unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h - mirror - Git at Google

 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
 // Copyright (C) 2009, 2010, 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
 // Copyright (C) 2011, 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_EXPONENTIAL
 #define EIGEN_MATRIX_EXPONENTIAL

 #include "StemFunction.h"

 #include "./InternalHeaderCheck.h"

 namespace Eigen {
 namespace internal {

 /** \brief Scaling operator.
  *
  * This struct is used by CwiseUnaryOp to scale a matrix by \f$ 2^{-s} \f$.
  */
 template <typename RealScalar>
 struct MatrixExponentialScalingOp
 {
   /** \brief Constructor.
    *
    * \param[in] squarings  The integer \f$ s \f$ in this document.
    */
   MatrixExponentialScalingOp(int squarings) : m_squarings(squarings) { }


   /** \brief Scale a matrix coefficient.
    *
    * \param[in,out] x  The scalar to be scaled, becoming \f$ 2^{-s} x \f$.
    */
   inline const RealScalar operator() (const RealScalar& x) const
   {
     using std::ldexp;
     return ldexp(x, -m_squarings);
   }

   typedef std::complex<RealScalar> ComplexScalar;

   /** \brief Scale a matrix coefficient.
    *
    * \param[in,out] x  The scalar to be scaled, becoming \f$ 2^{-s} x \f$.
    */
   inline const ComplexScalar operator() (const ComplexScalar& x) const
   {
     using std::ldexp;
     return ComplexScalar(ldexp(x.real(), -m_squarings), ldexp(x.imag(), -m_squarings));
   }

   private:
     int m_squarings;
 };

 /** \brief Compute the (3,3)-Pad&eacute; approximant to the exponential.
  *
  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
  */
 template <typename MatA, typename MatU, typename MatV>
 void matrix_exp_pade3(const MatA& A, MatU& U, MatV& V)
 {
   typedef typename MatA::PlainObject MatrixType;
   typedef typename NumTraits<typename traits<MatA>::Scalar>::Real RealScalar;
   const RealScalar b[] = {120.L, 60.L, 12.L, 1.L};
   const MatrixType A2 = A * A;
   const MatrixType tmp = b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
   U.noalias() = A * tmp;
   V = b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
 }

 /** \brief Compute the (5,5)-Pad&eacute; approximant to the exponential.
  *
  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
  */
 template <typename MatA, typename MatU, typename MatV>
 void matrix_exp_pade5(const MatA& A, MatU& U, MatV& V)
 {
   typedef typename MatA::PlainObject MatrixType;
   typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
   const RealScalar b[] = {30240.L, 15120.L, 3360.L, 420.L, 30.L, 1.L};
   const MatrixType A2 = A * A;
   const MatrixType A4 = A2 * A2;
   const MatrixType tmp = b[5] * A4 + b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
   U.noalias() = A * tmp;
   V = b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
 }

 /** \brief Compute the (7,7)-Pad&eacute; approximant to the exponential.
  *
  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
  */
 template <typename MatA, typename MatU, typename MatV>
 void matrix_exp_pade7(const MatA& A, MatU& U, MatV& V)
 {
   typedef typename MatA::PlainObject MatrixType;
   typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
   const RealScalar b[] = {17297280.L, 8648640.L, 1995840.L, 277200.L, 25200.L, 1512.L, 56.L, 1.L};
   const MatrixType A2 = A * A;
   const MatrixType A4 = A2 * A2;
   const MatrixType A6 = A4 * A2;
   const MatrixType tmp = b[7] * A6 + b[5] * A4 + b[3] * A2
     + b[1] * MatrixType::Identity(A.rows(), A.cols());
   U.noalias() = A * tmp;
   V = b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());

 }

 /** \brief Compute the (9,9)-Pad&eacute; approximant to the exponential.
  *
  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
  */
 template <typename MatA, typename MatU, typename MatV>
 void matrix_exp_pade9(const MatA& A, MatU& U, MatV& V)
 {
   typedef typename MatA::PlainObject MatrixType;
   typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
   const RealScalar b[] = {17643225600.L, 8821612800.L, 2075673600.L, 302702400.L, 30270240.L,
                           2162160.L, 110880.L, 3960.L, 90.L, 1.L};
   const MatrixType A2 = A * A;
   const MatrixType A4 = A2 * A2;
   const MatrixType A6 = A4 * A2;
   const MatrixType A8 = A6 * A2;
   const MatrixType tmp = b[9] * A8 + b[7] * A6 + b[5] * A4 + b[3] * A2
     + b[1] * MatrixType::Identity(A.rows(), A.cols());
   U.noalias() = A * tmp;
   V = b[8] * A8 + b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
 }

 /** \brief Compute the (13,13)-Pad&eacute; approximant to the exponential.
  *
  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
  */
 template <typename MatA, typename MatU, typename MatV>
 void matrix_exp_pade13(const MatA& A, MatU& U, MatV& V)
 {
   typedef typename MatA::PlainObject MatrixType;
   typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
   const RealScalar b[] = {64764752532480000.L, 32382376266240000.L, 7771770303897600.L,
                           1187353796428800.L, 129060195264000.L, 10559470521600.L, 670442572800.L,
                           33522128640.L, 1323241920.L, 40840800.L, 960960.L, 16380.L, 182.L, 1.L};
   const MatrixType A2 = A * A;
   const MatrixType A4 = A2 * A2;
   const MatrixType A6 = A4 * A2;
   V = b[13] * A6 + b[11] * A4 + b[9] * A2; // used for temporary storage
   MatrixType tmp = A6 * V;
   tmp += b[7] * A6 + b[5] * A4 + b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
   U.noalias() = A * tmp;
   tmp = b[12] * A6 + b[10] * A4 + b[8] * A2;
   V.noalias() = A6 * tmp;
   V += b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
 }

 /** \brief Compute the (17,17)-Pad&eacute; approximant to the exponential.
  *
  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
  *
  *  This function activates only if your long double is double-double or quadruple.
  */
 #if LDBL_MANT_DIG > 64
 template <typename MatA, typename MatU, typename MatV>
 void matrix_exp_pade17(const MatA& A, MatU& U, MatV& V)
 {
   typedef typename MatA::PlainObject MatrixType;
   typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
   const RealScalar b[] = {830034394580628357120000.L, 415017197290314178560000.L,
                           100610229646136770560000.L, 15720348382208870400000.L,
                           1774878043152614400000.L, 153822763739893248000.L, 10608466464820224000.L,
                           595373117923584000.L, 27563570274240000.L, 1060137318240000.L,
                           33924394183680.L, 899510451840.L, 19554575040.L, 341863200.L, 4651200.L,
                           46512.L, 306.L, 1.L};
   const MatrixType A2 = A * A;
   const MatrixType A4 = A2 * A2;
   const MatrixType A6 = A4 * A2;
   const MatrixType A8 = A4 * A4;
   V = b[17] * A8 + b[15] * A6 + b[13] * A4 + b[11] * A2; // used for temporary storage
   MatrixType tmp = A8 * V;
   tmp += b[9] * A8 + b[7] * A6 + b[5] * A4 + b[3] * A2
     + b[1] * MatrixType::Identity(A.rows(), A.cols());
   U.noalias() = A * tmp;
   tmp = b[16] * A8 + b[14] * A6 + b[12] * A4 + b[10] * A2;
   V.noalias() = tmp * A8;
   V += b[8] * A8 + b[6] * A6 + b[4] * A4 + b[2] * A2
     + b[0] * MatrixType::Identity(A.rows(), A.cols());
 }
 #endif

 template <typename MatrixType, typename RealScalar = typename NumTraits<typename traits<MatrixType>::Scalar>::Real>
 struct matrix_exp_computeUV
 {
   /** \brief Compute Pad&eacute; approximant to the exponential.
     *
     * Computes \c U, \c V and \c squarings such that \f$ (V+U)(V-U)^{-1} \f$ is a Pad&eacute;
     * approximant of \f$ \exp(2^{-\mbox{squarings}}M) \f$ around \f$ M = 0 \f$, where \f$ M \f$
     * denotes the matrix \c arg. The degree of the Pad&eacute; approximant and the value of squarings
     * are chosen such that the approximation error is no more than the round-off error.
     */
   static void run(const MatrixType& arg, MatrixType& U, MatrixType& V, int& squarings);
 };

 template <typename MatrixType>
 struct matrix_exp_computeUV<MatrixType, float>
 {
   template <typename ArgType>
   static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
   {
     using std::frexp;
     using std::pow;
     const float l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
     squarings = 0;
     if (l1norm < 4.258730016922831e-001f) {
       matrix_exp_pade3(arg, U, V);
     } else if (l1norm < 1.880152677804762e+000f) {
       matrix_exp_pade5(arg, U, V);
     } else {
       const float maxnorm = 3.925724783138660f;
       frexp(l1norm / maxnorm, &squarings);
       if (squarings < 0) squarings = 0;
       MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<float>(squarings));
       matrix_exp_pade7(A, U, V);
     }
   }
 };

 template <typename MatrixType>
 struct matrix_exp_computeUV<MatrixType, double>
 {
   typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
   template <typename ArgType>
   static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
   {
     using std::frexp;
     using std::pow;
     const RealScalar l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
     squarings = 0;
     if (l1norm < 1.495585217958292e-002) {
       matrix_exp_pade3(arg, U, V);
     } else if (l1norm < 2.539398330063230e-001) {
       matrix_exp_pade5(arg, U, V);
     } else if (l1norm < 9.504178996162932e-001) {
       matrix_exp_pade7(arg, U, V);
     } else if (l1norm < 2.097847961257068e+000) {
       matrix_exp_pade9(arg, U, V);
     } else {
       const RealScalar maxnorm = 5.371920351148152;
       frexp(l1norm / maxnorm, &squarings);
       if (squarings < 0) squarings = 0;
       MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<RealScalar>(squarings));
       matrix_exp_pade13(A, U, V);
     }
   }
 };

 template <typename MatrixType>
 struct matrix_exp_computeUV<MatrixType, long double>
 {
   template <typename ArgType>
   static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
   {
 #if   LDBL_MANT_DIG == 53   // double precision
     matrix_exp_computeUV<MatrixType, double>::run(arg, U, V, squarings);

 #else

     using std::frexp;
     using std::pow;
     const long double l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
     squarings = 0;

 #if LDBL_MANT_DIG <= 64   // extended precision

     if (l1norm < 4.1968497232266989671e-003L) {
       matrix_exp_pade3(arg, U, V);
     } else if (l1norm < 1.1848116734693823091e-001L) {
       matrix_exp_pade5(arg, U, V);
     } else if (l1norm < 5.5170388480686700274e-001L) {
       matrix_exp_pade7(arg, U, V);
     } else if (l1norm < 1.3759868875587845383e+000L) {
       matrix_exp_pade9(arg, U, V);
     } else {
       const long double maxnorm = 4.0246098906697353063L;
       frexp(l1norm / maxnorm, &squarings);
       if (squarings < 0) squarings = 0;
       MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
       matrix_exp_pade13(A, U, V);
     }

 #elif LDBL_MANT_DIG <= 106  // double-double

     if (l1norm < 3.2787892205607026992947488108213e-005L) {
       matrix_exp_pade3(arg, U, V);
     } else if (l1norm < 6.4467025060072760084130906076332e-003L) {
       matrix_exp_pade5(arg, U, V);
     } else if (l1norm < 6.8988028496595374751374122881143e-002L) {
       matrix_exp_pade7(arg, U, V);
     } else if (l1norm < 2.7339737518502231741495857201670e-001L) {
       matrix_exp_pade9(arg, U, V);
     } else if (l1norm < 1.3203382096514474905666448850278e+000L) {
       matrix_exp_pade13(arg, U, V);
     } else {
       const long double maxnorm = 3.2579440895405400856599663723517L;
       frexp(l1norm / maxnorm, &squarings);
       if (squarings < 0) squarings = 0;
       MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
       matrix_exp_pade17(A, U, V);
     }

 #elif LDBL_MANT_DIG <= 113  // quadruple precision

     if (l1norm < 1.639394610288918690547467954466970e-005L) {
       matrix_exp_pade3(arg, U, V);
     } else if (l1norm < 4.253237712165275566025884344433009e-003L) {
       matrix_exp_pade5(arg, U, V);
     } else if (l1norm < 5.125804063165764409885122032933142e-002L) {
       matrix_exp_pade7(arg, U, V);
     } else if (l1norm < 2.170000765161155195453205651889853e-001L) {
       matrix_exp_pade9(arg, U, V);
     } else if (l1norm < 1.125358383453143065081397882891878e+000L) {
       matrix_exp_pade13(arg, U, V);
     } else {
       const long double maxnorm = 2.884233277829519311757165057717815L;
       frexp(l1norm / maxnorm, &squarings);
       if (squarings < 0) squarings = 0;
       MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
       matrix_exp_pade17(A, U, V);
     }

 #else

     // this case should be handled in compute()
     eigen_assert(false && "Bug in MatrixExponential");

 #endif
 #endif  // LDBL_MANT_DIG
   }
 };

 template<typename T> struct is_exp_known_type : false_type {};
 template<> struct is_exp_known_type<float> : true_type {};
 template<> struct is_exp_known_type<double> : true_type {};
 #if LDBL_MANT_DIG <= 113
 template<> struct is_exp_known_type<long double> : true_type {};
 #endif

 template <typename ArgType, typename ResultType>
 void matrix_exp_compute(const ArgType& arg, ResultType &result, true_type) // natively supported scalar type
 {
   typedef typename ArgType::PlainObject MatrixType;
   MatrixType U, V;
   int squarings;
   matrix_exp_computeUV<MatrixType>::run(arg, U, V, squarings); // Pade approximant is (U+V) / (-U+V)
   MatrixType numer = U + V;
   MatrixType denom = -U + V;
   result = denom.partialPivLu().solve(numer);
   for (int i=0; i<squarings; i++)
     result *= result;   // undo scaling by repeated squaring
 }


 /* Computes the matrix exponential
  *
  * \param arg    argument of matrix exponential (should be plain object)
  * \param result variable in which result will be stored
  */
 template <typename ArgType, typename ResultType>
 void matrix_exp_compute(const ArgType& arg, ResultType &result, false_type) // default
 {
   typedef typename ArgType::PlainObject MatrixType;
   typedef typename traits<MatrixType>::Scalar Scalar;
   typedef typename NumTraits<Scalar>::Real RealScalar;
   typedef typename std::complex<RealScalar> ComplexScalar;
   result = arg.matrixFunction(internal::stem_function_exp<ComplexScalar>);
 }

 } // end namespace Eigen::internal

 /** \ingroup MatrixFunctions_Module
   *
   * \brief Proxy for the matrix exponential of some matrix (expression).
   *
   * \tparam Derived  Type of the argument to the matrix exponential.
   *
   * This class holds the argument to the matrix exponential 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::exp() and most of the time this is the only way it is used.
   */
 template<typename Derived> struct MatrixExponentialReturnValue
 : public ReturnByValue<MatrixExponentialReturnValue<Derived> >
 {
   public:
     /** \brief Constructor.
       *
       * \param src %Matrix (expression) forming the argument of the matrix exponential.
       */
     MatrixExponentialReturnValue(const Derived& src) : m_src(src) { }

     /** \brief Compute the matrix exponential.
       *
       * \param result the matrix exponential of \p src in the constructor.
       */
     template <typename ResultType>
     inline void evalTo(ResultType& result) const
     {
       const typename internal::nested_eval<Derived, 10>::type tmp(m_src);
       internal::matrix_exp_compute(tmp, result, internal::is_exp_known_type<typename Derived::RealScalar>());
     }

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

   protected:
     const typename internal::ref_selector<Derived>::type m_src;
 };

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

 template <typename Derived>
 const MatrixExponentialReturnValue<Derived> MatrixBase<Derived>::exp() const
 {
   eigen_assert(rows() == cols());
   return MatrixExponentialReturnValue<Derived>(derived());
 }

 } // end namespace Eigen

 #endif // EIGEN_MATRIX_EXPONENTIAL
	// This file is part of Eigen, a lightweight C++ template library
	// for linear algebra.
	//
	// Copyright (C) 2009, 2010, 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
	// Copyright (C) 2011, 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_EXPONENTIAL
	#define EIGEN_MATRIX_EXPONENTIAL

	#include "StemFunction.h"

	#include "./InternalHeaderCheck.h"

	namespace Eigen {
	namespace internal {

	/** \brief Scaling operator.
	*
	* This struct is used by CwiseUnaryOp to scale a matrix by \f$ 2^{-s} \f$.
	*/
	template <typename RealScalar>
	struct MatrixExponentialScalingOp
	{
	/** \brief Constructor.
	*
	* \param[in] squarings The integer \f$ s \f$ in this document.
	*/
	MatrixExponentialScalingOp(int squarings) : m_squarings(squarings) { }


	/** \brief Scale a matrix coefficient.
	*
	* \param[in,out] x The scalar to be scaled, becoming \f$ 2^{-s} x \f$.
	*/
	inline const RealScalar operator() (const RealScalar& x) const
	{
	using std::ldexp;
	return ldexp(x, -m_squarings);
	}

	typedef std::complex<RealScalar> ComplexScalar;

	/** \brief Scale a matrix coefficient.
	*
	* \param[in,out] x The scalar to be scaled, becoming \f$ 2^{-s} x \f$.
	*/
	inline const ComplexScalar operator() (const ComplexScalar& x) const
	{
	using std::ldexp;
	return ComplexScalar(ldexp(x.real(), -m_squarings), ldexp(x.imag(), -m_squarings));
	}

	private:
	int m_squarings;
	};

	/** \brief Compute the (3,3)-Padé approximant to the exponential.
	*
	* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
	* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
	*/
	template <typename MatA, typename MatU, typename MatV>
	void matrix_exp_pade3(const MatA& A, MatU& U, MatV& V)
	{
	typedef typename MatA::PlainObject MatrixType;
	typedef typename NumTraits<typename traits<MatA>::Scalar>::Real RealScalar;
	const RealScalar b[] = {120.L, 60.L, 12.L, 1.L};
	const MatrixType A2 = A * A;
	const MatrixType tmp = b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
	U.noalias() = A * tmp;
	V = b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
	}

	/** \brief Compute the (5,5)-Padé approximant to the exponential.
	*
	* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
	* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
	*/
	template <typename MatA, typename MatU, typename MatV>
	void matrix_exp_pade5(const MatA& A, MatU& U, MatV& V)
	{
	typedef typename MatA::PlainObject MatrixType;
	typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
	const RealScalar b[] = {30240.L, 15120.L, 3360.L, 420.L, 30.L, 1.L};
	const MatrixType A2 = A * A;
	const MatrixType A4 = A2 * A2;
	const MatrixType tmp = b[5] * A4 + b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
	U.noalias() = A * tmp;
	V = b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
	}

	/** \brief Compute the (7,7)-Padé approximant to the exponential.
	*
	* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
	* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
	*/
	template <typename MatA, typename MatU, typename MatV>
	void matrix_exp_pade7(const MatA& A, MatU& U, MatV& V)
	{
	typedef typename MatA::PlainObject MatrixType;
	typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
	const RealScalar b[] = {17297280.L, 8648640.L, 1995840.L, 277200.L, 25200.L, 1512.L, 56.L, 1.L};
	const MatrixType A2 = A * A;
	const MatrixType A4 = A2 * A2;
	const MatrixType A6 = A4 * A2;
	const MatrixType tmp = b[7] * A6 + b[5] * A4 + b[3] * A2
	+ b[1] * MatrixType::Identity(A.rows(), A.cols());
	U.noalias() = A * tmp;
	V = b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());

	}

	/** \brief Compute the (9,9)-Padé approximant to the exponential.
	*
	* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
	* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
	*/
	template <typename MatA, typename MatU, typename MatV>
	void matrix_exp_pade9(const MatA& A, MatU& U, MatV& V)
	{
	typedef typename MatA::PlainObject MatrixType;
	typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
	const RealScalar b[] = {17643225600.L, 8821612800.L, 2075673600.L, 302702400.L, 30270240.L,
	2162160.L, 110880.L, 3960.L, 90.L, 1.L};
	const MatrixType A2 = A * A;
	const MatrixType A4 = A2 * A2;
	const MatrixType A6 = A4 * A2;
	const MatrixType A8 = A6 * A2;
	const MatrixType tmp = b[9] * A8 + b[7] * A6 + b[5] * A4 + b[3] * A2
	+ b[1] * MatrixType::Identity(A.rows(), A.cols());
	U.noalias() = A * tmp;
	V = b[8] * A8 + b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
	}

	/** \brief Compute the (13,13)-Padé approximant to the exponential.
	*
	* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
	* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
	*/
	template <typename MatA, typename MatU, typename MatV>
	void matrix_exp_pade13(const MatA& A, MatU& U, MatV& V)
	{
	typedef typename MatA::PlainObject MatrixType;
	typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
	const RealScalar b[] = {64764752532480000.L, 32382376266240000.L, 7771770303897600.L,
	1187353796428800.L, 129060195264000.L, 10559470521600.L, 670442572800.L,
	33522128640.L, 1323241920.L, 40840800.L, 960960.L, 16380.L, 182.L, 1.L};
	const MatrixType A2 = A * A;
	const MatrixType A4 = A2 * A2;
	const MatrixType A6 = A4 * A2;
	V = b[13] * A6 + b[11] * A4 + b[9] * A2; // used for temporary storage
	MatrixType tmp = A6 * V;
	tmp += b[7] * A6 + b[5] * A4 + b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
	U.noalias() = A * tmp;
	tmp = b[12] * A6 + b[10] * A4 + b[8] * A2;
	V.noalias() = A6 * tmp;
	V += b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
	}

	/** \brief Compute the (17,17)-Padé approximant to the exponential.
	*
	* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
	* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
	*
	* This function activates only if your long double is double-double or quadruple.
	*/
	#if LDBL_MANT_DIG > 64
	template <typename MatA, typename MatU, typename MatV>
	void matrix_exp_pade17(const MatA& A, MatU& U, MatV& V)
	{
	typedef typename MatA::PlainObject MatrixType;
	typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
	const RealScalar b[] = {830034394580628357120000.L, 415017197290314178560000.L,
	100610229646136770560000.L, 15720348382208870400000.L,
	1774878043152614400000.L, 153822763739893248000.L, 10608466464820224000.L,
	595373117923584000.L, 27563570274240000.L, 1060137318240000.L,
	33924394183680.L, 899510451840.L, 19554575040.L, 341863200.L, 4651200.L,
	46512.L, 306.L, 1.L};
	const MatrixType A2 = A * A;
	const MatrixType A4 = A2 * A2;
	const MatrixType A6 = A4 * A2;
	const MatrixType A8 = A4 * A4;
	V = b[17] * A8 + b[15] * A6 + b[13] * A4 + b[11] * A2; // used for temporary storage
	MatrixType tmp = A8 * V;
	tmp += b[9] * A8 + b[7] * A6 + b[5] * A4 + b[3] * A2
	+ b[1] * MatrixType::Identity(A.rows(), A.cols());
	U.noalias() = A * tmp;
	tmp = b[16] * A8 + b[14] * A6 + b[12] * A4 + b[10] * A2;
	V.noalias() = tmp * A8;
	V += b[8] * A8 + b[6] * A6 + b[4] * A4 + b[2] * A2
	+ b[0] * MatrixType::Identity(A.rows(), A.cols());
	}
	#endif

	template <typename MatrixType, typename RealScalar = typename NumTraits<typename traits<MatrixType>::Scalar>::Real>
	struct matrix_exp_computeUV
	{
	/** \brief Compute Padé approximant to the exponential.
	*
	* Computes \c U, \c V and \c squarings such that \f$ (V+U)(V-U)^{-1} \f$ is a Padé
	* approximant of \f$ \exp(2^{-\mbox{squarings}}M) \f$ around \f$ M = 0 \f$, where \f$ M \f$
	* denotes the matrix \c arg. The degree of the Padé approximant and the value of squarings
	* are chosen such that the approximation error is no more than the round-off error.
	*/
	static void run(const MatrixType& arg, MatrixType& U, MatrixType& V, int& squarings);
	};

	template <typename MatrixType>
	struct matrix_exp_computeUV<MatrixType, float>
	{
	template <typename ArgType>
	static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
	{
	using std::frexp;
	using std::pow;
	const float l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
	squarings = 0;
	if (l1norm < 4.258730016922831e-001f) {
	matrix_exp_pade3(arg, U, V);
	} else if (l1norm < 1.880152677804762e+000f) {
	matrix_exp_pade5(arg, U, V);
	} else {
	const float maxnorm = 3.925724783138660f;
	frexp(l1norm / maxnorm, &squarings);
	if (squarings < 0) squarings = 0;
	MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<float>(squarings));
	matrix_exp_pade7(A, U, V);
	}
	}
	};

	template <typename MatrixType>
	struct matrix_exp_computeUV<MatrixType, double>
	{
	typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
	template <typename ArgType>
	static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
	{
	using std::frexp;
	using std::pow;
	const RealScalar l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
	squarings = 0;
	if (l1norm < 1.495585217958292e-002) {
	matrix_exp_pade3(arg, U, V);
	} else if (l1norm < 2.539398330063230e-001) {
	matrix_exp_pade5(arg, U, V);
	} else if (l1norm < 9.504178996162932e-001) {
	matrix_exp_pade7(arg, U, V);
	} else if (l1norm < 2.097847961257068e+000) {
	matrix_exp_pade9(arg, U, V);
	} else {
	const RealScalar maxnorm = 5.371920351148152;
	frexp(l1norm / maxnorm, &squarings);
	if (squarings < 0) squarings = 0;
	MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<RealScalar>(squarings));
	matrix_exp_pade13(A, U, V);
	}
	}
	};

	template <typename MatrixType>
	struct matrix_exp_computeUV<MatrixType, long double>
	{
	template <typename ArgType>
	static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
	{
	#if LDBL_MANT_DIG == 53 // double precision
	matrix_exp_computeUV<MatrixType, double>::run(arg, U, V, squarings);

	#else

	using std::frexp;
	using std::pow;
	const long double l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
	squarings = 0;

	#if LDBL_MANT_DIG <= 64 // extended precision

	if (l1norm < 4.1968497232266989671e-003L) {
	matrix_exp_pade3(arg, U, V);
	} else if (l1norm < 1.1848116734693823091e-001L) {
	matrix_exp_pade5(arg, U, V);
	} else if (l1norm < 5.5170388480686700274e-001L) {
	matrix_exp_pade7(arg, U, V);
	} else if (l1norm < 1.3759868875587845383e+000L) {
	matrix_exp_pade9(arg, U, V);
	} else {
	const long double maxnorm = 4.0246098906697353063L;
	frexp(l1norm / maxnorm, &squarings);
	if (squarings < 0) squarings = 0;
	MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
	matrix_exp_pade13(A, U, V);
	}

	#elif LDBL_MANT_DIG <= 106 // double-double

	if (l1norm < 3.2787892205607026992947488108213e-005L) {
	matrix_exp_pade3(arg, U, V);
	} else if (l1norm < 6.4467025060072760084130906076332e-003L) {
	matrix_exp_pade5(arg, U, V);
	} else if (l1norm < 6.8988028496595374751374122881143e-002L) {
	matrix_exp_pade7(arg, U, V);
	} else if (l1norm < 2.7339737518502231741495857201670e-001L) {
	matrix_exp_pade9(arg, U, V);
	} else if (l1norm < 1.3203382096514474905666448850278e+000L) {
	matrix_exp_pade13(arg, U, V);
	} else {
	const long double maxnorm = 3.2579440895405400856599663723517L;
	frexp(l1norm / maxnorm, &squarings);
	if (squarings < 0) squarings = 0;
	MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
	matrix_exp_pade17(A, U, V);
	}

	#elif LDBL_MANT_DIG <= 113 // quadruple precision

	if (l1norm < 1.639394610288918690547467954466970e-005L) {
	matrix_exp_pade3(arg, U, V);
	} else if (l1norm < 4.253237712165275566025884344433009e-003L) {
	matrix_exp_pade5(arg, U, V);
	} else if (l1norm < 5.125804063165764409885122032933142e-002L) {
	matrix_exp_pade7(arg, U, V);
	} else if (l1norm < 2.170000765161155195453205651889853e-001L) {
	matrix_exp_pade9(arg, U, V);
	} else if (l1norm < 1.125358383453143065081397882891878e+000L) {
	matrix_exp_pade13(arg, U, V);
	} else {
	const long double maxnorm = 2.884233277829519311757165057717815L;
	frexp(l1norm / maxnorm, &squarings);
	if (squarings < 0) squarings = 0;
	MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
	matrix_exp_pade17(A, U, V);
	}

	#else

	// this case should be handled in compute()
	eigen_assert(false && "Bug in MatrixExponential");

	#endif
	#endif // LDBL_MANT_DIG
	}
	};

	template<typename T> struct is_exp_known_type : false_type {};
	template<> struct is_exp_known_type<float> : true_type {};
	template<> struct is_exp_known_type<double> : true_type {};
	#if LDBL_MANT_DIG <= 113
	template<> struct is_exp_known_type<long double> : true_type {};
	#endif

	template <typename ArgType, typename ResultType>
	void matrix_exp_compute(const ArgType& arg, ResultType &result, true_type) // natively supported scalar type
	{
	typedef typename ArgType::PlainObject MatrixType;
	MatrixType U, V;
	int squarings;
	matrix_exp_computeUV<MatrixType>::run(arg, U, V, squarings); // Pade approximant is (U+V) / (-U+V)
	MatrixType numer = U + V;
	MatrixType denom = -U + V;
	result = denom.partialPivLu().solve(numer);
	for (int i=0; i<squarings; i++)
	result *= result; // undo scaling by repeated squaring
	}


	/* Computes the matrix exponential
	*
	* \param arg argument of matrix exponential (should be plain object)
	* \param result variable in which result will be stored
	*/
	template <typename ArgType, typename ResultType>
	void matrix_exp_compute(const ArgType& arg, ResultType &result, false_type) // default
	{
	typedef typename ArgType::PlainObject MatrixType;
	typedef typename traits<MatrixType>::Scalar Scalar;
	typedef typename NumTraits<Scalar>::Real RealScalar;
	typedef typename std::complex<RealScalar> ComplexScalar;
	result = arg.matrixFunction(internal::stem_function_exp<ComplexScalar>);
	}

	} // end namespace Eigen::internal

	/** \ingroup MatrixFunctions_Module
	*
	* \brief Proxy for the matrix exponential of some matrix (expression).
	*
	* \tparam Derived Type of the argument to the matrix exponential.
	*
	* This class holds the argument to the matrix exponential 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::exp() and most of the time this is the only way it is used.
	*/
	template<typename Derived> struct MatrixExponentialReturnValue
	: public ReturnByValue<MatrixExponentialReturnValue<Derived> >
	{
	public:
	/** \brief Constructor.
	*
	* \param src %Matrix (expression) forming the argument of the matrix exponential.
	*/
	MatrixExponentialReturnValue(const Derived& src) : m_src(src) { }

	/** \brief Compute the matrix exponential.
	*
	* \param result the matrix exponential of \p src in the constructor.
	*/
	template <typename ResultType>
	inline void evalTo(ResultType& result) const
	{
	const typename internal::nested_eval<Derived, 10>::type tmp(m_src);
	internal::matrix_exp_compute(tmp, result, internal::is_exp_known_type<typename Derived::RealScalar>());
	}

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

	protected:
	const typename internal::ref_selector<Derived>::type m_src;
	};

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

	template <typename Derived>
	const MatrixExponentialReturnValue<Derived> MatrixBase<Derived>::exp() const
	{
	eigen_assert(rows() == cols());
	return MatrixExponentialReturnValue<Derived>(derived());
	}

	} // end namespace Eigen

	#endif // EIGEN_MATRIX_EXPONENTIAL