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Use ReturnByValue to return result of ei_matrix_exponential() .
This commit is contained in:
Jitse Niesen
parent
a9096b626b
commit
25019f0836
@ -29,74 +29,32 @@
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template <typename Scalar> Scalar log2(Scalar v) { return std::log(v)/std::log(Scalar(2)); }
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#endif
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/** \ingroup MatrixFunctions_Module
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*
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* \brief Compute the matrix exponential.
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*
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* \param[in] M matrix whose exponential is to be computed.
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* \param[out] result pointer to the matrix in which to store the result.
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*
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* The matrix exponential of \f$ M \f$ is defined by
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* \f[ \exp(M) = \sum_{k=0}^\infty \frac{M^k}{k!}. \f]
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* The matrix exponential can be used to solve linear ordinary
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* differential equations: the solution of \f$ y' = My \f$ with the
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* initial condition \f$ y(0) = y_0 \f$ is given by
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* \f$ y(t) = \exp(M) y_0 \f$.
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*
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* The cost of the computation is approximately \f$ 20 n^3 \f$ for
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* matrices of size \f$ n \f$. The number 20 depends weakly on the
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* norm of the matrix.
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*
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* The matrix exponential is computed using the scaling-and-squaring
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* method combined with Padé approximation. The matrix is first
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* rescaled, then the exponential of the reduced matrix is computed
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* approximant, and then the rescaling is undone by repeated
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* squaring. The degree of the Padé approximant is chosen such
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* that the approximation error is less than the round-off
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* error. However, errors may accumulate during the squaring phase.
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*
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* Details of the algorithm can be found in: Nicholas J. Higham, "The
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* scaling and squaring method for the matrix exponential revisited,"
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* <em>SIAM J. %Matrix Anal. Applic.</em>, <b>26</b>:1179–1193,
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* 2005.
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*
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* Example: The following program checks that
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* \f[ \exp \left[ \begin{array}{ccc}
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* 0 & \frac14\pi & 0 \\
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* -\frac14\pi & 0 & 0 \\
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* 0 & 0 & 0
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* \end{array} \right] = \left[ \begin{array}{ccc}
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* \frac12\sqrt2 & -\frac12\sqrt2 & 0 \\
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* \frac12\sqrt2 & \frac12\sqrt2 & 0 \\
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* 0 & 0 & 1
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* \end{array} \right]. \f]
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* This corresponds to a rotation of \f$ \frac14\pi \f$ radians around
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* the z-axis.
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*
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* \include MatrixExponential.cpp
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* Output: \verbinclude MatrixExponential.out
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*
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* \note \p M has to be a matrix of \c float, \c double,
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* \c complex<float> or \c complex<double> .
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*/
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template <typename Derived>
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EIGEN_STRONG_INLINE void ei_matrix_exponential(const MatrixBase<Derived> &M,
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typename MatrixBase<Derived>::PlainMatrixType* result);
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/** \ingroup MatrixFunctions_Module
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* \brief Class for computing the matrix exponential.
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* \tparam MatrixType type of the argument of the exponential,
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* expected to be an instantiation of the Matrix class template.
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*/
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template <typename MatrixType>
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class MatrixExponential {
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public:
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/** \brief Compute the matrix exponential.
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*
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* \param M matrix whose exponential is to be computed.
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* \param result pointer to the matrix in which to store the result.
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*/
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MatrixExponential(const MatrixType &M, MatrixType *result);
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/** \brief Constructor.
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*
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* The class stores a reference to \p M, so it should not be
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* changed (or destroyed) before compute() is called.
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*
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* \param[in] M matrix whose exponential is to be computed.
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*/
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MatrixExponential(const MatrixType &M);
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/** \brief Computes the matrix exponential.
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*
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* \param[out] result the matrix exponential of \p M in the constructor.
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*/
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template <typename ResultType>
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void compute(ResultType &result);
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private:
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@ -109,7 +67,7 @@ class MatrixExponential {
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* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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*
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* \param A Argument of matrix exponential
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* \param[in] A Argument of matrix exponential
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*/
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void pade3(const MatrixType &A);
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@ -118,7 +76,7 @@ class MatrixExponential {
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* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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*
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* \param A Argument of matrix exponential
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* \param[in] A Argument of matrix exponential
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*/
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void pade5(const MatrixType &A);
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@ -127,7 +85,7 @@ class MatrixExponential {
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* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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*
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* \param A Argument of matrix exponential
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* \param[in] A Argument of matrix exponential
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*/
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void pade7(const MatrixType &A);
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@ -136,7 +94,7 @@ class MatrixExponential {
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* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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*
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* \param A Argument of matrix exponential
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* \param[in] A Argument of matrix exponential
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*/
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void pade9(const MatrixType &A);
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@ -145,7 +103,7 @@ class MatrixExponential {
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* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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*
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* \param A Argument of matrix exponential
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* \param[in] A Argument of matrix exponential
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*/
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void pade13(const MatrixType &A);
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@ -171,10 +129,10 @@ class MatrixExponential {
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void computeUV(float);
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typedef typename ei_traits<MatrixType>::Scalar Scalar;
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typedef typename NumTraits<typename ei_traits<MatrixType>::Scalar>::Real RealScalar;
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typedef typename NumTraits<Scalar>::Real RealScalar;
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/** \brief Pointer to matrix whose exponential is to be computed. */
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const MatrixType* m_M;
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/** \brief Reference to matrix whose exponential is to be computed. */
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const MatrixType& m_M;
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/** \brief Even-degree terms in numerator of Padé approximant. */
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MatrixType m_U;
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@ -199,8 +157,8 @@ class MatrixExponential {
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};
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template <typename MatrixType>
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MatrixExponential<MatrixType>::MatrixExponential(const MatrixType &M, MatrixType *result) :
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m_M(&M),
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MatrixExponential<MatrixType>::MatrixExponential(const MatrixType &M) :
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m_M(M),
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m_U(M.rows(),M.cols()),
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m_V(M.rows(),M.cols()),
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m_tmp1(M.rows(),M.cols()),
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@ -208,13 +166,20 @@ MatrixExponential<MatrixType>::MatrixExponential(const MatrixType &M, MatrixType
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m_Id(MatrixType::Identity(M.rows(), M.cols())),
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m_squarings(0),
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m_l1norm(static_cast<float>(M.cwiseAbs().colwise().sum().maxCoeff()))
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{
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/* empty body */
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}
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template <typename MatrixType>
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template <typename ResultType>
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void MatrixExponential<MatrixType>::compute(ResultType &result)
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{
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computeUV(RealScalar());
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m_tmp1 = m_U + m_V; // numerator of Pade approximant
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m_tmp2 = -m_U + m_V; // denominator of Pade approximant
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*result = m_tmp2.partialPivLu().solve(m_tmp1);
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result = m_tmp2.partialPivLu().solve(m_tmp1);
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for (int i=0; i<m_squarings; i++)
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*result *= *result; // undo scaling by repeated squaring
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result *= result; // undo scaling by repeated squaring
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}
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template <typename MatrixType>
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@ -286,13 +251,13 @@ template <typename MatrixType>
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void MatrixExponential<MatrixType>::computeUV(float)
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{
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if (m_l1norm < 4.258730016922831e-001) {
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pade3(*m_M);
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pade3(m_M);
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} else if (m_l1norm < 1.880152677804762e+000) {
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pade5(*m_M);
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pade5(m_M);
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} else {
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const float maxnorm = 3.925724783138660f;
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m_squarings = std::max(0, (int)ceil(log2(m_l1norm / maxnorm)));
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MatrixType A = *m_M / std::pow(Scalar(2), Scalar(static_cast<RealScalar>(m_squarings)));
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MatrixType A = m_M / std::pow(Scalar(2), Scalar(static_cast<RealScalar>(m_squarings)));
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pade7(A);
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}
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}
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@ -301,27 +266,126 @@ template <typename MatrixType>
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void MatrixExponential<MatrixType>::computeUV(double)
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{
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if (m_l1norm < 1.495585217958292e-002) {
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pade3(*m_M);
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pade3(m_M);
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} else if (m_l1norm < 2.539398330063230e-001) {
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pade5(*m_M);
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pade5(m_M);
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} else if (m_l1norm < 9.504178996162932e-001) {
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pade7(*m_M);
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pade7(m_M);
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} else if (m_l1norm < 2.097847961257068e+000) {
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pade9(*m_M);
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pade9(m_M);
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} else {
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const double maxnorm = 5.371920351148152;
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m_squarings = std::max(0, (int)ceil(log2(m_l1norm / maxnorm)));
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MatrixType A = *m_M / std::pow(Scalar(2), Scalar(m_squarings));
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MatrixType A = m_M / std::pow(Scalar(2), Scalar(m_squarings));
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pade13(A);
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}
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}
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/** \ingroup MatrixFunctions_Module
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*
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* \brief Proxy for the matrix exponential of some matrix (expression).
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*
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* \tparam Derived Type of the argument to the matrix exponential.
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*
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* This class holds the argument to the matrix exponential until it
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* is assigned or evaluated for some other reason (so the argument
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* should not be changed in the meantime). It is the return type of
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* ei_matrix_exponential() and most of the time this is the only way
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* it is used.
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*/
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template<typename Derived> struct MatrixExponentialReturnValue
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: public ReturnByValue<MatrixExponentialReturnValue<Derived> >
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{
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public:
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/** \brief Constructor.
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*
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* \param[in] src %Matrix (expression) forming the argument of the
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* matrix exponential.
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*/
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MatrixExponentialReturnValue(const Derived& src) : m_src(src) { }
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/** \brief Compute the matrix exponential.
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*
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* \param[out] result the matrix exponential of \p src in the
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* constructor.
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*/
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template <typename ResultType>
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inline void evalTo(ResultType& result) const
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{
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const typename ei_eval<Derived>::type srcEvaluated = m_src.eval();
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MatrixExponential<typename Derived::PlainMatrixType> me(srcEvaluated);
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me.compute(result);
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}
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int rows() const { return m_src.rows(); }
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int cols() const { return m_src.cols(); }
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protected:
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const Derived& m_src;
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};
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template<typename Derived>
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struct ei_traits<MatrixExponentialReturnValue<Derived> >
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{
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typedef typename Derived::PlainMatrixType ReturnMatrixType;
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};
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/** \ingroup MatrixFunctions_Module
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*
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* \brief Compute the matrix exponential.
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*
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* \param[in] M matrix whose exponential is to be computed.
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* \returns expression representing the matrix exponential of \p M.
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*
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* The matrix exponential of \f$ M \f$ is defined by
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* \f[ \exp(M) = \sum_{k=0}^\infty \frac{M^k}{k!}. \f]
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* The matrix exponential can be used to solve linear ordinary
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* differential equations: the solution of \f$ y' = My \f$ with the
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* initial condition \f$ y(0) = y_0 \f$ is given by
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* \f$ y(t) = \exp(M) y_0 \f$.
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*
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* The cost of the computation is approximately \f$ 20 n^3 \f$ for
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* matrices of size \f$ n \f$. The number 20 depends weakly on the
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* norm of the matrix.
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*
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* The matrix exponential is computed using the scaling-and-squaring
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* method combined with Padé approximation. The matrix is first
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* rescaled, then the exponential of the reduced matrix is computed
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* approximant, and then the rescaling is undone by repeated
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* squaring. The degree of the Padé approximant is chosen such
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* that the approximation error is less than the round-off
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* error. However, errors may accumulate during the squaring phase.
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*
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* Details of the algorithm can be found in: Nicholas J. Higham, "The
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* scaling and squaring method for the matrix exponential revisited,"
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* <em>SIAM J. %Matrix Anal. Applic.</em>, <b>26</b>:1179–1193,
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* 2005.
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*
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* Example: The following program checks that
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* \f[ \exp \left[ \begin{array}{ccc}
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* 0 & \frac14\pi & 0 \\
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* -\frac14\pi & 0 & 0 \\
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* 0 & 0 & 0
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* \end{array} \right] = \left[ \begin{array}{ccc}
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* \frac12\sqrt2 & -\frac12\sqrt2 & 0 \\
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* \frac12\sqrt2 & \frac12\sqrt2 & 0 \\
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* 0 & 0 & 1
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* \end{array} \right]. \f]
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* This corresponds to a rotation of \f$ \frac14\pi \f$ radians around
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* the z-axis.
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*
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* \include MatrixExponential.cpp
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* Output: \verbinclude MatrixExponential.out
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*
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* \note \p M has to be a matrix of \c float, \c double,
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* \c complex<float> or \c complex<double> .
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*/
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template <typename Derived>
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EIGEN_STRONG_INLINE void ei_matrix_exponential(const MatrixBase<Derived> &M,
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typename MatrixBase<Derived>::PlainMatrixType* result)
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MatrixExponentialReturnValue<Derived>
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ei_matrix_exponential(const MatrixBase<Derived> &M)
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{
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ei_assert(M.rows() == M.cols());
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MatrixExponential<typename MatrixBase<Derived>::PlainMatrixType>(M, result);
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return MatrixExponentialReturnValue<Derived>(M.derived());
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}
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#endif // EIGEN_MATRIX_EXPONENTIAL
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@ -12,7 +12,6 @@ int main()
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0, 0, 0;
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std::cout << "The matrix A is:\n" << A << "\n\n";
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MatrixXd B;
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ei_matrix_exponential(A, &B);
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MatrixXd B = ei_matrix_exponential(A);
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std::cout << "The matrix exponential of A is:\n" << B << "\n\n";
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}
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@ -61,7 +61,7 @@ void test2dRotation(double tol)
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std::cout << "test2dRotation: i = " << i << " error funm = " << relerr(C, B);
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VERIFY(C.isApprox(B, static_cast<T>(tol)));
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ei_matrix_exponential(angle*A, &C);
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C = ei_matrix_exponential(angle*A);
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std::cout << " error expm = " << relerr(C, B) << "\n";
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VERIFY(C.isApprox(B, static_cast<T>(tol)));
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}
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@ -86,7 +86,7 @@ void test2dHyperbolicRotation(double tol)
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std::cout << "test2dHyperbolicRotation: i = " << i << " error funm = " << relerr(C, B);
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VERIFY(C.isApprox(B, static_cast<T>(tol)));
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ei_matrix_exponential(A, &C);
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C = ei_matrix_exponential(A);
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std::cout << " error expm = " << relerr(C, B) << "\n";
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VERIFY(C.isApprox(B, static_cast<T>(tol)));
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}
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@ -110,7 +110,7 @@ void testPascal(double tol)
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std::cout << "testPascal: size = " << size << " error funm = " << relerr(C, B);
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VERIFY(C.isApprox(B, static_cast<T>(tol)));
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ei_matrix_exponential(A, &C);
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C = ei_matrix_exponential(A);
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std::cout << " error expm = " << relerr(C, B) << "\n";
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VERIFY(C.isApprox(B, static_cast<T>(tol)));
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}
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@ -137,10 +137,9 @@ void randomTest(const MatrixType& m, double tol)
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std::cout << "randomTest: error funm = " << relerr(identity, m2 * m3);
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VERIFY(identity.isApprox(m2 * m3, static_cast<RealScalar>(tol)));
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ei_matrix_exponential(m1, &m2);
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ei_matrix_exponential(-m1, &m3);
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std::cout << " error expm = " << relerr(identity, m2 * m3) << "\n";
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VERIFY(identity.isApprox(m2 * m3, static_cast<RealScalar>(tol)));
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m2 = ei_matrix_exponential(m1) * ei_matrix_exponential(-m1);
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std::cout << " error expm = " << relerr(identity, m2) << "\n";
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VERIFY(identity.isApprox(m2, static_cast<RealScalar>(tol)));
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}
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}
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@ -100,10 +100,9 @@ void testMatrixExponential(const MatrixType& A)
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typedef std::complex<RealScalar> ComplexScalar;
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for (int i = 0; i < g_repeat; i++) {
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MatrixType expA1, expA2;
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ei_matrix_exponential(A, &expA1);
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ei_matrix_function(A, StdStemFunctions<ComplexScalar>::exp, &expA2);
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VERIFY_IS_APPROX(expA1, expA2);
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MatrixType expA;
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ei_matrix_function(A, StdStemFunctions<ComplexScalar>::exp, &expA);
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VERIFY_IS_APPROX(ei_matrix_exponential(A), expA);
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}
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}
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@ -111,10 +110,10 @@ template<typename MatrixType>
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void testHyperbolicFunctions(const MatrixType& A)
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{
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for (int i = 0; i < g_repeat; i++) {
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MatrixType sinhA, coshA, expA;
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MatrixType sinhA, coshA;
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ei_matrix_sinh(A, &sinhA);
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ei_matrix_cosh(A, &coshA);
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ei_matrix_exponential(A, &expA);
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MatrixType expA = ei_matrix_exponential(A);
|
||||
VERIFY_IS_APPROX(sinhA, (expA - expA.inverse())/2);
|
||||
VERIFY_IS_APPROX(coshA, (expA + expA.inverse())/2);
|
||||
}
|
||||
@ -136,8 +135,7 @@ void testGonioFunctions(const MatrixType& A)
|
||||
for (int i = 0; i < g_repeat; i++) {
|
||||
ComplexMatrix Ac = A.template cast<ComplexScalar>();
|
||||
|
||||
ComplexMatrix exp_iA;
|
||||
ei_matrix_exponential(imagUnit * Ac, &exp_iA);
|
||||
ComplexMatrix exp_iA = ei_matrix_exponential(imagUnit * Ac);
|
||||
|
||||
MatrixType sinA;
|
||||
ei_matrix_sin(A, &sinA);
|
||||
|
||||
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Reference in New Issue
Block a user