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eigen/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h
Jitse Niesen 32f95ec267 Bug fix in MatrixExponential.h 
Initialize matrices for intermediate results to correct dimension
2009-09-01 10:50:54 +01:00

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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// Eigen is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 3 of the License, or (at your option) any later version.
//
// Alternatively, you can redistribute it and/or
// modify it under the terms of the GNU General Public License as
// published by the Free Software Foundation; either version 2 of
// the License, or (at your option) any later version.
//
// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public
// License and a copy of the GNU General Public License along with
// Eigen. If not, see <http://www.gnu.org/licenses/>.
#ifndef EIGEN_MATRIX_EXPONENTIAL
#define EIGEN_MATRIX_EXPONENTIAL
/** \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.
*
* The matrix exponential of \f$ M \f$ is defined by
* \f[ \exp(M) = \sum_{k=0}^\infty \frac{M^k}{k!}. \f]
* The matrix exponential can be used to solve linear ordinary
* differential equations: the solution of \f$ y' = My \f$ with the
* initial condition \f$ y(0) = y_0 \f$ is given by
* \f$ y(t) = \exp(M) y_0 \f$.
*
* The cost of the computation is approximately \f$ 20 n^3 \f$ for
* matrices of size \f$ n \f$. The number 20 depends weakly on the
* norm of the matrix.
*
* The matrix exponential is computed using the scaling-and-squaring
* method combined with Pad&eacute; approximation. The matrix is first
* rescaled, then the exponential of the reduced matrix is computed
* approximant, and then the rescaling is undone by repeated
* squaring. The degree of the Pad&eacute; approximant is chosen such
* that the approximation error is less than the round-off
* error. However, errors may accumulate during the squaring phase.
*
* Details of the algorithm can be found in: Nicholas J. Higham, "The
* scaling and squaring method for the matrix exponential revisited,"
* <em>SIAM J. %Matrix Anal. Applic.</em>, <b>26</b>:1179&ndash;1193,
* 2005.
*
* \note \p M has to be a matrix of \c float, \c double,
* \c complex<float> or \c complex<double> .
*/
template <typename Derived>
EIGEN_STRONG_INLINE void ei_matrix_exponential(const MatrixBase<Derived> &M,
typename MatrixBase<Derived>::PlainMatrixType* result);
/** \internal \brief Internal helper functions for computing the
* matrix exponential.
*/
namespace MatrixExponentialInternal {
#ifdef _MSC_VER
template <typename Scalar> Scalar log2(Scalar v) { return std::log(v)/std::log(Scalar(2)); }
#endif
/** \internal \brief Compute the (3,3)-Pad&eacute; approximant to
* the exponential.
*
* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
* approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
*
* \param M Argument of matrix exponential
* \param Id Identity matrix of same size as M
* \param tmp Temporary storage, to be provided by the caller
* \param M2 Temporary storage, to be provided by the caller
* \param U Even-degree terms in numerator of Pad&eacute; approximant
* \param V Odd-degree terms in numerator of Pad&eacute; approximant
*/
template <typename MatrixType>
EIGEN_STRONG_INLINE void pade3(const MatrixType &M, const MatrixType& Id, MatrixType& tmp,
MatrixType& M2, MatrixType& U, MatrixType& V)
{
typedef typename ei_traits<MatrixType>::Scalar Scalar;
const Scalar b[] = {120., 60., 12., 1.};
M2.noalias() = M * M;
tmp = b[3]*M2 + b[1]*Id;
U.noalias() = M * tmp;
V = b[2]*M2 + b[0]*Id;
}
/** \internal \brief Compute the (5,5)-Pad&eacute; approximant to
* the exponential.
*
* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
* approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
*
* \param M Argument of matrix exponential
* \param Id Identity matrix of same size as M
* \param tmp Temporary storage, to be provided by the caller
* \param M2 Temporary storage, to be provided by the caller
* \param U Even-degree terms in numerator of Pad&eacute; approximant
* \param V Odd-degree terms in numerator of Pad&eacute; approximant
*/
template <typename MatrixType>
EIGEN_STRONG_INLINE void pade5(const MatrixType &M, const MatrixType& Id, MatrixType& tmp,
MatrixType& M2, MatrixType& U, MatrixType& V)
{
typedef typename ei_traits<MatrixType>::Scalar Scalar;
const Scalar b[] = {30240., 15120., 3360., 420., 30., 1.};
M2.noalias() = M * M;
MatrixType M4 = M2 * M2;
tmp = b[5]*M4 + b[3]*M2 + b[1]*Id;
U.noalias() = M * tmp;
V = b[4]*M4 + b[2]*M2 + b[0]*Id;
}
/** \internal \brief Compute the (7,7)-Pad&eacute; approximant to
* the exponential.
*
* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
* approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
*
* \param M Argument of matrix exponential
* \param Id Identity matrix of same size as M
* \param tmp Temporary storage, to be provided by the caller
* \param M2 Temporary storage, to be provided by the caller
* \param U Even-degree terms in numerator of Pad&eacute; approximant
* \param V Odd-degree terms in numerator of Pad&eacute; approximant
*/
template <typename MatrixType>
EIGEN_STRONG_INLINE void pade7(const MatrixType &M, const MatrixType& Id, MatrixType& tmp,
MatrixType& M2, MatrixType& U, MatrixType& V)
{
typedef typename ei_traits<MatrixType>::Scalar Scalar;
const Scalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.};
M2.noalias() = M * M;
MatrixType M4 = M2 * M2;
MatrixType M6 = M4 * M2;
tmp = b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
U.noalias() = M * tmp;
V = b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id;
}
/** \internal \brief Compute the (9,9)-Pad&eacute; approximant to
* the exponential.
*
* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
* approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
*
* \param M Argument of matrix exponential
* \param Id Identity matrix of same size as M
* \param tmp Temporary storage, to be provided by the caller
* \param M2 Temporary storage, to be provided by the caller
* \param U Even-degree terms in numerator of Pad&eacute; approximant
* \param V Odd-degree terms in numerator of Pad&eacute; approximant
*/
template <typename MatrixType>
EIGEN_STRONG_INLINE void pade9(const MatrixType &M, const MatrixType& Id, MatrixType& tmp,
MatrixType& M2, MatrixType& U, MatrixType& V)
{
typedef typename ei_traits<MatrixType>::Scalar Scalar;
const Scalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240.,
2162160., 110880., 3960., 90., 1.};
M2.noalias() = M * M;
MatrixType M4 = M2 * M2;
MatrixType M6 = M4 * M2;
MatrixType M8 = M6 * M2;
tmp = b[9]*M8 + b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
U.noalias() = M * tmp;
V = b[8]*M8 + b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id;
}
/** \internal \brief Compute the (13,13)-Pad&eacute; approximant to
* the exponential.
*
* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
* approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
*
* \param M Argument of matrix exponential
* \param Id Identity matrix of same size as M
* \param tmp Temporary storage, to be provided by the caller
* \param M2 Temporary storage, to be provided by the caller
* \param U Even-degree terms in numerator of Pad&eacute; approximant
* \param V Odd-degree terms in numerator of Pad&eacute; approximant
*/
template <typename MatrixType>
EIGEN_STRONG_INLINE void pade13(const MatrixType &M, const MatrixType& Id, MatrixType& tmp,
MatrixType& M2, MatrixType& U, MatrixType& V)
{
typedef typename ei_traits<MatrixType>::Scalar Scalar;
const Scalar b[] = {64764752532480000., 32382376266240000., 7771770303897600.,
1187353796428800., 129060195264000., 10559470521600., 670442572800.,
33522128640., 1323241920., 40840800., 960960., 16380., 182., 1.};
M2.noalias() = M * M;
MatrixType M4 = M2 * M2;
MatrixType M6 = M4 * M2;
V = b[13]*M6 + b[11]*M4 + b[9]*M2;
tmp.noalias() = M6 * V;
tmp += b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
U.noalias() = M * tmp;
tmp = b[12]*M6 + b[10]*M4 + b[8]*M2;
V.noalias() = M6 * tmp;
V += b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id;
}
/** \internal \brief Helper class for computing Pad&eacute;
* approximants to the exponential.
*/
template <typename MatrixType, typename RealScalar = typename NumTraits<typename ei_traits<MatrixType>::Scalar>::Real>
struct computeUV_selector
{
/** \internal \brief Compute Pad&eacute; approximant to the exponential.
*
* Computes \p U, \p V and \p squarings such that \f$ (V+U)(V-U)^{-1} \f$
* is a Pad&eacute; of \f$ \exp(2^{-\mbox{squarings}}M) \f$
* around \f$ M = 0 \f$. The degree of the Pad&eacute;
* 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 Pad&eacute; approximant
* \param V Odd-degree terms in numerator of Pad&eacute; 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
}
} // end of namespace MatrixExponentialInternal
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
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