diff options
author | Jitse Niesen <jitse@maths.leeds.ac.uk> | 2009-08-12 15:44:22 +0100 |
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committer | Jitse Niesen <jitse@maths.leeds.ac.uk> | 2009-08-12 15:44:22 +0100 |
commit | f71f878babec8ffb01e563d2e234ecd56f7bc461 (patch) | |
tree | 61cbf1258aa1b74cabc23d71cb8cd5e74a549f8e /unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h | |
parent | 309d540d4a71527a4dba778dc5221b81fd18f540 (diff) |
Add support for matrix exponential of floats and complex numbers.
Diffstat (limited to 'unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h')
-rw-r--r-- | unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h | 331 |
1 files changed, 244 insertions, 87 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>:1179–1193, * 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)-Padé approximant to + * the exponential. + * + * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé + * approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$. + * + * \param M Argument of matrix exponential + * \param Id Identity matrix of same size as M + * \param tmp Temporary storage, to be provided by the caller + * \param M2 Temporary storage, to be provided by the caller + * \param U Even-degree terms in numerator of Padé approximant + * \param V Odd-degree terms in numerator of Padé approximant + */ + template <typename MatrixType> + EIGEN_STRONG_INLINE void pade3(const MatrixType &M, const MatrixType& Id, MatrixType& tmp, + MatrixType& M2, MatrixType& U, MatrixType& V) + { + typedef typename ei_traits<MatrixType>::Scalar Scalar; + const Scalar b[] = {120., 60., 12., 1.}; + M2 = (M * M).lazy(); + tmp = b[3]*M2 + b[1]*Id; + U = (M * tmp).lazy(); + V = b[2]*M2 + b[0]*Id; + } + + /** \internal \brief Compute the (5,5)-Padé approximant to + * the exponential. + * + * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé + * approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$. + * + * \param M Argument of matrix exponential + * \param Id Identity matrix of same size as M + * \param tmp Temporary storage, to be provided by the caller + * \param M2 Temporary storage, to be provided by the caller + * \param U Even-degree terms in numerator of Padé approximant + * \param V Odd-degree terms in numerator of Padé approximant + */ + template <typename MatrixType> + EIGEN_STRONG_INLINE void pade5(const MatrixType &M, const MatrixType& Id, MatrixType& tmp, + MatrixType& M2, MatrixType& U, MatrixType& V) + { + typedef typename ei_traits<MatrixType>::Scalar Scalar; const Scalar b[] = {30240., 15120., 3360., 420., 30., 1.}; - PlainMatrixType M2, M4; - M2 = (Meval * Meval).lazy(); - M4 = (M2 * M2).lazy(); - num = b[5]*M4 + b[3]*M2 + b[1]*Id; - U = (Meval * num).lazy(); + M2 = (M * M).lazy(); + MatrixType M4 = (M2 * M2).lazy(); + tmp = b[5]*M4 + b[3]*M2 + b[1]*Id; + U = (M * tmp).lazy(); V = b[4]*M4 + b[2]*M2 + b[0]*Id; - - } else if (l1norm < 9.504178996162932e-001) { - - // Use (7,7)-Pade + } + + /** \internal \brief Compute the (7,7)-Padé approximant to + * the exponential. + * + * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé + * approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$. + * + * \param M Argument of matrix exponential + * \param Id Identity matrix of same size as M + * \param tmp Temporary storage, to be provided by the caller + * \param M2 Temporary storage, to be provided by the caller + * \param U Even-degree terms in numerator of Padé approximant + * \param V Odd-degree terms in numerator of Padé approximant + */ + template <typename MatrixType> + EIGEN_STRONG_INLINE void pade7(const MatrixType &M, const MatrixType& Id, MatrixType& tmp, + MatrixType& M2, MatrixType& U, MatrixType& V) + { + typedef typename ei_traits<MatrixType>::Scalar Scalar; const Scalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.}; - PlainMatrixType M2, M4, M6; - M2 = (Meval * Meval).lazy(); - M4 = (M2 * M2).lazy(); - M6 = (M4 * M2).lazy(); - num = b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id; - U = (Meval * num).lazy(); + M2 = (M * M).lazy(); + MatrixType M4 = (M2 * M2).lazy(); + MatrixType M6 = (M4 * M2).lazy(); + tmp = b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id; + U = (M * tmp).lazy(); V = b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id; - - } else if (l1norm < 2.097847961257068e+000) { - - // Use (9,9)-Pade + } + + /** \internal \brief Compute the (9,9)-Padé approximant to + * the exponential. + * + * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé + * approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$. + * + * \param M Argument of matrix exponential + * \param Id Identity matrix of same size as M + * \param tmp Temporary storage, to be provided by the caller + * \param M2 Temporary storage, to be provided by the caller + * \param U Even-degree terms in numerator of Padé approximant + * \param V Odd-degree terms in numerator of Padé approximant + */ + template <typename MatrixType> + EIGEN_STRONG_INLINE void pade9(const MatrixType &M, const MatrixType& Id, MatrixType& tmp, + MatrixType& M2, MatrixType& U, MatrixType& V) + { + typedef typename ei_traits<MatrixType>::Scalar Scalar; const Scalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240., - 2162160., 110880., 3960., 90., 1.}; - PlainMatrixType M2, M4, M6, M8; - M2 = (Meval * Meval).lazy(); - M4 = (M2 * M2).lazy(); - M6 = (M4 * M2).lazy(); - M8 = (M6 * M2).lazy(); - num = b[9]*M8 + b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id; - U = (Meval * num).lazy(); + 2162160., 110880., 3960., 90., 1.}; + M2 = (M * M).lazy(); + MatrixType M4 = (M2 * M2).lazy(); + MatrixType M6 = (M4 * M2).lazy(); + MatrixType M8 = (M6 * M2).lazy(); + tmp = b[9]*M8 + b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id; + U = (M * tmp).lazy(); V = b[8]*M8 + b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id; - - } else { - - // Use (13,13)-Pade; scale matrix by power of 2 so that its norm - // is small enough - - const Scalar maxnorm = 5.371920351148152; + } + + /** \internal \brief Compute the (13,13)-Padé approximant to + * the exponential. + * + * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé + * 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 Padé approximant + * \param V Odd-degree terms in numerator of Padé 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 Padé + * approximants to the exponential. + */ + template <typename MatrixType, typename RealScalar = typename NumTraits<typename ei_traits<MatrixType>::Scalar>::Real> + struct computeUV_selector + { + /** \internal \brief Compute Padé approximant to the exponential. + * + * Computes \p U, \p V and \p 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. + * + * \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 Padé approximant + * \param V Odd-degree terms in numerator of Padé 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 |