Squashed 'third_party/eigen/' content from commit 61d72f6
Change-Id: Iccc90fa0b55ab44037f018046d2fcffd90d9d025
git-subtree-dir: third_party/eigen
git-subtree-split: 61d72f6383cfa842868c53e30e087b0258177257
diff --git a/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h b/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h
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+++ b/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h
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+// 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>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+#ifndef EIGEN_MATRIX_EXPONENTIAL
+#define EIGEN_MATRIX_EXPONENTIAL
+
+#include "StemFunction.h"
+
+namespace Eigen {
+
+/** \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 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);
+
+ /** \brief Computes the matrix exponential.
+ *
+ * \param[out] result the matrix exponential of \p M in the constructor.
+ */
+ template <typename ResultType>
+ void compute(ResultType &result);
+
+ private:
+
+ // Prevent copying
+ MatrixExponential(const MatrixExponential&);
+ MatrixExponential& operator=(const MatrixExponential&);
+
+ /** \brief Compute the (3,3)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ *
+ * \param[in] A Argument of matrix exponential
+ */
+ void pade3(const MatrixType &A);
+
+ /** \brief Compute the (5,5)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ *
+ * \param[in] A Argument of matrix exponential
+ */
+ void pade5(const MatrixType &A);
+
+ /** \brief Compute the (7,7)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ *
+ * \param[in] A Argument of matrix exponential
+ */
+ void pade7(const MatrixType &A);
+
+ /** \brief Compute the (9,9)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ *
+ * \param[in] A Argument of matrix exponential
+ */
+ void pade9(const MatrixType &A);
+
+ /** \brief Compute the (13,13)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ *
+ * \param[in] A Argument of matrix exponential
+ */
+ void pade13(const MatrixType &A);
+
+ /** \brief Compute the (17,17)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ *
+ * This function activates only if your long double is double-double or quadruple.
+ *
+ * \param[in] A Argument of matrix exponential
+ */
+ void pade17(const MatrixType &A);
+
+ /** \brief Compute Padé approximant to the exponential.
+ *
+ * Computes \c m_U, \c m_V and \c m_squarings such that
+ * \f$ (V+U)(V-U)^{-1} \f$ is a Padé of
+ * \f$ \exp(2^{-\mbox{squarings}}M) \f$ around \f$ M = 0 \f$. The
+ * degree of the Padé approximant and the value of
+ * squarings are chosen such that the approximation error is no
+ * more than the round-off error.
+ *
+ * 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 Padé approximant to the exponential.
+ *
+ * \sa computeUV(double);
+ */
+ void computeUV(float);
+
+ /** \brief Compute Padé 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 Padé approximant. */
+ MatrixType m_U;
+
+ /** \brief Even-degree terms in numerator of Padé 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)
+{
+#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
+}
+
+template <typename MatrixType>
+EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade3(const MatrixType &A)
+{
+ 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;
+}
+
+template <typename MatrixType>
+EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade5(const MatrixType &A)
+{
+ 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;
+}
+
+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)
+{
+ 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;
+}
+
+template <typename MatrixType>
+EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade13(const MatrixType &A)
+{
+ 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;
+}
+
+#if LDBL_MANT_DIG > 64
+template <typename MatrixType>
+EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade17(const MatrixType &A)
+{
+ 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;
+}
+#endif
+
+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) {
+ 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 / pow(Scalar(2), m_squarings);
+ pade7(A);
+ }
+}
+
+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) {
+ 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 / pow(Scalar(2), m_squarings);
+ pade13(A);
+ }
+}
+
+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
+ 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 / pow(Scalar(2), m_squarings);
+ pade13(A);
+ }
+#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);
+ }
+#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 / pow(Scalar(2), m_squarings);
+ pade17(A);
+ }
+#else
+ // this case should be handled in compute()
+ eigen_assert(false && "Bug in MatrixExponential");
+#endif // LDBL_MANT_DIG
+}
+
+/** \ingroup MatrixFunctions_Module
+ *
+ * \brief Proxy for the matrix exponential of some matrix (expression).
+ *
+ * \tparam Derived Type of the argument to the matrix exponential.
+ *
+ * This class holds the argument to the matrix exponential until it
+ * is assigned or evaluated for some other reason (so the argument
+ * should not be changed in the meantime). It is the return type of
+ * MatrixBase::exp() and most of the time this is the only way it is
+ * used.
+ */
+template<typename Derived> struct MatrixExponentialReturnValue
+: public ReturnByValue<MatrixExponentialReturnValue<Derived> >
+{
+ typedef typename Derived::Index Index;
+ public:
+ /** \brief Constructor.
+ *
+ * \param[in] 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.
+ */
+ 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);
+ }
+
+ Index rows() const { return m_src.rows(); }
+ Index cols() const { return m_src.cols(); }
+
+ protected:
+ const Derived& m_src;
+ private:
+ MatrixExponentialReturnValue& operator=(const MatrixExponentialReturnValue&);
+};
+
+namespace internal {
+template<typename Derived>
+struct traits<MatrixExponentialReturnValue<Derived> >
+{
+ typedef typename Derived::PlainObject ReturnType;
+};
+}
+
+template <typename Derived>
+const MatrixExponentialReturnValue<Derived> MatrixBase<Derived>::exp() const
+{
+ eigen_assert(rows() == cols());
+ return MatrixExponentialReturnValue<Derived>(derived());
+}
+
+} // end namespace Eigen
+
+#endif // EIGEN_MATRIX_EXPONENTIAL