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Add support for matrix exponential of floats and complex numbers.

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This commit is contained in:
Jitse Niesen 2009-08-12 15:44:22 +01:00
parent 309d540d4a
commit f71f878bab
2 changed files with 291 additions and 104 deletions
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331
unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h
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@ -25,13 +25,9 @@
#ifndef EIGEN_MATRIX_EXPONENTIAL #ifndef EIGEN_MATRIX_EXPONENTIAL
#define EIGEN_MATRIX_EXPONENTIAL #define EIGEN_MATRIX_EXPONENTIAL
#ifdef _MSC_VER /** \brief Compute the matrix exponential.
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 M matrix whose exponential is to be computed.
* \param result pointer to the matrix in which to store the result. * \param result pointer to the matrix in which to store the result.
* *
* The matrix exponential of \f$ M \f$ is defined by * 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&ndash;1193, * <em>SIAM J. %Matrix Anal. Applic.</em>, <b>26</b>:1179&ndash;1193,
* 2005. * 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> template <typename Derived>
void ei_matrix_exponential(const MatrixBase<Derived> &M, typename ei_plain_matrix_type<Derived>::type* result) EIGEN_STRONG_INLINE void ei_matrix_exponential(const MatrixBase<Derived> &M,
{ typename MatrixBase<Derived>::PlainMatrixType* 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; /** \internal \brief Internal helper functions for computing the
PlainMatrixType Id = PlainMatrixType::Identity(M.rows(), M.cols()); * matrix exponential.
typename ei_eval<Derived>::type Meval = M.eval(); */
RealScalar l1norm = Meval.cwise().abs().colwise().sum().maxCoeff(); namespace MatrixExponentialInternal {
int squarings = 0;
#ifdef _MSC_VER
// Choose degree of Pade approximant, depending on norm of M template <typename Scalar> Scalar log2(Scalar v) { return std::log(v)/std::log(Scalar(2)); }
if (l1norm < 1.495585217958292e-002) { #endif
// Use (3,3)-Pade /** \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$.
*
* \param M Argument of matrix exponential
* \param Id Identity matrix of same size as M
* \param tmp Temporary storage, to be provided by the caller
* \param M2 Temporary storage, to be provided by the caller
* \param U Even-degree terms in numerator of Pad&eacute; approximant
* \param V Odd-degree terms in numerator of Pad&eacute; approximant
*/
template <typename MatrixType>
EIGEN_STRONG_INLINE void pade3(const MatrixType &M, const MatrixType& Id, MatrixType& tmp,
MatrixType& M2, MatrixType& U, MatrixType& V)
{
typedef typename ei_traits<MatrixType>::Scalar Scalar;
const Scalar b[] = {120., 60., 12., 1.}; const Scalar b[] = {120., 60., 12., 1.};
PlainMatrixType M2; M2 = (M * M).lazy();
M2 = (Meval * Meval).lazy(); tmp = b[3]*M2 + b[1]*Id;
num = b[3]*M2 + b[1]*Id; U = (M * tmp).lazy();
U = (Meval * num).lazy();
V = b[2]*M2 + b[0]*Id; V = b[2]*M2 + b[0]*Id;
}
} else if (l1norm < 2.539398330063230e-001) {
/** \internal \brief Compute the (5,5)-Pad&eacute; approximant to
// Use (5,5)-Pade * the exponential.
*
* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
* approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
*
* \param M Argument of matrix exponential
* \param Id Identity matrix of same size as M
* \param tmp Temporary storage, to be provided by the caller
* \param M2 Temporary storage, to be provided by the caller
* \param U Even-degree terms in numerator of Pad&eacute; approximant
* \param V Odd-degree terms in numerator of Pad&eacute; approximant
*/
template <typename MatrixType>
EIGEN_STRONG_INLINE void pade5(const MatrixType &M, const MatrixType& Id, MatrixType& tmp,
MatrixType& M2, MatrixType& U, MatrixType& V)
{
typedef typename ei_traits<MatrixType>::Scalar Scalar;
const Scalar b[] = {30240., 15120., 3360., 420., 30., 1.}; const Scalar b[] = {30240., 15120., 3360., 420., 30., 1.};
PlainMatrixType M2, M4; M2 = (M * M).lazy();
M2 = (Meval * Meval).lazy(); MatrixType M4 = (M2 * M2).lazy();
M4 = (M2 * M2).lazy(); tmp = b[5]*M4 + b[3]*M2 + b[1]*Id;
num = b[5]*M4 + b[3]*M2 + b[1]*Id; U = (M * tmp).lazy();
U = (Meval * num).lazy();
V = b[4]*M4 + b[2]*M2 + b[0]*Id; V = b[4]*M4 + b[2]*M2 + b[0]*Id;
}
} else if (l1norm < 9.504178996162932e-001) {
/** \internal \brief Compute the (7,7)-Pad&eacute; approximant to
// Use (7,7)-Pade * the exponential.
*
* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
* approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
*
* \param M Argument of matrix exponential
* \param Id Identity matrix of same size as M
* \param tmp Temporary storage, to be provided by the caller
* \param M2 Temporary storage, to be provided by the caller
* \param U Even-degree terms in numerator of Pad&eacute; approximant
* \param V Odd-degree terms in numerator of Pad&eacute; approximant
*/
template <typename MatrixType>
EIGEN_STRONG_INLINE void pade7(const MatrixType &M, const MatrixType& Id, MatrixType& tmp,
MatrixType& M2, MatrixType& U, MatrixType& V)
{
typedef typename ei_traits<MatrixType>::Scalar Scalar;
const Scalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.}; const Scalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.};
PlainMatrixType M2, M4, M6; M2 = (M * M).lazy();
M2 = (Meval * Meval).lazy(); MatrixType M4 = (M2 * M2).lazy();
M4 = (M2 * M2).lazy(); MatrixType M6 = (M4 * M2).lazy();
M6 = (M4 * M2).lazy(); tmp = b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
num = b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id; U = (M * tmp).lazy();
U = (Meval * num).lazy();
V = b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id; V = b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id;
}
} else if (l1norm < 2.097847961257068e+000) {
/** \internal \brief Compute the (9,9)-Pad&eacute; approximant to
// Use (9,9)-Pade * the exponential.
*
* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
* approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
*
* \param M Argument of matrix exponential
* \param Id Identity matrix of same size as M
* \param tmp Temporary storage, to be provided by the caller
* \param M2 Temporary storage, to be provided by the caller
* \param U Even-degree terms in numerator of Pad&eacute; approximant
* \param V Odd-degree terms in numerator of Pad&eacute; approximant
*/
template <typename MatrixType>
EIGEN_STRONG_INLINE void pade9(const MatrixType &M, const MatrixType& Id, MatrixType& tmp,
MatrixType& M2, MatrixType& U, MatrixType& V)
{
typedef typename ei_traits<MatrixType>::Scalar Scalar;
const Scalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240., const Scalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240.,
2162160., 110880., 3960., 90., 1.}; 2162160., 110880., 3960., 90., 1.};
PlainMatrixType M2, M4, M6, M8; M2 = (M * M).lazy();
M2 = (Meval * Meval).lazy(); MatrixType M4 = (M2 * M2).lazy();
M4 = (M2 * M2).lazy(); MatrixType M6 = (M4 * M2).lazy();
M6 = (M4 * M2).lazy(); MatrixType M8 = (M6 * M2).lazy();
M8 = (M6 * M2).lazy(); tmp = b[9]*M8 + b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
num = b[9]*M8 + b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id; U = (M * tmp).lazy();
U = (Meval * num).lazy();
V = b[8]*M8 + b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id; V = b[8]*M8 + b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id;
}
} else {
/** \internal \brief Compute the (13,13)-Pad&eacute; approximant to
// Use (13,13)-Pade; scale matrix by power of 2 so that its norm * the exponential.
// is small enough *
* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
const Scalar maxnorm = 5.371920351148152; * 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., const Scalar b[] = {64764752532480000., 32382376266240000., 7771770303897600.,
1187353796428800., 129060195264000., 10559470521600., 670442572800., 1187353796428800., 129060195264000., 10559470521600., 670442572800.,
33522128640., 1323241920., 40840800., 960960., 16380., 182., 1.}; 33522128640., 1323241920., 40840800., 960960., 16380., 182., 1.};
M2 = (M * M).lazy();
squarings = std::max(0, (int)ceil(log2(l1norm / maxnorm))); MatrixType M4 = (M2 * M2).lazy();
PlainMatrixType A, A2, A4, A6; MatrixType M6 = (M4 * M2).lazy();
A = Meval / pow(Scalar(2), squarings); V = b[13]*M6 + b[11]*M4 + b[9]*M2;
A2 = (A * A).lazy(); tmp = (M6 * V).lazy();
A4 = (A2 * A2).lazy(); tmp += b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
A6 = (A4 * A2).lazy(); U = (M * tmp).lazy();
num = b[13]*A6 + b[11]*A4 + b[9]*A2; tmp = b[12]*M6 + b[10]*M4 + b[8]*M2;
V = (A6 * num).lazy(); V = (M6 * tmp).lazy();
num = V + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*Id; V += b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*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; /** \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 = 3.925724783138660;
*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 } // end of namespace MatrixExponentialInternal
den = -U + V; // denominator of Pade approximant
den.lu().solve(num, result);
// Undo scaling by repeated squaring template <typename Derived>
for (int i=0; i<squarings; i++) EIGEN_STRONG_INLINE void ei_matrix_exponential(const MatrixBase<Derived> &M,
*result *= *result; typename MatrixBase<Derived>::PlainMatrixType* result)
{
ei_assert(M.rows() == M.cols());
MatrixExponentialInternal::compute(M.eval(), result);
} }
#endif // EIGEN_MATRIX_EXPONENTIAL #endif // EIGEN_MATRIX_EXPONENTIAL

64
unsupported/test/matrixExponential.cpp
View File

@ -34,26 +34,47 @@ double binom(int n, int k)
return res; return res;
} }
void test2dRotation() template <typename T>
void test2dRotation(double tol)
{ {
Matrix2d A, B, C; Matrix<T,2,2> A, B, C;
double angle; T angle;
A << 0, 1, -1, 0;
for (int i=0; i<=20; i++) for (int i=0; i<=20; i++)
{ {
angle = pow(10, i / 5. - 2); angle = pow(10, i / 5. - 2);
A << 0, angle, -angle, 0;
B << cos(angle), sin(angle), -sin(angle), cos(angle); B << cos(angle), sin(angle), -sin(angle), cos(angle);
ei_matrix_exponential(A, &C); ei_matrix_exponential(angle*A, &C);
VERIFY(C.isApprox(B, 1e-14)); VERIFY(C.isApprox(B, tol));
} }
} }
void testPascal() template <typename T>
void test2dHyperbolicRotation(double tol)
{
Matrix<std::complex<T>,2,2> A, B, C;
std::complex<T> imagUnit(0,1);
T angle, ch, sh;
for (int i=0; i<=20; i++)
{
angle = (i-10) / 2.0;
ch = std::cosh(angle);
sh = std::sinh(angle);
A << 0, angle*imagUnit, -angle*imagUnit, 0;
B << ch, sh*imagUnit, -sh*imagUnit, ch;
ei_matrix_exponential(A, &C);
VERIFY(C.isApprox(B, tol));
}
}
template <typename T>
void testPascal(double tol)
{ {
for (int size=1; size<20; size++) for (int size=1; size<20; size++)
{ {
MatrixXd A(size,size), B(size,size), C(size,size); Matrix<T,Dynamic,Dynamic> A(size,size), B(size,size), C(size,size);
A.setZero(); A.setZero();
for (int i=0; i<size-1; i++) for (int i=0; i<size-1; i++)
A(i+1,i) = i+1; A(i+1,i) = i+1;
@ -62,11 +83,12 @@ void testPascal()
for (int j=0; j<=i; j++) for (int j=0; j<=i; j++)
B(i,j) = binom(i,j); B(i,j) = binom(i,j);
ei_matrix_exponential(A, &C); ei_matrix_exponential(A, &C);
VERIFY(C.isApprox(B, 1e-14)); VERIFY(C.isApprox(B, tol));
} }
} }
template<typename MatrixType> void randomTest(const MatrixType& m) template<typename MatrixType>
void randomTest(const MatrixType& m, double tol)
{ {
/* this test covers the following files: /* this test covers the following files:
Inverse.h Inverse.h
@ -80,16 +102,24 @@ template<typename MatrixType> void randomTest(const MatrixType& m)
m1 = MatrixType::Random(rows, cols); m1 = MatrixType::Random(rows, cols);
ei_matrix_exponential(m1, &m2); ei_matrix_exponential(m1, &m2);
ei_matrix_exponential(-m1, &m3); ei_matrix_exponential(-m1, &m3);
VERIFY(identity.isApprox(m2 * m3, 1e-13)); VERIFY(identity.isApprox(m2 * m3, tol));
} }
} }
void test_matrixExponential() void test_matrixExponential()
{ {
CALL_SUBTEST(test2dRotation()); CALL_SUBTEST(test2dRotation<double>(1e-14));
CALL_SUBTEST(testPascal()); CALL_SUBTEST(test2dRotation<float>(1e-5));
CALL_SUBTEST(randomTest(Matrix2d())); CALL_SUBTEST(test2dHyperbolicRotation<double>(1e-14));
CALL_SUBTEST(randomTest(Matrix3d())); CALL_SUBTEST(test2dHyperbolicRotation<float>(1e-5));
CALL_SUBTEST(randomTest(Matrix4d())); CALL_SUBTEST(testPascal<float>(1e-5));
CALL_SUBTEST(randomTest(MatrixXd(8,8))); CALL_SUBTEST(testPascal<double>(1e-14));
CALL_SUBTEST(randomTest(Matrix2d(), 1e-13));
CALL_SUBTEST(randomTest(Matrix<double,3,3,RowMajor>(), 1e-13));
CALL_SUBTEST(randomTest(Matrix4cd(), 1e-13));
CALL_SUBTEST(randomTest(MatrixXd(8,8), 1e-13));
CALL_SUBTEST(randomTest(Matrix2f(), 1e-4));
CALL_SUBTEST(randomTest(Matrix3cf(), 1e-4));
CALL_SUBTEST(randomTest(Matrix4f(), 1e-4));
CALL_SUBTEST(randomTest(MatrixXf(8,8), 1e-4));
} }