new unsupported module by Jitse Niesen: matrix exponential

unsupported/Eigen/CMakeLists.txt

@@ -1,4 +1,4 @@
-set(Eigen_HEADERS AdolcForward BVH IterativeSolvers MoreVectorization AutoDiff AlignedVector3)
+set(Eigen_HEADERS AdolcForward BVH IterativeSolvers MatrixFunctions MoreVectorization AutoDiff AlignedVector3)
 
 install(FILES
   ${Eigen_HEADERS}

unsupported/Eigen/MatrixFunctions

@@ -0,0 +1,54 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra. Eigen itself is part of the KDE project.
+//
+// Copyright (C) 2009 Jitse Niesen <jitse@maths.leeds.ac.uk>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#ifndef EIGEN_MATRIX_FUNCTIONS
+#define EIGEN_MATRIX_FUNCTIONS
+
+#include <Eigen/Core>
+#include <Eigen/Array>
+#include <Eigen/LU>
+
+namespace Eigen {
+
+/** \ingroup Unsupported_modules
+  * \defgroup MatrixFunctions_Module Matrix functions module
+  * \brief This module aims to provide various methods for the computation of
+  * matrix functions. Currently, there is only support for the matrix
+  * exponential. 
+  *
+  * \code
+  * #include <unsupported/Eigen/MatrixFunctions>
+  * \endcode
+  */
+//@{
+
+#include "src/MatrixFunctions/MatrixExponential.h"
+
+//@}
+
+}
+
+
+#endif // EIGEN_MATRIX_FUNCTIONS
+

unsupported/Eigen/src/MatrixFunctions/CMakeLists.txt

@@ -0,0 +1,6 @@
+FILE(GLOB Eigen_MatrixFunctions_SRCS "*.h")
+
+INSTALL(FILES
+  ${Eigen_MatrixFunctions_SRCS}
+  DESTINATION ${INCLUDE_INSTALL_DIR}/unsupported/Eigen/src/MatrixFunctions COMPONENT Devel
+  )

unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h

@@ -0,0 +1,155 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra. Eigen itself is part of the KDE project.
+//
+// Copyright (C) 2009 Jitse Niesen <jitse@maths.leeds.ac.uk>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#ifndef EIGEN_MATRIX_EXPONENTIAL
+#define EIGEN_MATRIX_EXPONENTIAL
+
+/** 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.
+ *
+ * The matrix exponential of \f$ M \f$ is defined by
+ * \f[ \exp(M) = \sum_{k=0}^\infty \frac{M^k}{k!}. \f]
+ * The matrix exponential can be used to solve linear ordinary
+ * differential equations: the solution of \f$ y' = My \f$ with the
+ * initial condition \f$ y(0) = y_0 \f$ is given by 
+ * \f$ y(t) = \exp(M) y_0 \f$.
+ *
+ * The cost of the computation is approximately \f$ 20 n^3 \f$ for
+ * matrices of size \f$ n \f$. The number 20 depends weakly on the
+ * norm of the matrix.
+ *
+ * The matrix exponential is computed using the scaling-and-squaring
+ * method combined with Pad&eacute; approximation. The matrix is first
+ * rescaled, then the exponential of the reduced matrix is computed
+ * approximant, and then the rescaling is undone by repeated
+ * squaring. The degree of the Pad&eacute; approximant is chosen such
+ * that the approximation error is less than the round-off
+ * error. However, errors may accumulate during the squaring phase.
+ * 
+ * Details of the algorithm can be found in: Nicholas J. Higham, "The
+ * scaling and squaring method for the matrix exponential revisited,"
+ * <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 .
+ */
+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;
+
+  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());
+  RealScalar l1norm = M.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 = (M * M).lazy();
+    num = b[3]*M2 + b[1]*Id;
+    U = (M * num).lazy();
+    V = b[2]*M2 + b[0]*Id;
+
+  } else if (l1norm < 2.539398330063230e-001) {
+
+    // Use (5,5)-Pade
+    const Scalar b[] = {30240., 15120., 3360., 420., 30., 1.};
+    PlainMatrixType M2, M4;
+    M2 = (M * M).lazy();
+    M4 = (M2 * M2).lazy();
+    num = b[5]*M4 + b[3]*M2 + b[1]*Id;
+    U = (M * num).lazy();
+    V = b[4]*M4 + b[2]*M2 + b[0]*Id;
+
+  } else if (l1norm < 9.504178996162932e-001) {
+
+    // Use (7,7)-Pade
+    const Scalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.};
+    PlainMatrixType M2, M4, M6;
+    M2 = (M * M).lazy();
+    M4 = (M2 * M2).lazy();
+    M6 = (M4 * M2).lazy();
+    num = b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
+    U = (M * num).lazy();
+    V = b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id;
+
+  } else if (l1norm < 2.097847961257068e+000) {
+
+    // Use (9,9)-Pade
+    const Scalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240.,
+                         2162160., 110880., 3960., 90., 1.};
+    PlainMatrixType M2, M4, M6, M8;
+    M2 = (M * M).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 = (M * num).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;
+    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 = M / pow(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;
+  }
+
+  num = U + V;			// numerator of Pade approximant
+  den = -U + V;			// denominator of Pade approximant
+  den.lu().solve(num, result);
+
+  // Undo scaling by repeated squaring
+  for (int i=0; i<squarings; i++)
+    *result *= *result;
+}
+
+#endif // EIGEN_MATRIX_EXPONENTIAL

unsupported/test/CMakeLists.txt

@@ -17,4 +17,5 @@ endif(ADOLC_FOUND)
 
 ei_add_test(autodiff)
 ei_add_test(BVH)
+ei_add_test(matrixExponential)
 ei_add_test(alignedvector3)

unsupported/test/matrixExponential.cpp

@@ -0,0 +1,95 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra. Eigen itself is part of the KDE project.
+//
+// Copyright (C) 2009 Jitse Niesen <jitse@maths.leeds.ac.uk>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#include <Eigen/StdVector>
+#include "main.h"
+#include <unsupported/Eigen/MatrixFunctions>
+
+double binom(int n, int k) 
+{
+  double res = 1;
+  for (int i=0; i<k; i++)
+    res = res * (n-k+i+1) / (i+1);
+  return res;
+}
+
+void test2dRotation()
+{
+  Matrix2d A, B, C;
+  double angle;
+
+  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(A, &C);
+    VERIFY(C.isApprox(B, 1e-14));
+  }
+}
+
+void testPascal()
+{
+  for (int size=1; size<20; size++)
+  {
+    MatrixXd 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;
+    B.setZero();
+    for (int i=0; i<size; i++)
+      for (int j=0; j<=i; j++)
+	B(i,j) = binom(i,j);
+    ei_matrix_exponential(A, &C);
+    VERIFY(C.isApprox(B, 1e-14));
+  }
+}
+
+template<typename MatrixType> void randomTest(const MatrixType& m)
+{
+  /* this test covers the following files:
+     Inverse.h
+  */
+  int rows = m.rows();
+  int cols = m.cols();
+  MatrixType m1(rows, cols), m2(rows, cols), m3(rows, cols),
+             identity = MatrixType::Identity(rows, rows);
+
+  for(int i = 0; i < g_repeat; i++) {
+    m1 = MatrixType::Random(rows, cols);
+    ei_matrix_exponential(m1, &m2);
+    ei_matrix_exponential(-m1, &m3);
+    VERIFY(identity.isApprox(m2 * m3, 1e-13));
+  }
+}
+
+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)));
+}