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-rw-r--r--unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h331
-rw-r--r--unsupported/test/matrixExponential.cpp62
2 files changed, 290 insertions, 103 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&ndash;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)-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$.
+ *
+ * \param M Argument of matrix exponential
+ * \param Id Identity matrix of same size as M
+ * \param tmp Temporary storage, to be provided by the caller
+ * \param M2 Temporary storage, to be provided by the caller
+ * \param U Even-degree terms in numerator of Pad&eacute; approximant
+ * \param V Odd-degree terms in numerator of Pad&eacute; approximant
+ */
+ template <typename MatrixType>
+ EIGEN_STRONG_INLINE void pade3(const MatrixType &M, const MatrixType& Id, MatrixType& tmp,
+ MatrixType& M2, MatrixType& U, MatrixType& V)
+ {
+ typedef typename ei_traits<MatrixType>::Scalar Scalar;
+ const Scalar b[] = {120., 60., 12., 1.};
+ M2 = (M * M).lazy();
+ tmp = b[3]*M2 + b[1]*Id;
+ U = (M * tmp).lazy();
+ V = b[2]*M2 + b[0]*Id;
+ }
+
+ /** \internal \brief Compute the (5,5)-Pad&eacute; approximant to
+ * the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
+ * approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
+ *
+ * \param M Argument of matrix exponential
+ * \param Id Identity matrix of same size as M
+ * \param tmp Temporary storage, to be provided by the caller
+ * \param M2 Temporary storage, to be provided by the caller
+ * \param U Even-degree terms in numerator of Pad&eacute; approximant
+ * \param V Odd-degree terms in numerator of Pad&eacute; approximant
+ */
+ template <typename MatrixType>
+ EIGEN_STRONG_INLINE void pade5(const MatrixType &M, const MatrixType& Id, MatrixType& tmp,
+ MatrixType& M2, MatrixType& U, MatrixType& V)
+ {
+ typedef typename ei_traits<MatrixType>::Scalar Scalar;
const Scalar b[] = {30240., 15120., 3360., 420., 30., 1.};
- PlainMatrixType M2, M4;
- M2 = (Meval * Meval).lazy();
- M4 = (M2 * M2).lazy();
- num = b[5]*M4 + b[3]*M2 + b[1]*Id;
- U = (Meval * num).lazy();
+ M2 = (M * M).lazy();
+ MatrixType M4 = (M2 * M2).lazy();
+ tmp = b[5]*M4 + b[3]*M2 + b[1]*Id;
+ U = (M * tmp).lazy();
V = b[4]*M4 + b[2]*M2 + b[0]*Id;
-
- } else if (l1norm < 9.504178996162932e-001) {
-
- // Use (7,7)-Pade
+ }
+
+ /** \internal \brief Compute the (7,7)-Pad&eacute; approximant to
+ * the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
+ * approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
+ *
+ * \param M Argument of matrix exponential
+ * \param Id Identity matrix of same size as M
+ * \param tmp Temporary storage, to be provided by the caller
+ * \param M2 Temporary storage, to be provided by the caller
+ * \param U Even-degree terms in numerator of Pad&eacute; approximant
+ * \param V Odd-degree terms in numerator of Pad&eacute; approximant
+ */
+ template <typename MatrixType>
+ EIGEN_STRONG_INLINE void pade7(const MatrixType &M, const MatrixType& Id, MatrixType& tmp,
+ MatrixType& M2, MatrixType& U, MatrixType& V)
+ {
+ typedef typename ei_traits<MatrixType>::Scalar Scalar;
const Scalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.};
- PlainMatrixType M2, M4, M6;
- M2 = (Meval * Meval).lazy();
- M4 = (M2 * M2).lazy();
- M6 = (M4 * M2).lazy();
- num = b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
- U = (Meval * num).lazy();
+ M2 = (M * M).lazy();
+ MatrixType M4 = (M2 * M2).lazy();
+ MatrixType M6 = (M4 * M2).lazy();
+ tmp = b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
+ U = (M * tmp).lazy();
V = b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id;
-
- } else if (l1norm < 2.097847961257068e+000) {
-
- // Use (9,9)-Pade
+ }
+
+ /** \internal \brief Compute the (9,9)-Pad&eacute; approximant to
+ * the exponential.
+ *
+ * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
+ * approximant of \f$ \exp(M) \f$ around \f$ M = 0 \f$.
+ *
+ * \param M Argument of matrix exponential
+ * \param Id Identity matrix of same size as M
+ * \param tmp Temporary storage, to be provided by the caller
+ * \param M2 Temporary storage, to be provided by the caller
+ * \param U Even-degree terms in numerator of Pad&eacute; approximant
+ * \param V Odd-degree terms in numerator of Pad&eacute; approximant
+ */
+ template <typename MatrixType>
+ EIGEN_STRONG_INLINE void pade9(const MatrixType &M, const MatrixType& Id, MatrixType& tmp,
+ MatrixType& M2, MatrixType& U, MatrixType& V)
+ {
+ typedef typename ei_traits<MatrixType>::Scalar Scalar;
const Scalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240.,
- 2162160., 110880., 3960., 90., 1.};
- PlainMatrixType M2, M4, M6, M8;
- M2 = (Meval * Meval).lazy();
- M4 = (M2 * M2).lazy();
- M6 = (M4 * M2).lazy();
- M8 = (M6 * M2).lazy();
- num = b[9]*M8 + b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
- U = (Meval * num).lazy();
+ 2162160., 110880., 3960., 90., 1.};
+ M2 = (M * M).lazy();
+ MatrixType M4 = (M2 * M2).lazy();
+ MatrixType M6 = (M4 * M2).lazy();
+ MatrixType M8 = (M6 * M2).lazy();
+ tmp = b[9]*M8 + b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id;
+ U = (M * tmp).lazy();
V = b[8]*M8 + b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id;
-
- } else {
-
- // Use (13,13)-Pade; scale matrix by power of 2 so that its norm
- // is small enough
-
- const Scalar maxnorm = 5.371920351148152;
+ }
+
+ /** \internal \brief Compute the (13,13)-Pad&eacute; approximant to
+ * the exponential.
+ *
+ * After exit, \f$ (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.};
-
- 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 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
- 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
diff --git a/unsupported/test/matrixExponential.cpp b/unsupported/test/matrixExponential.cpp
index 0ec4fae90..9a6c335d8 100644
--- a/unsupported/test/matrixExponential.cpp
+++ b/unsupported/test/matrixExponential.cpp
@@ -34,26 +34,47 @@ double binom(int n, int k)
return res;
}
-void test2dRotation()
+template <typename T>
+void test2dRotation(double tol)
{
- Matrix2d A, B, C;
- double angle;
+ Matrix<T,2,2> A, B, C;
+ T angle;
+ A << 0, 1, -1, 0;
for (int i=0; i<=20; i++)
{
angle = pow(10, i / 5. - 2);
- A << 0, angle, -angle, 0;
B << cos(angle), sin(angle), -sin(angle), cos(angle);
+ ei_matrix_exponential(angle*A, &C);
+ VERIFY(C.isApprox(B, tol));
+ }
+}
+
+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, 1e-14));
+ VERIFY(C.isApprox(B, tol));
}
}
-void testPascal()
+template <typename T>
+void testPascal(double tol)
{
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();
for (int i=0; i<size-1; i++)
A(i+1,i) = i+1;
@@ -62,11 +83,12 @@ void testPascal()
for (int j=0; j<=i; j++)
B(i,j) = binom(i,j);
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:
Inverse.h
@@ -80,16 +102,24 @@ template<typename MatrixType> void randomTest(const MatrixType& m)
m1 = MatrixType::Random(rows, cols);
ei_matrix_exponential(m1, &m2);
ei_matrix_exponential(-m1, &m3);
- VERIFY(identity.isApprox(m2 * m3, 1e-13));
+ VERIFY(identity.isApprox(m2 * m3, tol));
}
}
void test_matrixExponential()
{
- CALL_SUBTEST(test2dRotation());
- CALL_SUBTEST(testPascal());
- CALL_SUBTEST(randomTest(Matrix2d()));
- CALL_SUBTEST(randomTest(Matrix3d()));
- CALL_SUBTEST(randomTest(Matrix4d()));
- CALL_SUBTEST(randomTest(MatrixXd(8,8)));
+ CALL_SUBTEST(test2dRotation<double>(1e-14));
+ CALL_SUBTEST(test2dRotation<float>(1e-5));
+ CALL_SUBTEST(test2dHyperbolicRotation<double>(1e-14));
+ CALL_SUBTEST(test2dHyperbolicRotation<float>(1e-5));
+ CALL_SUBTEST(testPascal<float>(1e-5));
+ 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));
}
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