update

1 files changed, 323 insertions, 342 deletions

diff --git a/eigen/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h b/eigen/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h
index 88dba54..bb6d9e1 100644
--- a/eigen/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h
+++ b/eigen/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h
@@ -1,8 +1,8 @@
 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
-// Copyright (C) 2009, 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
-// Copyright (C) 2011 Chen-Pang He <jdh8@ms63.hinet.net>
+// 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
@@ -14,388 +14,374 @@
 #include "StemFunction.h"
 
 namespace Eigen {
+namespace internal {
 
-/** \ingroup MatrixFunctions_Module
-  * \brief Class for computing the matrix exponential.
-  * \tparam MatrixType type of the argument of the exponential,
-  * expected to be an instantiation of the Matrix class template.
-  */
-template <typename MatrixType>
-class MatrixExponential {
-
-  public:
+/** \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);
+  }
 
-    /** \brief Constructor.
-      * 
-      * The class stores a reference to \p M, so it should not be
-      * changed (or destroyed) before compute() is called.
-      *
-      * \param[in] M  matrix whose exponential is to be computed.
-      */
-    MatrixExponential(const MatrixType &M);
+  typedef std::complex<RealScalar> ComplexScalar;
 
-    /** \brief Computes the matrix exponential.
-      *
-      * \param[out] result  the matrix exponential of \p M in the constructor.
-      */
-    template <typename ResultType> 
-    void compute(ResultType &result);
+  /** \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:
-
-    // Prevent copying
-    MatrixExponential(const MatrixExponential&);
-    MatrixExponential& operator=(const MatrixExponential&);
-
-    /** \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$.
-     *
-     *  \param[in] A   Argument of matrix exponential
-     */
-    void pade3(const MatrixType &A);
-
-    /** \brief Compute the (5,5)-Pad&eacute; approximant to the exponential.
-     *
-     *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
-     *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
-     *
-     *  \param[in] A   Argument of matrix exponential
-     */
-    void pade5(const MatrixType &A);
-
-    /** \brief Compute the (7,7)-Pad&eacute; approximant to the exponential.
-     *
-     *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
-     *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
-     *
-     *  \param[in] A   Argument of matrix exponential
-     */
-    void pade7(const MatrixType &A);
-
-    /** \brief Compute the (9,9)-Pad&eacute; approximant to the exponential.
-     *
-     *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
-     *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
-     *
-     *  \param[in] A   Argument of matrix exponential
-     */
-    void pade9(const MatrixType &A);
-
-    /** \brief Compute the (13,13)-Pad&eacute; approximant to the exponential.
-     *
-     *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
-     *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
-     *
-     *  \param[in] A   Argument of matrix exponential
-     */
-    void pade13(const MatrixType &A);
-
-    /** \brief Compute the (17,17)-Pad&eacute; approximant to the exponential.
-     *
-     *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
-     *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
-     *
-     *  This function activates only if your long double is double-double or quadruple.
-     *
-     *  \param[in] A   Argument of matrix exponential
-     */
-    void pade17(const MatrixType &A);
-
-    /** \brief Compute Pad&eacute; approximant to the exponential.
-     *
-     * Computes \c m_U, \c m_V and \c m_squarings such that
-     * \f$ (V+U)(V-U)^{-1} \f$ is a Pad&eacute; of
-     * \f$ \exp(2^{-\mbox{squarings}}M) \f$ around \f$ M = 0 \f$. 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.
-     *
-     * The argument of this function should correspond with the (real
-     * part of) the entries of \c m_M.  It is used to select the
-     * correct implementation using overloading.
-     */
-    void computeUV(double);
-
-    /** \brief Compute Pad&eacute; approximant to the exponential.
-     *
-     *  \sa computeUV(double);
-     */
-    void computeUV(float);
-    
-    /** \brief Compute Pad&eacute; approximant to the exponential.
-     *
-     *  \sa computeUV(double);
-     */
-    void computeUV(long double);
-
-    typedef typename internal::traits<MatrixType>::Scalar Scalar;
-    typedef typename NumTraits<Scalar>::Real RealScalar;
-    typedef typename std::complex<RealScalar> ComplexScalar;
-
-    /** \brief Reference to matrix whose exponential is to be computed. */
-    typename internal::nested<MatrixType>::type m_M;
-
-    /** \brief Odd-degree terms in numerator of Pad&eacute; approximant. */
-    MatrixType m_U;
-
-    /** \brief Even-degree terms in numerator of Pad&eacute; approximant. */
-    MatrixType m_V;
-
-    /** \brief Used for temporary storage. */
-    MatrixType m_tmp1;
-
-    /** \brief Used for temporary storage. */
-    MatrixType m_tmp2;
-
-    /** \brief Identity matrix of the same size as \c m_M. */
-    MatrixType m_Id;
-
-    /** \brief Number of squarings required in the last step. */
     int m_squarings;
-
-    /** \brief L1 norm of m_M. */
-    RealScalar m_l1norm;
 };
 
-template <typename MatrixType>
-MatrixExponential<MatrixType>::MatrixExponential(const MatrixType &M) :
-  m_M(M),
-  m_U(M.rows(),M.cols()),
-  m_V(M.rows(),M.cols()),
-  m_tmp1(M.rows(),M.cols()),
-  m_tmp2(M.rows(),M.cols()),
-  m_Id(MatrixType::Identity(M.rows(), M.cols())),
-  m_squarings(0),
-  m_l1norm(M.cwiseAbs().colwise().sum().maxCoeff())
-{
-  /* empty body */
-}
-
-template <typename MatrixType>
-template <typename ResultType> 
-void MatrixExponential<MatrixType>::compute(ResultType &result)
+/** \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)
 {
-#if LDBL_MANT_DIG > 112 // rarely happens
-  if(sizeof(RealScalar) > 14) {
-    result = m_M.matrixFunction(StdStemFunctions<ComplexScalar>::exp);
-    return;
-  }
-#endif
-  computeUV(RealScalar());
-  m_tmp1 = m_U + m_V;   // numerator of Pade approximant
-  m_tmp2 = -m_U + m_V;  // denominator of Pade approximant
-  result = m_tmp2.partialPivLu().solve(m_tmp1);
-  for (int i=0; i<m_squarings; i++)
-    result *= result;   // undo scaling by repeated squaring
+  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());
 }
 
-template <typename MatrixType>
-EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade3(const MatrixType &A)
+/** \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)
 {
-  const RealScalar b[] = {120., 60., 12., 1.};
-  m_tmp1.noalias() = A * A;
-  m_tmp2 = b[3]*m_tmp1 + b[1]*m_Id;
-  m_U.noalias() = A * m_tmp2;
-  m_V = b[2]*m_tmp1 + b[0]*m_Id;
+  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());
 }
 
-template <typename MatrixType>
-EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade5(const MatrixType &A)
+/** \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)
 {
-  const RealScalar b[] = {30240., 15120., 3360., 420., 30., 1.};
-  MatrixType A2 = A * A;
-  m_tmp1.noalias() = A2 * A2;
-  m_tmp2 = b[5]*m_tmp1 + b[3]*A2 + b[1]*m_Id;
-  m_U.noalias() = A * m_tmp2;
-  m_V = b[4]*m_tmp1 + b[2]*A2 + b[0]*m_Id;
-}
+  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());
 
-template <typename MatrixType>
-EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade7(const MatrixType &A)
-{
-  const RealScalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.};
-  MatrixType A2 = A * A;
-  MatrixType A4 = A2 * A2;
-  m_tmp1.noalias() = A4 * A2;
-  m_tmp2 = b[7]*m_tmp1 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
-  m_U.noalias() = A * m_tmp2;
-  m_V = b[6]*m_tmp1 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
 }
 
-template <typename MatrixType>
-EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade9(const MatrixType &A)
+/** \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)
 {
-  const RealScalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240.,
-		      2162160., 110880., 3960., 90., 1.};
-  MatrixType A2 = A * A;
-  MatrixType A4 = A2 * A2;
-  MatrixType A6 = A4 * A2;
-  m_tmp1.noalias() = A6 * A2;
-  m_tmp2 = b[9]*m_tmp1 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
-  m_U.noalias() = A * m_tmp2;
-  m_V = b[8]*m_tmp1 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
+  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());
 }
 
-template <typename MatrixType>
-EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade13(const MatrixType &A)
+/** \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)
 {
-  const RealScalar b[] = {64764752532480000., 32382376266240000., 7771770303897600.,
-		      1187353796428800., 129060195264000., 10559470521600., 670442572800.,
-		      33522128640., 1323241920., 40840800., 960960., 16380., 182., 1.};
-  MatrixType A2 = A * A;
-  MatrixType A4 = A2 * A2;
-  m_tmp1.noalias() = A4 * A2;
-  m_V = b[13]*m_tmp1 + b[11]*A4 + b[9]*A2; // used for temporary storage
-  m_tmp2.noalias() = m_tmp1 * m_V;
-  m_tmp2 += b[7]*m_tmp1 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
-  m_U.noalias() = A * m_tmp2;
-  m_tmp2 = b[12]*m_tmp1 + b[10]*A4 + b[8]*A2;
-  m_V.noalias() = m_tmp1 * m_tmp2;
-  m_V += b[6]*m_tmp1 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
+  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 MatrixType>
-EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade17(const MatrixType &A)
+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};
-  MatrixType A2 = A * A;
-  MatrixType A4 = A2 * A2;
-  MatrixType A6 = A4 * A2;
-  m_tmp1.noalias() = A4 * A4;
-  m_V = b[17]*m_tmp1 + b[15]*A6 + b[13]*A4 + b[11]*A2; // used for temporary storage
-  m_tmp2.noalias() = m_tmp1 * m_V;
-  m_tmp2 += b[9]*m_tmp1 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
-  m_U.noalias() = A * m_tmp2;
-  m_tmp2 = b[16]*m_tmp1 + b[14]*A6 + b[12]*A4 + b[10]*A2;
-  m_V.noalias() = m_tmp1 * m_tmp2;
-  m_V += b[8]*m_tmp1 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
+                          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>
-void MatrixExponential<MatrixType>::computeUV(float)
+struct matrix_exp_computeUV<MatrixType, float>
 {
-  using std::frexp;
-  using std::pow;
-  if (m_l1norm < 4.258730016922831e-001) {
-    pade3(m_M);
-  } else if (m_l1norm < 1.880152677804762e+000) {
-    pade5(m_M);
-  } else {
-    const float maxnorm = 3.925724783138660f;
-    frexp(m_l1norm / maxnorm, &m_squarings);
-    if (m_squarings < 0) m_squarings = 0;
-    MatrixType A = m_M / Scalar(pow(2, m_squarings));
-    pade7(A);
+  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>
-void MatrixExponential<MatrixType>::computeUV(double)
+struct matrix_exp_computeUV<MatrixType, double>
 {
-  using std::frexp;
-  using std::pow;
-  if (m_l1norm < 1.495585217958292e-002) {
-    pade3(m_M);
-  } else if (m_l1norm < 2.539398330063230e-001) {
-    pade5(m_M);
-  } else if (m_l1norm < 9.504178996162932e-001) {
-    pade7(m_M);
-  } else if (m_l1norm < 2.097847961257068e+000) {
-    pade9(m_M);
-  } else {
-    const double maxnorm = 5.371920351148152;
-    frexp(m_l1norm / maxnorm, &m_squarings);
-    if (m_squarings < 0) m_squarings = 0;
-    MatrixType A = m_M / Scalar(pow(2, m_squarings));
-    pade13(A);
+  template <typename ArgType>
+  static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
+  {
+    using std::frexp;
+    using std::pow;
+    const double 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 double maxnorm = 5.371920351148152;
+      frexp(l1norm / maxnorm, &squarings);
+      if (squarings < 0) squarings = 0;
+      MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<double>(squarings));
+      matrix_exp_pade13(A, U, V);
+    }
   }
-}
-
+};
+  
 template <typename MatrixType>
-void MatrixExponential<MatrixType>::computeUV(long double)
+struct matrix_exp_computeUV<MatrixType, long double>
 {
-  using std::frexp;
-  using std::pow;
+  template <typename ArgType>
+  static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
+  {
 #if   LDBL_MANT_DIG == 53   // double precision
-  computeUV(double());
-#elif LDBL_MANT_DIG <= 64   // extended precision
-  if (m_l1norm < 4.1968497232266989671e-003L) {
-    pade3(m_M);
-  } else if (m_l1norm < 1.1848116734693823091e-001L) {
-    pade5(m_M);
-  } else if (m_l1norm < 5.5170388480686700274e-001L) {
-    pade7(m_M);
-  } else if (m_l1norm < 1.3759868875587845383e+000L) {
-    pade9(m_M);
-  } else {
-    const long double maxnorm = 4.0246098906697353063L;
-    frexp(m_l1norm / maxnorm, &m_squarings);
-    if (m_squarings < 0) m_squarings = 0;
-    MatrixType A = m_M / Scalar(pow(2, m_squarings));
-    pade13(A);
-  }
+    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 (m_l1norm < 3.2787892205607026992947488108213e-005L) {
-    pade3(m_M);
-  } else if (m_l1norm < 6.4467025060072760084130906076332e-003L) {
-    pade5(m_M);
-  } else if (m_l1norm < 6.8988028496595374751374122881143e-002L) {
-    pade7(m_M);
-  } else if (m_l1norm < 2.7339737518502231741495857201670e-001L) {
-    pade9(m_M);
-  } else if (m_l1norm < 1.3203382096514474905666448850278e+000L) {
-    pade13(m_M);
-  } else {
-    const long double maxnorm = 3.2579440895405400856599663723517L;
-    frexp(m_l1norm / maxnorm, &m_squarings);
-    if (m_squarings < 0) m_squarings = 0;
-    MatrixType A = m_M / pow(Scalar(2), m_squarings);
-    pade17(A);
-  }
+  
+    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 <= 112  // quadruple precison
-  if (m_l1norm < 1.639394610288918690547467954466970e-005L) {
-    pade3(m_M);
-  } else if (m_l1norm < 4.253237712165275566025884344433009e-003L) {
-    pade5(m_M);
-  } else if (m_l1norm < 5.125804063165764409885122032933142e-002L) {
-    pade7(m_M);
-  } else if (m_l1norm < 2.170000765161155195453205651889853e-001L) {
-    pade9(m_M);
-  } else if (m_l1norm < 1.125358383453143065081397882891878e+000L) {
-    pade13(m_M);
-  } else {
-    const long double maxnorm = 2.884233277829519311757165057717815L;
-    frexp(m_l1norm / maxnorm, &m_squarings);
-    if (m_squarings < 0) m_squarings = 0;
-    MatrixType A = m_M / Scalar(pow(2, m_squarings));
-    pade17(A);
-  }
+  
+    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 {
+      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"); 
+  
+    // this case should be handled in compute()
+    eigen_assert(false && "Bug in MatrixExponential"); 
+  
+#endif
 #endif  // LDBL_MANT_DIG
+  }
+};
+
+
+/* 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)
+{
+  typedef typename ArgType::PlainObject MatrixType;
+#if LDBL_MANT_DIG > 112 // rarely happens
+  typedef typename traits<MatrixType>::Scalar Scalar;
+  typedef typename NumTraits<Scalar>::Real RealScalar;
+  typedef typename std::complex<RealScalar> ComplexScalar;
+  if (sizeof(RealScalar) > 14) {
+    result = arg.matrixFunction(internal::stem_function_exp<ComplexScalar>);
+    return;
+  }
+#endif
+  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
 }
 
+} // 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.
+  * 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> >
@@ -404,31 +390,26 @@ template<typename Derived> struct MatrixExponentialReturnValue
   public:
     /** \brief Constructor.
       *
-      * \param[in] src %Matrix (expression) forming the argument of the
-      * matrix exponential.
+      * \param src %Matrix (expression) forming the argument of the matrix exponential.
       */
     MatrixExponentialReturnValue(const Derived& src) : m_src(src) { }
 
     /** \brief Compute the matrix exponential.
       *
-      * \param[out] result the matrix exponential of \p src in the
-      * constructor.
+      * \param result the matrix exponential of \p src in the constructor.
       */
     template <typename ResultType>
     inline void evalTo(ResultType& result) const
     {
-      const typename Derived::PlainObject srcEvaluated = m_src.eval();
-      MatrixExponential<typename Derived::PlainObject> me(srcEvaluated);
-      me.compute(result);
+      const typename internal::nested_eval<Derived, 10>::type tmp(m_src);
+      internal::matrix_exp_compute(tmp, result);
     }
 
     Index rows() const { return m_src.rows(); }
     Index cols() const { return m_src.cols(); }
 
   protected:
-    const Derived& m_src;
-  private:
-    MatrixExponentialReturnValue& operator=(const MatrixExponentialReturnValue&);
+    const typename internal::ref_selector<Derived>::type m_src;
 };
 
 namespace internal {