operator*() and most of the time this is the
+ * only way it is used.
+ */
+template<typename MatrixType, typename ExponentType, typename Derived> class MatrixPowerMultiplied
+: public ReturnByValue<MatrixPowerMultiplied<MatrixType, ExponentType, Derived> >
+{
+  public:
+    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.
+     * \param[in] b  %Matrix (expression), the multiplier.
+     */
+    MatrixPowerMultiplied(const MatrixType& A, const ExponentType& p, const Derived& b)
+    : m_A(A), m_p(p), m_b(b) { }
+
+    /**
+     * \brief Compute the matrix exponential.
+     *
+     * \param[out] result  \f$ A^p b \f$ where \c A ,\c p and \c b are as in
+     * the constructor.
+     */
+    template <typename ResultType>
+    inline void evalTo(ResultType& result) const
+    {
+      typedef typename Derived::PlainObject PlainObject;
+      const typename MatrixType::PlainObject Aevaluated = m_A.eval();
+      const PlainObject bevaluated = m_b.eval();
+      MatrixPower<MatrixType, ExponentType, PlainObject> mp(Aevaluated, m_p, bevaluated);
+      mp.compute(result);
+    }
+
+    Index rows() const { return m_b.rows(); }
+    Index cols() const { return m_b.cols(); }
+
+  private:
+    const MatrixType& m_A;
+    const ExponentType& m_p;
+    const Derived& m_b;
+
+    MatrixPowerMultiplied& operator=(const MatrixPowerMultiplied&);
+};
+
+/**
+ * \ingroup MatrixFunctions_Module
+ *
+ * \brief Proxy for the matrix power of some matrix (expression).
+ *
+ * \tparam Derived       type of the base, a matrix (expression).
+ * \tparam ExponentType  type of the exponent, a scalar.
+ *
+ * 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, typename ExponentType> class MatrixPowerReturnValue
+: public ReturnByValue<MatrixPowerReturnValue<Derived, ExponentType> >
+{
+  public:
+    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, const ExponentType& p)
+    : m_A(A), m_p(p) { }
+
+    /**
+     * \brief Return the matrix power multiplied by %Matrix \c b.
+     *
+     * The %MatrixPower class can optimize \f$ A^p b \f$ computing, and this
+     * method provides an elegant way to call it:
+     *
+     * \param[in] b  %Matrix (exporession), the multiplier.
+     */
+    template <typename OtherDerived>
+    const MatrixPowerMultiplied<Derived, ExponentType, OtherDerived> operator*(const MatrixBase<OtherDerived>& b) const
+    { return MatrixPowerMultiplied<Derived, ExponentType, OtherDerived>(m_A, m_p, b.derived()); }
+
+    /**
+     * \brief Compute the matrix power.
+     *
+     * \param[out] result  \f$ A^p \f$ where \c A and \c p are as in the
+     * constructor.
+     */
+    template <typename ResultType>
+    inline void evalTo(ResultType& result) const
+    {
+      typedef typename Derived::PlainObject PlainObject;
+      const PlainObject Aevaluated = m_A.eval();
+      const PlainObject Identity = PlainObject::Identity(m_A.rows(), m_A.cols());
+      MatrixPower<PlainObject, ExponentType> mp(Aevaluated, m_p, Identity);
+      mp.compute(result);
+    }
+
+    Index rows() const { return m_A.rows(); }
+    Index cols() const { return m_A.cols(); }
+
+  private:
+    const Derived& m_A;
+    const ExponentType& m_p;
+
+    MatrixPowerReturnValue& operator=(const MatrixPowerReturnValue&);
+};
+
+namespace internal {
+  template<typename MatrixType, typename ExponentType, typename Derived>
+  struct traits<MatrixPowerMultiplied<MatrixType, ExponentType, Derived> >
+  {
+    typedef typename Derived::PlainObject ReturnType;
+  };
+
+  template<typename Derived, typename ExponentType>
+  struct traits<MatrixPowerReturnValue<Derived, ExponentType> >
+  {
+    typedef typename Derived::PlainObject ReturnType;
+  };
+}
+
+template <typename Derived>
+template <typename ExponentType>
+const MatrixPowerReturnValue<Derived, ExponentType> MatrixBase<Derived>::pow(const ExponentType& p) const
+{
+  eigen_assert(rows() == cols());
+  return MatrixPowerReturnValue<Derived, ExponentType>(derived(), p);
+}
+
+} // end namespace Eigen
+
+#endif // EIGEN_MATRIX_POWER