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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra. Eigen itself is part of the KDE project.
//
// 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
#ifdef _MSC_VER
template <typename Scalar> Scalar log2(Scalar v) { return std::log(v)/std::log(Scalar(2)); }
#endif
/** 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 Padé 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 Padé 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>:1179–1193,
* 2005.
*
* \note Currently, \p M has to be a matrix of \c 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;
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());
RealScalar l1norm = M.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 = (M * M).lazy();
num = b[3]*M2 + b[1]*Id;
U = (M * num).lazy();
V = b[2]*M2 + b[0]*Id;
} else if (l1norm < 2.539398330063230e-001) {
// Use (5,5)-Pade
const Scalar b[] = {30240., 15120., 3360., 420., 30., 1.};
PlainMatrixType M2, M4;
M2 = (M * M).lazy();
M4 = (M2 * M2).lazy();
num = b[5]*M4 + b[3]*M2 + b[1]*Id;
U = (M * num).lazy();
V = b[4]*M4 + b[2]*M2 + b[0]*Id;
} else if (l1norm < 9.504178996162932e-001) {
// Use (7,7)-Pade
const Scalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.};
PlainMatrixType M2, M4, M6;
M2 = (M * M).lazy();
M4 = (M2 * M2).lazy();
M6 = (M4 * M2).lazy();
num = b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
U = (M * num).lazy();
V = b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id;
} else if (l1norm < 2.097847961257068e+000) {
// Use (9,9)-Pade
const Scalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240.,
2162160., 110880., 3960., 90., 1.};
PlainMatrixType M2, M4, M6, M8;
M2 = (M * M).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 = (M * num).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;
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 = M / 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;
}
num = U + V; // numerator of Pade approximant
den = -U + V; // denominator of Pade approximant
den.lu().solve(num, result);
// Undo scaling by repeated squaring
for (int i=0; i<squarings; i++)
*result *= *result;
}
#endif // EIGEN_MATRIX_EXPONENTIAL
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