Merge in jdh8's branch.

* Enable singular matrix power and complex exponents.
* Eliminate unnecessary copying for sparse Kronecker product.

Eigen/src/Core/MathFunctions.h

@@ -332,37 +332,45 @@ inline NewType cast(const OldType& x)
 * Implementation of atanh2                                                *
 ****************************************************************************/
 
-template<typename Scalar, bool IsInteger>
+template<typename Scalar>
-struct atanh2_default_impl
+struct atanh2_impl
 {
-  typedef Scalar retval;
+  static inline Scalar run(const Scalar& x, const Scalar& r)
-  typedef typename NumTraits<Scalar>::Real RealScalar;
-  static inline Scalar run(const Scalar& x, const Scalar& y)
   {
+    EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)
+    #if __cplusplus >= 201103L
+      using std::log1p;
+      return log1p(2 * x / (r - x)) / 2;
+    #else
       using std::abs;
       using std::log;
       using std::sqrt;
-    Scalar z = x / y;
+      Scalar z = x / r;
-    if (y == Scalar(0) || abs(z) > sqrt(NumTraits<RealScalar>::epsilon()))
+      if (r == 0 || abs(z) > sqrt(NumTraits<Scalar>::epsilon()))
-      return RealScalar(0.5) * log((y + x) / (y - x));
+        return log((r + x) / (r - x)) / 2;
+      else
+        return z + z*z*z / 3;
+    #endif
+  }
+};
+
+template<typename RealScalar>
+struct atanh2_impl<std::complex<RealScalar> >
+{
+  typedef std::complex<RealScalar> Scalar;
+  static inline Scalar run(const Scalar& x, const Scalar& r)
+  {
+    using std::log;
+    using std::norm;
+    using std::sqrt;
+    Scalar z = x / r;
+    if (r == Scalar(0) || norm(z) > NumTraits<RealScalar>::epsilon())
+      return RealScalar(0.5) * log((r + x) / (r - x));
     else
       return z + z*z*z / RealScalar(3);
   }
 };
 
-template<typename Scalar>
-struct atanh2_default_impl<Scalar, true>
-{
-  static inline Scalar run(const Scalar&, const Scalar&)
-  {
-    EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)
-    return Scalar(0);
-  }
-};
-
-template<typename Scalar>
-struct atanh2_impl : atanh2_default_impl<Scalar, NumTraits<Scalar>::IsInteger> {};
-
 template<typename Scalar>
 struct atanh2_retval
 {

Eigen/src/Core/MatrixBase.h

@@ -162,9 +162,6 @@ template<typename Derived> class MatrixBase
 #ifndef EIGEN_PARSED_BY_DOXYGEN
     template<typename ProductDerived, typename Lhs, typename Rhs>
     Derived& lazyAssign(const ProductBase<ProductDerived, Lhs,Rhs>& other);
-
-    template<typename MatrixPower, typename Lhs, typename Rhs>
-    Derived& lazyAssign(const MatrixPowerProduct<MatrixPower, Lhs,Rhs>& other);
 #endif // not EIGEN_PARSED_BY_DOXYGEN
 
     template<typename OtherDerived>
@@ -458,6 +455,7 @@ template<typename Derived> class MatrixBase
     const MatrixSquareRootReturnValue<Derived> sqrt() const;
     const MatrixLogarithmReturnValue<Derived> log() const;
     const MatrixPowerReturnValue<Derived> pow(const RealScalar& p) const;
+    const MatrixComplexPowerReturnValue<Derived> pow(const std::complex<RealScalar>& p) const;
 
 #ifdef EIGEN2_SUPPORT
     template<typename ProductDerived, typename Lhs, typename Rhs>

Eigen/src/Core/util/ForwardDeclarations.h

@@ -271,7 +271,7 @@ template<typename Derived> class MatrixFunctionReturnValue;
 template<typename Derived> class MatrixSquareRootReturnValue;
 template<typename Derived> class MatrixLogarithmReturnValue;
 template<typename Derived> class MatrixPowerReturnValue;
-template<typename Derived, typename Lhs, typename Rhs> class MatrixPowerProduct;
+template<typename Derived> class MatrixComplexPowerReturnValue;
 
 namespace internal {
 template <typename Scalar>

unsupported/Eigen/MatrixFunctions

@@ -13,8 +13,6 @@
 
 #include <cfloat>
 #include <list>
-#include <functional>
-#include <iterator>
 
 #include <Eigen/Core>
 #include <Eigen/LU>
@@ -230,22 +228,65 @@ const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(RealScalar p) con
 \endcode
 
 \param[in]  M  base of the matrix power, should be a square matrix.
-\param[in]  p  exponent of the matrix power, should be real.
+\param[in]  p  exponent of the matrix power.
 
 The matrix power \f$ M^p \f$ is defined as \f$ \exp(p \log(M)) \f$,
 where exp denotes the matrix exponential, and log denotes the matrix
 logarithm.
 
-The matrix \f$ M \f$ should meet the conditions to be an argument of
+If \p p is complex, the scalar type of \p M should be the type of \p
-matrix logarithm. If \p p is not of the real scalar type of \p M, it
+p . \f$ M^p \f$ simply evaluates into \f$ \exp(p \log(M)) \f$.
-is casted into the real scalar type of \p M.
+Therefore, the matrix \f$ M \f$ should meet the conditions to be an
+argument of matrix logarithm.
 
-This function computes the matrix power using the Schur-Pad&eacute;
+If \p p is real, it is casted into the real scalar type of \p M. Then
+this function computes the matrix power using the Schur-Pad&eacute;
 algorithm as implemented by class MatrixPower. The exponent is split
 into integral part and fractional part, where the fractional part is
 in the interval \f$ (-1, 1) \f$. The main diagonal and the first
 super-diagonal is directly computed.
 
+If \p M is singular with a semisimple zero eigenvalue and \p p is
+positive, the Schur factor \f$ T \f$ is reordered with Givens
+rotations, i.e.
+
+\f[ T = \left[ \begin{array}{cc}
+      T_1 & T_2 \\
+      0   & 0
+    \end{array} \right] \f]
+
+where \f$ T_1 \f$ is invertible. Then \f$ T^p \f$ is given by
+
+\f[ T^p = \left[ \begin{array}{cc}
+      T_1^p & T_1^{-1} T_1^p T_2 \\
+      0     & 0
+    \end{array}. \right] \f]
+
+\warning Fractional power of a matrix with a non-semisimple zero
+eigenvalue is not well-defined. We introduce an assertion failure
+against inaccurate result, e.g. \code
+#include <unsupported/Eigen/MatrixFunctions>
+#include <iostream>
+
+int main()
+{
+  Eigen::Matrix4d A;
+  A << 0, 0, 2, 3,
+       0, 0, 4, 5,
+       0, 0, 6, 7,
+       0, 0, 8, 9;
+  std::cout << A.pow(0.37) << std::endl;
+  
+  // The 1 makes eigenvalue 0 non-semisimple.
+  A.coeffRef(0, 1) = 1;
+
+  // This fails if EIGEN_NO_DEBUG is undefined.
+  std::cout << A.pow(0.37) << std::endl;
+
+  return 0;
+}
+\endcode
+
 Details of the algorithm can be found in: Nicholas J. Higham and
 Lijing Lin, "A Schur-Pad&eacute; algorithm for fractional powers of a
 matrix," <em>SIAM J. %Matrix Anal. Applic.</em>,

unsupported/Eigen/src/KroneckerProduct/KroneckerTensorProduct.h

@@ -14,9 +14,62 @@
 
 namespace Eigen {
 
-template<typename Scalar, int Options, typename Index> class SparseMatrix;
+/*!
+ * \ingroup KroneckerProduct_Module
+ *
+ * \brief The base class of dense and sparse Kronecker product.
+ *
+ * \tparam Derived is the derived type.
+ */
+template<typename Derived>
+class KroneckerProductBase : public ReturnByValue<Derived>
+{
+  private:
+    typedef typename internal::traits<Derived> Traits;
+    typedef typename Traits::Lhs Lhs;
+    typedef typename Traits::Rhs Rhs;
+    typedef typename Traits::Scalar Scalar;
+
+  protected:
+    typedef typename Traits::Index Index;
+
+  public:
+    /*! \brief Constructor. */
+    KroneckerProductBase(const Lhs& A, const Rhs& B)
+      : m_A(A), m_B(B)
+    {}
+
+    inline Index rows() const { return m_A.rows() * m_B.rows(); }
+    inline Index cols() const { return m_A.cols() * m_B.cols(); }
 
     /*!
+     * This overrides ReturnByValue::coeff because this function is
+     * efficient enough.
+     */
+    Scalar coeff(Index row, Index col) const
+    {
+      return m_A.coeff(row / m_B.rows(), col / m_B.cols()) *
+             m_B.coeff(row % m_B.rows(), col % m_B.cols());
+    }
+
+    /*!
+     * This overrides ReturnByValue::coeff because this function is
+     * efficient enough.
+     */
+    Scalar coeff(Index i) const
+    {
+      EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived);
+      return m_A.coeff(i / m_A.size()) * m_B.coeff(i % m_A.size());
+    }
+
+  protected:
+    typename Lhs::Nested m_A;
+    typename Rhs::Nested m_B;
+};
+
+/*!
+ * \ingroup KroneckerProduct_Module
+ *
  * \brief Kronecker tensor product helper class for dense matrices
  *
  * This class is the return value of kroneckerProduct(MatrixBase,
@@ -27,43 +80,26 @@ template<typename Scalar, int Options, typename Index> class SparseMatrix;
  * \tparam Rhs  Type of the rignt-hand side, a matrix expression.
  */
 template<typename Lhs, typename Rhs>
-class KroneckerProduct : public ReturnByValue<KroneckerProduct<Lhs,Rhs> >
+class KroneckerProduct : public KroneckerProductBase<KroneckerProduct<Lhs,Rhs> >
 {
   private:
-    typedef ReturnByValue<KroneckerProduct> Base;
+    typedef KroneckerProductBase<KroneckerProduct> Base;
-    typedef typename Base::Scalar Scalar;
+    using Base::m_A;
-    typedef typename Base::Index Index;
+    using Base::m_B;
 
   public:
     /*! \brief Constructor. */
     KroneckerProduct(const Lhs& A, const Rhs& B)
-      : m_A(A), m_B(B)
+      : Base(A, B)
     {}
 
     /*! \brief Evaluate the Kronecker tensor product. */
     template<typename Dest> void evalTo(Dest& dst) const;
-    
-    inline Index rows() const { return m_A.rows() * m_B.rows(); }
-    inline Index cols() const { return m_A.cols() * m_B.cols(); }
-
-    Scalar coeff(Index row, Index col) const
-    {
-      return m_A.coeff(row / m_B.rows(), col / m_B.cols()) *
-             m_B.coeff(row % m_B.rows(), col % m_B.cols());
-    }
-
-    Scalar coeff(Index i) const
-    {
-      EIGEN_STATIC_ASSERT_VECTOR_ONLY(KroneckerProduct);
-      return m_A.coeff(i / m_A.size()) * m_B.coeff(i % m_A.size());
-    }
-
-  private:
-    typename Lhs::Nested m_A;
-    typename Rhs::Nested m_B;
 };
 
 /*!
+ * \ingroup KroneckerProduct_Module
+ *
  * \brief Kronecker tensor product helper class for sparse matrices
  *
  * If at least one of the operands is a sparse matrix expression,
@@ -77,40 +113,28 @@ class KroneckerProduct : public ReturnByValue<KroneckerProduct<Lhs,Rhs> >
  * \tparam Rhs  Type of the rignt-hand side, a matrix expression.
  */
 template<typename Lhs, typename Rhs>
-class KroneckerProductSparse : public EigenBase<KroneckerProductSparse<Lhs,Rhs> >
+class KroneckerProductSparse : public KroneckerProductBase<KroneckerProductSparse<Lhs,Rhs> >
 {
   private:
-    typedef typename internal::traits<KroneckerProductSparse>::Index Index;
+    typedef KroneckerProductBase<KroneckerProductSparse> Base;
+    using Base::m_A;
+    using Base::m_B;
 
   public:
     /*! \brief Constructor. */
     KroneckerProductSparse(const Lhs& A, const Rhs& B)
-      : m_A(A), m_B(B)
+      : Base(A, B)
     {}
 
     /*! \brief Evaluate the Kronecker tensor product. */
     template<typename Dest> void evalTo(Dest& dst) const;
-    
-    inline Index rows() const { return m_A.rows() * m_B.rows(); }
-    inline Index cols() const { return m_A.cols() * m_B.cols(); }
-
-    template<typename Scalar, int Options, typename Index>
-    operator SparseMatrix<Scalar, Options, Index>()
-    {
-      SparseMatrix<Scalar, Options, Index> result;
-      evalTo(result.derived());
-      return result;
-    }
-
-  private:
-    typename Lhs::Nested m_A;
-    typename Rhs::Nested m_B;
 };
 
 template<typename Lhs, typename Rhs>
 template<typename Dest>
 void KroneckerProduct<Lhs,Rhs>::evalTo(Dest& dst) const
 {
+  typedef typename Base::Index Index;
   const int BlockRows = Rhs::RowsAtCompileTime,
             BlockCols = Rhs::ColsAtCompileTime;
   const Index Br = m_B.rows(),
@@ -124,9 +148,10 @@ template<typename Lhs, typename Rhs>
 template<typename Dest>
 void KroneckerProductSparse<Lhs,Rhs>::evalTo(Dest& dst) const
 {
+  typedef typename Base::Index Index;
   const Index Br = m_B.rows(),
               Bc = m_B.cols();
-  dst.resize(rows(),cols());
+  dst.resize(this->rows(), this->cols());
   dst.resizeNonZeros(0);
   dst.reserve(m_A.nonZeros() * m_B.nonZeros());
 
@@ -155,6 +180,7 @@ struct traits<KroneckerProduct<_Lhs,_Rhs> >
   typedef typename remove_all<_Lhs>::type Lhs;
   typedef typename remove_all<_Rhs>::type Rhs;
   typedef typename scalar_product_traits<typename Lhs::Scalar, typename Rhs::Scalar>::ReturnType Scalar;
+  typedef typename promote_index_type<typename Lhs::Index, typename Rhs::Index>::type Index;
 
   enum {
     Rows = size_at_compile_time<traits<Lhs>::RowsAtCompileTime, traits<Rhs>::RowsAtCompileTime>::ret,
@@ -193,6 +219,8 @@ struct traits<KroneckerProductSparse<_Lhs,_Rhs> >
           | EvalBeforeNestingBit | EvalBeforeAssigningBit,
     CoeffReadCost = Dynamic
   };
+
+  typedef SparseMatrix<Scalar> ReturnType;
 };
 
 } // end namespace internal
@@ -228,6 +256,16 @@ KroneckerProduct<A,B> kroneckerProduct(const MatrixBase<A>& a, const MatrixBase<
  * Computes Kronecker tensor product of two matrices, at least one of
  * which is sparse
  *
+ * \warning If you want to replace a matrix by its Kronecker product
+ *          with some matrix, do \b NOT do this:
+ * \code
+ * A = kroneckerProduct(A,B); // bug!!! caused by aliasing effect
+ * \endcode
+ * instead, use eval() to work around this:
+ * \code
+ * A = kroneckerProduct(A,B).eval();
+ * \endcode
+ *
  * \param a  Dense/sparse matrix a
  * \param b  Dense/sparse matrix b
  * \return   Kronecker tensor product of a and b, stored in a sparse

unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h

@@ -21,8 +21,8 @@ namespace Eigen {
   * expected to be an instantiation of the Matrix class template.
   */
 template <typename MatrixType>
-class MatrixExponential {
+class MatrixExponential : internal::noncopyable
-
+{
   public:
 
     /** \brief Constructor.
@@ -32,7 +32,7 @@ class MatrixExponential {
       *
       * \param[in] M  matrix whose exponential is to be computed.
       */
-    MatrixExponential(const MatrixType &M);
+    explicit MatrixExponential(const MatrixType &M);
 
     /** \brief Computes the matrix exponential.
       *
@@ -43,10 +43,6 @@ class MatrixExponential {
 
   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;
@@ -130,6 +126,7 @@ class MatrixExponential {
      */
     void computeUV(long double);
 
+    struct ScalingOp;
     typedef typename internal::traits<MatrixType>::Scalar Scalar;
     typedef typename NumTraits<Scalar>::Real RealScalar;
     typedef typename std::complex<RealScalar> ComplexScalar;
@@ -159,6 +156,43 @@ class MatrixExponential {
     RealScalar m_l1norm;
 };
 
+/** \brief Scaling operator.
+ *
+ * This struct is used by CwiseUnaryOp to scale a matrix by \f$ 2^{-s} \f$.
+ */
+template <typename MatrixType>
+struct MatrixExponential<MatrixType>::ScalingOp
+{
+  /** \brief Constructor.
+   *
+   * \param[in] squarings  The integer \f$ s \f$ in this document.
+   */
+  ScalingOp(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$.
+   */
+  inline const RealScalar operator() (const RealScalar& x) const
+  {
+    using std::ldexp;
+    return ldexp(x, -m_squarings);
+  }
+
+  /** \brief Scale a matrix coefficient.
+   *
+   * \param[in,out] x  The scalar to be scaled, becoming \f$ 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;
+};
+
 template <typename MatrixType>
 MatrixExponential<MatrixType>::MatrixExponential(const MatrixType &M) :
   m_M(M),
@@ -284,7 +318,6 @@ template <typename MatrixType>
 void MatrixExponential<MatrixType>::computeUV(float)
 {
   using std::frexp;
-  using std::pow;
   if (m_l1norm < 4.258730016922831e-001) {
     pade3(m_M);
   } else if (m_l1norm < 1.880152677804762e+000) {
@@ -293,7 +326,7 @@ void MatrixExponential<MatrixType>::computeUV(float)
     const float maxnorm = 3.925724783138660f;
     frexp(m_l1norm / maxnorm, &m_squarings);
     if (m_squarings < 0) m_squarings = 0;
-    MatrixType A = m_M / pow(Scalar(2), m_squarings);
+    MatrixType A = CwiseUnaryOp<ScalingOp, const MatrixType>(m_M, m_squarings);
     pade7(A);
   }
 }
@@ -302,7 +335,6 @@ template <typename MatrixType>
 void MatrixExponential<MatrixType>::computeUV(double)
 {
   using std::frexp;
-  using std::pow;
   if (m_l1norm < 1.495585217958292e-002) {
     pade3(m_M);
   } else if (m_l1norm < 2.539398330063230e-001) {
@@ -315,7 +347,7 @@ void MatrixExponential<MatrixType>::computeUV(double)
     const double maxnorm = 5.371920351148152;
     frexp(m_l1norm / maxnorm, &m_squarings);
     if (m_squarings < 0) m_squarings = 0;
-    MatrixType A = m_M / pow(Scalar(2), m_squarings);
+    MatrixType A = CwiseUnaryOp<ScalingOp, const MatrixType>(m_M, m_squarings);
     pade13(A);
   }
 }
@@ -324,7 +356,6 @@ template <typename MatrixType>
 void MatrixExponential<MatrixType>::computeUV(long double)
 {
   using std::frexp;
-  using std::pow;
 #if   LDBL_MANT_DIG == 53   // double precision
   computeUV(double());
 #elif LDBL_MANT_DIG <= 64   // extended precision
@@ -340,7 +371,7 @@ void MatrixExponential<MatrixType>::computeUV(long double)
     const long double maxnorm = 4.0246098906697353063L;
     frexp(m_l1norm / maxnorm, &m_squarings);
     if (m_squarings < 0) m_squarings = 0;
-    MatrixType A = m_M / pow(Scalar(2), m_squarings);
+    MatrixType A = CwiseUnaryOp<ScalingOp, const MatrixType>(m_M, m_squarings);
     pade13(A);
   }
 #elif LDBL_MANT_DIG <= 106  // double-double
@@ -358,7 +389,7 @@ void MatrixExponential<MatrixType>::computeUV(long double)
     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);
+    MatrixType A = CwiseUnaryOp<ScalingOp, const MatrixType>(m_M, m_squarings);
     pade17(A);
   }
 #elif LDBL_MANT_DIG <= 112  // quadruple precison
@@ -376,7 +407,7 @@ void MatrixExponential<MatrixType>::computeUV(long double)
     const long double maxnorm = 2.884233277829519311757165057717815L;
     frexp(m_l1norm / maxnorm, &m_squarings);
     if (m_squarings < 0) m_squarings = 0;
-    MatrixType A = m_M / pow(Scalar(2), m_squarings);
+    MatrixType A = CwiseUnaryOp<ScalingOp, const MatrixType>(m_M, m_squarings);
     pade17(A);
   }
 #else
@@ -407,7 +438,7 @@ template<typename Derived> struct MatrixExponentialReturnValue
       * \param[in] src %Matrix (expression) forming the argument of the
       * matrix exponential.
       */
-    MatrixExponentialReturnValue(const Derived& src) : m_src(src) { }
+    explicit MatrixExponentialReturnValue(const Derived& src) : m_src(src) { }
 
     /** \brief Compute the matrix exponential.
       *
@@ -427,8 +458,6 @@ template<typename Derived> struct MatrixExponentialReturnValue
 
   protected:
     const Derived& m_src;
-  private:
-    MatrixExponentialReturnValue& operator=(const MatrixExponentialReturnValue&);
 };
 
 namespace internal {

unsupported/Eigen/src/MatrixFunctions/MatrixFunction.h

@@ -34,7 +34,7 @@ namespace Eigen {
 template <typename MatrixType, 
 	  typename AtomicType,  
           int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex>
-class MatrixFunction
+class MatrixFunction : internal::noncopyable
 {  
   public:
 
@@ -65,7 +65,7 @@ class MatrixFunction
   * \brief Partial specialization of MatrixFunction for real matrices
   */
 template <typename MatrixType, typename AtomicType>
-class MatrixFunction<MatrixType, AtomicType, 0>
+class MatrixFunction<MatrixType, AtomicType, 0> : internal::noncopyable
 {  
   private:
 
@@ -111,8 +111,6 @@ class MatrixFunction<MatrixType, AtomicType, 0>
   private:
     typename internal::nested<MatrixType>::type m_A; /**< \brief Reference to argument of matrix function. */
     AtomicType& m_atomic; /**< \brief Class for computing matrix function of atomic blocks. */
-
-    MatrixFunction& operator=(const MatrixFunction&);
 };
 
       
@@ -120,7 +118,7 @@ class MatrixFunction<MatrixType, AtomicType, 0>
   * \brief Partial specialization of MatrixFunction for complex matrices
   */
 template <typename MatrixType, typename AtomicType>
-class MatrixFunction<MatrixType, AtomicType, 1>
+class MatrixFunction<MatrixType, AtomicType, 1> : internal::noncopyable
 {
   private:
 
@@ -176,8 +174,6 @@ class MatrixFunction<MatrixType, AtomicType, 1>
       * separation constant is set to 0.1, a value taken from the
       * paper by Davies and Higham. */
     static const RealScalar separation() { return static_cast<RealScalar>(0.1); }
-
-    MatrixFunction& operator=(const MatrixFunction&);
 };
 
 /** \brief Constructor. 
@@ -531,8 +527,6 @@ template<typename Derived> class MatrixFunctionReturnValue
   private:
     typename internal::nested<Derived>::type m_A;
     StemFunction *m_f;
-
-    MatrixFunctionReturnValue& operator=(const MatrixFunctionReturnValue&);
 };
 
 namespace internal {

unsupported/Eigen/src/MatrixFunctions/MatrixFunctionAtomic.h

@@ -21,7 +21,7 @@ namespace Eigen {
   * entries are close to each other.
   */
 template <typename MatrixType>
-class MatrixFunctionAtomic
+class MatrixFunctionAtomic : internal::noncopyable
 {
   public:
 
@@ -44,10 +44,6 @@ class MatrixFunctionAtomic
 
   private:
 
-    // Prevent copying
-    MatrixFunctionAtomic(const MatrixFunctionAtomic&);
-    MatrixFunctionAtomic& operator=(const MatrixFunctionAtomic&);
-
     void computeMu();
     bool taylorConverged(Index s, const MatrixType& F, const MatrixType& Fincr, const MatrixType& P);
 

unsupported/Eigen/src/MatrixFunctions/MatrixLogarithm.h

@@ -28,7 +28,7 @@ namespace Eigen {
   * \sa class MatrixFunctionAtomic, MatrixBase::log()
   */
 template <typename MatrixType>
-class MatrixLogarithmAtomic
+class MatrixLogarithmAtomic : internal::noncopyable
 {
 public:
 
@@ -71,10 +71,6 @@ private:
                                    std::numeric_limits<RealScalar>::digits<= 64?  8:  // extended precision
                                    std::numeric_limits<RealScalar>::digits<=106? 10:  // double-double
                                                                                  11;  // quadruple precision
-
-  // Prevent copying
-  MatrixLogarithmAtomic(const MatrixLogarithmAtomic&);
-  MatrixLogarithmAtomic& operator=(const MatrixLogarithmAtomic&);
 };
 
 /** \brief Compute logarithm of triangular matrix with clustered eigenvalues. */
@@ -429,7 +425,7 @@ public:
     *
     * \param[in]  A  %Matrix (expression) forming the argument of the matrix logarithm.
     */
-  MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { }
+  explicit MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { }
   
   /** \brief Compute the matrix logarithm.
     *
@@ -458,8 +454,6 @@ public:
   
 private:
   typename internal::nested<Derived>::type m_A;
-  
-  MatrixLogarithmReturnValue& operator=(const MatrixLogarithmReturnValue&);
 };
 
 namespace internal {

unsupported/Eigen/src/MatrixFunctions/MatrixPower.h

@@ -14,16 +14,48 @@ namespace Eigen {
 
 template<typename MatrixType> class MatrixPower;
 
+/**
+ * \ingroup MatrixFunctions_Module
+ *
+ * \brief Proxy for the matrix power of some matrix.
+ *
+ * \tparam MatrixType  type of the base, a matrix.
+ *
+ * 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
+ * MatrixPower::operator() and related functions and most of the
+ * time this is the only way it is used.
+ */
+/* TODO This class is only used by MatrixPower, so it should be nested
+ * into MatrixPower, like MatrixPower::ReturnValue. However, my
+ * compiler complained about unused template parameter in the
+ * following declaration in namespace internal.
+ *
+ * template<typename MatrixType>
+ * struct traits<MatrixPower<MatrixType>::ReturnValue>;
+ */
 template<typename MatrixType>
-class MatrixPowerRetval : public ReturnByValue< MatrixPowerRetval<MatrixType> >
+class MatrixPowerParenthesesReturnValue : public ReturnByValue< MatrixPowerParenthesesReturnValue<MatrixType> >
 {
   public:
     typedef typename MatrixType::RealScalar RealScalar;
     typedef typename MatrixType::Index Index;
 
-    MatrixPowerRetval(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p)
+    /**
+     * \brief Constructor.
+     *
+     * \param[in] pow  %MatrixPower storing the base.
+     * \param[in] p    scalar, the exponent of the matrix power.
+     */
+    MatrixPowerParenthesesReturnValue(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p)
     { }
 
+    /**
+     * \brief Compute the matrix power.
+     *
+     * \param[out] result
+     */
     template<typename ResultType>
     inline void evalTo(ResultType& res) const
     { m_pow.compute(res, m_p); }
@@ -34,11 +66,25 @@ class MatrixPowerRetval : public ReturnByValue< MatrixPowerRetval<MatrixType> >
   private:
     MatrixPower<MatrixType>& m_pow;
     const RealScalar m_p;
-    MatrixPowerRetval& operator=(const MatrixPowerRetval&);
 };
 
+/**
+ * \ingroup MatrixFunctions_Module
+ *
+ * \brief Class for computing matrix powers.
+ *
+ * \tparam MatrixType  type of the base, expected to be an instantiation
+ * of the Matrix class template.
+ *
+ * This class is capable of computing triangular real/complex matrices
+ * raised to a power in the interval \f$ (-1, 1) \f$.
+ *
+ * \note Currently this class is only used by MatrixPower. One may
+ * insist that this be nested into MatrixPower. This class is here to
+ * faciliate future development of triangular matrix functions.
+ */
 template<typename MatrixType>
-class MatrixPowerAtomic
+class MatrixPowerAtomic : internal::noncopyable
 {
   private:
     enum {
@@ -49,14 +95,14 @@ class MatrixPowerAtomic
     typedef typename MatrixType::RealScalar RealScalar;
     typedef std::complex<RealScalar> ComplexScalar;
     typedef typename MatrixType::Index Index;
-    typedef Array<Scalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> ArrayType;
+    typedef Block<MatrixType,Dynamic,Dynamic> ResultType;
 
     const MatrixType& m_A;
     RealScalar m_p;
 
-    void computePade(int degree, const MatrixType& IminusT, MatrixType& res) const;
+    void computePade(int degree, const MatrixType& IminusT, ResultType& res) const;
-    void compute2x2(MatrixType& res, RealScalar p) const;
+    void compute2x2(ResultType& res, RealScalar p) const;
-    void computeBig(MatrixType& res) const;
+    void computeBig(ResultType& res) const;
     static int getPadeDegree(float normIminusT);
     static int getPadeDegree(double normIminusT);
     static int getPadeDegree(long double normIminusT);
@@ -64,24 +110,45 @@ class MatrixPowerAtomic
     static RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p);
 
   public:
+    /**
+     * \brief Constructor.
+     *
+     * \param[in] T  the base of the matrix power.
+     * \param[in] p  the exponent of the matrix power, should be in
+     * \f$ (-1, 1) \f$.
+     *
+     * The class stores a reference to T, so it should not be changed
+     * (or destroyed) before evaluation. Only the upper triangular
+     * part of T is read.
+     */
     MatrixPowerAtomic(const MatrixType& T, RealScalar p);
-    void compute(MatrixType& res) const;
+    
+    /**
+     * \brief Compute the matrix power.
+     *
+     * \param[out] res  \f$ A^p \f$ where A and p are specified in the
+     * constructor.
+     */
+    void compute(ResultType& res) const;
 };
 
 template<typename MatrixType>
 MatrixPowerAtomic<MatrixType>::MatrixPowerAtomic(const MatrixType& T, RealScalar p) :
   m_A(T), m_p(p)
-{ eigen_assert(T.rows() == T.cols()); }
+{
+  eigen_assert(T.rows() == T.cols());
+  eigen_assert(p > -1 && p < 1);
+}
 
 template<typename MatrixType>
-void MatrixPowerAtomic<MatrixType>::compute(MatrixType& res) const
+void MatrixPowerAtomic<MatrixType>::compute(ResultType& res) const
 {
-  res.resizeLike(m_A);
+  using std::pow;
   switch (m_A.rows()) {
     case 0:
       break;
     case 1:
-      res(0,0) = std::pow(m_A(0,0), m_p);
+      res(0,0) = pow(m_A(0,0), m_p);
       break;
     case 2:
       compute2x2(res, m_p);
@@ -92,25 +159,24 @@ void MatrixPowerAtomic<MatrixType>::compute(MatrixType& res) const
 }
 
 template<typename MatrixType>
-void MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, MatrixType& res) const
+void MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, ResultType& res) const
 {
-  int i = degree<<1;
+  int i = 2*degree;
-  res = (m_p-degree) / ((i-1)<<1) * IminusT;
+  res = (m_p-degree) / (2*i-2) * IminusT;
+
   for (--i; i; --i) {
     res = (MatrixType::Identity(IminusT.rows(), IminusT.cols()) + res).template triangularView<Upper>()
-	.solve((i==1 ? -m_p : i&1 ? (-m_p-(i>>1))/(i<<1) : (m_p-(i>>1))/((i-1)<<1)) * IminusT).eval();
+	.solve((i==1 ? -m_p : i&1 ? (-m_p-i/2)/(2*i) : (m_p-i/2)/(2*i-2)) * IminusT).eval();
   }
   res += MatrixType::Identity(IminusT.rows(), IminusT.cols());
 }
 
 // This function assumes that res has the correct size (see bug 614)
 template<typename MatrixType>
-void MatrixPowerAtomic<MatrixType>::compute2x2(MatrixType& res, RealScalar p) const
+void MatrixPowerAtomic<MatrixType>::compute2x2(ResultType& res, RealScalar p) const
 {
   using std::abs;
   using std::pow;
-  
-  ArrayType logTdiag = m_A.diagonal().array().log();
   res.coeffRef(0,0) = pow(m_A.coeff(0,0), p);
 
   for (Index i=1; i < m_A.cols(); ++i) {
@@ -126,8 +192,9 @@ void MatrixPowerAtomic<MatrixType>::compute2x2(MatrixType& res, RealScalar p) co
 }
 
 template<typename MatrixType>
-void MatrixPowerAtomic<MatrixType>::computeBig(MatrixType& res) const
+void MatrixPowerAtomic<MatrixType>::computeBig(ResultType& res) const
 {
+  using std::ldexp;
   const int digits = std::numeric_limits<RealScalar>::digits;
   const RealScalar maxNormForPade = digits <=  24? 4.3386528e-1f:                           // sigle precision
 				    digits <=  53? 2.789358995219730e-1:                    // double precision
@@ -139,19 +206,6 @@ void MatrixPowerAtomic<MatrixType>::computeBig(MatrixType& res) const
   int degree, degree2, numberOfSquareRoots = 0;
   bool hasExtraSquareRoot = false;
 
-  /* FIXME
-   * For singular T, norm(I - T) >= 1 but maxNormForPade < 1, leads to infinite
-   * loop.  We should move 0 eigenvalues to bottom right corner.  We need not
-   * worry about tiny values (e.g. 1e-300) because they will reach 1 if
-   * repetitively sqrt'ed.
-   *
-   * If the 0 eigenvalues are semisimple, they can form a 0 matrix at the
-   * bottom right corner.
-   *
-   * [ T  A ]^p   [ T^p  (T^-1 T^p A) ]
-   * [      ]   = [                   ]
-   * [ 0  0 ]     [  0         0      ]
-   */
   for (Index i=0; i < m_A.cols(); ++i)
     eigen_assert(m_A(i,i) != RealScalar(0));
 
@@ -172,7 +226,7 @@ void MatrixPowerAtomic<MatrixType>::computeBig(MatrixType& res) const
   computePade(degree, IminusT, res);
 
   for (; numberOfSquareRoots; --numberOfSquareRoots) {
-    compute2x2(res, std::ldexp(m_p, -numberOfSquareRoots));
+    compute2x2(res, ldexp(m_p, -numberOfSquareRoots));
     res = res.template triangularView<Upper>() * res;
   }
   compute2x2(res, m_p);
@@ -237,19 +291,28 @@ template<typename MatrixType>
 inline typename MatrixPowerAtomic<MatrixType>::ComplexScalar
 MatrixPowerAtomic<MatrixType>::computeSuperDiag(const ComplexScalar& curr, const ComplexScalar& prev, RealScalar p)
 {
-  ComplexScalar logCurr = std::log(curr);
+  using std::ceil;
-  ComplexScalar logPrev = std::log(prev);
+  using std::exp;
-  int unwindingNumber = std::ceil((numext::imag(logCurr - logPrev) - M_PI) / (2*M_PI));
+  using std::log;
+  using std::sinh;
+
+  ComplexScalar logCurr = log(curr);
+  ComplexScalar logPrev = log(prev);
+  int unwindingNumber = ceil((numext::imag(logCurr - logPrev) - M_PI) / (2*M_PI));
   ComplexScalar w = numext::atanh2(curr - prev, curr + prev) + ComplexScalar(0, M_PI*unwindingNumber);
-  return RealScalar(2) * std::exp(RealScalar(0.5) * p * (logCurr + logPrev)) * std::sinh(p * w) / (curr - prev);
+  return RealScalar(2) * exp(RealScalar(0.5) * p * (logCurr + logPrev)) * sinh(p * w) / (curr - prev);
 }
 
 template<typename MatrixType>
 inline typename MatrixPowerAtomic<MatrixType>::RealScalar
 MatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev, RealScalar p)
 {
+  using std::exp;
+  using std::log;
+  using std::sinh;
+
   RealScalar w = numext::atanh2(curr - prev, curr + prev);
-  return 2 * std::exp(p * (std::log(curr) + std::log(prev)) / 2) * std::sinh(p * w) / (curr - prev);
+  return 2 * exp(p * (log(curr) + log(prev)) / 2) * sinh(p * w) / (curr - prev);
 }
 
 /**
@@ -272,15 +335,9 @@ MatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev
  * Output: \verbinclude MatrixPower_optimal.out
  */
 template<typename MatrixType>
-class MatrixPower
+class MatrixPower : internal::noncopyable
 {
   private:
-    enum {
-      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
-      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
-      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
-      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
-    };
     typedef typename MatrixType::Scalar Scalar;
     typedef typename MatrixType::RealScalar RealScalar;
     typedef typename MatrixType::Index Index;
@@ -294,7 +351,11 @@ class MatrixPower
      * The class stores a reference to A, so it should not be changed
      * (or destroyed) before evaluation.
      */
-    explicit MatrixPower(const MatrixType& A) : m_A(A), m_conditionNumber(0)
+    explicit MatrixPower(const MatrixType& A) :
+      m_A(A),
+      m_conditionNumber(0),
+      m_rank(A.cols()),
+      m_nulls(0)
     { eigen_assert(A.rows() == A.cols()); }
 
     /**
@@ -304,8 +365,8 @@ class MatrixPower
      * \return The expression \f$ A^p \f$, where A is specified in the
      * constructor.
      */
-    const MatrixPowerRetval<MatrixType> operator()(RealScalar p)
+    const MatrixPowerParenthesesReturnValue<MatrixType> operator()(RealScalar p)
-    { return MatrixPowerRetval<MatrixType>(*this, p); }
+    { return MatrixPowerParenthesesReturnValue<MatrixType>(*this, p); }
 
     /**
      * \brief Compute the matrix power.
@@ -322,21 +383,54 @@ class MatrixPower
 
   private:
     typedef std::complex<RealScalar> ComplexScalar;
-    typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, MatrixType::Options,
+    typedef Matrix<ComplexScalar, Dynamic, Dynamic, MatrixType::Options,
-              MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrix;
+              MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime> ComplexMatrix;
 
+    /** \brief Reference to the base of matrix power. */
     typename MatrixType::Nested m_A;
+
+    /** \brief Temporary storage. */
     MatrixType m_tmp;
-    ComplexMatrix m_T, m_U, m_fT;
+
+    /** \brief Store the result of Schur decomposition. */
+    ComplexMatrix m_T, m_U;
+    
+    /** \brief Store fractional power of m_T. */
+    ComplexMatrix m_fT;
+
+    /**
+     * \brief Condition number of m_A.
+     *
+     * It is initialized as 0 to avoid performing unnecessary Schur
+     * decomposition, which is the bottleneck.
+     */
     RealScalar m_conditionNumber;
 
-    RealScalar modfAndInit(RealScalar, RealScalar*);
+    /** \brief Rank of m_A. */
+    Index m_rank;
+    
+    /** \brief Rank deficiency of m_A. */
+    Index m_nulls;
+
+    /**
+     * \brief Split p into integral part and fractional part.
+     *
+     * \param[in]  p        The exponent.
+     * \param[out] p        The fractional part ranging in \f$ (-1, 1) \f$.
+     * \param[out] intpart  The integral part.
+     *
+     * Only if the fractional part is nonzero, it calls initialize().
+     */
+    void split(RealScalar& p, RealScalar& intpart);
+
+    /** \brief Perform Schur decomposition for fractional power. */
+    void initialize();
 
     template<typename ResultType>
-    void computeIntPower(ResultType&, RealScalar);
+    void computeIntPower(ResultType& res, RealScalar p);
 
     template<typename ResultType>
-    void computeFracPower(ResultType&, RealScalar);
+    void computeFracPower(ResultType& res, RealScalar p);
 
     template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
     static void revertSchur(
@@ -355,59 +449,102 @@ template<typename MatrixType>
 template<typename ResultType>
 void MatrixPower<MatrixType>::compute(ResultType& res, RealScalar p)
 {
+  using std::pow;
   switch (cols()) {
     case 0:
       break;
     case 1:
-      res(0,0) = std::pow(m_A.coeff(0,0), p);
+      res(0,0) = pow(m_A.coeff(0,0), p);
       break;
     default:
-      RealScalar intpart, x = modfAndInit(p, &intpart);
+      RealScalar intpart;
+      split(p, intpart);
+
+      res = MatrixType::Identity(rows(), cols());
       computeIntPower(res, intpart);
-      computeFracPower(res, x);
+      if (p) computeFracPower(res, p);
   }
 }
 
 template<typename MatrixType>
-typename MatrixPower<MatrixType>::RealScalar
+void MatrixPower<MatrixType>::split(RealScalar& p, RealScalar& intpart)
-MatrixPower<MatrixType>::modfAndInit(RealScalar x, RealScalar* intpart)
 {
-  typedef Array<RealScalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> RealArray;
+  using std::floor;
+  using std::pow;
 
-  *intpart = std::floor(x);
+  intpart = floor(p);
-  RealScalar res = x - *intpart;
+  p -= intpart;
 
-  if (!m_conditionNumber && res) {
+  // Perform Schur decomposition if it is not yet performed and the power is
+  // not an integer.
+  if (!m_conditionNumber && p)
+    initialize();
+
+  // Choose the more stable of intpart = floor(p) and intpart = ceil(p).
+  if (p > RealScalar(0.5) && p > (1-p) * pow(m_conditionNumber, p)) {
+    --p;
+    ++intpart;
+  }
+}
+
+template<typename MatrixType>
+void MatrixPower<MatrixType>::initialize()
+{
   const ComplexSchur<MatrixType> schurOfA(m_A);
+  JacobiRotation<ComplexScalar> rot;
+  ComplexScalar eigenvalue;
+
+  m_fT.resizeLike(m_A);
   m_T = schurOfA.matrixT();
   m_U = schurOfA.matrixU();
+  m_conditionNumber = m_T.diagonal().array().abs().maxCoeff() / m_T.diagonal().array().abs().minCoeff();
 
-    const RealArray absTdiag = m_T.diagonal().array().abs();
+  // Move zero eigenvalues to the bottom right corner.
-    m_conditionNumber = absTdiag.maxCoeff() / absTdiag.minCoeff();
+  for (Index i = cols()-1; i>=0; --i) {
+    if (m_rank <= 2)
+      return;
+    if (m_T.coeff(i,i) == RealScalar(0)) {
+      for (Index j=i+1; j < m_rank; ++j) {
+        eigenvalue = m_T.coeff(j,j);
+        rot.makeGivens(m_T.coeff(j-1,j), eigenvalue);
+        m_T.applyOnTheRight(j-1, j, rot);
+        m_T.applyOnTheLeft(j-1, j, rot.adjoint());
+        m_T.coeffRef(j-1,j-1) = eigenvalue;
+        m_T.coeffRef(j,j) = RealScalar(0);
+        m_U.applyOnTheRight(j-1, j, rot);
+      }
+      --m_rank;
+    }
   }
 
-  if (res>RealScalar(0.5) && res>(1-res)*std::pow(m_conditionNumber, res)) {
+  m_nulls = rows() - m_rank;
-    --res;
+  if (m_nulls) {
-    ++*intpart;
+    eigen_assert(m_T.bottomRightCorner(m_nulls, m_nulls).isZero()
+        && "Base of matrix power should be invertible or with a semisimple zero eigenvalue.");
+    m_fT.bottomRows(m_nulls).fill(RealScalar(0));
   }
-  return res;
 }
 
 template<typename MatrixType>
 template<typename ResultType>
 void MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p)
 {
-  RealScalar pp = std::abs(p);
+  using std::abs;
+  using std::fmod;
+  RealScalar pp = abs(p);
 
-  if (p<0)  m_tmp = m_A.inverse();
+  if (p<0) 
-  else      m_tmp = m_A;
+    m_tmp = m_A.inverse();
+  else     
+    m_tmp = m_A;
 
-  res = MatrixType::Identity(rows(), cols());
+  while (true) {
-  while (pp >= 1) {
+    if (fmod(pp, 2) >= 1)
-    if (std::fmod(pp, 2) >= 1)
       res = m_tmp * res;
-    m_tmp *= m_tmp;
     pp /= 2;
+    if (pp < 1)
+      break;
+    m_tmp *= m_tmp;
   }
 }
 
@@ -415,13 +552,18 @@ template<typename MatrixType>
 template<typename ResultType>
 void MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p)
 {
-  if (p) {
+  Block<ComplexMatrix,Dynamic,Dynamic> blockTp(m_fT, 0, 0, m_rank, m_rank);
   eigen_assert(m_conditionNumber);
-    MatrixPowerAtomic<ComplexMatrix>(m_T, p).compute(m_fT);
+  eigen_assert(m_rank + m_nulls == rows());
+
+  MatrixPowerAtomic<ComplexMatrix>(m_T.topLeftCorner(m_rank, m_rank), p).compute(blockTp);
+  if (m_nulls) {
+    m_fT.topRightCorner(m_rank, m_nulls) = m_T.topLeftCorner(m_rank, m_rank).template triangularView<Upper>()
+        .solve(blockTp * m_T.topRightCorner(m_rank, m_nulls));
+  }
   revertSchur(m_tmp, m_fT, m_U);
   res = m_tmp * res;
 }
-}
 
 template<typename MatrixType>
 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
@@ -464,7 +606,7 @@ class MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Deri
      * \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] p  real scalar, the exponent of the matrix power.
      */
     MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p)
     { }
@@ -485,25 +627,83 @@ class MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Deri
   private:
     const Derived& m_A;
     const RealScalar m_p;
-    MatrixPowerReturnValue& operator=(const MatrixPowerReturnValue&);
+};
+
+/**
+ * \ingroup MatrixFunctions_Module
+ *
+ * \brief Proxy for the matrix power of some matrix (expression).
+ *
+ * \tparam Derived  type of the base, a matrix (expression).
+ *
+ * This class holds the arguments to the matrix power until it is
+ * assigned or evaluated for some other reason (so the argument
+ * should not be changed in the meantime). It is the return type of
+ * MatrixBase::pow() and related functions and most of the
+ * time this is the only way it is used.
+ */
+template<typename Derived>
+class MatrixComplexPowerReturnValue : public ReturnByValue< MatrixComplexPowerReturnValue<Derived> >
+{
+  public:
+    typedef typename Derived::PlainObject PlainObject;
+    typedef typename std::complex<typename Derived::RealScalar> ComplexScalar;
+    typedef typename Derived::Index Index;
+
+    /**
+     * \brief Constructor.
+     *
+     * \param[in] A  %Matrix (expression), the base of the matrix power.
+     * \param[in] p  complex scalar, the exponent of the matrix power.
+     */
+    MatrixComplexPowerReturnValue(const Derived& A, const ComplexScalar& p) : m_A(A), m_p(p)
+    { }
+
+    /**
+     * \brief Compute the matrix power.
+     *
+     * Because \p p is complex, \f$ A^p \f$ is simply evaluated as \f$
+     * \exp(p \log(A)) \f$.
+     *
+     * \param[out] result  \f$ A^p \f$ where \p A and \p p are as in the
+     * constructor.
+     */
+    template<typename ResultType>
+    inline void evalTo(ResultType& res) const
+    { res = (m_p * m_A.log()).exp(); }
+
+    Index rows() const { return m_A.rows(); }
+    Index cols() const { return m_A.cols(); }
+
+  private:
+    const Derived& m_A;
+    const ComplexScalar m_p;
 };
 
 namespace internal {
 
 template<typename MatrixPowerType>
-struct traits< MatrixPowerRetval<MatrixPowerType> >
+struct traits< MatrixPowerParenthesesReturnValue<MatrixPowerType> >
 { typedef typename MatrixPowerType::PlainObject ReturnType; };
 
 template<typename Derived>
 struct traits< MatrixPowerReturnValue<Derived> >
 { typedef typename Derived::PlainObject ReturnType; };
 
+template<typename Derived>
+struct traits< MatrixComplexPowerReturnValue<Derived> >
+{ typedef typename Derived::PlainObject ReturnType; };
+
 }
 
 template<typename Derived>
 const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const
 { return MatrixPowerReturnValue<Derived>(derived(), p); }
 
+template<typename Derived>
+const MatrixComplexPowerReturnValue<Derived> MatrixBase<Derived>::pow(const std::complex<RealScalar>& p) const
+{ return MatrixComplexPowerReturnValue<Derived>(derived(), p); }
+
 } // namespace Eigen
 
 #endif // EIGEN_MATRIX_POWER

unsupported/Eigen/src/MatrixFunctions/MatrixSquareRoot.h

@@ -24,7 +24,7 @@ namespace Eigen {
   * \sa MatrixSquareRoot, MatrixSquareRootTriangular
   */
 template <typename MatrixType>
-class MatrixSquareRootQuasiTriangular
+class MatrixSquareRootQuasiTriangular : internal::noncopyable
 {
   public:
 
@@ -36,7 +36,7 @@ class MatrixSquareRootQuasiTriangular
       * The class stores a reference to \p A, so it should not be
       * changed (or destroyed) before compute() is called.
       */
-    MatrixSquareRootQuasiTriangular(const MatrixType& A) 
+    explicit MatrixSquareRootQuasiTriangular(const MatrixType& A) 
       : m_A(A) 
     {
       eigen_assert(A.rows() == A.cols());
@@ -253,10 +253,10 @@ void MatrixSquareRootQuasiTriangular<MatrixType>
   * \sa MatrixSquareRoot, MatrixSquareRootQuasiTriangular
   */
 template <typename MatrixType>
-class MatrixSquareRootTriangular
+class MatrixSquareRootTriangular : internal::noncopyable
 {
   public:
-    MatrixSquareRootTriangular(const MatrixType& A) 
+    explicit MatrixSquareRootTriangular(const MatrixType& A) 
       : m_A(A) 
     {
       eigen_assert(A.rows() == A.cols());
@@ -321,7 +321,7 @@ class MatrixSquareRoot
       * The class stores a reference to \p A, so it should not be
       * changed (or destroyed) before compute() is called.
       */
-    MatrixSquareRoot(const MatrixType& A); 
+    explicit MatrixSquareRoot(const MatrixType& A); 
     
     /** \brief Compute the matrix square root
       *
@@ -341,7 +341,7 @@ class MatrixSquareRoot<MatrixType, 0>
 {
   public:
 
-    MatrixSquareRoot(const MatrixType& A) 
+    explicit MatrixSquareRoot(const MatrixType& A) 
       : m_A(A) 
     {  
       eigen_assert(A.rows() == A.cols());
@@ -370,11 +370,11 @@ class MatrixSquareRoot<MatrixType, 0>
 // ********** Partial specialization for complex matrices **********
 
 template <typename MatrixType>
-class MatrixSquareRoot<MatrixType, 1>
+class MatrixSquareRoot<MatrixType, 1> : internal::noncopyable
 {
   public:
 
-    MatrixSquareRoot(const MatrixType& A) 
+    explicit MatrixSquareRoot(const MatrixType& A) 
       : m_A(A) 
     {  
       eigen_assert(A.rows() == A.cols());
@@ -422,7 +422,7 @@ template<typename Derived> class MatrixSquareRootReturnValue
       * \param[in]  src  %Matrix (expression) forming the argument of the
       * matrix square root.
       */
-    MatrixSquareRootReturnValue(const Derived& src) : m_src(src) { }
+    explicit MatrixSquareRootReturnValue(const Derived& src) : m_src(src) { }
 
     /** \brief Compute the matrix square root.
       *
@@ -442,8 +442,6 @@ template<typename Derived> class MatrixSquareRootReturnValue
 
   protected:
     const Derived& m_src;
-  private:
-    MatrixSquareRootReturnValue& operator=(const MatrixSquareRootReturnValue&);
 };
 
 namespace internal {

unsupported/Eigen/src/MatrixFunctions/StemFunction.h

@@ -16,7 +16,7 @@ namespace Eigen {
   * \brief Stem functions corresponding to standard mathematical functions.
   */
 template <typename Scalar>
-class StdStemFunctions
+class StdStemFunctions : internal::noncopyable
 {
   public:
 

unsupported/test/kronecker_product.cpp

@@ -107,31 +107,34 @@ void test_kronecker_product()
 
   SparseMatrix<double,RowMajor> SM_row_a(SM_a), SM_row_b(SM_b);
 
-  // test kroneckerProduct(DM_block,DM,DM_fixedSize)
+  // test DM_fixedSize = kroneckerProduct(DM_block,DM)
   Matrix<double, 6, 6> DM_fix_ab = kroneckerProduct(DM_a.topLeftCorner<2,3>(),DM_b);
 
   CALL_SUBTEST(check_kronecker_product(DM_fix_ab));
+  CALL_SUBTEST(check_kronecker_product(kroneckerProduct(DM_a.topLeftCorner<2,3>(),DM_b)));
 
   for(int i=0;i<DM_fix_ab.rows();++i)
     for(int j=0;j<DM_fix_ab.cols();++j)
        VERIFY_IS_APPROX(kroneckerProduct(DM_a,DM_b).coeff(i,j), DM_fix_ab(i,j));
 
-  // test kroneckerProduct(DM,DM,DM_block)
+  // test DM_block = kroneckerProduct(DM,DM)
   MatrixXd DM_block_ab(10,15);
   DM_block_ab.block<6,6>(2,5) = kroneckerProduct(DM_a,DM_b);
   CALL_SUBTEST(check_kronecker_product(DM_block_ab.block<6,6>(2,5)));
 
-  // test kroneckerProduct(DM,DM,DM)
+  // test DM = kroneckerProduct(DM,DM)
   MatrixXd DM_ab = kroneckerProduct(DM_a,DM_b);
   CALL_SUBTEST(check_kronecker_product(DM_ab));
+  CALL_SUBTEST(check_kronecker_product(kroneckerProduct(DM_a,DM_b)));
 
-  // test kroneckerProduct(SM,DM,SM)
+  // test SM = kroneckerProduct(SM,DM)
   SparseMatrix<double> SM_ab = kroneckerProduct(SM_a,DM_b);
   CALL_SUBTEST(check_kronecker_product(SM_ab));
   SparseMatrix<double,RowMajor> SM_ab2 = kroneckerProduct(SM_a,DM_b);
   CALL_SUBTEST(check_kronecker_product(SM_ab2));
+  CALL_SUBTEST(check_kronecker_product(kroneckerProduct(SM_a,DM_b)));
 
-  // test kroneckerProduct(DM,SM,SM)
+  // test SM = kroneckerProduct(DM,SM)
   SM_ab.setZero();
   SM_ab.insert(0,0)=37.0;
   SM_ab = kroneckerProduct(DM_a,SM_b);
@@ -140,8 +143,9 @@ void test_kronecker_product()
   SM_ab2.insert(0,0)=37.0;
   SM_ab2 = kroneckerProduct(DM_a,SM_b);
   CALL_SUBTEST(check_kronecker_product(SM_ab2));
+  CALL_SUBTEST(check_kronecker_product(kroneckerProduct(DM_a,SM_b)));
 
-  // test kroneckerProduct(SM,SM,SM)
+  // test SM = kroneckerProduct(SM,SM)
   SM_ab.resize(2,33);
   SM_ab.insert(0,0)=37.0;
   SM_ab = kroneckerProduct(SM_a,SM_b);
@@ -150,8 +154,9 @@ void test_kronecker_product()
   SM_ab2.insert(0,0)=37.0;
   SM_ab2 = kroneckerProduct(SM_a,SM_b);
   CALL_SUBTEST(check_kronecker_product(SM_ab2));
+  CALL_SUBTEST(check_kronecker_product(kroneckerProduct(SM_a,SM_b)));
 
-  // test kroneckerProduct(SM,SM,SM) with sparse pattern
+  // test SM = kroneckerProduct(SM,SM) with sparse pattern
   SM_a.resize(4,5);
   SM_b.resize(3,2);
   SM_a.resizeNonZeros(0);
@@ -169,7 +174,7 @@ void test_kronecker_product()
   SM_ab = kroneckerProduct(SM_a,SM_b);
   CALL_SUBTEST(check_sparse_kronecker_product(SM_ab));
 
-  // test dimension of result of kroneckerProduct(DM,DM,DM)
+  // test dimension of result of DM = kroneckerProduct(DM,DM)
   MatrixXd DM_a2(2,1);
   MatrixXd DM_b2(5,4);
   MatrixXd DM_ab2 = kroneckerProduct(DM_a2,DM_b2);

unsupported/test/matrix_functions.h

@@ -10,27 +10,48 @@
 #include "main.h"
 #include <unsupported/Eigen/MatrixFunctions>
 
+// For complex matrices, any matrix is fine.
+template<typename MatrixType, int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex>
+struct processTriangularMatrix
+{
+  static void run(MatrixType&, MatrixType&, const MatrixType&)
+  { }
+};
+
+// For real matrices, make sure none of the eigenvalues are negative.
+template<typename MatrixType>
+struct processTriangularMatrix<MatrixType,0>
+{
+  static void run(MatrixType& m, MatrixType& T, const MatrixType& U)
+  {
+    typedef typename MatrixType::Index Index;
+    const Index size = m.cols();
+
+    for (Index i=0; i < size; ++i) {
+      if (i == size - 1 || T.coeff(i+1,i) == 0)
+        T.coeffRef(i,i) = std::abs(T.coeff(i,i));
+      else
+        ++i;
+    }
+    m = U * T * U.transpose();
+  }
+};
+
 template <typename MatrixType, int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex>
 struct generateTestMatrix;
 
-// for real matrices, make sure none of the eigenvalues are negative
 template <typename MatrixType>
 struct generateTestMatrix<MatrixType,0>
 {
   static void run(MatrixType& result, typename MatrixType::Index size)
   {
-    MatrixType mat = MatrixType::Random(size, size);
+    result = MatrixType::Random(size, size);
-    EigenSolver<MatrixType> es(mat);
+    RealSchur<MatrixType> schur(result);
-    typename EigenSolver<MatrixType>::EigenvalueType eivals = es.eigenvalues();
+    MatrixType T = schur.matrixT();
-    for (typename MatrixType::Index i = 0; i < size; ++i) {
+    processTriangularMatrix<MatrixType>::run(result, T, schur.matrixU());
-      if (eivals(i).imag() == 0 && eivals(i).real() < 0)
-	eivals(i) = -eivals(i);
-    }
-    result = (es.eigenvectors() * eivals.asDiagonal() * es.eigenvectors().inverse()).real();
   }
 };
 
-// for complex matrices, any matrix is fine
 template <typename MatrixType>
 struct generateTestMatrix<MatrixType,1>
 {

unsupported/test/matrix_power.cpp

@@ -1,7 +1,7 @@
 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
-// Copyright (C) 2012 Chen-Pang He <jdh8@ms63.hinet.net>
+// Copyright (C) 2012, 2013 Chen-Pang He <jdh8@ms63.hinet.net>
 //
 // This Source Code Form is subject to the terms of the Mozilla
 // Public License v. 2.0. If a copy of the MPL was not distributed
@@ -9,33 +9,6 @@
 
 #include "matrix_functions.h"
 
-template <typename MatrixType, int IsComplex = NumTraits<typename MatrixType::Scalar>::IsComplex>
-struct generateTriangularMatrix;
-
-// for real matrices, make sure none of the eigenvalues are negative
-template <typename MatrixType>
-struct generateTriangularMatrix<MatrixType,0>
-{
-  static void run(MatrixType& result, typename MatrixType::Index size)
-  {
-    result.resize(size, size);
-    result.template triangularView<Upper>() = MatrixType::Random(size, size);
-    for (typename MatrixType::Index i = 0; i < size; ++i)
-      result.coeffRef(i,i) = std::abs(result.coeff(i,i));
-  }
-};
-
-// for complex matrices, any matrix is fine
-template <typename MatrixType>
-struct generateTriangularMatrix<MatrixType,1>
-{
-  static void run(MatrixType& result, typename MatrixType::Index size)
-  {
-    result.resize(size, size);
-    result.template triangularView<Upper>() = MatrixType::Random(size, size);
-  }
-};
-
 template<typename T>
 void test2dRotation(double tol)
 {
@@ -53,7 +26,7 @@ void test2dRotation(double tol)
 
     C = Apow(std::ldexp(angle,1) / M_PI);
     std::cout << "test2dRotation: i = " << i << "   error powerm = " << relerr(C,B) << '\n';
-    VERIFY(C.isApprox(B, static_cast<T>(tol)));
+    VERIFY(C.isApprox(B, tol));
   }
 }
 
@@ -75,12 +48,26 @@ void test2dHyperbolicRotation(double tol)
 
     C = Apow(angle);
     std::cout << "test2dHyperbolicRotation: i = " << i << "   error powerm = " << relerr(C,B) << '\n';
-    VERIFY(C.isApprox(B, static_cast<T>(tol)));
+    VERIFY(C.isApprox(B, tol));
+  }
+}
+
+template<typename T>
+void test3dRotation(double tol)
+{
+  Matrix<T,3,1> v;
+  T angle;
+
+  for (int i=0; i<=20; ++i) {
+    v = Matrix<T,3,1>::Random();
+    v.normalize();
+    angle = pow(10, (i-10) / 5.);
+    VERIFY(AngleAxis<T>(angle, v).matrix().isApprox(AngleAxis<T>(1,v).matrix().pow(angle), tol));
   }
 }
 
 template<typename MatrixType>
-void testExponentLaws(const MatrixType& m, double tol)
+void testGeneral(const MatrixType& m, double tol)
 {
   typedef typename MatrixType::RealScalar RealScalar;
   MatrixType m1, m2, m3, m4, m5;
@@ -97,19 +84,64 @@ void testExponentLaws(const MatrixType& m, double tol)
 
     m4 = mpow(x+y);
     m5.noalias() = m2 * m3;
-    VERIFY(m4.isApprox(m5, static_cast<RealScalar>(tol)));
+    VERIFY(m4.isApprox(m5, tol));
 
     m4 = mpow(x*y);
     m5 = m2.pow(y);
-    VERIFY(m4.isApprox(m5, static_cast<RealScalar>(tol)));
+    VERIFY(m4.isApprox(m5, tol));
 
     m4 = (std::abs(x) * m1).pow(y);
     m5 = std::pow(std::abs(x), y) * m3;
-    VERIFY(m4.isApprox(m5, static_cast<RealScalar>(tol)));
+    VERIFY(m4.isApprox(m5, tol));
+  }
+}
+
+template<typename MatrixType>
+void testSingular(MatrixType m, double tol)
+{
+  const int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex;
+  typedef typename internal::conditional< IsComplex, MatrixSquareRootTriangular<MatrixType>,
+          MatrixSquareRootQuasiTriangular<MatrixType> >::type SquareRootType;
+  typedef typename internal::conditional<IsComplex, TriangularView<MatrixType,Upper>, const MatrixType&>::type TriangularType;
+  typename internal::conditional< IsComplex, ComplexSchur<MatrixType>, RealSchur<MatrixType> >::type schur;
+  MatrixType T;
+
+  for (int i=0; i < g_repeat; ++i) {
+    m.setRandom();
+    m.col(0).fill(0);
+
+    schur.compute(m);
+    T = schur.matrixT();
+    const MatrixType& U = schur.matrixU();
+    processTriangularMatrix<MatrixType>::run(m, T, U);
+    MatrixPower<MatrixType> mpow(m);
+
+    SquareRootType(T).compute(T);
+    VERIFY(mpow(0.5).isApprox(U * (TriangularType(T) * U.adjoint()), tol));
+
+    SquareRootType(T).compute(T);
+    VERIFY(mpow(0.25).isApprox(U * (TriangularType(T) * U.adjoint()), tol));
+
+    SquareRootType(T).compute(T);
+    VERIFY(mpow(0.125).isApprox(U * (TriangularType(T) * U.adjoint()), tol));
+  }
+}
+
+template<typename MatrixType>
+void testLogThenExp(MatrixType m, double tol)
+{
+  typedef typename MatrixType::Scalar Scalar;
+  Scalar x;
+
+  for (int i=0; i < g_repeat; ++i) {
+    generateTestMatrix<MatrixType>::run(m, m.rows());
+    x = internal::random<Scalar>();
+    VERIFY(m.pow(x).isApprox((x * m.log()).exp(), tol));
   }
 }
 
 typedef Matrix<double,3,3,RowMajor>         Matrix3dRowMajor;
+typedef Matrix<long double,3,3>             Matrix3e;
 typedef Matrix<long double,Dynamic,Dynamic> MatrixXe;
  
 void test_matrix_power()
@@ -121,13 +153,46 @@ void test_matrix_power()
   CALL_SUBTEST_1(test2dHyperbolicRotation<float>(1e-5));
   CALL_SUBTEST_9(test2dHyperbolicRotation<long double>(1e-14));
 
-  CALL_SUBTEST_2(testExponentLaws(Matrix2d(),         1e-13));
+  CALL_SUBTEST_10(test3dRotation<double>(1e-13));
-  CALL_SUBTEST_7(testExponentLaws(Matrix3dRowMajor(), 1e-13));
+  CALL_SUBTEST_11(test3dRotation<float>(1e-5));
-  CALL_SUBTEST_3(testExponentLaws(Matrix4cd(),        1e-13));
+  CALL_SUBTEST_12(test3dRotation<long double>(1e-13));
-  CALL_SUBTEST_4(testExponentLaws(MatrixXd(8,8),      2e-12));
+
-  CALL_SUBTEST_1(testExponentLaws(Matrix2f(),         1e-4));
+  CALL_SUBTEST_2(testGeneral(Matrix2d(),         1e-13));
-  CALL_SUBTEST_5(testExponentLaws(Matrix3cf(),        1e-4));
+  CALL_SUBTEST_7(testGeneral(Matrix3dRowMajor(), 1e-13));
-  CALL_SUBTEST_8(testExponentLaws(Matrix4f(),         1e-4));
+  CALL_SUBTEST_3(testGeneral(Matrix4cd(),        1e-13));
-  CALL_SUBTEST_6(testExponentLaws(MatrixXf(2,2),      1e-3)); // see bug 614
+  CALL_SUBTEST_4(testGeneral(MatrixXd(8,8),      2e-12));
-  CALL_SUBTEST_9(testExponentLaws(MatrixXe(7,7),      1e-13));
+  CALL_SUBTEST_1(testGeneral(Matrix2f(),         1e-4));
+  CALL_SUBTEST_5(testGeneral(Matrix3cf(),        1e-4));
+  CALL_SUBTEST_8(testGeneral(Matrix4f(),         1e-4));
+  CALL_SUBTEST_6(testGeneral(MatrixXf(2,2),      1e-3)); // see bug 614
+  CALL_SUBTEST_9(testGeneral(MatrixXe(7,7),      1e-13));
+  CALL_SUBTEST_10(testGeneral(Matrix3d(),        1e-13));
+  CALL_SUBTEST_11(testGeneral(Matrix3f(),        1e-4));
+  CALL_SUBTEST_12(testGeneral(Matrix3e(),        1e-13));
+
+  CALL_SUBTEST_2(testSingular(Matrix2d(),         1e-13));
+  CALL_SUBTEST_7(testSingular(Matrix3dRowMajor(), 1e-13));
+  CALL_SUBTEST_3(testSingular(Matrix4cd(),        1e-13));
+  CALL_SUBTEST_4(testSingular(MatrixXd(8,8),      2e-12));
+  CALL_SUBTEST_1(testSingular(Matrix2f(),         1e-4));
+  CALL_SUBTEST_5(testSingular(Matrix3cf(),        1e-4));
+  CALL_SUBTEST_8(testSingular(Matrix4f(),         1e-4));
+  CALL_SUBTEST_6(testSingular(MatrixXf(2,2),      1e-3));
+  CALL_SUBTEST_9(testSingular(MatrixXe(7,7),      1e-13));
+  CALL_SUBTEST_10(testSingular(Matrix3d(),        1e-13));
+  CALL_SUBTEST_11(testSingular(Matrix3f(),        1e-4));
+  CALL_SUBTEST_12(testSingular(Matrix3e(),        1e-13));
+
+  CALL_SUBTEST_2(testLogThenExp(Matrix2d(),         1e-13));
+  CALL_SUBTEST_7(testLogThenExp(Matrix3dRowMajor(), 1e-13));
+  CALL_SUBTEST_3(testLogThenExp(Matrix4cd(),        1e-13));
+  CALL_SUBTEST_4(testLogThenExp(MatrixXd(8,8),      2e-12));
+  CALL_SUBTEST_1(testLogThenExp(Matrix2f(),         1e-4));
+  CALL_SUBTEST_5(testLogThenExp(Matrix3cf(),        1e-4));
+  CALL_SUBTEST_8(testLogThenExp(Matrix4f(),         1e-4));
+  CALL_SUBTEST_6(testLogThenExp(MatrixXf(2,2),      1e-3));
+  CALL_SUBTEST_9(testLogThenExp(MatrixXe(7,7),      1e-13));
+  CALL_SUBTEST_10(testLogThenExp(Matrix3d(),        1e-13));
+  CALL_SUBTEST_11(testLogThenExp(Matrix3f(),        1e-4));
+  CALL_SUBTEST_12(testLogThenExp(Matrix3e(),        1e-13));
 }