Add support for matrix exponential of floats and complex numbers.

2 files changed, 290 insertions, 103 deletions

diff --git a/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h b/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h
index 39d2809d4..dd25d7f3d 100644
--- a/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h
+++ b/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h
@@ -25,13 +25,9 @@
 #ifndef EIGEN_MATRIX_EXPONENTIAL
 #define EIGEN_MATRIX_EXPONENTIAL
 
-#ifdef _MSC_VER
-template <typename Scalar> Scalar log2(Scalar v) { return std::log(v)/std::log(Scalar(2)); }
-#endif
-
-/** Compute the matrix exponential. 
+/** \brief Compute the matrix exponential. 
  *
- * \param M      matrix whose exponential is to be computed.
+ * \param M      matrix whose exponential is to be computed. 
  * \param result pointer to the matrix in which to store the result.
  *
  * The matrix exponential of \f$ M \f$ is defined by
@@ -58,103 +54,264 @@ template <typename Scalar> Scalar log2(Scalar v) { return std::log(v)/std::log(S
  * <em>SIAM J. %Matrix Anal. Applic.</em>, <b>26</b>:1179&ndash;1193,
  * 2005. 
  *
- * \note Currently, \p M has to be a matrix of \c double .
+ * \note \p M has to be a matrix of \c float, \c double, 
+ * \c complex<float> or \c complex<double> .
  */
 template <typename Derived>
-void ei_matrix_exponential(const MatrixBase<Derived> &M, typename ei_plain_matrix_type<Derived>::type* result)
-{
-  typedef typename ei_traits<Derived>::Scalar Scalar;
-  typedef typename NumTraits<Scalar>::Real RealScalar;
-  typedef typename ei_plain_matrix_type<Derived>::type PlainMatrixType;
+EIGEN_STRONG_INLINE void ei_matrix_exponential(const MatrixBase<Derived> &M, 
+					       typename MatrixBase<Derived>::PlainMatrixType* result);
 
-  ei_assert(M.rows() == M.cols());
-  EIGEN_STATIC_ASSERT(NumTraits<Scalar>::HasFloatingPoint,NUMERIC_TYPE_MUST_BE_FLOATING_POINT)
 
-  PlainMatrixType num, den, U, V;
-  PlainMatrixType Id = PlainMatrixType::Identity(M.rows(), M.cols());
-  typename ei_eval<Derived>::type Meval = M.eval();
-  RealScalar l1norm = Meval.cwise().abs().colwise().sum().maxCoeff();
-  int squarings = 0;
-  
-  // Choose degree of Pade approximant, depending on norm of M
-  if (l1norm < 1.495585217958292e-002) {
-    
-    // Use (3,3)-Pade
-    const Scalar b[] = {120., 60., 12., 1.};
-    PlainMatrixType M2;
-    M2 = (Meval * Meval).lazy();
-    num = b[3]*M2 + b[1]*Id;
-    U = (Meval * num).lazy();
-    V = b[2]*M2 + b[0]*Id;
+/** \internal \brief Internal helper functions for computing the
+ *  matrix exponential.
+ */
+namespace MatrixExponentialInternal {
 
-  } else if (l1norm < 2.539398330063230e-001) {
+#ifdef _MSC_VER
+  template <typename Scalar> Scalar log2(Scalar v) { return std::log(v)/std::log(Scalar(2)); }
+#endif
 
-    // Use (5,5)-Pade
+  /** \internal \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(M) \f$ around \f$ M = 0 \f$.
+   *
+   *  \param M   Argument of matrix exponential
+   *  \param Id  Identity matrix of same size as M
+   *  \param tmp Temporary storage, to be provided by the caller
+   *  \param M2  Temporary storage, to be provided by the caller
+   *  \param U   Even-degree terms in numerator of Pad&eacute; approximant
+   *  \param V   Odd-degree terms in numerator of Pad&eacute; approximant
+   */
+  template <typename MatrixType>
+  EIGEN_STRONG_INLINE void pade3(const MatrixType &M, const MatrixType& Id, MatrixType& tmp, 
+				 MatrixType& M2, MatrixType& U, MatrixType& V)
+  {
+    typedef typename ei_traits<MatrixType>::Scalar Scalar;
+    const Scalar b[] = {120., 60., 12., 1.};
+    M2 = (M * M).lazy();
+    tmp = b[3]*M2 + b[1]*Id;
+    U = (M * tmp).lazy();
+    V = b[2]*M2 + b[0]*Id;
+  }
+  
+  /** \internal \brief Compute the (5,5)-Pad&eacute; approximant to
+   *  the exponential.
+   *  
+   *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
+   *  approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
+   *
+   *  \param M   Argument of matrix exponential
+   *  \param Id  Identity matrix of same size as M
+   *  \param tmp Temporary storage, to be provided by the caller
+   *  \param M2  Temporary storage, to be provided by the caller
+   *  \param U   Even-degree terms in numerator of Pad&eacute; approximant
+   *  \param V   Odd-degree terms in numerator of Pad&eacute; approximant
+   */
+  template <typename MatrixType>
+  EIGEN_STRONG_INLINE void pade5(const MatrixType &M, const MatrixType& Id, MatrixType& tmp, 
+				 MatrixType& M2, MatrixType& U, MatrixType& V)
+  {
+    typedef typename ei_traits<MatrixType>::Scalar Scalar;
     const Scalar b[] = {30240., 15120., 3360., 420., 30., 1.};
-    PlainMatrixType M2, M4;
-    M2 = (Meval * Meval).lazy();
-    M4 = (M2 * M2).lazy();
-    num = b[5]*M4 + b[3]*M2 + b[1]*Id;
-    U = (Meval * num).lazy();
+    M2 = (M * M).lazy();
+    MatrixType M4 = (M2 * M2).lazy();
+    tmp = b[5]*M4 + b[3]*M2 + b[1]*Id;
+    U = (M * tmp).lazy();
     V = b[4]*M4 + b[2]*M2 + b[0]*Id;
-
-  } else if (l1norm < 9.504178996162932e-001) {
-
-    // Use (7,7)-Pade
+  }
+  
+  /** \internal \brief Compute the (7,7)-Pad&eacute; approximant to
+   *  the exponential.
+   *  
+   *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
+   *  approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
+   *
+   *  \param M   Argument of matrix exponential
+   *  \param Id  Identity matrix of same size as M
+   *  \param tmp Temporary storage, to be provided by the caller
+   *  \param M2  Temporary storage, to be provided by the caller
+   *  \param U   Even-degree terms in numerator of Pad&eacute; approximant
+   *  \param V   Odd-degree terms in numerator of Pad&eacute; approximant
+   */
+  template <typename MatrixType>
+  EIGEN_STRONG_INLINE void pade7(const MatrixType &M, const MatrixType& Id, MatrixType& tmp, 
+				 MatrixType& M2, MatrixType& U, MatrixType& V)
+  {
+    typedef typename ei_traits<MatrixType>::Scalar Scalar;
     const Scalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.};
-    PlainMatrixType M2, M4, M6;
-    M2 = (Meval * Meval).lazy();
-    M4 = (M2 * M2).lazy();
-    M6 = (M4 * M2).lazy();
-    num = b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
-    U = (Meval * num).lazy();
+    M2 = (M * M).lazy();
+    MatrixType M4 = (M2 * M2).lazy();
+    MatrixType M6 = (M4 * M2).lazy();
+    tmp = b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
+    U = (M * tmp).lazy();
     V = b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id;
-
-  } else if (l1norm < 2.097847961257068e+000) {
-
-    // Use (9,9)-Pade
+  }
+  
+  /** \internal \brief Compute the (9,9)-Pad&eacute; approximant to
+   *  the exponential.
+   *  
+   *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
+   *  approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
+   *
+   *  \param M   Argument of matrix exponential
+   *  \param Id  Identity matrix of same size as M
+   *  \param tmp Temporary storage, to be provided by the caller
+   *  \param M2  Temporary storage, to be provided by the caller
+   *  \param U   Even-degree terms in numerator of Pad&eacute; approximant
+   *  \param V   Odd-degree terms in numerator of Pad&eacute; approximant
+   */
+  template <typename MatrixType>
+  EIGEN_STRONG_INLINE void pade9(const MatrixType &M, const MatrixType& Id, MatrixType& tmp, 
+				 MatrixType& M2, MatrixType& U, MatrixType& V)
+  {
+    typedef typename ei_traits<MatrixType>::Scalar Scalar;
     const Scalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240.,
-                         2162160., 110880., 3960., 90., 1.};
-    PlainMatrixType M2, M4, M6, M8;
-    M2 = (Meval * Meval).lazy();
-    M4 = (M2 * M2).lazy();
-    M6 = (M4 * M2).lazy();
-    M8 = (M6 * M2).lazy();
-    num = b[9]*M8 + b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
-    U = (Meval * num).lazy();
+  		      2162160., 110880., 3960., 90., 1.};
+    M2 = (M * M).lazy();
+    MatrixType M4 = (M2 * M2).lazy();
+    MatrixType M6 = (M4 * M2).lazy();
+    MatrixType M8 = (M6 * M2).lazy();
+    tmp = b[9]*M8 + b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
+    U = (M * tmp).lazy();
     V = b[8]*M8 + b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id;
-
-  } else {
-
-    // Use (13,13)-Pade; scale matrix by power of 2 so that its norm
-    // is small enough 
-
-    const Scalar maxnorm = 5.371920351148152;
+  }
+  
+  /** \internal \brief Compute the (13,13)-Pad&eacute; approximant to
+   *  the exponential.
+   *  
+   *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
+   *  approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
+   *
+   *  \param M   Argument of matrix exponential
+   *  \param Id  Identity matrix of same size as M
+   *  \param tmp Temporary storage, to be provided by the caller
+   *  \param M2  Temporary storage, to be provided by the caller
+   *  \param U   Even-degree terms in numerator of Pad&eacute; approximant
+   *  \param V   Odd-degree terms in numerator of Pad&eacute; approximant
+   */
+  template <typename MatrixType>
+  EIGEN_STRONG_INLINE void pade13(const MatrixType &M, const MatrixType& Id, MatrixType& tmp, 
+				  MatrixType& M2, MatrixType& U, MatrixType& V)
+  {
+    typedef typename ei_traits<MatrixType>::Scalar Scalar;
     const Scalar b[] = {64764752532480000., 32382376266240000., 7771770303897600., 
-			1187353796428800., 129060195264000., 10559470521600., 670442572800., 
-			33522128640., 1323241920., 40840800., 960960., 16380., 182., 1.};
-    
-    squarings = std::max(0, (int)ceil(log2(l1norm / maxnorm)));
-    PlainMatrixType A, A2, A4, A6;
-    A = Meval / pow(Scalar(2), squarings);
-    A2 = (A * A).lazy();
-    A4 = (A2 * A2).lazy();
-    A6 = (A4 * A2).lazy();
-    num = b[13]*A6 + b[11]*A4 + b[9]*A2;
-    V = (A6 * num).lazy();
-    num = V + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*Id;
-    U = (A * num).lazy();
-    num = b[12]*A6 + b[10]*A4 + b[8]*A2;
-    V = (A6 * num).lazy() + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*Id;
+  		      1187353796428800., 129060195264000., 10559470521600., 670442572800., 
+  		      33522128640., 1323241920., 40840800., 960960., 16380., 182., 1.};
+    M2 = (M * M).lazy();
+    MatrixType M4 = (M2 * M2).lazy();
+    MatrixType M6 = (M4 * M2).lazy();
+    V = b[13]*M6 + b[11]*M4 + b[9]*M2;
+    tmp = (M6 * V).lazy();
+    tmp += b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
+    U = (M * tmp).lazy();
+    tmp = b[12]*M6 + b[10]*M4 + b[8]*M2;
+    V = (M6 * tmp).lazy();
+    V += b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id;
+  }
+  
+  /** \internal \brief Helper class for computing Pad&eacute;
+   *  approximants to the exponential.
+   */
+  template <typename MatrixType, typename RealScalar = typename NumTraits<typename ei_traits<MatrixType>::Scalar>::Real>
+  struct computeUV_selector
+  {
+    /** \internal \brief Compute Pad&eacute; approximant to the exponential. 
+     *  
+     *  Computes \p U, \p V and \p 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.
+     *
+     *  \param M         Argument of matrix exponential
+     *  \param Id        Identity matrix of same size as M
+     *  \param tmp1      Temporary storage, to be provided by the caller
+     *  \param tmp2      Temporary storage, to be provided by the caller
+     *  \param U         Even-degree terms in numerator of Pad&eacute; approximant
+     *  \param V         Odd-degree terms in numerator of Pad&eacute; approximant
+     *  \param l1norm    L<sub>1</sub> norm of M
+     *  \param squarings Pointer to integer containing number of times
+     *                   that the result needs to be squared to find the
+     *                   matrix exponential 
+     */
+    static void run(const MatrixType &M, const MatrixType& Id, MatrixType& tmp1, MatrixType& tmp2, 
+		    MatrixType& U, MatrixType& V, float l1norm, int* squarings);
+  };
+  
+  template <typename MatrixType>
+  struct computeUV_selector<MatrixType, float>
+  {
+    static void run(const MatrixType &M, const MatrixType& Id, MatrixType& tmp1, MatrixType& tmp2, 
+		    MatrixType& U, MatrixType& V, float l1norm, int* squarings)
+    {
+      *squarings = 0;
+      if (l1norm < 4.258730016922831e-001) {
+        pade3(M, Id, tmp1, tmp2, U, V);
+      } else if (l1norm < 1.880152677804762e+000) {
+        pade5(M, Id, tmp1, tmp2, U, V);
+      } else {
+        const float maxnorm = 3.925724783138660;
+        *squarings = std::max(0, (int)ceil(log2(l1norm / maxnorm)));
+        MatrixType A = M / std::pow(typename ei_traits<MatrixType>::Scalar(2), *squarings);
+        pade7(A, Id, tmp1, tmp2, U, V);
+      }
+    }
+  };
+  
+  template <typename MatrixType>
+  struct computeUV_selector<MatrixType, double>
+  {
+    static void run(const MatrixType &M, const MatrixType& Id, MatrixType& tmp1, MatrixType& tmp2, 
+		    MatrixType& U, MatrixType& V, float l1norm, int* squarings)
+    {
+      *squarings = 0;
+      if (l1norm < 1.495585217958292e-002) {
+        pade3(M, Id, tmp1, tmp2, U, V);
+      } else if (l1norm < 2.539398330063230e-001) {
+        pade5(M, Id, tmp1, tmp2, U, V);
+      } else if (l1norm < 9.504178996162932e-001) {
+        pade7(M, Id, tmp1, tmp2, U, V);
+      } else if (l1norm < 2.097847961257068e+000) {
+        pade9(M, Id, tmp1, tmp2, U, V);
+      } else {
+        const double maxnorm = 5.371920351148152;
+        *squarings = std::max(0, (int)ceil(log2(l1norm / maxnorm)));
+        MatrixType A = M / std::pow(typename ei_traits<MatrixType>::Scalar(2), *squarings);
+        pade13(A, Id, tmp1, tmp2, U, V);
+      }
+    }
+  };
+  
+  /** \internal \brief Compute the matrix exponential. 
+   *
+   * \param M      matrix whose exponential is to be computed. 
+   * \param result pointer to the matrix in which to store the result.
+   */
+  template <typename MatrixType>
+  void compute(const MatrixType &M, MatrixType* result)
+  {
+    MatrixType num, den, U, V;
+    MatrixType Id = MatrixType::Identity(M.rows(), M.cols());
+    float l1norm = M.cwise().abs().colwise().sum().maxCoeff();
+    int squarings;
+    computeUV_selector<MatrixType>::run(M, Id, num, den, U, V, l1norm, &squarings);
+    num = U + V;			// numerator of Pade approximant
+    den = -U + V;			// denominator of Pade approximant
+    den.partialLu().solve(num, result);
+    for (int i=0; i<squarings; i++)
+      *result *= *result;		// undo scaling by repeated squaring
   }
 
-  num = U + V;			// numerator of Pade approximant
-  den = -U + V;			// denominator of Pade approximant
-  den.lu().solve(num, result);
+} // end of namespace MatrixExponentialInternal
 
-  // Undo scaling by repeated squaring
-  for (int i=0; i<squarings; i++)
-    *result *= *result;
+template <typename Derived>
+EIGEN_STRONG_INLINE void ei_matrix_exponential(const MatrixBase<Derived> &M, 
+					       typename MatrixBase<Derived>::PlainMatrixType* result)
+{
+  ei_assert(M.rows() == M.cols());
+  MatrixExponentialInternal::compute(M.eval(), result);
 }
 
 #endif // EIGEN_MATRIX_EXPONENTIAL
diff --git a/unsupported/test/matrixExponential.cpp b/unsupported/test/matrixExponential.cpp
index 0ec4fae90..9a6c335d8 100644
--- a/unsupported/test/matrixExponential.cpp
+++ b/unsupported/test/matrixExponential.cpp
@@ -34,26 +34,47 @@ double binom(int n, int k)
   return res;
 }
 
-void test2dRotation()
+template <typename T>
+void test2dRotation(double tol)
 {
-  Matrix2d A, B, C;
-  double angle;
+  Matrix<T,2,2> A, B, C;
+  T angle;
 
+  A << 0, 1, -1, 0;
   for (int i=0; i<=20; i++) 
   {
     angle = pow(10, i / 5. - 2);
-    A << 0, angle, -angle, 0;
     B << cos(angle), sin(angle), -sin(angle), cos(angle);
+    ei_matrix_exponential(angle*A, &C);
+    VERIFY(C.isApprox(B, tol));
+  }
+}
+
+template <typename T>
+void test2dHyperbolicRotation(double tol)
+{
+  Matrix<std::complex<T>,2,2> A, B, C;
+  std::complex<T> imagUnit(0,1);
+  T angle, ch, sh;
+
+  for (int i=0; i<=20; i++) 
+  {
+    angle = (i-10) / 2.0;
+    ch = std::cosh(angle);
+    sh = std::sinh(angle);
+    A << 0, angle*imagUnit, -angle*imagUnit, 0;
+    B << ch, sh*imagUnit, -sh*imagUnit, ch;
     ei_matrix_exponential(A, &C);
-    VERIFY(C.isApprox(B, 1e-14));
+    VERIFY(C.isApprox(B, tol));
   }
 }
 
-void testPascal()
+template <typename T>
+void testPascal(double tol)
 {
   for (int size=1; size<20; size++)
   {
-    MatrixXd A(size,size), B(size,size), C(size,size);
+    Matrix<T,Dynamic,Dynamic> A(size,size), B(size,size), C(size,size);
     A.setZero();
     for (int i=0; i<size-1; i++)
       A(i+1,i) = i+1;
@@ -62,11 +83,12 @@ void testPascal()
       for (int j=0; j<=i; j++)
 	B(i,j) = binom(i,j);
     ei_matrix_exponential(A, &C);
-    VERIFY(C.isApprox(B, 1e-14));
+    VERIFY(C.isApprox(B, tol));
   }
 }
 
-template<typename MatrixType> void randomTest(const MatrixType& m)
+template<typename MatrixType> 
+void randomTest(const MatrixType& m, double tol)
 {
   /* this test covers the following files:
      Inverse.h
@@ -80,16 +102,24 @@ template<typename MatrixType> void randomTest(const MatrixType& m)
     m1 = MatrixType::Random(rows, cols);
     ei_matrix_exponential(m1, &m2);
     ei_matrix_exponential(-m1, &m3);
-    VERIFY(identity.isApprox(m2 * m3, 1e-13));
+    VERIFY(identity.isApprox(m2 * m3, tol));
   }
 }
 
 void test_matrixExponential()
 {
-  CALL_SUBTEST(test2dRotation());
-  CALL_SUBTEST(testPascal());
-  CALL_SUBTEST(randomTest(Matrix2d()));
-  CALL_SUBTEST(randomTest(Matrix3d()));
-  CALL_SUBTEST(randomTest(Matrix4d()));
-  CALL_SUBTEST(randomTest(MatrixXd(8,8)));
+  CALL_SUBTEST(test2dRotation<double>(1e-14));
+  CALL_SUBTEST(test2dRotation<float>(1e-5));
+  CALL_SUBTEST(test2dHyperbolicRotation<double>(1e-14));
+  CALL_SUBTEST(test2dHyperbolicRotation<float>(1e-5));
+  CALL_SUBTEST(testPascal<float>(1e-5));
+  CALL_SUBTEST(testPascal<double>(1e-14));
+  CALL_SUBTEST(randomTest(Matrix2d(), 1e-13));
+  CALL_SUBTEST(randomTest(Matrix<double,3,3,RowMajor>(), 1e-13));
+  CALL_SUBTEST(randomTest(Matrix4cd(), 1e-13));
+  CALL_SUBTEST(randomTest(MatrixXd(8,8), 1e-13));
+  CALL_SUBTEST(randomTest(Matrix2f(), 1e-4));
+  CALL_SUBTEST(randomTest(Matrix3cf(), 1e-4));
+  CALL_SUBTEST(randomTest(Matrix4f(), 1e-4));
+  CALL_SUBTEST(randomTest(MatrixXf(8,8), 1e-4));
 }