Squashed 'third_party/eigen/' changes from 61d72f6..cf794d3
Change-Id: I9b814151b01f49af6337a8605d0c42a3a1ed4c72
git-subtree-dir: third_party/eigen
git-subtree-split: cf794d3b741a6278df169e58461f8529f43bce5d
diff --git a/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h b/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h
index 6825a78..e5ebbcf 100644
--- a/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h
+++ b/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h
@@ -1,8 +1,8 @@
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
-// Copyright (C) 2009, 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
-// Copyright (C) 2011 Chen-Pang He <jdh8@ms63.hinet.net>
+// Copyright (C) 2009, 2010, 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
+// Copyright (C) 2011, 2013 Chen-Pang He <jdh8@ms63.hinet.net>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
@@ -14,376 +14,374 @@
#include "StemFunction.h"
namespace Eigen {
+namespace internal {
-/** \ingroup MatrixFunctions_Module
- * \brief Class for computing the matrix exponential.
- * \tparam MatrixType type of the argument of the exponential,
- * expected to be an instantiation of the Matrix class template.
- */
-template <typename MatrixType>
-class MatrixExponential {
+/** \brief Scaling operator.
+ *
+ * This struct is used by CwiseUnaryOp to scale a matrix by \f$ 2^{-s} \f$.
+ */
+template <typename RealScalar>
+struct MatrixExponentialScalingOp
+{
+ /** \brief Constructor.
+ *
+ * \param[in] squarings The integer \f$ s \f$ in this document.
+ */
+ MatrixExponentialScalingOp(int squarings) : m_squarings(squarings) { }
- 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 Scale a matrix coefficient.
+ *
+ * \param[in,out] x The scalar to be scaled, becoming \f$ 2^{-s} x \f$.
+ */
+ inline const RealScalar operator() (const RealScalar& x) const
+ {
+ using std::ldexp;
+ return ldexp(x, -m_squarings);
+ }
- /** \brief Computes the matrix exponential.
- *
- * \param[out] result the matrix exponential of \p M in the constructor.
- */
- template <typename ResultType>
- void compute(ResultType &result);
+ typedef std::complex<RealScalar> ComplexScalar;
+
+ /** \brief Scale a matrix coefficient.
+ *
+ * \param[in,out] x The scalar to be scaled, becoming \f$ 2^{-s} x \f$.
+ */
+ inline const ComplexScalar operator() (const ComplexScalar& x) const
+ {
+ using std::ldexp;
+ return ComplexScalar(ldexp(x.real(), -m_squarings), ldexp(x.imag(), -m_squarings));
+ }
private:
-
- // Prevent copying
- MatrixExponential(const MatrixExponential&);
- MatrixExponential& operator=(const MatrixExponential&);
-
- /** \brief Compute the (3,3)-Padé approximant to the exponential.
- *
- * After exit, \f$ (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$. The
- * degree of the Padé approximant and the value of
- * squarings are chosen such that the approximation error is no
- * more than the round-off error.
- *
- * The argument of this function should correspond with the (real
- * part of) the entries of \c m_M. It is used to select the
- * correct implementation using overloading.
- */
- void computeUV(double);
-
- /** \brief Compute Padé approximant to the exponential.
- *
- * \sa computeUV(double);
- */
- void computeUV(float);
-
- /** \brief Compute Padé approximant to the exponential.
- *
- * \sa computeUV(double);
- */
- void computeUV(long double);
-
- typedef typename internal::traits<MatrixType>::Scalar Scalar;
- typedef typename NumTraits<Scalar>::Real RealScalar;
- typedef typename std::complex<RealScalar> ComplexScalar;
-
- /** \brief Reference to matrix whose exponential is to be computed. */
- typename internal::nested<MatrixType>::type m_M;
-
- /** \brief Odd-degree terms in numerator of Padé approximant. */
- MatrixType m_U;
-
- /** \brief Even-degree terms in numerator of Padé approximant. */
- MatrixType m_V;
-
- /** \brief Used for temporary storage. */
- MatrixType m_tmp1;
-
- /** \brief Used for temporary storage. */
- MatrixType m_tmp2;
-
- /** \brief Identity matrix of the same size as \c m_M. */
- MatrixType m_Id;
-
- /** \brief Number of squarings required in the last step. */
int m_squarings;
+};
- /** \brief L1 norm of m_M. */
- RealScalar m_l1norm;
+/** \brief Compute the (3,3)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ */
+template <typename MatA, typename MatU, typename MatV>
+void matrix_exp_pade3(const MatA& A, MatU& U, MatV& V)
+{
+ typedef typename MatA::PlainObject MatrixType;
+ typedef typename NumTraits<typename traits<MatA>::Scalar>::Real RealScalar;
+ const RealScalar b[] = {120.L, 60.L, 12.L, 1.L};
+ const MatrixType A2 = A * A;
+ const MatrixType tmp = b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
+ U.noalias() = A * tmp;
+ V = b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
+}
+
+/** \brief Compute the (5,5)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ */
+template <typename MatA, typename MatU, typename MatV>
+void matrix_exp_pade5(const MatA& A, MatU& U, MatV& V)
+{
+ typedef typename MatA::PlainObject MatrixType;
+ typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
+ const RealScalar b[] = {30240.L, 15120.L, 3360.L, 420.L, 30.L, 1.L};
+ const MatrixType A2 = A * A;
+ const MatrixType A4 = A2 * A2;
+ const MatrixType tmp = b[5] * A4 + b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
+ U.noalias() = A * tmp;
+ V = b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
+}
+
+/** \brief Compute the (7,7)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ */
+template <typename MatA, typename MatU, typename MatV>
+void matrix_exp_pade7(const MatA& A, MatU& U, MatV& V)
+{
+ typedef typename MatA::PlainObject MatrixType;
+ typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
+ const RealScalar b[] = {17297280.L, 8648640.L, 1995840.L, 277200.L, 25200.L, 1512.L, 56.L, 1.L};
+ const MatrixType A2 = A * A;
+ const MatrixType A4 = A2 * A2;
+ const MatrixType A6 = A4 * A2;
+ const MatrixType tmp = b[7] * A6 + b[5] * A4 + b[3] * A2
+ + b[1] * MatrixType::Identity(A.rows(), A.cols());
+ U.noalias() = A * tmp;
+ V = b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
+
+}
+
+/** \brief Compute the (9,9)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ */
+template <typename MatA, typename MatU, typename MatV>
+void matrix_exp_pade9(const MatA& A, MatU& U, MatV& V)
+{
+ typedef typename MatA::PlainObject MatrixType;
+ typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
+ const RealScalar b[] = {17643225600.L, 8821612800.L, 2075673600.L, 302702400.L, 30270240.L,
+ 2162160.L, 110880.L, 3960.L, 90.L, 1.L};
+ const MatrixType A2 = A * A;
+ const MatrixType A4 = A2 * A2;
+ const MatrixType A6 = A4 * A2;
+ const MatrixType A8 = A6 * A2;
+ const MatrixType tmp = b[9] * A8 + b[7] * A6 + b[5] * A4 + b[3] * A2
+ + b[1] * MatrixType::Identity(A.rows(), A.cols());
+ U.noalias() = A * tmp;
+ V = b[8] * A8 + b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
+}
+
+/** \brief Compute the (13,13)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ */
+template <typename MatA, typename MatU, typename MatV>
+void matrix_exp_pade13(const MatA& A, MatU& U, MatV& V)
+{
+ typedef typename MatA::PlainObject MatrixType;
+ typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
+ const RealScalar b[] = {64764752532480000.L, 32382376266240000.L, 7771770303897600.L,
+ 1187353796428800.L, 129060195264000.L, 10559470521600.L, 670442572800.L,
+ 33522128640.L, 1323241920.L, 40840800.L, 960960.L, 16380.L, 182.L, 1.L};
+ const MatrixType A2 = A * A;
+ const MatrixType A4 = A2 * A2;
+ const MatrixType A6 = A4 * A2;
+ V = b[13] * A6 + b[11] * A4 + b[9] * A2; // used for temporary storage
+ MatrixType tmp = A6 * V;
+ tmp += b[7] * A6 + b[5] * A4 + b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
+ U.noalias() = A * tmp;
+ tmp = b[12] * A6 + b[10] * A4 + b[8] * A2;
+ V.noalias() = A6 * tmp;
+ V += b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
+}
+
+/** \brief Compute the (17,17)-Padé approximant to the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
+ * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
+ *
+ * This function activates only if your long double is double-double or quadruple.
+ */
+#if LDBL_MANT_DIG > 64
+template <typename MatA, typename MatU, typename MatV>
+void matrix_exp_pade17(const MatA& A, MatU& U, MatV& V)
+{
+ typedef typename MatA::PlainObject MatrixType;
+ typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
+ const RealScalar b[] = {830034394580628357120000.L, 415017197290314178560000.L,
+ 100610229646136770560000.L, 15720348382208870400000.L,
+ 1774878043152614400000.L, 153822763739893248000.L, 10608466464820224000.L,
+ 595373117923584000.L, 27563570274240000.L, 1060137318240000.L,
+ 33924394183680.L, 899510451840.L, 19554575040.L, 341863200.L, 4651200.L,
+ 46512.L, 306.L, 1.L};
+ const MatrixType A2 = A * A;
+ const MatrixType A4 = A2 * A2;
+ const MatrixType A6 = A4 * A2;
+ const MatrixType A8 = A4 * A4;
+ V = b[17] * A8 + b[15] * A6 + b[13] * A4 + b[11] * A2; // used for temporary storage
+ MatrixType tmp = A8 * V;
+ tmp += b[9] * A8 + b[7] * A6 + b[5] * A4 + b[3] * A2
+ + b[1] * MatrixType::Identity(A.rows(), A.cols());
+ U.noalias() = A * tmp;
+ tmp = b[16] * A8 + b[14] * A6 + b[12] * A4 + b[10] * A2;
+ V.noalias() = tmp * A8;
+ V += b[8] * A8 + b[6] * A6 + b[4] * A4 + b[2] * A2
+ + b[0] * MatrixType::Identity(A.rows(), A.cols());
+}
+#endif
+
+template <typename MatrixType, typename RealScalar = typename NumTraits<typename traits<MatrixType>::Scalar>::Real>
+struct matrix_exp_computeUV
+{
+ /** \brief Compute Padé approximant to the exponential.
+ *
+ * Computes \c U, \c V and \c squarings such that \f$ (V+U)(V-U)^{-1} \f$ is a Padé
+ * approximant of \f$ \exp(2^{-\mbox{squarings}}M) \f$ around \f$ M = 0 \f$, where \f$ M \f$
+ * denotes the matrix \c arg. The degree of the Padé approximant and the value of squarings
+ * are chosen such that the approximation error is no more than the round-off error.
+ */
+ static void run(const MatrixType& arg, MatrixType& U, MatrixType& V, int& squarings);
};
template <typename MatrixType>
-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())
+struct matrix_exp_computeUV<MatrixType, float>
{
- /* empty body */
-}
+ template <typename ArgType>
+ static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
+ {
+ using std::frexp;
+ using std::pow;
+ const float l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
+ squarings = 0;
+ if (l1norm < 4.258730016922831e-001f) {
+ matrix_exp_pade3(arg, U, V);
+ } else if (l1norm < 1.880152677804762e+000f) {
+ matrix_exp_pade5(arg, U, V);
+ } else {
+ const float maxnorm = 3.925724783138660f;
+ frexp(l1norm / maxnorm, &squarings);
+ if (squarings < 0) squarings = 0;
+ MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<float>(squarings));
+ matrix_exp_pade7(A, U, V);
+ }
+ }
+};
template <typename MatrixType>
-template <typename ResultType>
-void MatrixExponential<MatrixType>::compute(ResultType &result)
+struct matrix_exp_computeUV<MatrixType, double>
{
-#if LDBL_MANT_DIG > 112 // rarely happens
- if(sizeof(RealScalar) > 14) {
- result = m_M.matrixFunction(StdStemFunctions<ComplexScalar>::exp);
- return;
+ typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
+ template <typename ArgType>
+ static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
+ {
+ using std::frexp;
+ using std::pow;
+ const RealScalar l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
+ squarings = 0;
+ if (l1norm < 1.495585217958292e-002) {
+ matrix_exp_pade3(arg, U, V);
+ } else if (l1norm < 2.539398330063230e-001) {
+ matrix_exp_pade5(arg, U, V);
+ } else if (l1norm < 9.504178996162932e-001) {
+ matrix_exp_pade7(arg, U, V);
+ } else if (l1norm < 2.097847961257068e+000) {
+ matrix_exp_pade9(arg, U, V);
+ } else {
+ const RealScalar maxnorm = 5.371920351148152;
+ frexp(l1norm / maxnorm, &squarings);
+ if (squarings < 0) squarings = 0;
+ MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<RealScalar>(squarings));
+ matrix_exp_pade13(A, U, V);
+ }
}
+};
+
+template <typename MatrixType>
+struct matrix_exp_computeUV<MatrixType, long double>
+{
+ template <typename ArgType>
+ static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
+ {
+#if LDBL_MANT_DIG == 53 // double precision
+ matrix_exp_computeUV<MatrixType, double>::run(arg, U, V, squarings);
+
+#else
+
+ using std::frexp;
+ using std::pow;
+ const long double l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
+ squarings = 0;
+
+#if LDBL_MANT_DIG <= 64 // extended precision
+
+ if (l1norm < 4.1968497232266989671e-003L) {
+ matrix_exp_pade3(arg, U, V);
+ } else if (l1norm < 1.1848116734693823091e-001L) {
+ matrix_exp_pade5(arg, U, V);
+ } else if (l1norm < 5.5170388480686700274e-001L) {
+ matrix_exp_pade7(arg, U, V);
+ } else if (l1norm < 1.3759868875587845383e+000L) {
+ matrix_exp_pade9(arg, U, V);
+ } else {
+ const long double maxnorm = 4.0246098906697353063L;
+ frexp(l1norm / maxnorm, &squarings);
+ if (squarings < 0) squarings = 0;
+ MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
+ matrix_exp_pade13(A, U, V);
+ }
+
+#elif LDBL_MANT_DIG <= 106 // double-double
+
+ if (l1norm < 3.2787892205607026992947488108213e-005L) {
+ matrix_exp_pade3(arg, U, V);
+ } else if (l1norm < 6.4467025060072760084130906076332e-003L) {
+ matrix_exp_pade5(arg, U, V);
+ } else if (l1norm < 6.8988028496595374751374122881143e-002L) {
+ matrix_exp_pade7(arg, U, V);
+ } else if (l1norm < 2.7339737518502231741495857201670e-001L) {
+ matrix_exp_pade9(arg, U, V);
+ } else if (l1norm < 1.3203382096514474905666448850278e+000L) {
+ matrix_exp_pade13(arg, U, V);
+ } else {
+ const long double maxnorm = 3.2579440895405400856599663723517L;
+ frexp(l1norm / maxnorm, &squarings);
+ if (squarings < 0) squarings = 0;
+ MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
+ matrix_exp_pade17(A, U, V);
+ }
+
+#elif LDBL_MANT_DIG <= 112 // quadruple precison
+
+ if (l1norm < 1.639394610288918690547467954466970e-005L) {
+ matrix_exp_pade3(arg, U, V);
+ } else if (l1norm < 4.253237712165275566025884344433009e-003L) {
+ matrix_exp_pade5(arg, U, V);
+ } else if (l1norm < 5.125804063165764409885122032933142e-002L) {
+ matrix_exp_pade7(arg, U, V);
+ } else if (l1norm < 2.170000765161155195453205651889853e-001L) {
+ matrix_exp_pade9(arg, U, V);
+ } else if (l1norm < 1.125358383453143065081397882891878e+000L) {
+ matrix_exp_pade13(arg, U, V);
+ } else {
+ const long double maxnorm = 2.884233277829519311757165057717815L;
+ frexp(l1norm / maxnorm, &squarings);
+ if (squarings < 0) squarings = 0;
+ MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
+ matrix_exp_pade17(A, U, V);
+ }
+
+#else
+
+ // this case should be handled in compute()
+ eigen_assert(false && "Bug in MatrixExponential");
+
#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++)
+#endif // LDBL_MANT_DIG
+ }
+};
+
+template<typename T> struct is_exp_known_type : false_type {};
+template<> struct is_exp_known_type<float> : true_type {};
+template<> struct is_exp_known_type<double> : true_type {};
+#if LDBL_MANT_DIG <= 112
+template<> struct is_exp_known_type<long double> : true_type {};
+#endif
+
+template <typename ArgType, typename ResultType>
+void matrix_exp_compute(const ArgType& arg, ResultType &result, true_type) // natively supported scalar type
+{
+ typedef typename ArgType::PlainObject MatrixType;
+ MatrixType U, V;
+ int squarings;
+ matrix_exp_computeUV<MatrixType>::run(arg, U, V, squarings); // Pade approximant is (U+V) / (-U+V)
+ MatrixType numer = U + V;
+ MatrixType denom = -U + V;
+ result = denom.partialPivLu().solve(numer);
+ for (int i=0; i<squarings; i++)
result *= result; // undo scaling by repeated squaring
}
-template <typename MatrixType>
-EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade3(const MatrixType &A)
+
+/* Computes the matrix exponential
+ *
+ * \param arg argument of matrix exponential (should be plain object)
+ * \param result variable in which result will be stored
+ */
+template <typename ArgType, typename ResultType>
+void matrix_exp_compute(const ArgType& arg, ResultType &result, false_type) // default
{
- const RealScalar b[] = {120., 60., 12., 1.};
- m_tmp1.noalias() = A * A;
- m_tmp2 = b[3]*m_tmp1 + b[1]*m_Id;
- m_U.noalias() = A * m_tmp2;
- m_V = b[2]*m_tmp1 + b[0]*m_Id;
+ typedef typename ArgType::PlainObject MatrixType;
+ typedef typename traits<MatrixType>::Scalar Scalar;
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ typedef typename std::complex<RealScalar> ComplexScalar;
+ result = arg.matrixFunction(internal::stem_function_exp<ComplexScalar>);
}
-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
-}
+} // end namespace Eigen::internal
/** \ingroup MatrixFunctions_Module
*
@@ -391,11 +389,9 @@
*
* \tparam Derived Type of the argument to the matrix exponential.
*
- * This class holds the argument to the matrix exponential until it
- * is assigned or evaluated for some other reason (so the argument
- * should not be changed in the meantime). It is the return type of
- * MatrixBase::exp() and most of the time this is the only way it is
- * used.
+ * This class holds the argument to the matrix exponential until it is assigned or evaluated for
+ * some other reason (so the argument should not be changed in the meantime). It is the return type
+ * of MatrixBase::exp() and most of the time this is the only way it is used.
*/
template<typename Derived> struct MatrixExponentialReturnValue
: public ReturnByValue<MatrixExponentialReturnValue<Derived> >
@@ -404,31 +400,26 @@
public:
/** \brief Constructor.
*
- * \param[in] src %Matrix (expression) forming the argument of the
- * matrix exponential.
+ * \param src %Matrix (expression) forming the argument of the matrix exponential.
*/
MatrixExponentialReturnValue(const Derived& src) : m_src(src) { }
/** \brief Compute the matrix exponential.
*
- * \param[out] result the matrix exponential of \p src in the
- * constructor.
+ * \param result the matrix exponential of \p src in the constructor.
*/
template <typename ResultType>
inline void evalTo(ResultType& result) const
{
- const typename Derived::PlainObject srcEvaluated = m_src.eval();
- MatrixExponential<typename Derived::PlainObject> me(srcEvaluated);
- me.compute(result);
+ const typename internal::nested_eval<Derived, 10>::type tmp(m_src);
+ internal::matrix_exp_compute(tmp, result, internal::is_exp_known_type<typename Derived::Scalar>());
}
Index rows() const { return m_src.rows(); }
Index cols() const { return m_src.cols(); }
protected:
- const Derived& m_src;
- private:
- MatrixExponentialReturnValue& operator=(const MatrixExponentialReturnValue&);
+ const typename internal::ref_selector<Derived>::type m_src;
};
namespace internal {