mirror of
https://gitlab.com/libeigen/eigen.git
synced 2025-09-17 03:43:13 +08:00
Re-factor matrix exponential.
Put all routines in a class. I think this is a cleaner design.
This commit is contained in:
Jitse Niesen
parent
220ff54131
commit
2a6db40f10
@ -25,6 +25,10 @@
|
||||
#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
|
||||
|
||||
/** \brief Compute the matrix exponential.
|
||||
*
|
||||
* \param M matrix whose exponential is to be computed.
|
||||
@ -61,260 +65,243 @@ template <typename Derived>
|
||||
EIGEN_STRONG_INLINE void ei_matrix_exponential(const MatrixBase<Derived> &M,
|
||||
typename MatrixBase<Derived>::PlainMatrixType* result);
|
||||
|
||||
/** \brief Class for computing the matrix exponential.*/
|
||||
template <typename MatrixType>
|
||||
class MatrixExponential {
|
||||
|
||||
/** \internal \brief Internal helper functions for computing the
|
||||
* matrix exponential.
|
||||
*/
|
||||
namespace MatrixExponentialInternal {
|
||||
public:
|
||||
|
||||
/** \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.
|
||||
*/
|
||||
MatrixExponential(const MatrixType &M, MatrixType *result);
|
||||
|
||||
#ifdef _MSC_VER
|
||||
template <typename Scalar> Scalar log2(Scalar v) { return std::log(v)/std::log(Scalar(2)); }
|
||||
#endif
|
||||
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$.
|
||||
*
|
||||
* \param A Argument of matrix exponential
|
||||
*/
|
||||
void pade3(const MatrixType &A);
|
||||
|
||||
/** \brief Compute the (5,5)-Padé approximant to the exponential.
|
||||
*
|
||||
* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
|
||||
* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
|
||||
*
|
||||
* \param A Argument of matrix exponential
|
||||
*/
|
||||
void pade5(const MatrixType &A);
|
||||
|
||||
/** \brief Compute the (7,7)-Padé approximant to the exponential.
|
||||
*
|
||||
* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
|
||||
* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
|
||||
*
|
||||
* \param A Argument of matrix exponential
|
||||
*/
|
||||
void pade7(const MatrixType &A);
|
||||
|
||||
/** \brief Compute the (9,9)-Padé approximant to the exponential.
|
||||
*
|
||||
* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
|
||||
* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
|
||||
*
|
||||
* \param A Argument of matrix exponential
|
||||
*/
|
||||
void pade9(const MatrixType &A);
|
||||
|
||||
/** \brief Compute the (13,13)-Padé approximant to the exponential.
|
||||
*
|
||||
* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
|
||||
* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
|
||||
*
|
||||
* \param A Argument of matrix exponential
|
||||
*/
|
||||
void pade13(const MatrixType &A);
|
||||
|
||||
/** \brief Compute Padé approximant to the exponential.
|
||||
*
|
||||
* Computes \c m_U, \c m_V and \c m_squarings such that
|
||||
* \f$ (V+U)(V-U)^{-1} \f$ is a Padé of
|
||||
* \f$ \exp(2^{-\mbox{squarings}}M) \f$ around \f$ M = 0 \f$. The
|
||||
* degree of the Padé approximant and the value of
|
||||
* squarings are chosen such that the approximation error is no
|
||||
* more than the round-off error.
|
||||
*
|
||||
* The argument of this function should correspond with the (real
|
||||
* part of) the entries of \c m_M. It is used to select the
|
||||
* correct implementation using overloading.
|
||||
*/
|
||||
void computeUV(double);
|
||||
|
||||
/** \brief Compute Padé approximant to the exponential.
|
||||
*
|
||||
* \sa computeUV(double);
|
||||
*/
|
||||
void computeUV(float);
|
||||
|
||||
/** \internal \brief Compute the (3,3)-Padé approximant to
|
||||
* the exponential.
|
||||
*
|
||||
* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
|
||||
* approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
|
||||
*
|
||||
* \param M Argument of matrix exponential
|
||||
* \param Id Identity matrix of same size as M
|
||||
* \param tmp Temporary storage, to be provided by the caller
|
||||
* \param M2 Temporary storage, to be provided by the caller
|
||||
* \param U Even-degree terms in numerator of Padé approximant
|
||||
* \param V Odd-degree terms in numerator of Padé approximant
|
||||
*/
|
||||
template <typename MatrixType>
|
||||
EIGEN_STRONG_INLINE void pade3(const MatrixType &M, const MatrixType& Id, MatrixType& tmp,
|
||||
MatrixType& M2, MatrixType& U, MatrixType& V)
|
||||
{
|
||||
typedef typename ei_traits<MatrixType>::Scalar Scalar;
|
||||
const Scalar b[] = {120., 60., 12., 1.};
|
||||
M2.noalias() = M * M;
|
||||
tmp = b[3]*M2 + b[1]*Id;
|
||||
U.noalias() = M * tmp;
|
||||
V = b[2]*M2 + b[0]*Id;
|
||||
}
|
||||
|
||||
/** \internal \brief Compute the (5,5)-Padé approximant to
|
||||
* the exponential.
|
||||
*
|
||||
* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
|
||||
* approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
|
||||
*
|
||||
* \param M Argument of matrix exponential
|
||||
* \param Id Identity matrix of same size as M
|
||||
* \param tmp Temporary storage, to be provided by the caller
|
||||
* \param M2 Temporary storage, to be provided by the caller
|
||||
* \param U Even-degree terms in numerator of Padé approximant
|
||||
* \param V Odd-degree terms in numerator of Padé approximant
|
||||
*/
|
||||
template <typename MatrixType>
|
||||
EIGEN_STRONG_INLINE void pade5(const MatrixType &M, const MatrixType& Id, MatrixType& tmp,
|
||||
MatrixType& M2, MatrixType& U, MatrixType& V)
|
||||
{
|
||||
typedef typename ei_traits<MatrixType>::Scalar Scalar;
|
||||
const Scalar b[] = {30240., 15120., 3360., 420., 30., 1.};
|
||||
M2.noalias() = M * M;
|
||||
MatrixType M4 = M2 * M2;
|
||||
tmp = b[5]*M4 + b[3]*M2 + b[1]*Id;
|
||||
U.noalias() = M * tmp;
|
||||
V = b[4]*M4 + b[2]*M2 + b[0]*Id;
|
||||
}
|
||||
|
||||
/** \internal \brief Compute the (7,7)-Padé approximant to
|
||||
* the exponential.
|
||||
*
|
||||
* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
|
||||
* approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
|
||||
*
|
||||
* \param M Argument of matrix exponential
|
||||
* \param Id Identity matrix of same size as M
|
||||
* \param tmp Temporary storage, to be provided by the caller
|
||||
* \param M2 Temporary storage, to be provided by the caller
|
||||
* \param U Even-degree terms in numerator of Padé approximant
|
||||
* \param V Odd-degree terms in numerator of Padé approximant
|
||||
*/
|
||||
template <typename MatrixType>
|
||||
EIGEN_STRONG_INLINE void pade7(const MatrixType &M, const MatrixType& Id, MatrixType& tmp,
|
||||
MatrixType& M2, MatrixType& U, MatrixType& V)
|
||||
{
|
||||
typedef typename ei_traits<MatrixType>::Scalar Scalar;
|
||||
const Scalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.};
|
||||
M2.noalias() = M * M;
|
||||
MatrixType M4 = M2 * M2;
|
||||
MatrixType M6 = M4 * M2;
|
||||
tmp = b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
|
||||
U.noalias() = M * tmp;
|
||||
V = b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id;
|
||||
}
|
||||
|
||||
/** \internal \brief Compute the (9,9)-Padé approximant to
|
||||
* the exponential.
|
||||
*
|
||||
* After exit, \f$ (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.,
|
||||
typedef typename NumTraits<typename ei_traits<MatrixType>::Scalar>::Real RealScalar;
|
||||
|
||||
/** \brief Pointer to matrix whose exponential is to be computed. */
|
||||
const MatrixType* m_M;
|
||||
|
||||
/** \brief Even-degree terms in numerator of Padé approximant. */
|
||||
MatrixType m_U;
|
||||
|
||||
/** \brief Odd-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. */
|
||||
float m_l1norm;
|
||||
};
|
||||
|
||||
template <typename MatrixType>
|
||||
MatrixExponential<MatrixType>::MatrixExponential(const MatrixType &M, MatrixType *result) :
|
||||
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(static_cast<float>(M.cwise().abs().colwise().sum().maxCoeff()))
|
||||
{
|
||||
computeUV(RealScalar());
|
||||
m_tmp1 = m_U + m_V; // numerator of Pade approximant
|
||||
m_tmp2 = -m_U + m_V; // denominator of Pade approximant
|
||||
m_tmp2.partialLu().solve(m_tmp1, result);
|
||||
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 Scalar 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 Scalar 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 Scalar 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 Scalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240.,
|
||||
2162160., 110880., 3960., 90., 1.};
|
||||
M2.noalias() = M * M;
|
||||
MatrixType M4 = M2 * M2;
|
||||
MatrixType M6 = M4 * M2;
|
||||
MatrixType M8 = M6 * M2;
|
||||
tmp = b[9]*M8 + b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
|
||||
U.noalias() = M * tmp;
|
||||
V = b[8]*M8 + b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id;
|
||||
}
|
||||
|
||||
/** \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.,
|
||||
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 Scalar b[] = {64764752532480000., 32382376266240000., 7771770303897600.,
|
||||
1187353796428800., 129060195264000., 10559470521600., 670442572800.,
|
||||
33522128640., 1323241920., 40840800., 960960., 16380., 182., 1.};
|
||||
M2.noalias() = M * M;
|
||||
MatrixType M4 = M2 * M2;
|
||||
MatrixType M6 = M4 * M2;
|
||||
V = b[13]*M6 + b[11]*M4 + b[9]*M2;
|
||||
tmp.noalias() = M6 * V;
|
||||
tmp += b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
|
||||
U.noalias() = M * tmp;
|
||||
tmp = b[12]*M6 + b[10]*M4 + b[8]*M2;
|
||||
V.noalias() = M6 * tmp;
|
||||
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 = 3.925724783138660f;
|
||||
*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(M.rows(), M.cols());
|
||||
MatrixType den(M.rows(), M.cols());
|
||||
MatrixType U(M.rows(), M.cols());
|
||||
MatrixType V(M.rows(), M.cols());
|
||||
MatrixType Id = MatrixType::Identity(M.rows(), M.cols());
|
||||
float l1norm = static_cast<float>(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
|
||||
}
|
||||
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;
|
||||
}
|
||||
|
||||
} // end of namespace MatrixExponentialInternal
|
||||
template <typename MatrixType>
|
||||
void MatrixExponential<MatrixType>::computeUV(float)
|
||||
{
|
||||
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;
|
||||
m_squarings = std::max(0, (int)ceil(log2(m_l1norm / maxnorm)));
|
||||
MatrixType A = *m_M / std::pow(Scalar(2), m_squarings);
|
||||
pade7(A);
|
||||
}
|
||||
}
|
||||
|
||||
template <typename MatrixType>
|
||||
void MatrixExponential<MatrixType>::computeUV(double)
|
||||
{
|
||||
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;
|
||||
m_squarings = std::max(0, (int)ceil(log2(m_l1norm / maxnorm)));
|
||||
MatrixType A = *m_M / std::pow(Scalar(2), m_squarings);
|
||||
pade13(A);
|
||||
}
|
||||
}
|
||||
|
||||
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);
|
||||
MatrixExponential<typename MatrixBase<Derived>::PlainMatrixType>(M, result);
|
||||
}
|
||||
|
||||
#endif // EIGEN_MATRIX_EXPONENTIAL
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user